What the structure fixes and what it leaves free [fdip-009G]
What the structure fixes and what it leaves free [fdip-009G]
Which features follow from a chosen finite structure, and which remain additional inputs?
Part VII compared the discrete physics with a continuous space that was supplied. The supplied-circle limit controls one fixed Fourier mode. Part XIII has since built cochains and their Hodge theory, a distance read from a commutator, and exact coarse-graining of a hierarchical precision. Whether the physics of continuous space comes back at large scales without being supplied can now be asked. Its answer is left to a separate plan. This section does not answer it, and it draws no continuum limit.
Wilson's finite groups stay first. The tested claims are the two finite maps of Proposition [fdip-00A0]: conjugation \(SU(2)\to SO(3)\) and the quotient \(2T\to A_4\), both with kernel \(\{1,-1\}\). Those maps fix the named kernels. The larger proposals remain open tests. The absence of a construction is not a no-go theorem and not a physical identification.
Several programmes outside this forest have not been read: a spectral derivation of the Standard Model, causal sets, loop quantum gravity, and the string landscape. This note does not record what they fix or leave free.
The Euler product, for \(\operatorname {Re}s>1\), is the product over primes not dividing \(2\det Q\) of \(\,(1-\hat Z_p(Q)\,p^{-s})^{-1}\). It equals \(L(s,\chi _D)\) times the product, over primes dividing \(2\det Q\) but not \(D\), of \(\,(1-\chi _D(p)\,p^{-s})\). The Gaussian induced action takes the weight to be half the logarithm of a determinant. The cycle scaling sends \((\log \det Q_n)/2-n\log n\) to a limit on the cycle. These are not the same object. This note does not identify the Euler product with that logarithm or with that limit.
The operator-algebra notes replace a supplied conditional expectation by the composition law for nested expectations (Theorem [fdip-008U]), and they replace a supplied fixed-point algebra by the image of the average (Theorem [fdip-008X]). The crossing block sets the reflection rank (Theorem [fdip-008W]). None of these is a continuum reconstruction.
On every square torus the Betti numbers are \((1,2,1)\) (Theorem [fdip-008Z]), and the harmonic cochains realize them (Theorem [fdip-0090]). The ordered compositions of exterior multiplication and contraction span the endomorphisms of the exterior algebra (Theorem [fdip-0092]). The cup product obeys the graded Leibniz rule and is not graded-commutative on cochains (Theorem [fdip-0093]). The note does not claim that the resulting Clifford representation is injective.
The Connes distance is computed on the stated finite triples (Theorem [fdip-0095]). The edge-source triple makes the distance the graph distance when each vertex has at most one incoming edge, and on one grid it lies strictly between the Euclidean distance and the graph distance (Theorem [fdip-0096]). On a triangle-free graph the gauge-dependent part of the fourth power of the massless covariant Laplacian is 8 times the sum of \(\cos \varphi _c-1\) over the undirected 4-cycles (Theorem [fdip-00A7]). The comparison of that trace with the logarithm of a massive determinant remains speculative. Neither finite trace is identified with a continuum limit.
Nested residue classes diagonalize a hierarchical precision (Theorem [fdip-0099]). Block means of that precision keep a hierarchical second-moment matrix (Theorem [fdip-009A]). For geometric coefficients, relevance of the top coefficient holds only for \(0<\lambda <1\). A hierarchical Gaussian has site fourth moment \(3\sigma ^4\) (Theorem [fdip-009B]). No non-Gaussian hierarchical fixed point is proved. These matrix relations do not by themselves produce a Gaussian measure on functions on the \(p\)-adic integers.
Surface gluing of the finite-group state sum is convolution of class functions (Theorem [fdip-00A8]). That convolution is not identified with the spatial transfer operator. On a finite incidence presentation, invariant tensors count the gauge orbits (Theorem [fdip-00A9]). One square in \(M_4(\mathbb C)\) cuts out 6 contexts with no global point, while each context separately has a character (Theorem [fdip-009E]). Whether that obstruction organises the selectors remains open (§ [fdip-009D]). Subgroup action groupoids form a groupoid-valued presheaf on the opposite of the subgroup poset (Theorem [fdip-009F]). No stack is proved.