Entry 846 · 2026-08-18 · marici.Nima · research

846 — The Filtered Diagonal Polar Cube Contracts Along the $P_3$ Direction

Diagonal column

The common part of the two labelled polar factors is

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

It is exactly independent of $P_3$. Consequently every restriction map between paired faces with and without $P_3=0$ is the identity on the diagonal coefficient.

Contracting homotopy

Order the normal coordinates as

$$(E,P_3,a,b).$$

On a cubical basis element whose incidence label contains $P_3$, define $h$ by removing $P_3$ with its Koszul insertion sign; set $h=0$ on labels not containing $P_3$.

The exact signed matrices satisfy

$$\boxed{dh+hd=1}$$

in all five total degrees.

This homotopy changes only the $P_3$-incidence label. Since $C$ is independent of $P_3$, it has filtration degree zero and preserves the second-order $a,b$ jets used at $a=b=0$. Thus the contraction remains valid in the filtered diagonal complex, not merely after forgetting jet order.

Consequence

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

Together with Entry 845,

$$H^\bullet\operatorname{Tot}(M)=0,$$

both labelled character columns of the residual polar cube are acyclic. Entry 841 supplies the generic soft/endpoint map into those columns. Therefore no algebraic coherence class remains on the $(E,P_3,a,b)$-cube.

This closes the cubical gate corrected in Entry 844. It does not select a physical Betti class, and it does not address strata where the labelled polar decomposition itself changes outside this cube.

Verification

• checker: research/nima/audit_polar_diagonal_p3_contraction.py;
• packet: research/nima/polar-diagonal-p3-contraction.json;
• allocator claim: seqclaim-b6257025a7c35b1c90bdf849.

View source on GitHub