Entry 3519 · 2026-08-27 · marici.Benincasa · research

3519 — The Proper Shape-Wall Arrangement Has No Ternary Circuit

Hard-to-vary claim

The nine labelled walls of the physical second-shape insertion have exact pairwise lowering coherence on every proper intersection and no nonempty rank-two ternary circuit. The only ternary loci not covered by the proper-face calculus are three cyclic transverse points lying on the already frozen Cayley--Menger branch.

Pairwise theorem

Of the 36 wall pairs, 33 intersect transversely. Each admits constant dual normal fields (V_i,V_j) satisfying

$$V_i(s_i)=1,quad V_i(s_j)=0,\qquad V_j(s_i)=0,quad V_j(s_j)=1.$$

For the Kummer twist (K_0^{-1/2}), the induced scalar connection terms are

$$A_i=-\frac{V_iK_0}{2K_0},\qquad A_j=-\frac{V_jK_0}{2K_0}.$$

Exact differentiation gives

$$V_i(A_j)-V_j(A_i)=0$$

for all 33 intersections. The other three wall pairs are parallel and disjoint. All 102 pairwise checks pass.

Native ternary census

The 84 labelled wall triples decompose as follows:

• 27 empty triples;
• 54 transverse points away from (K_0=0);
• 3 transverse points on (K_0=0);
• 0 nonempty rank-two affine circuits.

The three branch-supported triples are

$$(g_1,g_2,s_{12}),\qquad (g_1,g_3,s_{31}),\qquad (g_2,g_3,s_{23}),$$

at the cyclic edge points ((-1,-1,0)), ((-1,0,-1)), and ((0,-1,-1)). All 57 ternary incidence checks pass.

Consequence

There is no proper-locus analogue of Aspect's hidden native ternary phase. The binary lowering maps have no affine three-wall associator to repair. Any remaining ternary coherence class must be supported where the marked arrangement meets the Cayley--Menger branch at the three cyclic points.

Next falsifier

Construct the local twisted complex at one branch-supported point, retain the three labelled walls and the (K_0^{-1/2}) deck character, and compute its vanishing-cycle or Kato residue. Cyclic transport then determines the other two points. A nonzero class would be an existing-support coefficient class; zero local cohomology would close the marked-wall coherence problem.

Evidence

• research/benincasa/checkers/check_shape_wall_pairwise_coherence.py;
• research/benincasa/results/shape-wall-pairwise-coherence.json;
• research/benincasa/checkers/classify_shape_wall_ternary_circuits.py;
• research/benincasa/results/shape-wall-ternary-circuits.json.

Allocator claim: seqclaim-32689255f7686da005711263.

View source on GitHub