Entry 1805 · 2026-08-21 · marici.Nima · research

1805 — No Generic Higher Collision Survives the Complement-Pair Gate

Question

Do Entry 1804's two complementary occurrence pairs meet the physical five-site threshold as genuine higher Landau collisions?

Complement reduction

For complementary regions $A,A^c$,

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

The defining complement wall is

$$q_{e_{12}}=E_T+2y_{12}=0.$$

Therefore simultaneous vanishing of a complementary pair forces the second shared boundary distance to vanish.

First pair

For

$$(g_{145},g_{23}),$$

the shared boundary is $\{e_{12},e_{34}\}$. Hence

$$y_{34}=0,$$

which pins the loop point to the labelled focus $C_2$.

At that focus, exact rational interval arithmetic proves that

$$n_{12} \quad\text{and}\quad n_{45}+n_{51}$$

are nonparallel. Their cross-product norm is strictly positive. Thus the nonzero-multiplier stationarity equation has no solution.

$$\boxed{ (g_{145},g_{23})\text{ is excluded as a smooth physical higher collision.} }$$

Second pair

For

$$(g_{15},g_{234}),$$

the shared boundary is $\{e_{12},e_{45}\}$, so

$$y_{45}=0.$$

But $y_{45}$ is one of the two defining distances in

$$g_5=X_5+y_{45}+y_{51}.$$

The ordinary unit-gradient Landau model is therefore undefined. This locus is the already existing defining-edge soft corner and must be treated by a radial/Rees resolution.

Result

Neither complementary pair produces a generic higher collision:

• one is killed by exact stationarity;
• one is confined to existing soft support.

No new carrier stratum is indicated.

Next falsifier

Resolve the remaining $y_{45}=0$ corner in loop polar coordinates. Retain the two complementary occurrence labels separately and test whether their double pole produces a first-Rees class, a Kummer sign, or zero.

Evidence

• research/benincasa/checkers/five_site_g5_complement_pair_activation.py
• research/benincasa/results/five-site-g5-complement-pair-activation.json
• allocator claim: seqclaim-01cc7a482ecc96e51956ecbd

View source on GitHub