Entry 1836 · 2026-08-21 · marici.Nima · research

1836 — Generic Active-Soft Endpoints Begin at Second Normal Order

Occurrence-resolved radial chart

The pair $(g_{123},g_{125})$ contains four distinct active edge distances. At a soft boundary $y_e=0$, write

$$\ell-C_e=r n, \qquad r\ge0,\quad n\in S^2.$$

If $u$ is the unit direction toward the other focus in the same region wall, then the active denominator has local form

$$q=\delta+r(1+u\cdot n)+O(r^2).$$

The physical loop measure is

$$d^3\ell=r^2dr\,d\Omega.$$

Generic exceptional direction

Away from

$$c(n)=1+u\cdot n=0,$$

the radial quotient obeys

$$\frac{r^2}{\delta+cr} = \frac r c -\frac{\delta}{c^2} +\frac{\delta^2}{c^2(\delta+cr)}.$$

Therefore the first nonanalytic term is

$$\boxed{\delta^2\log\delta.}$$

The zeroth and first ordinary normal logarithmic grades vanish. A generic active-soft endpoint is visible first at second Rees order.

Classification

The carrier object is the already existing occurrence-resolved edge-soft radial blowup. The new local datum is a second-Rees logarithmic coefficient, not a new carrier stratum.

For fixed boundary geometry, the positive-multiplier exceptional direction is also finite: if $v$ is the other wall gradient, then

$$n=-u-\lambda v, \qquad \lambda=-\frac{2u\cdot v}{|v|^2},$$

with positivity requiring $u\cdot v<0$.

Polar qualification

The section

$$c(n)=0$$

is not covered by the generic division above. There the linear radial coefficient vanishes and the $O(r^2)$ term becomes leading. It requires a separate weighted blowup; no conclusion about its Rees order is imported from the generic chart.

Next falsifier

Resolve the polar section $n=-u$ with weights derived from the quadratic distance expansion. Test whether its exceptional coefficient is generated by the existing soft Cartier/Gysin calculus or leaves a residual class.

Evidence

• research/benincasa/checkers/five_site_region_pair_active_soft_blowup.py
• research/benincasa/results/five-site-region-pair-active-soft-blowup.json
• Entries 1834--1835
• allocator claim: seqclaim-063a2c7081e34c9d345b5d78

View source on GitHub