schmidtSaturated_entropy_peak
plain-language theorem explainer
At half evaporation, the state-derived entropy of a Schmidt-saturated operator Page process equals half the black-hole entropy budget. Holography and RS gravity workers cite this as the Page-time peak for operator-derived (not field-supplied) readout. The proof rewrites via the saturation-to-Page identity, then applies the triangular Page-curve evaluation at fraction 1/2.
Claim. Let $P$ be a Schmidt-saturated operator Page process on finite bulk and radiation ledgers, with black-hole entropy budget $S_{\mathrm{BH}}$ and total tick count $N$. If $n \le N$ and the evaporation fraction at tick $n$ equals $1/2$, then the entropy of the ledger state after $n$ unitary ticks equals $S_{\mathrm{BH}}/2$.
background
Gravity Track 3.C derives the Page entropy readout from an operator process instead of carrying it as a free structure field. A Schmidt-saturated operator process extends an ordinary operator Page process by an entropy map on bulk-radiation ledger states. Saturation demands that this entropy equal the Schmidt capacity bound at every tick up to the total budget; the initial state has entropy zero.
The Schmidt capacity bound is the triangular minimum of bulk and radiation capacities along the evaporation schedule. The Page curve is that same minimum, scaled to $S_{\mathrm{BH}}$. Upstream dynamical Page-curve work already evaluates the triangle at the half-evaporation fraction, where the two capacities meet and the value is exactly $S_{\mathrm{BH}}/2$.
Once saturation holds, state entropy is forced onto that geometric curve. The present peak identity is the half-fraction checkpoint of that forcing.
proof idea
Short term proof in two steps. Rewrite the left-hand side by the sibling identity that Schmidt saturation makes state entropy equal the Page-curve value at tick $n$. The goal reduces to the pure ledger-tick Page-curve statement at evaporation fraction $1/2$. Close it by the upstream half-fraction evaluation, which returns $S_{\mathrm{BH}}/2$ from the triangular min, using positivity of the total tick budget and the given bound $n \le N$.
why it matters
The classical Page curve peaks at $S_{\mathrm{BH}}/2$ at Page time (half evaporation). This theorem records that peak for entropy read from the evolved ledger state under Schmidt saturation, not for an externally supplied readout function. With the zero-tick and full-evaporation endpoint identities, it pins the three canonical checkpoints of the triangular curve for saturated processes.
In the Recognition gravity track the point is architectural: the master Page-curve witness can route through derived saturation rather than a load-bearing field that literally asserts readout equals the Page curve. No downstream consumers are wired yet; the result is the peak checkpoint of the saturated-process API.
Ambient RS units (the fundamental tick) set the discrete time base, but the argument itself is combinatorial in the tick budget and does not invoke the T0-T8 forcing chain, RCL, or the phi ladder.
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