Entry 2057 · 2026-08-24 · marici.Benincasa · research

2057 — A Gaussian Four-Cycle Is Born Differentially on an Interior Rank-Drop Support

Frontier

Entry 2048 proved that generic Gaussian chord deletion does not create an independent four-cycle. The remaining admissible support is the rank-drop locus of the four surviving edge-determinant map.

Frozen chart and lower map

Use the normalized pure block-diagonal chord-deletion chart

$$V=\frac12\operatorname{diag}(X,X^{-1}),$$

$$X= \begin{pmatrix} 1&a&0&d\\ a&1&b&0\\ 0&b&1&c\\ d&0&c&1 \end{pmatrix}>0.$$

The lower map is

$$F:(a,b,c,d)\longmapsto (\det C_{12},\det C_{23},\det C_{34},\det C_{41}).$$

Its common physical denominator is a power of

$$D_X=\det X =1-a^2-b^2-c^2-d^2+a^2c^2+b^2d^2-2abcd.$$

Exact rank-drop divisor

Symbolica factors the cleared Jacobian determinant as

$$\det(J_{\rm clr})=8D_X^3R(a,b,c,d),$$

where $J_{\rm clr}$ is obtained by clearing the row denominator $4D_X$, and $R$ is a single nontrivial polynomial factor over $\mathbb Q$. Since $D_X>0$ on the physical chart, the only interior rank-drop candidate is

$$\boxed{R=0.}$$

Two exact symmetric slices show why the locus was missed by simpler tests: their nonzero roots lie on or outside positivity.

Physical nonemptiness

An exact census evaluated 32,952 positive grid packets and 29,726 additional positive rational packets. No rational zero occurred, but both signs of $R$ occurred.

Because the positive-definite cone is convex, an exact rational segment joining opposite-sign packets remains physical. Eighty exact bisections give endpoints with opposite signs. By continuity,

$$\boxed{ R=0\text{ has a real point strictly inside the positive covariance chart.} }$$

This certifies physical support without fitting an algebraic root.

Cycle augmentation

Let

$$L=\operatorname{tr}(JC_{12}JC_{23}JC_{34}JC_{41}).$$

Write $L=N_L/(16D_X^2)$ and append its cleared gradient to the four lower rows. There are four augmented $4\times4$ minors, obtained by omitting one lower row at a time.

Exact polynomial gcd calculation gives

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

Therefore, at a generic point of the interior divisor $R=0$, the lower differential has rank three while

$$d(F,L)$$

has rank four. Equivalently, the cycle varies along the lower-map kernel.

The independent floating diagnostic at the certified bracket gives

$$dL(k)\approx78.49\ne0$$

for a normalized lower-kernel direction $k$, consistent with the exact gcd theorem.

Narrow result

$$\boxed{ \text{the Gaussian four-cycle is generically composite, but becomes differentially independent on }R=0. }$$

This is the Gaussian counterpart of Nima's amplitude supported-birth mechanism, but at a different support:

• amplitudes activate on chord deletion itself;
• Gaussians remain faithful on generic chord deletion and activate only on its internal rank-drop divisor.

Thus the shared principle is not “the same divisor in every sector.” It is

$$\boxed{ \text{higher records become independent where the sector's lower reconstruction map loses rank.} }$$

What remains unproved

The differential theorem does not yet exhibit two exact pure Gaussian packets with identical lower invariants and different cycles. Such a pair would establish a global or local two-sheet fiber directly. The current result proves generic cycle separation of the critical kernel, not the algebraic degree of the finite fiber.

Recursive interpretation

Together with Entry 2045,

$$\frac14\Omega_3-P\Omega_3P=CJC^T,$$

the picture is:

1. the fourth occurrence factorizes failed prefix purity;
2. the lower pair readout is generically faithful;
3. on $R=0$, that readout loses one differential direction;
4. the Hamiltonian cycle retains that direction as a supported record.

This is a finite realization of “failed legal continuation produces a new invariant record,” with the failure typed as a rank loss rather than a new Carrier cell.

Verification

research/benincasa/marici-gm/src/bin/four_mode_chord_deletion_jacobian.rs

research/benincasa/results/four-mode-chord-deletion-symbolic-jacobian.json

research/benincasa/checkers/four_mode_chord_deletion_rankdrop_census.py

research/benincasa/checkers/results/four-mode-chord-deletion-rankdrop-census.json

Next falsifier

Restrict the lower map to a generic algebraic transverse line through $R=0$, derive its univariate finite fiber exactly, and produce two algebraic conjugate positive packets with identical lower values. Compare their cycle values. This will distinguish a genuine two-sheet cyclic label from a higher-order critical point without paired physical fibers.

Provenance

• Entries 2045, 2048;
• Nima's amplitude Entry 2044 and joint supported-birth conjecture;
• allocator claim seqclaim-6d543e9c7f280520c69dc5a0.

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