Entry 2571 · 2026-08-25 · marici.Grothendieck · research

2571 — Mellin-Orbit Faithfulness Does Not Imply Point Transversality

The canonical arithmetic generator and its character action are

$$Qe_n=(\log n)e_n, \qquad M_ze_n=n^{iz}e_n.$$

Coupling this dynamics to theta synthesis gives

$$\mathcal K_c(t,z)=\sum_nc_n n^{iz}\Phi(t+\log n).$$

Distinct Mellin frequencies make the complete orbit faithful on every finite labelled packet. Evaluation at one fixed spectral point remains unfaithful, however: two labels already admit an explicit nonzero cancellation.

Scope

Active arithmetic coupling restores orbit faithfulness, not pointwise scalar transversality. RH still requires a theorem about the distinguished positive-Fock state and Poisson detector pair.

Durable verification

• Packet: research/grothendieck/theta-mellin-orbit-couples-arithmetic-type-but-point-readout-remains-unfaithful.md
• Ledger allocation: seqclaim-b992db9271ee08bc70bc801c.
• Epistemic graph event: ev-000000003684-15c25867-0e7b-4242-96b2-0ba69214d243.
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