S_rad_phase2
plain-language theorem explainer
In the second half of black-hole evaporation (t in [1/2, 1]), radiation entropy equals remaining bulk Hilbert-space capacity S_BH(1-t). Gravity and information theorists tracking the dynamical Page curve cite this for the purifying descent phase. The proof is a two-step rewrite: Schmidt purification equates radiation entropy to the unitarity Page curve, which the phase-2 lemma collapses to bulk capacity.
Claim. For real $t$ with $\tfrac{1}{2} \le t \le 1$, the radiation entropy at evaporation fraction $t$ equals the bulk capacity $S_{\mathrm{BH}}\cdot(1-t)$.
background
This module derives the triangular Page curve from Schmidt-balanced ledger dynamics rather than postulating it by hand. Evaporation is parameterized by $t \in [0,1]$, the fraction of total entropy transferred from bulk to radiation. Bulk capacity falls linearly as $S_{\mathrm{BH}}(1-t)$; radiation capacity rises as $S_{\mathrm{BH}} t$.
Because the joint bulk-radiation state remains pure under unitary evolution, Schmidt's theorem forces equal subsystem entropies, each bounded by the minimum of the two capacities. Radiation entropy is taken to saturate that bound, so the curve is $\min$ of the two monotone capacities, peaking at $t=1/2$.
Upstream, bulk capacity is defined by $S_{\mathrm{BH}}\cdot(1-t)$. The phase-2 unitarity lemma states that for $t \in [1/2,1]$ the unitarity Page curve equals bulk capacity (bulk-bound descent, the information-purifying regime).
proof idea
Term-mode proof by two rewrites. First apply the structure's Schmidt-purification identity, which identifies radiation entropy with the unitarity Page curve at the black-hole entropy. Then invoke the phase-2 unitarity lemma (nonnegative black-hole entropy, $t \ge 1/2$, $t \le 1$), which reduces that curve to bulk capacity on the second half of the interval. No further arithmetic is required.
why it matters
Closes the phase-2 half of the dynamical Page-curve identity: after the peak at $t=1/2$, radiation entropy descends with remaining bulk capacity. That descent is the purifying regime forced by Schmidt balance on a pure joint state, matching the module claim that the triangular shape emerges as $\min$ of bulk and radiation capacities rather than the kinematic ansatz of Session 101.
No downstream dependents are recorded yet; the sibling suite (capacity-sum invariant, tick-parameterized forms, full ledger-tick Page curve) is the natural landing zone. Within Recognition Science gravity track 3.C this is structural closure (0 sorry), not a forcing-chain (T0-T8) step, but it supplies the information-theoretic backbone for unitary black-hole evaporation.
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