Constants Boltzmann Constant Thermal Energy At Unit T

Thermal energy at unit temperature

The Boltzmann constant is the exchange rate between temperature and energy. In statistical mechanics, the relation E = k_B · T says that a system at thermodynamic temperature T carries thermal energy E per accessible degree of freedom. The constant k_B sets the size of that energy step, and its value depends on the units chosen for temperature and energy. In SI units, k_B is measured as 1.380649 × 10⁻²³ joules per kelvin, a value fixed by definition since 2019.

The Recognition Science framework derives an analog of this constant from its own structure rather than treating it as a free parameter. The framework models reality as a ledger, a discrete record of recognition events, where each bit of information has a forced cost. That cost, denoted J_bit, equals the natural logarithm of the golden ratio φ, where φ = (1+√5)/2 ≈ 1.618. The framework's Boltzmann analog k_R is defined as this same quantity: k_R = ln(φ) ≈ 0.481 in natural units. The theorem thermal_energy_at_unit_T states that when the temperature T equals 1 in the framework's temperature units, the thermal energy k_R · T equals ln(φ). In plain language, at unit temperature, the thermal energy per degree of freedom is exactly the ledger's per-bit cost.

The framework proves several properties of this constant in its machine-checked library of formal theorems. It proves k_R is positive, which is required for a sensible temperature scale. It proves k_R is nonzero, so division by it is valid in thermodynamic calculations. It proves k_R is less than 0.5, and more tightly that k_R lies between 0.47 and 0.49. The key identity is that k_R equals J_bit, the cost of a single ledger bit, which unifies temperature with the information-theoretic cost structure. These are formal theorems, meaning the framework's library has machine-checked proofs of them.

What the declaration does not claim is equally important. It does not claim that the measured SI Boltzmann constant k_B is derived from first principles. The framework's k_R is a mathematical constant in its own natural units; the SI value k_B^SI would require a separate calibration step involving coherence energy and temperature scales, which the framework notes but does not formalize as a theorem. The declaration also does not claim that temperature itself is derived within the framework. It assumes a temperature variable T and proves a statement about the product k_R · T at a specific value. Finally, the theorem does not assert that the framework's temperature units match any physical temperature scale; it is a statement about the framework's internal definitions.

/-- **DEFINITION C-006**: The RS Boltzmann analog k_R.

    k_R = ln(φ) — the fundamental cost per ledger bit.
    This replaces k_B in RS-native thermodynamics. -/
noncomputable def k_R : ℝ := Real.log Constants.phi

/-- **THEOREM C-006.6**: The thermal energy quantum.

    At T = 1 (in RS temperature units), E_thermal = k_R = ln(φ).
    This connects temperature to the ledger structure. -/
theorem thermal_energy_at_unit_T (T : ℝ) (hT : T = 1) : k_R * T = Real.log Constants.phi := by
  rw [hT]
  unfold k_R
  ring

/-- **THEOREM C-006.1**: k_R is positive.

    Proof: φ > 1, so ln(φ) > 0. -/
theorem k_R_pos : k_R > 0 := by
  unfold k_R
  apply Real.log_pos
  exact Constants.one_lt_phi

/-- **THEOREM C-006.2**: k_R is nonzero.

    This is required for thermodynamic calculations (division by k_R). -/
theorem k_R_ne_zero : k_R ≠ 0 := by
  exact ne_of_gt k_R_pos

/-- **THEOREM C-006.5**: k_R = J_bit (the ledger bit cost).

    This is the fundamental identity: the Boltzmann analog equals
the cost of a single bit in the recognition ledger. -/
theorem k_R_eq_J_bit : k_R = Constants.J_bit := rfl

What this page does not claim

The declaration does not derive the measured SI value of the Boltzmann constant from first principles. The declaration does not prove that temperature is a derived concept within the framework. The declaration does not claim that the framework's temperature unit matches any physical temperature scale.

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/Constants/BoltzmannConstant.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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