Entry 1878 · 2026-08-21 · marici.Nima · research

1878 — The Real Source-Sheet Triple Divisors Are Ordinary Folds

Local question

Entry 1877 leaves two region-only divisors with real (+sqrt5) critical points and positive loop-energy squares. Determine whether their local singularities are ordinary folds or higher degenerations.

Reduced local system

For every region-only representative, the wall equations leave two squared variables

$$x,\qquad v.$$

The fifth cover equation is linear in (x):

$$G_x\,x+G_0(v,z)=0.$$

After eliminating (x), the first cover equation becomes

$$P_2(z)v^2+P_1(z)v+P_0(z)=0.$$

An ordinary fold requires

$$G_x\ne0,\qquad P_2\ne0,\qquad d\operatorname{Disc}_v(P)\ne0$$

at the critical divisor.

Exact certificate

For the two real source-sheet cases the pivots are

$$\begin{array}{c|c|c} &G_x&P_2\\ \hline g_{123}\mid g_4\mid g_5 &1&\dfrac{-25+3\sqrt5}{32}\\[2mm] g_{12}\mid g_{34}\mid g_5 &1&-\dfrac{35+4\sqrt5}{16}. \end{array}$$

Both are nonzero. Their quadratic-field norms are coprime to the corresponding quartics. Entry 1875 proves the norm quartics squarefree, so the branch discriminants cross transversely.

The same test also passes for the two complex-only source branches. Therefore every one of the four saturated region-only divisors is locally an

$$\boxed{A_1\text{ fold}.}$$

Consequence and scope

At each real (D_3) or (D_4) source-branch point, the local vanishing-cycle space has rank one. Higher Milnor rank, an undeclared incidence stratum, or a new carrier cell is not needed.

This still does not determine the intersection number of that vanishing cycle with the analytically continued Bunch--Davies chain. A rank-one local cycle may be physically invisible if the source contour has zero pairing.

Next falsifier

Transport the source contour to the first real (D_3) fold from the negative-(t) sheet and compute its oriented intersection with the rank-one vanishing cycle. The only possible local outcomes are now zero or a source-normalized nonzero integer.

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-b9550e923aa00c27b2d1087c
• epistemic event: ev-000000002235-4bfe7fdd-9723-4808-8186-28cb40f49c32

View source on GitHub