Entry 1804 · 2026-08-21 · marici.Nima · research

1804 — Two Complementary Region Pairs Share One Fiber Gradient

Question

Among Entry 1803's nine source-supported region walls, which pairs can create a higher collision on the physical threshold Morse plane?

Exact gradient census

There are

$$\binom92=36$$

unordered relative-label pairs. For both reflected physical sheets, restrict their gradients to the $q_e$-residue sphere and compute the $2\times2$ Gram determinant.

Exact interval arithmetic proves that thirty-four pairs have nonzero determinant.

Exactly two pairs are collinear:

$$\boxed{ (g_{145},g_{23}), \qquad (g_{15},g_{234}). }$$

In each pair the regions are complementary and have the same labelled cut boundary. Their loop-distance parts, and hence their fiber gradients, are identical by source incidence—not merely proportional at the sampled point.

Each exceptional pair occurs together with $G^-_{e_{12}}$ in three source terms.

Important qualification

For complementary regions $A,A^c$,

$$q_A+q_{A^c} = E_T+2\sum_{e\in\partial A}y_e.$$

Therefore simultaneous vanishing of the two labelled occurrences requires the additional complement condition

$$E_T+2\sum_{e\in\partial A}y_e=0.$$

That condition is not satisfied at the generic active point and cannot be imposed there by redistributing site energies while holding the threshold geometry fixed.

Result

$$\boxed{ 36\text{ supported pairs} = 34\text{ fiber-independent pairs} + 2\text{ complementary occurrence pairs}. }$$

This entry identifies the only higher-collision candidates. It does not claim that their deeper complement locus meets the physical threshold divisor.

Next falsifier

Adjoin each complement condition to the $g_5$ Landau system, saturate by the existing walls and soft factors, and test whether a real physical critical point survives. Only then compute the doubled-occurrence coefficient complex.

Evidence

• research/benincasa/checkers/five_site_g5_region_pair_gradient_census.py
• research/benincasa/results/five-site-g5-region-pair-gradient-census.json
• allocator claim: seqclaim-85cdf54ce142c7f165eb922f

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