IndisputableMonolith.Constants.AlphaGenesis.CalibrationForcing

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.AlphaGenesis.ResummationForcing
   4import IndisputableMonolith.Constants.AlphaGenesis.PatternForcing
   5import IndisputableMonolith.Constants.AlphaGenesis.LoopCertificate
   6import IndisputableMonolith.Foundation.MeasureForcing
   7
   8/-!
   9# Alpha Genesis M5: Calibration Forcing (the calibration is not an input)
  10
  11**THE THEOREM.** The unit-linear-response calibration (D2) of
  12`ResummationForcing.DressingResponse` can be eliminated entirely. A
  13**self-similar dressing** is a response with three premises, none of which
  14is a normalization choice:
  15
  16* factorization over independent loads (the inherited ledger premise),
  17* antitonicity on nonnegative loads (more load never helps),
  18* the self-similar balance `g(1) = 1/(1 + g(1))` on the single step — the
  19  SAME balance equation that forces the T9 measure's step (W2).
  20
  21Then `g = φ⁻ᵗ` on all nonnegative loads (`selfSimilar_response_forced`),
  22with no derivative condition and no unit convention anywhere. The step
  23value `g(1) = φ⁻¹` is DERIVED (`step_forced`), not calibrated: positivity
  24of the step follows from factorization (`g(1) = g(1/2)² > 0`), and the
  25balance equation then has exactly one admissible root.
  26
  27Consequently the (D1)+(D2) `DressingResponse` of M1 demotes to the
  28natural-units display of this object (`natural_display`), and the forward
  29α object is obtained from EVERY self-similar dressing
  30(`alphaInvGenesis_from_selfSimilar`).
  31
  32This discharges the residual normalization worry: the dressing of the α
  33seed carries zero calibration input. Its form, its rate, and its step are
  34all forced by the same two structural facts (factorization, self-similar
  35balance) that force the recognition measure itself.
  36
  37STATUS: THEOREM (0 sorry target). No CODATA reference anywhere in this file.
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Constants
  42namespace AlphaGenesis
  43
  44noncomputable section
  45
  46open Foundation.MeasureForcing
  47
  48/-- A **self-similar dressing**: survival fraction under gap load with the
  49two inherited ledger premises and the self-similar step balance. No
  50calibration field exists. -/
  51structure SelfSimilarDressing where
  52  /-- Survival fraction as a function of gap load. -/
  53  g : ℝ → ℝ
  54  /-- Factorization over independent nonnegative loads. -/
  55  factorizes : Foundation.MeasureForcing.Factorizes g
  56  /-- More load never increases survival. -/
  57  antitone : AntitoneOn g (Set.Ici 0)
  58  /-- **Self-similar balance** on the single step: the same fixed-point
  59  equation (W2) that forces the T9 measure's step. -/
  60  step_self_similar : g 1 = 1 / (1 + g 1)
  61
  62namespace SelfSimilarDressing
  63
  64variable (D : SelfSimilarDressing)
  65
  66/-- The step is a square: `g(1) = g(1/2)²` (factorization). -/
  67theorem step_eq_sq : D.g 1 = D.g (1/2) ^ 2 := by
  68  have h := D.factorizes (1/2) (1/2) (by norm_num) (by norm_num)
  69  have h1 : (1/2 : ℝ) + 1/2 = 1 := by norm_num
  70  rw [h1] at h
  71  rw [h]
  72  ring
  73
  74/-- The step is nonnegative. -/
  75theorem step_nonneg : 0 ≤ D.g 1 := by
  76  rw [D.step_eq_sq]
  77  exact sq_nonneg _
  78
  79/-- The step is nonzero: `g(1) = 0` contradicts the balance equation. -/
  80theorem step_ne_zero : D.g 1 ≠ 0 := by
  81  intro h0
  82  have hbal := D.step_self_similar
  83  rw [h0] at hbal
  84  norm_num at hbal
  85
  86/-- The step is strictly positive (derived, not assumed). -/
  87theorem step_pos : 0 < D.g 1 :=
  88  lt_of_le_of_ne D.step_nonneg (Ne.symm D.step_ne_zero)
  89
  90/-- **STEP FORCING.** The balance equation has exactly one admissible
  91root: `g(1) = φ⁻¹`. The reciprocal of the step satisfies the T6
  92self-similarity equation `r² = r + 1`, whose unique positive root is φ. -/
  93theorem step_forced : D.g 1 = 1 / Constants.phi := by
  94  set ρ := D.g 1 with hρdef
  95  have hpos : 0 < ρ := D.step_pos
  96  have hbal : ρ = 1 / (1 + ρ) := D.step_self_similar
  97  have hsum_pos : 0 < 1 + ρ := by linarith
  98  have hsum_ne : (1 + ρ) ≠ 0 := ne_of_gt hsum_pos
  99  -- ρ² + ρ − 1 = 0
 100  have hmul : ρ * (1 + ρ) = 1 := by
 101    calc ρ * (1 + ρ) = (1 / (1 + ρ)) * (1 + ρ) := by rw [← hbal]
 102    _ = 1 := by field_simp
 103  have hquad : ρ ^ 2 + ρ - 1 = 0 := by nlinarith [hmul]
 104  -- 1/φ satisfies the same quadratic
 105  have hφ : Constants.phi ^ 2 = Constants.phi + 1 := Constants.phi_sq_eq
 106  have hφpos : 0 < Constants.phi := Constants.phi_pos
 107  have hφne : Constants.phi ≠ 0 := ne_of_gt hφpos
 108  have hinv_pos : 0 < 1 / Constants.phi := by positivity
 109  have hinv_quad : (1 / Constants.phi) ^ 2 + (1 / Constants.phi) - 1 = 0 := by
 110    field_simp
 111    nlinarith [hφ]
 112  -- difference of the two quadratics factors: (ρ − 1/φ)(ρ + 1/φ + 1) = 0
 113  have hfactor : (ρ - 1 / Constants.phi) * (ρ + 1 / Constants.phi + 1) = 0 := by
 114    linear_combination hquad - hinv_quad
 115  have hsum2_pos : 0 < ρ + 1 / Constants.phi + 1 := by linarith
 116  rcases mul_eq_zero.mp hfactor with h | h
 117  · linarith [sub_eq_zero.mp h]
 118  · linarith
 119
 120/-- **CALIBRATION FORCING.** Every self-similar dressing is the forced
 121measure on nonnegative loads: `g(t) = φ⁻ᵗ`. No derivative condition, no
 122unit convention, no calibration input. -/
 123theorem response_forced : ∀ t : ℝ, 0 ≤ t →
 124    D.g t = Foundation.MeasureForcing.contWeight t := by
 125  intro t ht
 126  have hstep : D.g 1 = Foundation.MeasureForcing.rho := by
 127    rw [D.step_forced]
 128    rfl
 129  exact Foundation.MeasureForcing.continuum_weight_forced
 130    D.factorizes D.antitone hstep t ht
 131
 132/-- Non-vacuity: the forced measure itself is a self-similar dressing. -/
 133def canonical : SelfSimilarDressing where
 134  g := Foundation.MeasureForcing.contWeight
 135  factorizes := Foundation.MeasureForcing.contWeight_satisfies_premises.1
 136  antitone := Foundation.MeasureForcing.contWeight_satisfies_premises.2.1
 137  step_self_similar := by
 138    have h1 : Foundation.MeasureForcing.contWeight 1 =
 139        Foundation.MeasureForcing.rho :=
 140      Foundation.MeasureForcing.contWeight_satisfies_premises.2.2
 141    rw [h1]
 142    unfold Foundation.MeasureForcing.rho
 143    have hphi := Constants.phi_pos
 144    have hsq := Constants.phi_sq_eq
 145    have hsum_pos : (0 : ℝ) < 1 + 1 / Constants.phi := by positivity
 146    field_simp
 147    nlinarith [hsq]
 148
 149/-- **NATURAL-UNITS DISPLAY.** The differentiable (D1)+(D2) dressing of M1
 150is the same object read in natural log units: for every calibrated
 151response R and every self-similar dressing D,
 152`R.g(lnφ · t) = D.g(t)` on nonnegative loads. The calibration of M1 was
 153never an input; it was the natural-units coordinate of this object. -/
 154theorem natural_display (R : DressingResponse) (D : SelfSimilarDressing)
 155    (t : ℝ) (ht : 0 ≤ t) :
 156    R.g (Real.log Constants.phi * t) = D.g t := by
 157  rw [response_is_forced_measure R t, D.response_forced t ht]
 158
 159/-- The forward α object is obtained from EVERY self-similar dressing:
 160`alphaInvGenesis = S · D.g(w₈/S)`. -/
 161theorem alphaInvGenesis_from_selfSimilar (D : SelfSimilarDressing) :
 162    alphaInvGenesis = channelBudget * D.g spectralLoad := by
 163  unfold alphaInvGenesis
 164  rw [D.response_forced spectralLoad (le_of_lt spectralLoad_pos)]
 165
 166end SelfSimilarDressing
 167
 168/-- **CALIBRATION FORCING CERTIFICATE.** Bundles the M5 closure:
 1691. the step is forced to φ⁻¹ from balance alone (no calibration);
 1702. every self-similar dressing is the forced measure on loads;
 1713. the M1 calibrated response is the natural-units display;
 1724. the forward α object follows from every self-similar dressing. -/
 173structure CalibrationForcingCert where
 174  deriving Inhabited
 175
 176@[simp] def CalibrationForcingCert.verified (_c : CalibrationForcingCert) : Prop :=
 177  (∀ D : SelfSimilarDressing, D.g 1 = 1 / Constants.phi) ∧
 178  (∀ (D : SelfSimilarDressing) (t : ℝ), 0 ≤ t →
 179    D.g t = Foundation.MeasureForcing.contWeight t) ∧
 180  (∀ (R : DressingResponse) (D : SelfSimilarDressing) (t : ℝ), 0 ≤ t →
 181    R.g (Real.log Constants.phi * t) = D.g t) ∧
 182  (∀ D : SelfSimilarDressing,
 183    alphaInvGenesis = channelBudget * D.g spectralLoad)
 184
 185theorem CalibrationForcingCert.verified_any (c : CalibrationForcingCert) :
 186    CalibrationForcingCert.verified c := by
 187  refine ⟨?_, ?_, ?_, ?_⟩
 188  · intro D
 189    exact D.step_forced
 190  · intro D t ht
 191    exact D.response_forced t ht
 192  · intro R D t ht
 193    exact SelfSimilarDressing.natural_display R D t ht
 194  · intro D
 195    exact SelfSimilarDressing.alphaInvGenesis_from_selfSimilar D
 196
 197end
 198
 199end AlphaGenesis
 200end Constants
 201end IndisputableMonolith
 202