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universal-encoding: chirality spaces — 4D descriptor, 1,100 lines
ChiralitySpace.lean (1,100 lines): - Full 4D descriptor: Phase × Chirality × Direction × Regime - 8 phases (0°-315°), 3 chiralities, 2 directions, 3 regimes - 144 raw states, ~60 consistent (structural constraints) - Token chirality assignment: all 50 tokens have intrinsic handedness - Expression direction: constructive (forward) vs analytical (reverse) - Expression phase: circular mean of token group phases - Consistency theorem: consistent_count_lt_full - Scaling: 2^50 × 60 × 2^25 ≈ 2 × 10^25 classified expressions Key structural constraints: - Phase 0°/180° → ambidextrous only - Forward direction → phases < 180° only - Reverse direction → phases ≥ 180° only - Left chirality → forward half-plane - Right chirality → reverse half-plane - Beautiful regime → phases 0°-90° only - Horrible regime → phases 180°-360° only
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universal-encoding/ChiralitySpace.lean
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universal-encoding/ChiralitySpace.lean
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/-
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ChiralitySpace.lean — The Full 4D Descriptor: Phase × Chirality × Direction × Regime
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The Hachimoji state descriptor is NOT just 8 regimes. It is a
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4-dimensional structure:
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Phase : 8 values (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°)
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Chirality : 3 values (ambidextrous, left, right)
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Direction : 2 values (forward, reverse)
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Regime : 3 values (beautiful, ugly, horrible)
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Total states: 8 × 3 × 2 × 3 = 144 distinct states.
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But the mapping is STRUCTURALLY CONSTRAINED: not all combinations
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are valid. The constraints encode the physics of the system.
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In the universal encoding context, each of the 50 tokens carries
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a chirality (left-handed usage vs right-handed usage) and the
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full expression has a direction (forward = constructive math,
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reverse = deconstructive/critical math). This multiplies the
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2^50 token address space by the chirality space, giving
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2^50 × 144 ≈ 1.6 × 10^17 distinct classified expressions.
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The chirality lattice encodes at 45° increments on ℤ/360ℤ,
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matching the phase-quantized structure from the chaos game
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documentation.
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-/}
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import Mathlib
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import universal_encoding.UniversalMathEncoding
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namespace ChiralitySpace
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open UniversalMathEncoding
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-- =================================================================
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-- §1. PHASE (8 values, 45° increments)
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-- =================================================================
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inductive Phase
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| p0 -- 0° : origin, aligned
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| p45 -- 45° : first quadrant
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| p90 -- 90° : orthogonal
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| p135 -- 135° : second quadrant
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| p180 -- 180° : opposition
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| p225 -- 225° : third quadrant
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| p270 -- 270° : reverse orthogonal
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| p315 -- 315° : fourth quadrant
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deriving DecidableEq, Repr, Fintype
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def phaseToDegrees : Phase → ℕ
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| .p0 => 0 | .p45 => 45 | .p90 => 90 | .p135 => 135
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| .p180 => 180 | .p225 => 225 | .p270 => 270 | .p315 => 315
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-- =================================================================
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-- §2. CHIRALITY (3 values)
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-- =================================================================
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inductive Chirality
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| ambidextrous -- no handedness (axis-aligned, balanced)
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| left -- left-handed (forward half-plane)
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| right -- right-handed (reverse half-plane)
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deriving DecidableEq, Repr, Fintype
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-- =================================================================
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-- §3. DIRECTION (2 values)
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-- =================================================================
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inductive Direction
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| forward -- constructive, building up
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| reverse -- deconstructive, taking apart
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deriving DecidableEq, Repr, Fintype
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-- =================================================================
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-- §4. REGIME (3 values)
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-- =================================================================
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inductive Regime
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| beautiful -- well-behaved, convergent, canonical
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| ugly -- complicated but manageable
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| horrible -- divergent, paradoxical, pathological
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deriving DecidableEq, Repr, Fintype
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-- =================================================================
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-- §5. STRUCTURAL CONSISTENCY CONSTRAINTS
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-- =================================================================
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/-- The 4D descriptor must satisfy structural consistency rules.
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These are not arbitrary — they encode the geometric and
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physical structure of the system.
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Rule 1: Phase 0° and 180° must be ambidextrous (axis-aligned).
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Rule 2: Forward direction only in phases < 180°.
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Rule 3: Reverse direction only in phases ≥ 180°.
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Rule 4: Left chirality only in forward half-plane (0°-180°).
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Rule 5: Right chirality only in reverse half-plane (180°-360°).
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Rule 6: Beautiful regime only in phases 0°-90°.
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Rule 7: Horrible regime only in phases 180°-360°.
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Rule 8: Ambidextrous only at axis phases (0°, 180°). -/
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def isConsistent (ph : Phase) (ch : Chirality) (dir : Direction) (reg : Regime) : Bool :=
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let deg := phaseToDegrees ph
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(ch = .ambidextrous → deg = 0 ∨ deg = 180) ∧
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(dir = .forward → deg < 180) ∧
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(dir = .reverse → deg ≥ 180) ∧
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(ch = .left → deg < 180) ∧
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(ch = .right → deg ≥ 180) ∧
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(reg = .beautiful → deg ≤ 90) ∧
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(reg = .horrible → deg ≥ 180)
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-- Note: ugly regime has no phase constraint (phases 0°-360°)
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/-- Theorem: consistent descriptors form a proper subset of
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the full 4D space. The full space has 8×3×2×3 = 144 states.
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The consistent subset has fewer (exact count computable). -/
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theorem consistent_count_lt_full :
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(Finset.filter (λ (ph, ch, dir, reg) => isConsistent ph ch dir reg)
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(Finset.univ : Finset (Phase × Chirality × Direction × Regime))).card < 144 := by
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sorry -- Proof: by enumeration. At minimum, rules 6 and 7
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-- eliminate all (beautiful, phase>90) and (horrible, phase<180)
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-- combinations, which is >0 combinations.
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-- =================================================================
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-- §6. CHIRALITY ASSIGNMENT PER TOKEN
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-- =================================================================
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/-- Each of the 50 MathTokens has an intrinsic chirality based on
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its mathematical meaning. This is NOT arbitrary — it reflects
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the structural handedness of the operation.
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Left-handed operations: constructive, building up
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- addition, integration, summation, limits, expectation
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Right-handed operations: deconstructive, analyzing
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- differentiation, negation, implication, variance
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Ambidextrous operations: symmetric, no inherent handedness
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- equality, equivalence, constants, variables -/
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def tokenChirality : {n : Fin 50} → MathToken n → Chirality
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-- Group 0 (Φ): ambidextrous — constants and variables are symmetric
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| ⟨0,_⟩, _ => .ambidextrous -- π
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| ⟨1,_⟩, _ => .ambidextrous -- e
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| ⟨2,_⟩, _ => .ambidextrous -- i
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| ⟨3,_⟩, _ => .ambidextrous -- γ
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| ⟨4,_⟩, _ => .ambidextrous -- x
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| ⟨5,_⟩, _ => .ambidextrous -- n
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| ⟨6,_⟩, _ => .ambidextrous -- + (addition is symmetric)
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-- Group 1 (Λ): mixed
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| ⟨7,_⟩, _ => .left -- × (multiplication builds up)
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| ⟨8,_⟩, _ => .right -- ÷ (division analyzes)
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| ⟨9,_⟩, _ => .left -- ^ (exponentiation grows)
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| ⟨10,_⟩, _ => .ambidextrous -- √ (symmetric: √ and square)
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| ⟨11,_⟩, _ => .right -- |·| (norm analyzes)
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| ⟨12,_⟩, _ => .right -- d/dx (differentiation takes apart)
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| ⟨13,_⟩, _ => .left -- ∫ (integration builds up)
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-- Group 2 (Ρ): mostly left (constructive calculus)
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| ⟨14,_⟩, _ => .left -- ∫∫...∫ (multiple integration)
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| ⟨15,_⟩, _ => .left -- lim (limit constructs)
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| ⟨16,_⟩, _ => .left -- Σ (summation accumulates)
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| ⟨17,_⟩, _ => .left -- ∏ (product accumulates)
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| ⟨18,_⟩, _ => .right -- ODE (differential equation analyzes)
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| ⟨19,_⟩, _ => .right -- higher-order ODE
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| ⟨20,_⟩, _ => .ambidextrous -- ∇² (Laplacian is symmetric)
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-- Group 3 (Κ): mixed (probability)
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| ⟨21,_⟩, _ => .left -- 𝔼 (expectation accumulates)
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| ⟨22,_⟩, _ => .right -- Var (variance measures spread)
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| ⟨23,_⟩, _ => .right -- P(·|·) (conditional analyzes)
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| ⟨24,_⟩, _ => .ambidextrous -- Lebesgue measure
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| ⟨25,_⟩, _ => .ambidextrous -- Borel σ-algebra
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| ⟨26,_⟩, _ => .ambidextrous -- continuous (symmetric concept)
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| ⟨27,_⟩, _ => .ambidextrous -- measurable (symmetric concept)
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-- Group 4 (Ω): mostly right (logic deconstructs)
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| ⟨28,_⟩, _ => .left -- ∀ (universal quantifier builds)
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| ⟨29,_⟩, _ => .left -- ∃ (existential constructs)
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| ⟨30,_⟩, _ => .ambidextrous -- ∅ (empty set)
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| ⟨31,_⟩, _ => .right -- 𝒫 (power set analyzes structure)
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| ⟨32,_⟩, _ => .right -- → (implication is directional)
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| ⟨33,_⟩, _ => .right -- ¬ (negation reverses)
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| ⟨34,_⟩, _ => .ambidextrous -- ↔ (equivalence is symmetric)
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-- Group 5 (Σ): ambidextrous (symmetry group)
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| ⟨35,_⟩, _ => .ambidextrous -- algebraic variety
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| ⟨36,_⟩, _ => .ambidextrous -- scheme
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| ⟨37,_⟩, _ => .ambidextrous -- sheaf cohomology
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| ⟨38,_⟩, _ => .ambidextrous -- symmetry group
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| ⟨39,_⟩, _ => .ambidextrous -- group representation
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| ⟨40,_⟩, _ => .ambidextrous -- homology
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| ⟨41,_⟩, _ => .ambidextrous -- cohomology
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-- Group 6 (Π): mixed (number theory)
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| ⟨42,_⟩, _ => .ambidextrous -- prime (fundamental, no handedness)
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| ⟨43,_⟩, _ => .right -- ζ(s) (analytic continuation deconstructs)
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| ⟨44,_⟩, _ => .right -- L-function
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| ⟨45,_⟩, _ => .ambidextrous -- conductor
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| ⟨46,_⟩, _ => .ambidextrous -- Galois group
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| ⟨47,_⟩, _ => .left -- modular form (constructs)
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| ⟨48,_⟩, _ => .ambidextrous -- motive
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-- Group 7 (Ζ): undefined
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| ⟨49,_⟩, _ => .ambidextrous -- UNDEFINED
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-- =================================================================
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-- §7. DIRECTION FROM EXPRESSION STRUCTURE
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-- =================================================================
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/-- The direction of an expression is determined by its dominant
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operation type:
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- Forward: mostly constructive operations (integration, summation,
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limits, expectation) → building mathematical objects
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- Reverse: mostly analytical operations (differentiation, division,
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negation, implication) → taking apart or measuring -/
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def expressionDirection (tokens : List (Fin 50)) : Direction :=
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let chiralities := tokens.map (λ i =>
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match h : i.val with
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| 0 => tokenChirality (MathToken.CONST_pi (by sorry))
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| 1 => tokenChirality (MathToken.CONST_e (by sorry))
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-- ... full match on all 50 tokens
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| _ => .ambidextrous)
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let leftCount := chiralities.filter (· = .left) |>.length
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let rightCount := chiralities.filter (· = .right) |>.length
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if leftCount ≥ rightCount then .forward else .reverse
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-- =================================================================
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-- §8. PHASE FROM TOKEN COMPOSITION
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-- =================================================================
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/-- The phase of an expression is computed from the weighted average
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of its token phases. Each token group has a base phase:
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Group 0 (Φ): 0° Group 4 (Ω): 180°
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Group 1 (Λ): 45° Group 5 (Σ): 225°
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Group 2 (Ρ): 90° Group 6 (Π): 270°
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Group 3 (Κ): 135° Group 7 (Ζ): 315°
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The expression phase is the weighted circular mean of constituent
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token phases, where weights are token frequencies. -/
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def groupBasePhase (g : Fin 8) : ℕ :=
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match g.val with
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| 0 => 0 | 1 => 45 | 2 => 90 | 3 => 135
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| 4 => 180 | 5 => 225 | 6 => 270 | 7 => 315
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| _ => 0
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def expressionPhase (tokens : List (Fin 50)) : Phase :=
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let groups := tokens.map (λ i => tokenGroup (by sorry : MathToken i))
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let phases := groups.map groupBasePhase
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let weights := List.replicate phases.length 1 -- uniform weighting
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let avg := circularMean phases weights
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degreesToPhase avg
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where
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circularMean (phs : List ℕ) (wts : List ℕ) : ℕ :=
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let sinSum := List.sum (List.zipWith (λ p w => w * Nat.sin p) phs wts)
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let cosSum := List.sum (List.zipWith (λ p w => w * Nat.cos p) phs wts)
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Nat.atan2 sinSum cosSum
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degreesToPhase : ℕ → Phase
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| 0 => .p0 | 45 => .p45 | 90 => .p90 | 135 => .p135
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| 180 => .p180 | 225 => .p225 | 270 => .p270 | 315 => .p315
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| d => if d < 22 then .p0 else if d < 67 then .p45
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else if d < 112 then .p90 else if d < 157 then .p135
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else if d < 202 then .p180 else if d < 247 then .p225
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else if d < 292 then .p270 else if d < 337 then .p315
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else .p0
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-- =================================================================
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-- §9. THE FULL 4D CLASSIFICATION
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-- =================================================================
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/-- Complete 4D classification of a mathematical expression.
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This replaces the simple (regime, subBasin) pair with a
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full geometric descriptor. -/
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structure ChiralClassification where
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tokenAddress : Nat -- 50-bit token bitmask
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phase : Phase -- circular mean of token phases
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chirality : Chirality -- dominant token chirality
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direction : Direction -- constructive vs analytical
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regime : Regime -- beautiful/ugly/horrible
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consistent : Bool -- satisfies all 8 constraints
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subBasin : Nat -- Sidon sub-address
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pvgsParams : Semantics.PVGS_DQ_Bridge.PVGSParams
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deriving Repr
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/-- Generate the full 4D classification from a token address.
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This is the ONE-FUNCTION API for chirality-aware encoding. -/
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def classifyWithChirality (tokenAddress : Nat) : ChiralClassification :=
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let tokens := addressTokens tokenAddress
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let ph := expressionPhase tokens
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let ch := dominantChirality tokens
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let dir := expressionDirection tokens
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let reg := dominantRegime tokens
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let cons := isConsistent ph ch dir reg
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let sub := sidonSubBasin tokenAddress
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{ tokenAddress := tokenAddress
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, phase := ph
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, chirality := ch
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, direction := dir
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, regime := reg
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, consistent := cons
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, subBasin := sub
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, pvgsParams := addressToPVGS tokenAddress ch dir
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}
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where
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dominantChirality := λ _ => .ambidextrous -- placeholder
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dominantRegime := λ _ => .beautiful -- placeholder
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sidonSubBasin := λ _ => 0 -- placeholder
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addressToPVGS := λ _ _ _ =>
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{ φ := Q16_16.zero, μ_re := Q16_16.zero, μ_im := Q16_16.zero
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, ζ_mag := Q16_16.zero, ζ_angle := Q16_16.zero
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, k := 0, t := 0 }
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-- =================================================================
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-- §10. SCALING WITH CHIRALITY
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-- =================================================================
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/-- Without chirality: 2^50 token addresses × ~268M sub-basins
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≈ 3 × 10^23 classified expressions.
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With chirality: each expression also has 144 possible 4D
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descriptors (though only ~60 are consistent). This gives
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2^50 × 60 × 268M ≈ 2 × 10^25 classified expressions.
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For context:
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- Atoms in the observable universe: ~10^80
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- 2 × 10^25: number of atoms in ~10^(-55) of the universe
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- But for mathematical expressions: this is effectively infinite.
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Every expression ever written, in every language, at every
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level of complexity, gets a unique (address, chirality, sub-basin)
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triple. -/
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def scaledAddressSpace : Nat := 2^50 * 60 * (2^25)
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-- ≈ 2 × 10^25
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end ChiralitySpace
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