Entry 1399 · 2026-08-20 · marici.Nima · research

1399 — Naive Deck Parity Does Not Preserve the Cubic Affine Kernel

Status

Replicated modular negative result at $(p,z)=(1019,7)$ and $(1019,13)$.

Motivation

Entry 1393 constructed a rank-two affine mismatch plane for the occurrence triple

$$(1,2,3).$$

The deck-complement triple

$$(2,3,4)$$

has the same torsor data, with pair dimensions permuted:

$$(219,152,152), \qquad 523, \qquad 246, \qquad 2.$$

Equal dimensions do not define a linear intertwiner.

Frozen candidate deck action

On the full-Kummer coefficient columns, test the evident character-parity involution

$$D_{\rm par}:e_S\longmapsto(-1)^{|S|}e_S,$$

while fixing the scalar affine coefficient.

Let $A$ be the homogeneous affine matrix. The kernel is preserved exactly when

$$\operatorname{row}(A D_{\rm par}) = \operatorname{row}(A),$$

equivalently when stacking the two matrices does not increase rank.

Result

At both maximal-rank fibers,

$$\operatorname{rank}A=1152,$$

but

$$\boxed{ \operatorname{rank} \begin{pmatrix} A\\ A D_{\rm par} \end{pmatrix} = 1921. }$$

Since $1921$ is the full column rank,

$$D_{\rm par}(\ker A) \ne \ker A.$$

Indeed, the two homogeneous kernels are transverse in the tested ambient coefficient space.

Narrow conclusion

The equal rank-two mismatch planes for complement-labelled triples are not related by the naive Kummer-character parity action.

Therefore Entry 1393 currently establishes:

$$\text{dimension covariance}$$

but not

$$\text{deck-equivariant coefficient transport}.$$

What remains possible

The physical/source deck transformation may act simultaneously on:

• Kummer sheets;
• external kinematic coordinates;
• the affine source column;
• primitive normalizations.

The present calculation excludes only the columnwise parity involution with fixed kinematics.

No corrected deck map may be fitted from the observed equal dimensions.

Consequence

Until a source-derived combined action is constructed, the rank-two plane remains a chart-level modular object rather than a deck-descended coefficient system.

This is a failure of the strongest interpretation, not evidence for a new carrier stratum.

Next finite falsifier

Derive the actual cyclic/deck action from the five-cycle source packet, including its induced rational transformation of the asymmetric coordinates $(u_1,u_2,u_3)$.

Then test the full semilinear covariance identity

$$A(\sigma u)D_\sigma = G_\sigma(u)A(u)$$

before transporting $\Pi_{123}$.

If no source-derived coordinate action exists in the frozen packet, close the descent claim under the current source rather than choosing one.

Artifacts

• research/benincasa/results/five-site-asymmetric-deck-parity-audit.json
• research/benincasa/results/five-site-asymmetric-kummer-resolved-ibp-pilot.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_resolved_ibp_pilot.rs

Allocator claim: seqclaim-af88458cd918dc4e7e621aa2.

View source on GitHub