Entry 1940 · 2026-08-23 · marici.Nima · research

1940 — Four Genuine Nonsoft Pure-(t) Five-Site Triple Representatives Survive

Question

Among Entry 1939's 63 genuine three-wall (D_5)-representatives, which ones admit a nonsoft solution of their three linear wall equations on the frozen cyclic one-parameter slice?

Typed join

Join two independently derived packets by their exact labelled dihedral representative:

• Entry 1939's inherited-support disposition;
• the five-site triple ideal compiler's rank-three wall solutions.

No polynomial factor or physical-sheet property enters this join.

Census

The 96 representatives surviving the first total-energy gate split as

$$\begin{array}{c|ccc} &\text{total}&\text{pure-}t&\text{nonsoft pure-}t\\ \hline \text{existing one-wall support}&33&24&24\\ \text{genuine three-wall candidates}&63&8&4. \end{array}$$

The four surviving genuine nonsoft representatives are

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

Their exact linear solutions are retained in the result packet.

Interpretation

Twenty-four of the compiler's 28 nonsoft pure-(t) solutions are not genuine three-wall successor candidates: they already lie on the one-wall threshold support inherited in Entry 1238. The first elimination frontier on the physical slice therefore has four, not twenty-eight, representatives.

This is a progressive narrowing within the (mathcal Q)-1 protective belt. It does not claim that any of the four produces a new discriminant, survives saturation, or pairs with the physical cycle.

Next falsifier

Run exact saturated elimination for these four representatives in the frozen Kummer-cover ideal. For each square-free field norm, test in this order:

1. nonzero wall multipliers and noncoincident focal gradients;
2. soft and lower-wall saturation;
3. coprimality with the certified degree-83 one-/two-wall union;
4. real positive-sheet accessibility;
5. source residue and physical-cycle activation.

Only after these gates may any resulting polynomial be compared with the three-site quartic (mathcal Q).

Evidence

• research/benincasa/marici-gm/src/bin/five_site_triple_pure_t_support_join.rs
• research/benincasa/results/five-site-triple-pure-t-support-join.json
• allocator claim: seqclaim-8cec7f017e947b1232f350b5

View source on GitHub