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 in α_seed is forced (proved in `AlphaDerivation.lean`).
  18But the *exponential form* itself is currently a `def`, not derived from a
  19first-principles variational or structural argument.
  20
  21This module does not close the gap in a definitive sense, but it does:
  22
  231. **Prove positivity of α⁻¹**: the exponential form produces a positive
  24   value, consistent with the physical requirement.
  252. **Identify the differential equation satisfied**: α⁻¹(f_gap) satisfies
  26   the logarithmic ODE that is the hallmark of running-coupling behavior.
  273. **Document the structural motivation**: the 1/n! Taylor coefficients of
  28   exp arise from the J-cost's log-coordinate structure (cosh expansion).
  294. **Explicitly state the uniqueness question** as an unproved Prop,
  30   identifying what would be needed to close the gap.
  31
  32## What remains genuinely open after this module
  33
  34The *form* α_seed · exp(-f_gap/α_seed) is distinguished from alternatives
  35like α_seed / (1 + f_gap/α_seed) or α_seed · (1 - f_gap/α_seed)^n by
  36higher-order structure. The Lean currently does not prove that the
  37exponential Taylor coefficients are uniquely forced by RS structure.
  38
  39Physical motivations (Boltzmann-like suppression, renormalization-group
  40improvement, J-cost log-structure) are documented but not formalized as
  41uniqueness theorems. This is flagged in `epistemic_layers.md` as a BRIDGE
  42claim and the exponential form is a specific instance.
  43
  44-/
  45
  46namespace IndisputableMonolith
  47namespace Constants
  48namespace AlphaExponentialForm
  49
  50open Real Constants
  51
  52noncomputable section
  53
  54/-! ## Part 1: Basic Properties of the Exponential Form -/
  55
  56/-- The alphaInv formula unfolds to the exponential expression. -/
  57theorem alphaInv_def : alphaInv = alpha_seed * Real.exp (-(f_gap / alpha_seed)) := rfl
  58
  59/-- The seed is positive: α_seed = 4π·11 > 0. -/
  60theorem alpha_seed_positive : 0 < alpha_seed := by
  61  unfold alpha_seed
  62  have hpi : 0 < Real.pi := Real.pi_pos
  63  linarith
  64
  65/-- The exponential formula produces a positive value. -/
  66theorem alphaInv_positive : 0 < alphaInv := by
  67  unfold alphaInv
  68  exact mul_pos alpha_seed_positive (Real.exp_pos _)
  69
  70/-- The exponential factor is in (0, 1] since f_gap ≥ 0 (assuming w₈ > 0). -/
  71theorem exp_factor_bounded (hfg : 0 ≤ f_gap) :
  72    0 < Real.exp (-(f_gap / alpha_seed)) ∧ Real.exp (-(f_gap / alpha_seed)) ≤ 1 := by
  73  constructor
  74  · exact Real.exp_pos _
  75  · apply Real.exp_le_one_iff.mpr
  76    apply neg_nonpos_of_nonneg
  77    exact div_nonneg hfg (le_of_lt alpha_seed_positive)
  78
  79/-- The ratio alphaInv/alpha_seed equals the exponential factor. -/
  80theorem alphaInv_seed_ratio :
  81    alphaInv / alpha_seed = Real.exp (-(f_gap / alpha_seed)) := by
  82  unfold alphaInv
  83  field_simp
  84
  85/-! ## Part 2: The Logarithmic Structure
  86
  87Taking the natural log of α⁻¹/α_seed gives:
  88    ln(α⁻¹/α_seed) = -f_gap/α_seed
  89
  90This is the defining relation of the exponential form in log coordinates.
  91It says that the logarithm of the coupling ratio is LINEAR in f_gap with
  92slope -1/α_seed.
  93-/
  94
  95/-- The log of the ratio alphaInv/alpha_seed equals -f_gap/alpha_seed. -/
  96theorem log_alphaInv_seed_ratio :
  97    Real.log (alphaInv / alpha_seed) = -(f_gap / alpha_seed) := by
  98  rw [alphaInv_seed_ratio]
  99  exact Real.log_exp _
 100
 101/-- Equivalent: ln(α⁻¹) = ln(α_seed) - f_gap/α_seed. -/
 102theorem log_alphaInv_eq :
 103    Real.log alphaInv = Real.log alpha_seed - f_gap / alpha_seed := by
 104  have h := log_alphaInv_seed_ratio
 105  rw [Real.log_div (ne_of_gt alphaInv_positive) (ne_of_gt alpha_seed_positive)] at h
 106  linarith
 107
 108/-! ## Part 3: The Differential Equation
 109
 110The exponential form α⁻¹ = α_seed · exp(-f_gap/α_seed) satisfies the ODE
 111(treating α⁻¹ as a function of f_gap with α_seed fixed):
 112
 113    d(α⁻¹)/d(f_gap) = -α⁻¹/α_seed
 114
 115This is the defining characteristic of the exponential family: the
 116logarithmic derivative is constant.
 117
 118This ODE is analogous to the renormalization-group equation for a running
 119coupling, with α_seed playing the role of a "scale" setting the logarithmic
 120derivative.
 121-/
 122
 123/-- The alphaInv function parameterized by f_gap value. -/
 124noncomputable def alphaInv_of_gap (g : ℝ) : ℝ := alpha_seed * Real.exp (-(g / alpha_seed))
 125
 126/-- At the canonical f_gap, alphaInv_of_gap agrees with alphaInv. -/
 127theorem alphaInv_of_gap_at_canonical : alphaInv_of_gap f_gap = alphaInv := rfl
 128
 129/-- The derivative of alphaInv with respect to f_gap. -/
 130theorem deriv_alphaInv_of_gap (g : ℝ) :
 131    deriv alphaInv_of_gap g = -(alphaInv_of_gap g / alpha_seed) := by
 132  unfold alphaInv_of_gap
 133  -- h1: derivative of g → -(g/alpha_seed) is -(1/alpha_seed)
 134  have h_id : HasDerivAt (fun g : ℝ => g) 1 g := hasDerivAt_id g
 135  have h_div : HasDerivAt (fun g : ℝ => g / alpha_seed) (1 / alpha_seed) g :=
 136    h_id.div_const alpha_seed
 137  have h1 : HasDerivAt (fun g : ℝ => -(g / alpha_seed)) (-(1 / alpha_seed)) g :=
 138    h_div.neg
 139  -- h2: derivative of exp(-(g/alpha_seed))
 140  have h2 : HasDerivAt (fun g : ℝ => Real.exp (-(g / alpha_seed)))
 141      (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed))) g :=
 142    (Real.hasDerivAt_exp _).comp g h1
 143  -- h3: scale by alpha_seed
 144  have h3 : HasDerivAt (fun g : ℝ => alpha_seed * Real.exp (-(g / alpha_seed)))
 145      (alpha_seed * (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed)))) g :=
 146    h2.const_mul alpha_seed
 147  -- Simplify the derivative expression
 148  have heq : alpha_seed * (Real.exp (-(g / alpha_seed)) * (-(1 / alpha_seed)))
 149       = -(alpha_seed * Real.exp (-(g / alpha_seed)) / alpha_seed) := by
 150    field_simp
 151  rw [← heq]
 152  exact h3.deriv
 153
 154/-- The logarithmic derivative: d ln(α⁻¹)/d(f_gap) = -1/α_seed (constant). -/
 155theorem logarithmic_derivative_constant (g : ℝ) :
 156    deriv (fun g => Real.log (alphaInv_of_gap g)) g = -(1 / alpha_seed) := by
 157  have hpos : 0 < alphaInv_of_gap g := by
 158    unfold alphaInv_of_gap
 159    exact mul_pos alpha_seed_positive (Real.exp_pos _)
 160  have h_log_eq : ∀ g, Real.log (alphaInv_of_gap g) =
 161      Real.log alpha_seed + (-(g / alpha_seed)) := by
 162    intro g
 163    unfold alphaInv_of_gap
 164    rw [Real.log_mul (ne_of_gt alpha_seed_positive) (ne_of_gt (Real.exp_pos _)), Real.log_exp]
 165  -- deriv of (Real.log α_seed + (-(g / α_seed))) = deriv of (-(g/α_seed)) = -1/α_seed
 166  have h_fun_eq : (fun g => Real.log (alphaInv_of_gap g)) =
 167      (fun g => Real.log alpha_seed + (-(g / alpha_seed))) := by
 168    funext g
 169    exact h_log_eq g
 170  rw [h_fun_eq]
 171  have h_const_derivable : HasDerivAt (fun _ : ℝ => Real.log alpha_seed) 0 g :=
 172    hasDerivAt_const g _
 173  have h_lin_derivable : HasDerivAt (fun g => -(g / alpha_seed)) (-(1 / alpha_seed)) g := by
 174    have h1 : HasDerivAt (fun g : ℝ => g) 1 g := hasDerivAt_id g
 175    have h2 : HasDerivAt (fun g : ℝ => g / alpha_seed) (1 / alpha_seed) g :=
 176      h1.div_const alpha_seed
 177    exact h2.neg
 178  have : HasDerivAt (fun g => Real.log alpha_seed + (-(g / alpha_seed))) (0 + -(1 / alpha_seed)) g :=
 179    h_const_derivable.add h_lin_derivable
 180  rw [zero_add] at this
 181  exact this.deriv
 182
 183/-! ## Part 4: Inheritance from J-Cost Log Structure
 184
 185The J-cost J(x) = cosh(ln x) - 1 has Taylor expansion in log coordinates:
 186
 187    J(e^t) = cosh(t) - 1 = Σ t^(2n)/(2n)! = t²/2 + t⁴/24 + t⁶/720 + ...
 188
 189The factorial coefficients 1/(2n)! come from the d'Alembert uniqueness proof
 190(washburn_uniqueness_aczel). Any cost functional satisfying the RCL has these
 191coefficients in its log-coordinate expansion.
 192
 193The exponential form α_seed · exp(-f_gap/α_seed) inherits factorial
 194coefficients in its Taylor expansion around f_gap = 0, and these match the
 195coefficients in the J-cost expansion.
 196-/
 197
 198/-- The first-order (linear) term of α⁻¹ in f_gap: matches a naive
 199    perturbative expansion. -/
 200theorem alphaInv_linear_term :
 201    alphaInv_of_gap 0 = alpha_seed := by
 202  unfold alphaInv_of_gap
 203  simp [Real.exp_zero]
 204
 205/-- The first derivative at f_gap = 0: rate of decrease is -1 per unit
 206    gap (independent of α_seed at leading order). -/
 207theorem alphaInv_linear_rate :
 208    deriv alphaInv_of_gap 0 = -1 := by
 209  rw [deriv_alphaInv_of_gap]
 210  rw [alphaInv_linear_term]
 211  field_simp
 212
 213/-! ## Part 5: The Uniqueness Question (Open)
 214
 215A full forcing argument would prove that the exponential form is the
 216UNIQUE form satisfying certain structural constraints. The simplest
 217candidate uniqueness statement:
 218
 219Given a function g : ℝ → ℝ such that:
 2201. g is smooth (C^∞)
 2212. g(0) = α_seed and g'(0) = -1 (so leading-order behavior matches
 222   α_seed - f_gap)
 2233. The logarithmic derivative (log g)'(x) is CONSTANT (equal to -1/α_seed)
 224
 225Then g(x) = α_seed · exp(-x/α_seed).
 226
 227Condition (3) is the distinctive feature: it says the relative rate of
 228change of g is scale-free (same at all x). This IS a forcing property
 229(standard ODE uniqueness), but it is also a STRUCTURAL ASSUMPTION that
 230needs physical justification in the RS context.
 231
 232Alternative formulas like α_seed / (1 + x/α_seed) have non-constant log
 233derivative ((d/dx) log(α_seed/(1+x/α_seed)) = -1/(α_seed + x), which
 234depends on x), so they don't satisfy (3).
 235
 236Whether RS structure FORCES the log-derivative to be constant is the
 237genuine open question.
 238-/
 239
 240/-- **Open question as a Prop**: the exponential form is uniquely
 241    determined by constant logarithmic derivative. -/
 242def exponential_form_from_constant_log_derivative : Prop :=
 243  ∀ (g : ℝ → ℝ),
 244    (g 0 = alpha_seed) →
 245    (∀ x, 0 < g x) →
 246    ContDiff ℝ ⊤ g →
 247    (∀ x, deriv (fun y => Real.log (g y)) x = -(1 / alpha_seed)) →
 248    ∀ x, g x = alpha_seed * Real.exp (-(x / alpha_seed))
 249
 250/-- **OPEN STATUS**: This uniqueness claim follows from standard ODE theory
 251    (if log g' is constant = k, then g(x) = g(0) · e^(kx), which is unique
 252    under Picard-Lindelöf). We leave it unproved here as it is provable in
 253    principle but requires ODE machinery.
 254
 255    The *physical* question — WHY the log derivative should be constant
 256    in the RS derivation — is the true remaining gap. -/
 257theorem exponential_form_uniqueness_ode_principle :
 258    True := trivial
 259
 260/-! ## Summary of What This Module Proves
 261
 2621. **Structural properties** of the exponential form:
 263   * `alphaInv_def`: α⁻¹ is the exponential expression (unfold)
 264   * `alphaInv_positive`: α⁻¹ > 0
 265   * `exp_factor_bounded`: the exponential factor is in (0, 1] when f_gap ≥ 0
 266
 2672. **Log-coordinate structure**:
 268   * `log_alphaInv_seed_ratio`: ln(α⁻¹/α_seed) = -f_gap/α_seed (linear in f_gap)
 269   * `log_alphaInv_eq`: ln(α⁻¹) = ln(α_seed) - f_gap/α_seed
 270
 2713. **Differential structure**:
 272   * `deriv_alphaInv_of_gap`: dα⁻¹/df_gap = -α⁻¹/α_seed (the defining ODE)
 273   * `logarithmic_derivative_constant`: d ln(α⁻¹)/df_gap = -1/α_seed
 274     (constant logarithmic rate)
 275
 2764. **Leading-order consistency**:
 277   * `alphaInv_linear_term`: at f_gap=0, α⁻¹ = α_seed
 278   * `alphaInv_linear_rate`: at f_gap=0, dα⁻¹/df_gap = -1
 279
 280## What's NOT proved
 281
 2821. **Uniqueness from structural principles**: the formula is defined, not
 283   derived. `exponential_form_from_constant_log_derivative` states a
 284   candidate uniqueness but is not proved.
 2852. **Forcing of the constant log-derivative**: why RS requires
 286   (d/df_gap) ln(α⁻¹) to be constant (and specifically = -1/α_seed)
 287   remains a BRIDGE claim between the formalism and physics.
 288
 289## Residual Openness
 290
 291The exponential form of α⁻¹ is best understood as a STRUCTURAL CHOICE
 292inherited from the J-cost's log-coordinate behavior, rather than as a
 293derived consequence. It is a natural choice given:
 294- The J-cost's exponential/cosh structure in log coordinates.
 295- The requirement of positivity for all f_gap values.
 296- The linear leading-order behavior α⁻¹ ≈ α_seed - f_gap.
 297- The scale-free running (constant logarithmic derivative).
 298
 299But NONE of these individually force the exponential form uniquely without
 300additional assumptions. The integer 44 IS forced (proved in
 301`alpha_44_forcing.md`). The exponential form is PLAUSIBLE but not uniquely
 302forced in the current Lean.
 303
 304This is documented in `epistemic_layers.md` as a BRIDGE claim.
 305
 306-/
 307
 308end
 309
 310end AlphaExponentialForm
 311end Constants
 312end IndisputableMonolith
 313