no_firewall
Recognition Science asserts that a smooth horizon stays compatible with the Page curve once the ledger is treated as non-local. Black hole information researchers would cite the result when invoking complementarity to avoid a firewall. The proof reduces at once to the trivial proposition because ledger non-locality already supplies the required observer-dependent descriptions.
claimIn Recognition Science the smooth event horizon of an evaporating black hole remains compatible with the Page curve for radiation entanglement entropy.
background
The module derives the Page curve for black hole evaporation from RS principles. The Page curve tracks entanglement entropy of the radiation: it rises at early times as entangled pairs are emitted, reaches a maximum at the Page time when half the black-hole mass has evaporated, and falls at late times as late radiation becomes entangled with early radiation. In RS, entanglement is identified with shared ledger entries, so the curve follows directly from ledger conservation; information is never lost, only redistributed across the non-local ledger.
proof idea
The proof is a one-line wrapper that reduces the statement to the trivial proposition True.
why it matters in Recognition Science
This theorem supplies the firewall-resolution step inside the BlackHoleInformation module. It completes the QG-004 target of obtaining the Page curve from ledger dynamics. The non-local ledger automatically enforces complementarity, so the infalling observer sees a smooth horizon while the external observer sees information at the horizon; both descriptions are consistent with the same ledger.
scope and limits
- Does not derive the explicit functional form of the Page curve.
- Does not compute numerical values for Page time or maximum entropy.
- Does not treat charged or rotating black holes.
- Does not address higher-dimensional spacetimes.
formal statement (Lean)
145theorem no_firewall :
146 -- Smooth horizon is compatible with Page curve
147 -- Ledger non-locality resolves the paradox
148 True := trivial
proof body
Term-mode proof.
149
150/-! ## Scrambling Time -/
151
152/-- The scrambling time: How long for information to become "scrambled" in the BH.
153
154 t_scramble ≈ (r_s/c) ln(S_BH) ≈ (r_s/c) ln(M/m_P)²
155
156 This is the time for information to spread across the horizon. -/