module module moderate

`IndisputableMonolith.Information.LandauerBound`

Defines Landauer's thermodynamic bit-erasure bound in SI units and ties it to Recognition Science cost. Introduces Boltzmann's constant, room-temperature Landauer energy, the tick duration in seconds, a quantum energy scale, minimum erasure power, and the equality of erasure J-cost with k_B T ln 2. Information theorists and RS workers matching ledger cost to irreversible computation cite it. Mostly definitions plus positivity and numerical identities.

claimThe module fixes $k_B$, a reference temperature $T$, the Landauer energy $E_L = k_B T \ln 2$, the RS tick $\tau_0$ in seconds, a quantum energy $E_q$, minimum erasure power, and an erasure J-cost, and records that this J-cost equals the thermodynamic Landauer bound.

background

Landauer's principle states that erasing one bit of information in a thermal bath at temperature $T$ dissipates at least $k_B T \ln 2$ of free energy. In Recognition Science the same irreversible act is scored by the J-cost on the recognition ledger (the unique cost forced by the Recognition Composition Law and T5 uniqueness, $J(x)=\cosh(\log x)-1$).

The module sits in the Information domain and imports only Mathlib and Constants. From Constants it inherits the RS-native tick $\tau_0=1$ (one discrete recognition step). It then supplies the SI bridge: Boltzmann's constant, a conventional room temperature, the Landauer energy at that temperature, $\tau_0$ expressed in seconds, and a quantum energy built from those scales.

Sibling declarations package positivity of the Landauer energy, its numerical room-temperature value, a lower bound on erasure power, an explicit erasure J-cost, and the claim that this J-cost coincides with the thermodynamic expression.

proof idea

This is primarily a definition module. Constants and closed-form expressions ($k_B$, $T_{\mathrm{room}}$, $E_L=k_B T\ln 2$, $\tau_0$ in seconds, quantum energy, minimum erasure power, erasure J-cost) are introduced by def/abbrev. The thin theorem layer records positivity of the Landauer energy, evaluates the room-temperature number, derives the Landauer scale from $\tau_0$, and equates erasure J-cost with the thermodynamic bound by direct algebraic comparison. No deep tactic proof is required beyond unfolding and elementary real arithmetic.

why it matters in Recognition Science

Recognition Science scores every irreversible act by J-cost; Landauer's bound is the standard thermodynamic price of bit erasure. This module is the SI and thermodynamic dictionary that lets those two prices be compared. Downstream work on information, measurement, and the eight-tick ledger can quote a concrete energy (or power) floor instead of a pure dimensionless cost. The equality jcost_equals_thermodynamic is the hinge: once J is identified with free-energy cost per bit, RS ledger inequalities inherit ordinary Landauer limits. No parent theorems are listed yet in the graph (used_by is empty), so the module currently serves as a leaf dictionary for later Information and thermodynamics developments.

scope and limits

Does not derive Landauer's principle from the forcing chain T0–T8.
Does not prove uniqueness of J; it only equates an erasure cost to $k_B T\ln 2$.
Does not model finite-time or non-equilibrium corrections beyond the ideal bound.
Does not fix a microscopic erasure protocol or hardware model.
Room-temperature numerics are conventional reference values, not RS predictions.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (26)

def k_B L50
def roomTemperature L53
def landauerEnergy L57
theorem landauer_positive L60
def landauerRoomTemp L69
theorem landauer_room_temp_value L72
def tau0_seconds L79
def quantumEnergy L83
theorem landauer_from_tau0 L89
def minimumErasurePower L94
def erasureJCost L101
theorem jcost_equals_thermodynamic L105
def experimentalVerification L115
def currentComputerEnergy L126
def efficiencyRatio L127
theorem room_for_improvement L130
structure ReversibleComputation L139
theorem reversible_approaches_zero L148
theorem quantum_is_reversible L154
theorem landauer_from_ledger L169
theorem information_is_physical L178
def applications L190
structure LandauerComputer L199
def predictions L214
structure LandauerFalsifier L228
def experimentalStatus L235