IndisputableMonolith.Meta.Axioms

   1import Mathlib
   2
   3/-!
   4# Axiom Registry for Recognition Science
   5
   6This module documents all axioms used in the Recognition Science formalization,
   7categorizing them by their epistemic status:
   8
   91. **Physical Postulates**: Foundational claims that bridge logic and physics.
  10   These cannot be "proven" in Lean because they define what physics *is*.
  11
  122. **Well-Known Mathematical Facts**: Standard theorems that could be proven
  13   with sufficient effort, but are accepted as axioms for pragmatic reasons.
  14
  153. **Open Problems**: Claims that are conjectured but not yet derived.
  16   These represent the research frontier.
  17
  18## Summary Table
  19
  20| Axiom | Category | Location | Status |
  21|-------|----------|----------|--------|
  22| Meta-Principle | Physical Postulate | Recognition.lean | Foundational |
  23| recognition_requires_distinguishability | Physical Postulate | LedgerNecessity.lean | Foundational |
  24| recognition_exchange_invariance_axiom | Physical Postulate | ConstraintForcing.lean | Foundational |
  25| recognition_identity_axiom | Physical Postulate | ConstraintForcing.lean | Foundational |
  26| ode_cosh_uniqueness | Mathematical Fact | FunctionalEquation.lean | Standard ODE |
  27| ode_zero_uniqueness | Mathematical Fact | FunctionalEquation.lean | Standard ODE |
  28| dAlembert_to_ODE | Mathematical Fact | FunctionalEquation.lean | Aczél (1966) |
  29| dAlembert_cosh_solution | Mathematical Fact | FunctionalEquation.lean | Aczél (1966) |
  30| cosh_strictly_convex | Mathematical Fact | Convexity.lean | Calculus |
  31| wallpaper_groups_count | Mathematical Fact | AlphaDerivation.lean | Fedorov (1891) |
  32| linking_dimension_theorem | Mathematical Fact | LedgerUniqueness.lean | Zeeman (1963) |
  33| pi_computable | Mathematical Fact | ConstructiveNote.lean | Computability |
  34| phi_computable | Mathematical Fact | ConstructiveNote.lean | Computability |
  35| missing_shift_exists | Open Problem | ElectronMass.lean | T9 Frontier |
  36| electron_mass_follows_Z_structure | Open Problem | ElectronMass.lean | T9 Frontier |
  37
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Meta
  42namespace AxiomRegistry
  43
  44/-! ## Category 1: Physical Postulates
  45
  46These axioms define the physical content of Recognition Science.
  47They cannot be proven because they are the *definitions* of what we mean
  48by "recognition" and "physics".
  49-/
  50
  51/-- **The Meta-Principle (MP)**: Nothing cannot recognize itself.
  52
  53This is the foundational axiom of Recognition Science. It is a logical
  54tautology (the empty set cannot stand in relation to itself) that has
  55physical consequences: existence requires recognition, and recognition
  56requires distinction.
  57
  58**Status**: Foundational postulate. Not provable within the system.
  59**Location**: IndisputableMonolith/Recognition.lean
  60-/
  61def meta_principle_status : String := "Physical Postulate (Foundational)"
  62
  63/-- **Recognition Requires Distinguishability**: MP implies that recognizable
  64states must be distinguishable.
  65
  66This bridges the logical MP to physical consequences: if states cannot
  67be distinguished, they cannot be recognized, violating MP.
  68
  69**Status**: Physical postulate deriving from MP.
  70**Location**: IndisputableMonolith/Verification/Necessity/LedgerNecessity.lean
  71-/
  72def recognition_distinguishability_status : String := "Physical Postulate (Derived from MP)"
  73
  74/-- **Exchange Invariance**: The cost for A to recognize B equals the cost
  75for B to recognize A.
  76
  77This is a symmetry requirement on the recognition relation. It follows
  78from the assumption that recognition is a symmetric relation.
  79
  80**Status**: Physical postulate (symmetry requirement).
  81**Location**: IndisputableMonolith/Verification/T5/ConstraintForcing.lean
  82-/
  83def exchange_invariance_status : String := "Physical Postulate (Symmetry)"
  84
  85/-- **Identity Cost Zero**: The cost of self-recognition is zero.
  86
  87This defines the baseline of the cost function: recognizing oneself
  88requires no "effort" in the recognition sense.
  89
  90**Status**: Physical postulate (normalization).
  91**Location**: IndisputableMonolith/Verification/T5/ConstraintForcing.lean
  92-/
  93def identity_cost_status : String := "Physical Postulate (Normalization)"
  94
  95/-! ## Category 2: Well-Known Mathematical Facts
  96
  97These axioms are standard results from mathematics that could be proven
  98in Lean with sufficient effort. We accept them as axioms with citations.
  99-/
 100
 101/-- **ODE Uniqueness (cosh)**: The unique solution to H'' = H with
 102H(0) = 1, H'(0) = 0 is H = cosh.
 103
 104**Reference**: Any ODE textbook; Picard-Lindelöf theorem.
 105**Status**: Standard mathematical fact.
 106**Location**: IndisputableMonolith/Cost/FunctionalEquation.lean
 107-/
 108def ode_cosh_status : String := "Mathematical Fact (ODE Uniqueness)"
 109
 110/-- **ODE Uniqueness (zero)**: The unique solution to f'' = f with
 111f(0) = 0, f'(0) = 0 is f = 0.
 112
 113**Reference**: Picard-Lindelöf theorem.
 114**Status**: Standard mathematical fact.
 115**Location**: IndisputableMonolith/Cost/FunctionalEquation.lean
 116-/
 117def ode_zero_status : String := "Mathematical Fact (ODE Uniqueness)"
 118
 119/-- **d'Alembert Functional Equation**: Continuous solutions to
 120H(t+u) + H(t-u) = 2H(t)H(u) with H(0) = 1, H''(0) = 1 equal cosh.
 121
 122**Reference**: Aczél, J. (1966). "Lectures on Functional Equations
 123and Their Applications". Academic Press.
 124**Status**: Standard mathematical fact.
 125**Location**: IndisputableMonolith/Cost/FunctionalEquation.lean
 126-/
 127def dAlembert_status : String := "Mathematical Fact (Functional Equations)"
 128
 129/-- **cosh Strict Convexity**: cosh is strictly convex on ℝ.
 130
 131**Reference**: cosh'' = cosh > 0 everywhere.
 132**Status**: Standard calculus result.
 133**Location**: IndisputableMonolith/Cost/Convexity.lean
 134-/
 135def cosh_convex_status : String := "Mathematical Fact (Calculus)"
 136
 137/-- **17 Wallpaper Groups**: There are exactly 17 distinct 2D crystallographic groups.
 138
 139**Reference**: Fedorov, E. S. (1891). Also Pólya (1924).
 140**Status**: Standard crystallography result.
 141**Location**: IndisputableMonolith/Constants/AlphaDerivation.lean
 142-/
 143def wallpaper_status : String := "Mathematical Fact (Crystallography)"
 144
 145/-- **Linking in D=3**: Non-trivial topological linking exists only in 3 dimensions.
 146
 147**Reference**: Zeeman, E. C. (1963). "Unknotting combinatorial balls".
 148**Status**: Standard topology result.
 149**Location**: IndisputableMonolith/Meta/LedgerUniqueness.lean
 150-/
 151def linking_status : String := "Mathematical Fact (Topology)"
 152
 153/-- **Computability of π and φ**: π and φ are computable real numbers.
 154
 155**Reference**: Standard computability theory; many explicit algorithms exist.
 156**Status**: Well-known computability result.
 157**Location**: IndisputableMonolith/Meta/ConstructiveNote.lean
 158-/
 159def computability_status : String := "Mathematical Fact (Computability)"
 160
 161/-! ## Category 3: Open Problems
 162
 163These axioms represent claims that are conjectured but not yet derived.
 164They mark the research frontier of Recognition Science.
 165-/
 166
 167/-- **Electron Mass Residue**: The electron mass follows the Z-structure
 168with a shift that is not yet derived.
 169
 170**Status**: Open problem (T9 frontier).
 171**Location**: IndisputableMonolith/Physics/ElectronMass.lean
 172-/
 173def electron_mass_status : String := "Open Problem (T9 Frontier)"
 174
 175/-- **Missing Shift**: There exists a topological shift relating the
 176Z-gap prediction to the observed electron mass residue.
 177
 178**Status**: Open problem. The shift is approximately 34.7, suggesting
 179a missing integer (possibly 2×17).
 180**Location**: IndisputableMonolith/Physics/ElectronMass.lean
 181-/
 182def missing_shift_status : String := "Open Problem (Active Research)"
 183
 184/-! ## Axiom Count Summary -/
 185
 186/-- Total axioms by category. -/
 187def axiom_summary : String :=
 188  "Physical Postulates: 4\n" ++
 189  "Mathematical Facts: 8\n" ++
 190  "Open Problems: 2\n" ++
 191  "Total: 14"
 192
 193end AxiomRegistry
 194end Meta
 195end IndisputableMonolith
 196