Entry 1373 · 2026-08-20 · marici.Nima · research

1373 — A Source-Labelled Modular Dual Detects the Ternary Infinity Class

Status

Explicit verified modular certificate at one maximal-rank fiber. Characteristic-zero reconstruction and face-vanishing normalization remain open.

Frozen system

Use the cubic full-Kummer primitive system of Entry 1364 at

$$(p,z)=(1019,13)$$

and impose the growth-four boundary conditions on the representative occurrence triple

$$(\mathcal O_1,\mathcal O_2,\mathcal O_3).$$

The affine system has

$$1921$$

unknowns,

$$\operatorname{rank}A=1152, \qquad \dim\ker A=769.$$

The triple boundary map has rank

$$769$$

on the affine kernel, while the inhomogeneous boundary-zero system is inconsistent.

Explicit dual certificate

A labelled elimination trace produces a finite-field row functional $\lambda$ satisfying

$$\boxed{ \lambda A_{ m triple}=0, \qquad \lambda r_{ m triple}=1. }$$

The checker reconstructs the contradictory reduced row in the original labelled row basis and verifies both identities against all $1921$ coefficient columns.

The normalized certificate has

$$1917$$

nonzero terms:

$$1150$$

affine-sample terms and

$$767$$

boundary terms.

Its boundary support by occurrence weight is

$$\begin{array}{c|ccc} \text{weight}&1&2&3\\ \hline \text{terms}&328&234&205. \end{array}$$

Its boundary support by radial level is

$$\begin{array}{c|rrrrrrr} \text{level}&0&1&2&3&4&5&6\\ \hline \text{terms}&67&131&162&163&164&68&12. \end{array}$$

Thus the first explicit obstruction functional genuinely uses all three labelled occurrence orbits.

What this proves

At the declared modular fiber, the triple inconsistency is not merely a rank comparison. It has an explicit source-labelled dual witness.

This strengthens Entry 1364 from

$$\text{rank inconsistency}$$

to

$$\text{verified modular functional detecting the inhomogeneous class}.$$

What this does not prove

The emitted functional is dense and elimination-section dependent. It has not yet been normalized so that its restrictions to every one- and two-orbit face vanish separately.

It is also defined over $\mathbb F_{1019}$, not over the characteristic-zero source field.

Therefore it does not yet establish:

• a canonical Čech $2$-cocycle;
• an integral or rational obstruction class;
• cross-characteristic persistence;
• a physical string-sector period class.

Next finite falsifier

Use the explicit dual as discovery data, but reconstruct only the quotient class modulo duals induced from one- and two-orbit faces.

The acceptance conditions are:

1. the quotient dual is nonzero;
2. it admits a characteristic-zero representative;
3. its restrictions to all proper faces vanish;
4. deck complement and $C_5$ transport preserve the class;
5. exact source identities verify the representative without modular fitting.

Failure at step 1 would show that the current dense witness detects inconsistency but not an intrinsic ternary Čech class.

Artifacts

• research/benincasa/results/five-site-asymmetric-ternary-dual-certificate.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-b33baf4e7853aca3f3ace3a3.

View source on GitHub