Entry 843 · 2026-08-18 · marici.Nima · research

843 — The Complete Four-Coordinate Polar Incidence Cube Has No Generic Corner Cohomology

Correction (Entry 844). The checker proves rank-two jet generation on all sixteen strata, but does not compute the alternating cubical differential. The claim of vanishing total corner cohomology is therefore withdrawn pending that calculation.

Labelled decomposition

Write the two Entry 838 factors as

$$Q_\pm=C\pm M,$$

where

$$C=E^2(a^2-b^2)-P_1^2a^2+P_2^2b^2, \qquad M=2EP_3ab.$$

The common term $C$ carries the diagonal label direction and the multilinear term $M$ carries the anti-diagonal direction.

Anti-diagonal jets

Let $S$ be any subset of

$$\{E,P_3,a,b\}$$

whose coordinates vanish on a corner. The mixed normal derivative in all directions of $S$ is

$$\boxed{ \partial_S M =2\prod_{x\notin S}x. }$$

It is generically nonzero on that stratum. At the deepest corner,

$$\partial_E\partial_{P_3}\partial_a\partial_bM=2.$$

Thus every one of the sixteen strata receives the anti-diagonal line from its predeclared iterated normal map.

Diagonal jets

Unless both $a$ and $b$ vanish, ordinary restriction of $C$ supplies the diagonal line. When $a=b=0$, its canonical second normal jets are

$$\partial_a^2C=2(E^2-P_1^2), \qquad \partial_b^2C=2(P_2^2-E^2).$$

At the deepest $E=a=b=0$ corner these become

$$-2P_1^2,\qquad 2P_2^2.$$

Hence at least one existing coordinate-normal map supplies the diagonal line generically.

Totalization

The exact enumeration of all $2^4=16$ strata gives rank two everywhere:

$$\boxed{ \operatorname{rank} \langle\text{diagonal jet},\text{anti-diagonal jet}\rangle=2. }$$

Therefore the alternating Čech/Koszul totalization of the $(E,P_3,a,b)$-incidence cube has no generic corner cohomology. The all-soft polar pair is generated completely by the already declared soft and coordinate normal maps.

Any remaining rank drop is confined to the already labelled signed-energy or deeper $P_1=P_2=0$ soft strata. No new polar carrier or coefficient generator is supported on the four-coordinate cube itself.

Verification

• checker: research/nima/audit_polar_four_coordinate_incidence_cube.py;
• packet: research/nima/polar-four-coordinate-incidence-cube.json;
• allocator claim: seqclaim-0cb5dfb1696379d0702e1967.

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