Constants Boltzmann Constant K R Eq J Bit

The bit cost identity

The Boltzmann constant k_B is the exchange rate between temperature and energy in statistical mechanics. The equation E = k_B · T means that a degree of thermal motion carries a fixed amount of energy per unit of temperature. In the International System of Units, its value is 1.380649 × 10⁻²³ joules per kelvin, defined exactly since 2019.

In Recognition Science, the framework's ledger (a discrete record of recognition events) assigns a cost to each bit of information. The framework proves that this cost, written J_bit, equals ln(φ), the natural logarithm of the golden ratio φ = (1+√5)/2 ≈ 1.618. The declaration k_R_eq_J_bit states that the framework's analog of the Boltzmann constant, called k_R, is exactly this bit cost: k_R = J_bit = ln(φ) ≈ 0.481 in natural units.

This identity is a theorem in the framework's machine-checked library of formal theorems. It follows from the definition of k_R as the logarithm of φ, and the definition of J_bit as the same quantity, so the proof is a direct equality. The framework's library also proves that k_R is positive, nonzero, and less than one half, which supports its use as a temperature scale.

In Recognition Science, the Boltzmann constant is not a free parameter. The framework derives it from the ledger's self-similarity scale, φ, which is itself forced by the cost function. The framework models temperature as the average cost per degree of freedom, so the relationship E = k_R · T emerges from the cost structure rather than being assumed.

This does not claim that the measured SI value of k_B is derived. The framework's k_R is a dimensionless quantity in natural units; converting it to joules per kelvin requires a calibration factor that the framework does not fix. The identity k_R = J_bit is a statement about the framework's internal consistency, not a prediction of the measured constant.

/-- **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

/-- **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-006.1**: 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-006.3**: k_R < 0.5.

    Since φ < 1.62 < e^0.5 ≈ 1.6487, we have ln(φ) < 0.5.
    
    **Proof**: From φ < 1.62 and the monotonicity of ln:
    ln(φ) < ln(1.62). 
    
    Numerically, ln(1.62) ≈ 0.482 < 0.5.
    
    **Status**: The bound follows from φ < 1.62 and ln monotonicity.
    **Numerical proof**: Taylor bound exp(0.5) > 1.645 > 1.62 via Real.exp_bound. -/
theorem k_R_lt_half : k_R < (0.5 : ℝ) := by
  unfold k_R
  have h1 : Constants.phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  -- ln(φ) < ln(1.62) by monotonicity
  have h2 : Real.log Constants.phi < Real.log (1.62 : ℝ) := by
    apply Real.log_lt_log
    all_goals nlinarith [Constants.phi_pos]
  -- Numerical bound: ln(1.62) < 0.5 via 1.62 < exp(0.5)
  have h3 : Real.log (1.62 : ℝ) < (0.5 : ℝ) := by
    have h_exp : Real.exp (0.5 : ℝ) > (1.62 : ℝ) := by
      -- Taylor bound: exp(0.5) > 1 + 0.5 + 0.125 + 0.02083 = 1.6458 > 1.62
      -- Verified using Real.exp_bound with n=4
      have h1 : |(0.5 : ℝ)| ≤ 1 := by norm_num [abs_of_nonneg]
      have h2 := Real.exp_bound h1 (by norm_num : (0 : ℕ) < 4)
      norm_num [Finset.sum_range_succ, Nat.factorial, abs] at h2 ⊢
      nlinarith [Real.exp_pos 0.5]
    have h_ln : Real.log (1.62 : ℝ) < (0.5 : ℝ) := by
      have h1 : Real.log (Real.exp (0.5 : ℝ)) = (0.5 : ℝ) := Real.log_exp (0.5 : ℝ)
      have h2 : Real.log (1.62 : ℝ) < Real.log (Real.exp (0.5 : ℝ)) := by
        apply Real.log_lt_log
        all_goals nlinarith [h_exp, Real.exp_pos 0.5]
      linarith [h1]
    linarith
  linarith

What this page does not claim

The measured SI value of the Boltzmann constant is not derived, only a dimensionless analog in natural units. The framework does not claim that the identity k_R = J_bit predicts the numerical value of k_B in joules per kelvin. The declaration does not establish that the ledger itself is a physical object, only that the framework models it as one.

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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