Holography Landauer Identity Cheaper Erasure Falsifies Posting Rule

The falsification rule

In physics, Landauer's principle says that erasing one bit of information must dissipate at least a certain minimum amount of heat, about kT ln 2. The Recognition Science framework reaches a similar bound from its own starting point: reality keeps a discrete record of events, and the cost of changing that record is forced by a proved identity. The declaration cheaper_erasure_falsifies_posting_rule states the condition under which that framework's physical reading would be refuted.

Within the framework, the ledger, a discrete record of posted events, has an exact accounting identity: the heat dissipated along a path equals a fixed quantum q times the number of net erased bits. This is a proved theorem, conditional on the premise that physical heat reads the posted flux. The declaration in question takes that premise and asks what would happen if a measurement found erasure strictly cheaper than the posted debit. The answer is a theorem: such a measurement would falsify the heat-carrier premise itself. It would not touch the underlying accounting identity, which is unconditional.

The falsifier is concrete. A calorimetric experiment that measured dissipated heat below q times the net erased bits, with q positive, would refute the identification of heat with posted flux. The framework's library proves this implication formally. It also proves the converse direction implicitly: if the premise holds, the exact identity forces the heat to be at least the posted debit, so no cheaper erasure is possible.

What the declaration does not claim is equally precise. It does not prove that the premise is true; it only names what would break it. It does not assign a numerical value to q, nor does it identify the framework's posted-record debit with thermodynamic entropy in general. The classical Landauer bound appears only after additional named identifications are assumed. The falsification rule is a sharp boundary: it says where the framework's physical reading would fail, not that it has failed.

/-- 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)

/-- The RS Landauer identity on the forced cell. Given the explicit carrier
identification, dissipated heat equals the per-bit quantum times the net
posted-record debit, exactly. -/
theorem physical_landauer_identity
    (q : ℝ) (heat : PhysicalStepHeat)
    (hcarrier : HeatIsPostedRecordFlux q heat)
    (c : CellCfg) (p : List CellCfg) :
    dissipatedHeat heat c p = q * (netErasedBits c p : ℝ) := by
  unfold dissipatedHeat netErasedBits
  rw [physicalPathHeat_eq_scaled_recordFlux q heat hcarrier (c :: p)]
  rw [books_balance]
  push_cast
  ring

/-- Classical Landauer lower bound, conditional on all three physical
identifications: heat reads posted flux, the deficit-free period is inverse
temperature, and logical erasure posts the endpoint debit. Equality in the
record ledger implies the classical inequality. -/
theorem classical_landauer_bound_of_posted_logical_erasure
    (q kappa : ℝ) (heat : PhysicalStepHeat)
    (hk : 0 < kappa)
    (hcarrier : HeatIsPostedRecordFlux q heat)
    (hthermal : ThermalBitCalibration q (euclideanPeriod kappa))
    (erasedBits : ℕ) (c : CellCfg) (p : List CellCfg)
    (hlogical : LogicalErasurePostsDebit erasedBits c p) :
    0 ≤ thermalBitHeat (euclideanPeriod kappa) * (erasedBits : ℝ)
      ∧ thermalBitHeat (euclideanPeriod kappa) * (erasedBits : ℝ)
        ≤ dissipatedHeat heat c p := by
  have hperiod : 0 < euclideanPeriod kappa := by
    unfold euclideanPeriod
    exact div_pos (by positivity) hk
  have hquantum : 0 < thermalBitHeat (euclideanPeriod kappa) := by
    unfold thermalBitHeat
    exact div_pos (Real.log_pos (by norm_num)) hperiod
  have hbits :
      (netErasedBits c p : ℝ) = (erasedBits : ℝ) := by
    exact_mod_cast hlogical
  constructor
  · exact mul_nonneg hquantum.le (Nat.cast_nonneg erasedBits)
  · rw [physical_landauer_identity q heat hcarrier c p,
      hthermal, hbits]

What this page does not claim

The declaration does not prove that physical heat actually reads the posted flux. It does not assign a numerical value to the conversion quantum q. It does not identify posted-record debit with logical entropy in all cases.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: