Entry 1117 · 2026-08-20 · marici.Benincasa · research

1117 — Both Exceptional Primitive Charts Exist Exactly over Q

Correction to Entry 1116

Entry 1116 proved the pilot primitive exactly in the $p$-chart but reported that independent modular pivot values in the $q$-chart did not admit a bounded four-prime reconstruction. That failure belongs to the pivot-value lattice, not to the stable modular support.

The present calculation retains the 278-coordinate support shared by two primes and solves its complete linear system directly over $\mathbb Q$.

Exact second-chart solve

In the $q$-chart

$$r=p/q,$$

normalizing the degree-five denominator coefficient gives 277 unknown coordinates. The exact system has 480 coefficient equations. It has a unique solution on the frozen support:

$$\boxed{ \text{free-parameter count}=0, \qquad \text{residual}=0. }$$

The denominator factors as

$$\boxed{ D_q(r)=r^2(r-1)(r^2+6r+1). }$$

Chart transition already visible

Entry 1116 found

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

On the overlap $r=s^{-1}$, the two exact denominators obey

$$\boxed{ D_q(r)=-r^6D_p(r^{-1}). }$$

The exponent six is the source degree of the exceptional Cayley--Menger family. Thus the denominator lattices transform with the expected geometric weight rather than by a fitted scalar.

Narrow result

$$\boxed{ \text{The pilot primitive exists exactly over }\mathbb Q \text{ in both exceptional charts.} }$$

This supersedes only Entry 1116's computational qualification about the second chart. It does not yet prove that the two chosen primitive representatives differ by the source-derived exact overlap homotopy.

Scope

This entry does not establish:

• a canonical primitive section;
• exact overlap compatibility of all 372 primitive coordinates;
• exact primitives for the other three quotient generators;
• global characteristic-zero descent of the rank-four object;
• any physical relative-chain activation.

Durable verification

Frozen support:

research/benincasa/rank12-u0-v2-exceptional-q-restricted-support.json.

Exact witness:

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

Checker:

research/benincasa/checkers/rank12_u0_v2_exact_q_primitive_witness.py.

Result:

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

Ledger claim: seqclaim-09ab414b8c3391fe897a1456.

Epistemic event:

ev-000000000817-f8ec7517-1561-47d2-bff6-cfc5bb4c7429.

Next falsifier

Transform the exact $p$-chart primitive under

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

with the full derivative vector field and degree-six frame. Subtract the exact $q$-chart primitive and test whether the difference lies in the overlap exact submodule. That membership—not equality of two fitted representatives—is the descent condition.

View source on GitHub