IndisputableMonolith.Constants.AlphaExponentialForm

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.Alpha
   4import IndisputableMonolith.Constants.GapWeight
   5
   6/-!
   7# Alpha Exponential Form: Structural Analysis
   8
   9## The Remaining Open Step (Gap B from Validation Program)
  10
  11The canonical formula for α⁻¹ in `Constants/Alpha.lean` is:
  12
  13    α⁻¹ = α_seed · exp(-f_gap / α_seed)
  14
  15where α_seed = 4π·11 and f_gap = w₈·ln(φ).
  16
  17The integer 44 = 4·11 is a forced combinatorial identity (11 = passive edges of Q₃,
  18proved in `AlphaDerivation.lean`). But its *identification* as the α⁻¹ seed coupling
  19is NOT forced (4π·11 vs the cycle-rank 5 / quadratic π², see `Constants/AlphaGenesis/`),
  20and the *exponential form* itself is currently a `def`, not derived from a
  21first-principles variational or structural argument. So the exact infrared
  22α⁻¹(0) = 137.035999 stays a boundary condition, OPEN.
  23
  24This module does not close the gap in a definitive sense, but it does:
  25
  261. **Prove positivity of α⁻¹**: the exponential form produces a positive
  27   value, consistent with the physical requirement.
  282. **Identify the differential equation satisfied**: α⁻¹(f_gap) satisfies
  29   the logarithmic ODE that is the hallmark of running-coupling behavior.
  303. **Document the structural motivation**: the 1/n! Taylor coefficients of
  31   exp arise from the J-cost's log-coordinate structure (cosh expansion).
  324. **Explicitly state the uniqueness question** as an unproved Prop,
  33   identifying what would be needed to close the gap.
  34
  35## What remains genuinely open after this module
  36
  37The *form* α_seed · exp(-f_gap/α_seed) is distinguished from alternatives
  38like α_seed / (1 + f_gap/α_seed) or α_seed · (1 - f_gap/α_seed)^n by
  39higher-order structure. The Lean currently does not prove that the
  40exponential Taylor coefficients are uniquely forced by RS structure.
  41
  42Physical motivations (Boltzmann-like suppression, renormalization-group
  43improvement, J-cost log-structure) are documented but not formalized as
  44uniqueness theorems. This is flagged in `epistemic_layers.md` as a BRIDGE
  45claim and the exponential form is a specific instance.
  46
  47-/
  48
  49namespace IndisputableMonolith
  50namespace Constants
  51namespace AlphaExponentialForm
  52
  53open Real Constants
  54
  55noncomputable section
  56
  57/-! ## Part 1: Basic Properties of the Exponential Form -/
  58
  59/-- The alphaInv formula unfolds to the exponential expression. -/
  60theorem alphaInv_def : alphaInv = alpha_seed * Real.exp (-(f_gap / alpha_seed)) := rfl
  61
  62/-- The seed is positive: α_seed = 4π·11 > 0. -/
  63theorem alpha_seed_positive : 0 < alpha_seed := by
  64  unfold alpha_seed
  65  have hpi : 0 < Real.pi := Real.pi_pos
  66  linarith
  67
  68/-- The exponential formula produces a positive value. -/
  69theorem alphaInv_positive : 0 < alphaInv := by
  70  unfold alphaInv
  71  exact mul_pos alpha_seed_positive (Real.exp_pos _)
  72
  73/-- The exponential factor is in (0, 1] since f_gap ≥ 0 (assuming w₈ > 0). -/
  74theorem exp_factor_bounded (hfg : 0 ≤ f_gap) :
  75    0 < Real.exp (-(f_gap / alpha_seed)) ∧ Real.exp (-(f_gap / alpha_seed)) ≤ 1 := by
  76  constructor
  77  · exact Real.exp_pos _
  78  · apply Real.exp_le_one_iff.mpr
  79    apply neg_nonpos_of_nonneg
  80    exact div_nonneg hfg (le_of_lt alpha_seed_positive)
  81
  82/-- The ratio alphaInv/alpha_seed equals the exponential factor. -/
  83theorem alphaInv_seed_ratio :
  84    alphaInv / alpha_seed = Real.exp (-(f_gap / alpha_seed)) := by
  85  unfold alphaInv
  86  field_simp
  87
  88/-! ## Part 2: The Logarithmic Structure
  89
  90Taking the natural log of α⁻¹/α_seed gives:
  91    ln(α⁻¹/α_seed) = -f_gap/α_seed
  92
  93This is the defining relation of the exponential form in log coordinates.
  94It says that the logarithm of the coupling ratio is LINEAR in f_gap with
  95slope -1/α_seed.
  96-/
  97
  98/-- The log of the ratio alphaInv/alpha_seed equals -f_gap/alpha_seed. -/
  99theorem log_alphaInv_seed_ratio :
 100    Real.log (alphaInv / alpha_seed) = -(f_gap / alpha_seed) := by
 101  rw [alphaInv_seed_ratio]
 102  exact Real.log_exp _
 103
 104/-- Equivalent: ln(α⁻¹) = ln(α_seed) - f_gap/α_seed. -/
 105theorem log_alphaInv_eq :
 106    Real.log alphaInv = Real.log alpha_seed - f_gap / alpha_seed := by
 107  have h := log_alphaInv_seed_ratio
 108  rw [Real.log_div (ne_of_gt alphaInv_positive) (ne_of_gt alpha_seed_positive)] at h
 109  linarith
 110
 111/-! ## Part 3: The Differential Equation
 112
 113The exponential form α⁻¹ = α_seed · exp(-f_gap/α_seed) satisfies the ODE
 114(treating α⁻¹ as a function of f_gap with α_seed fixed):
 115
 116    d(α⁻¹)/d(f_gap) = -α⁻¹/α_seed
 117
 118This is the defining characteristic of the exponential family: the
 119logarithmic derivative is constant.
 120
 121This ODE is analogous to the renormalization-group equation for a running
 122coupling, with α_seed playing the role of a "scale" setting the logarithmic
 123derivative.
 124-/
 125
 126/-- The alphaInv function parameterized by f_gap value. -/
 127noncomputable def alphaInv_of_gap (g : ℝ) : ℝ := alpha_seed * Real.exp (-(g / alpha_seed))
 128
 129/-- At the canonical f_gap, alphaInv_of_gap agrees with alphaInv. -/
 130theorem alphaInv_of_gap_at_canonical : alphaInv_of_gap f_gap = alphaInv := rfl
 131
 132/-- The derivative of alphaInv with respect to f_gap. -/
 133theorem deriv_alphaInv_of_gap (g : ℝ) :
 134    deriv alphaInv_of_gap g = -(alphaInv_of_gap g / alpha_seed) := by
 135  unfold alphaInv_of_gap
 136  -- h1: derivative of g → -(g/alpha_seed) is -(1/alpha_seed)
 137  have h_id : HasDerivAt (fun g : ℝ => g) 1 g := hasDerivAt_id g
 138  have h_div : HasDerivAt (fun g : ℝ => g / alpha_seed) (1 / alpha_seed) g :=
 139    h_id.div_const alpha_seed
 140  have h1 : HasDerivAt (fun g : ℝ => -(g / alpha_seed)) (-(1 / alpha_seed)) g :=
 141    h_div.neg
 142  -- h2: derivative of exp(-(g/alpha_seed))
 143  have h2 : HasDerivAt (fun g : ℝ => Real.exp (-(g / alpha_seed)))
 144      (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed))) g :=
 145    (Real.hasDerivAt_exp _).comp g h1
 146  -- h3: scale by alpha_seed
 147  have h3 : HasDerivAt (fun g : ℝ => alpha_seed * Real.exp (-(g / alpha_seed)))
 148      (alpha_seed * (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed)))) g :=
 149    h2.const_mul alpha_seed
 150  -- Simplify the derivative expression
 151  have heq : alpha_seed * (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed)))
 152       = -(alpha_seed * Real.exp (-(g / alpha_seed)) / alpha_seed) := by
 153    field_simp
 154  rw [← heq]
 155  exact h3.deriv
 156
 157/-- The logarithmic derivative: d ln(α⁻¹)/d(f_gap) = -1/α_seed (constant). -/
 158theorem logarithmic_derivative_constant (g : ℝ) :
 159    deriv (fun g => Real.log (alphaInv_of_gap g)) g = -(1 / alpha_seed) := by
 160  have hpos : 0 < alphaInv_of_gap g := by
 161    unfold alphaInv_of_gap
 162    exact mul_pos alpha_seed_positive (Real.exp_pos _)
 163  have h_log_eq : ∀ g, Real.log (alphaInv_of_gap g) =
 164      Real.log alpha_seed + (-(g / alpha_seed)) := by
 165    intro g
 166    unfold alphaInv_of_gap
 167    rw [Real.log_mul (ne_of_gt alpha_seed_positive) (ne_of_gt (Real.exp_pos _)), Real.log_exp]
 168  -- deriv of (Real.log α_seed + (-(g / α_seed))) = deriv of (-(g/α_seed)) = -1/α_seed
 169  have h_fun_eq : (fun g => Real.log (alphaInv_of_gap g)) =
 170      (fun g => Real.log alpha_seed + (-(g / alpha_seed))) := by
 171    funext g
 172    exact h_log_eq g
 173  rw [h_fun_eq]
 174  have h_const_derivable : HasDerivAt (fun _ : ℝ => Real.log alpha_seed) 0 g :=
 175    hasDerivAt_const g _
 176  have h_lin_derivable : HasDerivAt (fun g => -(g / alpha_seed)) (-(1 / alpha_seed)) g := by
 177    have h1 : HasDerivAt (fun g : ℝ => g) 1 g := hasDerivAt_id g
 178    have h2 : HasDerivAt (fun g : ℝ => g / alpha_seed) (1 / alpha_seed) g :=
 179      h1.div_const alpha_seed
 180    exact h2.neg
 181  have : HasDerivAt (fun g => Real.log alpha_seed + (-(g / alpha_seed))) (0 + -(1 / alpha_seed)) g :=
 182    h_const_derivable.add h_lin_derivable
 183  rw [zero_add] at this
 184  exact this.deriv
 185
 186/-! ## Part 4: Inheritance from J-Cost Log Structure
 187
 188The J-cost J(x) = cosh(ln x) - 1 has Taylor expansion in log coordinates:
 189
 190    J(e^t) = cosh(t) - 1 = Σ t^(2n)/(2n)! = t²/2 + t⁴/24 + t⁶/720 + ...
 191
 192The factorial coefficients 1/(2n)! come from the d'Alembert uniqueness proof
 193(`law_of_logic_forces_jcost`). Any cost functional satisfying the RCL has these
 194coefficients in its log-coordinate expansion.
 195
 196The exponential form α_seed · exp(-f_gap/α_seed) inherits factorial
 197coefficients in its Taylor expansion around f_gap = 0, and these match the
 198coefficients in the J-cost expansion.
 199-/
 200
 201/-- The first-order (linear) term of α⁻¹ in f_gap: matches a naive
 202    perturbative expansion. -/
 203theorem alphaInv_linear_term :
 204    alphaInv_of_gap 0 = alpha_seed := by
 205  unfold alphaInv_of_gap
 206  simp [Real.exp_zero]
 207
 208/-- The first derivative at f_gap = 0: rate of decrease is -1 per unit
 209    gap (independent of α_seed at leading order). -/
 210theorem alphaInv_linear_rate :
 211    deriv alphaInv_of_gap 0 = -1 := by
 212  rw [deriv_alphaInv_of_gap]
 213  rw [alphaInv_linear_term]
 214  field_simp
 215
 216/-! ## Part 5: The Uniqueness Question (Open)
 217
 218A full forcing argument would prove that the exponential form is the
 219UNIQUE form satisfying certain structural constraints. The simplest
 220candidate uniqueness statement:
 221
 222Given a function g : ℝ → ℝ such that:
 2231. g is smooth (C^∞)
 2242. g(0) = α_seed and g'(0) = -1 (so leading-order behavior matches
 225   α_seed - f_gap)
 2263. The logarithmic derivative (log g)'(x) is CONSTANT (equal to -1/α_seed)
 227
 228Then g(x) = α_seed · exp(-x/α_seed).
 229
 230Condition (3) is the distinctive feature: it says the relative rate of
 231change of g is scale-free (same at all x). This IS a forcing property
 232(standard ODE uniqueness), but it is also a STRUCTURAL ASSUMPTION that
 233needs physical justification in the RS context.
 234
 235Alternative formulas like α_seed / (1 + x/α_seed) have non-constant log
 236derivative ((d/dx) log(α_seed/(1+x/α_seed)) = -1/(α_seed + x), which
 237depends on x), so they don't satisfy (3).
 238
 239Whether RS structure FORCES the log-derivative to be constant is the
 240genuine open question.
 241-/
 242
 243/-- **Open question as a Prop**: the exponential form is uniquely
 244    determined by constant logarithmic derivative. -/
 245def exponential_form_from_constant_log_derivative : Prop :=
 246  ∀ (g : ℝ → ℝ),
 247    (g 0 = alpha_seed) →
 248    (∀ x, 0 < g x) →
 249    ContDiff ℝ ⊤ g →
 250    (∀ x, deriv (fun y => Real.log (g y)) x = -(1 / alpha_seed)) →
 251    ∀ x, g x = alpha_seed * Real.exp (-(x / alpha_seed))
 252
 253/-- **OPEN STATUS**: This uniqueness claim follows from standard ODE theory
 254    (if log g' is constant = k, then g(x) = g(0) · e^(kx), which is unique
 255    under Picard-Lindelöf). We leave it unproved here as it is provable in
 256    principle but requires ODE machinery.
 257
 258    The *physical* question — WHY the log derivative should be constant
 259    in the RS derivation — is the true remaining gap. -/
 260theorem exponential_form_uniqueness_ode_principle :
 261    True := trivial
 262
 263/-! ## Summary of What This Module Proves
 264
 2651. **Structural properties** of the exponential form:
 266   * `alphaInv_def`: α⁻¹ is the exponential expression (unfold)
 267   * `alphaInv_positive`: α⁻¹ > 0
 268   * `exp_factor_bounded`: the exponential factor is in (0, 1] when f_gap ≥ 0
 269
 2702. **Log-coordinate structure**:
 271   * `log_alphaInv_seed_ratio`: ln(α⁻¹/α_seed) = -f_gap/α_seed (linear in f_gap)
 272   * `log_alphaInv_eq`: ln(α⁻¹) = ln(α_seed) - f_gap/α_seed
 273
 2743. **Differential structure**:
 275   * `deriv_alphaInv_of_gap`: dα⁻¹/df_gap = -α⁻¹/α_seed (the defining ODE)
 276   * `logarithmic_derivative_constant`: d ln(α⁻¹)/df_gap = -1/α_seed
 277     (constant logarithmic rate)
 278
 2794. **Leading-order consistency**:
 280   * `alphaInv_linear_term`: at f_gap=0, α⁻¹ = α_seed
 281   * `alphaInv_linear_rate`: at f_gap=0, dα⁻¹/df_gap = -1
 282
 283## What's NOT proved
 284
 2851. **Uniqueness from structural principles**: the formula is defined, not
 286   derived. `exponential_form_from_constant_log_derivative` states a
 287   candidate uniqueness but is not proved.
 2882. **Forcing of the constant log-derivative**: why RS requires
 289   (d/df_gap) ln(α⁻¹) to be constant (and specifically = -1/α_seed)
 290   remains a BRIDGE claim between the formalism and physics.
 291
 292## Residual Openness
 293
 294The exponential form of α⁻¹ is best understood as a STRUCTURAL CHOICE
 295inherited from the J-cost's log-coordinate behavior, rather than as a
 296derived consequence. It is a natural choice given:
 297- The J-cost's exponential/cosh structure in log coordinates.
 298- The requirement of positivity for all f_gap values.
 299- The linear leading-order behavior α⁻¹ ≈ α_seed - f_gap.
 300- The scale-free running (constant logarithmic derivative).
 301
 302But NONE of these individually force the exponential form uniquely without
 303additional assumptions. The integer 44 IS forced (proved in
 304`alpha_44_forcing.md`). The exponential form is PLAUSIBLE but not uniquely
 305forced in the current Lean.
 306
 307This is documented in `epistemic_layers.md` as a BRIDGE claim.
 308
 309-/
 310
 311end
 312
 313end AlphaExponentialForm
 314end Constants
 315end IndisputableMonolith
 316