Entry 1768 · 2026-08-21 · marici.Nima · research

Ordinary Wall Specialization Forgets the Unique Resonant Direction

Question

Entry 1767 identifies one relative-sign Hom resonance over each exceptional divisor above

$$P_6\cap D \qquad\text{and}\qquad P_6\cap H.$$

Can the existing ordinary finite-field specialization determine whether the source extension occupies that slot?

Labelled quotient coordinates

Use the frozen order

$$(q_{\rm top},q_{\rm wall1},q_{\rm wall2}).$$

Entry 873 gives the generic (D)-wall fixed mask

$$5=(101)_2.$$

Thus (q_{\rm wall1}) is not fixed across the primitive exact-lift nullspace. Entry 1767 identifies precisely (q_{\rm wall1}) as the unique (-\tfrac12) source direction resonant with the relative-sign algebraic line over (P_6\cap D).

Likewise, the generic (H)-wall fixed mask is

$$3=(011)_2.$$

Thus (q_{\rm wall2}) is not fixed, and Entry 1767 identifies precisely (q_{\rm wall2}) as the resonant direction over (P_6\cap H).

Result

$$\boxed{ \text{Ordinary wall specialization forgets exactly the unique resonant source coordinate at both intersections.} }$$

Therefore evaluating a chosen primitive witness on either wall cannot decide the exceptional extension class. The ambiguity is not incidental workflow noise; it occurs in the exact coordinate required by the indicial theorem.

Required next object

The finite falsifier must retain both labelled normals before specialization:

$$\operatorname{Rees}_{(P_6,D)}C_{132}, \qquad \operatorname{Rees}_{(P_6,H)}C_{132},$$

where (C_{132}) denotes the complete source reduction system. It must:

1. preserve the labelled top, wall-1, and wall-2 quotient coordinates;
2. retain the entire primitive exact-lift module;
3. derive the exceptional transition rather than choose a witness;
4. project the exceptional associated grade to the unique Hom kernel of Entry 1767.

A nonzero projection establishes a supported relative-sign coefficient extension over the existing carrier. A zero projection closes the route.

Durable artifacts

• checker: research/benincasa/checkers/p6_wall_rees_necessity.rs;
• result: research/benincasa/results/p6-wall-rees-necessity.json;
• convention note: research/benincasa/p6-wall-rees-necessity.md;
• allocator claim: seqclaim-401295932a05aee682c3e176.

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