Entry 1869 · 2026-08-21 · marici.Nima · research

1869 — The First Triple-Wall Pilot Produces a New Nonsoft Quartic Candidate

Representative

Take the first nonzero triple-wall representative from Entry 1868:

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

Its three wall equations solve

$$y_1=-\frac52t, \qquad y_2=-\frac32t, \qquad y_3=-\frac12t.$$

Set $z=t^2$ and $v=y_4^2$. Differences of the five Kummer equations then solve $u_1,u_2,u_3$, leaving one exact quadratic

$$P(v,z)=0$$

over $\mathbb Q(\sqrt5)$.

Critical discriminant

The repeated-root condition in $v$ has discriminant

$$\frac5{64} \left[ -200+515z+24\sqrt5-114\sqrt5z +96\sqrt5z^2-320z^2 \right].$$

Its rational field norm, up to the nonzero source-fixed unit, is

$$\boxed{ D_3(z) = 7424-35728z+61041z^2-44032z^3+11264z^4. }$$

Exact exclusion tests

At the good prime $2147483647$:

• $D_3$ is square-free;
• $D_3$ is coprime to all eight factor blocks in Entry 1866;
• $D_3$ is coprime to the $y_4=0$ condition;
• $D_3$ is coprime to the $y_5=0$ condition.

The last test uses the exact repeated-root specialization

$$y_5^2=0 \quad\Longleftrightarrow\quad 780-375z-140\sqrt5+173\sqrt5z=0,$$

up to an invertible quadratic-field constant. Therefore the quartic is not a soft or previously compiled one-/two-wall factor.

Remaining saturation gate

The discriminant detects criticality of the triple-wall pullback. It does not yet prove that all three wall multipliers are nonzero. A generic root could still belong to an uncompiled lower-wall critical component with one vanished multiplier.

Hence the established statement is

$$\boxed{ D_3(z)\text{ is a new nonsoft triple-wall candidate, not yet certified period singular support.} }$$

Next falsifier

Compute the source-ordered null vector of the full fiber Jacobian on $P=\partial_vP=0$. Saturate by each of its three wall-multiplier coordinates. If all three are units on the generic quartic locus, adjoin $D_3$ to the Picard--Fuchs candidate denominator. Otherwise classify the surviving lower-wall factor and do not enlarge the bound.

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--1868
• allocator claim: seqclaim-e6b6290cf809258efcfd1a01

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