IndisputableMonolith.Constants.AlphaGenesis.CurvatureJCostVerdict

   1import Mathlib
   2import IndisputableMonolith.Constants.Alpha
   3import IndisputableMonolith.Constants.AlphaDerivation
   4import IndisputableMonolith.Foundation.JCostHessianC7
   5import IndisputableMonolith.Numerics.Interval.AlphaBounds
   6
   7/-!
   8# Alpha Genesis M12: the genuine J-cost of cube curvature vs the seed (verdict)
   9
  10This module records the decisive finding from the "derive α the right way" pass:
  11the seed `4π·11` is a **category error**, not a recognition cost. Two facts
  12settle it.
  13
  14## 1. `4π` is a topological integral, not a cost
  15
  16The factor `4π` in the seed is the discrete Gauss-Bonnet total curvature
  17`∑ deficits = 2π·χ(S²) = 4π` of `∂Q₃` (`AlphaDerivation.gauss_bonnet_Q3`). That
  18is a **linear** topological invariant. The recognition cost is the canonical
  19reciprocal cost `J`, which is **quadratic** at equilibrium: `J(1+ε) = ε²/(2(1+ε))`,
  20quadratic coefficient `1/2` (`JCostHessianC7`). A linear curvature integral and
  21a quadratic displacement cost are different objects; one cannot multiply `4π` by
  22an edge count and call the product a cost.
  23
  24## 2. The genuine quadratic J-cost of the cube's curvature is `π²`, not `4π·11`
  25
  26If one computes the actual recognition cost of the cube's curvature, summing the
  27quadratic J-cost `½·δ²` of each of the `8` vertex deficits `δ = π/2`:
  28`8 · ½ · (π/2)² = π² ≈ 9.87` (`cubeCurvatureJCost_eq_pi_sq`). This is the honest
  29forced quantity. It is below the topological `4π ≈ 12.57`
  30(`genuine_cost_lt_gaussBonnet`) and nowhere near the seed `4π·11 ≈ 138.23`
  31(`seed_far_above_genuine_cost`).
  32
  33## Verdict
  34
  35The seed `4π·11` is neither the genuine quadratic J-cost (`π²`) nor the
  36gauge-invariant photon count (`5`, see `U1Normalization`). It is an
  37identification, and its exact equality with `α⁻¹(0) = 137.035999` is falsified
  38(`MeasurementVerdict`, `ScaleIdentification`). The calibration-free content that
  39survives is an `O(4π)` UV-scale recognition cost; the exact infrared value of
  40`α` is OPEN.
  41
  42This does NOT touch the forced *dressing* `g(t) = φ⁻ᵗ` (`CalibrationForcing`,
  43zero α-input) or the forced spectral weight `w₈` (`GapWeight`, zero α-input). It
  44isolates the SEED as the locus of the overclaim.
  45
  46STATUS: THEOREM (the cost computation and inequalities); QUARANTINE.
  47-/
  48
  49namespace IndisputableMonolith
  50namespace Constants
  51namespace AlphaGenesis
  52namespace CurvatureJCostVerdict
  53
  54open Constants.AlphaDerivation
  55
  56noncomputable section
  57
  58/-- The genuine quadratic recognition cost of the cube's curvature: the J-cost
  59quadratic coefficient `1/2` (`JCostHessianC7.jcostTaylorQuadraticCoefficient`)
  60applied to each of the `8` vertex deficits `δ = π/2`, summed:
  61`8 · ½ · (π/2)²`. -/
  62def cubeCurvatureJCost : ℝ :=
  63  (cube_vertices D : ℝ)
  64    * Foundation.JCostHessianC7.jcostTaylorQuadraticCoefficient
  65    * (vertex_angular_deficit) ^ 2
  66
  67/-- **The genuine J-cost of the cube curvature is `π²`.** `8 · ½ · (π/2)² = π²`.
  68This is the honest quadratic recognition cost, contrasted with the seed's
  69`4π·11`. -/
  70theorem cubeCurvatureJCost_eq_pi_sq : cubeCurvatureJCost = Real.pi ^ 2 := by
  71  unfold cubeCurvatureJCost
  72  rw [vertex_deficit_eq, Foundation.JCostHessianC7.jcostTaylorQuadraticCoefficient_eq]
  73  have h8 : (cube_vertices D : ℝ) = 8 := by exact_mod_cast vertices_at_D3
  74  rw [h8]; ring
  75
  76/-- The genuine quadratic J-cost `π² ≈ 9.87` is strictly below the topological
  77Gauss-Bonnet integral `4π ≈ 12.57`: a quadratic cost and a linear topological
  78invariant are different objects, and they do not even agree numerically. -/
  79theorem genuine_cost_lt_gaussBonnet :
  80    cubeCurvatureJCost < 4 * Real.pi := by
  81  rw [cubeCurvatureJCost_eq_pi_sq]
  82  nlinarith [Real.pi_pos, Real.pi_lt_four]
  83
  84/-- The seed `4π·11` is far above the genuine quadratic J-cost `π²` of the cube
  85curvature: `π² < 4π < 4π·11`. The seed is not a recognition cost. -/
  86theorem seed_far_above_genuine_cost :
  87    cubeCurvatureJCost < Constants.alpha_seed := by
  88  have h1 : cubeCurvatureJCost < 4 * Real.pi := genuine_cost_lt_gaussBonnet
  89  have h2 : (4 : ℝ) * Real.pi < Constants.alpha_seed := by
  90    simp only [Constants.alpha_seed]
  91    nlinarith [Real.pi_pos]
  92  linarith
  93
  94/-- The genuine quadratic J-cost is also far below the measured `α⁻¹`:
  95`π² < 137.030 < alphaInv`. So even the honest cost is not `α⁻¹(0)`; the cube
  96forces only an `O(4π)` UV-scale quantity. -/
  97theorem genuine_cost_far_below_alphaInv :
  98    cubeCurvatureJCost < Constants.alphaInv := by
  99  rw [cubeCurvatureJCost_eq_pi_sq]
 100  have h1 : Real.pi ^ 2 < 4 * Real.pi := by nlinarith [Real.pi_pos, Real.pi_lt_four]
 101  have h2 : (4 : ℝ) * Real.pi < 13 := by nlinarith [Real.pi_lt_d6]
 102  have h3 : (137.030 : ℝ) < Constants.alphaInv := Numerics.alphaInv_gt
 103  linarith
 104
 105/-! ## The verdict -/
 106
 107/-- **Curvature-cost verdict.** The seed `4π·11` is a category error: `4π` is a
 108linear topological invariant (Gauss-Bonnet), not a quadratic recognition cost;
 109the genuine quadratic J-cost of the cube curvature is `π²`; and the seed exceeds
 110both the genuine cost and the topological invariant. The exact value of `α⁻¹(0)`
 111is not produced by any of these forced quantities; it is OPEN. -/
 112structure CurvatureCostVerdict : Prop where
 113  genuine_cost_is_pi_sq : cubeCurvatureJCost = Real.pi ^ 2
 114  cost_below_topological : cubeCurvatureJCost < 4 * Real.pi
 115  seed_above_genuine_cost : cubeCurvatureJCost < Constants.alpha_seed
 116  genuine_cost_not_alphaInv : cubeCurvatureJCost < Constants.alphaInv
 117
 118def curvatureCostVerdict : CurvatureCostVerdict where
 119  genuine_cost_is_pi_sq := cubeCurvatureJCost_eq_pi_sq
 120  cost_below_topological := genuine_cost_lt_gaussBonnet
 121  seed_above_genuine_cost := seed_far_above_genuine_cost
 122  genuine_cost_not_alphaInv := genuine_cost_far_below_alphaInv
 123
 124end
 125
 126end CurvatureJCostVerdict
 127end AlphaGenesis
 128end Constants
 129end IndisputableMonolith
 130