Proposition. Conditional propagators and induced vertices [fdip-0060]
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  • November 1, 2026

For an interacting selected background, put \(n=\dim H\) and \(r=\operatorname {rank}P\), and choose orthonormal basis maps \(B:\mathbb R^r\to X\) and \(B_\perp :\mathbb R^{n-r}\to Y\). Thus a full field is \(\phi =Bx+B_\perp y\); in this proposition \(x,y\) denote coordinates. Set \[\begin {aligned} A&=B^TQB,&K&=B^TQB_\perp ,\\ D&=B_\perp ^TQB_\perp >0,&S&=A-KD^{-1}K^T. \end {aligned}\] The eliminated coordinates, conditionally on \(x\), have law \[y=\mu (x)+\eta ,\qquad \mu (x)=-D^{-1}K^Tx,\qquad \eta \sim N(0,D^{-1}).\] In particular the full conditional mean and noise covariance are \[h(x)=Bx+B_\perp \mu (x),\qquad C=B_\perp D^{-1}B_\perp ^T.\] The map \(h\) is the harmonic extension: \(B^Th(x)=x\) and \(B_\perp ^TQh(x)=0\). It is the unique minimizer of the quadratic energy among fields with these retained coordinates.

Write \(F_\lambda (x)=F_{b,\mathcal V}(\lambda ,Bx)\). Then \[\begin {aligned} a(x)&=\mathbb E\mathcal V(h(x)+B_\perp \eta ),\\ b(x)&=\operatorname {Var}\bigl (\mathcal V(h(x)+B_\perp \eta )\bigr ),\\ F_\lambda (x)&=\lambda a(x)-\frac {\lambda ^2}{2}b(x) +O(\lambda ^3). \end {aligned}\] Thus external mean factors in the conditional moment expansion use \(h(x)\), and each internal pairing of eliminated coordinates uses \(D^{-1}\). These are the background-dependent Feynman rules for this fiber integral. In full-field components the internal pairing is \(C\); the eliminated mean \(\mu \) alone omits the retained part of \(h\). For a coordinate selector retaining \(R\) and eliminating \(U=V\setminus R\), the formulas become \[\mu (x)=-Q_{UU}^{-1}Q_{UR}x,\qquad \operatorname {Cov}(\eta )=Q_{UU}^{-1}.\]

Proof. 

  • November 1, 2026

Orthogonality preserves the volume and gives \[\begin {aligned} \langle \phi ,Q\phi \rangle &=x^TAx+2x^TKy+y^TDy\\ &=x^TSx+(y+D^{-1}K^Tx)^TD(y+D^{-1}K^Tx). \end {aligned}\] The second term is minimized uniquely at \(y=\mu (x)\). Normalizing its density proves the conditional law at every \(x\). Writing \(v_x(y)=Bx+B_\perp y\), it also gives \[\begin {aligned} c_D&=\frac {(2\pi )^{(n-r)/2}}{\sqrt {\det D}},\\ \int _{\mathbb R^{n-r}}&e^{-\frac 12\langle v_x(y),Qv_x(y)\rangle } e^{-\lambda \mathcal V(v_x(y))}\,dy\\ &=c_De^{-\frac 12x^TSx}e^{-F_\lambda (x)}. \end {aligned}\] The last equality means equality after integrating each polynomial coefficient. Hence it retains the scalar and degree-zero terms as well as the induced polynomial vertices.

The moment series starts with \(1-\lambda a+\lambda ^2\mathbb E[\mathcal V^2\mid x]/2+O(\lambda ^3)\). Applying \(-\log (1+z)=-z+z^2/2+O(z^3)\) gives the displayed variance coefficient. In each moment, expand around \(h(x)\). Odd centered Gaussian moments vanish; every even moment is a sum over pairings, with one \(D^{-1}\) entry per pair in eliminated coordinates. This is [etingof2024mathematical, v2, Theorem 3.1 and Example 3.2, p. 29] applied to the positive form \(D\). For a quartic \(\mathcal V(\phi )=T(\phi ,\phi ,\phi ,\phi )/24\), the first coefficient has degrees \(4,2,0\). In the variance, pairings with no contraction between the two copies cancel, leaving degrees among \(6,4,2,0\), as shown in Proposition [fdip-0046]. All these claims concern finite polynomial coefficients.

For example, in full vertex coordinates with the symmetric four-tensor \(T_{ijkl}\) and \(h=h(x)\), Wick's three pairings give \[\begin {aligned} 24a(x)={}&T(h,h,h,h) +6\sum _{i,j,k,l}T_{ijkl}h_i h_j C_{kl}\\ &+3\sum _{i,j,k,l}T_{ijkl}C_{ij}C_{kl}. \end {aligned}\] The covariance \(C\) can be singular on \(H\): it is supported on \(Y\), and the formula follows by applying Wick to the nonsingular \(D^{-1}\) in eliminated coordinates. No nondegeneracy on all of \(H\) is needed for this expression.

TauCeti's integral_mul_mul_mul_multivariateGaussian proves the three-pairing identity for 4 centered Gaussian coordinates, allowing repeated indices and any positive-semidefinite covariance. Setting that covariance to \(C\) supplies the fourth-moment step here. Contraction with \(T\), the formal logarithm and the higher-moment pairing arguments are the additional deductions above.

The massive lattice analogue is [friedli2017statistical, Theorem 8.26, p. 399]: on a finite domain of \(\mathbb Z^d\), the conditional field has the massive harmonic mean determined by its boundary values and a killed-walk Green covariance. That theorem uses its normalized lattice Laplacian; it does not supply the arbitrary-projector argument above. The massless finite-domain analogue is Theorem 8.17, p. 391 of the same source.

A formal conditional-distribution counterpart in TauCeti is condDistrib_multivariateGaussian. It assumes a positive-semidefinite joint covariance and a positive-definite observed covariance block, and identifies the regular conditional law with an explicit Gaussian kernel almost everywhere in the observed marginal. To specialize, order the coordinates as \(Y\oplus X\) and use \(G=Q^{-1}>0\). From \(QG=I\), \[\begin {aligned} DG_{YX}+K^TG_{XX}&=0,\\ DG_{YY}+K^TG_{XY}&=I. \end {aligned}\] Therefore \(G_{YX}G_{XX}^{-1}=-D^{-1}K^T\) and \(G_{YY}-G_{YX}G_{XX}^{-1}G_{XY}=D^{-1}\), the theorem's mean and covariance formulas. Its almost-everywhere identification is distinct from the explicit density kernel used here at every \(x\). The perturbative logarithm and vertex calculations are proved above.