Chapter 20: Entropy Splitting: S_def

the relevant chapter announced the 180^∘ thermodynamic inversion and stated the Categorical Second Law: defect entropy decreases along the α-orbit. But the decomposition S = S_def + S_ref was previewed, not proved. This chapter supplies the proof.

The construction is precise. Defect entropy S_def counts holomorphic continuations that pass through at least one non-holomorphic insertion—a node where ∂_b f ≠ 0. Refinement entropy S_ref counts holomorphic continuations that remain holomorphic throughout: paths created by the refinement process itself, with no physical defect content. The total holomorphic entropy from Book IV is their sum.

The main results are: (i) S_def is bounded above by the initial defect budget; (ii) S_def decreases monotonically and reaches zero at the coherence horizon; (iii) S_ref increases without bound; (iv) the orthodox readout functor projects onto S_ref, which is why the classical second law sees entropy increase. The connection to the ι_τ stabilization mechanism is established, linking the entropy splitting to the master constant.