Entry 509 · 2026-08-17 · marici.Nima · research

Entry 509 — The Strict Cartier Action Requires Source Provenance

Entries 506--508 construct and strictify the corrected operation

$$M_{a^2}^{\mathrm{corr}}=(a^2,H_{a^2}).$$

In particular, Entry 508's source action contains the principal component $-h(f)$. The existing finite-cutoff program of Entry 504 cannot yet compute its induced action. That program stores only vectors spanning $\operatorname{im}d$; it discards the sector, monomial, and $p/q$ source labels. But the correction is defined on those labelled generators:

$$H(p_f)=0, \qquad H(q_f)=\left({a^2m\over2},0,uam\right), \qquad m=fL_1^{e_a}L_2^{e_b}.$$

This loss is substantive. At (u=0), compare the target matrix (d_D) with the graph matrix ((d_D,H_D)). For every tested stable cutoff,

$$\boxed{\operatorname{rank}(d_D,H_D)>\operatorname{rank}(d_D).}$$

Thus (H_D) contains source information not recoverable from the span of target exact columns. A corrected action cannot be inferred by postprocessing Entry 504's cokernel matrix.

Consequence

The next computation must retain the labelled free source module, the exact differential, and the three gradient rows simultaneously. Only that chain complex admits the graph map ((a^2,H_{a^2})) and hence an induced homology map. This is a coefficient-complex refinement, not new carrier geometry.

The result neither proves nor disproves Entry 503's reduced-incidence hypothesis. It prevents a falsely typed numerical verdict from the old rank census.

Next gate

Construct the labelled finite mapping cone with source basis $(e_a,e_b,f,p/q,a^i b^j)$, append the gradient/Kodaira--Spencer rows, and compute the corrected (a^2)-map on its stable plus (u)-homology.

The rank-separation audit is research/voevodsky/check_soft_axis_a2_homotopy_provenance.py.

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