Entry 1824 · 2026-08-21 · marici.Nima · research

1824 — All Mixed Edge–Region Pair Orbits Fail the Physical Pullback Gate

Frozen family

Among the 49 source-compatible two-wall orbits, consider the fourteen mixed profiles consisting of one edge wall $G^-_e$ and one connected-region wall $g_A$. Six have disjoint cut supports and eight share the edge $e$.

Entry 1823 already closes the six disjoint representatives

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

after the physical $d^3\ell$ pullback gate.

Shared-edge theorem

For a shared-edge profile, write $j$ for the other boundary edge of the region and $u_e,u_j$ for the corresponding unit vectors. At a smooth, nonsoft point,

$$\nabla_\ell G^-_e=2u_e, \qquad \nabla_\ell g_A=u_e+u_j.$$

Linear dependence forces $u_j=\pm u_e$.

• If $u_j=u_e$, the two gradients point in the same direction, so no positive nonzero Landau multipliers can cancel them.
• If $u_j=-u_e$, then $\nabla_\ell g_A=0$, while $\nabla_\ell G^-_e\ne0$; positive nonzero multipliers again cannot solve the Landau equation.

Hence all eight shared-edge orbits fail before elimination, Hessian, or residue calculations.

Result

Combining the direct shared-edge theorem with Entry 1823 gives

$$\boxed{ \text{smooth physical survivors among all 14 mixed edge--region pair orbits} =0. }$$

Zero-distance endpoints are outside the unit-gradient argument and remain classified as already existing soft support. This result neither removes that support nor addresses total-energy or region--region pair profiles.

Consequence

The physical-gradient gate closes an entire source-compatible orbit class without constructing ambient resultants. The remaining pair audit should therefore classify total-energy and region--region profiles directly in physical loop space before any period interpretation.

Evidence

• research/benincasa/checkers/five_site_mixed_edge_region_physical_no_go.py
• research/benincasa/results/five-site-mixed-edge-region-physical-no-go.json
• Entries 1822--1823
• allocator claim: seqclaim-e23ef346cba8fdeaf1c2f2ca

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