Entry 1833 · 2026-08-21 · marici.Nima · research

1833 — The Deformed Region-Pair Coefficient Germ Is Cyclically Natural

Equivariant Landau equations

Let $S=(g_{123},g_{125})$, and let $\sigma$ generate the labelled five-cycle. The physical Landau map satisfies

$$F_{\sigma S}(\sigma X,\sigma\ell,t,\lambda) = \sigma F_S(X,\ell,t,\lambda).$$

Entry 1832 proves that the solution of $F_S=0$ is locally unique. Applying $\sigma$ to that solution and using uniqueness therefore gives

$$\boxed{ s_{\sigma S}(\sigma X)=\sigma s_S(X). }$$

This identity holds for the whole local implicit germ, not merely its value on the homogeneous slice.

First jets and thresholds

Differentiating gives the labelled first-jet identity

$$Ds_{\sigma S}|_{\sigma X}\circ\sigma = \sigma\circ Ds_S|_X.$$

In particular,

$$t_{\sigma S}(\sigma X)=t_S(X), \qquad \lambda_{\sigma S}(\sigma X)=\lambda_S(X).$$

Thus no fitted transition matrix or occurrence-dependent normalization is needed to transport the critical-value discriminant.

Coefficient-family assembly

Entry 1830's ordered double-Leray orientation is constant on the same orbit. Combining it with the unique equivariant deformation gives a local $C_5$-equivariant family of logarithmic coefficient lines:

$$\boxed{ \mathcal L_{ m loc} \simeq \mathbb Q[C_5] \text{ over a neighbourhood of the labelled carrier orbit}. }$$

This promotes Entry 1829's fiberwise cyclic packet to a coherent local family. No new carrier datum is required.

Remaining boundary

The result is local in external kinematic space. It does not supply the global Bunch--Davies relative chain, continue through other discriminant components, or prove global sewing with different occurrence orbits.

Next falsifier

Compute intersections of this local discriminant family with the already frozen total-energy and soft divisors. Test whether nearby cycles and excess terms are generated by the existing support calculus. A residual class that requires a new incidence stratum would be the first carrier-level challenge from this physical orbit.

Evidence

• research/benincasa/checkers/five_site_region_pair_cyclic_deformation_naturality.py
• research/benincasa/results/five-site-region-pair-cyclic-deformation-naturality.json
• Entries 1829--1832
• allocator claim: seqclaim-3dd29df907596fa40476ba59

View source on GitHub