Entry 1870 · 2026-08-21 · marici.Nima · research

1870 — Multiplier Saturation Certifies the First Triple-Wall Quartic

Remaining gate from Entry 1869

Entry 1869 derives the nonsoft quartic

$$D_3(z)=7424-35728z+61041z^2-44032z^3+11264z^4$$

for

$$G_{\setminus e_{12}}\mid g_{1345}\mid g_{145},$$

but leaves open whether one of the three wall multipliers vanishes identically on that divisor.

Normalization-free multiplier test

Let $\alpha_1,\alpha_2,\alpha_3$ be the Kummer-equation multipliers for the three solved square variables, normalized only by

$$\alpha_1+\alpha_2+\alpha_3=1.$$

The routing-column equations imply

$$\alpha_2=(H^{-1}u)_1, \qquad \alpha_3=(H^{-1}u)_2.$$

Using the source wall ordering, the three wall multipliers vanish precisely with

$$\alpha_2, \qquad \alpha_3, \qquad 5-8\alpha_2-6\alpha_3.$$

Instead of substituting the rational repeated root $v=-P_1/(2P_2)$, the checker clears its denominator first. This produces three polynomial zero tests over $\mathbb Q(\sqrt5)[z]$, independent of the null-vector normalization.

Exact result

The rational field norm of each multiplier-zero polynomial is coprime to $D_3$ at the good prime $2147483647$. Together with Entry 1869's square-free and soft-branch tests, this proves that on the generic quartic locus:

• $y_1,\ldots,y_5\ne0$;
• the routing Gram pivot remains a unit;
• all three source-ordered wall multipliers are nonzero.

Therefore

$$\boxed{ D_3(z)=0 \text{ is a saturated triple-wall Landau divisor.} }$$

Updated finite bound

Since $D_3$ is coprime to Entry 1866's degree-83 polynomial, the certified Landau denominator bound becomes

$$\boxed{ \deg_z D_{\rm cand}=87. }$$

This does not yet prove that the selected scalar period has nontrivial local monodromy at every root of $D_3$; numerator cancellation or trivial cycle pairing remains possible. It does prove that the source Landau geometry permits the divisor without lower-support contamination.

Next falsifier

Transport the pilot through every dihedral-equivalent occurrence and verify the orientation/unit packet. Then run the same elimination on the next geometric profile and compare its saturated factor with the degree-87 union.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_landau_pilot.rs
• research/benincasa/results/five-site-cyclic-triple-landau-pilot.json
• Entries 1866 and 1869
• allocator claim: seqclaim-a8f1bc5254d1fdf0845d5629

View source on GitHub