Holography Coefficient Bridge Single Event Entropy Eq H

The single-event entropy identity

The declaration single_event_entropy_eq_H is a theorem in the framework's machine-checked library of formal theorems. It states that the entropy carried by a single recognition event equals a fixed quantity called the forced entropy. In plain terms: when the framework's ledger, a discrete record of events, registers exactly one event, the entropy associated with that event is not a free parameter. The result proves this equality by direct computation from the definitions, with no additional assumptions.

The context is a larger argument about black-hole entropy. The Bekenstein-Hawking formula S = A/4 relates a black hole's entropy to its horizon area. The framework's library is investigating whether this 1/4 coefficient can be derived from its discrete model. The result single_event_entropy_eq_H is a small but necessary step: it shows that a single event carries exactly the forced per-event entropy H. This matters because the coefficient question then reduces to how many recognition events attach to a closed plaquette, the smallest unit of the model's surface.

The result does not claim that a black hole's entropy is actually A/4. That would require an additional physical input, the open selector: the assertion that one closed plaquette realizes exactly one recognition event. The library proves both possible branches conditionally. If the multiplicity is 1, the coefficient is the Bekenstein 1/4. If the multiplicity is 3, the coefficient becomes 4/3. The result single_event_entropy_eq_H is agnostic between these branches; it only fixes the per-event entropy once the multiplicity is specified.

What the result changes is the shape of the remaining problem. Before it, the coefficient seemed to require computing a number from a lattice model. After it, the coefficient is reduced to a single yes-or-no question: does one closed plaquette correspond to one recognition event, or to three? The framework proves everything downstream of that choice. The choice itself remains open, honestly labeled as the crux of the research program.

/-- Entropy payoff of the Bekenstein branch: a single-event plaquette carries exactly the
forced per-event entropy `H = forcedEntropy`, so `S_pixel = H`, `H` cancels against the
per-event capacity, and the pure geometric `1/4` survives. -/
theorem single_event_entropy_eq_H :
    RecognitionEventCapacity.eventAccess 1 = RecognitionEventCapacity.forcedEntropy := by
  simp [RecognitionEventCapacity.eventAccess]

/-- Entropy payoff of the Bekenstein branch: a single-event plaquette carries exactly the
forced per-event entropy `H = forcedEntropy`, so `S_pixel = H`, `H` cancels against the
per-event capacity, and the pure geometric `1/4` survives. -/
theorem single_event_entropy_eq_H :
    RecognitionEventCapacity.eventAccess 1 = RecognitionEventCapacity.forcedEntropy := by
  simp [RecognitionEventCapacity.eventAccess]

What this page does not claim

The theorem does not prove the Bekenstein-Hawking formula S = A/4. The theorem does not determine whether the event multiplicity is 1 or 3. The theorem does not assign a numerical value to the forced entropy H.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Holography/CoefficientBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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