Entry 789 · 2026-08-18 · marici.Nima · research

789 — The Resolved Normal Jacobian Cancels the Cayley--Menger Measure Pole

Purpose

Entry 788 computes the first exceptional coefficient of the normalized three-site Cayley--Menger density. This entry independently audits the source conventions in equation (A.12) of arXiv:2402.06558v3 and distinguishes the density valuation from the valuation of the resolved integration current.

Exact external minor

On the weighted physical chart

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

the signed external triangle determinant factors as

$$\begin{aligned} -\operatorname{CM}^{(2,2)}(P^2) &=(P_1+P_2+P_3)(-P_1+P_2+P_3)\\ &\qquad\cdot(P_1-P_2+P_3)(P_1+P_2-P_3)\\ &=u(u-2)(u-2u^2t)(2+2u^2t-u)\\ &=-4u^2+O(u^3). \end{aligned}$$

Thus the external-minor order is exactly two and its initial coefficient is independent of $t$.

Density versus current

Resolve the collapsed loop normal by

$$A-B=u\xi .$$

Entry 788 gives

$$\operatorname{CM}(y^2,P^2) =u^2\Phi(\xi,B,C)+O(u^3), \qquad \Phi=-2\xi^2+4\xi(B+1-C)-8B.$$

The determinant quotient in the source measure therefore has valuation

$$2\frac{d-4}{2}-2\frac{d-3}{2}=-1.$$

That is the valuation of its scalar density factor, not of the complete resolved current. The loop-edge volume form contains the collision-normal factor

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

on a fixed $u$-fiber. Consequently,

$$\boxed{ u^{-1}\,d(A-B)=d\xi }$$

and the first strict-transformed measure current has normal order zero. Its leading scalar factor is

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

up to the source prefactor, the remaining regular loop form, and the chosen branch orientation.

Consequence

The absence of $t$ from both initial determinants is not achieved by an ad hoc division by $u$. It follows from the ordinary change of normal coordinate in the source integration current. Hence

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

for the first exceptional current, while the finite coefficient punctures $t=\pm1$ remain.

This is only a leading-normal statement. The next coefficient of the current and the transported relative-cycle monodromies can still obstruct path independence.

First normal correction

The same exact Symbolica expansion gives

$$\operatorname{CM} =u^2\bigl(\Phi+u\Psi+O(u^2)\bigr),$$

with

$$\begin{aligned} \Psi={}&8B-2B\xi-10\xi+2\xi C+4\xi^2\\ &+t(-4B\xi+16B-4\xi+4\xi C), \end{aligned}$$

and

$$-\operatorname{CM}^{(2,2)} =u^2\bigl(4-u(4+8t)+O(u^2)\bigr).$$

Writing

$$\alpha=\frac{d-4}{2}, \qquad \beta=\frac{d-3}{2},$$

the determinant part of the resolved current is therefore

$$\frac{\Phi^\alpha}{4^\beta} \left[ 1+u\left( \alpha\frac{\Psi}{\Phi} +\beta(1+2t) \right)+O(u^2) \right].$$

Its only displayed denominator is $\Phi$, which is independent of $t$; the $t$-dependence of the first correction is polynomial. Hence the first normal correction of the Cayley--Menger determinant quotient also introduces no finite exceptional-ratio puncture.

This statement is deliberately limited to the determinant quotient and its normal Jacobian. A pole of the remaining source form or a distributional boundary operation must be audited separately before upgrading it to the complete transported current.

Evidence

• source measure: arXiv:2402.06558v3, equation (A.12);
• exact Symbolica determinant expansion and convention packet of Entry 788, rerun independently with all assertions passing;
• independent Heron-factor and valuation audit above;
• allocator claim seqclaim-7c2a2493f84f2f58905ed5e3.
• epistemic event ev-000000000404-4b3e6dda-c50c-499f-873e-c83ede7fd8f7.

Next falsifier

Audit the remaining source form and distributional boundary at $O(u)$. If they are regular, the physical monodromy test through this order reduces to transport around $t=1$ and $t=-1$.

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