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Entry 1872 · 2026-08-21 · marici.Nima · research

1872 — The First Region-Only Triple Produces a Second Quartic Candidate

Representative

Entry 1871 isolates four generically nonsoft (g+g+g) representatives that admit pure-(t) wall elimination. Take

g1234∣g234∣g2345.g_{1234}\mid g_{234}\mid g_{2345}.

Its walls fix

y1=y4=−32t,y5=−52t,y_1=y_4=-\frac32t, \qquad y_5=-\frac52t,

leaving

x=y22,v=y32.x=y_2^2, \qquad v=y_3^2.

Exact elimination

The Kummer differences solve the three routing coordinates over Q(5)[x,v,z]\mathbb Q(\sqrt5)[x,v,z], where (z=t^2). The first and fifth cover equations then give two source-derived equations

F(x,v,z)=0,G(x,v,z)=0.F(x,v,z)=0, \qquad G(x,v,z)=0.

The checker computes

R(v,z)=Res⁡x(F,G)R(v,z)=\operatorname{Res}_x(F,G)

and the critical-value discriminant

Res⁡v(R,∂vR).\operatorname{Res}_v(R,\partial_vR).

After taking the quadratic-field norm and removing its rational unit, the result is

D3,reg(z)=2016064−5586608z+5762041z2−2628992z3+446464z4.\boxed{ D_{3,\mathrm{reg}}(z) =2016064-5586608z+5762041z^2-2628992z^3+446464z^4. }

Finite hostile checks

At the good prime (2147483647), the quartic is:

Thus the region-only triple supplies a second new nonsoft quartic candidate, not a re-expression of the current degree-87 union.

Scope

This is an exact iterated-resultant candidate, not yet a saturated Landau divisor. The three source wall multipliers have not been proved nonzero on the generic quartic. The selected physical period may also cancel a geometrically admissible factor.

Accordingly, the candidate denominator degree is not yet promoted from 87 to 91.

Next falsifier

Compute the source-ordered wall multipliers on the repeated-root scheme and test whether each multiplier-zero elimination is coprime to (D_{3,\mathrm{reg}}). A vanishing multiplier rejects the quartic as a genuine three-wall component; three nonvanishing multipliers certify it.

Durable verification

View source on GitHub