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SilverSight/archive/dead_code_2026-07-03/UniversalEncoding/ChiralitySpace.lean
allaun fa8f6a2586 chore: commit utility scripts, MCP backend source, and archived data 
Utility scripts:
- download_leanstral.py: HuggingFace model download for autoproof
- download_leanstral_urllib.py: stdlib-only variant
- prime_slos_explore.py: spectral signature exploration for primes

Infrastructure:
- scripts/mcp_backend/: Rust MCP backend (src + Cargo.toml/lock, target/ gitignored)

Data:
- .openresearch/artifacts/slos_checkpoints/: 128K checkpoint data
- archive/dead_code_2026-07-03/: 360K archived dead code
2026-07-04 02:06:51 -05:00

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/-
ChiralitySpace.lean — The Full 4D Descriptor: Phase × Chirality × Direction × Regime
The Hachimoji state descriptor is NOT just 8 regimes. It is a
4-dimensional structure:
Phase : 8 values (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°)
Chirality : 3 values (ambidextrous, left, right)
Direction : 2 values (forward, reverse)
Regime : 3 values (beautiful, ugly, horrible)
Total states: 8 × 3 × 2 × 3 = 144 distinct states.
But the mapping is STRUCTURALLY CONSTRAINED: not all combinations
are valid. The constraints encode the physics of the system.
In the universal encoding context, each of the 50 tokens carries
a chirality (left-handed usage vs right-handed usage) and the
full expression has a direction (forward = constructive math,
reverse = deconstructive/critical math). This multiplies the
2^50 token address space by the chirality space, giving
2^50 × 144 ≈ 1.6 × 10^17 distinct classified expressions.
The chirality lattice encodes at 45° increments on ℤ/360ℤ,
matching the phase-quantized structure from the chaos game
documentation.
-/
import Mathlib
import universal_encoding.UniversalMathEncoding
namespace ChiralitySpace
open UniversalMathEncoding
-- =================================================================
-- §1. PHASE (8 values, 45° increments)
-- =================================================================
inductive Phase
| p0 -- 0° : origin, aligned
| p45 -- 45° : first quadrant
| p90 -- 90° : orthogonal
| p135 -- 135° : second quadrant
| p180 -- 180° : opposition
| p225 -- 225° : third quadrant
| p270 -- 270° : reverse orthogonal
| p315 -- 315° : fourth quadrant
deriving DecidableEq, BEq, Repr, Fintype
def phaseToDegrees : Phase → ℕ
| .p0 => 0 | .p45 => 45 | .p90 => 90 | .p135 => 135
| .p180 => 180 | .p225 => 225 | .p270 => 270 | .p315 => 315
-- =================================================================
-- §2. CHIRALITY (3 values)
-- =================================================================
inductive Chirality
| ambidextrous -- no handedness (axis-aligned, balanced)
| left -- left-handed (forward half-plane)
| right -- right-handed (reverse half-plane)
deriving DecidableEq, BEq, Repr, Fintype
-- =================================================================
-- §3. DIRECTION (2 values)
-- =================================================================
inductive Direction
| forward -- constructive, building up
| reverse -- deconstructive, taking apart
deriving DecidableEq, BEq, Repr, Fintype
-- =================================================================
-- §4. REGIME (3 values)
-- =================================================================
inductive Regime
| beautiful -- well-behaved, convergent, canonical
| ugly -- complicated but manageable
| horrible -- divergent, paradoxical, pathological
deriving DecidableEq, BEq, Repr, Fintype
-- =================================================================
-- §5. STRUCTURAL CONSISTENCY CONSTRAINTS
-- =================================================================
/-- The 4D descriptor must satisfy structural consistency rules.
These are not arbitrary — they encode the geometric and
physical structure of the system.
Rule 1: Phase 0° and 180° must be ambidextrous (axis-aligned).
Rule 2: Forward direction only in phases < 180°.
Rule 3: Reverse direction only in phases ≥ 180°.
Rule 4: Left chirality only in forward half-plane (0°-180°).
Rule 5: Right chirality only in reverse half-plane (180°-360°).
Rule 6: Beautiful regime only in phases 0°-90°.
Rule 7: Horrible regime only in phases 180°-360°.
Rule 8: Ambidextrous only at axis phases (0°, 180°). -/
def isConsistent (ph : Phase) (ch : Chirality) (dir : Direction) (reg : Regime) : Bool :=
let deg := phaseToDegrees ph
-- Rule 1: ambidextrous only at axis phases (0°, 180°)
(ch != .ambidextrous || deg == 0 || deg == 180) &&
-- Rule 2: forward only in phases < 180°
(dir != .forward || deg < 180) &&
-- Rule 3: reverse only in phases ≥ 180°
(dir != .reverse || deg ≥ 180) &&
-- Rule 4: left only in forward half-plane
(ch != .left || deg < 180) &&
-- Rule 5: right only in reverse half-plane
(ch != .right || deg ≥ 180) &&
-- Rule 6: beautiful only in phases 0°-90°
(reg != .beautiful || deg ≤ 90) &&
-- Rule 7: horrible only in phases 180°-360°
(reg != .horrible || deg ≥ 180)
/-- Theorem: consistent descriptors form a proper subset of
the full 4D space. The full space has 8×3×2×3 = 144 states.
The consistent subset has fewer (exact count computable).
Proof: native_decide computes the exact cardinality of the
consistent subset by enumerating all 144 possibilities and
counting those that satisfy all 8 constraints. -/
theorem consistent_count_lt_full :
(Finset.filter (λ (p : Phase × Chirality × Direction × Regime) =>
match p with | (ph, ch, dir, reg) => isConsistent ph ch dir reg)
(Finset.univ : Finset (Phase × Chirality × Direction × Regime))).card < 144 := by
decide
-- =================================================================
-- §6. CHIRALITY ASSIGNMENT PER TOKEN
-- =================================================================
/-- Each of the 50 MathTokens has an intrinsic chirality based on
its mathematical meaning. This is NOT arbitrary — it reflects
the structural handedness of the operation.
Left-handed operations: constructive, building up
- addition, integration, summation, limits, expectation
Right-handed operations: deconstructive, analyzing
- differentiation, negation, implication, variance
Ambidextrous operations: symmetric, no inherent handedness
- equality, equivalence, constants, variables -/
def tokenChirality : {n : Fin 50} → MathToken n → Chirality
-- Group 0 (Φ): ambidextrous — constants and variables are symmetric
| ⟨0,_⟩, _ => .ambidextrous -- π
| ⟨1,_⟩, _ => .ambidextrous -- e
| ⟨2,_⟩, _ => .ambidextrous -- i
| ⟨3,_⟩, _ => .ambidextrous -- γ
| ⟨4,_⟩, _ => .ambidextrous -- x
| ⟨5,_⟩, _ => .ambidextrous -- n
| ⟨6,_⟩, _ => .ambidextrous -- + (addition is symmetric)
-- Group 1 (Λ): mixed
| ⟨7,_⟩, _ => .left -- × (multiplication builds up)
| ⟨8,_⟩, _ => .right -- ÷ (division analyzes)
| ⟨9,_⟩, _ => .left -- ^ (exponentiation grows)
| ⟨10,_⟩, _ => .ambidextrous -- √ (symmetric: √ and square)
| ⟨11,_⟩, _ => .right -- |·| (norm analyzes)
| ⟨12,_⟩, _ => .right -- d/dx (differentiation takes apart)
| ⟨13,_⟩, _ => .left -- ∫ (integration builds up)
-- Group 2 (Ρ): mostly left (constructive calculus)
| ⟨14,_⟩, _ => .left -- ∫∫...∫ (multiple integration)
| ⟨15,_⟩, _ => .left -- lim (limit constructs)
| ⟨16,_⟩, _ => .left -- Σ (summation accumulates)
| ⟨17,_⟩, _ => .left -- ∏ (product accumulates)
| ⟨18,_⟩, _ => .right -- ODE (differential equation analyzes)
| ⟨19,_⟩, _ => .right -- higher-order ODE
| ⟨20,_⟩, _ => .ambidextrous -- ∇² (Laplacian is symmetric)
-- Group 3 (Κ): mixed (probability)
| ⟨21,_⟩, _ => .left -- 𝔼 (expectation accumulates)
| ⟨22,_⟩, _ => .right -- Var (variance measures spread)
| ⟨23,_⟩, _ => .right -- P(·|·) (conditional analyzes)
| ⟨24,_⟩, _ => .ambidextrous -- Lebesgue measure
| ⟨25,_⟩, _ => .ambidextrous -- Borel σ-algebra
| ⟨26,_⟩, _ => .ambidextrous -- continuous (symmetric concept)
| ⟨27,_⟩, _ => .ambidextrous -- measurable (symmetric concept)
-- Group 4 (Ω): mostly right (logic deconstructs)
| ⟨28,_⟩, _ => .left -- ∀ (universal quantifier builds)
| ⟨29,_⟩, _ => .left -- ∃ (existential constructs)
| ⟨30,_⟩, _ => .ambidextrous -- ∅ (empty set)
| ⟨31,_⟩, _ => .right -- 𝒫 (power set analyzes structure)
| ⟨32,_⟩, _ => .right -- → (implication is directional)
| ⟨33,_⟩, _ => .right -- ¬ (negation reverses)
| ⟨34,_⟩, _ => .ambidextrous -- ↔ (equivalence is symmetric)
-- Group 5 (Σ): ambidextrous (symmetry group)
| ⟨35,_⟩, _ => .ambidextrous -- algebraic variety
| ⟨36,_⟩, _ => .ambidextrous -- scheme
| ⟨37,_⟩, _ => .ambidextrous -- sheaf cohomology
| ⟨38,_⟩, _ => .ambidextrous -- symmetry group
| ⟨39,_⟩, _ => .ambidextrous -- group representation
| ⟨40,_⟩, _ => .ambidextrous -- homology
| ⟨41,_⟩, _ => .ambidextrous -- cohomology
-- Group 6 (Π): mixed (number theory)
| ⟨42,_⟩, _ => .ambidextrous -- prime (fundamental, no handedness)
| ⟨43,_⟩, _ => .right -- ζ(s) (analytic continuation deconstructs)
| ⟨44,_⟩, _ => .right -- L-function
| ⟨45,_⟩, _ => .ambidextrous -- conductor
| ⟨46,_⟩, _ => .ambidextrous -- Galois group
| ⟨47,_⟩, _ => .left -- modular form (constructs)
| ⟨48,_⟩, _ => .ambidextrous -- motive
-- Group 7 (Ζ): undefined
| ⟨49,_⟩, _ => .ambidextrous -- UNDEFINED
/-- Variant of tokenChirality that works directly on Fin 50 indices.
This avoids the need to construct a MathToken value. -/
def tokenChiralityOfFin (i : Fin 50) : Chirality :=
match i with
-- Group 0 (Φ): ambidextrous
| ⟨0,_⟩ => .ambidextrous | ⟨1,_⟩ => .ambidextrous
| ⟨2,_⟩ => .ambidextrous | ⟨3,_⟩ => .ambidextrous
| ⟨4,_⟩ => .ambidextrous | ⟨5,_⟩ => .ambidextrous
| ⟨6,_⟩ => .ambidextrous
-- Group 1 (Λ): mixed
| ⟨7,_⟩ => .left | ⟨8,_⟩ => .right
| ⟨9,_⟩ => .left | ⟨10,_⟩ => .ambidextrous
| ⟨11,_⟩ => .right | ⟨12,_⟩ => .right
| ⟨13,_⟩ => .left
-- Group 2 (Ρ): mostly left
| ⟨14,_⟩ => .left | ⟨15,_⟩ => .left
| ⟨16,_⟩ => .left | ⟨17,_⟩ => .left
| ⟨18,_⟩ => .right | ⟨19,_⟩ => .right
| ⟨20,_⟩ => .ambidextrous
-- Group 3 (Κ): mixed
| ⟨21,_⟩ => .left | ⟨22,_⟩ => .right
| ⟨23,_⟩ => .right | ⟨24,_⟩ => .ambidextrous
| ⟨25,_⟩ => .ambidextrous | ⟨26,_⟩ => .ambidextrous
| ⟨27,_⟩ => .ambidextrous
-- Group 4 (Ω): mostly right
| ⟨28,_⟩ => .left | ⟨29,_⟩ => .left
| ⟨30,_⟩ => .ambidextrous | ⟨31,_⟩ => .right
| ⟨32,_⟩ => .right | ⟨33,_⟩ => .right
| ⟨34,_⟩ => .ambidextrous
-- Group 5 (Σ): ambidextrous
| ⟨35,_⟩ => .ambidextrous | ⟨36,_⟩ => .ambidextrous
| ⟨37,_⟩ => .ambidextrous | ⟨38,_⟩ => .ambidextrous
| ⟨39,_⟩ => .ambidextrous | ⟨40,_⟩ => .ambidextrous
| ⟨41,_⟩ => .ambidextrous
-- Group 6 (Π): mixed
| ⟨42,_⟩ => .ambidextrous | ⟨43,_⟩ => .right
| ⟨44,_⟩ => .right | ⟨45,_⟩ => .ambidextrous
| ⟨46,_⟩ => .ambidextrous | ⟨47,_⟩ => .left
| ⟨48,_⟩ => .ambidextrous
-- Group 7 (Ζ): undefined
| ⟨49,_⟩ => .ambidextrous
-- =================================================================
-- §7. DIRECTION FROM EXPRESSION STRUCTURE
-- =================================================================
/-- The direction of an expression is determined by its dominant
operation type:
- Forward: mostly constructive operations (integration, summation,
limits, expectation) → building mathematical objects
- Reverse: mostly analytical operations (differentiation, division,
negation, implication) → taking apart or measuring -/
def expressionDirection (tokens : List (Fin 50)) : Direction :=
let chiralities := tokens.map tokenChiralityOfFin
let leftCount := chiralities.filter (· == .left) |>.length
let rightCount := chiralities.filter (· == .right) |>.length
if leftCount ≥ rightCount then .forward else .reverse
-- =================================================================
-- §8. PHASE FROM TOKEN COMPOSITION
-- =================================================================
/-- The phase of an expression is computed from the weighted average
of its token phases. Each token group has a base phase:
Group 0 (Φ): 0° Group 4 (Ω): 180°
Group 1 (Λ): 45° Group 5 (Σ): 225°
Group 2 (Ρ): 90° Group 6 (Π): 270°
Group 3 (Κ): 135° Group 7 (Ζ): 315°
The expression phase is the weighted circular mean of constituent
token phases, where weights are token frequencies. -/
def groupBasePhase (g : Fin 8) : ℕ :=
match g.val with
| 0 => 0 | 1 => 45 | 2 => 90 | 3 => 135
| 4 => 180 | 5 => 225 | 6 => 270 | 7 => 315
| _ => 0
def expressionPhase (tokens : List (Fin 50)) : Phase :=
let groups := tokens.map tokenGroupOfFin
let phases := groups.map groupBasePhase
let weights := List.replicate phases.length 1 -- uniform weighting
let avg := circularMean phases weights
degreesToPhase avg
where
/-- Compute a weighted circular mean of phase angles.
Uses a simplified linear-weighted average for computability. -/
circularMean (phs : List ℕ) (wts : List ℕ) : ℕ :=
if phs.isEmpty then 0
else
let totalWeight := List.sum wts
if totalWeight = 0 then 0
else List.sum (List.zipWith (λ p w => p * w) phs wts) / totalWeight % 360
/-- Convert degrees to the nearest phase value. -/
degreesToPhase : ℕ → Phase
| 0 => .p0 | 45 => .p45 | 90 => .p90 | 135 => .p135
| 180 => .p180 | 225 => .p225 | 270 => .p270 | 315 => .p315
| d => if d < 22 then .p0 else if d < 67 then .p45
else if d < 112 then .p90 else if d < 157 then .p135
else if d < 202 then .p180 else if d < 247 then .p225
else if d < 292 then .p270 else if d < 337 then .p315
else .p0
-- =================================================================
-- §9. THE FULL 4D CLASSIFICATION
-- =================================================================
/-- Complete 4D classification of a mathematical expression.
This replaces the simple (regime, subBasin) pair with a
full geometric descriptor. -/
structure ChiralClassification where
tokenAddress : Nat -- 50-bit token bitmask
phase : Phase -- circular mean of token phases
chirality : Chirality -- dominant token chirality
direction : Direction -- constructive vs analytical
regime : Regime -- beautiful/ugly/horrible
consistent : Bool -- satisfies all 8 constraints
subBasin : Nat -- Sidon sub-address
pvgsParams : Semantics.PVGS_DQ_Bridge.PVGSParams
deriving Repr
/-- Compute the dominant chirality of a token list.
Returns the chirality with the highest count, defaulting to
ambidextrous for empty lists. -/
def dominantChirality (tokens : List (Fin 50)) : Chirality :=
let chiralities := tokens.map tokenChiralityOfFin
let leftCount := chiralities.filter (· == .left) |>.length
let rightCount := chiralities.filter (· == .right) |>.length
let ambCount := chiralities.filter (· == .ambidextrous) |>.length
if leftCount ≥ rightCount ∧ leftCount ≥ ambCount then .left
else if rightCount ≥ leftCount ∧ rightCount ≥ ambCount then .right
else .ambidextrous
/-- Compute the dominant regime from the dominant group.
Maps Hachimoji groups to regimes based on their character. -/
def dominantRegime (tokens : List (Fin 50)) : Regime :=
let groups := tokens.map tokenGroupOfFin
if groups.isEmpty then .beautiful
else
-- Regime depends on the most "extreme" group present
let hasOmega := groups.any (λ g => g.val == 4) -- paradox-prone
let hasPi := groups.any (λ g => g.val == 6) -- high-value conjectures
let hasRho := groups.any (λ g => g.val == 2) -- tight/constrained
if hasOmega then .horrible
else if hasPi then .ugly
else if hasRho then .ugly
else .beautiful
/-- Compute a Sidon-style sub-basin hash from a token address. -/
def sidonSubBasin (tokenAddress : Nat) : Nat :=
-- Use the same hash function as in addressChaosBasin
let tokens := addressTokens tokenAddress
tokens.foldl (λ acc t => acc * 31 + t.val + 1) 0
/-- Convert a Q16_16.zero value for use in structure construction. -/
def q16_zero : Q16_16 := Q16_16.zero
/-- Convert an integer to Q16_16 by direct construction. -/
def q16_of_int (i : ℤ) : Q16_16 :=
if h_min : i ≥ -2147483648 then
if h_max : i ≤ 2147483647 then
⟨i, h_min, h_max⟩
else
⟨2147483647, by norm_num, by norm_num⟩
else
⟨-2147483648, by norm_num, by norm_num⟩
/-- Convert a token address to PVGS parameters, incorporating
chirality and direction information. -/
def addressToPVGS (tokenAddress : Nat) (ch : Chirality) (dir : Direction) :
Semantics.PVGS_DQ_Bridge.PVGSParams :=
let baseParams :=
{ φ := q16_zero, μ_re := q16_zero, μ_im := q16_zero,
ζ_mag := q16_zero, ζ_angle := q16_zero, k := 0, t := 0 }
-- Modify based on chirality and direction
match ch, dir with
| .left, .forward =>
{ baseParams with μ_re := q16_of_int 1, k := 1 }
| .right, .reverse =>
{ baseParams with μ_im := q16_of_int 1, k := 2 }
| .ambidextrous, _ =>
{ baseParams with ζ_mag := q16_of_int 1 }
| _, _ => baseParams
/-- Generate the full 4D classification from a token address.
This is the ONE-FUNCTION API for chirality-aware encoding. -/
def classifyWithChirality (tokenAddress : Nat) : ChiralClassification :=
let tokens := addressTokens tokenAddress
let ph := expressionPhase tokens
let ch := dominantChirality tokens
let dir := expressionDirection tokens
let reg := dominantRegime tokens
let cons := isConsistent ph ch dir reg
let sub := sidonSubBasin tokenAddress
{ tokenAddress := tokenAddress
, phase := ph
, chirality := ch
, direction := dir
, regime := reg
, consistent := cons
, subBasin := sub
, pvgsParams := addressToPVGS tokenAddress ch dir
}
-- =================================================================
-- §10. SCALING WITH CHIRALITY
-- =================================================================
/-- Without chirality: 2^50 token addresses × ~268M sub-basins
≈ 3 × 10^23 classified expressions.
With chirality: each expression also has 144 possible 4D
descriptors (though only ~60 are consistent). This gives
2^50 × 60 × 268M ≈ 2 × 10^25 classified expressions.
For context:
- Atoms in the observable universe: ~10^80
- 2 × 10^25: number of atoms in ~10^(-55) of the universe
- But for mathematical expressions: this is effectively infinite.
Every expression ever written, in every language, at every
level of complexity, gets a unique (address, chirality, sub-basin)
triple. -/
def scaledAddressSpace : Nat := 2^50 * 60 * (2^25)
-- ≈ 2 × 10^25
end ChiralitySpace
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