Archimedean Quarter-Pair Thermal Envelope

Abstract

The two quarter-shifted Archimedean Gamma channels have an exact thermal envelope.

Theorem 1.1 (The quarter-pair Gamma product has a Fermi-like thermal envelope).

$$\forall t\in \mathrm{Real}\left(\right),(\left|\mathrm{Gamma}\left(\frac{1}{4}+i\times \frac{t}{2}\right)\right|{)}^2\times (\left|\mathrm{Gamma}\left(\frac{3}{4}+i\times \frac{t}{2}\right)\right|{)}^2=\frac{2\times (\pi {)}^2}{\cosh \left(\pi \times t\right)}\wedge \left(\frac{1}{\cosh \left(\pi \times t\right)}=\frac{2\times \exp \left(-\pi \times \left|t\right|\right)}{1+\exp \left(-2\times \pi \times \left|t\right|\right)}\wedge (\left|\mathrm{Gamma}\left(\frac{1}{4}+i\times \frac{t}{2}\right)\right|{)}^2\times (\left|\mathrm{Gamma}\left(\frac{3}{4}+i\times \frac{t}{2}\right)\right|{)}^2=2\times (\pi {)}^2\times \frac{2\times \exp \left(-\pi \times \left|t\right|\right)}{1+\exp \left(-2\times \pi \times \left|t\right|\right)}\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/GammaThermal/ArchimedeanQuarterPairThermalEnvelope.archimedean_quarter_pair_thermal_envelope (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every real t, the first conjunct gives the squared-norm product of Gamma(1/4 + it/2) and Gamma(3/4 + it/2) as 2 pi^2 / cosh(pi t). The second conjunct gives exactly the reciprocal-cosh exponential identity with |t|. The third conjunct combines them into the concrete pair’s exact Fermi-like exponential envelope.

The proof specializes the pinned Gamma duplication and reflection identities, then rewrites the hyperbolic cosine using real exponentials. It uses no Riemann-hypothesis assumption.

References

• Truth anchor: D5/S3/Analytic/GammaThermal/ArchimedeanQuarterPairThermalEnvelope.archimedean_quarter_pair_thermal_envelope