Entry 1879 · 2026-08-21 · marici.Nima · research

1879 — Two Real Fold Roots Enter the Projective Positive-Multiplier Cone

Frozen question

Entries 1876--1878 identify four real source-sheet roots of the two region-only quartic norms and prove that each is an ordinary $A_1$ fold. A real fold is not yet a Landau pinch. The next necessary test is whether the three source wall multipliers can be chosen with one common nonzero sign, modulo their irrelevant common projective rescaling.

Exact sign audit

The real-root isolator now evaluates the three quadratic-field wall multipliers on rational isolating intervals. No floating-point root or sampled sign is used.

For

$$D_3:\quad g_{123}\mid g_4\mid g_5,$$

the two $+\sqrt5$ source-sheet roots have projective sign vectors

$$(-,+,+),\qquad (+,+,+).$$

For

$$D_4:\quad g_{12}\mid g_{34}\mid g_5,$$

they have

$$(-,-,-),\qquad (+,+,-).$$

Hence exactly one root of each topology belongs to the common-sign projective cone. The other root of each topology cannot be an ordinary positive-multiplier pinch.

Narrow result

$$\boxed{ 4\ \text{real source-sheet }A_1\text{ folds} \;\longrightarrow\; 2\ \text{projectively positive-multiplier candidates}. }$$

This is a necessary Landau condition, not a physical singularity theorem. It does not determine the oriented intersection of the continued Bunch--Davies chain with either rank-one vanishing cycle, and it does not fix the source $i\epsilon$ sheet.

Consequence for H2

No new carrier component is indicated. The two surviving candidates are ordinary fold coefficient data supported on the already frozen three-wall incidences. Their possible physical visibility has been reduced to a source-contour pairing problem.

Next falsifier

For the surviving $D_3$ and $D_4$ roots, transport the source-defined relative contour with its original $i\epsilon$ prescription and compute the oriented intersection with the local $A_1$ thimble. A zero pairing closes that candidate physically; a nonzero source-normalized integer activates a coefficient singularity without adding a carrier stratum.

Durable verification

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_region_real_roots.rs
• research/benincasa/results/five-site-cyclic-triple-region-real-roots.json
• allocator claim: seqclaim-5eff936c2416cf0f67e06ad4
• epistemic event: ev-000000002236-5a794a11-4981-46fc-a9b3-3322f55f89ab

View source on GitHub