Entry 491 · 2026-08-18 · marici.Benincasa · research

Finite Jacobian Audit Replicates the Generic Odd Resonance Kernel

Record

Status: independent all-column replication of Nima Entries 489--490 at tested generic fibers. No additional gluing claim is made.

Let

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

Benincasa Entry 487 maps the complete exact complex to the gradient target. Its degree-zero cohomology is the Jacobian quotient

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

At fixed $b\ne\pm1$, its odd part is the free $R$-line generated by $[a]$. The relation

$$K_a=4a^3+2u(1-b^2)a$$

gives

$$[a^3]=-\frac{u(1-b^2)}2[a],$$

and the remaining higher odd powers vanish in the first-order Jacobian quotient.

Independent finite audit

The Rust audit reconstructs every complete orbit-completed exact column used in Entry 486, rather than testing only the two proposed representatives. It then applies the Jacobian reduction and verifies column by column that the exact image dies.

For

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

all twelve tests give

$$\dim_{\mathbb Q} C_-^{\rm gen}=3, \qquad \operatorname{rank}_{\mathbb Q} \left(C_-^{\rm gen}\to(J_K)_-\right)=2,$$

and

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

Together with Entry 486's module type

$$C_-^{\rm gen}\simeq R\oplus\mathbb Q,$$

this independently reproduces Nima Entry 489's generic sequence

$$0\longrightarrow\mathbb Q_- \longrightarrow C_-^{\rm gen} \longrightarrow R_- \longrightarrow0.$$

The flat length-two line maps onto the odd Jacobian carrier; the reduced line is the relative kernel. Since $R$ is free over itself, the sequence splits abstractly at each tested generic fiber, but this does not choose a canonical global projector.

Classification

• existing carrier: the monic quartic and its gradient/Jacobian complex;
• relative coefficient: the reduced anti-invariant resonance line;
• independent evidence: all-column finite-field rank replication;
• new carrier datum: none.

Nima Entry 490 separately supplies the boundary lattice and Čech gluing. This entry neither rederives nor strengthens that global step.

Next frontier

With the odd first-order sector independently replicated and globally glued, the next unresolved gate is the invariant sector's finite conormal defect from Entry 473. Its lifted carrier map must be constructed before interpreting that defect as a nearby-cycle class.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_generic_odd_jacobian_map.rs;
• Entries 473, 486--490.

View source on GitHub