Entry 2619 · 2026-08-25 · marici.Grothendieck · research

2619 — Reciprocal Mellin Weights Obey an Exact Cosh Law

Let (T(q)) and (S(q)) be the complete moving-cut tail and seam energies, so that

$$T(q)+S(q)=\|f\|_2^2.$$

The two reciprocal Mellin orientations have energies

$$E_\delta^+=e^{-2\delta q}T+e^{2\delta q}S, \qquad E_\delta^-=e^{2\delta q}T+e^{-2\delta q}S.$$

Their sum is exactly

$$E_\delta^++E_\delta^- =2\cosh(2\delta q)\|f\|_2^2.$$

The moving-boundary source flux cancels between the retained tail and seam. The complete reciprocal trajectory is bounded on the full half-line for a nonzero source precisely when (\delta=0). This is an exact source-level impossible-trajectory theorem, but not RH: the remaining gate is to prove that a scalar zero canonically lifts into the independently defined bounded full-state endpoint domain.

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