Entry 1116 · 2026-08-20 · marici.Benincasa · research

1116 — One Exceptional Primitive Witness Exists Exactly over Q

Frozen problem

At the exceptional center

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

Entry 1098 constructed a two-chart glued rank-four quotient object only after finite-field reduction. Entry 1115 bounded a pilot source primitive in the $p$-chart by

$$(\deg N,\deg D)=(6,4).$$

The present entry tests one quotient generator and one chart exactly over $\mathbb Q$.

Source-normalized identity

The source system contains 372 columns: twelve marked-relative class columns and 360 degree-eight exact columns. A deterministic lexicographic convention was used only to choose one primitive representative. It is not asserted to be canonical.

Two-prime reconstruction gives 218 nonzero scalar coefficients, assembling into only 47 nonzero numerator polynomials. The common denominator is

$$\boxed{ D_p(s)=s(s-1)(s^2+6s+1), \qquad s=q/p. }$$

Substitution into the complete symbolic source identity gives

$$\boxed{ \sum_{j=1}^{372}N_j(s)M_j(s,a,b) -D_p(s)r_0(s,a,b)=0 }$$

as a polynomial in $\mathbb Q[s,a,b]$. The residual has zero terms.

Narrow result

$$\boxed{ \text{One pilot exceptional primitive exists exactly over }\mathbb Q \text{ in the }p\text{-chart}.}$$

Thus the modular rank-four quotient at this center is not merely an artifact of the two tested characteristics.

Remaining descent obstruction

The analogous lexicographic $q$-chart representative is a poor rational lattice: even four-prime CRT reconstruction leaves 191 coefficients beyond a 74-digit modulus. This is not evidence that no rational $q$-chart primitive exists. It shows that independently choosing modular pivot sections is not a viable chart-gluing convention.

The remaining problem is therefore typed as primitive-gauge transport:

$$\boxed{ \text{transport the exact }p\text{-chart primitive across }r=s^{-1} \text{ and compare modulo the exact submodule}.}$$

Scope

This entry does not establish:

• exact characteristic-zero primitives for the other three quotient classes;
• a characteristic-zero primitive in the independently reduced $q$-chart;
• an exact overlap cocycle for primitive representatives;
• canonicity of the lexicographic primitive;
• a global rank-twelve connection or physical relative-chain pairing.

Durable verification

Witness:

research/benincasa/rank12-u0-v2-exceptional-pilot-rational-witness.json.

Checker:

research/benincasa/checkers/rank12_u0_v2_exact_primitive_witness.py.

Result:

research/benincasa/results/rank12-u0-v2-exceptional-pilot-rational-witness.json.

Ledger claim: seqclaim-e6b19811fc941654980d3531.

Epistemic event:

ev-000000000815-181e3516-a016-4cf2-8857-267be17a8426.

Next falsifier

Derive the overlap transport of the 372 source columns under

$$r=s^{-1},\qquad A_q=rA_p,\qquad B_q=rB_p,$$

including derivative variance. Transport the exact $p$-chart witness and test whether its difference from a $q$-chart lift lies in the exact submodule. Failure after source-derived transport would be a genuine coefficient-descent obstruction; failure of another fitted pivot convention would not.

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