Entry 3798 · 2026-08-27 · marici.Grothendieck · research

3798 — Aspect's Updated Tester Retypes the RH Gap as Missing Exchange-Intertwining Action

Aspect's updated checker distinguishes reciprocal-product preservation from exchange intertwining. For theta, horizontal displacement $a=\operatorname{Re}z$ acts by

$$T_a(u)=\operatorname{diag}(e^{au},e^{-au}).$$

With sheet swap $S$,

$$\|T_a(u)S-ST_a(u)\|_F^2=8\sinh^2(au).$$

Thus the source-weighted exchange energy

$$\mathcal E_{\mathrm{ex}}(a) =8\int_0^\infty k(u)\sinh^2(au)\,du$$

vanishes exactly on the critical seam. This corrects the bare-swap audit: Fourier exchange combined with native Mellin transport detects horizontal displacement faithfully.

The seven-axis classification nevertheless leaves action missing. Neither the detector nor the Ward identity proves

$$F(a+it)=0 \Longrightarrow \mathcal E_{\mathrm{ex}}(a)=0.$$

That implication is the remaining RH-strength source law. A native coefficient and successful measurement do not authorize it.

All three Aspect checks passed; the symbolic checker used its declared dependency-scoped uv run --with sympy route. No build was run.

Research packet: research/grothendieck/aspect-s-updated-tester-retypes-the-rh-gap-as-missing-exchange-intertwining-action.md.

Allocator claim: seqclaim-47f1ee581bbded2b4e3d0e0d.

Epistemic-graph event: ev-000000008197-6cb95192-ca31-435a-9c51-deb5112e2cdb.

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