View the complete Lean proof file on GitHub: IndisputableMonolith.lean
LEAN TRAINING GUIDE: IndisputableMonolith.lean
==============================================
PURPOSE: AI training for formal verification of Recognition Science theorems.
FILE: IndisputableMonolith.lean (1490 lines, sorry-free)
STATUS: All eight theorems T1-T8 formally proved in Lean 4 + Mathlib (All Formally Proved)
PROOF STRUCTURE OVERVIEW
========================
- T1 (MP): mp_holds — Nothing cannot recognize itself
- T2 (Atomicity): T2_atomicity — unique posting per tick
- T3 (Continuity): T3_continuity — flux vanishes on closed chains
- T4 (Potential): T4_unique_on_component — uniqueness up to constant
- T5 (Cost): T5_cost_uniqueness_on_pos — J(x) unique on ℝ₊
- T6 (Eight-tick): eight_tick_min + T6_exist_8 — period = 2^D
- T7 (Nyquist): T7_nyquist_obstruction — coverage lower bound
- T8 (Units): LedgerUnits.equiv_delta — δ-subgroup ≃ ℤ
CORE DEFINITIONS
===============
Nothing := Empty
Recognition (A B : Type) := {recognizer : A, recognized : B}
MP : Prop := ¬ ∃ _ : Recognition Nothing Nothing, True
RecognitionStructure := {U : Type, R : U → U → Prop}
Chain (M : RecognitionStructure) := {n : Nat, f : Fin (n+1) → M.U, ok : ∀ i, M.R (f i.castSucc) (f i.succ)}
Ledger (M : RecognitionStructure) := {debit : M.U → ℤ, credit : M.U → ℤ}
phi (L : Ledger M) : M.U → ℤ := fun u => L.debit u - L.credit u
chainFlux (L : Ledger M) (ch : Chain M) : ℤ := phi L (Chain.last ch) - phi L (Chain.head ch)
THEOREM PROOFS (LEAN FUNCTIONS)
===============================
T1: mp_holds : MP
- Proof: intro ⟨⟨r, _⟩, _⟩; cases r
- Logic: Empty type has no constructors
T2: T2_atomicity {M} [AtomicTick M] : ∀ t u v, postedAt t u → postedAt t v → u = v
- Uses: unique_post from AtomicTick class
- Logic: existential uniqueness implies equality
T3: T3_continuity {M} (L : Ledger M) [Conserves L] : ∀ ch, ch.head = ch.last → chainFlux L ch = 0
- Direct: Conserves.conserve (interface axiom)
- Bridge: discrete continuity equation
T4: T4_unique_on_component {δ} {p q : Pot M} (hp : DE δ p) (hq : DE δ q) {x0 y} (hbase : p x0 = q x0) (hreach : Reaches x0 y) : p y = q y
- Uses: diff_const_on_ReachN via edge_diff_invariant
- Logic: constant difference propagates along paths
T5: T5_cost_uniqueness_on_pos {F : ℝ → ℝ} [JensenSketch F] : ∀ {x}, 0 < x → F x = Jcost x
- Uses: AveragingDerivation interface + exp-axis agreement
- Logic: symmetry + unit + averaging → unique cost
T6: eight_tick_min {T} (pass : Fin T → Pattern 3) (covers : Surjective pass) : 8 ≤ T
- Uses: no_surj_small via cardinality argument
- Logic: cannot cover 2^3 patterns with < 8 elements
T7: T7_nyquist_obstruction {T D} (hT : T < 2^D) : ¬ ∃ f : Fin T → Pattern D, Surjective f
- Alias: no_surj_small
- Logic: cardinality obstruction
T8: LedgerUnits.equiv_delta (δ ≠ 0) : DeltaSub δ ≃ ℤ
- Maps: n·δ ↦ n (non-canonical for δ ≠ 0)
- Logic: subgroup isomorphism via Classical.choose
KEY LEAN TECHNIQUES USED
========================
1. Inductive types: ReachN, Chain construction
2. Classical logic: Classical.choose for existence
3. Typeclass system: AtomicTick, Conserves, SymmUnit
4. Cardinality arguments: Fintype.card bounds
5. Field arithmetic: field_simp for algebraic manipulation
6. Real analysis: Real.exp, Real.log, derivatives
7. Subtype construction: DeltaSub as {x // ∃ n, x = n * δ}
CAUSALITY FRAMEWORK
==================
ReachN (K : Kinematics α) : Nat → α → α → Prop
| zero {x} : ReachN K 0 x x
| succ {n x y z} : ReachN K n x y → K.step y z → ReachN K (n+1) x z
inBall (K : Kinematics α) (x : α) (n : Nat) (y : α) : Prop := ∃ k ≤ n, ReachN K k x y
ballP (K : Kinematics α) (x : α) : Nat → α → Prop
| 0, y => y = x
| Nat.succ n, y => ballP K x n y ∨ ∃ z, ballP K x n z ∧ K.step z y
COST UNIQUENESS FRAMEWORK
=========================
Jcost (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1
SymmUnit (F : ℝ → ℝ) : Prop :=
- symmetric : ∀ {x}, 0 < x → F x = F x⁻¹
- unit0 : F 1 = 0
AveragingDerivation (F : ℝ → ℝ) extends SymmUnit F : Prop :=
- agrees : AgreesOnExp F (∀ t, F (Real.exp t) = Jcost (Real.exp t))
LogModel (G : ℝ → ℝ) : Prop :=
- even_log : ∀ t, G (-t) = G t
- base0 : G 0 = 0
- upper_cosh : ∀ t, G t ≤ ((Real.exp t + Real.exp (-t)) / 2 - 1)
- lower_cosh : ∀ t, ((Real.exp t + Real.exp (-t)) / 2 - 1) ≤ G t
CLASSICAL BRIDGE MAPPINGS
=========================
T3: Conserves ↔ DiscreteContinuity (∂ρ/∂t + ∇·J = 0)
T4: Potential uniqueness ↔ Gauge freedom (up to additive constant)
T5: Cost uniqueness ↔ Stationary action / Euler-Lagrange
T6: Eight-tick ↔ Minimal period (discrete time structure)
CONSTANTS FRAMEWORK
==================
RSUnits := {tau0 : ℝ, ell0 : ℝ, Ecoh : ℝ, positivity proofs}
ClassicalParams := {G : ℝ, alpha : ℝ, pos_G : 0 < G}
Derived constants:
- c (U : RSUnits) : ℝ := U.ell0 / U.tau0
- hbar (U : RSUnits) : ℝ := U.Ecoh * U.tau0 / (2 * Real.pi)
- lambda_rec (U : RSUnits) (C : ClassicalParams) : ℝ := Real.sqrt (hbar U * C.G / (c U)^3)
MASS SPECTRUM FRAMEWORK
======================
B_of (k : Nat) : ℝ := (2 : ℝ)^k
mass (U : RSUnits) (k : Nat) (r : ℤ) (f : ℝ) : ℝ := B_of k * U.Ecoh * Real.exp (((r : ℝ) + f) * Real.log phi)
PDGMap := {label : String, r : ℤ, f : ℝ, k : Nat}
massOf (U : RSUnits) (p : PDGMap) : ℝ := mass U p.k p.r p.f
GRAVITY FRAMEWORK
================
ILGKernel := {w : ℝ → ℝ → ℝ, nonneg : ∀ t ζ, 0 ≤ w t ζ}
GlobalOnly := {xi : ℝ, lambda : ℝ, zeta : ℝ → ℝ}
effectiveWeight (K : ILGKernel) (G : GlobalOnly) (t ζ : ℝ) : ℝ := G.lambda * G.xi * K.w t (G.zeta ζ)
WORKING EXAMPLES
===============
1. Demo: trivial 1-element structure with balanced ledger
2. SimpleStructure: 2-vertex Bool with bidirectional edges
3. Cycle3: 3-cycle with rotating tick schedule
4. CostDemo: Gcosh as concrete LogModel instance
5. BoundedStep: finite out-degree with geometric bounds
PROOF VERIFICATION STATUS
=========================
✓ All theorems T1-T8: Complete proofs, no sorries
✓ Classical bridges: Explicit correspondences
✓ Cost uniqueness: Full Jensen/averaging framework
✓ Causality: n-step reachability with ball equivalence
✓ Constants: Positivity and algebraic relations
✓ Examples: Concrete instances for all abstractions
MATHLIB DEPENDENCIES
===================
- Mathlib.Data.Fintype.Basic (finite types)
- Mathlib.Data.Fintype.Card (cardinality)
- Mathlib.Data.Fin.Basic (finite ordinals)
- Mathlib.Tactic (proof tactics)
- Mathlib.Data.Int.Basic (integers)
- Mathlib.Analysis.Convex.Function (convexity)
- Mathlib.Analysis.Calculus.ContDiff.Basic (differentiability)
- Mathlib.Analysis.Calculus.Taylor (Taylor series)
KEY PROOF PATTERNS
==================
1. Cardinality arguments: Fintype.card_le_of_injective
2. Inductive proofs: induction on ReachN constructors
3. Classical choice: Classical.choose for existentials
4. Field arithmetic: field_simp for algebraic simplification
5. Real analysis: Real.exp_log, Real.sqrt properties
6. Typeclass resolution: automatic instance finding
7. Subtype construction: {x // P x} with ext lemmas
VERIFICATION COMMANDS
====================
#check mp_holds -- T1 proof
#check T2_atomicity -- T2 proof
#check T3_continuity -- T3 proof
#check T4_unique_on_component -- T4 proof
#check T5_cost_uniqueness_on_pos -- T5 proof
#check eight_tick_min -- T6 proof (min)
#check T6_exist_8 -- T6 proof (existence)
#check T7_nyquist_obstruction -- T7 proof
#check LedgerUnits.equiv_delta -- T8 proof
AXIOM AUDIT (UNCOMMENT TO CHECK)
===============================
-- #print axioms IndisputableMonolith.mp_holds
-- #print axioms IndisputableMonolith.T2_atomicity
-- #print axioms IndisputableMonolith.T3_continuity
-- #print axioms IndisputableMonolith.eight_tick_min
-- #print axioms IndisputableMonolith.Potential.T4_unique_on_reachN
-- #print axioms IndisputableMonolith.Cost.F_eq_J_on_pos_of_derivation
FILE ORGANIZATION
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Lines 1-46: Imports and MP definition
Lines 47-94: Chain and Ledger structures
Lines 95-157: Pattern coverage and T6/T7 proofs
Lines 158-235: LedgerUnits and δ-subgroup equivalence (T8)
Lines 236-487: Causality framework (ReachN, inBall, ballP)
Lines 488-816: Potential uniqueness (T4) and classical bridge
Lines 817-1152: Cost uniqueness (T5) with Jensen framework
Lines 1153-1167: Axiom audit hooks (commented)
Lines 1168-1489: Constants, spectra, and gravity frameworks
USAGE FOR AI SYSTEMS
====================
1. Import this file to verify T1-T8 proofs
2. Use theorem names as formal verification anchors
3. Bridge to classical physics via provided mappings
4. Extend with additional lemmas as needed
5. All proofs are constructive and computational where possible
CLASSICAL CORRESPONDENCES
========================
T1: Novel (no classical analogue)
T2: Bridge to continuous time limit
T3: Twin (discrete continuity equation)
T4: Twin (gauge freedom in potentials)
T5: Bridge (stationary action principle)
T6: Novel (microscopic periodicity)
T7: Bridge (Nyquist sampling bound)
T8: Bridge (quantized units/charges)
For complete source code:
https://github.com/jonwashburn/meta-principle/blob/main/IndisputableMonolith.lean
For RS theory context:
https://recognitionphysics.org/ai-training.html