Entry 1791 · 2026-08-21 · marici.Nima · research

1791 — The Physical Five-Site Threshold Persists Off the Cyclic Slice

Question

Entry 1790 proves a nonzero physical anomalous threshold on the frozen cyclic one-parameter family. Is it an accidental singularity created by imposing cyclic symmetry, or does it intersect the slice transversely and persist as a codimension-one divisor in nearby five-site kinematics?

Frozen Jacobian gates

Two independent Jacobians control the local Landau system.

First, Entries 1779, 1784, and 1790 give an invertible internal transverse Hessian:

$$\det H_\perp\neq0, \qquad \operatorname{sign}(H_\perp)=(+,+).$$

Thus the critical loop point varies uniquely under sufficiently small kinematic perturbations.

Second, let (L_{g_5}(x)) be the exact saturated degree-six eliminant. On the isolating interval of the physically active root (x_*), exact Sturm sign evaluation gives

$$\boxed{ L_{g_5}'(x_*)\neq0. }$$

Therefore the cyclic slice crosses the projected Landau discriminant transversely.

Persistence theorem

Apply the analytic implicit-function theorem to the full source Landau equations before elimination. Invertibility of the internal Hessian solves for the critical loop variables, while the nonzero slice derivative supplies a nonvanishing normal coordinate on the resulting discriminant.

Hence there exists a neighbourhood (U) of the cyclic kinematic point and a smooth local hypersurface

$$\mathscr D_{g_5}\subset U$$

whose intersection with the frozen cyclic slice is (x=x_*).

$$\boxed{ \text{The physically active }g_5\text{ threshold is a genuine local codimension-one divisor, not a slice artifact.} }$$

Stability of the physical data

The following conditions are open and therefore persist after shrinking (U):

• positivity of the seven Cayley–Menger principal minors;
• positivity of the two Landau multipliers;
• positive-definiteness of the transverse Hessian;
• nonvanishing of all unused source denominators;
• nonvanishing of the complete source double residue.

Consequently the nonzero physical Picard–Lefschetz variation of Entry 1790 persists on an open real patch of (mathscr D_{g_5}).

The five cyclic occurrences assemble as a (C_5)-equivariant divisor orbit. They are not collapsed to one unlabelled equation.

Architectural consequence

This removes the principal specialization loophole from the five-site result. The new singularity belongs to the generic integrated coefficient geometry over the existing marked/routing carrier. It still supplies no evidence for a new carrier incidence generator.

Next falsifier

Compute the local monodromy and variation morphism of the rank-one vanishing cycle line over (mathscr D_{g_5}), including its intersections with existing one-wall and total-energy divisors. Test whether those intersections are generated by the existing nearby-cycle/Gysin calculus or create a supported extension class.

Evidence

• research/benincasa/checkers/five_site_g5_picard_lefschetz.py
• research/benincasa/results/five-site-g5-picard-lefschetz.json
• allocator claim: seqclaim-46489c4e6a3b5f2d9da219b2

View source on GitHub