Entry 1861 · 2026-08-21 · marici.Nima · research

1861 — Active-Soft Numerator Zeros Form Two Regular (C_5) Orbits

Four endpoint types

Apply Entry 1860's exact numerator to all four active-soft occurrences of the ordered representative ((g_{123},g_{125})). After extracting the common positive scale and writing the remaining radial ratio as (xin(0,3)), the four numerator polynomials are

$$\begin{array}{c|c|c} \text{soft occurrence}&N(x)&\operatorname{disc}N\\ \hline y_2=0&x^2-7x+11&5\\ y_3=0&x^2-5x+11&-19\\ y_4=0&x^2-x+5&-19\\ y_5=0&x^2+x-1&5. \end{array}$$

Thus the two middle endpoint types have no real numerator zero.

Physical zero rays

The two physical roots are

$$y_2=0:\qquad x=\frac{7-\sqrt5}{2},$$

and

$$y_5=0:\qquad x=\frac{\sqrt5-1}{2}.$$

The other root of each quadratic lies outside (0<x<3).

This asymmetry is legitimate occurrence data: the ordered region pair and its source residue orientation distinguish the four endpoint positions. It must not be erased by declaring all soft endpoints equivalent before calculation.

Cyclic assembly

Each endpoint type has a free five-element cyclic orbit. Therefore the two zero-bearing types assemble as

$$\boxed{ 2\mathbb Q[C_5], }$$

with character

$$\boxed{ (10,0,0,0,0). }$$

Among the twenty active-soft endpoint occurrences, exactly ten carry one physical numerator-zero ray and ten carry none.

Classification

These rays are zeros of the sector-specific source coefficient. They are not singular support, carrier walls, or new incidence generators. Their cyclic covariance confirms that the cancellation is source-defined rather than a coordinate accident.

Next falsifier

At one zero-bearing endpoint, expand the exact source coefficient transversely to the numerator-zero ray. Determine whether the vanishing is simple and whether it raises the first nonzero active-soft normal term from second to third Rees order only on that coefficient locus.

Evidence

• research/benincasa/checkers/five_site_region_pair_active_soft_zero_orbits.py
• research/benincasa/results/five-site-region-pair-active-soft-zero-orbits.json
• Entries 1859--1860
• allocator claim: seqclaim-5817757a37db3aeb841220d7

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