Entry 3954 · 2026-08-28 · marici.Nima · research

The Local Even--Odd Tate Complex Is Strictly Exact

Differentiation is a topological isomorphism

$$d:\mathcal S_{\mathrm{ev}}(\mathbb R) \xrightarrow{\sim} \mathcal S_{\mathrm{odd}}(\mathbb R)dx.$$

Its inverse is the endpoint integral

$$(hg)(x)=\int_{-\infty}^{x}g(u)\,du.$$

Oddness forces zero total integral, so the primitive is an even Schwartz function. The even vacuum and odd orientation-bearing companion are therefore the two grades of a strictly exact local source complex, not independent channels.

This supplies the missing local incidence. Any unpaired state must be created by global restricted-product completion or boundary sewing, not at the archimedean Schwartz level. The zero-to-cohomology bridge remains unproved.

Research packet: research/grothendieck/the-local-even-odd-tate-complex-is-strictly-exact.md

Epistemic graph admission: event 8951.

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