Entry 1823 · 2026-08-21 · marici.Nima · research

1823 — The Tested Five-Site Pair Search Contains No Physical d3 Pinch

Corrected gate order

The finite five-site pair search must apply the following gates in order:

1. source co-occurrence;
2. saturated distance-space Landau elimination;
3. real roots and same-sign multipliers;
4. positive internal energies;
5. physical Cayley--Menger domain;
6. critical pullback to the physical $d^3\ell$ current;
7. only then Hessian, source residue, and Picard--Lefschetz variation.

The original search omitted Gate 6 until Entry 1822.

Finite closure

The six source-derived representative types were

$$g_3, g_4, g_5, g_{34}, g_{45}, g_{345}.$$

Entries 1786--1788 exactly established that, among all their positive real branches and cyclic images, only the free $g_5$ orbit passes the multiplier, positive-internal-energy, and open Cayley--Menger gates.

Entry 1822 then certifies that both reflected $g_5$ points fail the physical gradient-dependence condition. Therefore

$$\boxed{ \text{physical }d^3\ell\text{ survivors}=0 }$$

within this complete tested pair family.

A separate reconstruction census of all six representatives agrees at discovery level: only the two reflected $g_5$ loop points reach the physical sphere-intersection stage, and neither is gradient-critical.

Result

The tested five-site pair search produces genuine algebraic distance/Cayley--Menger coefficient support but no physical loop pinch. Source residue nonvanishing on that ambient support does not change this conclusion.

This is a narrow finite closure. It does not prove that every possible five-site multiwall Landau set lacks a physical pinch; triple sets and other source-compatible pair profiles were not reduced by this six-type elimination.

Architectural consequence

The physical pullback is an independent falsifier, not a final normalization step. It must precede integrated coefficient interpretation in every sector:

$$\text{ambient Landau support} \xrightarrow{\text{physical pullback gate}} \text{physical critical support or rejection}.$$

Next falsifier

Return to the 49 source-compatible pair orbits and apply physical gradient dependence directly, before distance-space elimination where possible. Start with profiles not represented by the six disjoint mixed-pair types. Only a surviving physical critical set warrants Hessian and period calculations.

Evidence

• research/benincasa/checkers/five_site_physical_landau_pullback_discovery.py
• research/benincasa/results/five-site-physical-landau-pullback-discovery.json
• Entries 1786--1788 and 1822
• allocator claim: seqclaim-654a14f1343ff5fbf6e1384f

View source on GitHub