Holography Landauer Identity Posted Debit Lower Bound

The posted debit lower bound

The ledger, a discrete record of events, sits at the center of Recognition Science. The framework's library of formal theorems proves an exact accounting identity on this ledger: the heat dissipated during a process equals a fixed quantum times the net decrease in posted record potential. This is not a statistical approximation; it is an exact equality, derived from the framework's core forcing chain and checked by a computer.

From this exact identity, the declaration posted_debit_lower_bound derives a practical inequality. For any process that decreases the posted record (erases bits) and any nonnegative heat quantum, the theorem proves two things: the posted debit is nonnegative, and the dissipated heat is at least that quantum times the debit. In plain terms, you cannot erase a recorded bit for less than a fixed, positive cost. The inequality is saturated because the stronger exact identity holds, meaning the bound is tight, not a loose estimate.

The physical reading of this bound requires three explicit modeling choices, named as premises in the library. First, measured boundary heat is identified with the posted flux times a conversion quantum. Second, the proved deficit-free period is identified with inverse temperature, fixing the quantum as log 2 divided by that temperature. Third, a logical reset of n bits is identified with posting the same endpoint debit. Only with all three identifications in place does the framework reach the classical Landauer bound: erasing n bits of logical information dissipates at least n times kT log 2 of heat.

What the declaration does not claim is as important as what it proves. The lower bound alone does not identify posted weight loss with logical entropy erasure; that identification is a separate, named premise. The inverse temperature is not selected by the theorem; it is supplied by the carrier. And the theorem does not fix any numerical value for the quantum in joules or kelvin; those units remain outside the certificate. The bound is a structural consequence of the ledger accounting, conditional on the physical identifications that connect it to measurable heat.

The framework also names a concrete falsifier for the heat-carrier premise. If a physical erasure is measured strictly cheaper than its posted debit, that measurement refutes the identification of heat with posted flux. This keeps the physical reading honest: the bound is not a metaphysical guarantee but a testable consequence of a specific modeling choice.

/-- 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

/-- Posted-debit lower-bound form. For a record-decreasing process and a
nonnegative heat quantum, the posted debit is nonnegative and dissipated heat
is at least `q` times that debit. The inequality is saturated because the
stronger exact identity holds. This theorem alone does not identify posted
weight loss with logical entropy erasure. -/
theorem posted_debit_lower_bound
    (q : ℝ) (heat : PhysicalStepHeat)
    (hcarrier : HeatIsPostedRecordFlux q heat)
    (c : CellCfg) (p : List CellCfg)
    (hq : 0 ≤ q) (herase : 0 ≤ netErasedBits c p) :
    0 ≤ q * (netErasedBits c p : ℝ)
      ∧ q * (netErasedBits c p : ℝ) ≤ dissipatedHeat heat c p := by
  have hbits : 0 ≤ (netErasedBits c p : ℝ) := by
    exact_mod_cast herase
  constructor
  · exact mul_nonneg hq hbits
  · exact le_of_eq (physical_landauer_identity q heat hcarrier c p).symm

/-- 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]

/-- 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 lower bound does not by itself identify posted weight loss with logical entropy erasure. The theorem does not select a numerical value for the inverse temperature or the heat quantum in SI units. No claim is made that the framework's ledger model is the only way to derive a Landauer-type bound.

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.

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