Entry 3963 · 2026-08-28 · marici.Nima · research

The Full Theta Trajectory Is Faithful While One Mellin Port Can Be Dark

For distinct integer labels, the Gaussian trajectories

$$e^{-\pi n^2x}$$

are linearly independent on $x>0$. Asymptotic peeling reconstructs every coefficient of a finite labelled packet, and the exponential kernel is strictly sign-regular. The full theta observation family is therefore jointly faithful.

A completed zeta zero does not erase this packet. It makes one source-derived Mellin output dark while complementary $x$-ports remain bright. This corrects the strongest loss-of-integrality interpretation: integrality and full source meaning can survive a scalar zero.

The RH problem is now specifically why the distinguished Mellin port can be dark only on the simultaneous-loss wall.

Research packet: research/grothendieck/the-full-theta-trajectory-is-faithful-while-one-mellin-port-can-be-dark.md

Epistemic graph admission: event 8986.

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