Cryostatics

Closed-loop micro-Stirling and thermoacoustic pulse-tube cryocoolers are advanced as the enabling thermal substrate for deep-cryogenic photonic integrated circuits,eliminating expendable liquid cryogens. This document develops the thermodynamic, geometric, and optoelectronic basis of that architecture and subjects every quantitativelycheckable claim to symbolic and numerical validation in Wolfram Language. Of nineteenregistered claims, twelve are validated, four are corrected, two are falsified, and one isunregistrable. The principal corrections are: (i) the microring thermal stability budgetis governed by the cavity linewidth, not the free spectral range, tightening the requiredstability from tens of kelvin to ∼ 2.3 K at Q = 104; (ii) cryogenic co-packaged opticsincreases net facility heat rejection by a factor of 15.5 at 77 K and 353 at 4.2 K, so thearchitecture must be justified on performance rather than efficiency grounds; and (iii) theclaimed harmonic mapping between angular rotors and hydrogen spectra carries a 4.11%fractional defect, which is spectroscopically fatal. A closed-form regenerator design ruleis derived: axial parasitic conduction constrains a silicon matrix to A/L ≤ 8.10 µm for a350 mW cold load at 77 K, relieved 106× by dielectric substitution.