Constants Boltzmann Constant K R Bounds

The k_R bounds

The Boltzmann constant k_B is the number that converts temperature into energy in the equation E = k_B · T. In the International System of Units, its measured value is about 1.380649 × 10⁻²³ joules per kelvin. Recognition Science (RS) does not treat this constant as a free parameter. Instead, the framework derives an analog, called k_R, from a more basic structure: the cost of a single recognition event in its ledger, a discrete record of events. That cost, denoted J_bit, is the natural logarithm of the golden ratio φ, where φ = (1 + √5)/2 ≈ 1.618. The derivation is a theorem in the framework's machine-checked library of formal theorems, not a numerical fit.

The specific declaration k_R_bounds is a theorem that proves a simple numerical fact about this derived constant: k_R lies strictly between 0.47 and 0.49. In symbols, the theorem states 0.47 < k_R < 0.49. This is not an approximation or a measurement; it is a proved result that follows from the definition of k_R and the known bounds on φ. Since φ is between 1.61 and 1.62, and the natural logarithm is a monotonic function, the logarithm of φ must fall between the logarithms of those two bounds, which are about 0.476 and 0.482. The theorem k_R_bounds formalizes this reasoning in the machine-checked library, along with companion theorems k_R_pos and k_R_lt_half that prove k_R is positive and less than 0.5.

What the theorem does not claim is equally important. It does not claim that the SI value of the Boltzmann constant, 1.380649 × 10⁻²³ J/K, is derived from the golden ratio. The framework's k_R is a dimensionless number in its own natural units, and converting it to SI units requires a separate calibration factor involving coherence scales, which the pack does not specify. The theorem also does not claim that temperature itself is a fundamental quantity; rather, the framework models temperature as the average cost per degree of freedom in the ledger. Finally, k_R_bounds is a statement about a mathematical constant, not an empirical measurement of the physical world. It proves a property of the framework's internal definition, and any connection to the measured Boltzmann constant is a separate empirical check, not part of the theorem.

For a reader, the practical consequence is that the framework's thermodynamics has a fixed, computable energy scale. At a temperature of T = 1 in RS units, the thermal energy per degree of freedom is exactly k_R = ln(φ) ≈ 0.481, as proved by the theorem thermal_energy_at_unit_T. This gives the framework a concrete, testable prediction: if the ledger model is correct, then the ratio between the SI Boltzmann constant and the framework's coherence energy and temperature scales must equal ln(φ). That is a quantitative claim that could, in principle, be checked against experiment, and it is the kind of statement that separates a derived constant from an adjustable one.

/-- **THEOREM C-006.4**: Bounds on k_R from φ bounds.

    With φ ∈ (1.61, 1.62), we get k_R ∈ (0.47, 0.49).
    
    **Proof sketch**: 
    - φ > 1.61 implies ln(φ) > ln(1.61) > 0.47
    - φ < 1.62 implies ln(φ) < ln(1.62) < 0.49
    - The numerical bounds follow from the monotonicity of ln
    - Direct computation: ln(1.61) ≈ 0.476, ln(1.62) ≈ 0.482
    
    **Status**: The bounds follow from φ bounds and ln monotonicity.
    **Numerical verification**: Uses Real.exp_bound for Taylor series bounds. -/
theorem k_R_bounds : (0.47 : ℝ) < k_R ∧ k_R < (0.49 : ℝ) := by
  unfold k_R
  have h1 : (1.61 : ℝ) < Constants.phi := Constants.phi_gt_onePointSixOne
  have h2 : Constants.phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  constructor
  · -- Lower bound: ln(φ) > ln(1.61) > 0.47 via exp(0.47) < 1.61
    have h_log_mono : Real.log (1.61 : ℝ) < Real.log Constants.phi := by
      apply Real.log_lt_log
      all_goals nlinarith [Constants.phi_pos]
    -- Numerical verification: exp(0.47) < 1.61 using exp_bound'
    have h_lower : (0.47 : ℝ) < Real.log (1.61 : ℝ) := by
      have h_exp : Real.exp (0.47 : ℝ) < (1.61 : ℝ) := by
        -- Upper bound via Taylor remainder
        have h1 : (0.47 : ℝ) ≥ 0 := by norm_num
        have h2 : (0.47 : ℝ) ≤ 1 := by norm_num
        have h3 := Real.exp_bound' h1 h2 (by norm_num : (0 : ℕ) < 4)
        norm_num [Finset.sum_range_succ, Nat.factorial] at h3 ⊢
        nlinarith [Real.exp_pos 0.47]
      have h_ln : (0.47 : ℝ) < Real.log (1.61 : ℝ) := by
        have h1 : Real.log (Real.exp (0.47 : ℝ)) = (0.47 : ℝ) := Real.log_exp (0.47 : ℝ)
        have h2 : Real.log (Real.exp (0.47 : ℝ)) < Real.log (1.61 : ℝ) := by
          apply Real.log_lt_log
          all_goals nlinarith [h_exp, Real.exp_pos 0.47]
        linarith [h1]
      linarith
    linarith
  · -- Upper bound: ln(φ) < ln(1.62) < 0.49 via 1.62 < exp(0.49)
    have h_log_mono : Real.log Constants.phi < Real.log (1.62 : ℝ) := by
      apply Real.log_lt_log
      all_goals nlinarith [Constants.phi_pos]
    -- Numerical verification: 1.62 < exp(0.49) using exp_bound lower bound
    have h_upper : Real.log (1.62 : ℝ) < (0.49 : ℝ) := by
      have h_exp : Real.exp (0.49 : ℝ) > (1.62 : ℝ) := by
        -- Lower bound: exp(x) > sum of first n terms (all terms positive for x > 0)
        have h1 : |(0.49 : 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 0.49]
      have h_ln : Real.log (1.62 : ℝ) < (0.49 : ℝ) := by
        have h1 : Real.log (Real.exp (0.49 : ℝ)) = (0.49 : ℝ) := Real.log_exp (0.49 : ℝ)
        have h2 : Real.log (1.62 : ℝ) < Real.log (Real.exp (0.49 : ℝ)) := by
          apply Real.log_lt_log
          all_goals nlinarith [h_exp, Real.exp_pos 0.49]
        linarith [h1]
      linarith
    linarith

/-- **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-006.5**: 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

/-- **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

What this page does not claim

The theorem does not derive the SI value of the Boltzmann constant, 1.380649 × 10⁻²³ J/K, from the golden ratio. The theorem does not claim that temperature is a fundamental quantity in the framework. The theorem does not assert any empirical agreement with the measured Boltzmann constant.

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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