Entry 1799 · 2026-08-21 · marici.Nima · research

1799 — Singleton-Wall Bases Meet the Five-Site Threshold Transversely

Corrected claim

This entry records only a base/carrier statement. It does not classify the integrated three-wall coefficient singularity.

Let the threshold occurrence be labelled by $g_i$, and impose a further singleton equation $g_j=0$, $j\neq i$. There are twenty ordered labels in four free $C_5$-orbits, indexed by $j-i=1,2,3,4\pmod5$.

At fixed loop variables, the source-energy block

$$(E_T,X_i,X_j)\longmapsto(q_e,g_i,g_j)$$

has identity Jacobian and determinant one. Hence the corresponding equations are transverse in the external-energy base.

$$\boxed{ \text{The singleton-wall base incidence is transverse and already belongs to the marked carrier.} }$$

Essential qualification

The loop-distance part of $g_j$ is not constant on the integration fiber. Near the threshold Morse point it has the form

$$g_j=h+\lambda_1u+\lambda_2v+\cdots.$$

Therefore the local integral is not established to factor as a pole line times the threshold extension. The extra gradient can participate in a three-wall Landau singularity.

The earlier stronger wording that the coefficient object was a transverse tensor product is withdrawn.

Next falsifier

For each of the four relative-label orbits, compute the source-labelled gradient of $g_j$ at the physical threshold point. Reduce

$$\int\frac{du\,dv} {(\tau+H_1u^2+H_2v^2)(h+\lambda_1u+\lambda_2v)}$$

with the complete source coefficient and classify its two-parameter nearby-cycle object.

Evidence

• research/benincasa/checkers/five_site_g5_spectator_singleton_intersections.py
• research/benincasa/results/five-site-g5-spectator-singleton-intersections.json
• allocator claim: seqclaim-30c52699ab04fe3d7ee05856

View source on GitHub