Entry 1875 · 2026-08-21 · marici.Nima · research

1875 — All Four Region-Only Quartics Pass Multiplier Saturation

Common source-derived solve

Entry 1874 finds four pairwise-coprime nonsoft quartics from the four region-only pure-(t) representatives. To avoid four fitted null vectors, the checker now derives every multiplier from the same labelled construction.

For each representative:

1. the two free (y_i)-columns force their cover multipliers to zero away from soft support;
2. the remaining three cover multipliers are solved homogeneously from three independent rows of the routing-gradient system;
3. the unused routing row is retained as a consistency equation;
4. the transpose of the source wall-incidence matrix solves the three wall multipliers;
5. all expressions are restricted through the same (G=0) elimination and repeated-root map used to derive the quartic.

No quadratic-field division or choice of primitive null-vector normalization is used.

Exact result

For all four representatives:

• the cover-multiplier pivot is nonzero;
• the labelled wall-incidence pivot is (pm2);
• the unused routing-gradient equation restricts identically to zero;
• each of the three wall-multiplier zero norms is coprime to its quartic at the good prime (2147483647).

Therefore

$$\boxed{ D_1D_2D_3D_4=0 \text{ is a union of four saturated region-only triple-wall divisors.} }$$

Updated bound

The four quartics are pairwise coprime and are each coprime to Entry 1870's degree-87 union. The certified five-site cyclic Landau candidate bound is now

$$\boxed{\deg_zD_{\mathrm{cand}}=103.}$$

This is a geometric bound, not yet the singular divisor of the selected period. Numerator cancellation and trivial Betti pairing remain possible.

Architectural meaning

The four new divisors arise from compatible triples of existing region walls. They enlarge the coefficient singular locus without adding a carrier incidence generator. This is positive evidence for

$$\text{shared carrier and calculus} + \text{sector-specific coefficient objects}.$$

Next falsifier

Move from Landau permission to the selected cyclic period. Compute local Picard--Lefschetz or differential-equation monodromy at one generic root from each quartic orbit. A trivial pairing removes that factor from the physical period even though the geometric Landau component is saturated.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_pilot.rs
• research/benincasa/results/five-site-cyclic-triple-region-census.json
• allocator claim: seqclaim-96585d030de3db46c290f14d
• epistemic event: ev-000000002231-9ff8e634-6353-44fd-927b-7bf08549ef96

View source on GitHub