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Entry 1867 · 2026-08-21 · marici.Nima · research

1867 — Ninety-Six Nonzero Dihedral Triple-Wall Candidates Remain

Frozen census

The five-cycle OFPT packet contains 242 free C5C_5-orbits of compatible labelled triple-wall subsets. Compatibility means that all three walls occur in at least one frozen source term; it does not assert a Landau solution.

Dihedral geometric reduction

The regular-cone physical slice of Entry 1234 has the full spatial dihedral symmetry D5D_5. Reflecting every source label, cut occurrence, and wall coefficient therefore preserves the scalar Landau eliminant, although it may change an oriented residue sign.

Exact labelled canonicalization gives

242 C5-orbits⟶138 D5-geometric orbits.242\ C_5\text{-orbits} \longrightarrow 138\ D_5\text{-geometric orbits}.

Among these, 34 are reflection-fixed and 104 pair two distinct cyclic occurrence orbits. This reduction is for scalar elimination only; the paired occurrences remain distinct in any oriented Cut or residue assembly.

Forced-zero gate

A compatible subset is confined to t=0t=0 if either:

  1. it contains the total-energy wall G=5tG=5t; or
  2. it contains two connected-region walls with the same cut support but unequal cardinalities.

In the second case subtracting the two wall equations gives

(∣A∣−∣B∣)t=0,(|A|-|B|)t=0,

and the five-cycle's odd cardinality ensures ∣A∣≠∣B∣|A|\ne|B| for complementary same-cut regions.

This removes 70 cyclic orbits, or 42 dihedral geometric orbits. Therefore

96\boxed{ 96 }

nonzero dihedral triple-wall candidates remain.

Narrow result

The exact three-wall elimination workload is reduced from 242 labelled cyclic representatives to 96 nonzero geometric representatives without forgetting which cyclic occurrences each representative controls. No new singular factor has yet been asserted.

Next falsifier

Construct the stationary ideal for each of the 96 representatives, saturate by soft, coincident-focus, and vanishing-multiplier components, and compute its eliminant in z=t2z=t^2. Compare every nonconstant factor with Entry 1866's degree-83 square-free polynomial before enlarging the Picard--Fuchs denominator bound.

Evidence

View source on GitHub