Remark. How the selector changes induced interactions [fdip-0062]

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A comparison of selectors should hold the full interaction fixed. On the alternating cycle of Example [fdip-0061], now use \[\mathcal V_{\mathrm {tot}}(\phi )=\frac 1{24}\sum _{v=0}^3\phi _v^4.\] For opposite retained vertices \(R=\{0,2\}\), add \((x_0^4+x_2^4)/24\) to that example's first coefficient. The eliminated covariance is \(s^{-1}I_2\). For adjacent retained vertices \(R=\{0,1\}\), the eliminated coordinates are \((\phi _2,\phi _3)\), and \[\begin {aligned} D&=\begin {pmatrix}s&-1\\-1&s\end {pmatrix},\\ D^{-1}&=\frac 1{s^2-1}\begin {pmatrix}s&1\\1&s\end {pmatrix},\\ \mu (x)&=\frac {t}{s^2-1} \begin {pmatrix}1&s\\s&1\end {pmatrix} \begin {pmatrix}x_0\\x_1\end {pmatrix}. \end {aligned}\] Here \(s>1\), so the denominator is positive. These formulas follow from \(Q_{UR}=\left (\begin {smallmatrix}0&-t\\-t&0\end {smallmatrix}\right )\) and the conditional rules.

The adjacent first coefficient is explicitly \[\begin {aligned} a_{\mathrm {adj}}(x)={}&\frac {x_0^4+x_1^4}{24}\\ &+\frac 1{24}\sum _{j=2,3} \bigl (\mu _j(x)^4+6\mu _j(x)^2v+3v^2\bigr ),\\ v={}&\frac {s}{s^2-1}. \end {aligned}\] In particular its constant term is \(s^2/[4(s^2-1)^2]\), whereas the opposite selector gives \(1/(4s^2)\). They are unequal for \(s>1\). The adjacent propagator also has a nonzero cross entry \(1/(s^2-1)\) between eliminated vertices, which contributes pairings between their quartics in the variance coefficient. Such a cross entry is absent for opposite elimination. Thus the same precision and full interaction yield different conditional contraction rules and induced vertices when the selector changes.

For a restricting selector, \(PQ=QP\) makes the cross block \(K\) zero. The Gaussian law then factors between \(X\) and \(Y\), and the eliminated conditional mean is zero. Any remaining coupling between the two subspaces comes from \(\mathcal V\), not from a Gaussian cross term. In particular, if \(\mathcal V(x+y)=\mathcal V_X(x)+\mathcal V_Y(y)\), then \[F_\lambda (x)=\lambda \mathcal V_X(x) -\log \mathbb E_{N(0,D^{-1})}e^{-\lambda \mathcal V_Y}.\] The second term is a formal scalar independent of \(x\). If the interaction has mixed terms, new retained vertices can still be induced even though the Gaussian selector restricts. Group-algebra selectors restrict under the stated equivariance hypothesis; a subrepresentation selector need not.

These formulas answer the finite perturbative question: the propagator and induced interaction depend on the supplied background and selector. They do not establish convergence of the formal series, a continuum interacting measure, or a physical mechanism that selects the background.