Part 15 of 15. The most speculative and the final. Lie-group thermodynamics casts the Fisher metric as a geometric specific heat, and the j-function of classical number theory (the partition function of Monstrous Moonshine) shows up as an entropic potential on a Koszul-Souriau statistical manifold built from the Lie algebra of the Monster. The rigor runs out somewhere inside that sentence, and we will be honest about where.
Explore: The heatmap below shows the value of over the complex upper half-plane. The dashed white line outlines the "fundamental domain". According to our conjecture, you are looking at the free energy landscape of a thermodynamic system defined on the Lie algebra of the Monster group.
Lie-group thermodynamics (developed by Souriau in 1969) extends statistical mechanics so that the "temperature" isn't a single number, but a vector in a Lie algebra. The normalising constant for this probability distribution is the Koszul-Souriau partition function . Its log is the free energy.
Because this is just a specialized exponential family, the entire Information Geometry toolkit applies. The Fisher Information Metric becomes a "geometric specific heat" — the second derivative of the free energy.
The Frontier Conjecture: If we apply this machinery to the infinite-dimensional Lie algebra of the Monster group, the Koszul-Souriau free energy is exactly the -function of Monstrous Moonshine. The numerical coincidences of Moonshine become moments of a thermal state, and modular symmetries become thermodynamic invariances.
Information geometry began in 1945 with Rao's observation that the Fisher information matrix is a Riemannian metric tensor on the space of distributions. It sat as a textbook footnote for forty years until Amari turned it into a discipline, and it has been the quiet framework behind much of machine learning since.
Monstrous Moonshine has been fashionable since McKay noticed and . It has been studied by some of the best pure mathematicians of the twentieth century. The statistical-manifold reading sketched in Parts 13–15 is one corner that does not seem to have been mapped.
This reading does not solve Moonshine; it is suggestive. The partition functions of conformal field theories are probability distributions; their moduli spaces are statistical manifolds; their Weil-Petersson metrics are Fisher metrics; their KL divergences are thermodynamic lengths; their RG flows are natural gradients; their modular invariants may be entropic potentials. If that turns out to be right, a piece of Moonshine is a corollary of Rao's 1945 observation.
You now have the toolkit — Fisher, KL, exponential families, Pythagorean, Bregman, natural gradient, EM, Voronoi — and a sketched dictionary for applying it to the Moonshine ecosystem. The dictionary is incomplete; the problem is open.