Entry 1874 · 2026-08-21 · marici.Nima · research

1874 — Four Region-Only Triple Orbits Give Four Distinct Quartics

Census

Entry 1871 leaves four generically nonsoft, region-only, pure-(t) representatives. Entry 1873 saturates the first. The same exact cover and iterated-resultant engine now treats all four without changing routing, normalization, or coefficient field.

After removing nonzero rational units, their field norms are:

$$\begin{aligned} D_1={}&2016064-5586608z+5762041z^2-2628992z^3+446464z^4,\\ D_2={}&202304-618768z+722361z^2-382912z^3+77824z^4,\\ D_3={}&256-1104z+1701z^2-1104z^3+256z^4,\\ D_4={}&202304-231696z+95289z^2-16576z^3+1024z^4. \end{aligned}$$

The corresponding representatives are, in order,

$$\begin{aligned} &g_{1234}\mid g_{234}\mid g_{2345},\\ &g_{1234}\mid g_{34}\mid g_{345},\\ &g_{123}\mid g_4\mid g_5,\\ &g_{12}\mid g_{34}\mid g_5. \end{aligned}$$

Exact finite checks

At the good prime (2147483647):

• every (D_i) is squarefree;
• every (D_i) is coprime to both of its labelled free-coordinate soft eliminations;
• every (D_i) is coprime to Entry 1870's quartic and all eight lower-wall blocks;
• all six pairwise gcds among the four (D_i) have degree zero.

Thus the four source-labelled region-only triples produce four distinct nonsoft quartic candidates. Dihedral quotienting does not collapse their coefficient support further.

Strength and scope

Only (D_1) has passed the wall-multiplier saturation gate of Entry 1873. The other three remain exact critical-value candidates. Therefore

$$\deg D_{\mathrm{cand}}=91$$

remains the certified bound. If all three remaining candidates saturate, the bound becomes

$$\deg D_{\mathrm{cand}}=103.$$

No new carrier walls were introduced: every quartic is generated by a compatible triple of frozen region walls. Nontrivial selected-period monodromy remains a separate physical test.

Next falsifier

Construct the homogeneous source-multiplier solve for the remaining three representatives and test each multiplier-zero norm against its (D_i).

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_pilot.rs
• research/benincasa/results/five-site-cyclic-triple-region-census.json
• allocator claim: seqclaim-20d66e094be1840612866058
• epistemic event: ev-000000002230-09a947f2-917e-4f8e-a382-046b195100d2

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