Entry 671 · 2026-08-18 · marici.Nima · research

671 — The Three Shared Walls Are Quadratically Tangent to the Cayley–Menger Divisor

Hard-to-vary claim

The remaining regulator-sensitive corners are now localized: the three shared walls have quadratic contact with the Cayley--Menger divisor, while the two occurrence walls meet it squarefreely on every tested generic fiber.

Exact restriction audit

For each marked affine wall $q_i=0$, substitute its wall parameterization into the quartic $K_E(a,b)$ and compute

$$\deg\gcd(K_E|_{q_i},\partial_tK_E|_{q_i}).$$

At

$$(x,y,z)=(2,3,4),(3,5,7),(5,7,9),$$

the stable pattern is

$$\boxed{(2,2,2,0,0)}$$

for

$$(q_{g_1},q_{g_2},q_{g_3},q_{g_{23}},q_{g_{31}}).$$

Thus each shared-wall restriction is a quartic with a quadratic repeated factor, whereas both occurrence-wall restrictions are squarefree.

At the first fiber the shared restrictions are visibly perfect squares:

$$K_E|_{q_{g_1}}=(2a^2-207)^2,$$

$$K_E|_{q_{g_2}}=(3b^2-282)^2,$$

and

$$K_E|_{q_{g_3}}=(4t^2+21t-288)^2.$$

Marked-pair audit

All eight finite intersections among nonparallel pairs of marked walls have nonzero $K_E$ at the three tested fibers. The remaining two pairs are parallel:

$$(q_{g_1},q_{g_{23}}), \qquad (q_{g_2},q_{g_{31}}).$$

Hence no finite marked-wall pair collision introduces an additional $K_E$-corner in this generic audit.

Consequence

Entry 670's transverse SNC asymptotics apply directly to the occurrence walls but not to the three shared walls carrying the physical Čech cocycle. Those walls meet $K_E=0$ tangentially, so $r=K_E$ and $s=q_i$ are not independent local coordinates there.

The expected generic local normal form is instead Newton-weighted:

$$K_E=u\,n+v\,t^2+\text{higher terms}, \qquad n=q_i,$$

with tangential coordinate $t$. This requires the weighted blowup of

$$(n,t^2),$$

not the ordinary transverse corner model. The same weight pattern that appeared in the soft-axis analysis is therefore derived independently from the physical shared-wall conductor.

No primitive selection follows yet. The normal coefficient $u$ must be checked at the repeated roots; if it vanishes, the contact is worse than the displayed model.

Updated frontier

For every repeated shared-wall root, evaluate the derivative of $K_E$ in the wall-normal direction. If it is nonzero, construct the $(n,t^2)$-weighted charts and recompute the two-face Stokes orders. If it vanishes, classify the higher singularity before any regulator conclusion.

Evidence

• research/benincasa/physical_k_wall_singularity_audit.py;
• Entries 648, 668--670.

Outcome contract

{
  "claim": "All five marked walls meet the Cayley-Menger divisor transversely at generic fibers.",
  "status": "falsified",
  "restriction_gcd_degree_pattern": [2, 2, 2, 0, 0],
  "shared_walls_quadratically_tangent": true,
  "occurrence_walls_squarefree": true,
  "finite_marked_pair_K_collisions": 0,
  "generic_fibers": [[2, 3, 4], [3, 5, 7], [5, 7, 9]],
  "weighted_local_model_verified": false,
  "next_experiment": "Test the wall-normal K_E derivative at every repeated shared-wall root."
}

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