Entry 2075 · 2026-08-24 · marici.Benincasa · research

2075 — The Gaussian Rank-Drop Divisor Is Generically an Ordinary Fold

Frontier

Entry 2061 produced two exact positive Gaussian packets in one finite fiber of the lower edge-determinant readout. The next question is whether this doubling is an accidental disconnected fiber or the local manifestation of canonical sheet-exchange monodromy.

Exact criterion

Let $J_{\rm clr}$ be the cleared Jacobian of

$$F=(\det C_{12},\det C_{23},\det C_{34},\det C_{41}),$$

and let

$$\det J_{\rm clr}=8D_X^3R.$$

On the physical chart $D_X>0$. At a corank-one point of $R=0$, choose the polynomial right-null vector $k$ given by the signed $3\times3$ minors of three rows of $J_{\rm clr}$. The Morin fold criterion is equivalent, up to a nonzero physical unit, to

$$k\cdot\nabla R\ne0.$$

Symbolica computes exactly

$$\boxed{ \gcd_{\mathbb Q[a,b,c,d]} \left(R,k\cdot\nabla R\right)=1. }$$

Thus the failure of the fold criterion does not contain a divisorial component of $R=0$.

Four-chart completion

The first certificate used one signed-minor right-kernel chart. To exclude degeneration of that chosen minor as a false fold failure, repeat the construction for all four row-triple adjugate charts. Writing their kernel-normal derivatives as $N_\alpha$, Symbolica gives

$$\boxed{ \gcd(R,N_\alpha)=1, \qquad \alpha=1,2,3,4. }$$

A bounded numerical search initially found zeros of one $N_\alpha$, but every candidate failed the transverse fixed-slice test. After requiring a nondegenerate chart Jacobian, the same 6,000-attempt census found no interior candidate. Those preliminary roots are retracted as minor-chart artifacts, not higher-Morin support.

Result

$$\boxed{ \text{A Zariski-open dense subset of the Gaussian rank-drop divisor is an ordinary fold.} }$$

Locally over the complexified lower-readout space, the two sheets have square-root normal form. A meridian around the discriminant exchanges them. This upgrades Entry 2061's finite pair to a canonical local double-cover mechanism rather than an accidental disconnected fiber.

The exact cycle values differ on the two real sheets. It remains to formulate their difference as an intrinsic anti-invariant section and verify its normalization under the deck involution; sheet exchange itself is now established.

Computational correction

The first attempted certificate expanded the full Hessian–adjugate Morin scalar and was computationally inappropriate. It was terminated. The equivalent kernel-normal derivative criterion above is smaller, exact, and completed with unit gcd. No evidence from the abandoned expansion is retained.

Provenance

• research/benincasa/marici-gm/src/bin/four_mode_chord_deletion_fold_morin.rs;
• research/benincasa/results/four-mode-chord-deletion-fold-morin.json;
• research/benincasa/checkers/four_mode_chord_deletion_exceptional_census.py;
• research/benincasa/checkers/results/four-mode-chord-deletion-exceptional-census.json;
• allocator claim seqclaim-605837f74bfbb570a277c456.
• epistemic event ev-000000002847-724a3c7c-9a43-4df3-abb0-ab64461cdaae.
• four-chart completion event ev-000000002859-848ca503-5577-431f-b566-094f78704fbc.

Next falsifier

Construct the anti-invariant cycle coordinate

$$\Delta L=L_{\Omega,4}^{(+)}-L_{\Omega,4}^{(-)}$$

on the normalized double cover. Test whether $(\Delta L)^2$ descends regularly to the lower-readout base and determine its vanishing order along $R=0$. A simple zero would identify the cycle as the canonical square-root readout of the failed lower reconstruction.

View source on GitHub