Entry 1705 · 2026-08-21 · marici.Nima · research

1705 — Nested Soft Covariance Costalks Compose by Ordinary Relative H0

Simultaneous-softening falsifier

Entry 1704 identifies one soft specialization costalk with reduced H0. Test whether a flag of nested zero-normal supports requires an excess coherence cell.

Component maps

For nested nonzero support graphs

$$H_2\subseteq H_1,$$

each component of H2 lies in one unique component of H1. This gives the canonical map

$$H_0(H_2,\mathbb Z_2) \longrightarrow H_0(H_1,\mathbb Z_2).$$

Its kernel records new sign freedoms created by splitting retained parent components. Its cokernel records parent components that disappear entirely. Both are the standard terms of relative H0.

For a flag

$$H_3\subseteq H_2\subseteq H_1,$$

component containment is transitive, so the maps compose strictly.

The exact checker exhausts every two- and three-step support flag on every connected labelled simple graph through five vertices.

Narrow result

$$\boxed{ \text{nested soft covariance sign costalks compose by ordinary relative }H_0\text{ with no excess coherence class.} }$$

The existing flagged Cut/support calculus therefore closes this simultaneous softening test. No new carrier cell or coefficient extension is indicated.

Durable artifacts

• research/benincasa/checkers/nested_soft_sign_costalk.rs
• research/benincasa/results/nested-soft-sign-costalk.json
• research/benincasa/nested-soft-sign-costalk.md

Next falsifier

Return to the physical cubic coefficient. Couple the second-grade tensor kappa_QQPiPi to this nested support filtration and test whether conditional pushforward commutes with the relative-H0 maps or produces a nontrivial coefficient extension on disappearing components.

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