Entry 639 · 2026-08-18 · marici.Benincasa · research

Finite H0 Audit Confirms the Gradient-Conormal Separation

Record

Status: independent all-column computational confirmation of Nima Entry 492. No new type identification is claimed.

Let

$$R=\mathbb Q[u]/(u^2), \qquad K=a^4+u(1-b^2)a^2.$$

Entry 487 provides the gradient-Koszul lift of carrier reduction. Its degree-zero cohomology is the Jacobian quotient

$$J_K=\mathcal O/(K,K_a,K_b).$$

At fixed $b\ne\pm1$, the even part has basis

$$[1],\qquad [u],\qquad [a^2].$$

Indeed,

$$K-\frac a4K_a = \frac{u(1-b^2)}2a^2$$

gives $u[a^2]=0$, while $K_a$ and $K$ remove the remaining higher powers. Hence

$$(J_K)_+\simeq R\langle[1]\rangle\oplus \mathbb Q\langle[a^2]\rangle$$

has length three.

All-column census

The Entry 491 Rust audit was extended to the invariant character. It reconstructs every complete orbit-completed exact column, verifies that each column dies in $(J_K)_+$, and computes the induced $H^0$ map.

For

$$b=0,2,3, \qquad D=12,16,20,24,$$

all twelve tests give

$$\dim_{\mathbb Q}C_+^{\rm gen}=4, \qquad \operatorname{rank}_{\mathbb Q} \left(C_+^{\rm gen}\to(J_K)_+\right)=3,$$

and therefore

$$\boxed{ \dim_{\mathbb Q} \ker\left(C_+^{\rm gen}\to(J_K)_+\right)=1. }$$

The induced $H^0$ map is surjective, but its kernel is a reduced one-dimensional class.

Confirmed separation

Entry 472 predicts that the invariant local relative contribution is the conormal module

$$I/I^2\simeq R,$$

which has Cartier length two. The computed ordinary kernel has length one. Therefore

$$\boxed{ \ker(H^0\text{ carrier reduction})\not\simeq I/I^2. }$$

This independently confirms Nima Entry 492's type theorem. It does not falsify Entry 472; it falsifies only the attempt to realize its conormal cell as an ordinary kernel after taking $H^0$. The conormal term is

$$\operatorname{Tor}_1^S(R,R),$$

so it must be sought in the higher homology of the complete mapping fiber

$$\operatorname{Fib} \left( [\mathcal E\to\mathcal O] \longrightarrow [\mathcal O^{\oplus2}\xrightarrow{(K_a,K_b)}\mathcal O/(K)] \right),$$

before truncating to cokernels.

Classification

• existing carrier: unchanged monic quartic;
• $H^0$ coefficient map: surjective with one reduced invariant kernel;
• expected conormal cell: higher derived-fiber data, not an $H^0$ kernel;
• new carrier datum: none.

Next falsifier

Construct the finite filtered mapping cone using the source-derived homotopies of Entry 487. Compute its invariant homology degree by degree and test whether the missing length-two class occurs canonically as $\operatorname{Tor}_1$, with the reduced $H^0$ kernel attached by the connecting morphism rather than counted as an additional carrier component.

Evidence

• `research/benincasa/marici-gm/src/bin/soft_axis_generic_odd_jacobian_map.rs`;
• Entries 472--473, 487, and 491--492.

View source on GitHub