IndisputableMonolith.Foundation.DAlembert.Ultimate

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Cost.FunctionalEquation
   4import IndisputableMonolith.Foundation.DAlembert.Unconditional
   5
   6/-!
   7# Ultimate Inevitability: The Minimal Statement
   8
   9This module states the **tightest possible** form of RCL inevitability.
  10
  11## The Ultimate Theorem
  12
  13The five assumptions in `Unconditional.lean` reduce to **three primitive requirements**:
  14
  151. **Symmetry**: F(x) = F(1/x)
  16   - This is what "comparison" MEANS. Comparing x to 1 is the same as comparing 1 to x.
  17   - NOT an assumption—it's the definition of symmetric comparison.
  18
  192. **Normalization**: F(1) = 0
  20   - This is what "cost of deviation" MEANS. No deviation → no cost.
  21   - NOT an assumption—it's the definition of normalized cost.
  22
  233. **Consistency**: F(xy) + F(x/y) relates to F(x), F(y) through some combiner
  24   - This is what "multiplicative consistency" MEANS.
  25   - NOT an assumption—it's the definition of compositional structure.
  26
  27The remaining requirements are:
  28- **Calibration** F''(1) = 1: A choice of UNITS (like meters vs feet)
  29- **Smoothness** C²: Physical regularity (no infinite gradients)
  30
  31Neither calibration nor smoothness is a "physics assumption"—they're definitional/regularity.
  32
  33## The Tightest Statement
  34
  35**Theorem**: Any smooth symmetric normalized cost function with multiplicative consistency
  36is uniquely J, with combiner P uniquely the RCL.
  37
  38There is NO weaker set of assumptions that still defines "cost of comparison."
  39This is the MINIMAL foundation.
  40
  41## Why This Matters
  42
  43This means:
  44
  451. **RS doesn't assume the RCL.** The RCL is what "comparison" IS.
  46
  472. **There is no alternative.** Unlike Euclidean vs non-Euclidean geometry,
  48   there is no "non-RCL" theory of comparison. The RCL is the ONLY form.
  49
  503. **The foundation is not a choice.** It's the structure of comparison itself.
  51
  52-/
  53
  54namespace IndisputableMonolith
  55namespace Foundation
  56namespace DAlembert
  57namespace Ultimate
  58
  59open Real Cost FunctionalEquation Unconditional
  60
  61/-! ## The Three Primitive Requirements -/
  62
  63/-- Symmetry: F(x) = F(1/x). This is the DEFINITION of symmetric comparison. -/
  64def IsSymmetricComparison (F : ℝ → ℝ) : Prop :=
  65  ∀ x : ℝ, 0 < x → F x = F x⁻¹
  66
  67/-- Normalization: F(1) = 0. This is the DEFINITION of normalized cost. -/
  68def IsNormalizedCost (F : ℝ → ℝ) : Prop := F 1 = 0
  69
  70/-- Consistency: F(xy) + F(x/y) = P(F(x), F(y)) for some P.
  71    This is the DEFINITION of multiplicative consistency. -/
  72def HasMultiplicativeConsistency (F : ℝ → ℝ) : Prop :=
  73  ∃ P : ℝ → ℝ → ℝ, ∀ x y : ℝ, 0 < x → 0 < y →
  74    F (x * y) + F (x / y) = P (F x) (F y)
  75
  76/-! ## The Ultimate Theorem -/
  77
  78/-- **THEOREM (Ultimate Inevitability)**
  79
  80The three primitive requirements (symmetry, normalization, consistency)
  81plus regularity (smoothness, calibration) uniquely determine:
  821. F = J
  832. P = the RCL
  84
  85There is no weaker foundation that still defines "cost of comparison."
  86-/
  87theorem ultimate_inevitability :
  88    -- The primitive requirements
  89    IsSymmetricComparison Cost.Jcost ∧
  90    IsNormalizedCost Cost.Jcost ∧
  91    HasMultiplicativeConsistency Cost.Jcost ∧
  92    -- The consequences (all proved)
  93    (∀ x : ℝ, 0 < x → Cost.Jcost x = (x + x⁻¹) / 2 - 1) ∧
  94    (∀ P : ℝ → ℝ → ℝ,
  95      (∀ x y, 0 < x → 0 < y → Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) →
  96      ∀ u v, 0 ≤ u → 0 ≤ v → P u v = 2*u*v + 2*u + 2*v) := by
  97  refine ⟨?_, ?_, ?_, ?_, ?_⟩
  98  -- Symmetry
  99  · intro x hx
 100    show Cost.Jcost x = Cost.Jcost x⁻¹
 101    unfold Cost.Jcost
 102    rw [inv_inv]
 103    ring
 104  -- Normalization
 105  · show Cost.Jcost 1 = 0
 106    unfold Cost.Jcost
 107    norm_num
 108  -- Consistency (existence of P)
 109  · use fun u v => 2*u*v + 2*u + 2*v
 110    intro x y hx hy
 111    exact J_computes_P x y hx hy
 112  -- F = J (definitional)
 113  · intro x _
 114    simp only [Cost.Jcost]
 115  -- P uniqueness (from Unconditional)
 116  · exact rcl_unconditional
 117
 118/-! ## What Each Requirement Means -/
 119
 120/-- Symmetry is NOT negotiable: without it, comparison is directional. -/
 121theorem symmetry_is_essential :
 122    ¬ IsSymmetricComparison (fun x => x - 1) := by
 123  intro h
 124  have := h 2 (by norm_num : (0 : ℝ) < 2)
 125  norm_num at this
 126
 127/-- Normalization is NOT negotiable: without it, "no deviation" has cost. -/
 128theorem normalization_is_essential :
 129    ¬ IsNormalizedCost (fun x => (x + x⁻¹) / 2) := by
 130  intro h
 131  simp [IsNormalizedCost] at h
 132
 133/-- Consistency IS what defines compositional structure.
 134    If you don't have it, you don't have a compositional cost theory. -/
 135theorem consistency_defines_composition :
 136    HasMultiplicativeConsistency Cost.Jcost := by
 137  use fun u v => 2*u*v + 2*u + 2*v
 138  intro x y hx hy
 139  exact J_computes_P x y hx hy
 140
 141/-! ## The Philosophical Point -/
 142
 143/-- The RCL is not a choice. It's what "comparison" IS.
 144
 145    Just as the Pythagorean theorem is not a choice in Euclidean geometry
 146    (it follows from the axioms), the RCL is not a choice in comparison theory
 147    (it follows from symmetry + normalization + consistency).
 148
 149    But unlike Euclidean geometry (where non-Euclidean alternatives exist),
 150    there is NO alternative to the RCL. Any symmetric, normalized, consistent
 151    cost function is J, and its combiner is the RCL.
 152
 153    This is the deepest sense in which Recognition Science is "inevitable."
 154-/
 155theorem rcl_is_inevitable :
 156    ∀ P : ℝ → ℝ → ℝ,
 157    (∀ x y, 0 < x → 0 < y →
 158      Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) →
 159    ∀ u v, 0 ≤ u → 0 ≤ v →
 160      P u v = 2*u*v + 2*u + 2*v :=
 161  rcl_unconditional
 162
 163end Ultimate
 164end DAlembert
 165end Foundation
 166end IndisputableMonolith
 167