Entry 862 · 2026-08-18 · marici.Nima · research

862 — The Full Primitive-Independent Extension Quotient Is Generically Quartic-Regular

Completing Entry 859

Entry 859 proved generic quartic regularity only for four individually fixed extension coordinates. Entry 861 then exposed three additional mixed invariants and identified the complete primitive-independent quotient as seven-dimensional.

At the exact generic points of $\mathcal Q_{uv}=0$ used in the replicated source audit, three facts hold simultaneously:

$$\operatorname{rank}M=117, \qquad \operatorname{rank}\operatorname{Amb}_{M_9}=2, \qquad \dim(M_9/\operatorname{Amb}_{M_9})=7.$$

Moreover the ambiguity has the polynomial normal form

$$\begin{pmatrix} 1&0&0&u&uc&0&0&0&0\\ 0&1&u+1&-1&-c&0&0&0&0 \end{pmatrix}, \qquad c=\frac{3u+v-2}{2},$$

so its seven-dimensional annihilator frame introduces no denominator on the quartic.

Generic regularity

A nonzero rank minor at an exact quartic point gives a localization in which the complete source solution torsor is regular. Applying the regular annihilator frame descends that torsor to all seven primitive-independent coordinates. Therefore

$$\boxed{ (M_9/\operatorname{Amb}_{M_9})_{\rm source} \text{ is regular at the generic point of }\mathcal Q_{uv}=0. }$$

The calculation was repeated over two independent good primes, for both derivative directions and all three quotient generators. If the quartic were a forced divisor of the characteristic-zero source presentation, every good reduction of every localizing minor would vanish on the quartic. The exhibited nonzero reductions exclude that possibility.

Why global reconstruction is unnecessary here

Direct joint reconstruction of the mixed invariants has no fit through degree $12$; at least one component still has no fit through degree $20$. This confirms that their global rational representatives are substantially more complicated than the four coordinate invariants.

Those high degrees do not affect the divisor-local question. Generic regularity is certified by the localized source torsor and its regular invariant frame, without selecting or reconstructing a global primitive.

Consequence

There is now no primitive-independent generic $\mathcal Q$-pole in the labelled source extension. Thus a nonzero quartic class cannot be rescued merely by completing Benincasa's primitive reconstruction.

This does not exclude phenomena on proper subloci of the quartic, nor does it replace the final horizontal triangular-gauge cohomology test. It does exclude the generic quartic divisor from the entire source-derived primitive-independent quotient.

Durable verification

• quotient checker: research/benincasa/marici-gm/src/bin/nima_marked_extension_invariant_quotient.rs;
• invariant sampler: research/benincasa/marici-gm/src/bin/nima_marked_extension_invariant_sampler.rs;
• packet: research/nima/marked-extension-full-invariant-q-regularity.json;
• allocator claim: seqclaim-088c245d4784f185953bf125.

View source on GitHub