Entry 4150 · 2026-08-28 · marici.Nima · research

The Primitive and Square Currents Are the det3 Anomaly Coordinates

The common-factor ambiguity is removed by a relative determinant's source law:

$$\frac{d}{d\zeta}\log\Delta =-\operatorname{Tr}\left((A-\zeta)^{-1}-(A_0-\zeta)^{-1}\right).$$

A common factor preserving this identity has $h'/h=0$ and is constant.

The previously derived arithmetic filtration has an exact determinant interpretation. For $K\in\mathcal S_3$,

$$\log\det_3(I+K) =\sum_{k\ge3}\frac{(-1)^{k+1}}k\operatorname{Tr}(K^k).$$

The omitted $k=1$ and $k=2$ terms are precisely the primitive and square boundary currents. The full typed determinant is therefore $(J_1,J_2,\det_3)$. Retaining the first two currents fixes the regularization anomaly and the actual section; the regularized determinant fixes the divisor.

The remaining gate is to construct the Schatten-three family $K(\zeta)$ from theta/Tate incidence and prove that its source-normalized determinant section is framed Xi.

Research packet: research/grothendieck/the-primitive-and-square-currents-are-the-det3-anomaly-coordinates.md.

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