pageCurveFromUnitarity
plain-language theorem explainer
The Page curve is the pointwise minimum of bulk and radiation Hilbert-space entropy capacities under linear evaporation transfer. Gravity Track 3.C cites it as the unique saturation of the Schmidt bound on a pure bulk⊗radiation state. The body is a one-line min of the two linear capacity maps; the triangular shape is not postulated.
Claim. For black-hole entropy $S_{\mathrm{BH}}$ and evaporation fraction $t$, the unitarity Page curve is $\min\bigl(S_{\mathrm{BH}}(1-t),\, S_{\mathrm{BH}}\, t\bigr)$: the minimum of bulk capacity (linear decrease from $S_{\mathrm{BH}}$ to $0$) and radiation capacity (linear increase from $0$ to $S_{\mathrm{BH}}$).
background
Module Gravity.PageCurveDynamical (Track 3.C) replaces the Session-101 kinematic triangular ansatz with a derived curve. Evaporation is parameterized by $t\in[0,1]$: fraction of total entropy moved from bulk to radiation. The joint state on $H_{\mathrm{bulk}}\otimes H_{\mathrm{rad}}$ stays pure under unitary evolution from a pure initial bulk state.
Bulk capacity is $S_{\mathrm{BH}}(1-t)$ (Hilbert-space entropy budget of the shrinking bulk). Radiation capacity is $S_{\mathrm{BH}}, t$ (budget of the growing Hawking radiation space). Schmidt's theorem forces equal reduced entropies, each bounded by $\min(\log d_{\mathrm{bulk}},\log d_{\mathrm{rad}})$. Under maximal Schmidt balance the radiation entropy saturates that bound.
The module status is structural theorem: zero sorry, zero RS-internal axiom. Capacity sum $S_{\mathrm{BH}}(1-t)+S_{\mathrm{BH}}t=S_{\mathrm{BH}}$ is the conserved total.
proof idea
Definition, not a proved theorem. The body is the one-line expression $\min(\mathrm{bulkCapacity},\mathrm{radiationCapacity})$, i.e. $\min(S_{\mathrm{BH}}(1-t), S_{\mathrm{BH}} t)$. No tactics or lemmas; the mathematical content is the choice of that min as the entropy bound forced by Schmidt purification under linear capacity transfer.
why it matters
This is the dynamical recipe's step 5: the Page curve is no longer hand-drawn; it is the min of two monotone capacities. Downstream, canonicalProcess sets radiation entropy exactly to this curve (maximally saturating Schmidt-balanced evolution). The one-statement theorem packages the min-form, capacity-sum invariant, endpoints at $0$ and $1$, and the forced peak $S_{\mathrm{BH}}/2$ at $t=1/2$.
Information preservation is the evaluation at full evaporation: the curve returns to zero because bulk capacity vanishes, the unitarity signature. Tick-level bridges equate ledger-tick Page values to this continuum curve at the induced evaporation fraction, and the master cert records the curve definition as a witness field. Fork D (Track3 tick-capacity endpoint) closes by evaluating the same Schmidt min at tick-induced fractions.
In the broader RS gravity stack this is the structural core of Track 3.C: triangular Page shape forced by purification plus linear transfer, not by ansatz.
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