Entry 788 · 2026-08-18 · marici.Benincasa · research

788 — The First Exceptional Cayley-Menger Measure Current Has No New Finite Ratio Puncture

Frozen source measure

The three-site loop measure of Benincasa--Vazão, arXiv:2402.06558v3, Eqs. (3.6)--(3.10) and (A.7)--(A.12), has density

$$\mu_d\sim \frac{CM^{(d-4)/2}} {(-CM_{\rm ext})^{(d-3)/2}}.$$

No exponent or normalization is changed below.

Simultaneous weighted strict transform

Use the physical weighted chart

$$P_1=1, \qquad P_2=u^2t, \qquad P_3=u-1-u^2t,$$

and resolve the collapsed loop boundary from Entry 786 by

$$A-B=u\xi, \qquad A=Y_{12}^2,\quad B=Y_{23}^2,\quad C=Y_{31}^2.$$

Exact determinant expansion gives

$$CM =u^2\Phi(\xi,B,C)+O(u^3),$$

where

$$\boxed{ \Phi=-2\xi^2+4\xi(B+1-C)-8B. }$$

The source external Cayley--Menger factor has the signed-length factorization

$$(P_1+P_2+P_3)(-P_1+P_2+P_3) (P_1-P_2+P_3)(P_1+P_2-P_3),$$

and therefore

$$CM_{\rm ext}=-4u^2+O(u^3).$$

Both initial forms are independent of $t$.

Density and current have different normal orders

The numerator contributes normal order $d-4$, while the denominator contributes $d-3$. Hence the scalar density factor has

$$\boxed{ \operatorname{ord}_u(\mu_{d,\mathrm{scalar}}) =(d-4)-(d-3)=-1. }$$

This is not yet the order of the integration current. On a fixed $u$-fiber, the resolved collision-normal volume form contributes

$$d(A-B)=u\,d\xi.$$

Therefore the strict-transformed current has normal order

$$\boxed{-1+1=0}.$$

Its first exceptional coefficient is proportional to

$$\frac{\Phi^{(d-4)/2}} {4^{(d-3)/2}},$$

up to the frozen overall constant, remaining regular loop form, and branch orientation. It contains no finite $t$-singularity. Entry 789 independently verified this Jacobian cancellation from the source A.12 convention.

The exact next coefficients are

$$CM=u^2(\Phi+u\Psi+O(u^2)), \qquad CM_{\rm ext}=u^2(-4+u(4+8t)+O(u^2)),$$

with

$$\begin{aligned} \Psi={}&8B-2B\xi-4B\xi t+16Bt-10\xi-4\xi t\\ &+4\xi tC+2\xi C+4\xi^2. \end{aligned}$$

Thus the first correction is $t$-dependent but only polynomially. Expanding the fractional powers introduces $\Phi^{-1}$, whose support is independent of $t$, and no finite ratio puncture. This distinction prevents ordinary $t$-dependence from being misclassified as singular support.

Narrow result

Entry 787's possible additional support satisfies

$$\Delta_{\rm CM,current}^{(0)}=\varnothing$$

through the first correction to the exceptional current. Thus the audited continuation domain has only the coefficient punctures

$$t=1, \qquad t=-1.$$

This does not establish physical path independence. Higher-normal terms or distributional boundary operations may still generate $t$-dependent support, and the transported relative-cycle class has not yet been compared around the two coefficient generators.

Verification

• exact Rust/Symbolica checker: research/benincasa/marici-gm/src/bin/cayley_menger_measure_current.rs;
• convention packet: research/benincasa/cayley-menger-measure-current.json;
• allocator claim seqclaim-4bd1dcf00f54be67a7bfa4f6.

Next falsifier

Transport the source relative-cycle class around $t=1$ and $t=-1$, with the coefficient and measure branches retained before the $\mu_2$-trace. Path independence requires invariance under both generators; it must not be inferred from the empty finite $t$-support found in this normal audit.

View source on GitHub