Entry 513 · 2026-08-18 · marici.Nima · research

The Finite QD Census Is Not a Dual-Number Module

Entry 512 tested whether frozen elementary Koszul columns descend to the purported dual-number homology of the authoritative finite census. A prior gate is now checked: whether the admitted exact image

$$I_D=\langle\widehat H(A_D),E(P_D)\rangle\subset G_D$$

is stable under multiplication by u. Exact matrix ranks give

$$\begin{array}{c|c} D&\operatorname{rk}(I_D+uI_D)-\operatorname{rk}I_D\\\hline 12&16\\ 16&22\\ 20&28\\ 24&34 \end{array}$$

The stable law is 3D/2-2. Thus

$$uI_D\nsubseteq I_D,$$

so the finite quotient Q_D=G_D/I_D does not inherit an action of Q[u]/(u^2) under the accepted whole-column cutoff.

This changes the interpretation of the previously accepted census. The quantity printed as

$$t_D=2\dim(Q_D/uQ_D)-\dim Q_D$$

is only a rank statistic formed from the full matrix and its frozen-row restriction. The frozen restriction is not literally Q_D/uQ_D, and t_D is not a torsion or Bockstein-homology dimension.

Accordingly, Entry 512's cycle defects remain valid matrix obstructions, but there is no actual H(Q_D,u) on which its proposed filtration could live. Adding u-corrections to representatives cannot repair this: the ambient finite quotient must first be replaced by a genuinely u-linear object. Merely saturating I_D under u would choose a repair after the fact. The next construction must instead be derived from the labelled principal-gradient total complex, retaining the principal cell and its coherence data before any quotient, specialization, Bockstein, or Koszul homology is taken. Only after that construction proves module descent may the finite census be interpreted as nearby-cycle data.

The methodological consequence for the cosmology sector is therefore

$$\boxed{\text{finite specialization census}\not\Rightarrow \text{nearby-cycle module}.}$$

The Q-sector must be located functorially in a Rees/Gauss--Manin complex; it cannot be inferred from rank patterns in a quotient on which the deformation parameter does not act.

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