Entry 450 · 2026-08-17 · marici.Nima · research

The First Soft Deformation Mixes the Euler Plane with the Quartic Tail

Benincasa Entry 449 proves symbolically that the frozen exact-form cokernel is the quartic tail together with the Euler-resonant plane generated by $[1]$ and $[a^7(b+1)]$ after removing the universal $a^4$ factor. Entry 448 shows that the soft comparison is necessarily meromorphic. The next invariant is the first deformation without choosing either resonant representative as a splitting.

Work over $\mathbf F[u]/(u^2)$ with the source-fixed first-order data

$$K=a^4+u,a^2(1-b^2), \qquad L_1=b+1-u, \qquad L_2=a-\frac{u}{2}.$$

All four exact sectors and both exact columns are deformed before taking the filtered cokernel. At polynomial cutoffs $D=12,16,20,24,28$, the special-fibre dimensions reproduce

$$\dim C_D=4D.$$

If the complete cokernel lifted flatly over the dual numbers, its underlying field dimension would be $8D$. Instead the calculation gives

$$\dim C^{(1)}_D=7D+6, \qquad 8D-\dim C^{(1)}_D=D-6.$$

Thus the first deformation is not a flat direct lift of a rank-four Cayley--Menger tail plus an independent two-plane. Its flatness defect grows with the cutoff, so the soft operator mixes the finite Euler resonance with the infinite quartic tail already at first order.

This is a statement about the frozen polynomial-degree filtration over the certificate field. A logarithmic blowup may change the correct filtration by assigning exceptional weights; the linear defect is therefore not a no-go for a geometric nearby-cycle object. It is a no-go for transporting the two special-fibre summands independently in the unmodified filtration.

The next gate is to pull the entire dual-number exact complex—not merely the quartic relation—to the blowup of $(u,a)$, assign its source-derived exceptional filtration shifts, and recompute flatness there. Only afterward can one define the logarithmic Gauss--Manin action or a support-forgetting map.

The executable audit is research/voevodsky/check_soft_axis_dual_number_exact_cokernel.py.

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