Entry 669 · 2026-08-18 · marici.Nima · research

669 — No Minimal Source Syzygy Is Uniformly Improved at All Five K–Wall Corners

Hard-to-vary claim

None of the three minimal degree-seven source-logarithmic vector fields has an extra Cayley--Menger normal factor at all five codimension-two strata

$$K_E=q_i=0.$$

Thus Entry 668's generic smooth-boundary vanishing does not extend by a single uniformly improved minimal primitive across the complete marked divisor.

Corner condition

Write a complete logarithmic syzygy as

$$V(K_E)=n_0K_E, \qquad V(q_i)=\lambda_iq_i.$$

On the $K_E$-face, the primitive flux retains a marked factor

$$K_E^{\epsilon+1/2}\frac{n_0}{q_i}.$$

A source-derived sufficient condition for removing this corner pole is

$$n_0|_{K_E=0}\in(q_i),$$

or equivalently

$$n_0\in(K_E,q_i).$$

The checker imposes this ideal-membership condition coefficientwise. It does not replace it by the stronger global divisibility condition $q_i\mid n_0$, although both ranks are retained as a cross-check.

Exact finite-field result

Within the stable three-dimensional degree-seven syzygy space, the dimensions satisfying the individual corner conditions are

$$\boxed{(1,1,2,2,2)}$$

for

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

The same dimensions arise from global wall divisibility, but that equality is observed rather than assumed.

Imposing all five ideal-membership conditions simultaneously gives

$$\boxed{ \dim\{V:n_0\in\bigcap_i(K_E,q_i)\}=0. }$$

Both statements reproduce at the generic fibers

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

over $\mathbb F_{2305843009213693951}$.

Consequence

The codimension-two audit does not break the three-dimensional ambiguity by selecting a distinguished minimal primitive. Instead, the strongest uniform corner-regularity condition excludes all three directions.

This does not prove the physical IBP identities invalid. Corner residues may cancel between boundary faces, be integrable without pointwise vanishing, or require a relative-chain/Stokes prescription. It does prove that generic $K_E$-face vanishing cannot be promoted to complete-divisor vanishing merely by choosing one of the existing degree-seven syzygies.

At the meta level, regulator geometry has changed roles: it is not a normalization that selects one primitive. It is an obstruction showing that the primitive must be interpreted in a combined corner complex.

Updated frontier

Construct the oriented two-face corner functional for each $K_E=q_i=0$, adding the $K_E$-face and $q_i$-face contributions before taking $\epsilon\to0$. Test whether their polar terms cancel for the source-prescribed combination. Only this summed Stokes calculation can distinguish a genuine corner anomaly from failure of pointwise uniform vanishing.

Evidence

• research/benincasa/check_shared_wall_log_syzygy.rs, schema v4;
• Entries 651--652 and 668.

Outcome contract

{
  "claim": "A nonzero minimal degree-seven source syzygy has improved K-normal order at every marked K_E=q_i corner.",
  "status": "falsified",
  "minimal_syzygy_dimension": 3,
  "individual_corner_subspace_dimensions": [1, 1, 2, 2, 2],
  "all_corner_intersection_dimension": 0,
  "generic_fibers": [[2, 3, 4], [3, 5, 7]],
  "pointwise_corner_vanishing_required_for_physical_IBP": "not established",
  "next_experiment": "Compute the summed oriented two-face Stokes functional at every K_E=q_i corner."
}

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