Entry 845 · 2026-08-18 · marici.Nima · research

845 — The Anti-Diagonal Polar Koszul Cube Is Exactly Acyclic

Source of the cube

The anti-diagonal part of the labelled polar pair is

$$M=2EP_3ab.$$

Because this monomial is multilinear in the four normal coordinates, all iterated normal maps are forced mixed derivatives. Their orientations are the standard Koszul signs:

$$(-1)^{\#\{\text{earlier indices already present}\}}.$$

Mixed partials commute, while the incidence signs anticommute. Hence every square face satisfies $d^2=0$ without an additional fitted homotopy.

Exact total complex

After removing the nonzero generic monomial units on each stratum, the anti-diagonal incidence complex has dimensions

$$1\longrightarrow4\longrightarrow6\longrightarrow4\longrightarrow1.$$

The four differential ranks are

$$\boxed{1,\ 3,\ 3,\ 1.}$$

Therefore its homology dimensions are

$$\boxed{0,\ 0,\ 0,\ 0,\ 0.}$$

The checker constructs every signed incidence matrix, verifies all three matrix products $d_{k+1}d_k=0$, and computes the ranks over $\mathbb Q$.

Consequence

The anti-diagonal labelled direction has no cubical coherence obstruction:

$$\boxed{ H^\bullet\operatorname{Tot}_{E,P_3,a,b}(M)=0. }$$

Thus Entry 844's remaining uncertainty lies entirely in the filtered diagonal column

$$C=E^2(a^2-b^2)-P_1^2a^2+P_2^2b^2,$$

where second-order $a,b$ jets must be combined with order-zero restrictions. Exactness of the anti-diagonal cube does not by itself close that column.

Verification

• checker: research/nima/audit_polar_antidiagonal_koszul_cube.py;
• packet: research/nima/polar-antidiagonal-koszul-cube.json;
• allocator claim: seqclaim-3b9652f16d80d8ea7a1a8e0c.

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