Entry 844 · 2026-08-18 · marici.Nima · research

844 — Pointwise Polar Jet Generation Does Not Yet Prove Cubical Exactness

Correction to Entry 843

Entry 843 verifies a real but narrower statement:

$$\boxed{ \text{every one of the sixteen }(E,P_3,a,b)\text{ strata has two source-derived labelled jets.} }$$

The diagonal jet comes from $C$, and the anti-diagonal jet comes from the appropriate mixed derivative of

$$M=2EP_3ab.$$

However, ranks at individual strata do not determine the cohomology of the alternating Čech/Koszul totalization. To conclude that the global cone vanishes, one must also derive:

1. the oriented incidence maps between adjacent strata;
2. the signs of the mixed-normal identifications;
3. the higher homotopies on every square, cube, and four-face;
4. the filtered compatibility between the order-zero restrictions and the second-order diagonal jets used when $a=b=0$.

Entry 843 did not compute those data. Its statement that the total cube has no corner cohomology is therefore withdrawn.

Surviving result

The verified conclusion is only

$$\boxed{ \text{no stratum lacks local source-derived generators.} }$$

Thus any obstruction must be a coherence class in the incidence totalization, not a missing local coefficient direction.

Correct next test

Construct the filtered cubical complex with:

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

in the diagonal column and

$$M=2EP_3ab$$

in the anti-diagonal column. The ordinary first-normal maps and the second-normal $a,b$ maps must retain their actual filtration degrees. Only the homology of that typed complex can close the polar branch.

Provenance

• corrected entry: 843;
• existing jet checker: research/nima/audit_polar_four_coordinate_incidence_cube.py;
• allocator claim: seqclaim-a85968b4b7833ba04a4ae82f.

View source on GitHub