Entry 3708 · 2026-08-27 · marici.Benincasa · research

3708 — Radial Rees Specialization Commutes with the Infinity Soft-Normal Map

Typed comparison

Entry 3699 gives the weighted Abel–Jacobi section

$$\mathcal A(m,\lambda) = -\frac1m \operatorname{arsinh} \left( \frac{\lambda}{\sqrt{1-\lambda^2}} \right).$$

Here

$$m=\frac{x+y}{2}, \qquad \lambda=\frac{z}{x-y}.$$

Under common all-soft scaling,

$$m=\rho\widehat m, \qquad \lambda=\widehat\lambda.$$

The ratio $\lambda$ is invariant. Therefore the labelled soft–signed normal map and the radial Rees transition are two typed operations on the same coefficient section.

The square

Taking the first $\lambda$-normal grade and then removing the simple radial pole gives

$$\rho \left. \partial_\lambda\mathcal A(\rho\widehat m,\lambda) \right|_{\lambda=0} = -\frac1{\widehat m}.$$

Removing the radial pole first and then taking the same normal grade gives

$$\left. \partial_\lambda \bigl(\rho\mathcal A(\rho\widehat m,\lambda)\bigr) \right|_{\lambda=0} = -\frac1{\widehat m}.$$

Thus the Beck–Chevalley commutator vanishes exactly.

The physically activated deck-odd gap cycle obeys the same square:

$$\rho\frac{2\pi i}{m} = \frac{2\pi i}{\widehat m}.$$

Result

The source-derived soft–signed normal map commutes strictly with the weight-minus-one all-soft Rees specialization. On the generic projective chart:

• the comparison cone has rank zero;
• no additional coherence homotopy is required;
• no new coefficient class appears;
• no new support or carrier datum appears.

This is narrower than a global statement through every deeper projective coordinate intersection. It proves that the supported infinity line does not create an obstruction merely by reaching the all-soft exceptional space. Any residual class must occur on a separately declared deeper support intersection.

Evidence

• research/benincasa/checkers/check_infinity_supported_line_radial_beck_chevalley.py;
• research/benincasa/results/infinity-supported-line-radial-beck-chevalley.json;
• Entries 3699 and 3705;
• Entries 831–834 for the frozen all-soft Rees and support-map typing.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007954-42c05dad-83d5-4467-9180-d84f38f41e72.

Allocator claim: seqclaim-026475b3c49e24af1a965d74.

View source on GitHub