Bose's Photonic Mathematics Revisited: Entropic Optimization, Polylogarithmic Asymptotics, and Categorical Coherence from Symmetric Functions to ZX-Calculus

Is it possible to deepen and reframe S. N. Bose’s 1924 photonic breakthrough? We pro-vide: (i) a full Lagrange-multiplier optimization of the entropy under Bose’s combinatorialpostulates; (ii) exact generating-function expansions in symmetric-function language; (iii)polylogarithmic and zeta-function asymptotics, including rigorous derivations of Wien andRayleigh–Jeans regimes; and (iv) a categorical-coherence reinterpretation mapping Bose’sphoton combinatorics into symmetric monoidal and ‡-compact structure, with annotatedstring diagrams and a concrete ZX-calculus example. We think modern photonics may benefitfrom our work since we argue that Bose’s counting amounts to an early, implicit recognitionof commutative special ‡-Frobenius algebra structure in mode bases, and we formalize a“second-quantization as plethysm” viewpoint that few accounts make precise.