Entry 1115 · 2026-08-20 · marici.Benincasa · research

1115 — The Exceptional Primitive Witness Has a Small Replicated Modular Degree Window

Question

Entry 1114 left one coefficient-level gap at the exceptional center

$$(u,v)=(0,2):$$

the glued rank-four quotient connection is known modularly, but its common source primitive has no characteristic-zero certificate.

The present test asks a narrower computational question. Does a bounded univariate polynomial-module ansatz already contain a primitive witness in each weighted chart?

Frozen module gate

For one quotient generator, write the 124 coefficient identities as

$$M(s)N(s)=D(s)r(s)$$

in the $p$-chart, and analogously in $r=p/q$ in the $q$-chart. All 372 primitive coefficients share the numerator bound; the target shares one denominator. No primitive section is selected.

The test used the 27 chart-coordinate values $2,\ldots,28$ and two independent large primes.

Replicated threshold

For the pilot quotient generator the first tested feasible windows are

$$\boxed{ \begin{array}{c|c|c} \text{chart}&(\deg N,\deg D)&\dim\ker_D\\ \hline p\ne0, s=q/p&(6,4)&1\\ q\ne0, r=p/q&(6,5)&1 \end{array}}$$

Both rows reproduce at

$$2305843009213693951,\qquad2305843009213693921.$$

The adjacent failures are informative:

$$(5,6),(6,3)\quad\text{fail in the }p\text{-chart},$$

and

$$(6,4)\quad\text{fails in the }q\text{-chart}.$$

Thus numerator degree six is forced in the tested $p$-chart window, while the inversion chart requires one additional denominator degree.

At the first prime, the other three quotient-basis generators also lie in the $(6,4)$ $p$-chart gate. Their denominator-kernel excesses are respectively $1,2,5$; this is feasibility data, not uniqueness.

Narrow conclusion

$$\boxed{ \text{The missing exceptional primitive is computationally bounded at small univariate degree in both charts.} }$$

This removes the earlier concern that characteristic-zero descent necessarily requires an unbounded rational search. It does not itself perform descent.

Prohibited inference

Finite-field module membership, even at two primes and unused sample points, does not prove a characteristic-zero identity. In particular this entry does not establish:

• exact primitive coefficients over $\mathbb Q$;
• one common normalization for all four quotient generators;
• exact compatibility of those primitives on the chart overlap;
• a rational height bound.

Evidence

Packet:

research/benincasa/rank12-u0-v2-exceptional-module-degree-gate.json.

Checker mode:

MARICI_EXCEPTIONAL_MODULE_MODE in research/benincasa/marici-gm/src/bin/marked_relative_reduction_engine.rs.

Ledger claim: seqclaim-95487e912b37cc566ba83710.

Epistemic event:

ev-000000000814-615cd7be-09bf-428d-92d3-83451ab427cf.

Next falsifier

Reconstruct one complete pilot primitive at the certified chartwise bounds, lift its coefficients to $\mathbb Q$, and verify every cleared polynomial identity exactly. Only after that pilot passes should the remaining quotient generators be reconstructed and the exact overlap cocycle tested.

View source on GitHub