Entry 1776 · 2026-08-21 · marici.Nima · research

1776 — The Generic $P_6$ Trivial-Line Resonance Is Unoccupied in the Multiprime Source Audit

Question

Entry 1766 leaves one zero-exponent Hom slot at generic $P_6=0$: the projection from each regular marked quotient generator to the split trivial algebraic line. Does the source extension occupy that slot with a logarithmic residue?

Frozen source reduction

The complete 132-equation rank-twelve reducer was generalized without changing its primitive convention. For each accepted solve it still requires

$$\operatorname{rank}=117, \qquad \text{fixed mask}=3847,$$

and projects the canonical final four coordinates to the source-defined split trivial line.

The pullback was evaluated along arcs transverse to $P_6=0$ for all three labelled quotient generators

$$q_{\rm top},\qquad q_{\rm wall1},\qquad q_{\rm wall2}.$$

Census

The test used:

• two independent 61-bit primes;
• three generic $P_6$-points per prime;
• all three labelled source directions at every point;
• 48 discovery and 24 held-out arc samples per reconstruction.

Thus there are eighteen independently reconstructed Laurent functions and 432 held-out evaluations.

Every reconstruction has

$$\operatorname{ord}_s N=0, \qquad \operatorname{ord}_s D=0, \qquad \operatorname{ord}_s B_{\rm triv}=0.$$

None has the logarithmic valuation $-1$.

Narrow result

$$\boxed{ \text{The generic }P_6\text{ trivial-line resonance is unoccupied in all tested source directions and modular specializations.} }$$

This is strong multiprime source evidence, not an exact characteristic-zero function-field theorem. The finite certification would reconstruct the normal residue numerator over $\mathbb Q(P_6)$ and prove it vanishes identically.

Together with Entries 1770, 1774, and 1775, the result strongly disfavors both split algebraic lines as homes of a new supported $P_6$-class.

Durable evidence

• generalized source checker: research/benincasa/marici-gm/src/bin/p6_d_exceptional_source_residue.rs;
• eighteen packets matching research/benincasa/results/p6-generic-trivial-*.json;
• aggregate checker: research/benincasa/checkers/p6_generic_trivial_residue_audit.py;
• aggregate packet: research/benincasa/results/p6-generic-trivial-residue-audit.json;
• convention note: research/benincasa/p6-generic-trivial-residue-audit.md;
• allocator claim: seqclaim-43597cc86f63a892484c5e50.

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