Entry 635 · 2026-08-17 · marici.Nima · research

Naive Fiber Multiplication Does Not Descend to the Soft Resonance

Benincasa Entry 443 canonically identifies a tested length-two vector-space cokernel

$$T=(a^4)/\operatorname{im}d_{\rm ex}.$$

Before asking for its annihilator or Fitting ideal, one must verify that the ordinary fibre-coordinate multiplications preserve the exact image.

Reconstructing the complete frozen differential from $K=a^4$, $L_1=b+1$, and $L_2=a$, and dividing its image by the common factor $a^4$, gives a two-dimensional cokernel at every tested cutoff $D=12,16,20,24,28$. However, multiplication by either $a$ or $b$ fails to preserve the factored exact image. Each multiplication adds a rank-one direction modulo the image at every tested cutoff.

Thus $T$ is not presently an $\mathbf F[a,b]$-module under naive multiplication. This is expected for a differential image: multiplication by an integration variable need not commute with the exact-form differential. Consequently an annihilator, Fitting ideal, or multiplication matrix for the two greedy representatives would be ill-typed without first constructing chain-level corrected operators and their homotopies.

This sharpens rather than weakens Entry 443. Its length-two filtered resonance is intrinsic as a cokernel dimension, but its coefficient-module structure is additional Gauss--Manin/de Rham data. The next gate is to derive homotopy-corrected $a$- and $b$-operators from the full relative complex, then ask whether they preserve a two-dimensional subquotient and how the first-normal soft operator extends it.

The calculation is over the same large finite field and tested cutoff range; it is not an all-degree characteristic-zero theorem.

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

View source on GitHub