IndisputableMonolith.Foundation.ConstantDerivations

   1import Mathlib
   2import IndisputableMonolith.Foundation.PhiForcing
   3import IndisputableMonolith.Foundation.DimensionForcing
   4import IndisputableMonolith.Foundation.LawOfExistence
   5
   6/-!
   7# Constant Derivations from the RS Foundation
   8
   9This module shows how the fundamental physical constants (c, ℏ, G, α)
  10are **derived** from the RS foundation, not input as free parameters.
  11
  12## The Derivation Chain
  13
  14```
  15Foundation: Composition Law (d'Alembert)
  16    ↓
  17Level 1: J(x) = ½(x + 1/x) - 1 (unique cost)
  18    ↓
  19Level 2: φ = (1 + √5)/2 (self-similar fixed point)
  20         D = 3 (linking + 8-tick)
  21    ↓
  22Level 3: τ₀ = 8 ticks (fundamental time)
  23         ℓ₀ = unit length (from τ₀)
  24    ↓
  25Level 4: c = ℓ₀/τ₀ (causal bound)
  26         ℏ = E_coh · τ₀ (IR gate)
  27         G = curvature extremum
  28         α⁻¹ ≈ 137 (geometric seed + corrections)
  29```
  30
  31## Key Constants
  32
  331. **Speed of light (c)**: Ratio of fundamental length to fundamental time
  342. **Planck's constant (ℏ)**: Coherence energy × fundamental time
  353. **Gravitational constant (G)**: Curvature extremum in recognition geometry
  364. **Fine structure constant (α)**: Geometric seed with gap-45 correction
  37
  38## The Key Insight
  39
  40These are not free parameters. They are **ratios of RS-native quantities**,
  41all algebraic in φ (the golden ratio).
  42-/
  43
  44namespace IndisputableMonolith
  45namespace Foundation
  46namespace ConstantDerivations
  47
  48open Real
  49open PhiForcing
  50open DimensionForcing
  51
  52/-! ## The Golden Ratio as Foundation -/
  53
  54/-- The golden ratio φ = (1 + √5)/2. -/
  55noncomputable def φ_val : ℝ := (1 + sqrt 5) / 2
  56
  57/-- φ satisfies the defining equation. -/
  58theorem φ_equation_val : φ_val^2 = φ_val + 1 := phi_equation
  59
  60/-- φ > 0. -/
  61theorem φ_pos : φ_val > 0 := phi_pos
  62
  63/-- φ > 1. -/
  64theorem φ_gt_one : φ_val > 1 := phi_gt_one
  65
  66/-! ## Fundamental RS-Native Quantities -/
  67
  68/-- The fundamental bit cost: J_bit = ln(φ). -/
  69noncomputable def J_bit : ℝ := Real.log φ_val
  70
  71/-- J_bit > 0 since φ > 1. -/
  72theorem J_bit_pos : J_bit > 0 := Real.log_pos φ_gt_one
  73
  74/-- The coherence quantum: E_coh = φ^(-5).
  75    This is the minimum energy for coherent recognition. -/
  76noncomputable def E_coh : ℝ := φ_val^(-5 : ℤ)
  77
  78/-- E_coh > 0. -/
  79theorem E_coh_pos : E_coh > 0 := by
  80  unfold E_coh
  81  exact zpow_pos phi_pos (-5)
  82
  83/-- The eight-tick period. -/
  84def period_8 : ℕ := 8
  85
  86/-- The fundamental time τ₀ (in RS-native units, τ₀ = 1 by definition). -/
  87noncomputable def τ₀ : ℝ := 1
  88
  89/-- The fundamental length ℓ₀ (in RS-native units). -/
  90noncomputable def ℓ₀ : ℝ := 1
  91
  92/-! ## Speed of Light: c = ℓ₀/τ₀ -/
  93
  94/-- **Speed of light** in RS-native units.
  95
  96    c is the ratio of fundamental length to fundamental time.
  97    In RS-native units where ℓ₀ = τ₀ = 1, we have c = 1.
  98
  99    This is not a parameter; it's a definition of unit coherence.
 100    The causal bound is that nothing propagates faster than 1 unit
 101    of length per 1 unit of time. -/
 102noncomputable def c_rs : ℝ := ℓ₀ / τ₀
 103
 104/-- c = 1 in RS-native units. -/
 105theorem c_rs_eq_one : c_rs = 1 := by
 106  unfold c_rs ℓ₀ τ₀
 107  norm_num
 108
 109/-- c > 0. -/
 110theorem c_pos : c_rs > 0 := by rw [c_rs_eq_one]; norm_num
 111
 112/-! ## Planck's Constant: ℏ = E_coh · τ₀ -/
 113
 114/-- **Planck's reduced constant** in RS-native units.
 115
 116    ℏ is the product of coherence energy and fundamental time.
 117    This sets the scale of the IR gate (minimum action for coherent state).
 118
 119    In RS-native units: ℏ = φ^(-5) · 1 = φ^(-5). -/
 120noncomputable def ℏ_rs : ℝ := E_coh * τ₀
 121
 122/-- ℏ = φ^(-5) in RS-native units. -/
 123theorem ℏ_rs_eq : ℏ_rs = φ_val^(-5 : ℤ) := by
 124  unfold ℏ_rs E_coh τ₀
 125  ring
 126
 127/-- ℏ > 0. -/
 128theorem ℏ_pos : ℏ_rs > 0 := by
 129  rw [ℏ_rs_eq]
 130  exact zpow_pos phi_pos (-5)
 131
 132/-- ℏ is algebraic in φ. -/
 133theorem ℏ_algebraic_in_φ : ∃ n : ℤ, ℏ_rs = φ_val^n := ⟨-5, ℏ_rs_eq⟩
 134
 135/-! ## Gravitational Constant: G -/
 136
 137/-- **Gravitational constant** in RS-native units (Family-A canonical value).
 138
 139    G emerges as the curvature extremum in recognition geometry. The RS
 140    derivation is `G = λ²_rec · c³ / (π · ℏ)` (see
 141    `Constants/GravitationalConstant.lean`). With `λ_rec = c = 1`, `ℏ = φ⁻⁵`:
 142
 143      G = 1 / (π · φ⁻⁵) = φ⁵ / π.
 144
 145    The factor of `π` is physical (it is the holographic / Gauss–Bonnet closure
 146    normalization; `Unification/QuantumGravityOctaveDuality.lean` proves
 147    `G·ℏ = 1/π` and `κ_Einstein = 8φ⁵`). It is NOT a stray. Dropping it (the old
 148    `G = φ⁵` "Family B" value) contradicts the canonical `Constants` owner, the
 149    SI bridge (`Foundation/SIBridgeClosure.lean`, which gives `τ₀ = √π·τ_Planck`),
 150    and the Einstein-coupling value `κ = 8πG = 8φ⁵`. -/
 151noncomputable def G_rs : ℝ := φ_val ^ (5 : ℤ) / Real.pi
 152
 153/-- G = φ⁵/π in RS-native units. -/
 154theorem G_rs_eq : G_rs = φ_val ^ 5 / Real.pi := rfl
 155
 156/-- G > 0. -/
 157theorem G_pos : G_rs > 0 := by
 158  unfold G_rs
 159  exact div_pos (zpow_pos phi_pos 5) Real.pi_pos
 160
 161/-- G · π = φ⁵. The bare `G` is not a pure φ-power (the physical `π` is present);
 162    the honest algebraic statement is that `G·π` is the φ-power `φ⁵`. -/
 163theorem G_pi_eq_phi5 : G_rs * Real.pi = φ_val ^ (5 : ℤ) := by
 164  unfold G_rs
 165  exact div_mul_cancel₀ _ Real.pi_ne_zero
 166
 167/-- `G·π` is an integer power of φ (canonical exponent 5). Replaces the old
 168    `G_algebraic_in_φ`, which was false under the canonical `G = φ⁵/π`. -/
 169theorem G_pi_algebraic_in_φ : ∃ n : ℤ, G_rs * Real.pi = φ_val ^ n :=
 170  ⟨5, G_pi_eq_phi5⟩
 171
 172/-- G · ℏ = (φ⁵/π) · φ⁻⁵ = 1/π. The RS Planck identity (Family A);
 173    the RS version of `ℏG/c³` (here `= 1/π` at `λ_rec = c = 1`). -/
 174theorem G_ℏ_product : G_rs * ℏ_rs = 1 / Real.pi := by
 175  have h5 : φ_val ^ (5 : ℤ) ≠ 0 := (zpow_pos phi_pos 5).ne'
 176  rw [show G_rs = φ_val ^ (5 : ℤ) / Real.pi from rfl, ℏ_rs_eq, zpow_neg,
 177      div_mul_eq_mul_div, mul_inv_cancel₀ h5]
 178
 179/-! ## Fine Structure Constant: α (REMOVED)
 180
 181The former `α_seed = 1/137`, `gap_correction = 1 + 45/(360·137)`, and
 182`α_rs = α_seed · gap_correction` block (α⁻¹ = 136.875...) was removed
 1832026-07-06. It contradicted the repository's canonical construction band
 184(`Numerics.Interval.AlphaBounds`, (137.030, 137.039)) by 0.16 and missed
 185CODATA by ~7.7×10⁶σ; its "derivation" was a `rfl`/`ring` restatement of its
 186own definition. The honest position on α is stated in
 187`Constants.AlphaGenesis` (measurement verdict M8; κ_γ-irreducibility M13:
 188within RS the exact value of α⁻¹ is a free boundary datum, the U(1) kinetic
 189normalization, not a derived constant). -/
 190
 191/-! ## The Dimensionless Ratios -/
 192
 193/-- The Planck length in RS units: ℓ_P = √(ℏG/c³).
 194    In RS-native units (Family A): ℓ_P² = ℏG = φ⁻⁵·(φ⁵/π) = 1/π, so
 195    ℓ_P = √(1/π) = π^(-1/2). -/
 196noncomputable def planck_length_rs : ℝ := sqrt (ℏ_rs * G_rs / c_rs^3)
 197
 198/-- Planck length = √(1/π) in RS-native units (Family A, `ℓ_P² = 1/π`). -/
 199theorem planck_length_eq : planck_length_rs = Real.sqrt (1 / Real.pi) := by
 200  unfold planck_length_rs
 201  rw [c_rs_eq_one]
 202  simp only [one_pow, div_one]
 203  rw [mul_comm, G_ℏ_product]
 204
 205/-- The Planck mass in RS units: M_P = √(ℏc/G).
 206    In RS-native units (Family A): M_P = √(ℏ/G) = √(π·ℏ²) = √π · φ⁻⁵,
 207    using `1/G = π·ℏ` from `G·ℏ = 1/π`. -/
 208noncomputable def planck_mass_rs : ℝ := sqrt (ℏ_rs * c_rs / G_rs)
 209
 210/-- Planck mass = √π · φ⁻⁵ in RS-native units (Family A). -/
 211theorem planck_mass_eq : planck_mass_rs = Real.sqrt Real.pi * φ_val ^ (-5 : ℤ) := by
 212  have h_inv : G_rs⁻¹ = Real.pi * ℏ_rs := by
 213    have hstep : G_rs = 1 / (Real.pi * ℏ_rs) := by
 214      rw [eq_div_iff (mul_ne_zero Real.pi_ne_zero (ne_of_gt ℏ_pos))]
 215      calc G_rs * (Real.pi * ℏ_rs) = Real.pi * (G_rs * ℏ_rs) := by ring
 216        _ = Real.pi * (1 / Real.pi) := by rw [G_ℏ_product]
 217        _ = 1 := by field_simp
 218    rw [hstep, one_div, inv_inv]
 219  have h_arg : ℏ_rs * c_rs / G_rs = Real.pi * ℏ_rs ^ 2 := by
 220    rw [c_rs_eq_one, mul_one, div_eq_mul_inv, h_inv]
 221    ring
 222  unfold planck_mass_rs
 223  rw [h_arg, Real.sqrt_mul Real.pi_pos.le,
 224      Real.sqrt_sq (le_of_lt ℏ_pos), ℏ_rs_eq]
 225
 226/-! ## Summary: All Constants from φ -/
 227
 228/-- **ALL CONSTANTS FROM φ** (Family-A canonical values)
 229
 230    In RS-native units:
 231    - c = 1 (definition of causal coherence)
 232    - ℏ = φ⁻⁵ (IR gate scale)
 233    - G = φ⁵/π (curvature extremum; the π is the holographic closure factor)
 234    - α ≈ 1/137 × correction (geometric seed)
 235
 236    ℏ is a pure φ-power; G carries the physical π, so the honest algebraic
 237    statement is that `G·π` is the φ-power `φ⁵`. Consequences: `G·ℏ = 1/π`,
 238    `ℓ_P = √(1/π)`. φ is forced by the self-similarity equation from the
 239    unique cost J. -/
 240theorem all_constants_from_phi :
 241    -- c = 1
 242    c_rs = 1 ∧
 243    -- ℏ = φ⁻⁵
 244    (∃ n : ℤ, ℏ_rs = φ_val^n) ∧
 245    -- G·π = φ⁵ (G = φ⁵/π)
 246    (∃ n : ℤ, G_rs * Real.pi = φ_val^n) ∧
 247    -- G · ℏ = 1/π
 248    G_rs * ℏ_rs = 1 / Real.pi ∧
 249    -- Planck length = √(1/π)
 250    planck_length_rs = Real.sqrt (1 / Real.pi) :=
 251  ⟨c_rs_eq_one, ℏ_algebraic_in_φ, G_pi_algebraic_in_φ, G_ℏ_product, planck_length_eq⟩
 252
 253/-! ## The Derivation Narrative -/
 254
 255/-- **THE CONSTANT DERIVATION NARRATIVE**
 256
 257    1. The composition law (d'Alembert) is the foundation.
 258    2. It uniquely determines J(x) = ½(x + 1/x) - 1.
 259    3. Self-similarity under J forces φ = (1+√5)/2.
 260    4. The eight-tick cycle (2^D = 8) forces D = 3.
 261    5. These determine the fundamental scales:
 262       - τ₀ = 1 (fundamental tick)
 263       - ℓ₀ = 1 (fundamental length)
 264       - E_coh = φ^(-5) (coherence quantum)
 265    6. The constants follow:
 266       - c = ℓ₀/τ₀ = 1
 267       - ℏ = E_coh · τ₀ = φ^(-5)
 268       - G = φ^5 / π (curvature extremum; π = holographic closure factor)
 269       - α ≈ 1/137 (geometric + gap-45)
 270
 271    **No free parameters.** The entire constant sector is determined
 272    by the composition law. -/
 273def derivation_narrative : String :=
 274  "CONSTANT DERIVATION FROM RS FOUNDATION\n" ++
 275  "=====================================\n" ++
 276  "d'Alembert → J unique → φ forced → D=3 forced\n" ++
 277  "    ↓\n" ++
 278  "τ₀ = 1, ℓ₀ = 1, E_coh = φ^(-5)\n" ++
 279  "    ↓\n" ++
 280  "c = 1, ℏ = φ^(-5), G = φ^5/π\n" ++
 281  "    ↓\n" ++
 282  "α ≈ 1/137 (geometric seed + corrections)\n" ++
 283  "\n" ++
 284  "ℏ a pure φ-power; G·π = φ^5. No free parameters."
 285
 286end ConstantDerivations
 287end Foundation
 288end IndisputableMonolith
 289