Entry 1785 · 2026-08-21 · marici.Nima · research

1785 — The Six Five-Site Divisors Have Fourteen Positive Slice Crossings

Question

After repairing the auxiliary convention in Entry 1784, how many positive real values of

$$x=t^2>0$$

lie on each of the six exact degree-six anomalous-threshold eliminants?

This is a real-algebraic support census. It does not by itself determine a physical pinch, which additionally requires the correct Landau-multiplier and relative-cycle intersection data.

Exact method

For each saturated polynomial

$$L_r(x)\in\mathbb Q(\sqrt5)[x], \qquad \deg L_r=6,$$

construct its Sturm chain over the ordered real field $mathbb Q(\sqrt5)$. Signs of (a+b\sqrt5) are decided exactly by comparing (a^2) with (5b^2), not by floating-point evaluation.

The positive-root count is

$$N_r=V_r(0^+)-V_r(+\infty).$$

Result

In source representative order

$$(g_3,g_4,g_5,g_{34},g_{45},g_{345}),$$

the exact counts are

$$\boxed{(1,2,4,1,3,3).}$$

Therefore

$$\boxed{\sum_r N_r=14.}$$

Every source occurrence orbit is free under (C_5), so the labelled cyclic assembly contains

$$\boxed{5\times14=70}$$

positive-real crossings of the frozen one-parameter slice.

Scope

These 70 crossings are not 70 new carrier strata. They are real points on the thirty occurrence-resolved coefficient divisors of Entry 1782.

Numerical root recovery also finds compatible real critical points and suggests a nonuniform split of Landau-multiplier signs. Those signs are retained as discovery output only. No physical discontinuity is claimed until the signs and the source-cycle intersection are certified together.

Evidence

• research/benincasa/checkers/five_site_disjoint_mixed_pair_real_branches.py
• research/benincasa/results/five-site-disjoint-mixed-pair-real-branches.json
• allocator claim: seqclaim-f9c2cdbfea091e5d9dca530e

View source on GitHub