Holography Landauer Identity Physical Path Heat Eq Scaled Record Flux

The heat identity

Landauer's principle, proposed by Rolf Landauer in 1961, says that erasing one bit of information in a computer must dissipate at least a certain minimum amount of heat. The exact amount depends on temperature, and the principle is a foundation of the physics of information. The Recognition Science framework approaches this from a different direction: it starts with a discrete ledger, a record of events, and asks what the cost of recognition, the act of registering a change, must be. The framework's library, a machine-checked collection of formal theorems, contains a result that connects these two worlds.

The theorem in question, physicalPathHeat_eq_scaled_recordFlux, states a simple identity: along any finite path of cell configurations, the total physical heat equals a conversion quantum q multiplied by the total posted record flux. In plainer terms, if you add up the heat assigned to each step of a process, it is exactly proportional to the net change in the record's potential, the ledger's measure of how much has been posted. This is not a statement about joules or kelvin; it is a statement about the structure of the ledger itself. The identity is proved in the library as a theorem, meaning it follows from the framework's axioms without any additional assumptions.

The physical reading of this identity, however, requires two named premises. The first, HeatIsPostedRecordFlux, is a model choice: it declares that the heat you measure at the boundary of a system is exactly the posted flux multiplied by a single conversion quantum q. The second, ThermalBitCalibration, identifies the framework's proved deficit-free period with inverse temperature, so that one bit carries energy log 2 divided by beta. These are not theorems; they are definitions, choices about how to map the abstract ledger onto physical quantities. The theorem itself is unconditional, but calling the generic heat observable physical is conditional on these two premises.

What the theorem does not claim is just as important. It does not select a value for the inverse temperature; that is left open. It does not provide an SI value for q, the conversion quantum, so it says nothing about joules or kelvin. It does not, by itself, identify posted weight loss with logical entropy erasure. That identification requires a third premise, LogicalErasurePostsDebit, which states that resetting a certain number of logical bits produces the same endpoint debit in the ledger. The framework is explicit about this: the accounting identity is proved, but the bridge to classical logical erasure is a named model, not a theorem.

There is a named falsifier for the physical reading. If a calorimetric experiment measures erasure that is strictly cheaper than the posted debit, that is the CheaperThanPostedDebit condition, and it refutes the heat-carrier identification. The framework thus gives a precise, testable prediction: either the heat you measure obeys the identity, or the premise that heat reads posted flux is wrong. This is what makes the framework's claim scientific rather than merely formal. The identity is a theorem; its physical interpretation is a hypothesis with a named way to fail.

/-- A physical heat observable satisfying the carrier premise is exactly the
native posted flux times `q` on every path. -/
theorem physicalPathHeat_eq_scaled_recordFlux
    (q : ℝ) (heat : PhysicalStepHeat)
    (hcarrier : HeatIsPostedRecordFlux q heat) :
    ∀ p : List CellCfg,
      physicalPathHeat heat p = q * (pathHeatCell p : ℝ) := by
  intro p
  induction p with
  | nil =>
      simp [physicalPathHeat, pathHeatCell]
  | cons c rest ih =>
      cases rest with
      | nil =>
          simp [physicalPathHeat, pathHeatCell]
      | cons c' rest' =>
          simp only [physicalPathHeat, pathHeatCell]
          rw [hcarrier c c', ih]
          push_cast
          ring

/-- Named MODEL premise: physical boundary heat reads the signed posted-record
flux with conversion quantum `q`. No numerical unit is fixed here. -/
def HeatIsPostedRecordFlux (q : ℝ) (heat : PhysicalStepHeat) : Prop :=
  ∀ c c' : CellCfg, heat c c' = q * (stepHeatCell c c' : ℝ)

/-- Named MODEL premise for the thermal conversion. Applying it at
`euclideanPeriod kappa` identifies that proved period with inverse
temperature. In natural units one bit then carries `log 2 / beta`. -/
def ThermalBitCalibration (q beta : ℝ) : Prop :=
  q = thermalBitHeat beta

/-- Any physical erasure measured strictly cheaper than its posted debit
refutes the heat-carrier identification used by the physical reading. -/
theorem cheaper_erasure_falsifies_posting_rule
    (q : ℝ) (heat : PhysicalStepHeat) (c : CellCfg)
    (p : List CellCfg)
    (hcheap : CheaperThanPostedDebit q heat c p) :
    ¬ HeatIsPostedRecordFlux q heat := by
  intro hcarrier
  have hlt := hcheap.2.2
  rw [physical_landauer_identity q heat hcarrier c p] at hlt
  exact (lt_irrefl _ hlt)

What this page does not claim

The theorem does not provide a numerical value for the conversion quantum q in SI units. The identity does not by itself identify posted weight loss with logical entropy erasure. The framework does not claim that its physical interpretation is the only possible one.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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