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    simp only [Cost.Jcost]
 101    ring
 102  -- Normalization
 103  · simp only [Cost.Jcost, inv_one]
 104    ring
 105  -- Consistency (existence of P)
 106  · use fun u v => 2*u*v + 2*u + 2*v
 107    intro x y hx hy
 108    exact J_computes_P x y hx hy
 109  -- F = J (definitional)
 110  · intro x _
 111    simp only [Cost.Jcost]
 112  -- P uniqueness (from Unconditional)
 113  · exact rcl_unconditional
 114
 115/-! ## What Each Requirement Means -/
 116
 117/-- Symmetry is NOT negotiable: without it, comparison is directional. -/
 118theorem symmetry_is_essential :
 119    ¬ IsSymmetricComparison (fun x => x - 1) := by
 120  intro h
 121  have := h 2 (by norm_num : (0 : ℝ) < 2)
 122  simp at this
 123
 124/-- Normalization is NOT negotiable: without it, "no deviation" has cost. -/
 125theorem normalization_is_essential :
 126    ¬ IsNormalizedCost (fun x => (x + x⁻¹) / 2) := by
 127  intro h
 128  simp [IsNormalizedCost] at h
 129
 130/-- Consistency IS what defines compositional structure.
 131    If you don't have it, you don't have a compositional cost theory. -/
 132theorem consistency_defines_composition :
 133    HasMultiplicativeConsistency Cost.Jcost := by
 134  use fun u v => 2*u*v + 2*u + 2*v
 135  intro x y hx hy
 136  exact J_computes_P x y hx hy
 137
 138/-! ## The Philosophical Point -/
 139
 140/-- The RCL is not a choice. It's what "comparison" IS.
 141
 142    Just as the Pythagorean theorem is not a choice in Euclidean geometry
 143    (it follows from the axioms), the RCL is not a choice in comparison theory
 144    (it follows from symmetry + normalization + consistency).
 145
 146    But unlike Euclidean geometry (where non-Euclidean alternatives exist),
 147    there is NO alternative to the RCL. Any symmetric, normalized, consistent
 148    cost function is J, and its combiner is the RCL.
 149
 150    This is the deepest sense in which Recognition Science is "inevitable."
 151-/
 152theorem rcl_is_inevitable :
 153    ∀ P : ℝ → ℝ → ℝ,
 154    (∀ x y, 0 < x → 0 < y →
 155      Cost.Jcost (x*y) + Cost.Jcost (x/y) = P (Cost.Jcost x) (Cost.Jcost y)) →
 156    ∀ u v, 0 ≤ u → 0 ≤ v →
 157      P u v = 2*u*v + 2*u + 2*v :=
 158  rcl_unconditional
 159
 160end Ultimate
 161end DAlembert
 162end Foundation
 163end IndisputableMonolith
 164