Entry 1860 · 2026-08-21 · marici.Nima · research

1860 — The Active-Soft Second-Rees Coefficient Has One Physical Zero Ray

Representative endpoint

Choose the occurrence (y_2=0). On the two active walls,

$$y_4=-3t, \qquad y_5=-3t-y_3.$$

Write

$$\lambda=-t>0, \qquad x=\frac{y_3}{\lambda}, \qquad a=\frac{y_1}{\lambda}.$$

Positivity gives

$$0<x<3, \qquad a>0.$$

Exact source coefficient

Restricting the exact ten-term OFPT coefficient of Entry 1854 gives

$$\boxed{ C_{y_2=0} = \frac{2}{5}\lambda^{-8} \frac{x^2-7x+11}{ \begin{aligned} &(2x-5)(a-1)(x-1)(x-2)\\ &\quad\cdot(x+2)(x-a-2)(x-4)(x-5) \end{aligned}} }.$$

All denominator factors are restrictions of already frozen residual OFPT walls. Inside the positive (x)-chamber, the fixed-(x) poles are

$$x=1,\qquad x=2,\qquad x=\frac52,$$

with the additional shape-dependent walls (a=1) and (a=x-2) when compatible with (a>0).

Numerator zero

The numerator roots are

$$x_\pm=\frac{7\pm\sqrt5}{2}.$$

Only

$$\boxed{ x_-=\frac{7-\sqrt5}{2} \approx2.38196601125 }$$

lies in (0<x<3). At a generic value of (a), no denominator vanishes there. Hence this is a genuine source-numerator cancellation ray, not carrier support or a singularity.

Consequence for Entry 1836

Away from the residual OFPT walls, the polar section (c=0), and the single numerator-zero ray, the second-normal coefficient multiplying

$$\delta^2\log\delta$$

is nonzero. The first nonanalytic active-soft term is therefore genuinely present on a Zariski-open part of the supported endpoint locus.

Architectural conclusion

The endpoint exhibits all three distinctions simultaneously:

• carrier support from the existing soft and active-wall intersection;
• coefficient poles from existing residual source facets;
• a coefficient zero generated by source numerator cancellation.

No new carrier datum appears.

Next falsifier

Transport the numerator-zero ray through the four active-soft occurrences and the five cyclic wall-pair charts. Determine its labelled orbit and test whether the zero is covariant or depends on the chosen endpoint frame.

Evidence

• research/benincasa/checkers/five_site_region_pair_active_soft_source_coefficient.py
• research/benincasa/results/five-site-region-pair-active-soft-source-coefficient.json
• Entries 1836, 1854, and 1859
• allocator claim: seqclaim-b3185750e6e1d76fd11295ff

View source on GitHub