IndisputableMonolith.Constants.BoltzmannConstant

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-! 
   5# C-006: Boltzmann Constant k_R Derivation
   6
   7**Problem**: What determines the Boltzmann constant k_B?
   8Is temperature fundamental? What determines its relationship to energy?
   9
  10**RS Resolution**: In Recognition Science, the Boltzmann analog k_R is not a free parameter
  11but is derived from the fundamental ledger bit cost:
  12
  13    k_R = J_bit = ln(φ)
  14
  15where φ = (1+√5)/2 is the golden ratio forced by the ledger structure (T6).
  16
  17## Physical Interpretation
  18
  19In statistical mechanics, the Boltzmann constant k_B relates temperature to energy:
  20    E = k_B · T
  21
  22In RS, this relationship emerges naturally from the cost structure:
  23    - Each ledger bit has cost J_bit = ln(φ)
  24    - Temperature is the average cost per degree of freedom
  25    - Therefore: k_R = ln(φ) ≈ 0.481 in natural units
  26
  27**Key insight**: The Boltzmann constant is the "exchange rate" between thermal
  28energy (in temperature units) and actual energy, set by the ledger's
  29self-similarity scale φ.
  30
  31**SI Connection**: When converting to SI units:
  32    k_B^SI = k_R · (E_coh_SI / T_coh_SI)
  33where E_coh and T_coh are the coherence energy and temperature scales.
  34-/
  35
  36namespace IndisputableMonolith
  37namespace Constants
  38namespace BoltzmannConstant
  39
  40open Real
  41
  42/-! ## C-006: The RS Boltzmann Analog k_R -/
  43
  44/-- **DEFINITION C-006**: The RS Boltzmann analog k_R.
  45
  46    k_R = ln(φ) — the fundamental cost per ledger bit.
  47    This replaces k_B in RS-native thermodynamics. -/
  48noncomputable def k_R : ℝ := Real.log Constants.phi
  49
  50/-- **THEOREM C-006.1**: k_R is positive.
  51
  52    Proof: φ > 1, so ln(φ) > 0. -/
  53theorem k_R_pos : k_R > 0 := by
  54  unfold k_R
  55  apply Real.log_pos
  56  exact Constants.one_lt_phi
  57
  58/-- **THEOREM C-006.2**: k_R is nonzero.
  59
  60    This is required for thermodynamic calculations (division by k_R). -/
  61theorem k_R_ne_zero : k_R ≠ 0 := by
  62  exact ne_of_gt k_R_pos
  63
  64/-- **THEOREM C-006.3**: k_R < 0.5.
  65
  66    Since φ < 1.62 < e^0.5 ≈ 1.6487, we have ln(φ) < 0.5.
  67    
  68    **Proof**: From φ < 1.62 and the monotonicity of ln:
  69    ln(φ) < ln(1.62). 
  70    
  71    Numerically, ln(1.62) ≈ 0.482 < 0.5.
  72    
  73    **Status**: The bound follows from φ < 1.62 and ln monotonicity.
  74    **Numerical proof**: Taylor bound exp(0.5) > 1.645 > 1.62 via Real.exp_bound. -/
  75theorem k_R_lt_half : k_R < (0.5 : ℝ) := by
  76  unfold k_R
  77  have h1 : Constants.phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  78  -- ln(φ) < ln(1.62) by monotonicity
  79  have h2 : Real.log Constants.phi < Real.log (1.62 : ℝ) := by
  80    apply Real.log_lt_log
  81    all_goals nlinarith [Constants.phi_pos]
  82  -- Numerical bound: ln(1.62) < 0.5 via 1.62 < exp(0.5)
  83  have h3 : Real.log (1.62 : ℝ) < (0.5 : ℝ) := by
  84    have h_exp : Real.exp (0.5 : ℝ) > (1.62 : ℝ) := by
  85      -- Taylor bound: exp(0.5) > 1 + 0.5 + 0.125 + 0.02083 = 1.6458 > 1.62
  86      -- Verified using Real.exp_bound with n=4
  87      have h1 : |(0.5 : ℝ)| ≤ 1 := by norm_num [abs_of_nonneg]
  88      have h2 := Real.exp_bound h1 (by norm_num : (0 : ℕ) < 4)
  89      norm_num [Finset.sum_range_succ, Nat.factorial, abs] at h2 ⊢
  90      nlinarith [Real.exp_pos 0.5]
  91    have h_ln : Real.log (1.62 : ℝ) < (0.5 : ℝ) := by
  92      have h1 : Real.log (Real.exp (0.5 : ℝ)) = (0.5 : ℝ) := Real.log_exp (0.5 : ℝ)
  93      have h2 : Real.log (1.62 : ℝ) < Real.log (Real.exp (0.5 : ℝ)) := by
  94        apply Real.log_lt_log
  95        all_goals nlinarith [h_exp, Real.exp_pos 0.5]
  96      linarith [h1]
  97    linarith
  98  linarith
  99
 100/-- **THEOREM C-006.4**: Bounds on k_R from φ bounds.
 101
 102    With φ ∈ (1.61, 1.62), we get k_R ∈ (0.47, 0.49).
 103    
 104    **Proof sketch**: 
 105    - φ > 1.61 implies ln(φ) > ln(1.61) > 0.47
 106    - φ < 1.62 implies ln(φ) < ln(1.62) < 0.49
 107    - The numerical bounds follow from the monotonicity of ln
 108    - Direct computation: ln(1.61) ≈ 0.476, ln(1.62) ≈ 0.482
 109    
 110    **Status**: The bounds follow from φ bounds and ln monotonicity.
 111    **Numerical verification**: Uses Real.exp_bound for Taylor series bounds. -/
 112theorem k_R_bounds : (0.47 : ℝ) < k_R ∧ k_R < (0.49 : ℝ) := by
 113  unfold k_R
 114  have h1 : (1.61 : ℝ) < Constants.phi := Constants.phi_gt_onePointSixOne
 115  have h2 : Constants.phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
 116  constructor
 117  · -- Lower bound: ln(φ) > ln(1.61) > 0.47 via exp(0.47) < 1.61
 118    have h_log_mono : Real.log (1.61 : ℝ) < Real.log Constants.phi := by
 119      apply Real.log_lt_log
 120      all_goals nlinarith [Constants.phi_pos]
 121    -- Numerical verification: exp(0.47) < 1.61 using exp_bound'
 122    have h_lower : (0.47 : ℝ) < Real.log (1.61 : ℝ) := by
 123      have h_exp : Real.exp (0.47 : ℝ) < (1.61 : ℝ) := by
 124        -- Upper bound via Taylor remainder
 125        have h1 : (0.47 : ℝ) ≥ 0 := by norm_num
 126        have h2 : (0.47 : ℝ) ≤ 1 := by norm_num
 127        have h3 := Real.exp_bound' h1 h2 (by norm_num : (0 : ℕ) < 4)
 128        norm_num [Finset.sum_range_succ, Nat.factorial] at h3 ⊢
 129        nlinarith [Real.exp_pos 0.47]
 130      have h_ln : (0.47 : ℝ) < Real.log (1.61 : ℝ) := by
 131        have h1 : Real.log (Real.exp (0.47 : ℝ)) = (0.47 : ℝ) := Real.log_exp (0.47 : ℝ)
 132        have h2 : Real.log (Real.exp (0.47 : ℝ)) < Real.log (1.61 : ℝ) := by
 133          apply Real.log_lt_log
 134          all_goals nlinarith [h_exp, Real.exp_pos 0.47]
 135        linarith [h1]
 136      linarith
 137    linarith
 138  · -- Upper bound: ln(φ) < ln(1.62) < 0.49 via 1.62 < exp(0.49)
 139    have h_log_mono : Real.log Constants.phi < Real.log (1.62 : ℝ) := by
 140      apply Real.log_lt_log
 141      all_goals nlinarith [Constants.phi_pos]
 142    -- Numerical verification: 1.62 < exp(0.49) using exp_bound lower bound
 143    have h_upper : Real.log (1.62 : ℝ) < (0.49 : ℝ) := by
 144      have h_exp : Real.exp (0.49 : ℝ) > (1.62 : ℝ) := by
 145        -- Lower bound: exp(x) > sum of first n terms (all terms positive for x > 0)
 146        have h1 : |(0.49 : ℝ)| ≤ 1 := by norm_num [abs_of_nonneg]
 147        have h2 := Real.exp_bound h1 (by norm_num : (0 : ℕ) < 4)
 148        norm_num [Finset.sum_range_succ, Nat.factorial, abs] at h2 ⊢
 149        nlinarith [Real.exp_pos 0.49]
 150      have h_ln : Real.log (1.62 : ℝ) < (0.49 : ℝ) := by
 151        have h1 : Real.log (Real.exp (0.49 : ℝ)) = (0.49 : ℝ) := Real.log_exp (0.49 : ℝ)
 152        have h2 : Real.log (1.62 : ℝ) < Real.log (Real.exp (0.49 : ℝ)) := by
 153          apply Real.log_lt_log
 154          all_goals nlinarith [h_exp, Real.exp_pos 0.49]
 155        linarith [h1]
 156      linarith
 157    linarith
 158
 159/-- **THEOREM C-006.5**: k_R = J_bit (the ledger bit cost).
 160
 161    This is the fundamental identity: the Boltzmann analog equals
 162the cost of a single bit in the recognition ledger. -/
 163theorem k_R_eq_J_bit : k_R = Constants.J_bit := rfl
 164
 165/-- **THEOREM C-006.6**: The thermal energy quantum.
 166
 167    At T = 1 (in RS temperature units), E_thermal = k_R = ln(φ).
 168    This connects temperature to the ledger structure. -/
 169theorem thermal_energy_at_unit_T (T : ℝ) (hT : T = 1) : k_R * T = Real.log Constants.phi := by
 170  rw [hT]
 171  unfold k_R
 172  ring
 173
 174/-! ## C-006 Summary Certificate -/
 175
 176/-- **C-006 CERTIFICATE**: The Boltzmann analog k_R is DERIVED.
 177
 178    **Key Results**:
 179    1. k_R = ln(φ) — forced by the ledger structure (T6)
 180    2. k_R > 0 — required for positive temperature
 181    3. k_R < 0.5 — from φ < 1.62 < e^0.5
 182    4. k_R = J_bit — unified with ledger bit cost
 183    5. Thermal energy E = k_R · T connects to ledger cost
 184
 185    **Status**: DERIVED with 0 free parameters.
 186    **Origin**: Self-similar ledger structure → φ → ln(φ) = k_R.
 187    **SI Connection**: k_B^SI = k_R · (calibration factor from λ_rec). -/
 188def C006_certificate : String :=
 189  "═══════════════════════════════════════════════════════════\n" ++
 190  "  C-006: BOLTZMANN CONSTANT k_R — STATUS: DERIVED\n" ++
 191  "═══════════════════════════════════════════════════════════\n" ++
 192  "✓ k_R = ln(φ) — forced by ledger structure (T6)\n" ++
 193  "✓ k_R > 0 — positive temperature scale\n" ++
 194  "✓ k_R < 0.5 — bounded by φ < 1.62\n" ++
 195  "✓ k_R = J_bit — unified with bit cost\n" ++
 196  "✓ Thermal energy E = k_R · T\n" ++
 197  "ORIGIN: Self-similar ledger closure → φ → ln(φ) = k_R\n" ++
 198  "═══════════════════════════════════════════════════════════"
 199
 200end BoltzmannConstant
 201end Constants
 202end IndisputableMonolith
 203