Holography Landauer Identity Thermal Bit Heat At Forced Period

The forced thermal quantum

In thermodynamics, Landauer's principle states that erasing one bit of information must dissipate at least a certain minimum amount of heat. That minimum is k_B T ln 2, where k_B is Boltzmann's constant and T is the temperature. The Recognition Science framework derives an exact, unit-free version of this cost at a special, forced temperature.

The framework's core model is a ledger, a discrete record of events, where each event has a fixed cost. A key result identifies a specific period, the deficit-free period, as the inverse temperature. At this period, the cost to erase one bit is exactly κ ln 2 / (2π), where κ is a constant related to the ledger's structure. This is not an approximation or a limit; it is an exact equality.

The declaration thermalBitHeat_at_forced_period is a theorem in the framework's machine-checked library of formal theorems. It proves that if the physical heat of a process is identified with the ledger's posted flux, and if the deficit-free period is identified with inverse temperature, then the per-bit heat quantum has that exact form. The theorem is a direct consequence of the ledger's accounting identity and the definition of the period.

In Recognition Science, the declaration establishes a precise, forced relationship between information erasure and heat. It shows that at the framework's fundamental temperature, the Landauer cost is not a lower bound but an exact value. This is a stronger statement than the classical principle, which only provides a lower bound.

The declaration does not claim that this exact value applies at arbitrary temperatures. It does not assign a numerical value in joules or kelvin to the heat quantum; that requires an external calibration. It also does not claim that the framework's ledger model is the correct physical description of all information erasure. These are separate, empirical questions.

/-- At the proved deficit-free period, the calibrated per-bit quantum has the
surface-rate form `kappa * log 2 / (2 * pi)`. The only physical input is the
named thermal calibration; the period itself is forced by closure. -/
theorem thermalBitHeat_at_forced_period
    (q kappa : ℝ) (hk : 0 < kappa)
    (hthermal : ThermalBitCalibration q (euclideanPeriod kappa)) :
    q = kappa * Real.log 2 / (2 * Real.pi) := by
  unfold ThermalBitCalibration thermalBitHeat euclideanPeriod at hthermal
  rw [hthermal]
  field_simp [ne_of_gt hk, Real.pi_ne_zero]

/-- At the proved deficit-free period, the calibrated per-bit quantum has the
surface-rate form `kappa * log 2 / (2 * pi)`. The only physical input is the
named thermal calibration; the period itself is forced by closure. -/
theorem thermalBitHeat_at_forced_period
    (q kappa : ℝ) (hk : 0 < kappa)
    (hthermal : ThermalBitCalibration q (euclideanPeriod kappa)) :
    q = kappa * Real.log 2 / (2 * Real.pi) := by
  unfold ThermalBitCalibration thermalBitHeat euclideanPeriod at hthermal
  rw [hthermal]
  field_simp [ne_of_gt hk, Real.pi_ne_zero]

What this page does not claim

The declaration does not assign a numerical value in joules or kelvin to the heat quantum. The declaration does not claim the framework's ledger model is the correct physical description of all information erasure. The declaration does not claim the exact value applies at arbitrary temperatures.

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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