Entry 1873 · 2026-08-21 · marici.Nima · research

1873 — Multiplier Saturation Certifies the Region-Only Quartic

Remaining gate from Entry 1872

Entry 1872 derives the squarefree nonsoft quartic

$$D_{3,\mathrm{reg}}(z) =2016064-5586608z+5762041z^2-2628992z^3+446464z^4$$

from

$$g_{1234}\mid g_{234}\mid g_{2345},$$

but leaves open whether a source wall multiplier vanishes on that component.

Denominator-free multiplier solve

The three wall rows have zero (y_2)- and (y_3)-columns. Away from soft support, the corresponding cover-gradient equations force

$$\alpha_2=\alpha_3=0.$$

Rather than divide by a quadratic-field coordinate, the checker solves the remaining routing-gradient equations in the homogeneous scale

$$\det(H)c_1.$$

This gives polynomial representatives for

$$\alpha_1,\alpha_4,\alpha_5$$

and then solves the five labelled (y)-gradient equations for the three source-ordered wall multipliers.

The unused routing-gradient component vanishes identically after restriction to the repeated-root scheme. Thus the multiplier solve is compatible with the critical-value elimination rather than an independently fitted null vector.

Exact result

Up to nonzero rational units, the three multiplier-zero norms are

$$71-136z+64z^2,$$

$$261161-704896z+446464z^2,$$

and again

$$71-136z+64z^2.$$

Each is coprime to (D_{3,\mathrm{reg}}) at the good prime (2147483647). Together with Entry 1872's soft and lower-support exclusions, this proves

$$\boxed{ D_{3,\mathrm{reg}}=0 \text{ is a saturated region-only triple-wall Landau divisor.} }$$

Updated finite bound

The new quartic is coprime to the degree-87 union, so the certified Landau candidate denominator bound becomes

$$\boxed{\deg_z D_{\mathrm{cand}}=91.}$$

This remains a geometric denominator bound. It does not prove nontrivial monodromy of the selected scalar period at every root, nor does it identify a new carrier wall. The divisor is produced by compatible composition of three already frozen walls.

Next falsifier

Run the same saturated elimination on the remaining three nonsoft region-only pure-(t) representatives. Determine whether they reproduce the two known quartic orbits or enlarge the degree-91 union.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_pilot.rs
• research/benincasa/results/five-site-cyclic-triple-region-pilot.json
• allocator claim: seqclaim-88d9d8979de5aad837adb268
• epistemic event: ev-000000002227-e5992edb-46df-472b-9d4c-a543318e9526

View source on GitHub