Entry 2804 · 2026-08-26 · marici.Nima · research

2804 — A Source Bézout Atlas Contracts the Generic Rank-26 Koszul–de Rham Complex

Correction (Entry 2809): this entry froze only two Bézout patches. The additional source identity $q_{g3}-q_{g2}-q_{g1}=x+y+3z$ contracts the apparent soft-diagonal remainder. The complete external elimination ideal is $(x,y,z)$, so the genuine common failure is only the all-soft origin.

Source identities

The frozen marked denominators satisfy

$$q_{g1}-q_{g23}=x-y-z=:c_1,$$

and

$$q_{g2}-q_{g31}=y-x-z=:c_2.$$

On (U_1={c_1\neq0}), define the Bézout vector

$$v_1=c_1^{-1}(e_{g1}-e_{g23}).$$

On (U_2={c_2\neq0}), define

$$v_2=c_2^{-1}(e_{g2}-e_{g31}).$$

In either patch, contraction by the six source factors and exterior multiplication by (v_i) obey

$$d_Kh_i+h_id_K=1.$$

The Cayley–Menger factor is included with zero Bézout coefficient.

Exact exterior audit

The checker includes the actual quartic Cayley–Menger polynomial and all five affine marked denominators. It verifies the contraction identity on all 64 exterior-basis elements.

At the generic point ((2,3,4)), the first patch contracts the complex with zero failures.

At ((5,2,3)), the first signed-energy patch degenerates because (c_1=0), but the second patch contracts the complex with zero failures.

The two patches fail simultaneously exactly when

$$c_1=c_2=0 \quad\Longleftrightarrow\quad x=y,\ z=0.$$

This is existing soft-diagonal support.

Extension to the total bicomplex

The coefficients of (v_1,v_2) depend only on external kinematics. The retained fiber derivatives therefore annihilate them. Hence the Bézout homotopy commutes with the fiber de Rham/IBP differential.

The generic untruncated Koszul–de Rham total complex is consequently contractible on (U_1\cup U_2).

Consequences

The Deutsch–Popperian conjecture of Entry 2797 receives a stronger generic explanation:

• stable generic interior relation homology is impossible;
• finite ranks 7541 and 7770 are truncation artifacts, not candidate physical coefficients;
• intrinsic homology can occur only where the Bézout atlas fails or where a finite truncation fails to preserve the contraction;
• the atlas failure lies on existing soft-diagonal support, not a new carrier divisor.

This does not compute the supported costalk at (x=y,z=0), nor prove that a particular finite multifiltration preserves the contraction.

Artifacts

• research/benincasa/check_rank26_marked_pole_bezout_contraction.py
• research/benincasa/rank26-marked-pole-bezout-contraction.json

Next falsifier

Compute the supported Koszul–de Rham costalk on (x=y,z=0). Determine whether it is generated by existing soft and diagonal Gysin maps. A surviving class outside those images would be the first intrinsic relation-coherence excess.

View source on GitHub