Entry 1827 · 2026-08-21 · marici.Nima · research

1827 — A Disjoint Region Pair Survives the Physical Pullback Gate

Discovery and exact reduction

A bounded physical-loop scan of Entry 1826's seven disjoint-cut candidates finds one candidate orbit:

$$(g_{123},g_{125}), \qquad \partial g_{123}=\{2,4\},quad \partial g_{125}=\{1,3\}.$$

The source geometry has an involutive isometry exchanging these two labelled cut pairs. Its fixed line can be written

$$L(q)=M_{14}+q(M_{23}-M_{14}).$$

On this line the two wall sums agree identically. The physical gradient condition reduces to stationarity of one distance sum along the line.

Exact parameter

After clearing a positive factor, the squared stationarity equation is

$$(70+25\sqrt5)q^2 -(170+60\sqrt5)q +(85+30\sqrt5)=0.$$

It has exactly one root in $(0,1)$:

$$q_* = \frac{17+6\sqrt5-\sqrt{81+35\sqrt5}} {14+5\sqrt5}, \qquad 0.7067<q_*<0.7068.$$

The exact interval checker proves opposite signs at the isolating endpoints and a strictly negative derivative throughout the interval. Since $(q_*-1)q_*<0$, the root also passes the unsquared stationarity sign gate.

Physical Landau conditions

The two focal-line Gram determinants are strictly positive on the isolating interval. Therefore neither region gradient vanishes. The source isometry and fixed-line stationarity give

$$\nabla_\ell g_{125}=-\nabla_\ell g_{123},$$

so the positive multiplier ratio is exactly one. The wall equations set

$$t =-\frac{y_2+y_4}{3} =-\frac{y_1+y_3}{3}<0,$$

with all internal distances positive.

Hence

$$\boxed{ (g_{123},g_{125}) \text{ is an exact smooth physical two-region Landau survivor.} }$$

Scope and next gates

This certifies physical criticality only. It does not yet establish a nondegenerate pinch or a nonzero contribution of the frozen source form. The next mandatory gates are:

1. transverse Hessian/Morse nondegeneracy;
2. separation from every remaining source wall;
3. exact summed source residue;
4. only then Picard--Lefschetz and coefficient interpretation.

The bounded null results for the other six disjoint profiles remain discovery evidence, not exact exclusions.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_disjoint_region_pair_physical_discovery.rs
• research/benincasa/results/five-site-disjoint-region-pair-physical-discovery.json
• research/benincasa/marici-gm/src/bin/five_site_disjoint_region_pair_symmetry_polynomial.rs
• research/benincasa/results/five-site-disjoint-region-pair-symmetry-polynomial.json
• research/benincasa/checkers/five_site_disjoint_region_pair_exact_survivor.py
• research/benincasa/results/five-site-disjoint-region-pair-exact-survivor.json
• allocator claim: seqclaim-57284b81e2ce7eded47808f6

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