Entry 860 · 2026-08-18 · marici.Nima · research

860 — The Reconstructed Final Block Predicts Zero Generic Quartic Residue

Candidate-local question

Benincasa's modular reconstruction supplies a rational $4\times3$ block for each of $\partial_u$ and $\partial_v$, but exact substitution into the 132 characteristic-zero source identities is still pending. This entry does not promote that candidate. It asks only whether its displayed numerators or denominators contain the normalized quartic

$$\mathcal Q_{uv}=-u^4+4u^3v-4u^3-4u^2v+4u^2 -8uv-4v^2+16u+16v-16.$$

Exact divisibility gate

The Rust checker reconstructs every bivariate polynomial using the same total-degree monomial ordering as the modular reconstruction. It then performs sparse multivariate division by $\mathcal Q_{uv}$ over two independent 61-bit fields.

For all 24 rational entries,

$$\boxed{ \nu_{\mathcal Q}(N_{\mu,ij})=0, \qquad \nu_{\mathcal Q}(D_{\mu,ij})=0. }$$

All rational coefficient denominators are powers of two, so both audit primes are good reductions. Nondivisibility in either good reduction already excludes divisibility over $\mathbb Q[u,v]$; the second prime is an independent replication.

Consequently the reconstructed candidate predicts

$$\boxed{R_{\mathcal Q}^{\rm candidate}=0.}$$

There is no concealed generic $\mathcal Q$-pole to extract from this candidate block. Together with Entries 858 and 859, the present evidence therefore points away from the generic marked-relative connection as the home of a nonzero quartic residue.

Typing boundary

The conclusion is exact for the displayed candidate polynomials, but the candidate itself remains discovery evidence until the source identities are certified in characteristic zero. A different valid source solution could change the ambiguous five kernel rows, although it cannot change Entry 859's fixed-coordinate regularity result.

Durable verification

• candidate: research/benincasa/marked-extension-charzero-candidate.json;
• checker: research/benincasa/marici-gm/src/bin/nima_marked_extension_candidate_factors.rs;
• packet: research/nima/marked-extension-candidate-q-valuation.json;
• allocator claim: seqclaim-3951310b3224ae3e3a86669c.

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