In the undergraduate course on electromagnetism the field equations arrive in the language of vector calculus. With a charge density \(\rho \) and a current \(J\), and with the constants set to one, they read \[\operatorname {div} E=\rho ,\qquad \operatorname {div} B=0,\qquad \operatorname {curl} E=-\partial _t B,\qquad \operatorname {curl} B=\partial _t E+J.\] 4 equations, two vector fields, and a potential \(A\) that enters when one wants to solve them. There the potential is primitive and the fields are derived from it by a gradient, and the metric sits in the operators: the divergence, the curl and the cross product all know about lengths and angles. Every object is continuous as well: \(\rho \) is a density, and \(\partial _t\) presupposes a parameter along which to differentiate.

A course in mathematical physics rewrites the system in the language of differential forms. The potential becomes a \(1\)-form \(A\) and the field strength its exterior derivative \(F=dA\). The exterior derivative needs no metric, so \(dF=0\) is a consequence of \(d\circ d=0\) rather than an equation to impose, and gauge transformations \(A\mapsto A+d\chi \) leave \(F\) unchanged by construction, which is why gauge theory is written this way. The metric reappears in the Hodge star that the equations with sources require, \(d\star F=\star j\), while \(d\) raises the degree of a form. [hassani2013mathematical, Section 28.5] writes the field equations this way.

Readers who have met geometric calculus will recognize the same system in the unified language that Hestenes and Sobczyk proposed for mathematics and physics. The electric and magnetic fields are then grades of one object, the bivector \(F=E+IB\), where \(I\) is the pseudoscalar of the algebra, the product of the 4 frame vectors, with \(I^2=-1\), so the magnetic field enters as the dual bivector \(IB\), and the pair forms one bivector rather than the vector and pseudovector of vector calculus. The derivative is the vector derivative \(\nabla \): the derivative along each direction of a chosen frame, combined into a single vector-valued operator by the geometric product with the frame vectors. It acts by geometric multiplication, and a vector multiplied that way by a multivector splits into two parts, \[\nabla F=\nabla \cdot F+\nabla \wedge F,\] an inner product giving a divergence and an outer product giving a curl. With it the 4 equations become \(\nabla F=J\), the divergence part carrying the sources and the curl part vanishing, so each original equation returns as a definite grade. Hestenes and Sobczyk develop the calculus [hestenes2012clifford], Doran and Lasenby carry it through classical electrodynamics [doran2003geometric, Section 7.1], and Macdonald's textbook is the treatment [macdonald2012vector]. Here one algebra and one derivative serve both objects, and the metric is neither an operator nor a dual: it is inside the geometric product, which is why the language carries a chosen spacetime with it.

All three ways of writing these equations need a limit: in each, a derivative compares values at points whose separation shrinks. Those points are not bare: the continuum carries a metric, a topology and a smooth structure, and that is what gives a separation, a direction and a gradient their meaning. On a lattice only finitely many points remain, and a direction becomes a step from one site to a neighbour. What is left of the gradient is the difference of the values at the two ends of each step. A derivative of a product obeys the Leibniz rule; does a difference of values obey it as well, or does something else take its place? And which changes of the potential does a difference fail to see?

Let \(R\) be a commutative ring with identity. Choose finite sets \(V,E\) and maps \(s,h:E\to V\). The edge \(e\) goes from \(s(e)\) to \(h(e)\); loops and distinct edges with the same endpoints are allowed. Call the finite sets together with their endpoint maps the carrier. The field modules are the vertex-field module and the edge-field module \[\mathcal F_R(V)=\{f:V\to R\},\qquad \mathcal F_R(E)=\{a:E\to R\}.\] Addition, scalar multiplication and multiplication are pointwise: \[(f+g)(v)=f(v)+g(v),\quad (rf)(v)=rf(v),\quad (fg)(v)=f(v)g(v).\] These are modules and commutative unital \(R\)-algebras; the unit is the constant field \(1\).

The zero ring and an empty vertex set or edge set are allowed. On an empty set there is one field, and the constant \(1\) is also zero. No nontriviality, pairing, retained subset or positivity is assumed. The algebra convention is [stacks2017stacks, Section 22.2], Tag 09JF. The module and algebra laws are in Lemma 1.5.

The alternating four-cycle of Example 1.8 is an instance of these fields.

Let \(R\) be a commutative ring with identity, with the carrier of finite sets \(V,E\) and maps \(s,h:E\to V\), and the field modules over it. For a map \(a:E\to V\), its field pullback is \[a^*:\mathcal F_R(V)\to \mathcal F_R(E),\qquad (a^*f)(e)=f(a(e)).\] It preserves addition, scalars, products and the unit. The ring-valued incidence map is the typed difference \[d=h^*-s^*:\mathcal F_R(V)\longrightarrow \mathcal F_R(E).\] Thus \((df)(e)=f(h(e))-f(s(e))\). This is an \(R\)-linear map, with both coefficient rings equal to \(R\) and scalar homomorphism \(\sigma =\operatorname {id}_R\), as in Remark [ca-001G]. The two field modules need not coincide, so \(d^2\) has no meaning without an additional composable map. The pullback law, the linearity and the kernel of \(d\) are in Lemma 1.5.

For real oriented-graph incidence, see [spielman2009effective, Section 2.1, p. 4]; the ring-level claims follow from Lemma 1.5.

The potential of the preceding discussion is a field on the vertices: one value per site, added and scaled pointwise as in the vertex-field module. Along an edge the difference between the two endpoint values is what the edge carries, so the ring-valued incidence map takes a potential to the field strength read off the edges. On a cubic lattice whose steps are \(x\to x+\mu \), that difference is \(\varphi (x+\mu )-\varphi (x)\).

In the forms notation the potential is a \(0\)-form and the field strength a \(1\)-form, related by \(d\). The incidence map plays the role of \(d\) on \(0\)-forms, and it is not \(d\): the exterior derivative raises the degree of a form, while the incidence map runs from vertex fields to edge fields of one particular graph. A field pullback is composition with a map of the two finite sets, as in the substitution \(f'(x)=f(h(x))\).

Let \(R\) be a commutative unital ring, and let \(V,E,s,h\) and the field modules, field pullback and ring-valued incidence map be as in the two definitions above. The pointwise operations make both modules commutative unital \(R\)-algebras, the indicator fields are a basis of each, every field pullback preserves the operations, and \(d\) is \(R\)-linear. Call two vertices equivalent when they are equal or a finite sequence of edges has them as successive ends, ignoring orientation. The equivalence classes are the undirected components; a vertex that lies on no edge is a component of its own. Then \(df=0\) if and only if \(f\) is constant on each undirected component.

The kernel in Lemma 1.5 is the gauge freedom of the potential. Adding a constant on one undirected component leaves every edge difference unchanged, and the electric field is the negative of that difference. A vector potential is assigned to the edges, as the link variables of lattice gauge theory are, and its field strength is the difference of those values around a face. The identity \(d\circ d=0\) needs that second difference, from edge fields to face fields. The maps defined here run only from vertices to edges.

Let \(R\) be a commutative unital ring, and let \(d\) be the ring-valued incidence map of a finite carrier \(\,(V,E,s,h)\). Choose orders of the vertex and edge indicators. Let \(c_V:\mathcal F_R(V)\to R^{|V|}\) and \(c_E:\mathcal F_R(E)\to R^{|E|}\) send a field to its ordered values. The indicator expansion in Lemma 1.5 gives their inverses, where \(\delta _v\) is the indicator field of a vertex \(v\). The incidence matrix represents \(c_Edc_V^{-1}\): \[B_{ev}=(d\delta _v)(e)=\mathbf 1_{v=h(e)}-\mathbf 1_{v=s(e)}.\] Rows are indexed by edges and columns by vertices. A loop has a zero row; reversing an edge negates its row. This extends the oriented-graph sign convention of [spielman2009effective, Section 2.1, p. 4]. If coordinate reorderings are \(P_V,P_E\), then \(B'=P_EBP_V^{-1}\), since both sides send \(c'_V(f)\) to \(c'_E(df)\).

Let \(V=\mathbb Z/4\mathbb Z\) and \(E=\{e_0,e_1,e_2,e_3\}\), with \(s(e_i)=i\) and \(h(e_i)=i+1\) modulo \(4\). For \(t>0\), assign weights \[w_t(e_0)=w_t(e_2)=1,\qquad w_t(e_1)=w_t(e_3)=t.\] This is a weighted finite incidence presentation: the parameter \(t\) is supplied, and every edge weight is positive.

Each arrow points from source to head; the label gives its edge name and weight. A half-turn exchanges opposite vertices and edges.

On the real four-cycle, order vertices as \(0,1,2,3\) and edges as \(e_0,e_1,e_2,e_3\). The incidence matrix \(B\) of \(D\), and the coordinate array \(W_t\) of the weights, are \[B=\begin {pmatrix} -1&1&0&0\\ 0&-1&1&0\\ 0&0&-1&1\\ 1&0&0&-1 \end {pmatrix},\qquad W_t=\operatorname {diag}(1,t,1,t).\] Thus a field with ordered values \((a,b,c,d)\) has edge differences \((b-a,c-b,d-c,a-d)\). For \(f=\delta _1\), the edge values of \(Df\) are \((1,-1,0,0)\): the field rises along \(e_0\) and falls along \(e_1\).

The quadratic action of Example 7.4 uses these same edge orientations and weights, with a positive mass term.

Define the half-turn on both sets by \[r_V(i)=i+2,\qquad r_E(e_i)=e_{i+2}.\] Applying either map twice adds 4, so both are involutions, and \(r_V(0)=2\) shows that the map on vertices is not the identity. Moreover, \[s(r_E(e_i))=i+2=r_V(s(e_i)),\] \[h(r_E(e_i))=i+3=r_V(h(e_i)).\] The half-turn therefore preserves the directed incidence data. Since \(i\) and \(i+2\) have the same parity, it also preserves the weights: \(w_t(r_E(e_i))=w_t(e_i)\). In contrast, a one-step rotation sends \(e_0\) to \(e_1\) and fails to preserve weights when \(t\ne 1\).

After forgetting orientations and weights, the underlying graph is Mathlib's SimpleGraph.cycleGraph at 4 vertices, identifying \(\operatorname {Fin}(4)\) with the residue classes modulo \(4\). That declaration specifies the unweighted graph. The displayed matrix and weight-preserving half-turn follow from the substitutions above.

The incidence matrix \(B\) is the difference matrix of the preceding discussion, and \(L=B^{\mathsf T}B\) is the graph Laplacian. The matrix \(-B^{\mathsf T}\) is the discrete divergence. With the edge field \(E=-B\varphi \), Gauss's law \(\nabla \cdot E=\rho \) becomes \[\,(-B^{\mathsf T})(-B\varphi )=B^{\mathsf T}B\varphi =L\varphi =\rho ,\] the discrete Poisson equation. Computations, and the later Laplacian chapters, use \(d\) in this matrix form.

A coupling that varies from site to site enters a lattice action as a coefficient: the term \(m\sum _v\varphi (v)^2\) for a constant mass \(m\) becomes \(\sum _v m(v)\varphi (v)^2\), so a fixed field multiplies the field at each site. Multiplying by a fixed field and taking differences along the edges are then two linear maps of one module, and composing them in the two orders gives two different maps. What is the map that multiplies, and how badly does it fail to commute with a difference?

Let \(R\) be a commutative unital ring and \(V\) a finite set. On the vertex-field module, each \(f\in \mathcal F_R(V)\) determines a multiplication operator \[M_f:\mathcal F_R(V)\to \mathcal F_R(V),\qquad M_f(g)=fg.\] The assignment \(f\mapsto M_f\) is an injective \(R\)-algebra map, \[M_{f+g}=M_f+M_g,\quad M_{rf}=rM_f,\quad M_{fg}=M_fM_g,\quad M_1=I.\] Proposition 1.13 proves these identities from the ring laws.

The multiplication operator \(M_V\) is the potential term of a lattice Schrödinger operator, added to a difference operator. The assignment \(f\mapsto M_f\) is how a field acts on its own module, and \(M_f M_g=M_g M_f\). A shift need not commute with \(M_f\). That product is the operator product of Proposition 1.13, and Example 1.22 computes the failure on a cycle.

Let \(R\) be a commutative unital ring. For \(R\)-modules \(M,N,P\), use \(\operatorname {Hom}_R(M,N)\) for additive, scalar-preserving maps. Thus linearity fixes \(\sigma =\operatorname {id}_R\). Given \(A\in \operatorname {Hom}_R(M,N)\) and \(B\in \operatorname {Hom}_R(N,P)\), write \[BA=B\circ A:M\to P.\] Composition is associative and \(R\)-bilinear where the types match. In particular, the endomorphisms \(\operatorname {End}_R(M)=\operatorname {Hom}_R(M,M)\) form an associative unital \(R\)-algebra, with unit \(I_M\) and central scalar map \(r\mapsto rI_M\).

On the vertex-field module, pointwise products are the multiplication operators. The map \(f\mapsto M_f\) is an injective \(R\)-algebra homomorphism into \(\operatorname {End}_R(\mathcal F_R(V))\). The proof evaluates those algebra laws. Example 1.22 exhibits endomorphisms that do not commute.

The Hom-module and algebra conventions are [stacks2017stacks, Section 10.10; Section 22.2], Tags 0581 and 09JF; the field multiplication assertions are the direct calculation above. Mathlib's Module.End.mul_eq_comp and Module.End.mul_apply give this right-factor-first product. These declarations do not assert that arbitrary maps with different endpoints form one algebra, or that a formal generator presentation is faithful.

For the ring-valued incidence map \(d=h^*-s^*:\mathcal F_R(V)\to \mathcal F_R(E)\) over a commutative unital ring \(R\) and fields \(f,g\in \mathcal F_R(V)\), one has \[d(fg)=(df)(h^*g)+(s^*f)(dg).\] Every factor on the right is an edge field, and the products are pointwise in \(\mathcal F_R(E)\). Equivalently, \[d(fg)=(df)(s^*g)+(s^*f)(dg)+(df)(dg).\]

This is a direct endpoint calculation. Its cyclic specialization is Lemma 1.20. The forward-difference convention is [wingate2025lattice, Section 4, Equations (4.2)-(4.3), p. 3]; that real-lattice convention is not an attribution of the general ring-valued incidence statement.

An action for a field on the four-cycle of Example 1.8 sums the squares of its differences along the edges, weighted by the edge weights \(w(e)\): that sum is the discrete form of \(\int |\nabla \varphi |^2\). A pairing sums products of the values themselves, the discrete form of \(\int fg\) when each site has unit volume. Both expressions need coefficients that can be squared and added, and a sum of squares that vanishes only when every term does. Which coefficient rings admit such a sum, and what does positivity require of the weights?

Specialize the field modules to \(R=\mathbb R\). The finite sets \(V,E\) and endpoint maps \(s,h:E\to V\) form a finite incidence presentation; its real specialization adds the scalar order needed for positive pairings and weights. A real vertex field is a function \(f:V\to \mathbb R\), written \(f\in \mathbb R^V\) with \(\mathbb R^V=\mathcal F_{\mathbb R}(V)\) and \(\mathbb R^E=\mathcal F_{\mathbb R}(E)\).

Their counting inner products are \[\langle f,g\rangle _V=\sum _{v\in V}f(v)g(v),\qquad \langle a,b\rangle _E=\sum _{e\in E}a(e)b(e).\] These finite sums are bilinear and symmetric. A sum of squares vanishes exactly when all its entries vanish, so they are positive definite, including on the zero space when an indexing set is empty. The real field values remain continuous even though the carrier is finite.

The real incidence map is the specialization \[D=d_{\mathbb R}:\mathbb R^V\longrightarrow \mathbb R^E,\qquad (Df)(e)=f(h(e))-f(s(e)).\] Its linearity is the specialization of the ring-valued incidence map. Loops have zero incidence, and parallel edges remain distinct. For an oriented loopless graph this is the sign convention of [spielman2009effective, Section 2.1, p. 4]; the endpoint formula extends it to loops. Connectedness and positive weights are not needed to define \(D\).

A positive edge-weight function \(w:E\to \mathbb R_{>0}\) is additional data. It acts on edge fields by \((Wa)(e)=w(e)a(e)\). Different weights change the supplied data. No retained subset belongs to this definition. The alternating four-cycle of Example 1.8 is one weighted instance.

For the counting inner product, Mathlib's EuclideanSpace is the corresponding finite function space and PiLp.inner_apply gives the sum of coordinate inner products. These declarations concern the pairing, rather than the incidence presentation as a whole.

The counting inner products are \(\int fg\) with unit volume at each site or edge. For a positive weight \(w(e)\) on each edge, \[\sum _{e\in E}w(e)\,(Df)(e)^2\] is \(\int |\nabla \varphi |^2\), and \(w(e)\) is the coupling on that edge. The Gaussian laws later in the notes need this positivity. A general commutative ring gives the difference; an order that makes a sum of squares positive is further structure.

Label the sites of a cycle by \(\mathbb Z/n\mathbb Z\). The difference of neighbouring values is the derivative one writes when the step has length \(1\). The product rule a difference obeys has a correction term, and on a cycle the last site is again the first: which rule holds when the shift wraps around, and what does it leave uncorrected?

Let \(R\) be a commutative unital ring, let \(n\ge 1\), and take \(V=\mathbb Z/n\mathbb Z\) and \(E=\{e_i:i\in V\}\), with \(s(e_i)=i\) and \(h(e_i)=i+1\). The edge-label bijection gives an \(R\)-algebra isomorphism \[j:\mathcal F_R(E)\to \mathcal F_R(V),\qquad j(a)(i)=a(e_i).\] Its inverse sends \(f\) to the edge field \(e_i\mapsto f(i)\); both preserve the pointwise operations. With the ring-valued incidence map, the cyclic shift and forward difference are \[\,(Tf)(i)=f(i+1),\qquad \Delta =jd=T-I.\] The shift is an algebra automorphism, with \(\,(T^{-1}f)(i)=f(i-1)\). Vertex and edge fields are identified only through the displayed map \(j\).

Let \(R\) be a commutative unital ring and let \(n\ge 1\). On \(\mathcal F_R(\mathbb Z/n\mathbb Z)\), let \(T\) and \(\Delta \) be the cyclic shift and forward difference.

For \(f,g\) in that module, \[\Delta (fg)=(\Delta f)(Tg)+f\Delta g.\] Equivalently, \[\Delta (fg)=(\Delta f)g+f\Delta g+(\Delta f)(\Delta g).\] Wingate defines forward and backward differences with lattice spacing \(a\) in [wingate2025lattice, Section 4, Equations (4.2)-(4.3), p. 3]. Here \(a=1\); the ring-valued finite identities are direct calculations, with no continuum approximation.

At \(n=1\), the cycle has a loop and \(T=I\), \(\Delta =0\). At \(n=2\), its two edges have opposite orientations between the same vertices; it is not a single simple edge. The unshifted Leibniz rule fails by the correction \(\,(\Delta f)(\Delta g)\). On the four-cycle, \(f=\delta _1\) satisfies \(f^2=f\); at vertex \(0\), \(\,(\Delta (f^2))(0)=1\) and \(\,((\Delta f)f+f\Delta f)(0)=0\), while \(\,((\Delta f)(Tf)+f\Delta f)(0)=1\). Lemma 3.6 adds a real pairing and identifies the positive kinetic factorization.

Mathlib's fwdDiff_smul proves the three-term formula for an additive commutative monoid, a ring and a module over it. Take the index monoid \(\mathbb Z/n\mathbb Z\), step \(1\), and the regular \(R\)-module \(R\). Scalar multiplication is ring multiplication. Substituting \(Tg=g+\Delta g\) yields the shifted formula; this marker does not claim the endpoint incidence or counting-adjoint theorem.

Let \(R\) be a commutative unital ring and \(n\ge 1\) an integer. On \(\mathcal F_R(\mathbb Z/n\mathbb Z)\), let \(T\) and \(\Delta =T-I\) be the cyclic shift and forward difference, and let \(M_f\) be the multiplication operator of a field \(f\in \mathcal F_R(\mathbb Z/n\mathbb Z)\). As actual endomorphisms of this one module, \[TM_f=M_{Tf}T,\qquad \Delta M_f=M_{\Delta f}T+M_f\Delta .\]

The operator product is fixed in Proposition 1.13. For the difference convention, see [wingate2025lattice, Section 4, Equations (4.2)-(4.3), p. 3]. The proof concerns these represented maps; it asserts no universal presentation for formal generators. Example 1.22 is a clock and a shift for which the two products differ.

Let \(N\ge 1\) and let \(R\) be a commutative unital ring containing an element \(\omega \) with \(\omega ^N=1\). On \(\mathbb Z/N\mathbb Z\) set \(\chi (i)=\omega ^i\), let \(U=M_\chi \) be the multiplication operator of \(\chi \), and let \(T\) be the cyclic shift. Lemma 1.21 gives \(TU=M_{T\chi }T\). Since \(\,(T\chi )(i)=\chi (i+1)=\omega ^{i+1}=\omega \chi (i)\), one has \(M_{T\chi }=\omega U\), and therefore \(TU=\omega UT\). When \(\omega \ne 1\) the two products differ. Also \(T^N=I\) and \(U^N=M_{\chi ^N}=I\). If \(\omega \) is primitive of order \(N\), the operators \(T\), \(U\) and the powers of \(\omega \) generate a finite Heisenberg group. The four-cycle is the case \(N=4\).

The continuum form is the commutator of differentiation with multiplication by the coordinate. For a smooth function \(f\), \[\frac {d}{dx}(xf)-x\frac {d}{dx}f=f,\] so \(\bigl [\tfrac {d}{dx},x\bigr ]=1\). With \(p=-i\hbar \,d/dx\) this is \([p,x]=-i\hbar \).

The same relation holds over the field \(\mathbb F_p\) when \(N\) divides \(p-1\). Then \(X^N-1\) divides \(X^{p-1}-1=\prod _{c\in \mathbb F_p^\times }(X-c)\). The divisor is separable, because \(p\) does not divide \(N\) and \(X^N-1\) shares no root with its derivative, so it has \(N\) distinct roots in \(\mathbb F_p\). Those roots form a subgroup \(G\). Let \(m\) be the least common multiple of the orders of its elements, and factor \(m\) into prime powers. For each prime power \(q\) in that factorization, some element of \(G\) has order divisible by \(q\), and a power of it has order \(q\). The product of one such element for each prime power has order \(m\): the factors commute, and their orders are coprime. Every element of \(G\) is a root of \(X^m-1\), and a field has at most \(m\) roots of that polynomial, so \(N\le m\). Orders in \(G\) divide \(N\), so \(m\) divides \(N\). Hence \(m=N\), and \(G\) contains a primitive \(N\)-th root.

Let \(R\) be a commutative unital ring, \(V\) a finite set and \(H=\mathcal F_R(V)\) its field module. The counting bilinear form is \[b(f,g)=\sum _{v\in V}f(v)g(v).\] It is symmetric. Write \(H^\vee =\operatorname {Hom}_R(H,R)\) for the dual module, using ordinary \(R\)-linear maps: both coefficient rings are \(R\) and \(\sigma =\operatorname {id}_R\), as in Remark [ca-001G]. The perfect counting duality is the pair of inverse \(R\)-linear maps \[\begin {aligned} \Phi :H&\longrightarrow H^\vee ,&\Phi (g)(f)&=b(f,g),\\ \Psi :H^\vee &\longrightarrow H,&(\Psi \ell )(v)&=\ell (\delta _v). \end {aligned}\] Here \(\delta _v(u)\) is \(1\) when \(u=v\), and \(0\) otherwise. Perfection means precisely that \(\Phi \) is an isomorphism.

This proof includes the empty set and the zero ring. It uses no order, positivity or division. A perfect form can have nonzero vectors of zero self-pairing: over \(\mathbb F_2\), the counting form on two points has \(b((1,1),(1,1))=0\). Conversely, separation of vectors alone does not imply perfection over a ring: the form \(b(x,y)=2xy\) on \(\mathbb Z\) cannot represent the functional \(x\mapsto x\). The Hom-module convention is [stacks2017stacks, Section 10.10], Tag 0581; the explicit inverse above supplies the finite counting assertion.

Let \(R\) be a commutative unital ring and \(A\) an associative unital \(R\)-algebra. An ordinary \(R\)-linear involution on \(A\) is an \(R\)-linear map \(a\mapsto a^\dagger \), with \(\sigma =\operatorname {id}_R\), satisfying \[1^\dagger =1,\qquad (ab)^\dagger =b^\dagger a^\dagger ,\qquad (a^\dagger )^\dagger =a.\] Thus it reverses products, preserves sums and satisfies \((ra)^\dagger =ra^\dagger \). The involution on the coefficient ring is explicitly the identity.

These are ordinary product-reversal laws: no sign depending on degree occurs. Over \(\mathbb C\), this convention is bilinear and does not introduce complex conjugation. A Hermitian adjoint needs different coefficient data. Neither a grading nor freeness selects an involution; the counting adjoint below is constructed from a specified pairing.

Mathlib separates the involutive law InvolutiveStar, product reversal StarMul, additive compatibility StarRing and scalar compatibility StarModule. Its starRingOfComm permits the identity involution on a commutative ring. These interfaces support the stated convention; the represented example requires its own proof. For the library attribution, see [mathlib2020lean].

Let \(R\) be a commutative unital ring and \(A=M_n(R)\). Define \((X^T)_{ij}=X_{ji}\). This gives an ordinary linear involution: \[(rX+Y)^T=rX^T+Y^T,\qquad I^T=I,\qquad (XY)^T=Y^TX^T,\qquad (X^T)^T=X.\]

For instance, transposing \(\left (\begin {smallmatrix}a&b\\c&d\end {smallmatrix}\right )\) exchanges \(b\) and \(c\). Over \(\mathbb C\), transpose is complex-linear, whereas conjugate transpose is not: \((iI)^*=-iI\ne iI^*\) when \(n>0\). Conjugate transpose is nevertheless real-linear, so the coefficient ring matters. The counting adjoint recovers ordinary transpose from its specified bilinear pairing.

These are entrywise deductions from the definition. Mathlib's Matrix.transpose_add, Matrix.transpose_smul, Matrix.transpose_mul and Matrix.transpose_transpose record the corresponding identities; see [mathlib2020lean].

Let \(R\) be a commutative unital ring, \(V\) a finite set and \(H=\mathcal F_R(V)\). Equip \(H\) with the perfect counting duality \(b,\Phi ,\Psi \), with \(H^\vee =\operatorname {Hom}_R(H,R)\). For \(a\in \operatorname {End}_R(H)\), define \[\begin {aligned} a^\vee :H^\vee &\longrightarrow H^\vee , &a^\vee (\ell )&=\ell \circ a,\\ a^\dagger &=\Psi \circ a^\vee \circ \Phi :H\longrightarrow H. \end {aligned}\] The counting adjoint \(a^\dagger \) is the unique ordinary \(R\)-linear endomorphism satisfying \[b(f,a^\dagger g)=b(af,g)\qquad (f,g\in H).\] It gives an ordinary \(R\)-linear involution on the composition algebra \(\operatorname {End}_R(H)\). Products act with the right factor first.

The real incidence construction in Definition 3.4 starts from \(D:\mathbb R^V\to \mathbb R^E\) and returns an adjoint in the opposite direction. Here \(a:H\to H\) is an endomorphism over the commutative ring \(R\). Both use the characterizing identity that moves the operator between the two counting-pairing slots. The existence and laws below follow from Lemma 2.1, independently of the real incidence proof.

The construction also applies when \(V\) is empty or \(R\) is the zero ring. Its representability is supplied by the explicit inverse to \(\Phi \), not by a field hypothesis or a positivity argument. Dual precomposition reverses composition before any pairing is chosen; the Hom convention is [stacks2017stacks, Section 10.10], Tag 0581. The proof here adds the specified symmetric counting identification with the dual.

Let \(R\) be a commutative unital ring and choose a partition \(V=V_0\sqcup V_1\) of a finite set. Put \(H=\mathcal F_R(V)\) and \(K_i=\mathcal F_R(V_i)\) for \(i=0,1\). Restriction and extension by zero are ordinary \(R\)-linear maps \[\pi _i:H\to K_i,\qquad \iota _i:K_i\to H,\] with \((\pi _i f)(v)=f(v)\) for \(v\in V_i\), and \((\iota _i u)(v)=u(v)\) on \(V_i\), zero elsewhere. The support submodules are \(H_i=\iota _i(K_i)\subseteq H\). The support projections \(p_i=\iota _i\pi _i:H\to H\) satisfy \[\pi _i\iota _i=I_{K_i},\quad \pi _i\iota _j=0\ (i\ne j),\quad p_i^2=p_i,\quad p_ip_j=0\ (i\ne j),\quad p_0+p_1=I_H.\] Every \(f\in H\) has a unique expression \(f=f_0+f_1\) with \(f_i\in H_i\), namely \(f_i=p_if\).

The partition is additional data, not part of the general incidence-field definition. If one part is empty, its field module and support projection are zero. Nothing requires division by \(2\); the same construction works in characteristic \(2\) and over the zero ring. This is the direct finite-function realization of a module decomposition.

[stacks2017stacks, Section 10.10], Tag 0581 supplies the module convention.

Let \(R\) be a commutative unital ring and \(A\) an associative unital \(R\)-algebra. An internal parity grading consists of \(R\)-submodules \(A_0,A_1\subseteq A\), with operations inherited from \(A\), such that every \(a\in A\) has a unique decomposition \(a=a_0+a_1\), with \(a_i\in A_i\), and \[1\in A_0,\qquad A_iA_j\subseteq A_{i+j}.\] Subscripts are added in \(\mathbb Z/2\mathbb Z\). An element of \(A_i\) is homogeneous of parity \(i\); parity \(0\) is even and parity \(1\) is odd. The zero element lies in both submodules. Scalars \(r1\) belong to \(A_0\).

The multiplication condition controls the component of a product; it does not require \(ab=(-1)^{ij}ba\) for \(a\in A_i\), \(b\in A_j\). Nor does it require an odd element to square to zero. Those are additional equations, tested explicitly by the four-cycle.

The internal-submodule viewpoint and multiplication rule follow [wieser2022graded, Section 1, accepted-manuscript pp. 1-2], here specialized to the index group \(\mathbb Z/2\mathbb Z\). Mathlib's GradedRing and GradedAlgebra encode graded multiplication and compatible scalar data. The chosen operator decomposition still needs the proof in Proposition 2.7.

Let \(R\) be a commutative unital ring, \(V=V_0\sqcup V_1\) a finite partition, and \(H=\mathcal F_R(V)\). Use its support submodules \(H_i\) and support projections \(p_i\). All subscripts below are in \(\mathbb Z/2\mathbb Z\). For \(A=\operatorname {End}_R(H)\) with composition product, put \[A_k=\{a\in A:a(H_i)\subseteq H_{i+k}\text { for }i=0,1\}.\] These submodules give an internal parity grading. The two components of \(a\in A\) are \[a_0=p_0ap_0+p_1ap_1,\qquad a_1=p_1ap_0+p_0ap_1.\] All products are compositions of maps \(H\to H\); these formulas require no choice of ordered basis.

If one part is empty, \(A_1=0\); if both parts are empty, the whole algebra is zero. The argument also allows the zero ring and characteristic \(2\). Graded endomorphism composition appears in [stacks2017stacks, Section 22.25, Example 22.25.5], Tag 09L1. That example is integer-graded; the direct proof here establishes the two-parity, finite-function instance.

Let \(R\) be a commutative unital ring and \(V=V_0\sqcup V_1\) a finite partition. Equip \(H=\mathcal F_R(V)\) with its counting pairing \(b\), counting adjoint constructed in Proposition 2.4, and parity grading \(A_0,A_1\) on \(\operatorname {End}_R(H)\). Then \(p_i^\dagger =p_i\) for the support projections, and \[a\in A_k\quad \Longrightarrow \quad a^\dagger \in A_k.\] Every multiplication operator \(M_f\), for \(f\in H\), is even and satisfies \(M_f^\dagger =M_f\).

The degree statement depends on this orthogonal support decomposition and this symmetric counting pairing. It is not a law for an unspecified grading and involution. The involution on coefficients is the identity, including for complex coefficients; no Hermitian or positive conclusion is inferred. The graded composition convention is [stacks2017stacks, Section 22.25], Tag 09L1; the compatibility with the chosen pairing is proved above.

Let \(R\) be a commutative unital ring. On the four-cycle, take \(V=\mathbb Z/4\mathbb Z\), \(H=\mathcal F_R(V)\) and the additional partition \(V_0=\{0,2\}\), \(V_1=\{1,3\}\). The cyclic shift \(T:H\to H\) is \((Tf)(i)=f(i+1)\). It exchanges the two support submodules: if \(f\) is supported on \(V_j\), then \(Tf\) is supported on \(V_{j+1}\). Hence \(T\) is odd in the chosen operator grading.

For the counting adjoint constructed over \(R\) in Proposition 2.4, reindexing the finite sum proves \[b(Tf,g)=\sum _i f(i+1)g(i) =\sum _j f(j)g(j-1)=b(f,T^{-1}g),\] so \(T^\dagger =T^{-1}\). Also \(T^4=I\); thus the half-turn \(T^2\) is even and satisfies \((T^2)^\dagger =T^{-2}=T^2\). These unweighted counting identities do not assert that the one-step shift preserves the alternating weights. The weight-preserving half-turn and the one-step counterexample remain those in Example 1.8.

The arrows show the action on basis fields, opposite to the source-to-head orientation on vertices. Every arrow switches support parity; two steps give the half-turn. In particular, over \(\mathbb Z\), \(T^2\delta _0=\delta _2\ne 0\). Thus an odd operator need not square to zero. It need not satisfy a graded commutation law either: that law applied to \(T,T\) would give \(2T^2=0\), whereas \(2T^2\delta _0=2\delta _2\ne 0\).

Identify edge \(i\) with vertex \(i\), with source \(i\) and head \(i+1\), as in Lemma 1.20. Only after this identification does incidence become the endomorphism \(\Delta =T-I\). Its even component is \(-I\) and its odd component is \(T\). Over \(\mathbb Z\), both are nonzero, so \(\Delta \) is inhomogeneous, and \[\Delta ^2\delta _0=\delta _2-2\delta _3+\delta _0\ne 0.\] The formulas remain valid over every commutative unital ring, but the nonzero witnesses just used are specifically over \(\mathbb Z\); in the zero ring all these fields and maps vanish. This is a direct specialization of the shift-multiplication realization and partition construction, not a square-zero differential supplied by parity.

[wieser2022graded, Section 1, accepted-manuscript pp. 1-2] gives the grading convention; the shift, adjoint and failure witnesses are the computations here.

The perfect counting duality represents every linear functional by a unique field. To make the same construction on another carrier, let \(R\) be a commutative unital ring and \(M\) a unital \(R\)-module. Write \(M^\vee =\operatorname {Hom}_R(M,R)\) for its dual module. Every linear map here has coefficient map \(\operatorname {id}_R\), as in Remark [ca-001G].

A chosen perfect symmetric pairing on \(M\) is a bilinear form \(b_M:M\times M\to R\) such that \(b_M(x,y)=b_M(y,x)\) and the linear map \[\Phi _M:M\longrightarrow M^\vee ,\qquad \Phi _M(y)(x)=b_M(x,y)\] is an isomorphism. Thus each functional \(\ell \) has exactly one representative \(\Phi _M^{-1}\ell \), characterized by \(b_M(x,\Phi _M^{-1}\ell )=\ell (x)\) for every \(x\). Bilinearity makes \(\Phi _M\) linear. Symmetry identifies the other currying map \(y\mapsto (x\mapsto b_M(y,x))\) with \(\Phi _M\), so both are isomorphisms. This agrees with the two-sided perfect-pairing convention in Mathlib's LinearMap.IsPerfPair. The ordinary Hom convention is [stacks2017stacks, Section 10.10], Tag 0581.

Injectivity alone says that \(b_M(x,y)=0\) for every \(x\) forces \(y=0\); symmetry gives the analogous assertion in the other slot. It does not give every functional a representative; see a zero-radical pairing that is not perfect.

No finiteness, basis or projectivity hypothesis is imposed on \(M\); the specified duality is the assumption. The counting form on every finite function module satisfies it by the explicit inverse in Lemma 2.1. The zero module has its unique perfect pairing. If \(R\) is the zero ring, every unital module is zero since \(x=1x=0x=0\), so this case is included. Perfection does not assert positivity or determine a bilinear form from its diagonal values.

On the free rank-one \(\mathbb Z\)-module \(M=\mathbb Z\), the symmetric bilinear form \(b(x,y)=2xy\) separates vectors in both slots but is not a perfect pairing. Its currying map is \[\Phi :\mathbb Z\longrightarrow \operatorname {Hom}_{\mathbb Z} (\mathbb Z,\mathbb Z),\qquad \Phi (y)(x)=2xy.\]

If \(b(x,y)=0\) for every integer \(x\), testing against \(x=1\) gives \(2y=0\), hence \(y=0\). Symmetry gives the same conclusion in the other slot, so both radicals are zero and \(\Phi \) is injective. Every integer-linear functional \(\ell \) on \(\mathbb Z\) satisfies \(\ell (x)=x\ell (1)\), so it is determined by its value at \(1\). But \(\Phi (y)(1)=2y\) is even for every \(y\). Thus \(\Phi \) misses the linear functional \(x\mapsto x\), whose value at \(1\) is \(1\), and is not surjective. Freeness of \(M\) does not repair this failure of the chosen form.

The Hom convention is [stacks2017stacks, Section 10.10], Tag 0581; the calculation above proves the failure for this chosen form. For the perfect counting form and its explicit inverse, see Lemma 2.1.

Let \(R\) be a commutative unital ring and let \(M,N,P\) be unital \(R\)-modules with chosen perfect symmetric pairings \(b_M,b_N,b_P\) and duality maps \(\Phi _M,\Phi _N,\Phi _P\). For an ordinary \(R\)-linear map \(A:M\to N\), precomposition defines \[A^\vee :N^\vee \longrightarrow M^\vee ,\qquad A^\vee \ell =\ell \circ A.\] Its adjoint for chosen perfect pairings is \[A^\dagger =\Phi _M^{-1}A^\vee \Phi _N:N\longrightarrow M.\] Products denote composition, with the right factor first, as in composition and field multiplication. This is the unique linear map satisfying \[b_M(x,A^\dagger y)=b_N(Ax,y)\qquad (x\in M,\ y\in N).\]

For \(A,A':M\to N\), \(B:N\to P\) and \(r\in R\), \[\begin {aligned} (A+A')^\dagger &=A^\dagger +(A')^\dagger ,& (rA)^\dagger &=rA^\dagger ,\\ 0^\dagger &=0,&I_M^\dagger &=I_M,\\ (BA)^\dagger &=A^\dagger B^\dagger ,& (A^\dagger )^\dagger &=A. \end {aligned}\] If \(A\) is an isomorphism, then \(A^\dagger \) is an isomorphism and \[(A^{-1})^\dagger =(A^\dagger )^{-1}.\] An endomorphism \(L:M\to M\) is self-adjoint for this pairing when \(L^\dagger =L\). In particular, the inverse of an invertible self-adjoint endomorphism is self-adjoint.

In particular, for \(d:M\to N\) and a self-adjoint \(W:N\to N\), the composite \(L=d^\dagger Wd:M\to M\) satisfies \[L^\dagger =d^\dagger W^\dagger d=L,\qquad b_M(f,Lg)=b_N(df,Wdg).\] The first equality uses product reversal and double adjunction; the second is the characterizing identity for \(d\). Taking \(g=f\) gives its energy identity, without any order assumption on \(R\).

For a finite function module with its counting form, \(\Phi ^{-1}=\Psi \) from Lemma 2.1; the formula is exactly \(\Psi A^\vee \Phi \) from Proposition 2.4. Hence it gives the same ordinary \(R\)-linear involution on endomorphisms. The proof above extends that direct construction to different carriers.

The Hom convention is [stacks2017stacks, Section 10.10], Tag 0581.

TauCeti's Module.dualEndAlgEquiv expresses reversed dual composition for reflexive modules. It concerns dual endomorphisms, before the chosen symmetric identifications used here. Mathlib's LinearMap.BilinForm.toDual constructs a duality from a nondegenerate form in the finite-dimensional field case. Here the dualities are explicitly assumed over \(R\); the argument above proves the resulting adjoint laws. A nontrivial coefficient involution and Hermitian forms require different scalar conventions.

Let \(R\) be a commutative unital ring, \(V,E\) finite sets and \(s,h:E\to V\) endpoint maps. The ring-valued incidence map is \(d_R:H_V\to H_E\), where \(H_V=\mathcal F_R(V)\), \(H_E=\mathcal F_R(E)\) and \(d_Rf(e)=f(h(e))-f(s(e))\). Equip both modules with the symmetric counting forms \[b_V(f,g)=\sum _v f(v)g(v),\qquad b_E(a,c)=\sum _e a(e)c(e).\] Their perfect counting dualities give the counting adjoint \(d_R^\dagger :H_E\to H_V\) by Proposition 3.3. It is characterized by \(b_E(d_Rf,a)=b_V(f,d_R^\dagger a)\).

The return map can be read directly from the endpoints: \[(d_R^\dagger a)(v) =\sum _{h(e)=v}a(e)-\sum _{s(e)=v}a(e).\]

For arbitrary weights \(w:E\to R\), let \(W:H_E\to H_E\) multiply pointwise by \(w\): \((Wa)(e)=w(e)a(e)\). The weighted graph Laplacian is \[L=d_R^\dagger Wd_R:H_V\longrightarrow H_V.\] Commutativity makes \(W\) self-adjoint, since \(b_E(Wa,c)=\sum _e w(e)a(e)c(e)=b_E(a,Wc)\). The weighted edge form \[b_{E,w}(a,c)=b_E(a,Wc)=\sum _e w(e)a(e)c(e)\] is symmetric bilinear. It is perfect if and only if every \(w(e)\) is a unit, meaning an invertible element of \(R\).

Even when some weights are nonunits, the represented return map \(d_R^\dagger W\) exists and satisfies \[b_{E,w}(d_Rf,a)=b_V(f,d_R^\dagger Wa).\] Its existence does not make the weighted edge form perfect. Thus it does not by itself give a two-sided adjoint involution for that form. All these ring identities are direct deductions from the chosen counting dualities. Their positive-real incidence convention is [spielman2009effective, Section 2.1, p. 4].

Orientation reversal leaves \(L\) unchanged. Choose which edges to reverse, and put \(\varepsilon (e)=-1\) on those edges and \(1\) on the others, interpreted in \(R\). The sign map \(Ja(e)=\varepsilon (e)a(e)\) satisfies \(J^\dagger =J\), \(J^2=I\), \(JW=WJ\), and the reversed incidence map is \(d'_R=Jd_R\). Therefore \[(d'_R)^\dagger Wd'_R=d_R^\dagger JWJd_R=d_R^\dagger Wd_R.\] The corresponding edge field is \(a'=Ja\), and \(b_{E,w}(Ja,Jc)=b_{E,w}(a,c)\). The proof includes characteristic two, loops and parallel edges. Endpoints and weights are otherwise fixed.

For the real incidence adapter, start with the incidence map \(D:\mathbb R^V\to \mathbb R^E\) and positive weights \(w(e)>0\). The counting forms are the counting inner products, and write \(D^*=d_{\mathbb R}^\dagger \). The edge form becomes the weighted inner product \[\langle a,c\rangle _{E,w}=\sum _e w(e)a(e)c(e).\] It is symmetric bilinear, and its self-pairing is a finite sum of nonnegative terms. Strictly positive weights make that sum zero only at the zero edge field. Keeping the vertex pairing counting, the weighted adjoint is \(D_w^\dagger =D^*W\), since \[\langle Df,a\rangle _{E,w} =\langle Df,Wa\rangle _E=\langle f,D^*Wa\rangle _V.\] Hence \(L=D^*WD=D_w^\dagger D\); \(D^*D\) is the unit-weight case.

For the coordinate recovery, use the ordered-value isomorphisms \(c_V,c_E\) and incidence matrix \(B=c_EDc_V^{-1}\). The endpoint return formula gives \(c_VD^*c_E^{-1}=B^T\). The multiplication map \(W\) has diagonal array \(\operatorname {diag}(w(e))\); writing the same letter for that array yields \([D_w^\dagger ]=B^TW\) and \([L]=B^TWB\). For a single edge \(0\to 1\) of weight \(c>0\), the two adjoints send an edge value \(a\) to \((-a,a)\) and \((-ca,ca)\), respectively. Under orientation reversal the sign array \(S\) gives \(B'=SB\), \(S^T=S\), \(S^2=I\), \(SWS=W\), and \((B')^TWB'=B^TWB\). A loop has a zero incidence row before and after reversal.

Mathlib's LinearMap.toMatrix_adjoint identifies adjoint matrices with conjugate transposes in orthonormal bases. Over the reals this is transpose. The coordinate edge basis is orthonormal for counting, but not generally for the weighted pairing; the return-map calculation above accounts for the extra \(W\).

Use the alternating four-cycle, with vertices \(0,1,2,3\), edges \(e_i:i\to i+1\), and positive real weights \((1,t,1,t)\). For a field \(f=(a,b,c,d)\), the incidence values are \(Df=(b-a,c-b,d-c,a-d)\). The counting adjoint sends an edge field to \[D^*(u_0,u_1,u_2,u_3) =(u_3-u_0,u_0-u_1,u_1-u_2,u_2-u_3).\] First multiply by \(W_t=\operatorname {diag}(1,t,1,t)\), then apply \(D^*\). The resulting \(L_t=D^*W_tD\) has coordinate matrix \[L_t=\begin {pmatrix} 1+t&-1&0&-t\\ -1&1+t&-t&0\\ 0&-t&1+t&-1\\ -t&0&-1&1+t \end {pmatrix}.\]

For example, its first coordinate is \(t(a-d)-(b-a)=(1+t)a-b-td\); the other coordinates follow from the same return formula. Thus \[L_t\delta _0=(1+t,-1,0,-t),\qquad L_t\mathbf 1=0,\] and the counting energy is \[\langle f,L_tf\rangle =(b-a)^2+t(c-b)^2+(d-c)^2+t(a-d)^2.\] The counting adjoint \(D^*\) does not depend on \(t\). The weighted return map \(D^*W_t\), and hence \(L_t\), do. When \(t=1\) the two return maps coincide; otherwise the factors \(t\) must be retained.

This calculation substitutes the four-cycle data into Definition 3.4; its incidence convention follows [spielman2009effective, Section 2.1, p. 4]. The cyclic energy factorization gives the corresponding operator identity.

Let \(R\) be a commutative unital ring. Take the cyclic shift on \(\mathcal F_R(\mathbb Z/n\mathbb Z)\), \(n\ge 1\), with its perfect counting duality \(b\). For the counting adjoint, write \(A^*=A^\dagger \). Then \[T^*=T^{-1},\qquad \Delta ^*=T^{-1}-I.\] Writing \(\nabla ^-=I-T^{-1}\) for the backward difference gives \[\Delta ^*\Delta =-\nabla ^-\Delta =2I-T-T^{-1}.\] Here \(2=1+1\); its invertibility is not required. For the real difference and kinetic signs, see [wingate2025lattice, Section 4, Equations (4.3) and (4.5), p. 3], with lattice spacing one. The finite periodic ring identities follow by reindexing, independently of the product rule.

For the indexed cycle \(e_i:i\to i+1\), the edge-label isomorphism \(j(a)(i)=a(e_i)\) of the cyclic shift and edge-label identification preserves counting pairings: \(b_V(ja,jc)=\sum _i a(e_i)c(e_i)=b_E(a,c)\). Taking \(c=j^{-1}f\) shows that \(j^*=j^{-1}\), hence \(j^*j=I\). Thus \(\Delta =jd_R\) and \[\Delta ^*\Delta =d_R^\dagger j^*j d_R=d_R^\dagger d_R,\] the unit-weight graph Laplacian. At \(n=2\), the two oppositely oriented edges give \(b(f,d_R^\dagger d_Rf)=2(f(1)-f(0))^2\); they are not one simple edge. At \(n=1\), there is one loop, \(T=I\), and both incidence and kinetic maps vanish. An arbitrary incidence map still has distinct vertex and edge carriers; this factorization does not assert that \(\Delta \) is skew-adjoint or square-zero.

Let \(R\) be a commutative unital ring, \(V,E,s,h\) finite incidence data and \(w:E\to R\) arbitrary weights. With the counting forms and weighted graph Laplacian \(L=d_R^\dagger Wd_R\), one has \[L^\dagger =L,\qquad b_V(f,Lg)=\sum _e w(e)(d_Rf)(e)(d_Rg)(e).\] The kernel of \(d_R\) consists exactly of fields constant on each connected component of the underlying undirected graph, and \(\ker d_R\subseteq \ker L\). Two vertices are in the same component when a finite path joins them, with edge orientation ignored.

The energy is unchanged by edge-orientation reversal: each reversed edge negates both factors in its summand. Equivalently, the orientation-invariant return composite gives the same Laplacian.

Perfection and unit weights do not give the reverse kernel inclusion. Over \(\mathbb F_2\), take two vertices \(0,1\) and two distinct edges, both oriented \(0\to 1\), both of weight \(1\). The counting pairings are perfect and \(W=I\) is invertible. Nevertheless, \[\begin {aligned} d_R(a,b)&=(b-a,b-a),\\ d_R^\dagger (u_1,u_2)&=(-u_1-u_2,u_1+u_2),\\ L(a,b)&=(0,0). \end {aligned}\] For \(\delta _1=(0,1)\), one has \(d_R\delta _1=(1,1)\ne 0\), but its energy is \(1+1=0\). Hence \(\ker L=\mathbb F_2^2\) strictly contains \(\ker d_R=\{(a,a):a\in \mathbb F_2\}\). This failure comes from the absence of real positivity, not from nonunit weights or nonperfect forms.

Now specialize to \(R=\mathbb R\) and \(w(e)>0\). Write \(D=d_{\mathbb R}\). The Laplacian is symmetric and positive semidefinite, and \[\ker L=\ker D,\qquad \dim \ker L=k,\] where \(k\) is the number of connected components. The energy convention and connected-graph argument appear in [spielman2009effective, Section 2.1, pp. 4-5]; the ring identity above and the disconnected, loop and parallel-edge extensions are proved directly here.

If \(V\) is nonempty and connected, the kernel is the one-dimensional space spanned by the constant field \(\mathbf 1_V\). Its positive unit mode is \[|V|^{-1/2}\mathbf 1_V.\] An isolated vertex contributes one component and one kernel basis vector. If \(V\) is empty, the maps \(s,h:E\to V\) force \(E\) to be empty too: both the component count and kernel dimension are zero. Loops contribute zero energy, while distinct parallel edges each contribute their own positive summand.

For the alternating four-cycle, put \(u=1+t\). Direct multiplication gives \[L_t=\begin {pmatrix} u&-1&0&-t\\ -1&u&-t&0\\ 0&-t&u&-1\\ -t&0&-1&u \end {pmatrix}.\] For \(f=(a,b,c,d)\), the energy is \[\begin {aligned} f^TL_tf={}&(b-a)^2+t(c-b)^2\\ &+(d-c)^2+t(a-d)^2. \end {aligned}\] Every row of \(L_t\) sums to zero, and for \(t>0\) zero energy forces all 4 values equal. The comparison with the positive action identifies how this edge energy combines with a mass term.

The hypotheses have concrete consequences. With vertices \(0,1,2,3\) and just the unit-weight edges \(0\to 1\) and \(2\to 3\), the matrix has two diagonal blocks \(\bigl (\begin {smallmatrix}1&-1\\-1&1\end {smallmatrix}\bigr )\). Its kernel is \(\{(a,a,b,b):a,b\in \mathbb R\}\); connectivity is needed for a single constant mode.

On two vertices with one edge \(0\to 1\) of weight zero, the underlying graph is connected but \(L=0\), so its kernel is all of \(\mathbb R^2\). For nonnegative weights the same proof identifies the kernel using only the positive-weight edges. A zero weight also makes the proposed edge pairing degenerate. If instead this edge has weight \(-1\), then \[f^TLf=-(f(1)-f(0))^2,\] which is negative at \(f=(0,1)\). Positivity fails, and the edge form is not an inner product. This example is negative semidefinite; two disjoint edges with weights \(1\) and \(-1\) give an indefinite form, since fields supported on the separate edges have energies of opposite signs.

For unweighted simple graphs, Mathlib's SimpleGraph.posSemidef_lapMatrix proves positivity of its Laplacian. That model has no separate weighted edge set; its energy formula sums both adjacency directions and divides by two. The edge-once sum above also handles loops and parallel edges, and is proved directly here.

Let \(M\) be a unital module over a commutative unital ring \(R\), with a chosen perfect symmetric pairing \(b\). Let \(L:M\to M\) be self-adjoint, and let \(\mu \in R\). The endomorphism \[Q=L+\mu I_M:M\longrightarrow M\] is self-adjoint. Assume additionally that \(Q\) is an isomorphism, and put \(G=Q^{-1}\). Then \(G^\dagger =G\), and for every source \(j\in M\) the equation \[Qf=j\] has the unique solution \(f=Gj\).

This is a direct consequence of the adjoint and inverse laws and the two-sided inverse convention of [stacks2017stacks, Definition 4.2.4], Tag 0017.

For \(M=\mathcal F_R(V)\) on a finite set with its perfect counting duality, define the point kernel only after this inverse map exists: \[G(v,u)=(G\delta _u)(v).\] The indicator expansion and linearity give \[(Gf)(v)=\sum _{u\in V}G(v,u)f(u).\] Moreover, \[\begin {aligned} G(v,u)&=b(\delta _v,G\delta _u)\\ &=b(G\delta _v,\delta _u)=G(u,v). \end {aligned}\] Thus the symmetric array presents the already defined inverse. On the zero module the unique endomorphism is the identity and is invertible; for an empty carrier the kernel and these sums are empty.

A unit mass coefficient does not ensure invertibility over a ring. Over \(\mathbb Z\), take one unit-weight edge \(0\to 1\), the Laplacian \(L(a,b)=(a-b,b-a)\), and \(\mu =1\). Then \[Q(a,b)=(2a-b,2b-a).\] If \(Q(a,b)=(1,0)\), the second equation gives \(a=2b\), and the first gives \(3b=1\), impossible in \(\mathbb Z\). Thus \(Q\) is not onto, even though the mass and edge weight are units. The real massive Green operator acquires invertibility from positive energy and finite dimension. No positivity, spectrum or probability law is asserted over an arbitrary \(R\).

To eliminate part of a field, first choose which part is retained. Let \(R\) be a commutative unital ring and \(M,X,Y\) unital \(R\)-modules. A chosen binary splitting of \(M\) consists of ordinary \(R\)-linear maps \[\iota _X:X\to M,\quad \iota _Y:Y\to M,\qquad \pi _X:M\to X,\quad \pi _Y:M\to Y\] satisfying all 5 identities \[\begin {aligned} \pi _X\iota _X&=I_X,&\pi _Y\iota _Y&=I_Y,\\ \pi _X\iota _Y&=0,&\pi _Y\iota _X&=0,\\ \iota _X\pi _X+\iota _Y\pi _Y&=I_M. \end {aligned}\] Products apply the right factor first, as in composition and field multiplication. The maps \((x,y)\mapsto \iota _Xx+\iota _Yy\) and \(z\mapsto (\pi _Xz,\pi _Yz)\) are inverse linear maps by these identities. We consequently write \(M=X\oplus Y\) and \(z=(x,y)\) relative to this chosen identification.

Each summand is included in \(M\) and recovered by its projection. The composite \(\pi _X\iota _X\) is \(I_X\), whereas \(\iota _X\pi _X\) is an endomorphism of \(M\). In the partition realization, these are restriction and extension by zero; the latter composite is the support projection.

For an endomorphism \(Q:M\to M\), its components for the chosen splitting are \[\begin {aligned} A&=\pi _XQ\iota _X:X\to X,&B&=\pi _XQ\iota _Y:Y\to X,\\ E&=\pi _YQ\iota _X:X\to Y,&C&=\pi _YQ\iota _Y:Y\to Y. \end {aligned}\] Inserting \(I_M=\iota _X\pi _X+\iota _Y\pi _Y\) gives \[Q(x,y)=(Ax+By,Ex+Cy).\] If the eliminated component \(C\) is an isomorphism, define the split Schur map, relative to this splitting and pivot, by \[S=A-BC^{-1}E:X\longrightarrow X.\] Changing the splitting or the pivot changes the construction.

The module convention is [stacks2017stacks, Section 10.10], Tag 0581. Mathlib's CategoryTheory.Biprod.ofComponents gives components in a preadditive category with binary biproducts. Its arguments index source then target; after ordering summands \((Y,X)\), the arguments corresponding to \(Q\) are \((C,B,E,A)\).

[boyd2004convex, Appendix A.5.5, pp. 650-651] treats the real Schur complement. The source eliminates its first block: exchanging its variables to \((y,x)\) makes its block order \((C,E,B,A)\) and its residual \(A-BC^{-1}E\). The chosen-splitting definition above uses only linear maps; it does not assume a basis, pairing or positivity.

The ring elimination example computes the components and Schur map for a concrete chosen splitting.

Let \(R\) be a commutative unital ring and \(M=X\oplus Y\) a chosen binary splitting of unital \(R\)-modules. Let \(Q(x,y)=(Ax+By,Ex+Cy)\) have the typed components of this splitting, and assume \(C:Y\to Y\) is an isomorphism. Set \[K=C^{-1}E:X\to Y,\qquad S=A-BC^{-1}E:X\to X.\] Define three endomorphisms of \(M\) by \[\begin {aligned} U(x,y)&=(x+BC^{-1}y,y),\\ D(x,y)&=(Sx,Cy),\\ V(x,y)&=(x,y+Kx). \end {aligned}\] Then \(U,V\) are isomorphisms and, with right-factor-first composition, \[Q=UDV.\]

[boyd2004convex, Appendix C.4.1, pp. 672-673] gives real block-system elimination. The proof above extends it directly to modules. The source uses first-pivot order; swapping to the eliminated summand gives the convention here. No finite rank, projectivity or determinant is used.

Mathlib's CategoryTheory.Biprod.gaussian' supports elimination in a preadditive category with binary biproducts and an invertible first component. In the order \((Y,X)\), use \((C,B,E,A)\) as its component arguments. Its notation \(f\gg g\) applies \(f\) first, so its residual is \(A-BC^{-1}E\). For finite coordinate index types, Matrix.fromBlocks_eq_of_invertible₂₂ gives the corresponding matrix factorization over a commutative ring; its 4 arguments become \((A,B,E,C)\). These support the factorization, while the subsequent inverse and kernel statements have their own proofs.

Over a commutative unital ring \(R\), let \(M=X\oplus Y\) be a chosen splitting, with ordinary linear endomorphism \(Q(x,y)=(Ax+By,Ex+Cy)\). Assume that the eliminated component \(C\) is an isomorphism, and put \(S=A-BC^{-1}E\). Then \(Q\) is an isomorphism if and only if \(S\) is an isomorphism. When these conditions hold, all 4 components of the inverse are \[\begin {aligned} \pi _XQ^{-1}\iota _X&=S^{-1}:X\to X,\\ \pi _XQ^{-1}\iota _Y&=-S^{-1}BC^{-1}:Y\to X,\\ \pi _YQ^{-1}\iota _X&=-C^{-1}ES^{-1}:X\to Y,\\ \pi _YQ^{-1}\iota _Y&=C^{-1}+C^{-1}ES^{-1}BC^{-1}:Y\to Y. \end {aligned}\] In particular, the retained compression of \(Q^{-1}\) is \(S^{-1}\); it is not in general \(A^{-1}\).

The real symmetric inverse formula is [boyd2004convex, Appendix A.5.5, p. 650]; its first-pivot convention is exchanged here. The direct proof uses arbitrary unital \(R\)-modules and needs no pairing or symmetry. Mathlib's CategoryTheory.Biprod.isoElim' constructs the residual isomorphism when both the first component and the whole biproduct map are isomorphisms. With summands \((Y,X)\) and component arguments \((C,B,E,A)\), it supports the whole-to-residual direction; it does not by itself assert the 4 inverse formulas above.

For a self-adjoint precision with a chosen perfect pairing, inverse response supplies its separate response interpretation. The cycle calculation exhibits a retained inverse entry \(2/15\) where \(A^{-1}\) has entry zero. No probability law is needed for the map comparison.

Let \(R\) be a commutative unital ring and choose \(M=X\oplus Y\) with components \(Q(x,y)=(Ax+By,Ex+Cy)\). Assume \(C:Y\to Y\) is an isomorphism and set \(S=A-BC^{-1}E\). For every \(u\in X,v\in Y\), \[Q(x,y)=(u,v)\] is equivalent to the two equations \[y=C^{-1}(v-Ex),\qquad Sx=u-BC^{-1}v.\] No invertibility of \(S\) is required. In particular, the maps \[\begin {aligned} J_S:\ker S&\longrightarrow \ker Q,& x&\longmapsto (x,-C^{-1}Ex),\\ \pi _X:\ker Q&\longrightarrow \ker S,& (x,y)&\longmapsto x \end {aligned}\] are inverse \(R\)-linear maps. Here each kernel is the submodule of elements sent to zero.

[boyd2004convex, Appendix C.4.1, pp. 672-673] gives real linear-system elimination. The direct argument above extends it to modules. After exchanging the source variables to \((y,x)\) and the right side to \((v,u)\), its first-pivot equations become the displayed formulas. The kernel correspondence does not assume that the whole system is invertible.

The pivot hypothesis cannot be deduced from invertibility of \(Q\). Over a nonzero ring take \(X=Y=R\) and \(Q(x,y)=(y,x)\). Then \(Q^2=I\), but its component \(C=0:R\to R\) is not an isomorphism. These formulas cannot invert that prescribed pivot.

Zero summands cause no exception. If \(Y=0\), its unique endomorphism is its identity and is invertible; then \(S=A\) and elimination changes nothing. If \(X=0\), \(S\) is the unique invertible endomorphism of zero and \(Q=C\). Over the zero ring every unital module is zero, because \(x=1x=0x=0\); all statements remain valid.

Let \(X,Y\) be unital modules over a commutative unital ring \(R\), with chosen perfect symmetric pairings \(b_X,b_Y\). On the chosen sum \(M=X\oplus Y\), choose the orthogonal sum form \[b_M((x,y),(x',y'))=b_X(x,x')+b_Y(y,y').\] It is perfect and symmetric, and its injection/projection adjoints are \[\iota _X^\dagger =\pi _X,\quad \pi _X^\dagger =\iota _X,\qquad \iota _Y^\dagger =\pi _Y,\quad \pi _Y^\dagger =\iota _Y.\] Suppose \(Q:M\to M\) is self-adjoint, with components \(A,B,E,C\) and invertible \(C\). Then \[A^\dagger =A,\quad C^\dagger =C,\quad E=B^\dagger ,\qquad S^\dagger =S\quad \text {for }S=A-BC^{-1}E.\] For the triangular factors, \[V^\dagger =U,\qquad Q=V^\dagger DV,\qquad D(x,y)=(Sx,Cy).\] Consequently, with \(K=C^{-1}E\), \[\begin {aligned} b_M((x,y),Q(x,y)) &=b_X(x,Sx)\\ &\quad +b_Y(y+Kx,C(y+Kx)). \end {aligned}\]

This is a direct deduction from chosen dualities and triangular factorization. The real symmetric completion in [boyd2004convex, Appendix A.5.5, p. 650] is its real coordinate specialization. The chosen orthogonal sum is essential: an arbitrary perfect form on \(M\) need not have these injection/projection adjoints. No division by two or recovery of bilinear data from diagonal values is used; the identity remains valid in characteristic two.

TauCeti's LinearMap.BilinForm.prod gives the orthogonal product form, and LinearMap.BilinForm.nondegenerate_prod_iff identifies its radical nondegeneracy. The functional-splitting proof above also proves the surjectivity required for perfection. These declarations alone do not assert the paired-congruence result.

Let \(X,Y\) be finite-dimensional real vector spaces with positive definite inner products \(b_X,b_Y\). Give \(M=X\oplus Y\) their orthogonal sum, and let \(Q:M\to M\) be self-adjoint, with components \(A,B,E,C\). For a self-adjoint map \(T\) on any of these spaces, write \(T>0\) when \(b(z,Tz)>0\) for every nonzero \(z\).

If \(Q>0\), then \(C\) is positive definite and invertible, and \(S=A-BC^{-1}E\) is positive definite and invertible. Conversely, if \(C\) is invertible, then \[Q>0\quad \Longleftrightarrow \quad C>0\ \text {and}\ S>0.\]

[boyd2004convex, Appendix A.5.5, p. 650] gives the positive-definite Schur criterion. Here the eliminated summand is placed second.

This uses finite-dimensional real positivity in addition to the ring-level factorization. It explains the positivity assumption in real effective precision. It does not evaluate an integral or determine a normalization factor; those require the separately specified real measure and Gaussian law.

Use the four-cycle endpoints \(s(e_i)=i\), \(h(e_i)=i+1\) on \(\mathbb Z/4\mathbb Z\), but now let \(R\) be any commutative unital ring and take \(t,\mu \in R\). Give successive edges weights \((1,t,1,t)\). On the finite function modules with their counting pairings, set \[Q=d^\dagger Wd+\mu I,\qquad s=1+t+\mu .\] Here \(d\) is incidence and \(W\) multiplies each edge field by its weight. This extends the positive-real weight choice in Example 1.8; no weight is assumed positive or invertible.

First compute from endpoints. The difference is \((df)(e_i)=f(i+1)-f(i)\). Grouping the incoming and outgoing edge terms gives \[(d^\dagger Wdf)(i) =w(e_{i-1})(f(i)-f(i-1))+w(e_i)(f(i)-f(i+1)).\] Adding \(\mu f(i)\) yields \[\begin {aligned} (Qf)(0)&=sf(0)-f(1)-tf(3),\\ (Qf)(1)&=sf(1)-f(0)-tf(2),\\ (Qf)(2)&=sf(2)-tf(1)-f(3),\\ (Qf)(3)&=sf(3)-tf(0)-f(2). \end {aligned}\]

Choose the additional partition \(V_0=\{0,2\}\), \(V_1=\{1,3\}\), and put \(X=\mathcal F_R(V_0)\), \(Y=\mathcal F_R(V_1)\). Restriction and extension by zero give the chosen splitting. The resulting component maps are \[\begin {aligned} A&=sI_X,&C&=sI_Y,\\ (Ex)(1)&=-x(0)-tx(2),&(Ex)(3)&=-tx(0)-x(2),\\ (By)(0)&=-y(1)-ty(3),&(By)(2)&=-ty(1)-y(3). \end {aligned}\] The pivot \(C\) is invertible exactly when \(s\) is a unit. A unit gives \(C^{-1}=s^{-1}I_Y\). Conversely, if \(CJ=I_Y\), evaluation of \(CJ\delta _1=\delta _1\) at vertex \(1\) gives \(s(J\delta _1)(1)=1\); commutativity makes this a two-sided inverse of \(s\). In the zero ring \(0=1\) is a unit, consistently with this argument.

Assume this unit condition. Applying \(B\) to \(Ex\) gives \[\begin {aligned} (BEx)(0)&=(1+t^2)x(0)+2tx(2),\\ (BEx)(2)&=2tx(0)+(1+t^2)x(2). \end {aligned}\] Consequently the split Schur map is \[\begin {aligned} (Sx)(0)&=[s-s^{-1}(1+t^2)]x(0)-2ts^{-1}x(2),\\ (Sx)(2)&=[s-s^{-1}(1+t^2)]x(2)-2ts^{-1}x(0). \end {aligned}\] Here \(2=1+1\); neither \(2\), \(t\), \(\mu \) nor \(S\) is asserted to be invertible. In coordinates ordered \((0,2)\), the same map is \[[S]=\begin {pmatrix} s-s^{-1}(1+t^2)&-2ts^{-1}\\ -2ts^{-1}&s-s^{-1}(1+t^2) \end {pmatrix}.\] This array presents the map just computed from endpoints.

The half-turn exchanges \(0\leftrightarrow 2\) and \(1\leftrightarrow 3\). Let \(r_X,r_Y\) exchange the corresponding field values. The component formulas give \(Ar_X=r_XA\), \(Cr_Y=r_YC\), \(Er_X=r_YE\) and \(Br_Y=r_XB\); hence \(Sr_X=r_XS\). A one-step shift exchanges the two parts and fails to preserve the alternating weights when \(t\ne 1\). No retained partition is implicit in general incidence data.

Now specialize to \(R=\mathbb R\), \(t>0\), \(\mu =m^2>0\). The weighted energy identity gives, for \(f\ne 0\), \[b(f,Qf)=\sum _e w(e)(df(e))^2+m^2\sum _v f(v)^2>0.\] Thus \(Q\) is positive definite and invertible, and the positive-real criterion gives \(S>0\). At \(t=m=1\), \(s=3\) and \[[S]=\frac 13\begin {pmatrix}7&-2\\-2&7\end {pmatrix}.\] To solve \(Sx=\delta _2\), solve \(7x(0)-2x(2)=0\) and \(-2x(0)+7x(2)=3\). These give \(x(2)=7/15\) and \(x(0)=2/15\). Therefore \[(\pi _XQ^{-1}\iota _X\delta _2)(0) =(S^{-1}\delta _2)(0)=\frac {2}{15}.\] By contrast, fixing the eliminated field to zero leaves the retained map \(A=3I_X\), and \((A^{-1}\delta _2)(0)=0\). These are different map operations.

At real \(\mu =0,t>0\), the pivot \(C=(1+t)I_Y\) is still invertible. But \(s-s^{-1}(1+t^2)=2t/(1+t)\), so the nonzero constant retained field satisfies \(S1=0\). The kernel correspondence then recovers the constant full field in \(\ker Q\), since \(-C^{-1}E1=1\). An invertible pivot does not imply an invertible full precision.

[spielman2009effective, Section 2.1, pp. 4-5] gives the incidence sign convention for positive real weights. The arbitrary-ring identities use the direct counting construction of Definition 3.4.

[boyd2004convex, Appendix C.4.1, pp. 672-673] gives the real elimination convention. All coefficients and inverse entries here were computed from the specified endpoints and weights. They recover the algebraic values in real alternate-vertex elimination; the covariance and marginal-law interpretation there additionally uses the Gaussian measure. No massless Gaussian formula follows by substituting \(\mu =0\) into a massive normalization.

Let \(R\) be a commutative unital ring, allowing \(1=0\), and let \(A\) be a unital associative algebra over \(R\). Its scalar map is \(\eta :R\to A\), \(r\mapsto r1_A\), with central image. A normalized linear functional is an ordinary \(R\)-linear map \(\omega :A\to R\) satisfying \(\omega (1_A)=1_R\). Thus \(\omega (ra+sb)=r\omega (a)+s\omega (b)\); the coefficient map for linearity is \(\mathrm {id}_R\), not a nontrivial semilinear twist. This is a member of the dual module, with an additional normalization condition.

Normalization is equivalent to \(\omega \circ \eta =\mathrm {id}_R\). In particular, a normalized functional can exist only if \(\eta \) is injective. This necessary condition alone is not a claim of existence.

For example, the scalar map \(\mathbb Z\to \mathbb Z/2\mathbb Z\) is not injective, so this algebra admits no normalized \(\mathbb Z\)-linear functional. If \(A\) is the zero algebra, normalization is possible exactly when \(R\) is the zero ring. If \(R\) is the zero ring, every unital \(R\)-module is zero, since \(a=1_Ra=0_Ra=0\); its unique functional is normalized.

The algebra conventions are those of [stacks2017stacks, Section 22.2], Tag 09JF and [jadczyk2019notes, Definition 1.6, p. 4]. Mathlib's Module.Dual supplies the ordinary linear dual; normalization and the scalar left-inverse criterion are the additional conditions proved here. No order, involution or topology is part of this definition.

Let \(V\) be a finite vertex set and \(S\) a finite set of possible values. A configuration is a function \(\sigma :V\to S\); the configuration set is \(\Omega =S^V\). The set \(\Omega \) is distinct from the vertex set \(V\): an observable takes a whole configuration as its input. For a commutative unital ring \(R\), set \[A_\Omega =\operatorname {Map}(\Omega ,R).\] This is the pointwise function algebra: \((f+g)(\sigma )=f(\sigma )+g(\sigma )\), \((fg)(\sigma )=f(\sigma )g(\sigma )\) and \((rf)(\sigma )=rf(\sigma )\), with unit the constant function \(1_R\).

For each \(\sigma \in \Omega \), evaluation \(\operatorname {ev}_\sigma (f)=f(\sigma )\) is a unital \(R\)-algebra map and a normalized linear functional. The scalar map sends \(r\) to the constant function \(r\). The indicator \(\delta _\sigma \) is \(1_R\) at \(\sigma \) and \(0_R\) elsewhere. These are the same indicators used in counting duality for finite functions, now indexed by configurations.

If \(V=\varnothing \), then \(S^V\) contains the unique empty function, even when \(S=\varnothing \). If \(S=\varnothing \) and \(V\ne \varnothing \), then \(\Omega =\varnothing \); its function algebra is the zero algebra. There is then no evaluation at a configuration.

The underlying unital algebra convention is [stacks2017stacks, Section 22.2], Tag 09JF. Mathlib's Pi.algebra, Pi.evalAlgHom and Pi.constAlgHom provide the pointwise algebra, evaluation and constant algebra maps. The configuration interpretation and the empty-set cases above are direct consequences of these definitions.

For a finite set \(\Omega \) and commutative unital ring \(R\), every \(R\)-linear functional on the function algebra has a unique weight function \(p:\Omega \to R\), given by \[\omega (f)=\sum _{x\in \Omega }p(x)f(x),\qquad p(x)=\omega (\delta _x).\] Conversely, every such weight function defines an \(R\)-linear functional. It is normalized exactly when \(\sum _xp(x)=1_R\).

When \(\Omega =\varnothing \), the sole functional is the zero map from the zero algebra and its total weight is \(0_R\); it is normalized exactly when \(R\) is the zero ring. When \(\Omega \ne \varnothing \), a chosen evaluation is normalized over every \(R\). Uniform weights \(p(x)=c\) on a set of size \(n>0\) are normalized exactly when \((n1_R)c=1_R\). Such normalized uniform weights exist exactly when \(n1_R\) is a unit, and then the unique coefficient is \(c=(n1_R)^{-1}\). Thus failure of uniform normalization does not imply failure of all normalized functionals.

More generally, for raw weights \(w(x)\) with total \(Z=\sum _xw(x)\), a common rescaling \(p(x)=cw(x)\) is normalized exactly when \(cZ=1_R\). This is possible exactly when \(Z\) is a unit, and then \(c=Z^{-1}\) is unique. No division by a nonunit is defined by this formula. In the zero ring the sole element is a unit, so the same criterion applies. For real nonnegative weights, \(Z>0\) exactly when at least one weight is positive; all-zero weights cannot be normalized by rescaling.

This is the indicator-basis argument of finite function duality, with the unital algebra convention of [stacks2017stacks, Section 22.2], Tag 09JF. For real nonnegative normalized weights it gives the finite expectation. No positivity or division is needed for the ring-valued classification.

Let \(K=\mathbb R\) or \(K=\mathbb C\), with conjugation the identity in the real case. Let \(A\) be a unital associative \(K\)-algebra equipped with an additive operation \(a\mapsto a^*\) satisfying \[\begin {aligned} (\lambda a)^*&=\overline \lambda a^*,& (ab)^*&=b^*a^*,\\ (a^*)^*&=a,&1^*&=1. \end {aligned}\] There is no graded sign in the product law. A \(K\)-linear functional \(\omega :A\to K\) is Hermitian positive when \[\omega (a^*)=\overline {\omega (a)},\qquad \omega (a^*a)\in \mathbb R_{\geq 0}\quad (a\in A).\] A real algebraic state, respectively a complex algebraic state, is such a functional that is also normalized. The scalar field and involution are part of the assertion.

Over \(\mathbb C\), nonnegativity on all \(a^*a\) already implies Hermiticity, without assuming normalization. Over \(\mathbb R\), Hermiticity is a separate requirement, as the real dual-number example shows.

For these conventions, see [waldmann2004states, Section 3.1, p. 9; Definition 3.7 and Definition 3.9 with Remark 3.10(iv), p. 11]. Waldmann works over a complexification of an ordered ring. Here the coefficients are the ordinary fields \(\mathbb R\) and \(\mathbb C\); the real Hermiticity requirement is explicit. On p. 9, Definition 3.1 prints \(R=-P\sqcup \{0\}\sqcup R\); the intended final \(P\), rather than the printed \(R\), is a mathematical emendation, not a verbatim transcription.

The ordinary linear involution is not a substitute for conjugate scalar compatibility over \(\mathbb C\). With a complex-linear unital involution, \((i1)^*(i1)=-1\); a normalized functional takes this to \(-1\), contradicting square positivity. The earlier chosen bilinear adjoints likewise do not supply this Hermitian structure. No norm, continuity or operator representation is assumed.

In the real algebra \(A=\mathbb R[\varepsilon ]/(\varepsilon ^2)\), each element is uniquely \(x+y\varepsilon \). Define \((x+y\varepsilon )^*=x-y\varepsilon \). For any fixed real \(c\ne 0\), let \[\omega _c(x+y\varepsilon )=x+cy.\] Then \(\omega _c\) is normalized and nonnegative on every \(a^*a\), but is not a real algebraic state, because it is not Hermitian.

This direct calculation distinguishes square positivity from the extra real Hermiticity condition. Square positivity is defined in [waldmann2004states, Definition 3.9, p. 11]. It is separate from the characteristic-two group-algebra example: the present algebra and functional are over \(\mathbb R\).

Let \(\Omega \) be finite, \(K=\mathbb R\) or \(\mathbb C\), and give \(A_\Omega =\operatorname {Map}(\Omega ,K)\) the pointwise involution \(f^*(x)=\overline {f(x)}\). The weight functional \(\omega _p(f)=\sum _xp(x)f(x)\) is a real or complex algebraic state exactly when every \(p(x)\) is real and nonnegative and \(\sum _xp(x)=1\).

There is no state on the empty configuration set over \(\mathbb R\) or \(\mathbb C\), because the empty sum is zero. On two points, the weights \(p(0)=p(1)=1/2\) give a state that is not multiplicative: \(\omega _p(\delta _0\delta _1)=0\), whereas \(\omega _p(\delta _0)\omega _p(\delta _1)=1/4\). A state is not required to be an algebra homomorphism.

The positive-functional convention is [waldmann2004states, Definition 3.9, p. 11]; the indicator calculation above identifies it with finite probability and expectation on this particular algebra. It does not identify positivity cones on arbitrary star algebras; the distinctions on [waldmann2004states, p. 12] remain separate.

Let \(R\) be a commutative unital ring, \(A,B\) unital associative \(R\)-algebras, and let \(\iota :B\to A\) be a supplied unital algebra homomorphism. Precomposition defines an \(R\)-linear map in the opposite direction on ordinary duals: \[\begin {aligned} \iota ^\flat :\operatorname {Hom}_R(A,R)&\longrightarrow \operatorname {Hom}_R(B,R),\\ \omega &\longmapsto \omega \circ \iota . \end {aligned}\] It sends normalized functionals to normalized functionals. For another unital associative \(R\)-algebra \(C\) and a unital \(R\)-algebra homomorphism \(\kappa :C\to B\), \[(\iota \circ \kappa )^\flat =\kappa ^\flat \circ \iota ^\flat , \qquad (\mathrm {id}_A)^\flat =\mathrm {id}.\] When \(\iota \) is inclusion this is literal restriction; for an injective map it becomes restriction after identifying \(B\) with its image. No injectivity is needed for precomposition itself. A partition of a set by itself does not supply an algebra map between arbitrary algebras.

Linearity and preservation of the unit suffice for the normalization calculation; multiplication preservation additionally respects observable products. Without unitality, the zero multiplicative linear map between nonzero real algebras pulls every normalized functional back to zero.

Over \(K=\mathbb R\) or \(\mathbb C\), suppose both algebras carry the star laws in the state definition, and suppose additionally \(\iota (b^*)=\iota (b)^*\). Then \(\iota ^\flat \) sends algebraic states to algebraic states. For composition in this setting, let \(C\) also be a unital associative \(K\)-algebra with these star laws, and let \(\kappa :C\to B\) be a unital \(K\)-algebra homomorphism preserving star.

Star compatibility cannot be omitted. Give \(\mathbb C[z]\) the involution conjugating coefficients and fixing \(z\); give \(\mathbb C\) ordinary conjugation. Evaluation at \(z=i\) is a unital complex algebra map. The identity functional on \(\mathbb C\) is a state, but its pullback takes \(z^*z=z^2\) to \(-1\), so is not positive.

The dual-map construction is supported by Mathlib's LinearMap.dualMap, LinearMap.dualMap_apply, LinearMap.dualMap_id and LinearMap.dualMap_comp_dualMap; these do not require finite dimension. The star-preserving map structures StarAlgHom and StarAlgHom.comp concern supplied star operations; the genuine star laws above remain explicit hypotheses. For the algebra and positivity conventions see [stacks2017stacks, Section 22.2], Tag 09JF and [waldmann2004states, Definitions 3.7 and 3.9, p. 11]. The normalization and positivity assertions here are proved directly.

Let \(\rho :\Omega \to \Lambda \) be any map of finite sets and let \(R\) be a commutative unital ring. On the function algebras, \[\begin {aligned} \rho ^*:A_\Lambda &\longrightarrow A_\Omega ,& f&\longmapsto f\circ \rho \end {aligned}\] is a unital \(R\)-algebra map. Over \(\mathbb R\) or \(\mathbb C\) it preserves pointwise conjugation. For \(\tau :\Lambda \to \Gamma \), \((\tau \circ \rho )^*=\rho ^*\circ \tau ^*\), and the identity set map induces the identity algebra map.

The image of \(\rho ^*\) consists exactly of functions constant on each fiber of \(\rho \). If \(\rho \) is surjective, \(\rho ^*\) is injective. Conversely, injectivity of \(\rho ^*\) forces surjectivity of \(\rho \) provided \(R\) is nonzero.

For the weight functional determined by \(p\), its pullback as a functional along \(\rho ^*\) has weights \[q(y)=\sum _{\rho (x)=y}p(x).\] Thus normalized weights stay normalized; real nonnegative weights stay nonnegative. Empty fibers have weight zero.

For real nonnegative weights of total one, this is pushforward of finite probabilities, or a marginal law when \(\rho \) forgets coordinates. It uses no division by fiber mass, even when that mass is zero; it is not a conditional expectation. The pointwise algebra construction uses [stacks2017stacks, Section 22.2], Tag 09JF. Mathlib's AlgHom.pi constructs the family of evaluation maps \(f\mapsto f(\rho (x))\). Composition here changes the function's domain, not its coefficient ring.

Over arbitrary rings, not every unital algebra map between function algebras is induced by a set map. For \(R=\mathbb Z/6\mathbb Z\), define \(h:R^2\to R\) by \(h(a,b)=3a+4b\). It is \(R\)-linear and unital since \(3+4=1\). The identities \(3^2=3\), \(4^2=4\), and \(3\cdot 4=0\) show \(h(a,b)h(c,d)=3ac+4bd=h(ac,bd)\), so it is multiplicative. But \(h(1,0)=3\), whereas either coordinate evaluation gives \(0\) or \(1\). Thus this map is not the pullback of a map from a singleton to two points.

Use the 4 vertices \(V=\{0,1,2,3\}\) of the alternating cycle, the alphabet \(S=\{0,1\}\), and retained vertices \(J=\{0,2\}\). There are 16 configurations \(\Omega =S^V\) and 4 retained configurations \(\Lambda =S^J\). The map \(\rho (\sigma )=\sigma |_J\) is surjective: assign zero at vertices \(1,3\) to extend any retained configuration. Each fiber has 4 elements. The pullback identifies retained observables with those depending only on the values at vertices \(0,2\).

Choose the following real weights, independently of the edge energies: \[p(\sigma )=\begin {cases} 1/12,&\sigma _0=\sigma _2,\\ 1/24,&\sigma _0\ne \sigma _2. \end {cases}\] There are 8 configurations of each kind, so \(\sum _\sigma p(\sigma )=8/12+8/24=1\). By the finite-state criterion, this defines a real algebraic state. Summing the 4 weights in each fiber gives \[q(0,0)=q(1,1)=1/3,\qquad q(0,1)=q(1,0)=1/6.\] For \(F(\eta )=\eta _0\eta _2\), only \(\eta =(1,1)\) contributes, and hence \[\omega _q(F)=1/3=4\cdot (1/12)=\omega _p(F\circ \rho ).\] This is an explicit instance of the finite fiber-sum identity.

The law here is a chosen finite probability distribution. It is not obtained from the cycle's edge weights or its precision matrix, and the carrier \(S^V\) is distinct from the real configuration space \(\mathbb R^V\) used by the Gaussian law. The algebra convention remains [stacks2017stacks, Section 22.2], Tag 09JF; the positivity convention is [waldmann2004states, Definition 3.9, p. 11]. If the vertex set is empty, \(S^\varnothing \) has one outcome, even for empty \(S\). An empty alphabet with nonempty vertex set instead gives no outcomes and no real state. Thus zero vertices and an empty configuration set are different cases.

Let \(V\) be finite, \(J\subseteq V\), \(X=\mathbb R^V\) and \(Y=\mathbb R^J\). Let \(\pi :X\to Y\) restrict a configuration to \(J\). Write \(\mathcal P(X)\) and \(\mathcal P(Y)\) for the algebras of real polynomial functions generated by the coordinate evaluation functions, with pointwise operations and the identity involution. Then \[\begin {aligned} \pi ^*:\mathcal P(Y)&\longrightarrow \mathcal P(X),& P&\longmapsto P\circ \pi \end {aligned}\] is an injective unital real algebra map preserving these involutions. Every supplied real algebraic state on \(\mathcal P(X)\) therefore pulls back to one on \(\mathcal P(Y)\).

Here polynomial functions are actual functions, rather than formal polynomial expressions. The proof uses coordinate composition and the zero-fill section, not a finiteness assertion about the real set \(X\). The algebra and positivity conventions are [stacks2017stacks, Section 22.2], Tag 09JF and [waldmann2004states, Definition 3.9, p. 11]. The finite-map construction gives the same contravariant direction, but its finite indicator expansion is not used here.

For the Gaussian expectation, measure and polynomial integrability first establishes that the Gaussian polynomial state on \(\mathcal P(X)\) exists. Once supplied, it restricts by the algebraic result above. Identifying its retained law requires the separate Gaussian marginal calculation. This proposition neither extends that analytic calculation to new degenerate cases nor asserts that all integrable functions form an algebra. Polynomial functions are also distinct from the algebra of field endomorphisms.

For a finite set \(V\), put \(X_V=\operatorname {Map}(V,\mathbb R)\), the real vertex-field space. Its indicator fields \(\delta _v\) form the finite function basis, so \(x=\sum _{v\in V}x(v)\delta _v\). Use the counting inner product \(h_V(x,y)=\sum _vx(v)y(v)\) and its perfect duality. The carrier \(V\) is finite; the set \(X_V\) of configurations is generally infinite. For \(V=\varnothing \), \(X_V=\{0\}\).

The real field topology is the finite product of the usual topology on \(\mathbb R\). Its basic open sets are boxes \(\prod _{v\in V}O_v\) with every \(O_v\) open.

For a topological space \(Y\), a map \(f:Y\to X_V\) is continuous if and only if every coordinate \(y\mapsto f(y)(v)\) is continuous. Consequently every real linear map between finite field spaces is continuous, and every linear isomorphism is a homeomorphism.

A sigma-algebra on a set \(X\) is a family of subsets containing \(X\) and closed under complements and countable unions. It is also closed under countable intersections, by taking complements. The Borel sigma-algebra \(\mathcal B(X)\) of a topological space is the smallest sigma-algebra containing all its open sets; see [tao2011measure, Definition 1.4.16, p. 87, preliminary version].

On the real field space, define the coordinate-product sigma-algebra \(\Sigma _V\) to be generated by all cylinders \(\{x:x(v)\in B\}\), where \(v\in V\) and \(B\in \mathcal B(\mathbb R)\). This is the product construction of [tao2011measure, Section 1.7.4, pp. 194-195, preliminary version]. The Borel-product proposition identifies it with \(\mathcal B(X_V)\). We use this Borel structure without completion: no additional subsets of null sets are adjoined.

For the singleton \(X_\varnothing \), the Borel structure is \(\{\varnothing ,X_\varnothing \}\). A product of completed factor sigma-algebras need not be the completed product sigma-algebra; these constructions are distinguished in [tao2011measure, p. 94 and Exercise 1.7.19, p. 195, preliminary version].

For every finite real field space, \[\mathcal B(X_V)=\Sigma _V.\] Here \(\Sigma _V\) is the coordinate-product sigma-algebra. This finite-coordinate assertion, including dimension zero, follows by the direct argument below; compare [tao2011measure, Exercises 1.7.18-1.7.19, pp. 194-195, preliminary version].

Mathlib's Pi.borelSpace requires countable indices, second-countable topological factors and their Borel measurable structures. Finite real products meet these assumptions. The less restrictive pi_le_borel_pi gives only one inclusion, not the displayed equality.

A measurable map \(f:(X,\Sigma )\to (Y,\mathcal T)\) satisfies \(f^{-1}(B)\in \Sigma \) for every \(B\in \mathcal T\). A Borel observable on \(X_V\) is a measurable real-valued function for \(\mathcal B(X_V)\) and \(\mathcal B(\mathbb R)\). This is the preimage convention of [tao2011measure, Definition 1.4.32 and Remark 1.4.33, pp. 95-96, preliminary version].

Compositions of measurable maps are measurable because \((g\circ f)^{-1}(B)=f^{-1}(g^{-1}(B))\). The coordinate criterion supplies the maps used for real fields. Measurability does not imply integrability. The integrable-observable space is a vector space, whereas polynomial functions form a separate observable algebra; finite \(V\) does not make the configuration set \(X_V\) finite.

On the four-cycle, a configuration is a point \(x=(x_0,x_1,x_2,x_3)\in \mathbb R^4\). The product topology makes \((-1,1)^4\) open. The Borel sigma-algebra also contains the closed half-space \(H=\{x:x_0\le 0\}\), because its complement is open. Thus Borel sets include both open and closed coordinate conditions.

The coordinate observable \(p_0(x)=x_0\) is continuous: the preimage of an open interval is its coordinate cylinder. For every Borel \(A\subseteq \mathbb R\), the set \(p_0^{-1}(A)\) is Borel by the coordinate-product definition and Proposition 6.3, proving measurability. The indicator \(\mathbf 1_H\) is also measurable: its preimage of a Borel set is one of \(\varnothing ,H,H^c,\mathbb R^4\). It is not continuous at the origin, since its values on \((1/k,0,0,0)\), for positive integers \(k\), are all zero while its value at their limit is one.

Mathlib's continuous_apply and measurable_pi_apply give the coordinate projection facts for product topologies and product measurable spaces, respectively.

These deductions distinguish topology, measurable sets and measurable maps using the conventions of [tao2011measure, Definitions 1.4.16 and 1.4.32, pp. 87 and 95, preliminary version]. They say nothing by themselves about integrability. For example, the constant observable \(1\) is measurable, but its integral against the coordinate reference measure on \(\mathbb R^4\) is infinite.

Let \(V\) be finite and \((Y,\mathcal T)\) a measurable space. A map \(f:Y\to X_V\) is measurable if and only if every map \(y\mapsto f(y)(v)\) to the Borel real line is measurable. Continuous maps between finite real field spaces are Borel measurable. In particular this holds for linear maps, coordinate restriction, zero extension and polynomial functions. Compare [tao2011measure, Exercise 1.4.29 and Remark 1.4.33, pp. 95-96, preliminary version].

The product-measurability equivalence is measurable_pi_iff; it concerns measurable products, without asserting topology or integrability. For Borel spaces an inducing map also detects measurability through composition, as in Topology.IsInducing.measurable_comp_iff; no measurable range is required. Neither statement attaches a state to every observable.

Let \(\lambda _{\mathbb R}\) be Lebesgue measure restricted to \(\mathcal B(\mathbb R)\), normalized by \(\lambda _{\mathbb R}(\{x\in \mathbb R:a<x\le b\})=b-a\) for \(a<b\). The coordinate Lebesgue measure of a finite real field space is \[\lambda _V=\bigotimes _{v\in V}\lambda _{\mathbb R}\] on \(\mathcal B(X_V)=\Sigma _V\), characterized on Borel rectangles by \[\lambda _V\left (\prod _v A_v\right ) =\prod _v\lambda _{\mathbb R}(A_v).\] Products lie in \([0,\infty ]\), with \(0\cdot \infty =0\); the empty product is \(1\). The reference is fixed by counting coordinates, independently of any quadratic precision.

Existence follows by induction from the sigma-finite product-measure theorem, [tao2011measure, Proposition 1.7.11, pp. 197-199, preliminary version]. Start with \(\lambda _\varnothing =\delta _0\) on \(X_\varnothing =\{0\}\). Each factor is sigma-finite, and the cubes \([-k,k]^V\), \(k\ge 1\), cover with finite masses \((2k)^{|V|}\). Thus the product is sigma-finite. For nonempty \(V\) these masses grow without bound, so \(\lambda _V(X_V)=\infty \); it is not a probability measure. Finite-cube uniqueness handles its infinite total mass.

The real-spin reference convention appears in [friedli2017statistical, Definition 8.1 and Equation (8.3), p. 381, author version revised 22 August 2017]. The model's boundary conditions and density are additional data. No completion or normalizing density is part of \(\lambda _V\).

Mathlib constructs the finite product as MeasureTheory.Measure.pi; its rectangle law MeasureTheory.Measure.pi_pi uses sigma-finite factors. The product-volume convention is MeasureTheory.volume_pi. The empty cases MeasureTheory.Measure.pi_of_empty and MeasureTheory.Measure.pi_empty_univ give Dirac mass and total mass one, respectively.

Keep the four-cycle carrier \(V=\{0,1,2,3\}\). For real \(a_i\le b_i\), let \[B=\{x\in \mathbb R^4:a_i<x_i\le b_i\text { for }i=0,1,2,3\}.\] The coordinate Lebesgue measure gives \[\lambda _V(B)=\prod _{i=0}^3(b_i-a_i).\] Each factor is the length of its half-open interval, and the defining rectangle law multiplies them. If an endpoint pair is equal, the box is empty and the formula gives zero.

If every lower endpoint is zero and every upper endpoint is one, the box has volume \(1^4=1\). For lower endpoints \((0,0,-1,0)\) and upper endpoints \((2,1,1,3)\), its side lengths are \((2,1,2,3)\), so \[\lambda _V(B)=2\cdot 1\cdot 2\cdot 3=12.\] For \(a_i=-k\) and \(b_i=k\) in all 4 coordinates, with \(k>0\), the volume is \((2k)^4\). These volumes are unbounded as \(k\) increases, so the whole configuration space has infinite volume. Neither the edge weights nor a mass parameter enter these values. Reference volume is not a probability law; the later Gaussian density supplies a normalized weighting.

This is the rectangle formula of Definition 6.7, from [tao2011measure, Proposition 1.7.11, pp. 197-199, preliminary version]. Mathlib's MeasureTheory.Measure.pi_pi gives the finite-product rectangle law; Real.volume_Ioc gives each interval length.

Let \(V\) be finite. Suppose two Borel measures \(\mu ,\nu \) on \(X_V\) agree on all bounded half-open coordinate boxes with rational endpoints and on the empty set. Assume explicitly that their common mass on every rational cube \(C_k=\{x\in X_V:-k<x(v)\le k\text { for every }v\in V\}\), \(k\ge 1\), is finite. Then \(\mu =\nu \). In particular the hypothesis holds if both assign the prescribed finite Lebesgue volumes to those boxes. For empty \(V\), a box is the singleton, and agreement of its mass gives the conclusion.

This is the localized uniqueness argument behind [tao2011measure, Proposition 1.7.11, pp. 197-199, preliminary version]. Mathlib's MeasureTheory.Measure.pi_eq characterizes a sigma-finite product by all measurable rectangles. TauCeti's MeasureTheory.Measure.ext_pi assumes a finite total first measure; it cannot be applied directly to positive-dimensional \(\lambda _V\). Bare agreement of arbitrary infinite measures on rational boxes is not the lemma stated here.

For a bijection \(e:V\to W\), define \(U_e:X_V\to X_W\) by \((U_ex)(w)=x(e^{-1}w)\). It is a linear homeomorphism, with inverse \(U_{e^{-1}}\), preserving the counting pairing, and \[(U_e)_*\lambda _V=\lambda _W,\qquad f_*\mu (B):=\mu (f^{-1}B).\] For a disjoint decomposition \(V=R\sqcup D\), the restriction-pair map \(\sigma (x)=(x|_R,x|_D)\) has inverse pasting and satisfies \[\sigma _*\lambda _V=\lambda _R\otimes \lambda _D.\] Either factor may be empty, carrying singleton mass one.

These finite-coordinate proofs use the product convention of [tao2011measure, Section 1.7.4, pp. 194-199, preliminary version]. The corresponding sigma-finite APIs are MeasureTheory.measurePreserving_piCongrLeft and MeasureTheory.measurePreserving_piEquivPiSubtypeProd; the latter uses a decidable splitting predicate. A one-coordinate split is TauCeti.measurePreserving_piSplitAt, and the specified four-coordinate grouping is TauCeti.measurePreserving_finFourArrowPairPair.

Bare restriction \(\rho _R\) is measurable, but if \(D\ne \varnothing \) it is not reference-preserving: for a Borel \(A\subseteq X_R\) of finite positive mass, \(\rho _R^{-1}(A)\) has mass \(\lambda _R(A)\lambda _D(X_D)=\infty \). This includes \(R=\varnothing \), when the target singleton has mass one. The probability-factor assumption in TauCeti.measurePreserving_pi_comp_embedding excludes these unnormalized Lebesgue factors. Normalized marginals are a different measure statement.

Let \(T:X_V\to X_W\) be an invertible real linear map, with \(|V|=|W|=n\). Choose ordered indicator bases and normalized top forms \(\eta _V,\eta _W\), each taking value one on its chosen basis. Define \[T^*\eta _W=d(T)\eta _V,\qquad J(T)=|d(T)|.\] Reordering either basis changes \(d(T)\) only by a sign; \(J(T)>0\) is the reference-volume Jacobian. For Borel sets \(E,B\), \[\begin {aligned} \lambda _W(T(E))&=J(T)\lambda _V(E),\\ T_*\lambda _V(B)&=J(T)^{-1}\lambda _W(B). \end {aligned}\] For nonnegative Borel \(F:X_W\to [0,\infty ]\), \[\int _{X_V}F(Tx)\,d\lambda _V(x) =J(T)^{-1}\int _{X_W}F(y)\,d\lambda _W(y).\] For real signed \(F\), absolute integrability on either side is equivalent to that on the other; under this condition the same equality holds. Translation, and hence \(x\mapsto T(x)+b\), has the same coefficient.

For an endomorphism, \(d(T)=\det T\) only when source and target use the same normalized top form. Independently ordered indicator bases give \(d(T)=\varepsilon \det T\), \(\varepsilon \in \{1,-1\}\), although \(J(T)=|\det T|\). The identity on \(\mathbb R^2\) with reversed target order has \(d=-1\), intrinsic determinant \(1\), and \(J=1\). In dimension zero the top form is \(1\) and \(J=1\).

The image convention is stated in [tao2011measure, Exercises 1.2.20-1.2.22, p. 41, preliminary version]; the argument above proves its finite-field form. Mathlib's Real.map_linearMap_volume_pi_eq_smul_volume_pi is a finite-coordinate endomorphism theorem with nonzero determinant and reciprocal pushforward coefficient. Distinct carriers require the coordinate transport already proved, not an intrinsic signed determinant between unrelated spaces.

A linear isomorphism \(T:X_V\to X_W\) preserving the counting inner products satisfies \(J(T)=1\). It therefore preserves coordinate Lebesgue measure and the integrals in Theorem 6.11. An isometric embedding into a higher-dimensional space is not included.

This is a direct consequence of the preceding transformation theorem, with the convention of [tao2011measure, Exercise 1.2.21, p. 41, preliminary version]. Mathlib's LinearIsometryEquiv.measurePreserving concerns finite-dimensional real inner-product spaces with compatible Borel structures and their canonical volumes. An arbitrary perfect pairing or a general quadratic-form isometry does not supply these hypotheses.

Let \(H\) be a finite-dimensional real inner-product space, in particular a subspace of \((X_V,h_V)\) with its induced topology and inner product. Choose an orthonormal chart \(C:H\to \mathbb R^k\). The induced Euclidean measure is \(\lambda _H=(C^{-1})_*\lambda _{\{1,\ldots ,k\}}\) on its Borel sets. It is independent of the orthonormal chart. If \(H=\{0\}\), it is unit mass at zero. For an affine translate \(a+H\), use the pushforward of \(\lambda _H\) under \(h\mapsto a+h\).

For example, on \(H=\{(s,s):s\in \mathbb R\}\), the segment \(0\le s\le 1\) has induced length \(\sqrt 2\), ambient area zero, and raw \(s\)-coordinate length one. Indeed its unit vector is \((1,1)/\sqrt 2\), so its orthonormal coordinate runs from zero to \(\sqrt 2\). An arbitrary basis therefore needs its volume factor.

The argument uses the transformation convention of [tao2011measure, Exercise 1.2.21, p. 41, preliminary version]. Mathlib's OrthonormalBasis.measurePreserving_repr targets EuclideanSpace; PiLp.volume_preserving_ofLp and PiLp.volume_preserving_toLp identify its real finite PiLp volume with ordinary coordinate products. TauCeti's Module.Basis.volume_eq_smul_addHaar records the square-root Gram determinant factor for an arbitrary basis, while AffineIsometryEquiv.measurePreserving applies to finite-dimensional real Borel inner-product spaces. These compare the same canonical-volume convention.

On the Borel real line, let \(T(x)=2x\). The Jacobian theorem gives \(J(T)=2\), but its two measure directions have reciprocal coefficients: \[\lambda _{\mathbb R}(T[0,1])=2,\qquad T_*\lambda _{\mathbb R}([0,1])=\tfrac 12.\] The first set is \([0,2]\), whereas the second expression measures the preimage \([0,1/2]\). This directly checks the convention of [tao2011measure, Exercise 1.2.21, p. 41, preliminary version]. Mathlib's Real.map_volume_mul_left requires a nonzero scalar and gives the reciprocal absolute pushforward factor.

Put \(q_V(x)=x^2\) and \(q_W(y)=y^2/4\). Then \(q_W(Tx)=q_V(x)\) although \(J(T)=2\). Preserving a quadratic form alone does not preserve the supplied reference volume.

If a Borel weight \(w\ge 0\) has a finite positive integral \(Z=\int w\,d\lambda _V\), replacing the reference by \(c\lambda _V\), \(c>0\), replaces \(Z\) by \(cZ\). Consequently \((cZ)^{-1}w\,d(c\lambda _V)=Z^{-1}w\,d\lambda _V\). This conditional cancellation follows from linearity for indicators, simple functions and monotone limits. It proves neither existence nor finiteness of \(Z\); the reference and the action are separate data.

Let \(E=E_1\mathbin \perp E_2\) be an orthogonal direct sum of finite-dimensional real inner-product spaces. Use their canonical Euclidean volumes on the uncompleted Borel sigma-algebras. Addition \(a(x_1,x_2)=x_1+x_2\) is a measure-preserving isometric isomorphism: \[a_*(\lambda _{E_1}\otimes \lambda _{E_2})=\lambda _E.\] All these measures are sigma-finite. Nonnegative Borel functions therefore satisfy Tonelli's iterated-integral identities. For a real Borel function with finite absolute integral, Fubini holds and its sections are integrable almost everywhere.

This follows from canonical Euclidean volume and finite product splitting. Mathlib's LinearIsometryEquiv.measurePreserving supports the isometry step for finite-dimensional real Borel inner-product spaces with canonical volume; arbitrary perfect pairings do not supply these hypotheses.

Let \((E,h)\) be a real inner-product space of dimension \(n\geq 0\), and let \(T:E\to E\) be linear and self-adjoint. There is an orthonormal basis \(u_1,\ldots ,u_n\) with real eigenvalues \(q_i\). Put \(P_ix=h(u_i,x)u_i\). Then \[\begin {aligned} I&=\sum _iP_i,&T&=\sum _iq_iP_i,\\ P_iP_j&=\delta _{ij}P_i,&\det T&=\prod _iq_i. \end {aligned}\] Moreover \(h(x,Tx)>0\) for every nonzero \(x\) exactly when every \(q_i>0\). In that case \(T^{-1}=\sum _iq_i^{-1}P_i\) is self-adjoint and positive.

This gives the operator proof of the real symmetric decomposition in [boyd2004convex, Appendix A.5.2, pp. 646-647]. Mathlib's LinearMap.IsSymmetric.eigenvectorBasis supplies an orthonormal eigenbasis for a finite-dimensional inner-product space; here its scalar field is specialized to \(\mathbb R\). The declaration alone asserts neither Gaussian normalization nor a graph spectrum.

Let \((E,h)\) be a finite-dimensional real inner-product space, with \(n=\dim E\geq 0\), its finite-dimensional topology, Borel sigma-algebra and canonical Euclidean volume. The field specialization is \(E=X_V=\operatorname {Map}(V,\mathbb R)\) for a finite vertex set, including the empty set, with counting inner product \(h_V(x,y)=\sum _{v\in V}x(v)y(v)\) and coordinate Lebesgue measure. There are finitely many coordinates but, in positive dimension, continuously many configurations. Real-spin integration uses this reference convention in [friedli2017statistical, Definition 8.1 and Equation (8.3), author version p. 381]. The extension to \((E,h)\) is the same paired construction in an orthonormal chart; it asserts no ring-general probability law.

A positive precision parameter is a real linear endomorphism \(Q:E\to E\) that is self-adjoint for \(h\) and strictly positive: \(h(x,Qx)>0\) for every \(x\ne 0\). Define its symmetric bilinear form and associated quadratic form by \[b_Q(x,y)=h(x,Qy),\qquad q_Q(x)=b_Q(x,x).\] Linearity in both slots follows from linearity of \(Q\) and the pairing. Self-adjointness gives \(b_Q(y,x)=h(Qy,x)=h(y,Qx)=b_Q(x,y)\). Consequently \[q_Q(x+y)-q_Q(x)-q_Q(y)=2b_Q(x,y).\] Also \(q_Q(cx)=c^2q_Q(x)\) for every real \(c\). This is the quadratic form used here; positivity of its diagonal alone would not assert self-adjointness of an arbitrary endomorphism.

Its Euclidean quadratic action and Boltzmann weight are \[\begin {aligned} S_Q(x)&=\frac 12 q_Q(x)=\frac 12h(x,Qx),\\ w_Q(x)&=\exp (-S_Q(x)). \end {aligned}\] Thus \(q_Q=2S_Q\), \(S_Q(0)=0\), and \(S_Q(x)>0\) off zero. Dividing the polarization identity by two gives \[S_Q(x+y)-S_Q(x)-S_Q(y)=b_Q(x,y).\] The finite-coordinate formulas make \(S_Q\) continuous, hence Borel measurable, and so is its exponential. In particular \(0<w_Q(x)\le 1\), with equality \(w_Q(x)=1\) exactly when \(x=0\). This bound does not prove integrability on a space of infinite reference volume; Gaussian normalization supplies that separate argument.

The counting-array bridge recovers the familiar symmetric matrix and \(x^TQx\) notation from this operator definition. The variables and coefficients here are dimensionless: their chosen scales are absorbed into \(Q\). Wingate's Euclidean convention is \(\exp (-S)\); his lattice-spacing factors belong to his lattice model [wingate2025lattice, Sections 3-4, Equations (3.1) and (4.1)-(4.6), pp. 2-3]. This real nonnegative weight supplies no real-time evolution. The complete four-cycle action gives the concrete operator, array, half factor and test-field value.

Let \(Q:X_V\to X_V\) be a positive precision parameter. Choose an indicator enumeration and its counting chart \(C:X_V\to \mathbb R^n\). Write \(A=CQC^{-1}\) for the resulting array. Then \[h_V(x,Qy)=(Cx)^TA(Cy).\] Thus \(Q\) is self-adjoint exactly when \(A^T=A\), and is strictly positive exactly when \(z^TAz>0\) for every nonzero \(z\). The action is \(S_Q(x)=\tfrac 12(Cx)^TA(Cx)\).

In the ensuing Gaussian formulas, the customary matrix symbol \(Q\) denotes this counting array of the endomorphism. An arbitrary nonorthonormal chart need not make its array symmetric: its Gram matrix must also enter the pairing. This calculation is a coordinate recovery of Definition 7.2, not a new normalization result. The half-action convention agrees with [wingate2025lattice, Section 4, Equation (4.5), p. 3].

Use the four-cycle oriented \(0\to 1\to 2\to 3\to 0\), with edge weights \((1,t,1,t)\), \(t>0\). Let \(D:X_V\to X_E\) evaluate endpoint differences and \(W\) multiply by the supplied edge weights. With counting adjoint \(D^\dagger \), put \[L=D^\dagger WD,\qquad Q=L+m^2\operatorname {id},\qquad m>0.\] For every \(x,y\), the energy interface gives \[h_V(x,Ly)=\sum _e w(e)(Dx)(e)(Dy)(e).\] It is symmetric in \(x,y\), so \(Q\) is self-adjoint. On the diagonal all edge terms are nonnegative and \(m^2h_V(x,x)>0\) off zero. Thus \(Q\) is a positive precision parameter. At zero mass constant fields instead expose the Laplacian kernel.

In the specified vertex order \((0,1,2,3)\), write \(x=(a,b,c,d)\). The 4 differences are \(b-a,c-b,d-c,a-d\), so the action is exactly \[\begin {aligned} 2S_{t,m}(a,b,c,d) &=(b-a)^2+t(c-b)^2\\ &\quad +(d-c)^2+t(a-d)^2\\ &\quad +m^2(a^2+b^2+c^2+d^2). \end {aligned}\] Each edge occurs once, with one half of its weighted energy in \(S\). Every summand of \(2S\) is nonnegative, with strict positivity off zero from the mass term.

Set \(s=1+t+m^2\). The 4 edge outer products, equivalently expansion of the 4 squares, give diagonal coefficient \(s\), cross terms \(-2ab\), \(-2tbc\), \(-2cd\), \(-2tad\), and no \(ac\) or \(bd\) term. The counting array is therefore \[Q_{t,m}=\begin {pmatrix} s&-1&0&-t\\ -1&s&-t&0\\ 0&-t&s&-1\\ -t&0&-1&s \end {pmatrix}.\] Multiplication recovers \(x^TQ_{t,m}x=2S_{t,m}(x)\). For \(x=(0,1,0,0)\), the edge energy is \(1+t\) and the mass energy is \(m^2\); hence \(S_{t,m}(0,1,0,0)=s/2\), explicitly \(3/2\) when \(t=m=1\). No change of basis or reference measure occurs.

This is a direct calculation in the supplied cycle data, with the edge-once convention of [spielman2009effective, Section 2.1, pp. 4-5] and the positive kinetic sign and half-action of [wingate2025lattice, Section 4, Equation (4.5), p. 3]. All coefficients here are dimensionless. No Gaussian normalizer or covariance is inferred from the energy calculation.

For the real alternating four-cycle, take \(t,m>0\) and let \(L_t\) be the weighted Laplacian. Its positive precision parameter is exactly \[Q_{t,m}=L_t+m^2I.\] Indeed, the displayed array of \(L_t\) has diagonal \(1+t\) and the same off-diagonal entries as the action's precision. Adding \(m^2I\) changes that diagonal to \(1+t+m^2\), giving the displayed \(Q_{t,m}\) in Example 7.4.

For \(f=(a,b,c,d)\), the Euclidean quadratic action is therefore \[\begin {aligned} S_{t,m}(f) &=\frac 12\langle f,L_tf\rangle _V +\frac {m^2}{2}\langle f,f\rangle _V\\ &=\frac 12\bigl ((b-a)^2+t(c-b)^2\\ &\qquad \quad +(d-c)^2+t(a-d)^2\bigr )\\ &\qquad +\frac {m^2}{2}(a^2+b^2+c^2+d^2). \end {aligned}\] This is the same squared-difference formula as in Example 7.4: half of the edge-once energy plus the mass term. The first summand is nonnegative, and the mass summand is positive for \(f\ne 0\). This calculation compares the graph convention of [spielman2009effective, Section 2.1, pp. 4-5] with the positive kinetic sign and half-action convention of [wingate2025lattice, Section 4, Equation (4.5), p. 3]; the finite four-cycle equality follows by the substitutions above.

For real \(a\), consider the extended nonnegative Lebesgue integral \[I(a)=\int _{\mathbb R}e^{-a x^2/2}\,dx.\] For \(a>0\) it is finite and positive, and \[I(a)=\sqrt {2\pi /a},\qquad I(1)^2=2\pi .\] For \(a\leq 0\) it is infinite.

The calculation adapts Keith Conrad's improper-integral argument in [conradGaussianIntegral, Section 2, Second proof: Another change of variables, PDF pp. 1-2] to Lebesgue integration. The explicit tail bound and nonnegative Tonelli justify the adaptation; Tonelli is [tao2011measure, Theorem 1.7.15, pp. 200-201, preliminary version]. Mathlib's integrable_exp_neg_mul_sq and integral_gaussian support the real one-variable result with coefficient \(b=a/2>0\). The value theorem uses a totalized integral; its value at a nonintegrable parameter does not mean the nonnegative normalizer is finite.

Let \((E,h)\) be a finite-dimensional real inner-product space, \(n=\dim E\geq 0\), and let \(Q:E\to E\) be self-adjoint. With the canonical h-volume, put locally \[Z=\int _E\exp \bigl (-h(x,Qx)/2\bigr )\,d\lambda _E(x).\] This nonnegative integral is finite if and only if \(Q\) is strictly positive. In that case \[0<Z=\frac {(2\pi )^{n/2}}{\sqrt {\det Q}}<\infty .\] For \(n=0\), the determinant and normalizer are both one.

For the spectral input, see [boyd2004convex, Appendix A.5.2, pp. 646-647]. For the scalar integral, see Keith Conrad's [conradGaussianIntegral, Section 2, PDF pp. 1-2]. The density convention in [friedli2017statistical, Lemma B.54 and Proposition B.55, pp. 504-505] is a comparison, not an assumption of normalization: that source uses a covariance parameter, while \(Q\) here is precision. For the counting field space, the array bridge recovers the customary determinant formula without changing its reference volume.

Let \(Q\) be a positive precision parameter on a finite real inner-product space \((E,h)\), with canonical volume \(\lambda _E\). Its Gaussian partition function is \[Z_Q=\int _E w_Q\,d\lambda _E, \qquad w_Q(x)=e^{-h(x,Qx)/2}.\] The normalization theorem proves \[0<Z_Q=\frac {(2\pi )^{n/2}}{\sqrt {\det Q}}<\infty , \qquad n=\dim E.\] Its Gaussian probability law is the Borel measure \[\mu _Q(A)=Z_Q^{-1}\int _A w_Q\,d\lambda _E.\] In the counting field specialization this is exactly the density \(Z_Q^{-1}\exp (-\phi ^TQ\phi /2)\) against the product reference measure.

In an orthonormal eigenchart the density is the product of the centered scalar densities \(I(q_i)^{-1}e^{-q_i y_i^2/2}\). This identification uses the scalar value and normalization, before computing covariance. For \(n=0\), the action is zero at the unique point, \(Z_Q=1\), and \(\mu _Q=\delta _0\). Replacing the reference volume by \(c\lambda _E\), \(0<c<\infty \), multiplies \(Z_Q\) by \(c\) and leaves \(\mu _Q\) unchanged, as in the reference-rescaling example. A finite-sum partition function has the same normalizing role. A finite vertex set gives a finite-dimensional configuration space. Integrability of its Gaussian weight follows from the normalization theorem, using positive precision.

The density convention is compared with [friedli2017statistical, Lemma B.54 and Proposition B.55, pp. 504-505]; that source uses covariance rather than precision. Countable additivity uses [tao2011measure, Theorem 1.4.44, pp. 107-108, preliminary version]. The normalization failures distinguish infinite normalizers, zero weights and the zero-dimensional case.

The normalization theorem distinguishes infinite mass from zero mass. In positive dimension, zero precision has \(S=0\) and weight one, so its integral is the infinite Euclidean volume. Invertibility alone is insufficient: on \(\mathbb R^2\), \(Q=\operatorname {diag}(1,-1)\) has weight \(e^{-x^2/2+y^2/2}\). On \(|x|\leq 1\) the first factor is at least \(e^{-1/2}\), while the integral in \(y\) diverges. Neither weight defines the normalized Gaussian law on that full space.

For a general nonnegative weight \(w\equiv 0\), the integral is zero, so division cannot normalize it. This differs from zero precision. On \(E=0\), the unique endomorphism is invertible and vacuously positive, its Gaussian weight is one, and its normalizer is one. A singular law on a proper subspace requires that subspace's own reference measure; inserting a pseudoinverse does not normalize a divergent ambient density.

These calculations use the spectral and scalar ingredients of Theorem 7.7. For the scalar distinction, see Keith Conrad's [conradGaussianIntegral, Section 2, PDF pp. 1-2]; the explicit bounds are proved in Lemma 7.6.

For a finite-dimensional real vector space \(E\), the actual polynomial function algebra used in Proposition 5.10 is independent of coordinates. It is the real unital subalgebra of functions \(E\to \mathbb R\) generated by linear functionals. In a chosen basis, every linear functional is a finite linear combination of coordinate evaluations. Conversely every coordinate evaluation is linear. Thus these generators give exactly the coordinate polynomial algebra of that note. Substituting linear combinations in a finite polynomial gives a finite polynomial, so changing basis preserves the algebra.

Its operations and identity involution are pointwise, and its functions are continuous, hence Borel for the finite-dimensional topology. For \(E=0\), the algebra consists of constants. These are actual functions; formal polynomial expressions and their polynomial rings are related algebraic vocabulary, not a second choice of observable functions here.

This coordinate-independence argument is a direct deduction from the generators. The algebra and star conventions are those already used in Proposition 5.10. For these conventions, see [stacks2017stacks, Section 22.2], Tag 09JF and [waldmann2004states, Definition 3.9, p. 11]. Neither citation asserts Gaussian integrability.

Let \(Q\) be positive self-adjoint on a finite real inner-product space \((E,h)\), with weight \(w_Q=e^{-h(x,Qx)/2}\) and canonical volume. Every actual polynomial function satisfies \[\int _E |P(x)|w_Q(x)\,d\lambda _E(x)<\infty .\] This holds for every finite product and power of polynomials, including \(P^2\), and therefore also for integration against the normalized law.

The envelope estimate and finite-sum argument are direct deductions from Keith Conrad's [conradGaussianIntegral, Section 2, PDF pp. 1-2] and [tao2011measure, Theorem 1.7.15, pp. 200-201, preliminary version]. The orthogonal reduction agrees with [friedli2017statistical, Lemma B.54, p. 504]. TauCeti's TauCeti.integrable_eval_mul_gaussianEnvelope supports the one-variable polynomial envelope with mean zero and variance \(w=1/a>0\). The finite products, coordinate substitution and algebra closure above are separate arguments; no inverse-covariance identity is used.

For the Gaussian probability law \(\mu _Q\) on \((E,h)\), an integrable observable is a Borel function \(F:E\to \mathbb R\) with \(\int _E|F|\,d\mu _Q<\infty \). Its Gaussian expectation is \[\begin {aligned} \omega _Q(F)=\langle F\rangle _Q&=\int _E F\,d\mu _Q\\ &=Z_Q^{-1}\int _E F(x)w_Q(x)\,d\lambda _E(x). \end {aligned}\] The equality follows first for nonnegative simple functions from the density definition, then by monotone convergence, and finally by positive and negative parts for an absolutely integrable function.

These actual functions form a real vector space: the pointwise bound \(|aF+bG|\leq |a||F|+|b||G|\) proves closure, and integration is real linear on it. One may instead take almost-everywhere classes when using \(L^1\); polynomial observables here remain actual functions. All polynomial moments give its polynomial subalgebra; the state proposition proves normalization, positivity and Hermiticity there. The integrable-domain example explains why the full vector space need not be an algebra.

This extends finite expectation: absolute integrability replaces automatic convergence of a finite sum. The real-spin convention is [friedli2017statistical, Definition 8.1 and Equation (8.3), p. 381]; the density-integral extension uses [tao2011measure, Theorem 1.4.44, pp. 107-108, preliminary version]. Constants and oddness give the vanishing linear means. Covariance and variance extend the finite convention of Notation [ftip-008H]; the partition function, two-point expectation and other observable expectations remain distinct quantities.

For positive self-adjoint \(Q\) on a finite real inner-product space, its expectation restricts to a real algebraic state on the actual polynomial function algebra, with identity involution.

This is the direct integral realization of the convention in [waldmann2004states, Section 3.1, p. 9; Definitions 3.7 and 3.9, Remark 3.10(iv), p. 11], specialized to ordinary real coefficients as in Definition 5.4. A supplied unital star-compatible algebra map permits state pullback, in particular the coordinate restriction pullback. This asserts neither an automatic Clifford-algebra state nor a conditional expectation.

For the Gaussian expectation associated to positive self-adjoint \(Q\) on \((E,h)\), constants satisfy \(\omega _Q(c)=c\). Every absolutely integrable odd Borel function \(F\), meaning \(F(-x)=-F(x)\), has \(\omega _Q(F)=0\). In particular each linear observable \(\ell _j(x)=h(j,x)\) has mean zero.

This is a direct consequence of normalization and the change-of-variables convention used in [tao2011measure, Exercise 1.2.21, p. 41, preliminary version]. Oddness alone, without absolute integrability, does not define an expectation.

Let \(\mu _Q\) be the Gaussian law on a finite real inner-product space \((E,h)\), and write \(\omega _Q\) for its expectation. For Borel \(X,Y:E\to \mathbb R\) with \(X\), \(Y\) and \(XY\) absolutely integrable, define their covariance by \[\begin {aligned} \operatorname {Cov}_Q(X,Y) &=\omega _Q(XY)-\omega _Q(X)\omega _Q(Y)\\ &=\omega _Q\bigl ((X-\omega _Q(X))(Y-\omega _Q(Y))\bigr ). \end {aligned}\] Expansion and normalization prove the second equality; all 4 terms are integrable by the stated hypotheses. This extends the finite-sum covariance to integrals.

When \(X\) and \(X^2\) are integrable, its variance is \[\begin {aligned} \operatorname {Var}_Q(X)&=\operatorname {Cov}_Q(X,X)\\ &=\omega _Q\bigl ((X-\omega _Q(X))^2\bigr )\\ &=\omega _Q(X^2)-\omega _Q(X)^2\geq 0. \end {aligned}\] The inequality is positivity of the integral. In particular all these expressions exist for polynomials by Proposition 7.11, consistently with the polynomial state.

For \(\ell _j(x)=h(j,x)\), set \(K(j,k)=\operatorname {Cov}_Q(\ell _j,\ell _k)\). It is a finite symmetric bilinear form. There is a unique linear covariance operator \(C_Q:E\to E\), relative to the specified \(h\), such that \(h(j,C_Qk)=K(j,k)\).

For the counting field space, the coordinate means vanish by Proposition 7.14, and the former coordinate notation is \[\begin {aligned} \operatorname {Cov}_Q(u,v) &=\left \langle (\phi _u-\langle \phi _u\rangle _Q) (\phi _v-\langle \phi _v\rangle _Q)\right \rangle _Q\\ &=\langle \phi _u\phi _v\rangle _Q. \end {aligned}\] The partition function, this two-point expectation, and the expectation of a different observable are different quantities. The separate inverse-precision theorem computes this covariance.

The definitions extend Notation [ftip-008H] directly; the integral identities use the preceding moment and state results. Positivity uses the ordinary real convention of [waldmann2004states, Definition 3.9, p. 11] through Definition 5.4, not an additional inverse-covariance assumption.

The four-cycle covariance calculation evaluates these quantities for the running example.

Let \(Q\) be positive self-adjoint on \((E,h)\) with \(n=\dim E>0\). The Borel function \(F=e^{S_Q/2}\) is integrable under the Gaussian law, since \[\int _E Fw_Q\,d\lambda _E =\int _E e^{-S_Q/2}\,d\lambda _E=Z_{Q/2}<\infty \] by positive normalization. But \(F^2=e^{S_Q}\) is not integrable: \(F^2w_Q=1\), and \(\lambda _E(E)=\infty \). Indeed orthonormal cubes have masses \((2R)^n\) tending to infinity.

Thus actual integrable observables form a vector space but are not closed under multiplication. The same counterexample works for their almost-everywhere classes in \(L^1(\mu _Q)\). The polynomial algebra is an integrable algebra by Proposition 7.11. For \(n=0\), every real observable is constant on a singleton, so this failure is absent.

This is a direct comparison of the two normalizers, using Theorem 7.7 and the scalar ingredient in Keith Conrad's [conradGaussianIntegral, Section 2, PDF pp. 1-2]; it also retains \(e^{S_Q}\) as a concrete Borel nonintegrable observable in positive dimension.

Let \(V\) be a finite vertex set, with positive edge weights, and let \(L\) be its weighted graph Laplacian. For \(m>0\), define the massive precision and its Green operator by \[Q_m=L+m^2I,\qquad G_m=Q_m^{-1}.\] The inverse exists even when the graph is disconnected. Indeed, the energy identity gives, for \(f\ne 0\), \[\begin {aligned} \langle f,Q_mf\rangle _V &=\sum _e w(e)(Df)(e)^2\\ &\quad +m^2\sum _v f(v)^2>0. \end {aligned}\] Thus \(Q_mf=0\) implies \(f=0\). Finite dimension and rank-nullity make this injective endomorphism surjective, so it is an isomorphism. The inverse-response lemma now gives a self-adjoint inverse and the unique solution \(f=G_mj\) to \(Q_mf=j\). The endomorphism \(Q_m\) is a positive precision parameter; its counting array realizes it as a positive symmetric matrix. For \(j\ne 0\), write \(x=G_mj\ne 0\); then \(\langle j,G_mj\rangle _V=\langle Q_mx,x\rangle _V>0\). On an empty vertex set the same map argument gives the unique isomorphism of zero modules; there are no point-source columns.

By the real spectral theorem, choose an orthonormal eigenbasis \(u_a\) of the self-adjoint operator \(L\), with eigenvalues \(\lambda _a\ge 0\). The energy identity proves this last inequality by evaluation on each unit eigenvector. Put \(q_a=\lambda _a+m^2>0\). Then \[\begin {aligned} Q_m&=\sum _a q_a u_au_a^T,\\ G_m&=\sum _a q_a^{-1}u_au_a^T. \end {aligned}\] Since \(u_a^Tu_b=\delta _{ab}\) and \(\sum _a u_au_a^T=I\), multiplication gives \(Q_mG_m=G_mQ_m=I\). Also \(G_m^T=G_m\), and \(f^TG_mf=\sum _a q_a^{-1}(u_a^Tf)^2>0\) for nonzero \(f\).

The Green kernel is defined from the inverse map by \(G_m(v,w)=(G_m\delta _w)(v)\). These are its matrix entries in the counting coordinate basis. Its column at \(w\) is the unique field solving \[Q_mG_m(\cdot ,w)=\delta _w,\] where \(\delta _w(v)\) is one at \(v=w\) and zero elsewhere. For a source field \(j\), the unique solution of \(Q_mf=j\) is \(f(v)=\sum _wG_m(v,w)j(w)\). No vertex-volume factor occurs because the vertex pairing and basis are counting.

This convention has to be matched when comparing lattice Green functions. Wingate's lattice action uses a positive operator built from minus the backward-forward difference and a mass term [wingate2025lattice, Section 4, Equations (4.2)-(4.5), p. 3]. For unit spacing, that difference term has the sign of our \(L\).

In [friedli2017statistical, Theorem 8.26, p. 399], the massive lattice precision is \[Q=-\frac {1}{2d}\Delta +m^2I.\] Here \(\Delta f(x)=\sum _{y\sim x}(f(y)-f(x))\). The factor \(1/(2d)\) and the lattice boundary conditions belong to that normalization. Identifying their covariance with an inverse requires that precision, not an unscaled adjacency Laplacian.

Mathlib's Matrix.PosDef.isUnit proves invertibility of a positive definite matrix, and Matrix.PosDef.inv proves positivity of its inverse. These apply to the matrix step after the graph-energy calculation supplies positivity. The kernel convention and its graph interpretation are specified here.

Let \(Q\) be a positive self-adjoint precision on a finite real inner-product space \((E,h)\), with its Gaussian law. For \(j,k\in E\), put \(\ell _j(x)=h(j,x)\). Then \[\begin {aligned} \omega _Q(\ell _j)&=0,\\ \operatorname {Cov}_{\mu _Q}(\ell _j,\ell _k) &=h(j,Q^{-1}k). \end {aligned}\] Thus its covariance operator, relative to the specified pairing, is \(C_Q=Q^{-1}\). On the counting vertex space, \(\ell _{\delta _v}(\phi )=\phi (v)\), so \[\omega _Q(\phi (v)\phi (w)) =h_V(\delta _v,Q^{-1}\delta _w).\] For \(Q=L+m^2I\), this is the Green kernel, with no vertex-volume factor. The density convention agrees with [friedli2017statistical, Proposition B.55, p. 505, and Theorem 8.4, p. 385] after substituting covariance \(Q^{-1}\) for precision \(Q\).

For a nonorthonormal basis \(b_i\), the coordinate functional is \(h(b^i,\cdot )\). The dual basis satisfies \(h(b^i,b_j)=\delta _{ij}\). Its covariance array therefore has entries \(h(b^i,Q^{-1}b^j)\); it is not the unqualified array of the inverse operator in that basis. The counting basis is orthonormal, giving the coordinate formula above. The alternating-cycle example evaluates it explicitly.

To compare with formal Gaussian providers, put \(n=\dim E\), choose an orthonormal chart and let \(S\) represent \(Q^{-1}\) there. The positive inverse is positive definite, \(S^{-1}\) represents \(Q\), and \(\det S=(\det Q)^{-1}\). Consequently the provider density is \[\begin {aligned} &(2\pi )^{-n/2}(\det S)^{-1/2} e^{-y^TS^{-1}y/2}\\ &\qquad =Z_Q^{-1}e^{-h(x,Qx)/2}. \end {aligned}\] The orthonormal chart and the Euclidean-space coordinate wrapper preserve canonical volume. Thus the chart pushforward of \(\mu _Q\) is \(\mathrm {multivariateGaussian}(0,S)\) by TauCeti.Probability.multivariateGaussian_eq_withDensity. Mathlib's ProbabilityTheory.covariance_eval_multivariateGaussian then identifies its coordinate covariance with \(S\); its positive semidefinite hypothesis follows from positive definiteness. The parameter is covariance \(S=Q^{-1}\), not precision \(Q\). This density, determinant and volume identification is needed in addition to the covariance declaration; the integral proof above is independent of it.

Take the four-cycle on \(V=(0,1,2,3)\), with forward edges and weights \((1,t,1,t)\), \(t>0\). Use counting pairing and \(Q=L+m^2I\), \(m>0\). Put \(a=m^2\) and \(s=1+t+a\). The following functions on vertices form an orthonormal eigenbasis; the middle column lists the unnormalized rows \(2u_i^T\): \[\begin {array}{c|c|c} i&2u_i^T&\lambda _i\\\hline 0&(1,1,1,1)&0\\ 1&(1,-1,-1,1)&2\\ 2&(1,1,-1,-1)&2t\\ 3&(1,-1,1,-1)&2+2t \end {array}\]

Set \[\begin {aligned} q_0&=a,\\ q_1&=a+2,\\ q_2&=a+2t,\\ q_3&=a+2+2t,\\ \Delta &=a(a+2)(a+2t)(a+2+2t). \end {aligned}\] All 4 \(q_i\) are positive. With \(P_{u_i}=u_i u_i^T\), normalization and inverse covariance give \[\begin {aligned} \det Q&=\Delta ,& Z_Q&=4\pi ^2/\sqrt \Delta ,\\ G&=\sum _{i=0}^3q_i^{-1}P_{u_i}. \end {aligned}\] With \(O\) the array whose columns are \(u_i\), this is \(G=O\operatorname {diag}(q_i^{-1})O^T\).

Write \(d_i=q_i^{-1}\). Multiplying the signs of the relevant coordinates in each projector gives \[\begin {aligned} 4G_{00}&=d_0+d_1+d_2+d_3,\\ 4G_{01}&=d_0-d_1+d_2-d_3,\\ 4G_{02}&=d_0-d_1-d_2+d_3,\\ 4G_{03}&=d_0+d_1-d_2-d_3. \end {aligned}\] Thus the complete covariance, with \((A,B,C,D)=(G_{00},G_{01},G_{02},G_{03})\), is \[G=\begin {pmatrix} A&B&C&D\\B&A&D&C\\C&D&A&B\\D&C&B&A \end {pmatrix}.\] In particular the variance and adjacent two-point expectation are the first two formulas, with their indicated signs.

For the opposite entry, the common denominator \(4\Delta \) has numerator \[\begin {aligned} N&=(a+2)(a+2t)(a+2+2t)\\ &\quad -a(a+2t)(a+2+2t)\\ &\quad -a(a+2)(a+2+2t)\\ &\quad +a(a+2)(a+2t)\\ &=2[(a+2t)(a+2+2t)-a(a+2)]\\ &=8t(a+1+t). \end {aligned}\] Consequently \(G_{02}=2ts/\Delta \). At \((t,m)=(1,1)\), \(q=(1,3,3,5)\), \(\Delta =45\), \(Z_Q=4\pi ^2/(3\sqrt 5)\), and \[\begin {aligned} &(G_{00},G_{01},G_{02},G_{03})\\ &\qquad =(7/15,1/5,2/15,1/5). \end {aligned}\] At \((t,m)=(2,1)\), \(q=(1,3,5,7)\), \(\Delta =105\), and \[(G_{00},G_{01},G_{02},G_{03}) =(44,19,16,26)/105.\] These numbers are obtained by direct substitution in the 4 signed sums above. They give, in particular, the opposite covariance used in block elimination and the two distinct adjacent covariances used in relabeling.

More generally, weights \((b,c,b,c)\) with \(b,c>0\) keep the same 4 eigenvectors and give eigenvalues \(0,2b,2c,2b+2c\), by the same coordinate products. This is a real spectral calculation. The arithmetic columns are \(2u_i\), whose Gram matrix is \(4I\); they are not an orthonormal basis over an arbitrary coefficient ring.

The eigenvectors and signed sums are computed here from the weighted cycle.

[boyd2004convex, Section A.5.2, pp. 646-647] gives general spectral decomposition.

[friedli2017statistical, Proposition B.55, p. 505] gives the Gaussian density. Covariance equals inverse precision by Theorem 7.18.

Let \(L:E\to E\) be a self-adjoint positive semidefinite operator on a finite real inner-product space \((E,h)\). Set \[\begin {aligned} K&=\ker L,& H&=K^\perp ,\\ k&=\dim K,& r&=\dim H. \end {aligned}\] Write \(i:H\to E\) for inclusion, \(p_H:E\to H\) for orthogonal projection, \(P_H=ip_H\) and \(P_K=I-P_H\) for the endomorphism projections.

The spectral theorem gives \(E=K\mathbin \perp H\). The positive-eigenvalue vectors span \(H\), so \(L(H)\subset H\) and \(L_H=L|_H:H\to H\) is positive definite. Define the pseudodeterminant by \[\operatorname {pdet}L=\det L_H =\prod _{\lambda _a>0}\lambda _a.\] Its empty product is one, including \(L=0\). These definitions use the specified pairing and its induced subspace volume.

For a weighted incidence Laplacian with positive real edge weights, the kernel theorem identifies \(K\) with fields constant on each connected component. Indeed its energy is a sum of positive weights times squared edge differences; zero energy means every difference is zero, and propagation along undirected paths gives those constants. Loops contribute zero and isolated vertices are components. Thus \(H\) imposes zero sum separately on each component. For a connected nonempty graph, \(n=|V|\) gives the mean-zero field space \[\begin {aligned} u_0&=\mathbf 1_V/\sqrt n,\\ H&=\{\phi :\sum _v\phi (v)=0\},\\ P_H\phi &=\phi -\frac {\sum _v\phi (v)}{n}\mathbf 1_V. \end {aligned}\] The empty graph has \(E=K=H=0\); no division by \(n\) is used there.

The kernel complement and determinant product are direct consequences of the spectral decomposition in [boyd2004convex, Section A.5.2, pp. 646-647]; the graph kernel uses the incidence-energy identity proved in Theorem 3.7.

The massless four-cycle calculation makes this kernel complement and pseudodeterminant explicit.

With the kernel decomposition, define the Laplacian pseudoinverse by \[L^+=iL_H^{-1}p_H.\] This is the unique endomorphism \(A\) satisfying the 4 Moore-Penrose identities: \[\begin {aligned} LAL&=L,& ALA&=A,\\ (LA)^\dagger &=LA,&(AL)^\dagger &=AL. \end {aligned}\] It vanishes on \(K\), is the inverse of \(L_H\) on \(H\), and has spectral expression \(L^+=\sum _{\lambda _a>0}\lambda _a^{-1}P_{u_a}\), where \(P_u x=h(u,x)u\) for a unit eigenvector.

TauCeti's LinearMap.IsMoorePenroseInverse uses exactly these 4 conditions for real inner-product spaces. Its LinearMap.IsMoorePenroseInverse.unique supports uniqueness, and the finite-dimensional characterization LinearMap.isMoorePenroseInverse_iff_eq_moorePenroseInverse identifies this construction with \(L.\mathrm {moorePenroseInverse}\). The weaker inner-inverse identity \(LAL=L\) alone gives neither this uniqueness nor a constrained probability law.

The construction is a direct specialization of the orthogonal spectral decomposition in [boyd2004convex, Section A.5.2, pp. 646-647]. The four-identity and uniqueness arguments above establish the additional Moore-Penrose conclusion; no Gaussian law follows from that conclusion alone.

Let \(L\) be self-adjoint positive semidefinite on a finite real inner-product space \((E,h)\), with the kernel decomposition \(K=\ker L\), \(H=K^\perp \), \(k=\dim K\), \(r=\dim H\). The induced volume \(\lambda _H\) gives the centered probability law \[\begin {aligned} Z_H&=\frac {(2\pi )^{r/2}}{\sqrt {\operatorname {pdet}L}},\\ d\nu _H(x)&=Z_H^{-1}e^{-h(x,Lx)/2}\,d\lambda _H(x). \end {aligned}\] Its pushforward to \(E\) has covariance \(L^+\). If \(k>0\), the full-space massless integral is infinite, and the restriction of ambient volume to \(H\) is zero; neither replaces this induced measure.

For a connected nonempty graph with positive edge weights, put \(n=|V|\). Then \(H\) is exactly the mean-zero field space, \(r=n-1\), and the formula becomes \(Z_H=(2\pi )^{(n-1)/2}/\sqrt {\prod _{a=1}^{n-1}\lambda _a}\). At \(n=1\), \(H=0\), \(Z_H=1\), and the full one-dimensional integral still diverges. With \(k>1\) components, a single global sum-zero constraint removes only one dimension of the component-constant kernel: the sum functional is nonzero on that \(k\)-dimensional space, so its kernel has dimension \(k-1\). The restricted integral still diverges by the same kernel-box argument. Componentwise zero sums remove all these modes. The empty graph uses the zero-dimensional case above.

Massive factorization gives the exact divergent factor, while projected limits identify the finite limits of polynomial and difference observables. Vertex pinning uses another constraint and generally another coordinate covariance, as the two-vertex example shows. The distinction between free and fixed-boundary massless models also appears in [friedli2017statistical, Definition 8.1, Remark 8.2 and Exercise 8.2, p. 381]; the induced-volume and finite-graph proofs here specify the present law.

Let \(L\) have the kernel decomposition and let \(m>0\). The positive precision \(Q_m=L+m^2I\) has inverse and normalizer \[\begin {aligned} Q_m^{-1}&=m^{-2}P_K\\ &\quad +i(L_H+m^2I_H)^{-1}p_H,\\ Z_{Q_m}&=(\sqrt {2\pi }/m)^k Z_{H,m},\\ Z_{H,m}&=\frac {(2\pi )^{r/2}} {\sqrt {\prod _{\lambda _a>0}(\lambda _a+m^2)}}. \end {aligned}\] Under the addition isometry \(K\times H\to E\), its probability law is the product of the centered \(K\)-law with precision \(m^2I_K\) and the centered \(H\)-law with precision \(L_H+m^2I_H\). In particular its pushforward by \(p_H\) is exactly the second law.

This product calculation combines the orthogonal decomposition with the positive-covariance density of [friedli2017statistical, Proposition B.55, p. 505] after replacing covariance by inverse precision. The displayed kernel factor is derived here, including zero-dimensional factors.

For the kernel decomposition, let \(\nu _m\) be the projection onto \(H\) of the law with precision \(L+m^2I\), \(m>0\). Then \(\nu _m\) converges to the constrained law \(\nu _H\) in total variation, meaning \(\sup _{A\text { Borel}}|\nu _m(A)-\nu _H(A)|\to 0\). For every fixed polynomial \(P\) on \(H\), its expectations converge too.

For a fixed source \(j\in E\) and \(\ell _j(x)=h(j,x)\), the inverse-covariance formula and the kernel split give \[\begin {aligned} \operatorname {Var}_{Q_m}(\ell _j) &=m^{-2}\|P_Kj\|^2\\ &\quad +\sum _{\lambda _a>0} \frac {h(j,u_a)^2}{\lambda _a+m^2}. \end {aligned}\] Every term is nonnegative. The finite sum converges; the first term is finite in the limit precisely when \(P_Kj=0\). Thus the variance has a finite limit if and only if \(j\in H\), in which case it is \(h(j,L^+j)\).

For a connected nonempty graph with positive edge weights, put \(n=|V|\). The source \(u_0=\mathbf 1_V/\sqrt n\) has variance \(m^{-2}\) as a source. Its contribution to every Green-kernel entry is \(1/(nm^2)\), and \(G_m(v,v)\geq 1/(nm^2)\). A field difference has source \(\delta _v-\delta _w\in H\), so its variance converges to the constrained variance. On a disconnected graph this holds for vertices in the same component; a difference between components has a nonzero kernel projection. These observable limits do not supply a normalized full-space massless density.

The analytic ingredient is [tao2011measure, Theorem 1.4.49, pp. 111-112, preliminary version]. The three dominators above verify its hypotheses for these Gaussian laws and their fixed polynomial observables.

Let a finite nonempty graph be connected, with positive edge weights. Put \(n=|V|\) and fix a vertex \(p\). Its massless action defines a probability law on \(F_p=\{\phi :\phi (p)=0\}\) with induced counting volume. For any \(b\in \mathbb R\), translation by \(b\mathbf 1_V\) defines the law on the affine constraint \(\phi (p)=b\); its mean is \(b\mathbf 1_V\) and its covariance is that of the zero-pinned law.

These are laws on their own constraint spaces. They are not defined by conditioning on an event in an unconstrained massless probability, whose normalizer diverges. Fixed external boundary values also remove zero modes in finite lattice regions, as in [friedli2017statistical, Definition 8.1, Remark 8.2 and Exercise 8.2, p. 381]. That source's lattice boundary and volume conventions remain part of its model; the induced-volume construction here proves the finite-graph claim.

Join vertices \(0,1\) by one edge of weight \(w>0\). In counting coordinates its Laplacian is \[L=w\begin {pmatrix}1&-1\\-1&1\end {pmatrix}.\] The unit mean-zero vector \(u=(1,-1)/\sqrt 2\) has eigenvalue \(2w\). Thus the mean-zero law has \[\begin {aligned} Z_H&=\sqrt {2\pi }/\sqrt {2w},\\ L^+&=\frac 1{4w}\begin {pmatrix}1&-1\\-1&1\end {pmatrix}. \end {aligned}\] The factor \(1/\sqrt 2\) is required because the reference measure is the induced orthonormal volume on \(H\).

Pinning \(\phi _0=0\) leaves the unit vector \(\delta _1\); its action is \(w\phi _1^2/2\). The pinned law therefore has \[\begin {aligned} Z_F&=\sqrt {2\pi }/\sqrt w,\\ C_F&=\begin {pmatrix}0&0\\0&1/w\end {pmatrix}. \end {aligned}\] These ambient coordinate covariances differ. But for the difference \(\phi _0-\phi _1\), applying the two covariance forms to \((1,-1)\) gives \(1/w\) in both cases. This agreement of one observable does not identify the laws. For a single vertex the zero-pinned and mean-zero spaces are both zero, so that special case has no such distinction.

These are direct one-dimensional Gaussian calculations using the positive density in [friedli2017statistical, Proposition B.55, p. 505] and the two explicitly specified induced volumes.

The alternating-cycle eigenbasis has constant unit vector \(u_0=(1,1,1,1)/2\); its positive eigenvalues are \(2,2t,2+2t\). Hence the mean-zero law has \[\begin {aligned} \operatorname {pdet}L&=8t(1+t),\\ Z_H&=\frac {(2\pi )^{3/2}}{\sqrt {8t(1+t)}},\\ L^+&=\tfrac 12P_{u_1}+\tfrac 1{2t}P_{u_2} +\tfrac 1{2+2t}P_{u_3}. \end {aligned}\] Its covariance rows follow from the 4 signed sums of Example 7.19 by setting \(d_0=0\), \(d_1=1/2\), \(d_2=1/(2t)\), and \(d_3=1/(2+2t)\). This is the pseudoinverse, not an inverse on the full vertex space.

At \(t=1\), its first row is \[(5/16,-1/16,-3/16,-1/16).\] All diagonal entries are \(5/16\). Thus an opposite field difference has variance \(2(5/16)-2(-3/16)=1\), while an adjacent difference has variance \(2(5/16)-2(-1/16)=3/4\). Both are finite limits of the corresponding massive variances by Proposition 8.5.

The massive constant contribution to every covariance entry is \(1/(4m^2)\). Its divergence cancels in field differences, and kernel factorization gives the exact normalizer relation \[\begin {aligned} Z_{Q_m}&=\frac {\sqrt {2\pi }}{m}Z_{H,m},\\ mZ_{Q_m}&\longrightarrow \sqrt {2\pi }Z_H. \end {aligned}\] The full normalizer therefore does not tend to \(Z_H\); the constrained law uses its own three-dimensional reference volume.

This calculation applies the induced-volume construction of Proposition 8.3 to the positive spectral modes.

[friedli2017statistical, Proposition B.55, p. 505] supplies the positive Gaussian density. The zero-mode removal and its normalization factor are derived above.

Let \(E=X\mathbin \perp Y\) be a specified orthogonal sum of finite-dimensional real inner-product spaces. Write \(h_E,h_X,h_Y\) for their pairings, \(i_X,i_Y\) for inclusion and \(p_X,p_Y\) for orthogonal projection. The addition isometry is \(J(x,y)=i_Xx+i_Yy\). Use canonical Euclidean volume on uncompleted Borel sets. Then product volume gives \(J_*(\lambda _X\otimes \lambda _Y)=\lambda _E\). Zero summands are allowed; their volume is unit mass on a singleton.

Let \(Q:E\to E\) be a self-adjoint positive precision parameter. Its components for this splitting are \[\begin {aligned} A&=p_XQi_X:X\to X,\\ B&=p_XQi_Y:Y\to X,\\ C&=p_YQi_Y:Y\to Y. \end {aligned}\] The fourth component is \(p_YQi_X=B^\dagger \), where the adjoint uses the specified pairings. Both \(C\) and the effective precision, or real Schur complement, \[S=A-BC^{-1}B^\dagger :X\to X\] are positive definite and invertible. This is the real positive specialization of the split Schur map.

Put \(w_Q(z)=\exp (-h_E(z,Qz)/2)\) and \(q=\dim Y\). For every \(x\in X\), the exact fiber integral is \[\begin {aligned} I_Q(x)&=\int _Y w_Q(J(x,y))\,d\lambda _Y(y)\\ &=c_C\exp (-h_X(x,Sx)/2),\\ c_C&=\frac {(2\pi )^{q/2}}{\sqrt {\det C}}=Z_C>0. \end {aligned}\] Moreover \(\det Q=\det C\det S\). Thus elimination induces both \(S\) and a positive finite scalar \(c_C\).

Taking minus the logarithm of the positive fiber integral defines an exact effective action: \[\begin {aligned} \mathcal A_{\mathrm {eff}}(x) &=\tfrac 12h_X(x,Sx)-\log c_C\\ &=\tfrac 12h_X(x,Sx)+\tfrac 12\log \det C\\ &\quad -\tfrac q2\log (2\pi ). \end {aligned}\] The constant cancels from normalized retained expectations but belongs to the unnormalized weight. Rescaling the two reference volumes by \(a_X,a_Y>0\) multiplies \(c_C\) by \(a_Y\), \(Z_S\) by \(a_X\) and \(Z_Q\) by \(a_Xa_Y\); normalized laws remain unchanged. For \(Y=0\), \(c_C=1\); for \(X=0\), \(Z_S=1\). In each case the unique zero-space endomorphism is invertible and has determinant one.

For counting fields on \(V=R\sqcup U\), the delta bases identify \(X=\mathbb R^R\), \(Y=\mathbb R^U\), \(B=H\) and \(B^\dagger =H^T\). Coordinate volume splitting recovers the usual \(dx\,dy\) formula. The carrier, split, input precision and volumes are supplied; \((S,c_C)\) is induced relative to those choices. The singular-pivot boundary requires a separate argument.

[boyd2004convex, Appendix A.5.5, pp. 650-651] gives square completion with the first block as pivot. Exchanging its order to \((y,x)\) gives the present eliminated pivot \(C\).

[rasmussen2006gaussian, Appendix A.2, equation (A.4), p. 200] gives the Gaussian density with covariance as parameter. Substituting \(C^{-1}\) gives the normalizer used here. The invariant integration and determinant argument above reconstruct these ingredients with specified pairings and volume.

Mathlib's Matrix.schur_complement_eq₂₂ supports square completion in compatible finite coordinates, with a Hermitian invertible eliminated block. Its Matrix.det_fromBlocks₂₂ supports the determinant identity. TauCeti's Matrix.PosDef.gaussianCondCov supports strict Schur positivity for a real positive-definite block matrix. Despite its conditional-covariance name, this algebraic conclusion can be applied to the precision \(Q\). These ingredients do not supply the integral or its reference-volume convention.

The operation is algebraic: it acts on vertex and edge sets that are supplied, and carries neither a dynamical carrier nor an effective metric, which are additional data.

Use the orthogonal sum \(E=X\mathbin \perp Y\), positive precision \(Q\), effective precision \(S\) and exact scalar \(c_C\) of Gaussian elimination. Let \(\pi =p_X:E\to X\) and let \(\mu _Q,\mu _S\) be the independently defined Gaussian probability laws. Then \[Z_Q=c_CZ_S,\qquad \pi _*\mu _Q=\mu _S.\] Thus the retained marginal law is \(\mu _S\): for every Borel \(D\subseteq X\), \(\mu _Q(\pi ^{-1}D)=\mu _S(D)\). Here pushforward assigns \(D\) the mass of its inverse image, extending the finite pushforward convention to these Borel spaces.

Every real Borel function \(F:X\to \mathbb R\) satisfies the extended nonnegative identity \[\int _E|F\circ \pi |\,d\mu _Q =\int _X|F|\,d\mu _S\ \in [0,\infty ].\] Consequently \(F\in L^1(\mu _S)\) if and only if \(F\circ \pi \in L^1(\mu _Q)\). Under these equivalent conditions, \[\int _E F\circ \pi \,d\mu _Q=\int _X F\,d\mu _S.\] We use actual Borel functions; \(L^1\) here denotes their integrable vector space. It is not asserted to be an algebra.

For a finite real vector space \(W\), write \(\mathcal P(W)\) for its intrinsic polynomial function algebra, generated by linear functionals. The actual map on observables is \[\begin {aligned} \pi ^*:\mathcal P(X)&\longrightarrow \mathcal P(E),\\ P&\longmapsto P\circ \pi . \end {aligned}\] It is an injective unital real algebra homomorphism preserving the identity involutions. For the Gaussian polynomial states, \[\omega _S=\omega _Q\circ \pi ^*.\] The field projection goes \(E\to X\); this observable inclusion goes in the opposite direction.

The inverse-compression and covariance identities are \[\begin {aligned} p_XQ^{-1}i_X&=S^{-1},\\ \operatorname {Cov}_{\mu _Q}(\ell _j\circ \pi ,\ell _k\circ \pi ) &=h_X(j,S^{-1}k), \end {aligned}\] where \(\ell _j(x)=h_X(j,x)\). The means vanish. In counting coordinates this is \((Q^{-1})_{RR}=S^{-1}\).

Fixing the eliminated field is a different operation from averaging it. The conditional-law calculation gives its parameters; the four-cycle comparison exhibits a difference even when the fixed value is zero.

[rasmussen2006gaussian, Appendix A.2, equations (A.5)-(A.6), p. 200] gives Gaussian marginal and conditional parameters. The inverse block formula is in [rasmussen2006gaussian, Appendix A.3, equation (A.13), p. 202]. Its joint parameter is covariance \(\Sigma =Q^{-1}\). Retained covariance is therefore \(p_X\Sigma i_X=S^{-1}\), not \(S\).

In orthonormal coordinates, \(\Sigma ^{-1}=Q\) and \(\det \Sigma =(\det Q)^{-1}\) turn the provider density into \(Z_Q^{-1}w_Q\); the density and volume comparison identifies the probability laws. Mathlib's ProbabilityTheory.measurePreserving_restrict₂_multivariateGaussian then supports coordinate restriction with covariance \(\Sigma \). TauCeti's TauCeti.Probability.map_affine_multivariateGaussian supports rectangular affine images with positive-semidefinite covariance. Apply its linear map as the orthonormal presentation of \(p_X\) and its translation as zero. These concern Gaussian probability measures, not volume preservation by a projection.

Mathlib's MeasureTheory.integral_prod supports signed Fubini under integrability. None of these declarations supplies the raw factor \(c_C\) or removes the preceding absolute-value argument.

In the orthogonal positive setting of Gaussian elimination, let \(X\) be retained and \(Y\) eliminated. The components of \(Q\) are \(A,B,B^\dagger ,C\). Put \(T=C-B^\dagger A^{-1}B\) and \(c_A=(2\pi )^{\dim X/2}/\sqrt {\det A}\). The maps \(A,T\) are positive definite by exchanging the eliminated summand. For every \(y\in Y\), define a probability measure on \(X\) by \[\begin {aligned} &k_y(x)=c_A^{-1}\exp \bigl (-\tfrac 12\\ &\qquad \cdot h_X(x+A^{-1}By,A(x+A^{-1}By))\bigr ),\\ &\kappa (y,D)=\int _D k_y(x)\,d\lambda _X(x) \end {aligned}\] for Borel \(D\subseteq X\). These measures form a measurable conditional probability kernel for the retained field given the eliminated field. Its mean is \(-A^{-1}By\) and covariance operator is \(A^{-1}\).

The retained marginal instead has mean zero and covariance \(S^{-1}\) for \(S=A-BC^{-1}B^\dagger \). At \(y=0\), the conditional precision is \(A\), not \(S\). If \(B=0\), then \(S=A\) and the conditional mean is zero for every \(y\), so the laws coincide. The four-cycle exhibits their different off-diagonal covariances when the coupling is nonzero.

To compare covariance parameters, put \(\Sigma =Q^{-1}\) and write its components as \(\Sigma _{XX},\Sigma _{XY},\Sigma _{YY}\). The opposite-pivot inverse formulas give \[\begin {aligned} \Sigma _{YY}&=T^{-1},\\ \Sigma _{XY}&=-A^{-1}BT^{-1},\\ \Sigma _{XX}&=A^{-1}+A^{-1}BT^{-1}B^\dagger A^{-1}. \end {aligned}\] Consequently \[\begin {aligned} \Sigma _{XY}\Sigma _{YY}^{-1}&=-A^{-1}B,\\ \Sigma _{XX}-\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}&=A^{-1}. \end {aligned}\] The covariance Schur complement for conditioning differs from the precision Schur complement for marginalization.

[rasmussen2006gaussian, Appendix A.2, equation (A.6), p. 200] gives these conditional Gaussian parameters. The density construction above additionally checks normalization, parameter measurability and the conditional integral identity.

TauCeti's TauCeti.Probability.condDistrib_multivariateGaussian identifies a conditional distribution with its Gaussian kernel almost everywhere for the conditioning marginal, assuming a Gaussian joint law, positive-semidefinite covariance and a positive-definite conditioning covariance block. Here \(\Sigma =Q^{-1}>0\) and \(\Sigma _{YY}=T^{-1}>0\); the displayed identities supply the parameter conversion. The totalized Gaussian constructor or a Markov-kernel instance alone would not identify the conditional distribution.

Use the orthogonal spaces and components of Gaussian elimination, but assume only \(Q=Q^\dagger \geq 0\). Then \(C\geq 0\). If \(N=\ker C\ne 0\), every full fiber integral is infinite: \[\int _Y e^{-h_E(J(x,y),QJ(x,y))/2}\,d\lambda _Y(y)=\infty .\] Replacing \(C^{-1}\) by a pseudoinverse does not change that integral.

For example, on \(X=Y=\mathbb R\), \(Q(x,y)=(x,0)\) has weight \(e^{-x^2/2}\), independent of \(y\). Thus \(e^{-x^2/2}\int _{\mathbb R}dy=\infty \) for every \(x\).

A different model integrates only over \(Y_0\) with its induced Euclidean volume. Let \(j_0:Y_0\to Y\) and \(p_0:Y\to Y_0\) be orthogonal inclusion and projection. The restriction \(C_0=C|_{Y_0}\) is positive definite, and the Moore-Penrose construction is \[C^+=j_0C_0^{-1}p_0.\] For \(r=\dim Y_0=\operatorname {rank}C\), put \[\begin {aligned} S_0&=A-BC^+B^\dagger ,\\ c_0&=\frac {(2\pi )^{r/2}}{\sqrt {\operatorname {pdet}C}}. \end {aligned}\] Then \(S_0\geq 0\) and \[\begin {aligned} &\int _{Y_0}e^{-h_E(J(x,j_0y),QJ(x,j_0y))/2}\,d\lambda _{Y_0}(y)\\ &\qquad =c_0e^{-h_X(x,S_0x)/2}. \end {aligned}\] A normalized law on the full retained \(X\) exists exactly when \(S_0>0\).

The singular positive-semidefinite pivot, its range condition and pseudoinverse quadratic minimum are treated in [boyd2004convex, Appendix A.5.5, p. 651]. Exchanging the pivot gives the algebra used here. The divergence and induced-volume calculation above are additional arguments. The constrained Gaussian construction likewise changes the space and measure. Neither a general indefinite pivot nor a regularized value for the original divergent fiber is asserted.

In the alternating four-cycle, take \(t,m>0\) and \(Q=Q_{t,m}=L_t+m^2I\). Its counting-pairing energy is \[\begin {aligned} &\langle \phi ,Q\phi \rangle \\ &\quad =(\phi _1-\phi _0)^2+t(\phi _2-\phi _1)^2\\ &\qquad +(\phi _3-\phi _2)^2+t(\phi _0-\phi _3)^2\\ &\qquad +m^2\sum _v\phi _v^2. \end {aligned}\] It is positive for every nonzero field. Retain \(R=(0,2)\) and eliminate \(U=(1,3)\), with coordinates \(x=(\phi _0,\phi _2)\) and \(y=(\phi _1,\phi _3)\) in these orders. The maps \(x\mapsto x_0\delta _0+x_2\delta _2\) and \(y\mapsto y_1\delta _1+y_3\delta _3\) are isometries from ordinary \(\mathbb R^2\); their sum is the specified orthogonal splitting, and product volume is \(dx\,dy\). Write \(a=m^2\) and \(s=1+t+a\). Reordering the explicit Laplacian gives \[\begin {aligned} A=C&=sI_2,\\ B=H&=-\begin {pmatrix}1&t\\t&1\end {pmatrix}. \end {aligned}\] The elimination theorem therefore gives \[\begin {aligned} S&=\begin {pmatrix}\alpha &-\beta \\-\beta &\alpha \end {pmatrix},\\ \alpha &=s-\frac {1+t^2}{s},\qquad \beta =\frac {2t}{s}>0. \end {aligned}\] Indeed \(HH^T\) has diagonal \(1+t^2\) and off-diagonal \(2t\), so \(S=sI_2-s^{-1}HH^T\).

[boyd2004convex, Appendix A.5.5], pp. 650-651, gives the general Schur complement. Here the chosen blocks make its entries explicit.

Solid circles are retained; dashed circles are integrated out. Orientations are suppressed because the quadratic energy is orientation independent. The bottom edge represents the induced coupling \(\beta \); its precision also has the diagonal term computed below.

The vectors \((1,1)^T/\sqrt 2\) and \((1,-1)^T/\sqrt 2\) diagonalize \(S\). Their eigenvalues are \[\begin {aligned} \sigma _+&=\alpha -\beta =\frac {s^2-(1+t)^2}{s}\\ &=\frac {a(a+2+2t)}{s},\\ \sigma _-&=\alpha +\beta =\frac {s^2-(1-t)^2}{s}\\ &=\frac {(a+2)(a+2t)}{s}. \end {aligned}\] Both are positive. The factorizations follow from \(s=1+t+a\); they also check positivity without appealing again to the general theorem. Put \[\mathcal D=a(a+2)(a+2t)(a+2+2t).\] Then \[\begin {aligned} \det C&=s^2,\qquad \det S=\mathcal D/s^2,\\ \det Q&=\det C\det S=\mathcal D,\\ c_C&=\frac {2\pi }{s},\qquad Z_S=\frac {2\pi s}{\sqrt {\mathcal D}},\\ Z_Q&=c_CZ_S=\frac {(2\pi )^2}{\sqrt {\mathcal D}}. \end {aligned}\] This reproduces the four-mode determinant and partition function. The effective action is \[\mathcal A_{\mathrm {eff}}(x)=\tfrac 12x^TSx+\log \frac {s}{2\pi }.\] Its constant records the integrated block.

A retained observable checks more than normalization. For \(F(x)=x_0x_2\), polynomial integrability and the marginal identity give \[\begin {aligned} \langle \phi _0\phi _2\rangle _Q &=\langle x_0x_2\rangle _S =(S^{-1})_{02}\\ &=\frac {\beta }{\alpha ^2-\beta ^2} =\frac {2ts}{\mathcal D}. \end {aligned}\] The indices \(0,2\) here name retained vertices. Inverting the displayed two-by-two matrix gives its off-diagonal entry \(\frac {\beta }{\det S}\), which proves the last equalities. Independently, the four-cycle spectral table gives \[\begin {aligned} 4\langle \phi _0\phi _2\rangle _Q &=\frac 1a-\frac 1{a+2}\\ &\quad -\frac 1{a+2t}+\frac 1{a+2+2t}\\ &=\frac {8ts}{\mathcal D}. \end {aligned}\] For the last equality, pair the first two terms and the last two: their difference has numerator \(2((a+2t)(a+2+2t)-a(a+2))=8ts\) over \(\mathcal D\). At \(t=m=1\), one obtains \[\begin {aligned} S&=\begin {pmatrix}7/3&-2/3\\-2/3&7/3\end {pmatrix},\\ S^{-1}&=\frac 1{15}\begin {pmatrix}7&2\\2&7\end {pmatrix},\\ \langle \phi _0\phi _2\rangle _Q&=\frac {2}{15},\\ c_C&=\frac {2\pi }{3},\quad Z_S=\frac {2\pi }{\sqrt 5},\quad Z_Q=\frac {4\pi ^2}{3\sqrt 5}. \end {aligned}\] Fixing the eliminated fields to zero instead gives precision \(A=3I_2\), covariance \(I_2/3\) and zero off-diagonal covariance. At \(t=2,m=1\), the same formulas give \[\begin {aligned} S&=\begin {pmatrix}11/4&-1\\-1&11/4\end {pmatrix},\\ \det S&=105/16,\\ S^{-1}&=\frac 1{105}\begin {pmatrix}44&16\\16&44\end {pmatrix}. \end {aligned}\] These are the parameters used in the symmetry comparison.

In the orthonormal sector coordinates \(x_\pm =(x_0\pm x_2)/\sqrt 2\) and \(y_\pm =(y_1\pm y_3)/\sqrt 2\), the couplings are \(-(1+t)\) and \(-(1-t)\). Their two eliminated pivots are both \(s\); each contributes \(\sqrt {2\pi/s}\), whose product is \(2\pi/s\). The factors \(1/\sqrt 2\) make these changes orthogonal, so there is no extra volume factor. The retained variance is \(\frac 12(\sigma _+^{-1}+\sigma _-^{-1})\), and the opposite covariance is \(\frac 12(\sigma _+^{-1}-\sigma _-^{-1})\). This agrees with the sector calculation.

There was no edge between vertices \(0\) and \(2\) in the supplied cycle. Elimination induces one in the effective quadratic form: \[S=\beta \begin {pmatrix}1&-1\\-1&1\end {pmatrix} +\sigma _+I_2.\] Thus \(\beta \) is an effective edge weight and \(\sigma _+\) an effective mass coefficient. This is a finite instance of fill-in: eliminating a single vertex joined to retained vertices with weights \(w_i\) and diagonal precision \(c>0\) subtracts \(ww^T/c\), creating off-diagonal entries \(-w_iw_j/c\) where no retained edge existed. The formula is the same Schur complement with \(H=-w\) and \(C=(c)\). Locality in the original graph need not survive elimination.

The half-turn preserves both sets \(R,U\) and swaps the two coordinates within each. For the swap matrix \(J\), the displayed \(S\) satisfies \(JSJ=S\); the induced two-site form retains this symmetry. The density and orthogonal-volume invariance then show that the retained law is invariant as well.

At \(m=0\), \(C=(1+t)I_2\) remains positive. Square completion and integration over the eliminated block still give \[S=\frac {2t}{1+t}\begin {pmatrix}1&-1\\-1&1\end {pmatrix}, \qquad c_C=\frac {2\pi }{1+t}.\] The retained constant mode has zero energy, so its full integral diverges by the kernel-box argument. The finite fiber factor cannot be divided by a finite retained partition function.

The global mean-zero law instead uses the space \(H=\{\phi :\sum _v\phi _v=0\}\) with its own induced volume. Its constraint \(x_0+x_2+y_1+y_3=0\) couples the two parts; \(H\) is neither their unrestricted product nor the product of their separate mean-zero lines. At \(t=1\), write a field in the three positive orthonormal modes as \(\phi =z_1u_1+z_2u_2+z_3u_3\). The coordinates are independent centered Gaussians of precisions \(2,2,4\). Put \[r=\frac {z_1+z_2}{\sqrt 2},\qquad v=\frac {z_1-z_2}{\sqrt 2}.\] This orthogonal change preserves \(z_1^2+z_2^2=r^2+v^2\) and two-dimensional volume. Thus \(r,v,z_3\) still have independent centered Gaussian laws of precisions \(2,2,4\). The retained unit coordinates satisfy \[\begin {aligned} x_-&=(\phi _0-\phi _2)/\sqrt 2=r,\\ x_+&=(\phi _0+\phi _2)/\sqrt 2=z_3/\sqrt 2. \end {aligned}\] Integrating the normalized \(v\) factor and changing variable \(z_3=\sqrt 2x_+\) give independent retained Gaussian densities of precisions \(2\) and \(8\). This density calculation identifies the law, not merely its covariance. Returning orthogonally to \((x_0,x_2)\) gives \[\begin {aligned} \operatorname {Cov}(x)&=\frac 1{16} \begin {pmatrix}5&-3\\-3&5\end {pmatrix},\\ Q_{H\to R}&=\begin {pmatrix}5&3\\3&5\end {pmatrix}. \end {aligned}\] These matrices multiply to the identity. The projected normalizer for this precision is \(\frac {2\pi }{\sqrt {16}}=\frac \pi 2\).

In contrast, directly setting \(m=0,t=1\) in the Schur formula gives the singular precision \(\begin {pmatrix}1&-1\\-1&1\end {pmatrix}\). Separately constraining that retained model to \(x_0+x_2=0\) leaves unit direction \((1,-1)/\sqrt 2\) with precision \(2\), normalizer \(\sqrt \pi \) and ambient covariance \(\frac 14\begin {pmatrix}1&-1\\-1&1\end {pmatrix}\). These are three distinct specified measures. The pinned affine law is another constraint with its own induced volume; fixing a vertex is not the marginal operation above.

The arithmetic pivot boundary is separate from the positive-real case. At \(m^2=1,t=-2\), \(s=t+2=0\), whereas \(\det Q=1\cdot 3\cdot (-3)\cdot (-1)=9\). Thus the full map can be invertible while this eliminated pivot is not. At \(t=-1\), the pivot exists but the later ratio denominator \(t+1\) vanishes. Neither value satisfies the positive-weight hypotheses used for the Gaussian laws here.

Eliminating a field with a quartic interaction generally creates quadratic and sextic terms. The coefficients through second order can be found by conditional Gaussian moments, while preserving a supplied symmetry and composition of eliminations.

Use the finite orthogonal sum \(E=X\mathbin \perp Y\), positive precision \(Q\), components \(A,B,C\), Schur precision \(S\) and scalar \(c_C\) of Gaussian elimination. Let \(T:E^4\to \mathbb R\) be a symmetric four-linear form and put \[V(u)=\frac {T(u,u,u,u)}{4!}.\] Write \(\mathcal P(X)\) for the polynomial function algebra. For \(x\in X\), let \(\eta \) have centered Gaussian law with covariance \(C^{-1}\) on \(Y\), and set \[v_x=(x,-C^{-1}B^\dagger x+\eta ).\] This is the conditional law of the eliminated field given the retained value. It is the conditional kernel with its two named summands exchanged; its covariance is \(C^{-1}\), whereas that proposition's retained-given-eliminated kernel has covariance \(A^{-1}\).

For a formal variable \(\lambda \), use ordinary power-series coefficients in \(\lambda \) and define the fiber effective interaction by \[\begin {aligned} F_\lambda (x)&=-\log \mathbb E_\eta e^{-\lambda V(v_x)},\\ \mathbb E_\eta e^{-\lambda V(v_x)} &=\sum _{n\geq 0}\frac {(-\lambda )^n}{n!} \mathbb E_\eta V(v_x)^n. \end {aligned}\] These are formal series in \(\lambda \) with coefficients in \(\mathcal P(X)\); the logarithm is the formal logarithm of a series with constant coefficient one. Every coefficient is finite by polynomial integrability. The notation \(O(\lambda ^3)\) means a series divisible by \(\lambda ^3\), not an analytic error estimate. With \[\begin {aligned} a(x)&=\mathbb E[V\mid x],\\ b(x)&=\mathbb E[V^2\mid x]-a(x)^2, \end {aligned}\] the first two coefficients are \[F_\lambda =\lambda a-\frac {\lambda ^2}{2}b+O(\lambda ^3).\] The degree-zero terms are retained: coefficientwise integration gives \[\begin {aligned} &\int _Y e^{-\frac 12\langle (x,y),Q(x,y)\rangle } e^{-\lambda V(x,y)}\,dy\\ &\qquad =c_Ce^{-\frac 12\langle x,Sx\rangle }e^{-F_\lambda (x)}. \end {aligned}\] Here \(dy\) is the specified Euclidean volume. This fiber logarithm integrates eliminated variables; it does not take the Legendre transform of a source-generating function used for a one-particle irreducible (1PI) effective action in [etingof2024mathematical, v2, Proposition 3.23, p. 50].

For a specified orthogonal flag \(E=X\mathbin \perp Y\mathbin \perp Z\), first eliminate \(Z\) and then \(Y\). All conditional expectations below use the original Gaussian law, with the \(Y\)-given-\(X\) kernel obtained after eliminating \(Z\). Put \[\begin {aligned} a_1(x,y)&=\mathbb E[V\mid x,y],\\ b_1(x,y)&=\mathbb E[V^2\mid x,y]-a_1(x,y)^2. \end {aligned}\] Both stages together give \[\begin {aligned} F_\lambda (x) ={}&\lambda \mathbb E[a_1\mid x]\\ &-\frac {\lambda ^2}{2}\bigl ( \mathbb E[b_1\mid x]+\operatorname {Var}(a_1\mid x)\bigr )\\ &+O(\lambda ^3), \end {aligned}\] which agrees with direct elimination of \(Y\oplus Z\). The first coefficient has degrees among \(4,2,0\); the second has degrees among \(6,4,2,0\). Particular components may vanish.

Suppose a second such model is related by a linear isometric isomorphism \(L=L_X\oplus L_Y:E\to E'\) satisfying \(Q'L=LQ\) and \(V'\circ L=V\). Then \[F'_\lambda (L_Xx)=F_\lambda (x)\pmod {\lambda ^3}.\] Indeed the component identities give \(C'L_Y=L_YC\) and \((B')^\dagger L_X=L_YB^\dagger \). Thus \(L_Y\) transports the conditional mean and covariance, hence its Gaussian law and both moments \(a,b\). For two-stage elimination also require preservation of the supplied flag. In particular, a supplied finite group of these isometries preserves each induced coefficient. The result transports a given action; it does not select that action.

Wick contractions are [etingof2024mathematical, v2, Theorem 3.1 and Example 3.2, p. 29]. The perturbation convention in [etingof2024mathematical, v2, Section 3.3, p. 33] is \(B(u,u)/2-\sum _i g_iB_i(u,\ldots ,u)/i!\); here \(g_4=-\lambda \) and \(B_4=T\), at \(\hbar =1\). The conditional formulas and composition above are direct deductions using these contractions and the specified Gaussian kernels. The three-coordinate example exhibits every indicated degree.

On the real realization of \(\mathbb Q^3\), use coordinates \(x,y,z\), counting pairing \(g_0=I\) and the coordinate flag retaining \((x,y)\) and then \(x\). Fix the action \(q/2+\lambda V\), where \[\begin {aligned} q&=x^2+(y-x)^2+(z-y)^2,\\ V&=\frac {x^4+y^4+z^4}{4!},\\ Q&=\begin {pmatrix}2&-1&0\\-1&2&-1\\0&-1&1\end {pmatrix},\\ Q^{-1}&=\begin {pmatrix}1&1&1\\1&2&2\\1&2&3\end {pmatrix}. \end {aligned}\] The quadratic form is positive: the map \((x,y,z)\mapsto (x,y-x,z-y)\) is invertible. The fourth tensor of \(V\) has diagonal components one and all other components zero, with the displayed \(1/4!\) convention. Simultaneous sign reversal supplies an action of \(C_2\) preserving the pairing, action and flag.

For the fiber interaction after eliminating \(y,z\), the retained precision is \(S=1\), the scalar is \(c_C=2\pi \), and \[\begin {aligned} F_\lambda (x) ={}&\lambda \left (\frac {x^4}{8}+\frac {3x^2}{4}+\frac 58\right )\\ &-\lambda ^2\left (\frac {5x^6}{72}+\frac {53x^4}{48}\right .\\ &\hspace {38pt}\left .+\frac {97x^2}{24}+\frac {41}{24}\right ) +O(\lambda ^3). \end {aligned}\] In particular, a pure quartic interaction is not preserved by elimination. Adding a quadratic term and a constant still does not close this family: the second-order sextic coefficient is nonzero.

A source on a retained space \(X\) is a linear functional \(j\in X^*\), coupled by multiplying the weight by \(e^{j(x)}\). Under a retained linear isomorphism \(L_X:X\to X'\), a source \(j'\in (X')^*\) pulls back to \(j'\circ L_X\), since \(e^{j'(L_Xx)}=e^{(j'\circ L_X)(x)}\). In this example \(X=\mathbb R\) and \(j(x)=Jx\), with \(J\) treated as a formal scalar variable. Let \(\mu _Q\) be the normalized Gaussian law. Define \[\begin {aligned} W(J,\lambda ) ={}&\log \mathbb E_{\mu _Q}e^{Jx-\lambda V}\\ &-\log \mathbb E_{\mu _Q}e^{-\lambda V}. \end {aligned}\] Each coefficient in the two formal variables \(J,\lambda \) is a finite Gaussian moment. Thus \(W(0,\lambda )=0\). Coefficients use ordinary powers of \(\lambda \); 4 source derivatives multiply the coefficient of \(J^4\) by \(4!\). The connected four-point function in this scalar example is \[\begin {aligned} G_c^{(4)}(\lambda ) &=\left .\frac {\partial ^4W}{\partial J^4}\right |_{J=0}\\ &=-3\lambda +101\lambda ^2+O(\lambda ^3). \end {aligned}\]

These are formal expansions. For real \(\lambda \geq 0\), the original interaction is nonnegative, so its exact partition integral is positive and bounded by the Gaussian normalizer. Polynomial moments remain finite. A real source also gives a finite integral, since the linear tilt of a positive Gaussian is integrable. None of this asserts convergence of the formal series at zero. For every fixed \(\lambda >0\), exponentiating only the displayed second-order \(F_\lambda \) instead gives a nonintegrable retained weight: its leading term is \(-5\lambda ^2x^6/72\), so \(e^{-x^2/2-F_\lambda ^{(2)}(x)}\) grows at infinity. Here \(F_\lambda ^{(2)}\) denotes the displayed polynomial through \(\lambda ^2\).

The Gaussian moment rule is [etingof2024mathematical, v2, Theorem 3.1, p. 29]. The connected-logarithm interpretation is [etingof2024mathematical, v2, Theorem 3.11, p. 38], with \(g_4=-\lambda \) and the zero-source logarithm subtracted. The rational coefficients above are the explicit calculation for this supplied model. Stationary elimination retains its sextic term but omits the fluctuation-generated lower degrees.

For the three-coordinate quartic action, formal power-series stationarity around the Gaussian stationary point and integration of the eliminated fluctuations agree on some coefficients but not all. Write \(u=(y,z)\), \(u_0=(x,x)\) and \[H=C^{-1}=\begin {pmatrix}1&1\\1&2\end {pmatrix}.\] At fixed \(x\), the action is \[\begin {aligned} \mathcal A(x,u)={}&\frac {x^2}{2} +\frac 12(u-u_0)^TC(u-u_0)\\ &+\lambda V(x,u). \end {aligned}\] There is a unique formal stationary solution \(u_*(x,\lambda )=u_0+O(\lambda )\). Its stationary value is \[\begin {aligned} \mathcal A_{\mathrm {cl}}(x,\lambda ) &=\mathcal A(x,u_*(x,\lambda ))\\ &=\frac {x^2}{2}+\frac {\lambda x^4}{8} -\frac {5\lambda ^2x^6}{72}+O(\lambda ^3). \end {aligned}\] For real \(\lambda \geq 0\), the action is coercive and strictly convex in \(u\): the quadratic Hessian \(C\) is positive definite and the quartic summands are convex. It therefore has a unique minimizer. The calculation below uses only formal power-series stationarity around \(u_0\) at \(\lambda =0\).

To distinguish fluctuation terms, take \(\hbar >0\), let \(\eta _\hbar \) be centered Gaussian with covariance \(\hbar H\), and use the normalized fiber expression \[\begin {aligned} &F_{\lambda ,\hbar }(x)\\ &\quad =-\hbar \log \mathbb E\exp \left ( -\frac {\lambda }{\hbar }V(x,u_0+\eta _\hbar )\right ). \end {aligned}\] Expand formally in \(\lambda \); through the second order its coefficients are polynomials in \(\hbar \): \[\begin {aligned} F_{\lambda ,\hbar }(x) ={}&\lambda \left (\frac {x^4}{8}+\frac {3\hbar x^2}{4} +\frac {5\hbar ^2}{8}\right )\\ &-\lambda ^2\left (\frac {5x^6}{72} +\frac {53\hbar x^4}{48}\right .\\ &\hspace {38pt}\left .+\frac {97\hbar ^2x^2}{24} +\frac {41\hbar ^3}{24}\right )\\ &+O(\lambda ^3). \end {aligned}\] Indeed each Gaussian pairing contributes one \(\hbar \); the first coefficient is \(\mathbb E V\) and the second is \(-\operatorname {Var}(V)/(2\hbar )\). Applying this scaling to the explicit moment calculation proves the formula.

Consequently, through order \(\lambda ^2\), the coefficient of \(\hbar ^0\) in this fiber interaction is \[\begin {aligned} [\hbar ^0]F_{\lambda ,\hbar }(x) &=\frac {\lambda x^4}{8}-\frac {5\lambda ^2x^6}{72} +O(\lambda ^3)\\ &=\mathcal A_{\mathrm {cl}}(x,\lambda )-\frac {x^2}{2} +O(\lambda ^3). \end {aligned}\] The retained Gaussian action \(x^2/2\) must be added to recover the full stationary value. At \(\lambda =0\) the normalized fiber interaction is zero, whereas that stationary value is \(x^2/2\).

This is the tree contribution in the loop grading of [etingof2024mathematical, v2, Section 3.6 and Theorem 3.12, pp. 38-39]. We adapt Etingof's rescaled convention to the displayed weight \(\exp (-\lambda V/\hbar )\) with covariance \(\hbar H\): a Gaussian contraction contributes \(\hbar \) and an interaction vertex contributes \(\hbar ^{-1}\). The logarithm selects connected graphs; after multiplication by \(-\hbar \), a connected graph with \(e\) contracted edges and \(v\) interaction vertices has power \(\hbar ^{e-v+1}\). Its number of independent cycles is \(e-v+1\); zero cycles means a tree. Thus connected contractions need not be trees. In the displayed second-order coefficient, only the sextic term is at tree order; its quartic, quadratic and constant companions have positive powers of \(\hbar \). The source's stationary-action convention is applied to the shifted fiber action, whose unperturbed minimum is zero, which explains the subtraction of \(x^2/2\).

Reversing an oriented edge changes the sign of its difference, but does not erase its endpoints. Fix a commutative unital ring \(k\) and two finite incidence presentations \((V,E,s,h)\) and \((V',E',s',h')\). Equip them with weights \(w:E\to k\), \(w':E'\to k\), scalars \(\mu ,\mu '\in k\), and retained subsets \(R\subseteq V\), \(R'\subseteq V'\). Empty carriers, loops and distinct parallel edge labels are allowed.

Write \(C_2=\{+,-\}\) for the abstract group with \(-\cdot -=+\), and \(\sigma :C_2\to k^\times \) for \(\sigma (+)=1\), \(\sigma (-)=-1\). A signed endpoint arrow consists of bijections \(p:V\to V'\), \(q:E\to E'\) and signs \(\varepsilon :E\to C_2\) such that \[\begin {aligned} \varepsilon (e)=+&:\quad s'q(e)=ps(e),\quad h'q(e)=ph(e),\\ \varepsilon (e)=-&:\quad s'q(e)=ph(e),\quad h'q(e)=ps(e),\\ w'q&=w,\qquad \mu '=\mu ,\qquad pR=R'. \end {aligned}\] The signs are abstract orientation data, even when \(1=-1\) in \(k\). An arrow with all signs \(+\) is directed. For the real Gaussian backgrounds, directed arrows are their exact relabelings, with \(\mu =m^2\); positivity of mass then recovers \(m'=m\).

Composition applies the right arrow first. The composite with \((p',q',\varepsilon ')\) is \[\bigl (p'p,q'q,\ e\mapsto \varepsilon '(q(e))\varepsilon (e)\bigr ).\] Identity arrows have identity bijections and all signs \(+\). The inverse of \((p,q,\varepsilon )\) has bijections \(p^{-1},q^{-1}\) and the sign \(\varepsilon (e)\) at \(q(e)\). These operations make the declared objects and arrows a groupoid: composition is associative, identities exist, and every arrow is invertible, as for an action groupoid.

Let \(M=k^V\), \(N=k^E\) and similarly \(M',N'\), with incidence maps \((df)(e)=f(h(e))-f(s(e))\). The induced ordinary linear maps are \[\begin {aligned} (Pf)(p(v))&=f(v),&P&:M\to M',\\ (Ja)(q(e))&=\sigma (\varepsilon (e))a(e),&J&:N\to N'. \end {aligned}\] They are isomorphisms: \(P^{-1}f'=f'\circ p\) and \((J^{-1}a')(e)=\sigma (\varepsilon (e))a'(q(e))\). Evaluation at \(q(e)\), in the two endpoint cases, gives \[d'P=Jd:M\longrightarrow N'.\] Both differences vanish on a loop; the endpoint condition still remembers where that loop is attached.

For a left action of a group \(\Gamma \), take actions \(p_g,q_g\) on the vertex and edge sets. A signed endpoint action additionally requires \[\varepsilon _1(e)=+,\qquad \varepsilon _{gk}(e)=\varepsilon _g(q_k(e))\varepsilon _k(e).\] Thus the arrow assigned to \(gk\) is the composite of those assigned to \(g\) and \(k\). The induced maps satisfy \(P_gP_k=P_{gk}\) and \(J_gJ_k=J_{gk}\). Indeed evaluation at \(p_{gk}(v)\) gives \(f(v)\) on both sides of the first equation; at \(q_{gk}(e)\) the second has coefficient \(\sigma (\varepsilon _g(q_k(e)))\sigma (\varepsilon _k(e))a(e)\). Identity signs and maps give the identity operators. These are linear representations on two distinct carriers; no faithfulness is assumed.

A weaker signed-incidence symmetry requires the same actions, abstract sign cocycle and weight preservation, but only the incidence equation \(dP_g=J_gd\), instead of the endpoint equations. It is an incidence intertwiner and suffices for operator transport. It need not be a graph automorphism; loops and characteristic two distinguish the requirements. The concrete half-turn satisfies the stronger endpoint conditions.

The left-action and equivariance conventions are [milne2025group, Definition 4.1, pp. 57-58]. [milne2025group, Example 7.1(b), p. 101] gives a permutation representation over a field; the ring and signed endpoint construction above is proved directly. The incidence convention in [spielman2009effective, Section 2.1, pp. 4-5] concerns a connected undirected positive-weight graph with a chosen orientation; the endpoint-difference definition here also allows loops, parallel labels and arbitrary coefficients.

Mathlib's Representation.ofMulAction sends the free basis vector at \(v\) to that at \(g\cdot v\). Finsupp.linearEquivFunOnFinite identifies its coefficients with functions on a finite set, yielding inverse precomposition for a group action. Representation.IntertwiningMap bundles an intertwiner between already supplied representations. Neither supplies the endpoint equations or abstract sign cocycle above.

The signed endpoint equations retain information that an incidence operator alone can lose. Take vertices \(\{0,1\}\) and one loop \(e\) attached at \(0\). Its incidence map is zero over every ring. Swapping the vertices, fixing \(e\) and choosing sign \(+\) therefore intertwines incidence and preserves any weight on \(e\). Nevertheless \(s(q(e))=0\ne 1=p(s(e))\); the same obstruction holds for sign \(-\), since both endpoints of the loop are \(0\). This is a signed-incidence symmetry, but not a signed endpoint arrow.

A separate loss concerns composition. Let \(C_3=\{1,g,g^2\}\) fix a single loop and its vertex. Assign abstract signs \(\varepsilon _1=+\), \(\varepsilon _g=\varepsilon _{g^2}=-\). Every endpoint and zero-incidence equation holds separately. But the action cocycle at \(g\cdot g=g^2\) would require \[\varepsilon _{g^2}=\varepsilon _g\varepsilon _g=+,\] contrary to the assigned \(-\). Thus these arrows do not define a \(C_3\)-action. Over a ring of characteristic two the scalar maps all become the identity because \(-1=1\); even their representation equations then conceal the failed abstract cocycle. Orientation data remain subject to the abstract \(C_2\) law.

By contrast, one edge \(e:0\to 1\) admits the vertex swap with fixed edge label and sign \(-\). Its endpoints reverse as required, and using this arrow twice gives sign \(+\) and identity bijections. For fields, \[(dPf)(e)=f(0)-f(1)=-(df)(e)=(Jdf)(e).\] It is a valid signed symmetry. Parallel labels are still distinct: an edge bijection may permute them, but never identifies two labels. Each sign must satisfy its own endpoint rule and the composition law.

These calculations use the endpoint-difference construction of Definition 10.1.

[spielman2009effective, Section 2.1, pp. 4-5] allows an arbitrary orientation of each edge. The calculations explain its extension in the signed endpoint construction.

[milne2025group, Definition 4.1, p. 57] gives the group-law requirement. Neither source is being asserted to contain these loop or characteristic-two examples.

The alternating four-cycle has a symmetry for every \(t>0\), including unequal weights. Let \(C_2=\{1,r\}\) with \(r^2=1\). Its left actions on the distinct vertex and edge sets \(V=\mathbb Z/4\mathbb Z\) and \(E=\{e_0,e_1,e_2,e_3\}\) are \[r\cdot i=i+2,\qquad r\cdot e_i=e_{i+2}.\] Indices are modulo \(4\). These permutations square to the identity, preserve the endpoints \(e_i:i\to i+1\), and preserve the weights \((1,t,1,t)\). Milne's [milne2025group, Definition 4.1, p. 57] gives the left-action convention; his [milne2025group, Example 7.1(b), p. 101] constructs the associated permutation representation.

The half-turn exchanges \(0,2\) and \(1,3\), and exchanges \(e_0,e_2\) and \(e_1,e_3\). Directed arrows and their weights agree after rotation. Solid vertices form the retained set \(\{0,2\}\); the dashed vertices form its complement. Each set is preserved.

The induced maps on real fields use inverse precomposition: \[\begin {aligned} P:\mathbb R^V&\to \mathbb R^V,&(P\phi )_i&=\phi _{i-2},\\ U:\mathbb R^E&\to \mathbb R^E,&(Ua)(e_i)&=a(e_{i-2}). \end {aligned}\] Both maps have the coordinate array \[\begin {pmatrix} 0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0 \end {pmatrix},\] but their domains differ. They satisfy \(P^2=I\) and \(U^2=I\). Sending \(1\) to the identity and \(r\) to these maps gives two real linear representations of \(C_2\).

Write \(\delta _i(j)=\mathbf 1_{j=i}\) for the vertex coordinate basis and \(\eta _i(e_j)=\mathbf 1_{j=i}\) for the edge coordinate basis. Then \(P\delta _0=\delta _2\ne \delta _0\) and \(U\eta _0=\eta _2\ne \eta _0\). Since \(r\) is the only nonidentity element of \(C_2\), both representations are faithful. Finite coordinate reindexing preserves the counting pairings; weight preservation gives \[\begin {aligned} P^TP&=I,\qquad U^TU=I,\\ U^TWU&=W, \end {aligned}\] where \(W=\operatorname {diag}(1,t,1,t)\). The last identity also preserves the weighted edge pairing.

The two representations interact through the incidence matrix. With its edge-by-vertex convention from [spielman2009effective, Section 2.1, p. 4], \[\begin {aligned} (BP\phi )(e_i)&=\phi _{i+1-2}-\phi _{i-2}\\ &=(UB\phi )(e_i). \end {aligned}\] Thus \(BP=UB\): the incidence map is equivariant between the vertex and edge actions.

The same formulas define linear representations over any nonzero commutative unital ring, with any period-two weights in that ring. The delta witnesses still prove faithfulness because \(1\ne 0\). Over the zero ring the two field modules have one element, so this faithfulness assertion is excluded. The positive real weights above are the specialization used for Gaussian laws.

This is the all-positive case of a signed endpoint action. The general endpoint composition, abstract orientation cocycle and weaker signed-incidence symmetry have that defining address. The loop and reversal examples explain why endpoint and sign coherence cannot be inferred from the incidence intertwiner alone.

Mathlib's Representation.ofMulAction sends a coordinate basis vector at \(v\) to the one at \(g\cdot v\). Its finitely supported coefficient module is identified with all functions on a finite set by Finsupp.linearEquivFunOnFinite, giving the inverse-precomposition convention above. Representation.IntertwiningMap bundles a linear map with the equation \(B P_g=U_gB\). These interfaces do not provide the particular incidence identity or the coherent signed-edge action without their proofs.

Let \(k\) be a commutative unital ring. Give \(M,N,M',N'\) chosen perfect symmetric pairings, and let \(d:M\to N\), \(d':M'\to N'\) be ordinary linear maps. Suppose the linear isomorphisms \(P:M\to M'\), \(J:N\to N'\) preserve these pairings and satisfy \(d'P=Jd\). Let \(W:N\to N\), \(W':N'\to N'\) be self-adjoint maps with \(W'J=JW\). For a fixed \(\mu \in k\), put \[L=d^\dagger Wd,\qquad Q=L+\mu I_M,\] and define \(L',Q'\) by the primed data. Then \[\begin {aligned} (d')^\dagger &=Pd^\dagger J^{-1}:N'\to M',\\ L'&=PLP^{-1},\qquad Q'=PQP^{-1}. \end {aligned}\] If \(Q\) is an isomorphism, so is \(Q'\), and its inverse response satisfies \[G'=(Q')^{-1}=PQ^{-1}P^{-1}=PGP^{-1}.\] Consequently \(Qf=j\) implies \(Q'(Pf)=Pj\).

For signed endpoint arrows, the finite function modules have perfect counting duality, including empty carriers. Reindexing gives \[\begin {aligned} b_{M'}(Pf,Pg)&=\sum _{v\in V}f(v)g(v),\\ b_{N'}(Ja,Jb)&=\sum _{e\in E}\sigma (\varepsilon (e))^2a(e)b(e) =b_N(a,b). \end {aligned}\] Multiplication by transported weights obeys \(W'J=JW\), by evaluation at \(q(e)\). Thus the theorem applies without inverting any weight. It also applies to the weaker signed-incidence symmetries. The weighted edge pairing is \(b_W(a,b)=b_N(a,Wb)\); its adjoint of \(d\) into the counting vertex pairing is \(d^\dagger W\), whereas \(d^\dagger \) uses the counting edge pairing. The expression \(d^\dagger Wd\) does not require \(b_W\) to be perfect.

The inverse hypothesis cannot be replaced by a unit mass scalar. Over \(\mathbb Z\), a unit-weight edge on two vertices with \(\mu =1\) has \(Q=\begin {pmatrix}2&-1\\-1&2\end {pmatrix}\). Solving \(Q(x,y)=(1,0)\) would require \(x=2y\) and \(3y=1\), which is impossible in \(\mathbb Z\). This \(Q\) is not even surjective.

For automorphisms the conjugacies are commutation identities. They concern symmetric bilinear pairings over fixed coefficients. A grading is preserved only if \(P\) preserves its graded pieces; a star is preserved only if \(P\) commutes with that star. A nontrivial coefficient involution and Hermitian adjoint need their own conventions.

The proof uses the chosen duality construction in Proposition 3.3, with the Hom convention of [stacks2017stacks, Section 10.10], Tag 0581. For real graph incidence, the energy convention is [spielman2009effective, Section 2.1, pp. 4-5]; the displayed ring transport is a direct deduction, not a positivity theorem. Mathlib's Matrix.reindexLinearEquiv and Matrix.reindexRingEquiv support its coordinate reindexing specialization, without supplying the endpoint or pairing hypotheses.

Over a commutative unital ring \(k\), give \(M=X\oplus Y\) and \(M'=X'\oplus Y'\) chosen splittings. Suppose isomorphisms \(P:M\to M'\), \(P_X:X\to X'\), \(P_Y:Y\to Y'\) preserve the inclusions and projections: \[\begin {aligned} P\iota _X&=\iota '_XP_X,&P\iota _Y&=\iota '_YP_Y,\\ \pi '_XP&=P_X\pi _X,&\pi '_YP&=P_Y\pi _Y. \end {aligned}\] Let \(Q'=PQP^{-1}\). Write its 4 split components and those of \(Q\) as \(A:X\to X\), \(B:Y\to X\), \(E:X\to Y\), \(C:Y\to Y\), and their primed versions. Then \[\begin {aligned} A'&=P_XAP_X^{-1},&B'&=P_XBP_Y^{-1},\\ E'&=P_YEP_X^{-1},&C'&=P_YCP_Y^{-1}. \end {aligned}\] If \(C\) is invertible, the split Schur maps obey \[S'=P_XSP_X^{-1},\qquad S=A-BC^{-1}E.\] For \(K=C^{-1}E:X\to Y\), \(K'=(C')^{-1}E'\), \[K'P_X=P_YK,\qquad -K'(P_Xx)=P_Y(-Kx).\] Thus the eliminated-equation field \(y_*(x)=-Kx\), the unique solution of the lower equation with zero source, is transported with its sign intact.

When \(Q\) is also invertible, the inverse-compression theorem says \[S^{-1}=\pi _XQ^{-1}\iota _X.\] Conjugating and using the split maps gives \[(S')^{-1}=P_XS^{-1}P_X^{-1} =\pi '_X(Q')^{-1}\iota '_X.\] This uses an actual inverse. No pairing has been assumed; only for an orthogonal split and self-adjoint \(Q\) does one further have \(E=B^\dagger \), by Proposition 4.5. For an orthogonal split of finite-dimensional real inner-product spaces and a positive self-adjoint \(Q\), this field is also stationary for the quadratic action in the eliminated variable, by the real square-completion argument.

For a vertex bijection \(p:V\to V'\), let \(R'=pR\), \(X=k^R\), \(X'=k^{R'}\), and let \(P_R\) be inverse precomposition by \(p|_R\). At \(p(v)\), \(v\in R\), both sides of \[\pi _{R'}P=P_R\pi _R\] take the value \(f(v)\). The target space is \(k^{pR}\), even if the labels of \(R\) and \(pR\) differ. This equation supplies the moving projection square in Gaussian marginal transport.

If a group action preserves \(R\), its restrictions compose on \(k^R\) and the square becomes \(\pi _RP_g=P_{g,R}\pi _R\). Faithfulness of the full action need not survive restriction: a permutation moving only eliminated vertices acts trivially on retained fields. The opposite-pair half-turn has a separate nontrivial retained action.

Stable subsets and equivariance follow the convention of [milne2025group, p. 58]. The real Schur convention is [boyd2004convex, Appendix A.5.5, pp. 650-651], with the source's first pivot exchanged to the eliminated \(C\) block. The typed transport proof above uses only the chosen splitting, and does not by itself supply any integral or Gaussian normalizer.

Let \(k\) be a commutative unital ring. For a \(k\)-module \(M\), write \(\mathcal P_k(M)\) for the unital \(k\)-subalgebra of functions \(M\to k\) generated by ordinary linear functionals. This extends the intrinsic real polynomial functions using the same pointwise operations. These are actual functions: over finite fields, distinct formal polynomials can have identical evaluations.

For a linear isomorphism \(P:M\to M'\), pullback gives a unital algebra isomorphism \[P^*:\mathcal P_k(M')\longrightarrow \mathcal P_k(M),\qquad P^*F'=F'\circ P.\] An already supplied normalized functional \(\omega \) on \(\mathcal P_k(M)\) transports to \(\omega '=\omega \circ P^*\). For the moving split square, the retained functionals \(\omega _X=\omega \circ \pi ^*\) and \(\omega '_{X'}=\omega '\circ (\pi ')^*\) satisfy \[\omega '_{X'}=\omega _X\circ P_X^*.\] Here \(\pi :M\to X\), \(\pi ':M'\to X'\) are the specified projections, and \(\pi 'P=P_X\pi \).

Over \(\mathbb R\) with the identity involution, or over \(\mathbb C\) with pointwise conjugation, suppose the chosen observable algebras are closed under that involution. Pullback then commutes with star. If \(\omega \) is an algebraic state, \[\begin {aligned} \omega '(F'^*F') &=\omega \bigl ((P^*F')^*(P^*F')\bigr )\geq 0, \end {aligned}\] and the star compatibility of the functional is preserved too. Holomorphic polynomial functions on a complex vector space are not automatically closed under conjugation; closure is an extra hypothesis. One can instead generate an algebra by both linear functionals and their conjugates; precomposition preserves those generators as well. No positivity over an arbitrary ring or existence of a measure is asserted. For the real Gaussian, absolute moments first give its polynomial state; the transported law then identifies this algebraic transport with expectation. Integrable functions as a whole are not an algebra, and restriction here is neither conditional expectation nor partial trace.

The algebra and state conventions are [stacks2017stacks, Section 22.2], Tag 09JF and [waldmann2004states, Definitions 3.7 and 3.9, p. 11]. The composition and split-square calculations are direct extensions of Proposition 5.10 and Proposition 5.7; their linear-dual and star-algebra interfaces do not supply a Gaussian measure.

To compare two discrete Gaussian models, specify which data a change of labels must carry. A finite Gaussian background consists of a finite incidence presentation \((V,E,s,h)\), positive weights \(w:E\to \mathbb R_{>0}\), a mass \(m>0\), and a nonempty proper retained subset \(R\subsetneq V\). Put \(U=V\setminus R\); both retained and eliminated carriers are nonempty. The specified reference structures are the counting inner products on vertex and edge fields, the weighted edge inner product, and the induced Borel Euclidean volume \(\lambda _V\) on \(\mathbb R^V\). The orthogonal coordinate split identifies it with the product of the retained and eliminated volumes.

Let \(d:\mathbb R^V\to \mathbb R^E\) be incidence, \((df)(e)=f(h(e))-f(s(e))\), and let \(W\) multiply edge fields by \(w\). The derived operators are \[L=d^\dagger Wd,\qquad Q=L+m^2I>0,\] where \(d^\dagger \) is the counting adjoint. In counting bases, \(d\) has array \(D\) and \(L=D^TWD\). The action \(\mathcal A_Q(f)=b_V(f,Qf)/2\), finite positive normalizer \(Z_Q\), Gaussian law \(\mu _Q\), and retained marginal \(\mu _S\) are derived data. For the 4 split components, \(S=A-BC^{-1}E\); the real orthogonal specialization has \(E=B^\dagger \); the block \(B:Y\to X\) is distinct from incidence.

An exact relabeling to a primed background consists of bijections \(p:V\to V'\), \(q:E\to E'\) satisfying \[\begin {aligned} s'q&=ps,&h'q&=ph,\\ w'q&=w,&m'&=m,\qquad pR=R'. \end {aligned}\] It is the all-positive specialization of a signed endpoint arrow. It preserves the attachment of a loop even when its incidence is zero. The induced maps are \[(Pf)(p(v))=f(v),\qquad (Ja)(q(e))=a(e).\] They preserve both counting pairings, the weighted edge pairing, and by isometry of reference volume the reference volume. Allowing coherent endpoint reversals gives the larger signed relation; it does not change what an exact directed relabeling means.

Evaluation at \(q(e)\) gives \((d'Pf)(q(e))=f(h(e))-f(s(e))=(Jdf)(q(e))\). Weight preservation gives \(W'J=JW\). Thus paired transport yields \[L'=PLP^{-1},\qquad Q'=PQP^{-1}.\] Counting isometry gives \(P^{-1}=P^T\), \(J^{-1}=J^T\) in the coordinate bases, recovering \(D'=JDP^T\) and \(W'=JWJ^T\). The normalized-law theorem carries the laws and moving marginals. Identity, inverse and composition are the directed cases of the proved signed-arrow laws, so existence of an exact relabeling is an equivalence relation on backgrounds.

Two broader comparisons retain less data. The positive-scaling Laplacian relation forgets mass, state and partition, and compares nonzero Laplacians by \(L'=cPLP^{-1}\) with \(c>0\) and a vertex permutation. Its cycle trace ratio still separates two examples. An arbitrary invertible real field map \(y=T\phi \) instead changes precision to \(T^{-T}QT^{-1}\) and the counting pairing to \(y^TT^{-T}T^{-1}z\). Its reference Jacobian is \(dy=|\det T|\,d\phi \). Such a map can transport the normalized law by Jacobian compensation, while losing the reference pairing or retained projection. It is not thereby an enriched background arrow.

The covariance-parameter Gaussian density in [friedli2017statistical, Proposition B.55, p. 505] agrees with these conventions after the inverse-precision theorem identifies \(\Sigma =Q^{-1}\). Its change-of-variables argument does not specify the endpoint, mass or retained-subset data. Mathlib's Matrix.reindexLinearEquiv and Matrix.reindexRingEquiv support the coordinate reindexing identities. The extra background requirements are those stated here.

Let \(M,M'\) be finite-dimensional real positive inner-product spaces with their induced Borel Euclidean volumes \(\lambda ,\lambda '\). Let \(Q:M\to M\) be positive and self-adjoint, and let \(P:M\to M'\) be a linear isometry equivalence. Put \(Q'=PQP^{-1}\). Then \(Q'\) is positive and self-adjoint, and the Gaussian laws satisfy \[Z_{Q'}=Z_Q,\qquad P_*\mu _Q=\mu _{Q'}.\] Borel pushforward means \((P_*\mu )(A)=\mu (P^{-1}(A))\) for every Borel set \(A\), extending the finite convention. For every real or complex Borel function \(F'\), \[\begin {aligned} \int |F'|\,d\mu _{Q'}&=\int |F'\circ P|\,d\mu _Q,\\ F'\in L^1(\mu _{Q'})&\Longleftrightarrow F'\circ P\in L^1(\mu _Q). \end {aligned}\] The first identity allows infinity. Under the equivalent finite absolute-integral conditions, \[\omega _{Q'}(F')=\omega _Q(F'\circ P).\] In particular, the target observable transported from \(F\) is \(F'=F\circ P^{-1}\). Exact and signed arrows of real backgrounds supply such isometries.

Now let \(M=X\oplus Y\) and \(M'=X'\oplus Y'\) be orthogonal splittings that \(P\) carries by isometries \(P_X,P_Y\), so the split-square hypotheses hold. For the 4 components, \[\begin {aligned} A'&=P_XAP_X^{-1},&B'&=P_XBP_Y^{-1},\\ E'&=P_YEP_X^{-1},&C'&=P_YCP_Y^{-1}. \end {aligned}\] Here \(E=B^\dagger \), and \(C>0\). The retained precisions and eliminated Gaussian factors satisfy \[\begin {aligned} S'&=P_XSP_X^{-1},& S&=A-BC^{-1}E>0,\\ c_{C'}&=c_C,&c_C&=\frac {(2\pi )^{\dim Y/2}}{\sqrt {\det C}},\\ Z_{S'}&=Z_S,&(P_X)_*\mu _S&=\mu _{S'}. \end {aligned}\] Thus the complete fiber scalar and the additive effective-action constant \(-\log c_C\) are preserved, along with the normalized law.

For vertex bijections \(p:V\to V'\), take a nonempty proper \(R\subsetneq V\), set \(R'=pR\), and use the coordinate splits. The map \(P_R:\mathbb R^R\to \mathbb R^{R'}\) has the genuine moving target \(\mathbb R^{pR}\). At \(p(v)\), \(v\in R\), both \(\pi _{R'}Pf\) and \(P_R\pi _Rf\) have value \(f(v)\). Thus:

The lower map exists because the retained subset is transported. Keeping the target labels fixed can select an inequivalent marginal; the adjacent-versus-opposite example gives both the precision calculation and a failure of projection factorization.

These conclusions use positive precisions and the stated product volumes. Fixing an eliminated field, taking a conditional-density kernel, and taking a marginal are different operations. A singular full-fiber integral is not repaired by a pseudoinverse: changing the integration space changes the measure. The massless projected cycle law has its proved density on the specified retained image, and is not obtained by inserting a mean-zero constraint into the unconstrained marginal formula.

[friedli2017statistical, Proposition B.55, p. 505] gives the Gaussian change-of-variables convention.

[tao2011measure, Theorem 1.7.15, pp. 200-202] gives the nonnegative integration convention on the Borel product measures; absolute integrability precedes signed subtraction.

[boyd2004convex, Appendix A.5.5, pp. 650-651] gives the real block convention, with its first pivot exchanged to \(C\).

Mathlib's LinearIsometryEquiv.measurePreserving supplies volume preservation for finite-dimensional real linear isometry equivalences. TauCeti's TauCeti.Probability.map_affine_multivariateGaussian supplies covariance-parameter transport. Set mean and shift to zero, linear map to \(P\), and covariance to \(Q^{-1}\); the precision-density identification and the proved inverse identity align it with this law. These interfaces do not supply the endpoint data, declared partition or its counterexample.

Let a group \(\Gamma \) act by a signed-incidence symmetry on a finite incidence presentation with positive real weights and nonempty vertex set. Use the counting pairings, their induced Borel volume and a scalar mass \(m>0\). The positive massive precision and inverse-precision covariance are \[L=d^\dagger Wd,\qquad Q=L+m^2I,\qquad G=Q^{-1}.\] For every \(g\in \Gamma \), \[\begin {aligned} LP_g&=P_gL,&QP_g&=P_gQ,\\ GP_g&=P_gG,&P_gGP_g^\dagger &=G. \end {aligned}\] Thus \(Qf=j\) implies \(Q(P_gf)=P_gj\). The Gaussian law is invariant: \((P_g)_*\mu _Q=\mu _Q\). For every real or complex Borel observable \(F\), \[\begin {aligned} \int |F\circ P_g|\,d\mu _Q&=\int |F|\,d\mu _Q,\\ \omega _Q(F\circ P_g)&=\omega _Q(F) \end {aligned}\] with extended integrals in the first equality, and finite absolute integral required in the second.

Suppose further that a nonempty proper retained subset \(R\subsetneq V\) is stable under every \(p_g\). Its complement \(D=V\setminus R\) is also stable. The restricted maps \(P_{g,R},P_{g,D}\) give the orthogonal split correspondence. For the split components \(A,B,E,C\), with \(E=B^\dagger \), and \(S=A-BC^{-1}E>0\), one has \[\begin {aligned} P_{g,R}SP_{g,R}^{-1}&=S,\\ P_{g,R}S^{-1}P_{g,R}^{-1}&=S^{-1},\\ (P_{g,R})_*\mu _S&=\mu _S. \end {aligned}\] The retained equation \(Sx=k\) is equivariant as well. The full fiber factor \(c_C=(2\pi )^{|D|/2}/\sqrt {\det C}\) and the stationary eliminated field are compatible with the action: \[y_*(P_{g,R}x)=P_{g,D}y_*(x),\qquad y_*(x)=-C^{-1}Ex.\]

For the half-turn, retain \(R=(0,2)\) and eliminate \(D=(1,3)\). Both restrictions are the nonidentity swap \(J=\begin {pmatrix}0&1\\1&0\end {pmatrix}\). Here \(H\) is the array of the split component \(B\). The projection square is

The lower map swaps the two retained coordinates. Put \(a=m^2\) and \(s=1+t+a\). The explicit elimination gives \[\begin {aligned} A=C&=sI_2,\qquad H=-\begin {pmatrix}1&t\\t&1\end {pmatrix},\\ S&=\begin {pmatrix}\alpha &-\beta \\-\beta &\alpha \end {pmatrix},\\ \alpha &=s-\frac {1+t^2}{s},\qquad \beta =\frac {2t}{s}. \end {aligned}\] All three blocks commute with \(J\), hence \(JSJ=S\). Positivity follows also from the two eigenvalues \[\begin {aligned} \alpha -\beta &=\frac {a(a+2+2t)}s>0,\\ \alpha +\beta &=\frac {(a+2)(a+2t)}s>0. \end {aligned}\] Thus the same nontrivial action preserves the full and retained laws. The full covariance obeys \(G_{i+2,j+2}=G_{ij}\). Invariant state does not mean that each observable is pointwise invariant: \(\phi \mapsto \phi _0^2\) is sent to \(\phi \mapsto \phi _2^2\); they differ at \(\delta _0\), but their expectations agree.

At \(t=m=1\), direct inversion gives \[\begin {aligned} S&=\begin {pmatrix}7/3&-2/3\\-2/3&7/3\end {pmatrix},\\ S^{-1}&=\begin {pmatrix}7/15&2/15\\2/15&7/15\end {pmatrix}. \end {aligned}\] Both swapped squared coordinates have expectation \(7/15\). The observable \(\phi _0\phi _2\) is itself invariant under the half-turn and has expectation \(2/15\). These expectations agree before and after elimination.

The full and retained half-turns are both faithful: their nonidentity maps send \(\delta _0\) to \(\delta _2\). A general full faithful action can restrict trivially, for instance when it moves only eliminated vertices. These two assertions are compatible.

Three concrete obstructions distinguish the hypotheses: keeping fixed target labels after relabeling, breaking the alternating weights, and choosing a nonstable retained subset. The last is a failure of descent through that projection and does not exclude accidental symmetry of a marginal density. Extra mass fields or reference pairings would require their own preservation hypotheses.

[milne2025group, Definition 4.1 and p. 58] gives the group-action and stable-subset conventions.

[friedli2017statistical, Proposition B.55, p. 505] gives the Gaussian substitution convention. The incidence, stationary and half-turn computations above use the proved transport results and the exact cycle elimination.

Mathlib's LinearIsometryEquiv.measurePreserving supplies isometry-volume preservation. TauCeti's TauCeti.Probability.map_affine_multivariateGaussian supports covariance-parameter pushforward for the supplied \(P_g\) or retained projection after the precision-density bridge. These declarations do not assert the endpoint, weight or stable-subset hypotheses, or the concrete failures just cited.

A global relabeling need not preserve a marginal requested on fixed labels. In the four-cycle, take \(t=m=1\) and retain \(R=\{0,2\}\). Its effective precision is \[S_{\mathrm {opp}}= \begin {pmatrix}7/3&-2/3\\-2/3&7/3\end {pmatrix}.\] Exchange labels \(1,2\), fixing \(0,3\), and form \(Q'=PQP^T\). If the new retained labels are still \(\{0,2\}\), they select the original adjacent variables \(\{0,1\}\). In the old coordinate order \((0,1)\mid (2,3)\), the relevant blocks are \[\begin {aligned} A=C&=\begin {pmatrix}3&-1\\-1&3\end {pmatrix},\\ H&=\begin {pmatrix}0&-1\\-1&0\end {pmatrix},\\ C^{-1}=HC^{-1}H^T &=\frac 18\begin {pmatrix}3&1\\1&3\end {pmatrix}. \end {aligned}\] Subtraction gives \[S_{\mathrm {adj}}= \begin {pmatrix}21/8&-9/8\\-9/8&21/8\end {pmatrix}.\] The diagonal entries \(7/3\) and \(21/8\) differ, so no two-coordinate permutation conjugates these precisions. There is also no map on the old retained fields factoring \(\pi _RP\) through \(\pi _R\): the fields \(0\) and \(\delta _1\) have the same old projection, but their projections after \(P\) are \(0\) and \((0,1)\). Here \(\delta _1\) is the delta field at vertex \(1\), not the edge label \(e_1\). Transporting the retained subset to \(p(R)=\{0,1\}\) in the new labels restores the square and the marginal transport theorem.

The blocks are a direct calculation from the alternating-cycle precision, using the Schur convention of [boyd2004convex, Appendix A.5.5, pp. 650-651] after exchanging its first pivot to \(C\). The example separates a fixed set of labels from the moving retained space required by Proposition 10.5.

In the alternating four-cycle, take \(t,m>0\) and the half-turn. It fails to preserve the action after a specific positive change of weights. Replace \((1,t,1,t)\) by \((1,t,2,t)\) while keeping the proposed half-turn. Then \(L_{00}=1+t\) and \(L_{22}=2+t\), so \(PLP^T\ne L\) and \(\mathcal A(\delta _2)-\mathcal A(\delta _0)=1/2\). If its Gaussian law were invariant, second moments would give \(PGP^T=G\); inversion would give \(PQP^T=Q\), contradicting those diagonal entries. At \(t=m=1\), retain the opposite pair \(R=(0,2)\) and eliminate \(D=(1,3)\). The blocks in that order are \[\begin {aligned} A=C&=\operatorname {diag}(3,4),\\ H&=-\begin {pmatrix}1&1\\1&2\end {pmatrix},\\ S_{\mathrm {broken}} &=\begin {pmatrix}29/12&-5/6\\-5/6&8/3\end {pmatrix}. \end {aligned}\] Indeed \[HC^{-1}H^T=\begin {pmatrix}7/12&5/6\\5/6&4/3\end {pmatrix},\] and subtracting from \(A\) gives the displayed precision. Its diagonal entries differ by \(1/4\), so its marginal also fails swap invariance. Its positive Gaussian still exists.

[spielman2009effective, Section 2.1, pp. 4-5] gives the edge-once convention.

[boyd2004convex, Appendix A.5.5, pp. 650-651] gives the positive Gaussian elimination convention. The action difference and matrix products are direct calculations with these conventions. Weight preservation is sufficient in the general transport theorem; it need not be necessary for every presentation with redundant loops or parallel edges. This example establishes failure for the stated weights, not a universal necessity.

In the alternating four-cycle with weights \((1,t,1,t)\) and \(t,m>0\), consider the half-turn and the adjacent retained subset \(R=\{0,1\}\). The half-turn preserves the full state, but sends \(R\) to \(D=\{2,3\}\). No map \(T\) of retained fields can satisfy \(\pi _RP=T\pi _R\): the zero field forces \(T(0)=0\), while \(\delta _2\) has retained projection zero and projects to \((1,0)\) after the half-turn. The action transports the marginal to a different subset; it does not descend through this fixed projection. This does not exclude accidental symmetries of an individual two-site density. A general restricted action may also be trivial, unlike the opposite-pair swap. Extra mass fields or reference pairings would require their own preservation hypotheses.

The stable-subset requirement is the equivariance convention of [milne2025group, p. 58]. The two-field witness proves failure of factorization even through a nonlinear function. It differs from the fixed-label relabeling failure: here the full half-turn symmetry is unchanged, and the chosen projection fails to admit its action. The moving square of Proposition 10.5 remains valid on \(\mathbb R^{pR}\). Over any nonzero commutative unital coefficient ring \(k\), the same delta-field witness works on \(k^V\); over the zero ring all fields coincide.

A trace can be defined before choosing a pairing or displaying a matrix. Let \(k\) be a commutative unital ring and \(M\) a finite free \(k\)-module. A finite dual frame is a finite family \(m_i\in M\), \(\ell _i\in M^\vee \) such that \[x=\sum _i\ell _i(x)m_i\qquad (x\in M).\] A basis and its coefficient functionals supply such a frame; the counting carrier \(k^V\) uses its delta dual frame. For an endomorphism \(T:M\to M\), define its intrinsic trace by \[\operatorname {tr}_M(T)=\sum _i\ell _i(Tm_i).\] This scalar is independent of the frame. It is linear in \(T\), and for maps between finite free modules \(f:M\to N\), \(g:N\to M\), \[\operatorname {tr}_M(gf)=\operatorname {tr}_N(fg).\] In particular, for an isomorphism \(P:M\to M'\), \[\operatorname {tr}_{M'}(PTP^{-1})=\operatorname {tr}_M(T).\]

With a basis, \(\ell _i(Tm_i)\) is the \(i\)-th diagonal coefficient, so coordinate trace is recovered as \(\sum _iT_{ii}\). For the zero module the empty frame gives trace zero; the same formulas also make sense over the zero ring. The declared category here is finite free modules. No totalized trace on arbitrary modules and no finite-projective extension are needed.

The duality/evaluation convention is [stacks2017stacks, More on Algebra, Lemma 74.1], Tag 0FNJ. [stacks2017stacks, Exercises, Exercise 22.6], Tag 02DU asks for trace, naturality and cyclicity on finite locally free modules; the finite-free proof above is supplied here, rather than attributed to a proof in that exercise.

Mathlib's LinearMap.trace has a totalized definition. On the finite-free commutative-ring domain, its LinearMap.trace_comp_comm' is rectangular cyclicity and LinearMap.trace_conj' is conjugacy by a linear equivalence. The definition above uses only this finite-free domain. Matrix.trace_mul_comm expresses the coordinate rectangular identity; a trace pairing is not an assumption of either construction.

Let \(L:M\to M\) be an endomorphism of a finite free module over a commutative unital ring \(k\). On the domain \(\operatorname {tr}_M(L)\in k^\times \), define the trace ratio \[R(L)=\operatorname {tr}_M(L^2)\bigl (\operatorname {tr}_M(L)\bigr )^{-2},\] using intrinsic trace. If \(P:M\to M'\) is an isomorphism, \(c\in k^\times \), and \(L'=cPLP^{-1}\), then \(\operatorname {tr}_{M'}(L')\) is a unit and \(R(L')=R(L)\).

Consider a finite-dimensional positive inner-product space over \(\mathbb R\). A nonzero self-adjoint positive-semidefinite operator on it has positive trace. Indeed an orthonormal eigenbasis gives eigenvalues \(\lambda _i\geq 0\); if all were zero the operator would be zero. Hence \(\operatorname {tr}L=\sum _i\lambda _i>0\). This verifies the denominator for a nonzero positive-weight graph Laplacian.

[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the spectral and trace formulas. The exceptional examples show why nonzero alone is insufficient over other rings or without positivity.

For nonzero real positive-semidefinite Laplacians with their counting pairings, write \(L\approx L'\) when \(L'=cPLP^{-1}\) for a positive scalar and a vertex permutation. Identity uses \((1,I)\); inverse uses \((c^{-1},P^{-1})\); composition uses the products of the scalars and permutations. This is an equivalence relation with invariant \(R\). It forgets the mass, normalized state and retained partition of an enriched background. It also differs from arbitrary congruence of bilinear-form arrays: the ratio compares the associated endomorphisms. It is an invariant, not a classification assertion.

The ring calculation is a direct consequence of Definition 10.13, whose trace convention is [stacks2017stacks, Exercise 22.6], Tag 02DU. Mathlib's Matrix.trace_smul supplies the coordinate scalar identity; the unit domain and cancellation above are additional hypotheses and deductions. The cycle comparison computes the invariant and the associated normalized observables explicitly.

The unit condition in the trace ratio distinguishes several elementary cases. For one unit-weight edge on two vertices, the Laplacian over \(\mathbb Z\) is \[L=\begin {pmatrix}1&-1\\-1&1\end {pmatrix}.\] Its trace is \(2\), which is not a unit in \(\mathbb Z\); the defining formula therefore does not give a \(\mathbb Z\)-valued ratio. After inverting \(2\) it does: \(L^2=2L\) gives ratio \(1\). Reduction over \(\mathbb F_2\) instead gives the nonzero matrix \[\overline L=\begin {pmatrix}1&1\\1&1\end {pmatrix},\] whose trace is zero and whose square is zero. Nonzero incidence energy as an algebraic operator does not imply a unit trace.

Over \(\mathbb R\), the nonzero self-adjoint matrix \(\operatorname {diag}(1,-1)\) has trace zero. It is not positive semidefinite, so it does not contradict the positive-trace proof. For an incidence presentation containing only loops, \(d=0\) and \(L=d^\dagger Wd=0\); the trace is zero whatever the loop weights. The empty carrier likewise has zero trace by the empty dual-frame sum. In each case the ratio is undefined over a nonzero ring.

In the zero ring, \(0=1\) is its sole unit, and every unital module and scalar has one element. The formula for the ratio then has the unique value \(0\). This algebraic convention makes no assertion about a nonzero positive-real Laplacian.

[spielman2009effective, Section 2.1, pp. 4-5] gives the edge-once incidence convention used here, as extended algebraically in Definition 10.1. These computations also use the directly proved dual-frame trace.

[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the spectral criterion for the real positivity boundary. The ring reductions and zero-ring conclusion are the explicit calculations here, not probability statements from those sources.

The alternating cycles at \(t=1\) and \(t=2\) remain different even under the positive-scaling Laplacian comparison \(L'=cPLP^{-1}\), with \(c>0\) and \(P\) a vertex permutation. Use the trace ratio \[R(L)=\frac {\operatorname {tr}(L^2)}{(\operatorname {tr}L)^2},\] where the trace is the intrinsic endomorphism trace. The positive graph energy gives a nonzero positive-semidefinite \(L_t\) for \(t>0\), so its trace is positive and the ratio is defined. The cited proposition proves invariance by trace conjugacy and scalar cancellation. It compares counting-relative operators, not bilinear-form arrays under arbitrary congruence.

[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the symmetric spectral and trace formulas. The spectral cycle calculation supplies the concrete basis; the following traces and comparisons are direct computations.

For the alternating four-cycle, \(t>0\), the orthonormal spectral basis gives eigenvalues \(0,2,2t,2+2t\), including at \(t=1\). Consequently \[\begin {aligned} \operatorname {tr}L_t&=4(1+t),\\ \operatorname {tr}(L_t^2) &=4+4t^2+4(1+t)^2\\ &=8(t^2+t+1),\\ R(t):=R(L_t)&=\frac {t^2+t+1}{2(t+1)^2}. \end {aligned}\] In particular \[\begin {aligned} R(1)&=\frac 38,\qquad R(2)=\frac 7{18},\\ R(2)-R(1)&=\frac 1{72}>0. \end {aligned}\] There can be no \(c>0\) and vertex permutation \(P\) with \(L_2=cPL_1P^T\). Thus the associated finite Gaussian backgrounds cannot be exactly relabeled into each other either. This calculation separates the stated pair; it does not claim a classification of all backgrounds.

The reciprocal identity \(R(t)=R(1/t)\) is consistent with this comparison. A one-step rotation \(p(i)=i+1\pmod 4\) exchanges the alternating edge weights \(1,t\) and leaves the common diagonal \(1+t\) unchanged. Its coordinate permutation therefore satisfies \[PL_tP^T=tL_{1/t}.\] Since \(1/t>0\), the reciprocal Laplacians are equivalent under \(\approx \). Their equal ratios do not demonstrate a failure of completeness. On the strict enriched slice with weights \((1,t,1,t)\), fixed mass and fixed opposite retained pair, the weight product is \(t^2\), whereas at parameter \(1/t\) it is \(t^{-2}\). A weight-preserving edge bijection would preserve this product; for positive \(t\) equality forces \(t=1\). Thus the broader reciprocal comparison is not a strict background isomorphism for \(t\ne 1\).

A fixed-mass calculation shows what changes in the induced state. Set \(m=1\) and retain the same labeled vertices \(R=\{0,2\}\). The alternate-vertex elimination has \(s=t+2\), giving \[\begin {aligned} \alpha (t)&=s-\frac {1+t^2}{s}=\frac {4t+3}{t+2},\\ \beta (t)&=\frac {2t}{t+2},\\ S_t&=\begin {pmatrix}\alpha (t)&-\beta (t)\\ -\beta (t)&\alpha (t)\end {pmatrix}. \end {aligned}\] Hence the effective coupling changes from \(\beta (1)=2/3\) to \(\beta (2)=1\), and the two precisions are \[\begin {aligned} S_1&=\begin {pmatrix}7/3&-2/3\\-2/3&7/3\end {pmatrix},\\ S_2&=\begin {pmatrix}11/4&-1\\-1&11/4\end {pmatrix}. \end {aligned}\] Their determinants are \(5\) and \(105/16\). The off-diagonal entry of \(S_t^{-1}\) is \(\beta (t)/\det S_t\). By the marginal expectation identity, the same labeled observable therefore has different normalized expectations: \[\begin {aligned} \langle \phi _0\phi _2\rangle _{t=1,m=1}&=\frac {2}{15},\\ \langle \phi _0\phi _2\rangle _{t=2,m=1}&=\frac {16}{105}. \end {aligned}\] This comparison fixes the mass, coordinates, reference pairing and partition. The labeled observable is not itself asserted invariant under every relabeling; the trace ratio already established the broader inequivalence.

Scale invariance of \(R\) does not mean equality of fixed-mass states. On an eigenvector of \(L\) with eigenvalue \(\lambda >0\), replacing \(L\) by \(cL\) changes the covariance eigenvalue from \(1/(\lambda +m^2)\) to \(1/(c\lambda +m^2)\) when \(c\ne 1\). Scaling all of \(Q\) instead gives covariance \(c^{-1}G\) and Schur map \(cS\), by inversion and the Schur formula. Changing mass can also leave \(R(L)\) unchanged while changing the law. The exact finite differences above do not construct a physical map from an environment to the parameter \(t\).

The trace identities and their formalization correspondence are at Definition 10.13 and Proposition 10.14. Their unit-domain hypotheses specialize here to positive real trace; the values and normalized expectation comparisons above are additional cycle calculations.

Let \(M,M'\) be finite-dimensional real positive inner-product spaces with induced Borel Euclidean volumes \(\lambda ,\lambda '\). Let \(Q,Q'\) be positive self-adjoint precisions, and suppose a linear isomorphism \(T:M\to M'\) preserves their actions: \[\mathcal A_{Q'}(Tx)=\mathcal A_Q(x).\] It need not preserve the reference pairings. For its absolute Jacobian \(J(T)>0\), one has \[\begin {aligned} T_*\lambda &=J(T)^{-1}\lambda ',\\ Z_{Q'}&=J(T)Z_Q,\qquad T_*\mu _Q=\mu _{Q'}. \end {aligned}\] For every real or complex Borel observable \(F'\), \[\int |F'|\,d\mu _{Q'}=\int |F'\circ T|\,d\mu _Q\] as extended nonnegative integrals. Hence integrability is equivalent on the two sides, and, under that condition, \(\omega _{Q'}(F')=\omega _Q(F'\circ T)\).

In orthonormal coordinates, action preservation means \(T^TQ'T=Q\), so inversion gives \[(Q')^{-1}=TQ^{-1}T^T.\] This is covariance congruence, not generally operator similarity. The scalar Jacobian is computed in the specified reference volumes; independently ordered orthonormal frames may change the signed determinant but not its absolute value. The quadratic-isometry example shows that this normalized transport can lose both counting-pairing and retained-projection compatibility. Such maps are not enriched background arrows.

The volume convention is [tao2011measure, Exercises 1.2.20-22, p. 41]; the covariance-parameter Gaussian substitution is [friedli2017statistical, Proposition B.55, p. 505]. The inverse-precision covariance theorem identifies its covariance with \(Q^{-1}\) for these reference pairings. TauCeti's TauCeti.Probability.map_affine_multivariateGaussian supports the covariance congruence for a supplied finite real linear map, even rectangular. That broader covariance result does not assert invertibility, endpoint preservation or the retained square used here.

Let \(k\) be a commutative unital ring, \(G\) a group and \(M\) a unital \(k\)-module. A linear action \(\rho :G\to \operatorname {Aut}_k(M)\) extends uniquely to a unital \(k\)-algebra homomorphism \[\widehat \rho :k[G]\longrightarrow \operatorname {End}_k(M),\qquad \widehat \rho \left (\sum _g a_g[g]\right )=\sum _g a_g\rho (g).\] Products of endomorphisms compose right to left. This extends the Group Algebra universal property to the actual module action; no finiteness of \(G\) is assumed.

Consequently linear \(G\)-actions correspond to unital left \(k[G]\)-module structures whose restriction along \(c\mapsto c[1_G]\) is the given scalar action. Under this correspondence, a \(k\)-linear map \(f:M\to N\) is a module map exactly when \(f\rho _M(g)=\rho _N(g)f\) for every \(g\).

[sengupta2010representations, Sections 3.1-3.2, pp. 39-41] presents the field case, with finite support for an arbitrary group. The proof above is the ring adaptation of representation/module correspondence. Mathlib's Representation.asAlgebraHom and Representation.asAlgebraHom_single supply the linear extension and its value on basis monomials. They do not assert that an arbitrary faithful group action has a faithful algebra extension. For example, \(C_2\to \mathbb R^\times \), \(r\mapsto -1\), is injective, but its algebra extension annihilates the nonzero element \(1+[r]\).

Let \(k\) be a commutative unital ring in which \(2\) is a unit, put \(h=2^{-1}\), and let \(T:M\to M\) be linear with \(T^2=I\). The maps \(E_\varepsilon =h(I+\varepsilon T)\), for \(\varepsilon \in \{1,-1\}\), give \[\begin {aligned} E_\varepsilon ^2&=E_\varepsilon ,&E_+E_-&=E_-E_+=0,\\ E_++E_-&=I,&E_+-E_-&=T,\\ \operatorname {im}E_\varepsilon &=\ker (T-\varepsilon I)=M_\varepsilon . \end {aligned}\] Thus \(M\cong M_+\oplus M_-\), with map \(v\mapsto (E_+v,E_-v)\) and inverse addition. No finite generation, freeness or pairing is needed.

If an algebra acts by \(\alpha :A\to \operatorname {End}_k(M)\), these projectors are \(A\)-linear precisely when \[T\alpha (a)=\alpha (a)T\qquad (a\in A).\] Indeed this is exactly \(A\)-linearity of \(T\); its linear combinations with \(I\) are then \(A\)-linear. Conversely, either projector recovers \(T=2E_+-I=I-2E_-\). When \(T=\alpha (\omega )\), centrality of \(\omega \) is sufficient. It need not be necessary: on \(M=\mathbb R\), let \(A=\mathbb R\times M_2(\mathbb R)\) act by \(\alpha (a,B)=aI\), and put \(\omega =(1,\operatorname {diag}(1,-1))\). Then \(\omega ^2=1\) and \(T=I\), although \(\omega \) fails to commute with a matrix unit in the second factor.

For comparison, the central-idempotent product is an algebra statement. Central orthogonal idempotents \(e_+,e_-\) with sum one give the mutually inverse unital algebra maps \[a\longmapsto (e_+a,e_-a),\qquad (x,y)\longmapsto x+y.\] Each ideal \(e_\varepsilon A\) has unit \(e_\varepsilon \) and scalar map \(c\mapsto ce_\varepsilon \). Centrality makes each component multiplicative and the mixed products zero; the sum and orthogonality identities prove both inverse equations. This is stronger than the module splitting above.

In \(k[C_2]\), writing \(g=[r]\) for \(r^2=1\), the two central idempotents are \(e_\pm =h(1\pm g)\). The explicit product is \[\begin {aligned} \Phi (a+bg)&=(a+b,a-b),\\ \Phi ^{-1}(x,y)&=h(x+y)+h(x-y)g. \end {aligned}\] The coefficient product \((a+bg)(c+dg)=(ac+bd)+(ad+bc)g\) maps to \(((a+b)(c+d),(a-b)(c-d))\). Thus \(\Phi \) is multiplicative; its inverse equations use \(2h=1\), its unit image is \((1,1)\), and scalars map diagonally. The images of \(e_+,e_-\) are \((1,0),(0,1)\). An arbitrary coefficient ring need not be semisimple, and an abstract idempotent summand need not be free.

[sengupta2010representations, Section 4.5, pp. 63-64] discusses character projectors over a field with invertible group order; its subsequent ideal argument assumes semisimplicity. The displayed ring identities require only invertible two. Mathlib's LinearMap.IsIdempotentElem.isCompl supplies the abstract range/kernel complement. TauCeti's TauCeti.isIdempotentElem_halfOneAdd supplies the square-one idempotent identity, and MonoidAlgebra.cyclicTwoEquivProd supplies the \(C_2\) algebra product. Its generator is \(\operatorname {Multiplicative.ofAdd}(1:\mathbb Z/2\mathbb Z)\). None of these facts determines the ranks of a particular represented sector; those require the coordinate bases.

The half-turn acts on fields before a precision or probability law is chosen. Let \(k\) be a nonzero commutative ring and \(C_2=\{1,r\}\), with \(r^2=1\). In its Group Algebra, write \(g=[r]\) and \(1=[1]\). Every element has unique coefficients \(a+bg\), and \[\begin {aligned} (a+bg)(c+dg)&=(ac+bd)\\ &\quad +(ad+bc)g. \end {aligned}\] [sengupta2010representations, Sections 3.1-3.2, pp. 39-41] gives the group basis and linear extension over a field. The ring-module extension supplies the general ring theorem; the following formulas identify its particular faithful actions.

Put \(V_k=k^4\), in vertex order \((0,1,2,3)\), and \(W_k=k^R\), with \(R=(0,2)\) in that order. Let \(\delta _i(j)=\mathbf 1_{j=i}\) be the coordinate basis of \(V_k\); let \(\delta _i^R(j)=\mathbf 1_{j=i}\) for \(i,j\in R\) be the coordinate basis of \(W_k\). These coordinate vectors differ from the real spectral vectors \(u_a\) and from the edge labels \(e_i\). The half-turn and its retained action are \[\begin {aligned} P(\phi _0,\phi _1,\phi _2,\phi _3)&=(\phi _2,\phi _3,\phi _0,\phi _1),\\ J(x_0,x_2)&=(x_2,x_0). \end {aligned}\] Applying either permutation twice returns the original coordinates, so \(P^2=I\) and \(J^2=I\). Their unique unital \(k\)-algebra extensions are \[\begin {aligned} \widehat \rho _V:k[C_2]&\longrightarrow \operatorname {End}_k(V_k),\\ \widehat \rho _V(a+bg)&=aI+bP,\\ \widehat \rho _W:k[C_2]&\longrightarrow \operatorname {End}_k(W_k),\\ \widehat \rho _W(a+bg)&=aI+bJ. \end {aligned}\] The associated left module action is \(z\cdot v=\widehat \rho _V(z)v\), and similarly on \(W_k\). Both algebra actions are faithful, meaning that their defining maps are injective.

Indeed, \(P^2=I\) gives \[\begin {aligned} (aI+bP)(cI+dP) &=(ac+bd)I\\ &\quad +(ad+bc)P. \end {aligned}\] The same calculation uses \(J^2=I\) on \(W_k\). Addition and scalar multiplication are preserved term by term, and \(1\) maps to \(I\). These facts prove the module laws. A linear extension is unique because \(1,g\) are a basis. If \(aI+bP=0\), applying it to \(\delta _0\) gives \(a\delta _0+b\delta _2=0\); coordinates \(0,2\) give \(a=b=0\). Applying \(aI+bJ\) to \(\delta _0^R\) proves injectivity on \(W_k\) in the same way. No absence of zero divisors is required.

Coordinate projection \(\pi _R(\phi )=(\phi _0,\phi _2)\) sends \(\delta _0,\delta _2\) to \(\delta _0^R,\delta _2^R\) and kills \(\delta _1,\delta _3\). Since \(\pi _RP=J\pi _R\), it intertwines the whole algebra action: \[\pi _R\widehat \rho _V(z)=\widehat \rho _W(z)\pi _R \qquad (z\in k[C_2]).\] Thus it is a \(k[C_2]\)-module map by the all-element intertwining theorem. Both coordinate faithfulness witnesses above work with zero divisors; they use more than faithfulness of the group action alone.

Suppose now that \(2\) is a unit in \(k\), and put \(h=2^{-1}\). The element \(g\) is a central involution. Its two idempotents and algebra product are \[e_+=h(1+g),\qquad e_-=h(1-g).\] Their images under the two algebra actions are the following concrete module projectors. Abstract centrality in \(k[C_2]\) supplies commutation with the represented algebra; it does not make \(P\) central in all \(\operatorname {End}_k(V_k)\).

The represented projectors are \[E_\pm =h(I\pm P),\qquad F_\pm =h(I\pm J).\] They give the eigenspace decompositions \[\begin {aligned} V_k&=V_+\oplus V_-,&V_\pm &=\operatorname {im}E_\pm ,\\ W_k&=W_+\oplus W_-,&W_\pm &=\operatorname {im}F_\pm . \end {aligned}\] For either sign \(\varepsilon \in \{1,-1\}\), the concrete half-turn sector is \[\begin {aligned} V_\varepsilon &=\{(x,y,\varepsilon x,\varepsilon y):x,y\in k\},\\ W_\varepsilon &=\{(x,\varepsilon x):x\in k\}. \end {aligned}\] Their free bases are \[\begin {aligned} b_{\varepsilon ,0}&=\delta _0+\varepsilon \delta _2,\\ b_{\varepsilon ,1}&=\delta _1+\varepsilon \delta _3,\\ c_\varepsilon &=\delta _0^R+\varepsilon \delta _2^R. \end {aligned}\] The first two coordinates of \(V_\varepsilon \), and the first coordinate of \(W_\varepsilon \), recover the coefficients uniquely. Consequently the free ranks are two and one.

To verify the splitting, \(E_++E_-=I\) gives \(v=E_+v+E_-v\). An element in both images is fixed and annihilated by \(E_+\), so it is zero. Further, \(PE_\varepsilon =\varepsilon E_\varepsilon \), while \(Pv=\varepsilon v\) gives \(E_\varepsilon v=h(v+\varepsilon Pv)=v\). This identifies the images with the displayed coordinate sets; the proof for \(W_k\) is identical. Mathlib's LinearMap.IsIdempotentElem.isCompl supplies the abstract complement between an idempotent's range and kernel. The coordinate bases and ranks above identify its concrete summands here.

Projection respects these summands: \[\pi _RE_\varepsilon =F_\varepsilon \pi _R.\] Its restriction \(\pi _\varepsilon :V_\varepsilon \to W_\varepsilon \) sends \(b_{\varepsilon ,0}\) to \(c_\varepsilon \) and \(b_{\varepsilon ,1}\) to zero. It is surjective, with \[\ker \pi _\varepsilon =k b_{\varepsilon ,1}.\] The first coefficient of a sector vector proves the kernel formula and the displayed image of \(b_{\varepsilon ,0}\) proves surjectivity. For each sign, the square commutes, with the horizontal codomains restricted to the indicated sectors:

The horizontal maps select a sector; the right map removes one free coordinate within it. The algebra factors \(k\times k\), the full rank-two sectors, and the retained rank-one sectors are different levels of decomposition.

For the four-cycle over a nonzero commutative ring with invertible two, three decompositions record different structures. The support grading separates fields supported on \(\{0,2\}\) and \(\{1,3\}\). The one-step shift \(Tf(i)=f(i+1)\) exchanges these supports, so it is odd, whereas \(P=T^2\) and \(E_\pm =h(I\pm P)\) preserve them and are even. A single half-turn sign sector contains both \(b_{\varepsilon ,0}=\delta _0+\varepsilon \delta _2\) and \(b_{\varepsilon ,1}=\delta _1+\varepsilon \delta _3\), which have different support parities. A minus eigenvalue does not mean odd support.

The canonical group grading of \(k[C_2]\) has components \(k\,1\) of degree \(1_G\) and \(k\,g\) of degree \(r\). Their multiplication follows the group law, in particular \(g^2=1\). Over a ring these are rank-one free summands. The representation sends \(g\) to support-even \(P\); it therefore does not preserve these particular gradings under \(r\mapsto 1\in \mathbb Z/2\mathbb Z\). The projector ideals are different again: \(e_-^2=e_-\ne 0\), while an odd homogeneous component would have its square in the even component. The direct product, the group grading and the support grading cannot be interchanged. These are the distinct splitting levels applied to the concrete action.

The ordinary adjoint uses the counting pairing \(b(f,u)=\sum _i f_i u_i\), with identity coefficient involution. Delta vectors recover all coefficients, so the pairing is perfect. Reindexing a finite sum shows that every permutation action is an isometry, hence \[b(\rho (\sigma )f,u)=b(f,\rho (\sigma ^{-1})u).\] Uniqueness of the ordinary adjoint gives \(\rho (\sigma )^\dagger =\rho (\sigma ^{-1})\); in particular \(T^\dagger =T^{-1}\) and \(P^\dagger =P\). There is no complex conjugation or Koszul sign in this calculation.

The corresponding ordinary group-algebra operation is \[\left (\sum _g a_g[g]\right )^*=\sum _g a_g[g^{-1}].\] It is \(k\)-linear, fixes scalars, squares to the identity and reverses products, since \((gh)^{-1}=h^{-1}g^{-1}\) and all sums are finite. For an isometric representation with a perfect pairing, taking the adjoint of its finite linear extension therefore gives \(\widehat \rho (z^*)=\widehat \rho (z)^\dagger \). The isometry premise matters: the infinite cyclic group acting on \(\mathbb R\) by multiplication by \(2\) has adjoint \(2\) and inverse \(1/2\), so this identity fails for that representation. The counting action satisfies the premise.

The group basis and inversion are as in [sengupta2010representations, Sections 3.1-3.2, pp. 39-41]; the grading and adjoint comparisons above are direct calculations. The shift calculation supplies the support-degree and adjoint identities. Clifford parity of a quadratic quotient is a further structure; these ordinary formulas introduce neither a graded star nor a Hermitian extension.

Let \(H\) be a finite-dimensional real inner-product space, \(P:H\to H\) an orthogonal involution, and \(Q:H\to H\) a positive self-adjoint precision with \(QP=PQ\). Put \(H_\varepsilon =\ker (P-\varepsilon I)\) and \(E_\varepsilon =(I+\varepsilon P)/2\). Use induced Euclidean volumes on the uncompleted Borel sigma-algebras. The restrictions \(Q_\varepsilon =Q|_{H_\varepsilon }\) are positive self-adjoint. For their Gaussian laws, \[\begin {aligned} D:H&\longrightarrow H_+\times H_-,\\ z&\longmapsto (E_+z,E_-z),\\ D_*\mu _Q&=\mu _{Q_+}\otimes \mu _{Q_-},\qquad Z_Q=Z_{Q_+}Z_{Q_-}. \end {aligned}\] Thus the sector-valued random variables \(E_+,E_-\) are independent with the indicated marginal laws. The four-cycle recovery computes both blocks, their normalizers and their retained scalar laws explicitly.

For the actual polynomial function algebras, write \(\omega _R(p)=\int p\,d\mu _R\). Then \[\omega _Q\bigl ((p_+\circ E_+)(p_-\circ E_-)\bigr ) =\omega _{Q_+}(p_+)\omega _{Q_-}(p_-).\] Indeed linear precomposition takes polynomial generators to polynomial generators. Every polynomial, product and square has finite absolute Gaussian integral by Proposition 7.11, applied on the full space and on each sector. Tonelli applied first to the absolute product therefore justifies signed Fubini, which gives the formula from the product law. These are real polynomial states: they are linear, normalized and nonnegative on \(p^2\), with the identity star. No algebra structure on all integrable functions, or state on \(\operatorname {End}(H)\), is being used.

[friedli2017statistical, Proposition B.55, p. 505] uses covariance as its positive parameter. In an orthonormal chart, set \(\Sigma =Q^{-1}\). Its eigenvalues are positive, \(\Sigma ^{-1}=Q\) and \(\det \Sigma =(\det Q)^{-1}\), so its normalized density is exactly \(Z_Q^{-1}w_Q\). The inverse-precision covariance theorem identifies \(\Sigma \) as the covariance. The product proof above uses densities, not an inference from zero covariance for an unspecified law.

Mathlib's OrthonormalBasis.measurePreserving_repr supports passage to orthonormal coordinates. Index coordinates by the disjoint union of orthonormal bases of \(H_+\) and \(H_-\). Then \(\Sigma =\operatorname {diag}(Q_+^{-1},Q_-^{-1})\) is positive definite, hence positive semidefinite, with zero cross block and mean zero. After the density identification, TauCeti's TauCeti.Probability.indepFun_sumEquivProd_multivariateGaussian_iff gives the corresponding coordinate independence. It concerns this Gaussian covariance, not an arbitrary algebraic splitting or Schur fiber integral.

The integration convention is [tao2011measure, Theorem 1.7.15, pp. 200-201, preliminary version]. All probability assertions here are finite-dimensional and real, with strictly positive precision and specified volume. They supply no Lebesgue probability over a general coefficient ring; a singular precision needs a separately specified constrained space and law.

Let \(k\) be a commutative unital ring with invertible two and \(H=X\oplus Y\). Let \(P=P_X\oplus P_Y\), where \(P_X^2=I_X\), \(P_Y^2=I_Y\), and let \(Q:H\to H\) commute with \(P\). Write the typed split components as \[\begin {aligned} A&:X\to X,&B&:Y\to X,\\ E&:X\to Y,&C&:Y\to Y,\\ Q(x,y)&=(Ax+By,Ex+Cy). \end {aligned}\] If \(C\) is invertible, the split Schur map \(S=A-BC^{-1}E\) preserves both sign sectors of \(P_X\), and \[S|_{X_\varepsilon } =A_\varepsilon -B_\varepsilon C_\varepsilon ^{-1}E_\varepsilon .\] Here \(X_\varepsilon =\ker (P_X-\varepsilon I)\), \(Y_\varepsilon =\ker (P_Y-\varepsilon I)\), and each subscript means restriction to the indicated source and target sectors.

This is the sign-sector specialization of Schur transport, using the module projectors. Its block convention agrees with [boyd2004convex, Appendix A.5.5, pp. 650-651] after exchanging the source's first pivot to \(C\). The proof here is over modules; no pairing, self-adjointness, stationary field or probability law has been assumed. If the prescribed splitting is not preserved by \(P\), the required \(P_X,P_Y\) and these restrictions are not supplied.

Let \(H=X\mathbin \perp Y\) be a finite-dimensional real orthogonal sum, \(P=P_X\oplus P_Y\) an orthogonal involution, and \(Q\) a positive self-adjoint precision commuting with \(P\). Use induced Euclidean volumes and Borel Gaussian laws. For \(Q(x,y)=(Ax+By,B^\dagger x+Cy)\), put \(S=A-BC^{-1}B^\dagger \). The sign sectors are \(H_\varepsilon =X_\varepsilon \mathbin \perp Y_\varepsilon \). Let \(\pi (x,y)=x\) and let \(\pi _\varepsilon \) be its sector restriction. The typed restriction identity gives \[S_\varepsilon :=S|_{X_\varepsilon } =A_\varepsilon -B_\varepsilon C_\varepsilon ^{-1} B_\varepsilon ^\dagger .\] Then the following square commutes both for maps and for their pushforwards of the Gaussian laws:

Here \(D_H(z)=(E_+z,E_-z)\) and \(D_X(x)=(F_+x,F_-x)\) are the orthogonal sector decompositions. The full retained law is \(\pi _*\mu _Q=\mu _S\); both routes give \(\mu _{S_+}\otimes \mu _{S_-}\) on \(X_+\times X_-\).

Pullback on the polynomial function algebras reverses all 4 arrows. For every polynomial \(f\) on \(X_+\times X_-\), the commuting map square gives \[f\circ D_X\circ \pi =f\circ (\pi _+\times \pi _-)\circ D_H.\] Precomposition by each linear map is a unital algebra homomorphism, as in polynomial transport. For a polynomial \(p\in \operatorname {Pol}(X)\), nonnegative integration first gives \(\int |p\circ \pi |\,d\mu _Q=\int |p|\,d\mu _S\); absolute Gaussian moments make both sides finite for polynomials. Subtraction of positive and negative parts then gives \(\omega _S=\omega _Q\circ \pi ^*\) and the corresponding equalities on every side of the square. This is equality of normalized polynomial functionals, in addition to maps and probability laws.

The specified conditional kernel at eliminated field zero instead has precision \(A\); it need not equal the marginal of precision \(S\). If \(B=0\), then \(S=A\), so equality can occur. The coupled cycle gives a concrete difference. A nonstable retained set does not supply this square: for retention \(\{0,1\}\), \(\pi \delta _2=0\) but \(\pi P\delta _2=\delta _0\). A singular eliminated direction can have infinite fiber integral, as for \(Q(x,y)=(x,0)\) on \(\mathbb R^2\). A pseudoinverse does not make that integral finite. The massless cycle has a constant zero mode and requires a separately specified constrained law.

The Schur convention is [boyd2004convex, Appendix A.5.5, pp. 650-651], with the pivot exchanged to \(C\). The Gaussian density convention is [friedli2017statistical, Proposition B.55, p. 505], where covariance is \(\Sigma =Q^{-1}\). The sector square is the direct combination proved above; the formalization references discussed with the retained marginal theorem support Gaussian images and signed Fubini with their stated hypotheses, not the pointwise fiber constants.

For the alternating four-cycle with \(t,m>0\), the two real half-turn sectors are independent under the Gaussian law. Their precision and covariance blocks vary with \(t,m\), although the action and its projectors do not. Put \(a=m^2\) and \(s=1+t+a\). The counting quadratic form is \[\begin {aligned} \langle \phi ,Q\phi \rangle &=(\phi _1-\phi _0)^2+t(\phi _2-\phi _1)^2\\ &\quad +(\phi _3-\phi _2)^2+t(\phi _0-\phi _3)^2 +a\sum _i\phi _i^2. \end {aligned}\] It is strictly positive for nonzero \(\phi \), since \(a>0\). The weighted incidence construction is self-adjoint, and the half-turn reindexes the 4 edge terms without changing their weights. Thus \(QP=PQ\), as required by the intrinsic product law. In vertex order \((0,1,2,3)\), the massive precision is \[Q=\begin {pmatrix} s&-1&0&-t\\-1&s&-t&0\\0&-t&s&-1\\-t&0&-1&s \end {pmatrix}.\] Use the orthonormal vectors \[\begin {aligned} v_{\varepsilon ,0}&=(\delta _0+\varepsilon \delta _2)/\sqrt 2,\\ v_{\varepsilon ,1}&=(\delta _1+\varepsilon \delta _3)/\sqrt 2, \qquad \varepsilon \in \{1,-1\}. \end {aligned}\] Write \(\phi =\sum _\varepsilon (x_\varepsilon v_{\varepsilon ,0}+y_\varepsilon v_{\varepsilon ,1})\) and \(z_\varepsilon =(x_\varepsilon ,y_\varepsilon )\). These are normalized sector coordinates, rather than the coefficients in the unnormalized free bases of Definition 11.3. In particular \[\begin {aligned} \phi _0&=(x_++x_-)/\sqrt 2,&\phi _2&=(x_+-x_-)/\sqrt 2,\\ \phi _1&=(y_++y_-)/\sqrt 2,&\phi _3&=(y_+-y_-)/\sqrt 2. \end {aligned}\]

In the order \((v_{+,0},v_{+,1},v_{-,0},v_{-,1})\), the precision is \(\operatorname {diag}(Q_+,Q_-)\) and its inverse is \(\operatorname {diag}(G_+,G_-)\), where \[\begin {aligned} Q_+&=\begin {pmatrix}s&-(1+t)\\-(1+t)&s\end {pmatrix},\\ Q_-&=\begin {pmatrix}s&t-1\\t-1&s\end {pmatrix},\\ G_+&=\tfrac 1{a(a+2+2t)} \begin {pmatrix}s&1+t\\1+t&s\end {pmatrix},\\ G_-&=\tfrac 1{(a+2)(a+2t)} \begin {pmatrix}s&1-t\\1-t&s\end {pmatrix}. \end {aligned}\] The covariance in these coordinates is given by the displayed inverse blocks. If \(T\phi =(z_+,z_-)\), then \[T_*\mu _Q=\mu _{Q_+}\otimes \mu _{Q_-}.\] The product on the right is the product probability measure. In particular the sector random vectors \(z_+,z_-\) are independent.

[friedli2017statistical, Proposition B.55, p. 505] gives the Gaussian covariance and coordinate-change convention. The positive Gaussian normalization and inverse-precision covariance apply to each block. The product assertion follows from the explicit density calculation below, rather than from the algebraic direct sum alone.

Retain \(R=(0,2)\). Its orthonormal sector vectors are \[w_\varepsilon =(\delta _0^R+\varepsilon \delta _2^R)/\sqrt 2.\] The projection satisfies \[\begin {aligned} \pi _Rv_{\varepsilon ,0}&=w_\varepsilon ,\\ \pi _Rv_{\varepsilon ,1}&=0. \end {aligned}\] Thus the retained sector coordinates are exactly \(x_+,x_-\), with no additional factor of \(\sqrt 2\). In these coordinates, the retained precision has positive scalar blocks \[\begin {aligned} \sigma _+&=s-\frac {(1+t)^2}{s} =\frac {a(a+2+2t)}s,\\ \sigma _-&=s-\frac {(t-1)^2}{s} =\frac {(a+2)(a+2t)}s. \end {aligned}\] The retained coordinates are independent centered Gaussians with variances \(\sigma _+^{-1},\sigma _-^{-1}\). Decomposing into sectors and taking this marginal therefore give the same law in either order, as in the map and measure square.

Transforming the retained coordinates back gives \[\begin {aligned} \operatorname {Var}(\phi _0) =\operatorname {Var}(\phi _2) &=\tfrac 12(\sigma _+^{-1}+\sigma _-^{-1}),\\ \langle \phi _0\phi _2\rangle _Q &=\tfrac 12(\sigma _+^{-1}-\sigma _-^{-1}). \end {aligned}\] For \(m=t=1\), the sector blocks and their inverses are \[\begin {aligned} Q_+&=\begin {pmatrix}3&-2\\-2&3\end {pmatrix},&Q_-&=3I_2,\\ G_+&=\tfrac 15\begin {pmatrix}3&2\\2&3\end {pmatrix},&G_-&=I_2/3. \end {aligned}\] Hence \(\sigma _+=5/3\), \(\sigma _-=3\); the retained variance is \(7/15\) and the cross covariance is \(2/15\). Directly, \[\begin {aligned} S&=\tfrac 13\begin {pmatrix}7&-2\\-2&7\end {pmatrix},& S^{-1}&=\tfrac 1{15}\begin {pmatrix}7&2\\2&7\end {pmatrix},\\ Z_Q&=\frac {4\pi ^2}{3\sqrt 5},&Z_S&=\frac {2\pi }{\sqrt 5}. \end {aligned}\] The eliminated factor is \(2\pi/3\). The specified conditional kernel at eliminated field zero instead gives precision \(A=3I_2\) and covariance \(I_2/3\), whose zero cross covariance differs from \(2/15\). For \(m=1,t=2\), \[\begin {aligned} Q_+&=\begin {pmatrix}4&-3\\-3&4\end {pmatrix},\\ Q_-&=\begin {pmatrix}4&1\\1&4\end {pmatrix},\\ G_+&=\tfrac 17\begin {pmatrix}4&3\\3&4\end {pmatrix},\\ G_-&=\tfrac 1{15}\begin {pmatrix}4&-1\\-1&4\end {pmatrix}. \end {aligned}\] Now \(\sigma _+=7/4\), \(\sigma _-=15/4\); the retained variance is \(44/105\) and the cross covariance is \(16/105\). Here \[\begin {aligned} S&=\begin {pmatrix}11/4&-1\\-1&11/4\end {pmatrix},\\ S^{-1}&=\tfrac 1{105}\begin {pmatrix}44&16\\16&44\end {pmatrix},\\ Z_Q&=\frac {4\pi ^2}{\sqrt {105}},&Z_S&=\frac {8\pi }{\sqrt {105}}. \end {aligned}\] The eliminated factor is \(\pi/2\). Both examples have the same \(P,J,E_\pm ,F_\pm \), but different covariance matrices and hence different Gaussian states: their polynomial functionals differ on \(\phi _0\phi _2\).

The half-turn commutant explains why these two-dimensional precision blocks may mix coordinates while respecting the same symmetry.

Mathlib's OrthonormalBasis.measurePreserving_repr supplies the volume-preserving passage to orthonormal coordinates. TauCeti's TauCeti.Probability.indepFun_sumEquivProd_multivariateGaussian_iff characterizes independence of complementary coordinate blocks by vanishing cross covariance. Index each sector by \(\{0,1\}\) and the full coordinates by the disjoint union of these sets. Take mean zero and covariance \(\Sigma =\operatorname {diag}(G_+,G_-)\). The positive eigenvalues above make \(\Sigma \) positive definite, hence positive semidefinite, and its cross block is zero. After identifying the normalized sector density with this Gaussian law, the theorem gives independence. The formalization references discussed with the inverse-precision covariance theorem use \(G_\pm =Q_\pm ^{-1}\) as their parameters. The concrete matrices, density identification and sectorwise Schur calculation are supplied by the proofs above.

For the real half-turn, define its commutant by \(\operatorname {Comm}(P)=\{R\in \operatorname {End}_{\mathbb R}(V):RP=PR\}\). Restriction to the two sectors gives an algebra isomorphism \[\operatorname {Comm}(P) \cong \operatorname {End}(V_+)\times \operatorname {End}(V_-) \cong M_2(\mathbb R)\times M_2(\mathbb R).\] If \(R\) commutes with \(P\) and \(Pv=\varepsilon v\), then \(PRv=RPv=\varepsilon Rv\), so it preserves each sector. Conversely, a map preserving both sectors obeys \(PR(v_++v_-)=Rv_+-Rv_-=RP(v_++v_-)\). The inverse to restriction acts separately on the unique sector components; composition and the unit are componentwise. The explicit bases identify each endomorphism algebra with \(M_2(\mathbb R)\).

In sector coordinates \(P=\operatorname {diag}(I_2,-I_2)\). For \(R=\begin {pmatrix}A&B\\C&D\end {pmatrix}\), direct comparison gives \[RP=PR\quad \Longleftrightarrow \quad 2B=2C=0 \quad \Longleftrightarrow \quad B=C=0.\] The group-algebra image is the strictly smaller algebra \[\{\operatorname {diag}(xI_2,yI_2):x,y\in \mathbb R\}.\] Indeed \(aI+bP\) has \(x=a+b\), \(y=a-b\), and every pair is obtained from \(a=(x+y)/2\), \(b=(x-y)/2\). The map swapping \(b_{+,0},b_{+,1}\) and vanishing on \(V_-\) belongs to the commutant but is not scalar on \(V_+\), proving strict containment.

Each sign occurs twice, so an operator commuting with \(P\) may mix the two vectors in either sector. Symmetry therefore does not force \(Q_\pm \) to be scalar; the cycle precision blocks exhibit the freedom explicitly. The group-algebra/module convention comes from [sengupta2010representations, Sections 3.1-3.2, pp. 39-41]; the commutant calculation is direct. The interpretation boundary remains: this finite algebra calculation selects no continuous gauge group, physical mass spectrum or interaction.

The coefficient assumption in the two-sector decomposition is essential. Let \(k\) now be a nonzero commutative ring of characteristic two. Use the same \(C_2\), group-basis element \(g\), coordinate modules \(V_k,W_k\), and maps \(P,J\). The faithful algebra actions still exist, but the algebra is \[k[C_2]\cong k[z]/(z^2),\] and its idempotents are exactly the scalar idempotents of \(k\). In particular, if \(k\) is a field, it has no idempotents other than \(0,1\), so it cannot be isomorphic to \(k\times k\).

The Group Algebra still has its group basis, as in [sengupta2010representations, Sections 3.1-3.2, pp. 39-41]. The character projectors in [sengupta2010representations, Section 4.5, p. 63] require the group order to be invertible in the field. The following calculation identifies what replaces that product when the order is zero in \(k\).

On the vertex module, the same half-turn still satisfies \(P^2=I\), while \(-1=1\). Its two proposed eigenspaces now coincide: \[\begin {aligned} \ker (P-I)&=\ker (P+I)\\ &=\{(x,y,x,y):x,y\in k\}. \end {aligned}\] Its coordinate basis is \(\delta _0+\delta _2,\delta _1+\delta _3\): coordinates zero and one recover the coefficients. It is free of rank two (dimension two over a field); adding it to itself does not recover the full module, because \(\delta _0\) is not in it. On the retained module, \[\begin {aligned} \ker (J-I)&=\ker (J+I)\\ &=\{(x,x):x\in k\}, \end {aligned}\] whose basis is \(\delta _0^R+\delta _2^R\), with its coefficient recovered at zero. It has free rank one, and dimension one over a field. These equalities follow directly from the coordinate formulas for \(P,J\).

The nonzero nilpotent is visible in the faithful action as well. For \(N=P-I\), \[\begin {aligned} N^2&=P^2-2P+I=0,\\ N\delta _0&=\delta _2-\delta _0\ne 0. \end {aligned}\] The last vector has coefficient \(1\) at coordinate \(2\). Similarly, \(J-I\) is nonzero on \(\delta _0^R\) and has square zero: \((J-I)\delta _0^R=\delta _2^R-\delta _0^R\ne 0\). Both coordinate injectivity proofs for the algebra actions in Definition 11.3 still apply. Thus the nilpotent \(\eta \) has a faithful nonzero action on both modules. The expressions \((I\pm P)/2\) and \((I\pm J)/2\) are undefined because \(2=0\). The action and projection intertwiner survive, but the complementary projectors and eigenspace direct sum do not. This is a change of coefficient ring for the algebra and modules; the real Gaussian law is not being extended to characteristic two.

TauCeti's MonoidAlgebra.isIdempotentElem_cyclicTwo_iff proves the idempotent obstruction more generally for a commutative nonzero ring without zero divisors in which \(2\) is not a unit. A characteristic-two field meets these hypotheses. The generator is identified with \(\operatorname {Multiplicative.ofAdd} (1:\mathbb Z/2\mathbb Z)\) as in Theorem 11.2. Indeed, if \(a+bg\) is idempotent, coefficient comparison gives \(a^2+b^2=a\) and \(2ab=b\). Hence \(b(2a-1)=0\). If \(b\ne 0\), absence of zero divisors gives \(2a=1\), contradicting the nonunit hypothesis. Therefore \(b=0\) and \(a\in \{0,1\}\). This argument does not require characteristic two, but does require absence of zero divisors. The dual-number presentation and the coordinate nilpotent above are separate characteristic-two calculations.

Fix finite incidence data \((V,E,s,h)\), weights \(w_e>0\), a constant mass \(m>0\), and counting pairings on \(H=\mathbb R^V\) and \(\mathbb R^E\). A finite symmetry of a Gaussian background is a finite group \(G\) with a weight-preserving signed endpoint action \(g\mapsto (p_g,q_g,\varepsilon _g)\) on these data. No retained subset is selected here: in the signed-arrow definition one may take \(R=V\). An additional retained subset would have to be \(G\)-stable. The signs belong to the abstract \(C_2\) of that definition, with \(\sigma (+)=1\) and \(\sigma (-)=-1\). This unpartitioned data does not assert preservation of a nonempty proper retained set as in Definition 10.7.

The vertex and edge representations are \[\begin {aligned} (\rho (g)f)(p_g(v))&=f(v),\\ (J_ga)(q_g(e))&=\sigma (\varepsilon _g(e))a(e). \end {aligned}\] Thus \(\rho (g)f=f\circ p_g^{-1}\); in particular \(\rho (g)\delta _v=\delta _{p_g(v)}\). We call \(H\) with this action the vertex permutation module. With \((df)(e)=f(h(e))-f(s(e))\), the endpoint equations give \(d\rho (g)=J_gd\). The sign cocycle gives \(\rho (gk)=\rho (g)\rho (k)\) and \(J_{gk}=J_gJ_k\). These conventions are the real specialization of Definition 10.1, whose proof includes loops and parallel edge labels.

Write \(W=\operatorname {diag}(w_e)\) and \(Q=d^*Wd+m^2I\) for the massive precision. A reflection may reverse the chosen edge orientations; it is then a signed symmetry, not necessarily a directed relabeling in Definition 10.7. Forgetting those orientations preserves the energy, while still retaining incidence, weights and distinct edge labels. The underlying permutation-representation convention is [milne2025group, Example 7.1(b), p. 101]; the signed extension and its endpoint requirements are proved in Definition 10.1.

For a finite symmetry of a Gaussian background, \(\rho (g)\) and \(J_g\) are orthogonal, \(J_gW=WJ_g\), and \[\rho (g)^*Q\rho (g)=Q,\qquad Q\rho (g)=\rho (g)Q.\] The Green operator \(Q^{-1}\) also commutes with every \(\rho (g)\).

This specializes signed operator transport to positive real data. The incidence convention is [spielman2009effective, Section 2.1, pp. 4-5]; the argument above also covers loops and parallel labels.

For the four-cycle half-turn, \(G=C_2=\{1,r\}\) and \(\rho (r)=P\). Its two projectors are \((I+P)/2\) and \((I-P)/2\), with the concrete two-dimensional ranges. A larger symmetry group can split these ranges further; a sector for one chosen group is not an intrinsic label independent of that choice.

The group-algebra extension gives \[\widehat \rho :\mathbb R[G]\longrightarrow \operatorname {End}_{\mathbb R}(H), \qquad \widehat \rho \left (\sum _g a_g[g]\right )=\sum _g a_g\rho (g).\] It uses the group algebra and representation/module correspondence. Since every \(\rho (g)\) commutes with \(Q\), so does every \(\widehat \rho (a)\). The action groupoid organizes transported backgrounds, whereas this representation describes a fixed background's symmetry. The character-projector construction in [serre1977linear, Section 2.6, Theorem 8, pp. 21-22] recovers exactly the two half-turn projectors.

Let \(G\) be finite and \(k\) a field in which \(|G|\) is invertible. Every subrepresentation \(W\subseteq V\) has a \(G\)-stable complement. Consequently every finite-dimensional \(k[G]\)-module, and \(k[G]\) itself, is semisimple.

Over a splitting field of characteristic not dividing \(|G|\), an irreducible character \(\chi \) defines the central element \[ e_\chi =\frac {\chi (1)}{|G|}\sum _{g\in G}\chi (g^{-1})[g]. \] It is a nonzero primitive central idempotent. On a simple module with character \(\psi \), it acts as the identity if \(\psi =\chi \) and as zero otherwise. Distinct \(e_\chi \) are therefore orthogonal block projectors. See [lux2010representations, thm. 2.1.6 and cor. 2.1.7, pp. 88-90]. A constructed family of such projectors does not yield a complete block decomposition until a sum-to-1 statement is also known.

Let \(G\) be finite and let \(\rho \) be an orthogonal action on a finite-dimensional real inner-product space \(H\). Let \(Q\) be a linear operator commuting with every \(\rho (g)\). Write \(\operatorname {Irr}_{\mathbb C}(G)\) for the irreducible complex characters. For each orbit \(\Xi \) under \(\chi \mapsto \overline \chi \), put \[\begin {aligned} e_\chi &=\frac {\chi (1)}{|G|}\sum _{g\in G}\chi (g^{-1})[g],\\ \epsilon _\Xi &=\sum _{\chi \in \Xi }e_\chi , & P_\Xi &=\widehat \rho (\epsilon _\Xi ). \end {aligned}\] Then \(\epsilon _\Xi \) belongs to \(\mathbb R[G]\), is central, and is fixed by the real involution \([g]^*=[g^{-1}]\). The operators satisfy \[\begin {aligned} P_\Xi ^2&=P_\Xi =P_\Xi ^*,& QP_\Xi &=P_\Xi Q,\\ P_\Xi P_\Upsilon &=0\quad (\Xi \ne \Upsilon ),& \sum _\Xi P_\Xi &=I_H. \end {aligned}\] Thus \(H\) is the orthogonal direct sum of their ranges; a range may be zero. These ranges are its real isotypic sectors: each is the sum of all irreducible real subrepresentations of one isomorphism type. Complexification of a sector is the sum of the complex isotypic components with characters in \(\Xi \).

Serre's [serre1977linear, Section 2.6, Theorem 8, p. 21] gives the canonical complex components and their projectors; the real grouping and adjoint argument above are deductions from it. TauCeti's TauCeti.Representation.sum_primitiveCentralIdempotent proves the complex sum-to-one identity for a finite pairwise inequivalent family of irreducibles whose cardinality equals the number of conjugacy classes, over an algebraically closed field with invertible group order. It supports the completeness step, not the real adjoint or graph claims.

Let \(n\ge 3\), \(V=E=\mathbb Z/n\mathbb Z\), with edge \(e_v\) from \(v\) to \(v+1\), uniform weight \(w>0\) and mass \(m>0\). The generator \(r\) of \(C_n\) sends \(v\) to \(v+1\) and \(e_v\) to \(e_{v+1}\), with every edge sign positive. Its action on fields is \((\rho (r)f)(v)=f(v-1)\). The cyclic characters in [serre1977linear, Section 5.1, p. 35] and the real projectors yield the following direct calculation.

Set \(\theta _k=2\pi k/n\). The normalized constant vector \(u_0(v)=1/\sqrt n\) spans the trivial sector. For \(1\le k<n/2\), the sector \(H_k\) has orthonormal basis \[c_k(v)=\sqrt {2/n}\cos (\theta _kv),\qquad s_k(v)=\sqrt {2/n}\sin (\theta _kv).\] When \(n\) is even there is also \(u_{n/2}(v)=(-1)^v/\sqrt n\). These give all sectors indexed by \(0\le k\le \lfloor n/2\rfloor \). In the ordered basis \((c_k,s_k)\), the rotation is \[\rho (r)|_{H_k}=\begin {pmatrix} \cos \theta _k&-\sin \theta _k\\ \sin \theta _k&\cos \theta _k \end {pmatrix}.\] For \(0<\theta _k<\pi \) it has no real eigenline, so \(H_k\) is real irreducible. Its complex constituents have characters \(r\mapsto e^{i\theta _k}\) and \(r\mapsto e^{-i\theta _k}\).

The precision and each sector eigenvalue are \[\begin {aligned} (Qf)(v)&=(m^2+2w)f(v)-wf(v-1)-wf(v+1),\\ Q|_{H_k}&=q_kI,\\ q_k&=m^2+2w(1-\cos \theta _k). \end {aligned}\] Substitution of each sine, cosine and exceptional one-dimensional mode proves the formula. The orthogonal projector kernels are \[\begin {aligned} P_0(v,u)&=1/n,\\ P_k(v,u)&=\frac 2n\cos (\theta _k(v-u))\quad (0<k<n/2),\\ P_{n/2}(v,u)&=(-1)^{v-u}/n\quad (n\text { even}). \end {aligned}\] They follow by summing the outer products of the displayed orthonormal basis vectors, so their sum is the identity. For \(n=3\), this reads \(P_0=\mathbf 1\mathbf 1^T/3\) and \(P_1=I-P_0\); the eigenvalues are \(m^2\) and \(m^2+3w\), the latter twice. The later circle comparison rescales these same finite modes; no continuum limit is used in this calculation.

Take the cycle with \(n=4\), \(w=m=1\). Let \(D_4=\langle r,s:r^4=s^2=1,\ srs=r^{-1}\rangle \), of order 8, act on vertices by \(r(v)=v+1\), \(s(v)=-v\). On the oriented edges \(e_v:v\to v+1\), rotation sends \(e_v\) to \(e_{v+1}\) with positive sign, while reflection sends \(e_v\) to \(e_{-v-1}\) with negative sign. The latter edge runs from \(-v-1\) to \(-v\), the reverse of the transported endpoints. The 4 rotations restore each edge; two reflections restore its label and give positive sign. Reflecting, rotating, then reflecting sends \(e_v\) to \(e_{v-1}\) with two negative signs, hence agrees with the inverse rotation. The vertex maps obey the same relations, so these maps define a signed endpoint action. This uses the dihedral presentation in [serre1977linear, Section 5.3, p. 36]; the edge signs and precision calculation are explicit here.

The dashed diagonal is the reflection axis: \(s\) fixes vertices \(0,2\), exchanges \(1,3\), and sends \(e_0\) to \(e_3\) with reversed orientation. All 4 weights and the mass are one.

An orthonormal basis, grouped by real irreducible type, is \[\begin {aligned} u&=(1,1,1,1)/2,&a&=(1,-1,1,-1)/2,\\ c&=(1,0,-1,0)/\sqrt 2,&b&=(0,1,0,-1)/\sqrt 2. \end {aligned}\] The line \(\mathbb Ru\) is trivial. On \(\mathbb Ra\), rotation acts by \(-1\) and reflection by \(+1\); call this the alternating line. On \(K=\operatorname {span}(c,b)\), \[\rho (r)|_K=\begin {pmatrix}0&-1\\1&0\end {pmatrix},\qquad \rho (s)|_K=\begin {pmatrix}1&0\\0&-1\end {pmatrix}.\] There is no real rotation-invariant line, so \(K\) is irreducible. A real matrix commuting with the rotation is \(xI+y\rho (r)\); commuting also with the reflection forces \(y=0\). Thus its commuting algebra is \(\mathbb R I\), the real-type case. The two distinct lines and \(K\) are pairwise nonisomorphic, each occurring once.

Direct substitution in the cycle precision gives \[Qu=u,\qquad Qa=5a,\qquad Qc=3c,\qquad Qb=3b.\] The three orthogonal projectors are \(uu^T\), \(aa^T\), and \(cc^T+bb^T=I-uu^T-aa^T\); their ranges have dimensions \(1,1,2\) and precision eigenvalues \(1,5,3\), respectively. For the subgroup generated by \(r^2\), the positive sector is \(\operatorname {span}(u,a)\) and the negative sector is \(K\). The half-turn calculation therefore has two \(2\times 2\) blocks, whereas the full dihedral action separates its positive block into two lines. Changing the acting group changes the isotypic decomposition without changing this precision.

Let a finite group \(G\) act orthogonally on a finite-dimensional real space \(H\). In its isotypic decomposition, write \(H_\Xi \cong W_\Xi ^{\oplus m_\Xi }\), where \(W_\Xi \) is irreducible and \(m_\Xi >0\) is its multiplicity. The commutant consists of all linear maps commuting with \(G\). With multiplication given by composition, put \(D_\Xi =\operatorname {End}_G(W_\Xi )\). Then \[\begin {aligned} D_\Xi &\cong \mathbb R,\ \mathbb C\ \text {or}\ \mathbb H,\\ \operatorname {End}_G(H)&\cong \prod _\Xi M_{m_\Xi }(D_\Xi ). \end {aligned}\] The isomorphism can be chosen to carry adjunction to conjugate transpose. If \(H=\mathbb R^V\) is a vertex permutation representation, the real dimension of \(\operatorname {End}_G(H)\) is the number of orbits of the diagonal action \(g(u,v)=(gu,gv)\) on \(V\times V\). These orbits are called orbitals.

The scalar-side convention matters for \(\mathbb H\). Here \(D_\Xi \) acts on \(W_\Xi \) on the left by endomorphisms. If instead \(W_\Xi \) is presented as a left module for a simple matrix algebra, its commuting scalars act on the right and form an opposite algebra, as in Proposition [fcap-0042]. Quaternion conjugation identifies that opposite algebra with \(\mathbb H\); it does not license reversing the order of quaternion products.

Under the commutant identification, an equivariant self-adjoint positive definite precision \(Q\) is specified by \(Q_\Xi \in M_{m_\Xi }(D_\Xi )\) with \(Q_\Xi ^*=Q_\Xi >0\). Thus all such precisions form an open cone in a real vector space of dimension \[\sum _\Xi h(m_\Xi ,D_\Xi ),\qquad \begin {array}{c|ccc} D&\mathbb R&\mathbb C&\mathbb H\\\hline h(m,D)&m(m+1)/2&m^2&2m^2-m. \end {array}\] For a vertex permutation representation this dimension also counts \(G\)-orbits on unordered pairs \(\{u,v\}\), including \(u=v\). This is the full cone of equivariant precisions; graph Laplacians with prescribed edges and positive weights generally form a smaller family.

These coordinates fix the determinant convention without a noncommutative determinant ambiguity: \[\begin {aligned} \dim _{\mathbb R}H_\Xi &=m_\Xi r_\Xi d_{D_\Xi },\\ \det _{\mathbb R}(Q|_{H_\Xi }) &=\det _{\mathbb R}(Q_\Xi :D_\Xi ^{m_\Xi }\to D_\Xi ^{m_\Xi })^{r_\Xi }. \end {aligned}\] For a positive Hermitian block with eigenvalues \(\lambda _1,\ldots ,\lambda _m\), the latter real determinant is \(\prod _j\lambda _j^{d_D}\). The spectral assertion follows by taking an eigenvector of the real self-adjoint operator: its eigenspace is stable under right scalar multiplication on \(D^m\); its scalar line has an invariant orthogonal complement, and induction gives a \(D\)-orthonormal eigenbasis. In particular a complex determinant is squared, and the product of quaternionic Hermitian eigenvalues is raised to the fourth power, before taking the exponent \(r_\Xi \). This is the positive-matrix specialization of the real block classification in [serre1977linear, Section 13.2, pp. 108-109].

For a graph write \(L=d^*Wd\) and \(Q=L+m^2I\), with \(m^2>0\). The multiplicity matrices obey \(Q_\Xi =L_\Xi +m^2I\). A zero Laplacian eigenvalue therefore gives a strictly positive precision eigenvalue \(m^2\), not a normalized massless Gaussian.

Let \(\rho \) be an irreducible finite-dimensional representation of a finite group over an algebraically closed field of characteristic 0. Its Frobenius-Schur indicator \[ \nu _2(\rho )=\frac 1{|G|}\sum _{g\in G}\chi _\rho (g^2) \] takes exactly 1 of the values \(1\), \(0\), and \(-1\). The value \(1\) is equivalent to the existence of a nondegenerate invariant symmetric bilinear form; \(-1\) is equivalent to a nondegenerate invariant alternating form; and \(0\) is equivalent to the absence of a nonzero invariant bilinear form. See [james2001representations, def. 23.13, the proof of thm. 23.14, and thm. 23.16, pp. 273-277]. James and Liebeck state the alternatives using nonzero invariant forms; for an irreducible representation, a nonzero invariant symmetric or alternating form has zero radical and is therefore nondegenerate.

This trichotomy classifies the invariant-form behavior of the complex irreducible representation. It does not by itself compute a Schur index or a complete decomposition after descent to \(\mathbb R\).

Let \(W\) be an irreducible real representation of a finite group, \(D=\operatorname {End}_G(W)\), and \(\chi \) an irreducible constituent of \(W_{\mathbb C}=W\otimes _{\mathbb R}\mathbb C\). The Frobenius-Schur indicator gives exactly the following possibilities, where \(X_\chi \) denotes a complex module of character \(\chi \): \[\begin {array}{c|c|c|c} \nu _2(\chi )&D&W_{\mathbb C}&\dim _{\mathbb R}W\\\hline 1&\mathbb R&X_\chi &\chi (1)\\ 0&\mathbb C&X_\chi \oplus X_{\bar \chi }&2\chi (1)\\ -1&\mathbb H&X_\chi \oplus X_\chi &2\chi (1). \end {array}\] In the middle row \(\chi \ne \bar \chi \). In the last row \(\chi (1)\) is even and \(\dim _{\mathbb H}W=\chi (1)/2\). The form trichotomy alone is not the real descent argument; these assertions use the bridge proved below. They agree with [serre1977linear, Section 13.2, Proposition 39, p. 109].

For an equivariant centered real Gaussian on an isotypic sector, use the coordinates \(z_1,\ldots ,z_r\in D^m\) of Corollary 13.2. Its real density is proportional to \[\exp \left (-\frac 12\sum _{a=1}^r \operatorname {Re}(z_a^*Q_\Xi z_a)\right ).\] The real covariance is the Hermitian multiplicity matrix \(Q_\Xi ^{-1}\) acting on each column. For \(D=\mathbb C\), multiplication of a column by \(e^{i\theta }\) preserves this real law: this is circular symmetry. With the displayed real normalization the complex second moment is \(\mathbb E[z_a z_a^*]=2Q_\Xi ^{-1}\). For \(D=\mathbb H\), right multiplication \(z_a\mapsto z_a q\) by a unit quaternion likewise preserves the real quadratic form and Lebesgue measure. This is the asserted quaternionic symmetry; left multiplication by an arbitrary quaternion need not commute with the entries of \(Q_\Xi \). These coordinate symmetries need not be additional graph symmetries. The real Gaussian normalization and covariance are those of Theorem 10.8.

TauCeti's Representation.frobeniusSchurIndicator_eq_one_iff_isRealizableOverReal proves the indicator-one/real-realizability equivalence for an irreducible complex representation of a finite group. It supports the real-form step; the full commutant identification, all three real dimensions and the Gaussian coordinate statements are the arguments above.

Take two triangles with vertices \(x,x'\) indexed by \(x\in \mathbb Z/3\mathbb Z\). Give the upper triangle weight \(a=1\), the lower triangle weight \(b=2\), the edges \(\{x,x'\}\) weight \(u=1\), and the edges \(\{x,(x+1)'\}\) weight \(v=1/2\). The graph precision is \(Q=L+I\). Each edge is undirected for the energy; arbitrary chosen orientations are transported with the signed convention.

Thin solid edges have weight \(1\); the thick lower triangle has weight \(2\); dashed edges have weight \(1/2\). The outer dashed edge joins \(2\) to \(0'\), completing the shift modulo three. Crossings without a circle are not vertices.

The simultaneous rotation \(x\mapsto x+1\) generates \(C_3\). It is the entire weight-preserving vertex automorphism group. The weighted degrees \(7/2\) and \(11/2\) distinguish the two layers. The interlayer edges of weight \(1\) force the same permutation \(\pi \) on both layers, and those of weight \(1/2\) force \(\pi (x+1)=\pi (x)+1\). Thus only the three rotations remain.

The real representation is \(\mathbb R^6\cong \mathbb R^{\oplus 2}\oplus W^{\oplus 2}\), where \(W\) is the two-dimensional rotation representation and has \(D=\mathbb C\) by Proposition 13.4. The commutant is \(M_2(\mathbb R)\times M_2(\mathbb C)\), of real dimension \(4+8=12\). Its orbitals are the 4 choices of ordered layers and three differences \(y-x\). The symmetric dimension is \(3+4=7\): there are two diagonal orbits, two within-layer edge orbits and three interlayer pair orbits.

Use normalized constant vectors on the two layers for the trivial sector. For the other sector put \(\zeta =e^{2\pi i/3}\) and represent its upper values by \(f(x)=\sqrt {2/3}\operatorname {Re}(z\zeta ^{-x})\), and its lower values by the same formula with \(z'\). This is an isometry from \(\mathbb C^2\) with its real Euclidean inner product to the four-dimensional sector. Evaluating the edge energy gives \[\begin {aligned} Q_{\mathrm {triv}}&= \begin {pmatrix}5/2&-3/2\\-3/2&5/2\end {pmatrix},\\ Q_{\mathbb C}&= \begin {pmatrix} 11/2&-3/4+i\sqrt3/4\\ -3/4-i\sqrt3/4&17/2 \end {pmatrix}. \end {aligned}\] Indeed the diagonal Fourier entries are \(1+3a+u+v\) and \(1+3b+u+v\); the upper-right entry is \(-u-v\zeta ^{-1}\). The trivial eigenvalues are \(1,4\). The complex block has trace \(14\), determinant \(46\), and eigenvalues \(7-\sqrt 3,7+\sqrt 3\), each repeated twice over the reals. Hence \[\det _{\mathbb R}Q=4\cdot 46^2=8464.\] The square is required by the real determinant convention.

The imaginary entry records the shift in the declared Fourier coordinates. Rephasing one coordinate can make this single Hermitian matrix real; the intrinsic complex feature is the full commutant \(M_2(\mathbb C)\), not an obstruction to diagonalizing one positive operator. This calculation adapts the cyclic Fourier modes of Example 12.7, based on [serre1977linear, Section 5.1, p. 35].

Let \(T=2T\) be the 24 Hurwitz units, with \(\omega =(-1+i+j+k)/2\), and take the inverse-closed generating set \[S=\{i,-i,j,-j,k,-k,\omega ,\bar \omega \}.\] Give each unordered edge \(\{g,gs\}\) weight one, counting it once. There are no loops, the degree is 8, and left multiplication acts by signed graph symmetries. On \(\mathbb R^T\) the Laplacian is \[(Lf)(g)=8f(g)-\sum _{s\in S}f(gs).\] The underlying group construction is [voight2021quaternion, Section 11.2, pp. 166-168]. The following spectrum and blocks are calculations for this particular generating set.

The quotient \(T/Q_8\cong C_3\) gives the character \(\lambda (\omega )=\zeta =e^{2\pi i/3}\), \(\lambda |_{Q_8}=1\). Let \(\sigma \) be the defining two-dimensional complex quaternion representation, with \(\chi _\sigma (q)=2\operatorname {Re}q\), and let \(\tau \) act by quaternion conjugation on \(\operatorname {span}_{\mathbb R}\{i,j,k\}\). Its character is \(4(\operatorname {Re}q)^2-1\). The 7 conjugacy classes and their characters are \[\begin {array}{c|rrrrrrr} q&1&-1&i&\omega &-\omega &\omega ^2&-\omega ^2\\\hline |[q]|&1&1&6&4&4&4&4\\ \lambda ^j&1&1&1&\zeta ^j&\zeta ^j&\zeta ^{2j}&\zeta ^{2j}\\ \sigma \lambda ^j&2&-2&0&-\zeta ^j&\zeta ^j&-\zeta ^{2j}&\zeta ^{2j}\\ \tau &3&3&-1&0&0&0&0 \end {array}\] with \(j=0,1,2\). The class of \(i\) comprises \(\{\pm i,\pm j,\pm k\}\); each remaining class is determined by its \(Q_8\)-coset and real part. These descriptions follow by conjugating the explicit Hurwitz units.

Using the displayed sizes, the character inner products are \(\delta _{\alpha \beta }\) and the sum of squared degrees is \(3\cdot 1^2+3\cdot 2^2+3^2=24\). Thus the 7 characters are irreducible, inequivalent and complete. Squaring sends the first three classes to \(1,1,-1\), the two \(\pm \omega \) classes to \([\omega ^2]\), and the two \(\pm \omega ^2\) classes to \([\omega ]\). Substitution into \(|T|^{-1}\sum _g\chi (g^2)\) gives indicators \(1,0,0\) for \(1,\lambda ,\lambda ^2\), \(-1,0,0\) for \(\sigma ,\sigma \lambda ,\sigma \lambda ^2\), and \(1\) for \(\tau \). For example the numerator for \(\sigma \) is \(2+2-12-16=-24\); for \(\sigma \lambda \) it is \(4-12-8(\zeta +\zeta ^2)=0\). The real-type bridge therefore gives \[\operatorname {End}_T(\mathbb R^T) \cong \mathbb R\times \mathbb C\times M_3(\mathbb R) \times \mathbb H\times M_2(\mathbb C).\] Here the regular real multiplicity is \(m_\Xi =\dim _{D_\Xi }W_\Xi \): a real simple matrix algebra \(M_r(D)\) has \(r\) column copies of its defining module, as in Proposition [fcap-0042].

Fourier decomposition of right translation gives, up to the chosen dual-basis convention, the block \(8I-\sum _{s\in S}\pi (s)\). Its adjoint or transpose gives the same spectra and determinants, since \(S=S^{-1}\). The multiplicity matrices are as follows; in the table \(m\) is multiplicity, not the mass parameter: \[\begin {array}{c|c|c|c|c} W\text { type}&D&m&\dim _{\mathbb R}W&L_\Xi \\\hline 1&\mathbb R&1&1&0\\ \lambda ,\bar \lambda &\mathbb C&1&2&3\\ \tau &\mathbb R&3&3&11I_3-J_3\\ \sigma &\mathbb H&1&4&9\\ \sigma \lambda ,\sigma \bar \lambda &\mathbb C&2&4&\operatorname {diag}(6,9) \end {array}\] Here \(J_3\) is the all-ones matrix. For \(\tau \), the 6 \(\pm i,\pm j,\pm k\) contribute \(-2I_3\), while \(\tau (\omega )+\tau (\bar \omega )=J_3-I_3\); the block has eigenvalues \(8,11,11\). For \(\sigma \) the 6 contributions cancel and \(\sigma (\omega )+\sigma (\bar \omega )=-I_2\), giving \(9\). For \(\sigma \lambda \), diagonalize \(\sigma (\omega )\) with eigenvalues \(\zeta ,\zeta ^2\). The twisted adjacency sum has eigenvalues \(-1,2\), giving \(9,6\). The nonreal linear pair has adjacency eigenvalue \(6+\zeta +\zeta ^2=5\).

The real sector dimensions are \(1,2,9,4,8\), summing to \(24\). Consequently \[\operatorname {spec}(L)=\{0,3^{[2]},6^{[4]},8^{[3]},9^{[8]},11^{[6]}\},\] where brackets denote eigenvalue multiplicity. For a mass squared \(a>0\), set \(Q=L+aI\); every block receives the same shift and \[\begin {aligned} \det _{\mathbb R}Q={}&a(a+3)^2(a+6)^4\\ &\cdot (a+8)^3(a+9)^8(a+11)^6. \end {aligned}\] The complex and quaternionic exponents are real dimensions, not complex degrees copied without descent. The commutant dimension is \(24\); the symmetric dimension is \(13\), also the number of inversion orbits in \(T\), since only \(1,-1\) are self-inverse. These are finite graph calculations; the possible physical interpretations discussed in Remark [fgap-0018] are separate hypotheses.

Let a finite group \(G\) act orthogonally on a finite-dimensional real inner-product space \(H\), and let \(Q>0\) be self-adjoint and \(G\)-equivariant. Write the orthogonal isotypic projectors as \(E_\Xi \), put \(H_\Xi =E_\Xi H\), and omit the zero sectors. Use induced Euclidean volumes and uncompleted Borel sigma-algebras. For the normalized Gaussian law \(\mu _Q\), the map \[\mathcal D:H\longrightarrow \prod _\Xi H_\Xi , \qquad z\longmapsto (E_\Xi z)_\Xi \] satisfies \[\mathcal D_*\mu _Q=\bigotimes _\Xi \mu _{Q|_{H_\Xi }}. \qquad Z_Q=\prod _\Xi Z_{Q|_{H_\Xi }}.\] Thus the projected fields are independent. Each has covariance \((Q|_{H_\Xi })^{-1}\) on its own real sector.

In the multiplicity coordinates, set \(D_\Xi =\operatorname {End}_G(W_\Xi )\), \(r_\Xi =\dim _{D_\Xi }W_\Xi \), and \(d_\Xi =\dim _{\mathbb R}D_\Xi \in \{1,2,4\}\). Write \(Q_\Xi \in M_{m_\Xi }(D_\Xi )\) for the multiplicity matrix; it is not the entire real-sector matrix. Then \[\begin {aligned} \det _{\mathbb R}Q &=\prod _\Xi \det _{\mathbb R} (Q_\Xi :D_\Xi ^{m_\Xi }\to D_\Xi ^{m_\Xi })^{r_\Xi },\\ Z_{Q|_{H_\Xi }} &=\frac {(2\pi )^{m_\Xi r_\Xi d_\Xi/2}} {\det _{\mathbb R}(Q_\Xi )^{r_\Xi/2}}. \end {aligned}\] The determinants in these formulas are ordinary determinants of real linear operators, including when \(D_\Xi =\mathbb H\).

For the twisted prism, the trivial multiplicity matrix has determinant \(4\), while the complex matrix has complex determinant \(46\). Both have \(r_\Xi =1\), so \(\det _{\mathbb R}Q=4\cdot 46^2=8464\) and \(Z_Q=(2\pi )^3/92\). For the binary tetrahedral Cayley graph, write \(a=m^2>0\). The 5 real sectors give the following factors: \[\begin {array}{c|c|c} D_\Xi &r_\Xi &\det _{\mathbb R}(Q|_{H_\Xi })\\\hline \mathbb R&1&a\\ \mathbb C&1&(a+3)^2\\ \mathbb R&3&[(a+8)(a+11)^2]^3\\ \mathbb H&1&(a+9)^4\\ \mathbb C&2&[(a+6)(a+9)]^4 \end {array}\] Their product is \[\det _{\mathbb R}Q =a(a+3)^2(a+6)^4(a+8)^3(a+9)^8(a+11)^6.\] The exponents sum to \(24\), the real field dimension. The normalizer is \[Z_Q=\frac {(2\pi )^{12}}{\sqrt {\det _{\mathbb R}Q}}.\] These factors use the mass-shifted precision, not the singular Laplacian alone.

The density convention agrees with [friedli2017statistical, Theorem 8.4, p. 385] after setting its mean to zero and its covariance to \(\Sigma =Q^{-1}\). That theorem identifies the Gaussian density; the finite-group projectors and real determinant exponents are supplied by the proofs above and in Corollary 13.2.

Mathlib's ProbabilityTheory.HasGaussianLaw.iIndepFun_of_covariance_inner supports the independence step for a finite family of jointly Gaussian real Hilbert-space variables. Here the family is \(z\mapsto (E_\Xi z)_\Xi \), a linear image of the Gaussian field. For distinct sectors and \(u\in H_\Xi \), \(v\in H_\Psi \), its scalar cross covariance is \(\langle u,Q^{-1}v\rangle =0\), because \(Q^{-1}\) preserves the orthogonal sectors. Thus the declaration's zero-cross-covariance hypothesis matches this family. It does not assert the representation-theoretic decomposition or the real determinant exponents.

Let \(H=X\mathbin \perp Y\) be a finite-dimensional orthogonal real representation of a finite group \(G\), with both summands \(G\)-stable. Let the positive self-adjoint equivariant precision be \[Q=\begin {pmatrix}A&B\\B^*&C\end {pmatrix}, \qquad B:Y\to X.\] Then \(C>0\), and the retained precision \(S=A-BC^{-1}B^*:X\to X\) is positive, self-adjoint and \(G\)-equivariant. For every real irreducible type \(\Xi \), set \(X_\Xi =E_\Xi X\) and \(Y_\Xi =E_\Xi Y\). All block maps restrict to the indicated source and target sectors, and \[S|_{X_\Xi } =A|_{X_\Xi }-B_\Xi (C|_{Y_\Xi })^{-1}B_\Xi ^*.\] In common multiplicity coordinates, the same formula is \[S_\Xi =A_\Xi -B_\Xi C_\Xi ^{-1}B_\Xi ^*,\] where \(B_\Xi \) is an \(m_{X,\Xi }\times m_{Y,\Xi }\) matrix over \(D_\Xi \), and the products retain their displayed order.

Use induced Euclidean volumes and let \(\pi :H\to X\) be the orthogonal projection. For every fixed \(x\), translation in \(Y\) and the completed square give \[\int _Y e^{-\langle (x,y),Q(x,y)\rangle/2}\,d\lambda _Y(y) =Z_Ce^{-\langle x,Sx\rangle/2}.\] Nonnegative Tonelli and normalization therefore give \(Z_Q=Z_CZ_S\) and \(\pi _*\mu _Q=\mu _S\), as in positive elimination. Applying the same argument in \(X_\Xi \mathbin \perp Y_\Xi \), and then the sector product law, gives \[\begin {aligned} (\mathcal D_X)_*\pi _*\mu _Q &=\bigotimes _\Xi \mu _{S|_{X_\Xi }},\\ (\pi _\Xi )_*\mu _{Q|_{H_\Xi }} &=\mu _{S|_{X_\Xi }},\qquad Z_C=\prod _\Xi Z_{C|_{Y_\Xi }}. \end {aligned}\] Here \(\pi _\Xi :H_\Xi \to X_\Xi \); the identity \(\pi E_\Xi =E_\Xi \pi \) shows that decomposing and projecting commute as maps. Product measures then give the same commuting square as in Theorem 11.7, with all real isotypic sectors in place of the two sign sectors.

For vertex fields, the hypothesis holds when the retained set \(R\) is \(G\)-stable: take \(X=\mathbb R^R\) and \(Y=\mathbb R^{V\setminus R}\). It asserts equivariance of the induced marginal law. It does not assert that \(S=A\), the compressed precision. The cycle decimation gives a \(G\)-stable retained set with a nonzero Schur correction.

Take the unit-weight cycle on \(\mathbb Z/(2N)\), with \(N\ge 3\) and precision \(Q=L_{C_{2N}}+aI\), where \(a=m^2>0\). The group \(G=C_N\) acts by two-site rotations. It preserves the even retained vertices and the odd eliminated vertices. Write \(x_j=\phi _{2j}\), \(y_j=\phi _{2j+1}\) and \(q=2+a\); indices \(j\) lie in \(\mathbb Z/N\). For the shift \((Tz)_j=z_{j-1}\), the blocks are \[A=C=qI,\qquad B=-(I+T),\qquad B^*=-(I+T^{-1}).\] Thus the equivariant Schur formula gives \[\begin {aligned} S&=qI-q^{-1}(2I+T+T^{-1})\\ &=w'L_{C_N}+a'I,\\ w'&=\frac 1{2+a},\qquad a'=(2+a)-\frac 4{2+a} =\frac {a(a+4)}{a+2}>0. \end {aligned}\] Here \(L_{C_N}=2I-T-T^{-1}\). The restriction \(Q_{RR}=qI\) has no off-diagonal entries, while \(S\) has entries \(-1/q\) between adjacent retained vertices. The retained set is stable under \(G\), but the marginal precision is induced by elimination.

The six-cycle illustrates \(N=3\). Shaded even vertices remain; dashed odd vertices are integrated out. Each triangle edge has weight \(1/q\), and the mass squared changes from \(a\) to \(q-4/q\), with \(q=2+a\).

The finite Gaussian normalization is retained exactly: \(Z_Q=Z_CZ_S\) with \(Z_C=(2\pi/q)^{N/2}\), and the retained law is \(\mu _S\). The effective weight and mass are both part of this law. Dividing \(S\) by \(w'\) is useful for comparing its parameters, but changes the field covariance by a scalar unless the corresponding field rescaling is also made.

Put \(q=2\cosh \omega \) with \(\omega >0\). Then \[2+\frac {a'}{w'}=q^2-2=2\cosh (2\omega ).\] The doubled parameter has an exact finite-ring interpretation. For \(M\ge 3\), the covariance of precision \(w(2\cosh \eta \,I-T-T^{-1})\) on \(\mathbb Z/M\), with \(w,\eta >0\), is the periodic kernel \[\begin {aligned} K_{M,w,\eta }(r) &=\frac {e^{-\eta r}+e^{-\eta (M-r)}} {2w\sinh \eta \,(1-e^{-M\eta })},\\ &\hspace {24mm}0\le r\le M-1. \end {aligned}\] For the original and retained fields this gives \[(Q^{-1})_{0,2r} =K_{2N,1,\omega }(2r) =K_{N,w',2\omega }(r) =(S^{-1})_{0,r}.\] The equality is between covariances with their actual weights.

In the convention that correlation length is the reciprocal decay rate of these periodized exponential images, it changes from \(\xi =1/\omega \) fine lattice units to \(\xi '=1/(2\omega )=\xi / 2\) coarse lattice units. The finite ring has two winding directions, so its covariance is not a single exponential in separation. If the old spacing is \(\ell \), the new spacing is \(2\ell \); the physical decay length \(\ell / \omega =(2\ell )/(2\omega )\) is unchanged.

The bound \(N\ge 3\) keeps the reduced graph a simple cycle. At \(N=2\), \(T=T^{-1}\) and two eliminated vertices connect the same retained pair, producing two parallel edges of weight \(1/q\), or one edge of weight \(2/q\). At \(a=0\), the algebraic formula still gives \(w'=1/2\) and \(a'=0\), the two-resistors-in-series limit. The full precision then has a constant zero mode and no normalized full-space Gaussian law. Pinning or a mean-zero constraint must be specified as in constrained massless laws; the positive-mass covariance formula is not a finite limit on the full field space.

This is a direct cycle calculation within Kron reduction, the Schur reduction of graph Laplacians with possible positive diagonal terms. Closure follows from [dorfler2011kron, Lemma 2.1, pp. 5-6]. Successive and simultaneous eliminations agree by [dorfler2011kron, Lemma 3.3, p. 14]. In that source the diagonal mass corresponds to a shunt conductance or graph self-loop. In the incidence convention used here an edge with identical endpoints has zero difference; the mass is therefore kept as the separate term \(aI\). The source supplies the general reduction context, while the periodic covariance and scale statements are proved above.

The sector laws use an ordinary orthogonal finite-group action on a real field space and a strictly positive precision. These hypotheses allow independent Gaussian factors and equivariant elimination. They do not make every stable retained subspace a reducing subspace for the precision. A stable vertex set says that its coordinate projection commutes with the group action; cycle decimation shows that it need not commute with \(Q\). The resulting Schur precision can therefore differ from the compression \(Q_{RR}\). Commuting with the group action and commuting with \(Q\) are distinct conditions.

Three other representation-theoretic questions need additional data. Projective representations and twisted group algebras record a multiplier rather than an ordinary group action; the projective obstruction explains its role. Restriction to a normal subgroup and Clifford's homogeneity theorem concern constituents under a specified subgroup inclusion; this is Clifford theory of group representations, distinct from Clifford algebras. In positive characteristic, the averaging denominator or Gaussian measure used here may be absent. None of these is supplied by real isotypic factorization alone.

For the alternating four-cycle, let \(V=\mathbb R^4\) be the space of vertex fields and let \(\delta _i(j)=\delta _{ij}\) be its coordinate vectors. These are distinct from the oriented edges \(e_i\) and the spectral vectors \(u_a\) of Example 7.19. With \(t,m>0\), put \(a=m^2\) and \(s=1+t+a\). The massive precision gives the quadratic form \[\begin {aligned} q_{t,m}(v)&=v^TQ_{t,m}v,\\ Q_{t,m}&=\begin {pmatrix} s&-1&0&-t\\-1&s&-t&0\\0&-t&s&-1\\-t&0&-1&s \end {pmatrix}. \end {aligned}\] The Euclidean action is \(\mathcal A(v)=q_{t,m}(v)/2\). The factor \(1/2\) belongs to the action; the quadratic form used below is \(q_{t,m}\) itself.

Write \(q=q_{t,m}\) and \(Q=Q_{t,m}\). The unhalved polar form and the half-polar form are respectively \[\begin {aligned} B_q(u,v)&=q(u+v)-q(u)-q(v)\\ &=2u^TQv,\\ b_q(u,v)&=\tfrac 12 B_q(u,v)=u^TQv. \end {aligned}\] Thus \(q(v)=b_q(v,v)\), while \(B_q(v,v)=2q(v)\). This is the polarization dictionary: the symmetric bilinear form called \(B\) by Meinrenken is \(b_q\), not \(B_q\). His Clifford relation uses \(2B\) [meinrenken2013clifford, Definition 2.1, p. 27].

Mathlib's Matrix.toQuadraticForm' constructs this matrix quadratic form. Its underlying LinearMap.BilinMap.toQuadraticMap sends a bilinear map \(b\) to \(v\mapsto b(v,v)\); here \(b(u,v)=u^TQv\). The definition QuadraticMap.polar is unhalved. These generic constructions supply the conventions; the displayed matrix and positivity calculation specify this example.

The 4 coordinates are configuration directions, one per vertex. Positive definiteness therefore describes a form on field space. It supplies no Lorentzian spacetime metric. Changing \(t,m\) changes this form and the Gaussian state, even though the half-turn action stays fixed.

Let \(k\) be a commutative ring, \(M\) a \(k\)-module and \(q:M\to k\) a quadratic form. Use the tensor algebra and the square-quotient convention for the Clifford algebra: \[\operatorname {Cl}(q)=T_k(M)/I_q,\] where \(I_q\) is the two-sided ideal generated by \(v\otimes v-q(v)1\) for all \(v\in M\). The canonical linear map \(\iota _q:M\to \operatorname {Cl}(q)\) satisfies \(\iota _q(v)^2=q(v)1\). Scalars in this formula mean their images under the algebra's structure map. No invertibility of \(2\) is required for this construction.

For every unital associative \(k\)-algebra \(A\), a linear map \(f:M\to A\) extends to a unique unital \(k\)-algebra homomorphism \(\widehat f:\operatorname {Cl}(q)\to A\) if and only if \[f(v)^2=q(v)1_A\qquad (v\in M).\] This is the Clifford lift and its universal property.

Linearity and the square relation give the anticommutator: \[\begin {gathered} \iota _q(u)\iota _q(v)+\iota _q(v)\iota _q(u)\\ =\bigl (q(u+v)-q(u)-q(v)\bigr )1\\ =B_q(u,v)1. \end {gathered}\] Indeed, expand \((\iota _q(u)+\iota _q(v))^2\) and subtract the two squares. This uses no division by \(2\). When \(2\) is invertible, \(b_q=B_q/2\) exists and the relation becomes \(\iota _q(u)\iota _q(v)+\iota _q(v)\iota _q(u)=2b_q(u,v)1\). Conversely that relation with \(u=v\) recovers the square relation by cancellation of \(2\). Thus the two presentations agree in this case. Without that hypothesis, the anticommutator alone need not recover the squares.

Meinrenken gives the bilinear presentation in [meinrenken2013clifford, Definition 2.1, p. 27] and the universal property in [meinrenken2013clifford, Proposition 2.2 and Remark 2.2, p. 29]. His chapter assumes a finite-dimensional vector space over a characteristic-zero field (p. 23); Remark 2.2 includes the ordinary algebra formulation. The direct argument above supplies the stated commutative-ring generality. Mathlib's CliffordAlgebra.ι_sq_scalar, CliffordAlgebra.lift, CliffordAlgebra.lift_unique and CliffordAlgebra.ι_mul_ι_add_swap provide these respective generic square, extension, uniqueness and unhalved-polar statements.

Now take the real configuration form and put \(c_i=\iota _q(\delta _i)\). Write \(\{x,y\}=xy+yx\) for the anticommutator. The matrix entries give \[\begin {aligned} c_i^2&=s1\quad (i=0,1,2,3),\\ \{c_0,c_1\}&=\{c_2,c_3\}=-2\,1,\\ \{c_1,c_2\}&=\{c_3,c_0\}=-2t\,1,\\ \{c_0,c_2\}&=\{c_1,c_3\}=0. \end {aligned}\] These follow from \(q(\delta _i)=Q_{ii}=s\) and \(b_q(\delta _i,\delta _j)=Q_{ij}\). In particular the squares are positive in this convention. The opposite-sign convention would use \(-q\).

For this real form, both the scalar map and \(\iota _q\) are injective. Here is the exterior algebra argument of [meinrenken2013clifford, Proposition 2.1, p. 28], with its operator identities made explicit. On \(\bigwedge V\), let \(\epsilon (v)\omega =v\wedge \omega \), and define contraction by \[d_v(1)=0,\] \[\begin {gathered} d_v(w_1\wedge \cdots \wedge w_r)\\ =\sum _{j=1}^r(-1)^{j-1}b_q(v,w_j)\\ \quad \cdot w_1\wedge \cdots \widehat {w_j}\cdots \wedge w_r. \end {gathered}\] The alternating multilinear expression defines a linear operator on the exterior algebra. Exterior alternation gives \(\epsilon (v)^2=0\). In \(d_v^2\), each pair of removed positions occurs twice with opposite signs and the same scalar coefficient, so \(d_v^2=0\). Separating the first term of \(d_v(v\wedge \omega )\) gives \[d_v\epsilon (v)+\epsilon (v)d_v=b_q(v,v)I.\] Consequently \(C(v)=\epsilon (v)+d_v\) is linear in \(v\) and \(C(v)^2=q(v)I\). The universal property gives a unital algebra map \(\rho :\operatorname {Cl}(q)\to \operatorname {End}_{\mathbb R}(\bigwedge V)\). It satisfies \(\rho (\iota _q(v))1=v\) and \(\rho (r1)1=r1\). Since \(\bigwedge ^1 V=V\) and \(\bigwedge ^0V=\mathbb R\), these equations prove both injections. The contraction identities are also recorded in [meinrenken2013clifford, Section 2.1.3, pp. 25-26]. TauCeti's CliffordAlgebra.ι_injective and CliffordAlgebra.algebraMap_injective give the corresponding injections for an arbitrary quadratic form over a commutative ring with invertible \(2\).

The exterior action just used proves faithfulness on vectors and scalars; it does not assert that \(\rho \) is a faithful algebra representation. Nor does this construction select a physical spinor module, a positive expectation on the Clifford algebra, or a differential operator. The earlier Gaussian expectation acts on polynomial observables of vertex fields; an expectation on this new algebra would require a specified construction.

Let \(k\) be a commutative ring, \(M_1,M_2\) be \(k\)-modules with quadratic forms \(q_1,q_2\), and let \(f:M_1\to M_2\) be linear with \[q_2(f(v))=q_1(v)\qquad (v\in M_1).\] This form-preserving map induces a unique unital \(k\)-algebra map \[\operatorname {Cl}(f):\operatorname {Cl}(q_1)\longrightarrow \operatorname {Cl}(q_2)\] satisfying \(\operatorname {Cl}(f)(\iota _1(v))=\iota _2(f(v))\). This is the underlying Clifford map in isometry transport. It requires no inverse for \(f\). The defining square is:

The top and vertical arrows are linear maps; the bottom arrow is a unital algebra map. The square commutes as a square of linear maps. For composable form-preserving maps \(f,g\), \[\begin {aligned} \operatorname {Cl}(\operatorname {id})&=\operatorname {id},\\ \operatorname {Cl}(g\circ f)&=\operatorname {Cl}(g)\circ \operatorname {Cl}(f). \end {aligned}\] If \(f\) is a linear isomorphism, \(\operatorname {Cl}(f)\) is an algebra isomorphism with inverse \(\operatorname {Cl}(f^{-1})\).

Meinrenken derives this functoriality for bilinear-form-preserving maps in [meinrenken2013clifford, Section 2.2.2, p. 29]. Over his characteristic-zero field, polarization equates that condition with preservation of \(q\). The proof here applies directly to quadratic forms over commutative rings. Mathlib's CliffordAlgebra.map, CliffordAlgebra.map_apply_ι, CliffordAlgebra.map_id, CliffordAlgebra.map_comp_map and CliffordAlgebra.equivOfIsometry provide the extension, generator equation, identity, composition and inverse construction. Their input is a linear map preserving the quadratic form, or a linear equivalence for the last result.

Return to \(q=q_{t,m}\) on \(V=\mathbb R^4\). The same half-turn sends \(\delta _i\) to \(\delta _{i+2}\), with indices modulo \(4\). It satisfies \(P^2=I\) and \(P^TQP=Q\), so \(q(Pv)=q(v)\). Therefore \[\begin {aligned} \Theta &=\operatorname {Cl}(P),\\ \Theta (c_i)&=c_{i+2},\qquad \Theta ^2=\operatorname {id}. \end {aligned}\] The matrix equation follows by shifting both indices of \(Q\) by two: its diagonal, its paired off-diagonal entries \(-1\) and \(-t\), and its zeros are preserved. Composition gives \(\Theta ^2=\operatorname {Cl}(P^2)=\operatorname {id}\). Moreover \(\delta _0\ne \delta _2\), so injectivity of \(\iota _q\) gives \(c_0\ne c_2\). Thus \(\Theta \) is nontrivial and has order exactly two.

The real half-turn sectors \(V_\pm =\ker (P\mp I)\) are orthogonal for \(b_q\). Indeed, \(P^TQP=Q\) gives \(b_q(Pu,Pv)=b_q(u,v)\). For \(u\in V_+\) and \(v\in V_-\), the left side is \(b_q(u,-v)=-b_q(u,v)\). Over \(\mathbb R\) this forces \(b_q(u,v)=0\), and hence \[\iota _q(u)\iota _q(v)=-\iota _q(v)\iota _q(u).\] This is the generic orthogonality consequence CliffordAlgebra.ι_mul_ι_comm_of_isOrtho. The cross-sector vector generators anticommute; the direct sum of sectors does not make them commuting factors of an ordinary tensor-product algebra.

Preservation of the form is essential. Compare \(q_{1,1}\) and \(q_{2,1}\) while keeping the coordinate labels unchanged. At \(\delta _0\), \[q_{1,1}(\delta _0)=3,\qquad q_{2,1}(\delta _0)=4.\] Thus the identity linear map is not form-preserving. Write \(c_i^{(1)},c_i^{(2)}\) for the two sets of generators. A unital real algebra map with \(c_i^{(1)}\mapsto c_i^{(2)}\) would send \((c_0^{(1)})^2=3\,1\) to \((c_0^{(2)})^2=3\,1\), but the target square is \(4\,1\). Its injective scalar map rules this out. This excludes precisely the unchanged-label assignment. It does not exclude other isometries between the positive forms or an abstract isomorphism of their Clifford algebras.

The construction transports a supplied configuration form and its discrete symmetry. It supplies no spacetime derivative, Dirac operator or Lorentzian interpretation. Those structures need additional data and compatibility conditions.

To couple Clifford multiplication to spatial differences, fix an integer \(d\ge 1\) and periods \(N_j\ge 3\) for \(1\le j\le d\). The periodic grid and its scalar fields are \[\begin {aligned} \Lambda &=\prod _{j=1}^d\mathbb Z/N_j\mathbb Z,\\ M&=\prod _{j=1}^d N_j,\qquad H=\mathbb C^\Lambda . \end {aligned}\] Write \(e_j\in \Lambda \) for the unit step in coordinate \(j\). Addition is periodic and the lattice spacing is one. Extend the counting pairing to the Hermitian inner product \[\langle f,g\rangle _H =\sum _{x\in \Lambda }\overline {f(x)}g(x),\] conjugate-linear in the first slot.

The spatial quadratic form is \[q_{\mathrm {sp}}(z)=\sum _{j=1}^d z_j^2 \qquad (z\in \mathbb C^d).\] It is distinct from the Hermitian norm squared \(\sum _j|z_j|^2\) and from the real four-field configuration form. For example, at \(i\varepsilon _1\), where \(\varepsilon _j\) is the standard basis of \(\mathbb C^d\), the quadratic value is \(-1\) and the Hermitian norm squared is \(1\). The grid steps \(e_j\), these complex coordinate vectors, and the 4 configuration coordinates have different carriers.

Choose \(S=(\mathbb C^2)^{\otimes d}\) with its standard tensor basis and product Hermitian inner product. In the Pauli convention, take \[X=\begin {pmatrix}0&1\\1&0\end {pmatrix},\qquad Z=\begin {pmatrix}1&0\\0&-1\end {pmatrix},\] and put \(I=I_2\) and \[\gamma _j=Z^{\otimes (j-1)}\otimes X \otimes I^{\otimes (d-j)}.\] An empty tensor product contributes the scalar identity. Direct multiplication gives \(X^*=X\), \(Z^*=Z\), \(X^2=Z^2=I\) and \(XZ=-ZX\). Ordinary tensor products obey \[\begin {aligned} (A\otimes B)^*&=A^*\otimes B^*,\\ (A\otimes B)(C\otimes E)&=AC\otimes BE. \end {aligned}\] The product formula follows on pure tensors, which span; the adjoint formula follows from the product inner product on such tensors. Thus \(\gamma _j^*=\gamma _j\) and \(\gamma _j^2=I_S\). For \(i<j\), the only anticommuting slot is \(i\), where \(X\) meets \(Z\). Before it both factors are \(Z\); after it at least one factor is \(I\). Hence \[\gamma _i\gamma _j+\gamma _j\gamma _i =2\delta _{ij}I_S.\] Mathlib's Matrix.kronecker, Matrix.mul_kronecker_mul, Matrix.conjTranspose_kronecker and Matrix.one_kronecker_one give the corresponding matrix construction and identities. Iterating them requires identifying the product-index bases.

The complex-linear map \(c:\mathbb C^d\to \operatorname {End}_{\mathbb C}(S)\) defined by \(c(z)=\sum _j z_j\gamma _j\) satisfies \[\begin {aligned} c(z)^2&=\sum _j z_j^2 I_S\\ &\quad +\sum _{i<j}z_i z_j (\gamma _i\gamma _j+\gamma _j\gamma _i)\\ &=q_{\mathrm {sp}}(z)I_S. \end {aligned}\] The Clifford universal property therefore gives a unique unital complex-algebra map \[\begin {aligned} \rho :\operatorname {Cl}(q_{\mathrm {sp}})&\longrightarrow \operatorname {End}_{\mathbb C}(S),\\ \rho (\iota (\varepsilon _j))&=\gamma _j. \end {aligned}\] This makes \(S\) a Clifford module, in the module-map convention. The ordinary-algebra universal property is stated in [meinrenken2013clifford, Proposition 2.2 and Remark 2.2, p. 29]. Its bilinear form is \(b(z,w)=\sum _j z_jw_j\), with relation \(uv+vu=2b(u,v)1\); see [meinrenken2013clifford, Definition 2.1, p. 27]. The displayed square identity is precisely the input required by CliffordAlgebra.lift over \(\mathbb C\).

This chosen module is reducible. The involution \[J=I^{\otimes (d-1)}\otimes X\] commutes with every \(\gamma _j\): its last factor meets either \(I\) or \(X\), and its other factors are identities. Its two eigenspaces are \[S_\pm =(\mathbb C^2)^{\otimes (d-1)} \otimes \mathbb C(1,\pm 1).\] Both are nonzero and have dimension \(2^{d-1}\). Commutation with \(J\) makes them invariant under every generator and hence under the generated Clifford action. We use \(\dim S=2^d\); no minimality or faithfulness of this representation is needed.

Extend the cyclic shift coordinatewise: \[(T_j f)(x)=f(x+e_j).\] Its inverse shifts by \(-e_j\). Reindexing \(y=x+e_j\) gives \[\begin {aligned} \langle T_jf,g\rangle _H &=\sum _y\overline {f(y)}g(y-e_j)\\ &=\langle f,T_j^{-1}g\rangle _H. \end {aligned}\] Thus \(T_j^*=T_j^{-1}\), so \(T_j\) is unitary. The shifts and their inverses commute, since addition in \(\Lambda \) commutes. The centered differences are the endomorphisms \[\nabla _j=(T_j-T_j^{-1})/2.\] They commute and satisfy \(\nabla _j^*=-\nabla _j\). Their source and target are both \(H\), unlike a general vertex-to-edge incidence map.

Set \(K=S\otimes _{\mathbb C}H\), with \(\langle s\otimes f,t\otimes g\rangle _K =\langle s,t\rangle _S\langle f,g\rangle _H\). Identify \(s\otimes f\) with the field \(x\mapsto f(x)s\). The centered Dirac operator is \[D=\sum _{j=1}^d\gamma _j\otimes \nabla _j.\] Equivalently, \[\begin {aligned} (D\psi )(x) &=\frac 12\sum _{j=1}^d\gamma _j\psi (x+e_j)\\ &\quad -\frac 12\sum _{j=1}^d\gamma _j\psi (x-e_j). \end {aligned}\] For \(d=1\), this is \(X\) applied to the ordinary centered difference of a two-component field. Wingate's gauge-covariant centered difference reduces to this difference when all links are one and the spacing is one [wingate2025lattice, Section 9.1, Equation (9.2), p. 19]. The finite grid and tensor module here are specified directly; this construction introduces no Grassmann integration or gauge field. The Lorentz-sign gamma algebra is a different choice of form, scalar field and representation.

For the periodic grid and Clifford module, the centered Dirac operator is skew-adjoint and has square \[\begin {aligned} D^*&=-D,\\ D^2&=I_S\otimes \sum _{j=1}^d\nabla _j^2,\\ -D^2&=\frac 14 I_S\otimes \sum _{j=1}^d(2I_H-T_j^2-T_j^{-2}). \end {aligned}\] In particular \(-D^2=D^*D\) is positive semidefinite. Its complete finite Fourier decomposition has blocks \[\begin {aligned} D(p)&=i\sum _{j=1}^d\gamma _j\sin p_j,\\ -D(p)^2&=s(p)I_S,\\ s(p)&=\sum _{j=1}^d\sin ^2p_j, \end {aligned}\] where \(p_j=2\pi r_j/N_j\) and \(0\le r_j<N_j\). The sine convention agrees with the free propagator in [wingate2025lattice, Section 9.1, Equations (9.6)-(9.7), p. 20] after setting the spacing to one. The finite decomposition and square are proved below.

Mathlib's LinearMap.adjoint and LinearMap.adjoint_inner_right give the finite-dimensional adjoint and its characterizing identity; LinearMap.adjoint_comp reverses composition. The relevant formal inner-product structure is the counting Euclidean space, for example a \(\operatorname {PiLp}(2)\) model. The raw function type's default supremum norm cannot silently replace that structure. The translation and tensor identifications are the direct calculations above.

Mathlib's ZMod.dft, ZMod.dft_apply and ZMod.invDFT_apply provide the one-coordinate Fourier equivalence: its forward sum has a negative exponent and no normalizing factor, while its inverse has a positive exponent and factor \(N^{-1}\). Iterating over coordinates and scaling the forward transform by \(M^{-1/2}\) gives the conventions above. AddChar.wInner_cWeight_eq_boole gives generic character orthogonality for the normalized average inner product. Here the unscaled characters have counting norm \(\sqrt M\); the factor \(M^{-1/2}\) makes them orthonormal. Their explicit enumeration, the shift eigenvalues and the spinor blocks are supplied by this finite proof.

On the periodic grid, let \(e\) be the number of even periods among \(N_1,\ldots ,N_d\). The centered Dirac operator has exactly \(2^e\) zero momentum blocks and \[\dim _{\mathbb C}\ker D=2^{d+e}.\] This is the kernel dimension for the explicitly chosen reducible module \(S\), whose dimension is \(2^d\). The unit-weight graph Laplacian \[L_{\mathrm {nn}}=\sum _{j=1}^d (2I_H-T_j-T_j^{-1})\] has only the zero momentum in its scalar kernel. Moreover \(-D^2\ne I_S\otimes L_{\mathrm {nn}}\) on every permitted grid, including grids with all periods odd. These are finite consequences of the Fourier blocks; the centered-difference convention comes from [wingate2025lattice, Section 9.1, Equations (9.2), (9.6)-(9.7), pp. 19-20].

For \(d=1,N=4\), the alternating scalar field \(a(x)=(-1)^x\) has momentum \(p=\pi \). Its centered difference is zero, whereas \(L_{\mathrm {nn}}a=4a\). Consequently every field \(x\mapsto a(x)v\), \(v\in S\), is killed by \(D\). Together with the constant fields these give the four-dimensional kernel. Write \(K_2=(2I_H-T^2-T^{-2})/4\), so \(-D^2=I_S\otimes K_2\). The two scalar Laplacians have the following weighted graphs:

Left: the 4 unit-weight nearest-neighbor edges. Right: the two-step edges, with vertices rearranged to display the parity classes \(\{0,2\}\) and \(\{1,3\}\). At period \(4\), \(T^2=T^{-2}\), so \(K_2=(I_H-T^2)/2\): the two shift contributions of weight \(1/4\) combine into one undirected edge of weight \(1/2\) for each opposite pair. The alternating field is constant on each right-hand component, explaining its zero energy there.

For \(d=1,N=3\), momentum \(p=2\pi/3\) instead gives \[s(p)=\tfrac 34,\qquad \ell (p)=3.\] Only \(r=0\) is a zero momentum, yet the operators still differ. Thus extra exact zero momenta occur precisely when at least one period is even, whereas the square mismatch always occurs. The excluded case \(N=2\) has \(T=T^{-1}\), making its centered difference identically zero.

The finite kernel dimension above is not a physical species or chirality count. Friedan's Nielsen-Ninomiya theorem concerns free fermions on the infinite lattice \(\mathbb Z^d\) for odd \(d\), with continuous torus momenta. Its Hamiltonian is quadratic, phase invariant, translation invariant and local, with a Hermitian momentum symbol. The locality discussion reduces the Hamiltonian requirement to continuity; the internal-charge projections must be smooth. The zero set is finite and isolated, with specified relativistic Clifford-linear behavior near each zero. Under these hypotheses the total chiral index vanishes separately for each charge [friedan1982nielsen, Sections 1-2, pp. 481-484; Equations (6), (10)-(11)]. A finite sampled grid in arbitrary dimension with the chosen reducible module computes a different quantity. The odd-period example neither contradicts that theorem nor establishes a continuum limit. No Lorentzian model or factorization of the earlier weighted precision is supplied by this centered operator.

A classification of Gaussian backgrounds must retain the maps that transport their observables. Consider the alternating four-cycle with vertices \(V=\mathbb Z/4\mathbb Z\), forward edges \(e_i:i\to i+1\), and weights \((a,b,a,b)\), where \(a,b>0\). Choose \(m>0\) and one of \(R_0=\{0,2\}\), \(R_1=\{1,3\}\) as retained vertices. Write \(x=(a,b,m,R_j)\) and let \(X\) be the set of all these tuples. The tuple specifies a finite Gaussian background, together with its named half-turn.

More explicitly, retain the carrier, counting reference pairing, weight matrix \(W_x=\operatorname {diag}(a,b,a,b)\), and \[L_x=B^TW_xB,\qquad Q_x=L_x+m^2I.\] Here \((B\phi )_i=\phi _{i+1}-\phi _i\). For \(\phi \ne 0\), \[\begin {aligned} \phi ^TQ_x\phi &=\sum _iw_i(\phi _{i+1}-\phi _i)^2\\ &\quad +m^2\|\phi \|^2>0. \end {aligned}\] The normalized Gaussian \(\mu _{Q_x}\) is determined by this precision and coordinate Lebesgue measure. Also retain the measurable coordinate projection \(\pi _{R_j}:\mathbb R^V\to \mathbb R^{R_j}\) and the named action \(\rho (h)\delta _i=\delta _{i+2}\) of \(C_2=\{1,h\}\). The vertex basis \(\delta _i(v)=\mathbf 1_{v=i}\) is distinct from the edges \(e_i\) and the spectral vectors \(u_a\). These structures are specified or determined, rather than additional freely variable coordinates of \(X\).

Allow rotations and reflections of the underlying cycle. Put \[D_4=\langle r,s\mid r^4=s^2=1,\ srs=r^{-1}\rangle ,\] where \(r(i)=i+1\) and \(s(i)=-i\). A reflection reverses the forward orientation. Thus these arrows use the signed endpoint convention and extend the earlier strictly directed exact relabelings. For \(g=r^ks^\varepsilon \), with \(k\in \mathbb Z/4\) and \(\varepsilon \in \{0,1\}\), set \[\begin {aligned} g(i)&=k+(-1)^\varepsilon i,\\ P_g\delta _i&=\delta _{g(i)},\\ \tau _g(i)&=\begin {cases}k+i&\varepsilon =0,\\k-i-1&\varepsilon =1,\end {cases}\\ U_g\eta _i&=(-1)^\varepsilon \eta _{\tau _g(i)}. \end {aligned}\] Here \(\eta _i\) is the edge-field coordinate basis. All indices are modulo \(4\). The 8 vertex maps are distinct because their values at \(0,1\) determine \(k,\varepsilon \).

The edge indexed by \(\tau _g(i)\) has endpoints \(g(i),g(i+1)\), in forward order for a rotation and reverse order for a reflection. Consequently, for every field \(\phi \), \[\begin {aligned} (BP_g\phi )_{\tau _g(i)}&=(-1)^\varepsilon (B\phi )_i,\\ BP_g&=U_gB. \end {aligned}\] If \(k'\) denotes a second group element, endpoint transport and multiplication of the orientation signs give \[P_{k'g}=P_{k'}P_g,\qquad U_{k'g}=U_{k'}U_g.\] For an explicit group-law check, composition of \((k,\varepsilon )\) followed by \((\ell ,\eta )\) gives \[\begin {aligned} (\ell ,\eta )(k,\varepsilon ) &=(\ell +(-1)^\eta k,\eta +\varepsilon ). \end {aligned}\] The second coordinate is taken modulo two. Both coordinate maps are orthogonal, and their inverses are indexed by \(g^{-1}\).

Transport weights by \(W_{gx}=U_gW_xU_g^T\) and the retained set by \(R\mapsto gR\). The signs cancel in the diagonal weight matrix. The weights swap exactly when \(k+\varepsilon \) is odd, while \(R_j\mapsto R_{j+k\bmod 2}\). In particular, \[\begin {aligned} r(a,b,m,R_j)&=(b,a,m,R_{j+1}),\\ s(a,b,m,R_j)&=(b,a,m,R_j). \end {aligned}\] The family is closed under these maps, with mass unchanged. Using \(BP_g^T=U_g^TB\) gives \[\begin {aligned} L_{gx}&=B^TU_gW_xU_g^TB=P_gL_xP_g^T,\\ Q_{gx}&=P_gQ_xP_g^T. \end {aligned}\] Moreover \(g(i+2)=g(i)+2\), so every arrow commutes with the same named half-turn: \(P_g\rho (h)=\rho (h)P_g\).

Form the action groupoid \(D_4\ltimes X\) by setting \[\operatorname {Hom}(x,y)=\{g\in D_4:g x=y\}.\] The group element remains part of an arrow. Compose \(g:x\to y\) and \(k:y\to z\) as \(kg:x\to z\); the action law gives its target. The identity is \(1:x\to x\), the inverse is \(g^{-1}:gx\to x\), and the two inverse identities and associativity follow from the group laws. Thus every arrow is invertible, as in [stacks2017stacks, Definition 4.2.5], Tag 0018. The source, target and composition follow the set-level pattern in [stacks2017stacks, Lemma 78.15.1], Tag 0444; that lemma is stated for algebraic spaces, whereas the axioms here were checked directly for sets. \(X\) is infinite, but each orbit and its restricted groupoid is finite.

The stabilizer of the weights at \(a\ne b\) is \(\{1,r^2,rs,r^3s\}\cong C_2\times C_2\); preserving weights requires \(k+\varepsilon \) even. The stabilizer of either retained partition is \(\{1,r^2,s,r^2s\}\cong C_2\times C_2\), since preserving the partition requires \(k\) even. Intersecting these conditions gives the following automorphism groups. When \(a\ne b\), \[\operatorname {Aut}(x)=\{1,r^2\}\cong C_2.\] When \(a=b\), \[\begin {aligned} \operatorname {Aut}(x)&=\{1,r^2,s,r^2s\}\\ &\cong C_2\times C_2. \end {aligned}\] At equal weights the first condition disappears. In each four-element subgroup the two indicated generators are distinct commuting involutions. No further condition comes from the named half-turn, which commutes with all 8 elements. The kernel \(\{1,r^2\}\) of the parameter action is retained as distinct arrows: \(P_{r^2}\delta _0=\delta _2\ne \delta _0\).

For unequal weights the orbit consists of both weight orders and both partitions, at fixed mass. Write \(x_j=(a,b,m,R_j)\) and \(y_j=(b,a,m,R_j)\). The drawing shows selected generator arrows; it does not draw all arrows or isotropy loops.

At \(a=b\), put \(z_j=(a,a,m,R_j)\). There are two objects, each with automorphism group \(C_2\times C_2\):

The generators \(s,r\) reach every displayed object; there are exactly 4 objects when \(a\ne b\) and two when \(a=b\). It follows in both directions that the coarse isomorphism classes are \((m,\{a,b\})\). The coarse orbit set forgets the isotropy just computed, as in Example [fgap-000P]. On the strict slice \((1,t,m,R_0)\), isomorphism requires the same mass and \(\{1,t\}=\{1,t'\}\), whose products give \(t=t'\). Thus reciprocal parameters are not identified there unless \(t=1\). This is a set-level groupoid, within the boundary before stacks.

Mathlib's CategoryTheory.ActionCategory constructs the generic action category; a group action makes it a groupoid. CategoryTheory.ActionCategory.stabilizerIsoEnd identifies the stabilizer monoid with endomorphisms, which are invertible in the group case. MulAction.orbitRel and MulAction.stabilizer provide the generic coarse relation and subgroup. DihedralGroup.card gives 8 elements at \(n=4\), and DihedralGroup.center_eq_closure_of_even_ne_two gives the central half-turn. In its constructor convention our \(r^ks\) corresponds to \(\texttt {sr}(-k)\), while \(r^k\) corresponds to \(\texttt {r}(k)\). These declarations support the group constructions; the particular parameter action, signed incidence and stabilizer intersections are the calculations above.

On the enriched four-cycle groupoid, the trace ratio, with the cycle values, is a scalar on isomorphism classes, while covariance and effective precision are transported operators on their specified field spaces. For \(x=(a,b,m,R)\) write \(C_x=Q_x^{-1}\). If \(g:x\to y\), then \[\begin {aligned} C_y&=P_gC_xP_g^T,\\ (P_g)_*\mu _{Q_x}&=\mu _{Q_y},\\ \langle F\rangle _{Q_y}&=\langle F\circ P_g\rangle _{Q_x}. \end {aligned}\] The last identity holds for Borel \(F\in L^1(\mu _{Q_y})\); integrability is equivalent to that of \(F\circ P_g\) in the source law. These statements use the inverse-precision covariance and Gaussian transport, including their normalization.