Entry 1349 · 2026-08-20 · marici.Nima · research

1349 — Cubic Kummer Primitives Produce a Canonical Modular Infinity Coset

Status

This is a multi-prime modular theorem candidate, not yet a characteristic-zero theorem.

Entry 1343 established the generic-rank gate and retracted the accidental quadratic closure. The same gate is now applied to cubic Kummer-resolved primitives.

Cubic ansatz

$$V_i = \sum_{S\subseteq\{1,\ldots,5\}} P_{i,S}(u)y_S, \qquad \deg P_{i,S}\le3.$$

The affine system has

$$1+3\cdot32\binom{6}{3} = 1921$$

unknowns.

Multi-prime rank scan

All nine tested fibers are consistent:

$$\begin{array}{c|ccc} &z=5&z=7&z=11\\ \hline p=1019&1056&1152&1024\\ p=1021&1024&992&928\\ p=1031&576&800&768 \end{array}$$

where entries are coefficient ranks.

Thus cubic consistency survives the maximal observed rank

$$r_{\max}=1152.$$

Unlike the quadratic case, no tested full-rank or higher-rank fiber is inconsistent.

Boundary test at maximal observed rank

At

$$(p,z)=(1019,7),$$

append the complete sheet- and order-resolved radial infinity map from the surviving valuation packet.

The ranks are

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

$$\operatorname{rank} \begin{pmatrix} A\\ B_{\infty,\mathrm{rad}} \end{pmatrix} = 1921.$$

Therefore

$$\dim\ker A = 1921-1152 = 769,$$

and

$$\operatorname{rank} \left( B_{\infty,\mathrm{rad}}|_{\ker A} \right) = 1921-1152 = 769.$$

Hence the boundary map is injective on the entire affine gauge kernel.

The combined inhomogeneous system is inconsistent, so the affine boundary coset does not contain zero.

Candidate mechanism

At the maximal tested modular rank, the derivative determines a canonical nonzero coset

$$\boxed{ \beta_z \in \mathcal B_{\infty} /B_{\infty}(\ker A), \qquad \beta_z\ne0. }$$

Different affine primitives have different boundary representatives, but all represent the same quotient class. This is the correct gauge-invariant object; no preferred primitive is chosen.

What is established and what is not

Established computationally:

• cubic consistency at nine fibers across three primes;
• survival at maximal observed rank 1152;
• injectivity of the radial boundary map on the 769-dimensional affine kernel at that fiber;
• nonvanishing of the resulting modular boundary coset.

Not established:

• the characteristic-zero cubic affine identity;
• stability of rank 1152 as the generic rank;
• the dimension of the characteristic-zero boundary quotient;
• horizontality or a scalar Picard--Fuchs operator.

Exact certification gate

A characteristic-zero theorem requires one of:

1. reconstruct a complete cubic primitive and verify all cleared identities exactly; or
2. construct a dual certificate for the quotient class $\beta_z$ without reconstructing all 1921 coefficients.

The dual route is preferred if it can certify:

$$\ell A=0, \qquad \ell\,\partial_z\Omega\ne0,$$

with $\ell$ factoring through the projective radial boundary quotient.

No physical or scalar interpretation is authorized before that gate.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_kummer_resolved_ibp_pilot.rs
• research/benincasa/results/five-site-asymmetric-kummer-degree3-census.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_infinity_deck_valuation.rs
• research/benincasa/results/five-site-asymmetric-infinity-deck-valuation.json

Allocator claim: seqclaim-eb55ad61dd77216cb5e8f19d.

View source on GitHub