OperatorPageProcess
plain-language theorem explainer
Packages the operator-level Page process for Track 3.C: nonnegative black-hole entropy, a positive finite tick budget, a reversible linear tick on the closed bulk⊗radiation carrier, and an initial joint state. Gravity and holography workers cite it as the dynamical interface under which the triangular Page curve is read out. As a structure definition it only assembles those fields; no derivation is claimed here.
Claim. An operator-level Page process on finite bulk and radiation index types consists of a black-hole entropy $S_{\mathrm{BH}}\ge 0$, a positive total evaporation tick count $N\in\mathbb{N}_{>0}$, a reversible $\mathbb{C}$-linear tick (with two-sided inverse) on the closed carrier $\mathrm{Bulk}\otimes_{\mathbb{C}}\mathrm{Hawking}$, and an initial joint bulk-radiation state on that carrier.
background
Module Track 3.C upgrades the Session-101 kinematic triangular Page ansatz to a dynamical derivation: evaporation fraction $t\in[0,1]$ transfers capacity so bulk capacity is $S_{\mathrm{BH}}(1-t)$ and radiation capacity is $S_{\mathrm{BH}}t$. Purity of the joint state plus Schmidt balance forces radiation entropy to equal $\min$ of those two monotone bounds, which is the triangular Page curve with forced peak at $t=1/2$.
The closed carrier is the tensor product of a macroscopic bulk ledger (remaining black-hole degrees of freedom) with a Hawking-radiation ledger. The tick interface records algebraic unitarity only: a $\mathbb{C}$-linear map and inverse with both inverse laws; metric inner-product preservation is deferred. Ledger-closure notions from the holography pixel layer supply the broader balanced-loop context for these finite carriers.
proof idea
No proof body: this is a structure definition. It bundles four data fields (entropy scale with nonnegativity, positive tick budget, reversible tick operator, initial joint state) and inherits the algebraic inverse laws from the tick-operator structure. Downstream lemmas then define tick-indexed evaporation fraction, capacity transfer, iterated state evolution by applying the tick map, and the entropy-readout bridge.
why it matters
Gives Track 3.C the explicit bulk⊗radiation carrier and unitary tick surface requested for operator-level Page dynamics, without claiming a microscopic Hamiltonian entropy derivation. Downstream it feeds capacity-at-tick identities, the sum-invariant of bulk plus radiation capacity, evaporation-fraction-from-ticks, the entropy-readout structure that matches the ledger-tick Page curve, the one-statement operator-process interface theorem, and the MasterTheorem handoff endpoint for Agent D (carrier nonempty, tick nonempty, readout nonempty, and state evolution by the tick). Open remainder: readout from a concrete recognition update or Hamiltonian remains structural, not master-clause ready.
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