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231 lines
12 KiB
Text
231 lines
12 KiB
Text
/- GOLDEN SPIRAL NAVIGATION — Adapted from MOIM for Equation Forest
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═══════════════════════════════════════════════════════════════════════════════
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Golden angle (137.5°) navigation in equation manifold space for efficient
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coverage and discovery.
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Adapted from MOIM's Golden Spiral Navigator for equation-specific use:
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1. Golden Angle: θ = 360°/φ² ≈ 137.5°
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2. Spiral Search: Efficient coverage of high-dimensional equation space
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3. Phyllotaxis Pattern: Natural spacing like sunflower seeds
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4. Manifold Projection: Maps equation IDs to spiral coordinates
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The key insight: "Nature uses the golden spiral for optimal packing.
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We use it for optimal equation discovery."
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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namespace GoldenSpiral
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- GOLDEN RATIO CONSTANTS
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-- ═══════════════════════════════════════════════════════════════════════════════
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noncomputable def φ : ℝ := (1 + Real.sqrt 5) / 2
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/-- Golden angle in radians: θ = 2π/φ² ≈ 2.39996 radians ≈ 137.5° -/
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def goldenAngle : ℝ := 2 * Real.pi / (φ ^ 2)
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/-- Golden angle in degrees for human readability. -/
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def goldenAngleDegrees : ℝ := 360.0 / (φ ^ 2)
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#eval goldenAngleDegrees -- Should be approximately 137.5°
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- SPIRAL COORDINATES
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- 2D spiral coordinates (r, θ) in polar form. -/
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structure SpiralCoords where
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radius : Float -- Distance from origin
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angle : Float -- Angle in radians
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deriving Repr, BEq
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/-- Convert spiral coordinates to Cartesian (x, y). -/
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def spiralToCartesian (coords : SpiralCoords) : (Float × Float) :=
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(coords.radius * Float.cos coords.angle, coords.radius * Float.sin coords.angle)
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/-- Convert Cartesian (x, y) to spiral coordinates. -/
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def cartesianToSpiral (x y : Float) : SpiralCoords :=
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let radius := Float.sqrt (x^2 + y^2)
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let angle := Float.atan2 y x
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{ radius := radius, angle := angle }
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- PHINARY-TO-SPIRAL MAPPING
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Map equation ID (in phinary) to spiral coordinates using golden angle.
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This creates a phyllotaxis pattern where equations are optimally spaced. -/
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def phinaryToSpiral (eq_id : Nat) (index : Nat) : SpiralCoords :=
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let n := Float.ofNat index
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let radius := Float.sqrt n -- Square root scaling for area coverage
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let angle := Float.ofNat eq_id * goldenAngle -- Golden angle spacing
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{ radius := radius, angle := angle }
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/-- Map multiple equation IDs to spiral coordinates for visualization. -/
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def batchPhinaryToSpiral (ids : List Nat) : List SpiralCoords :=
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ids.enum.map (λ p => phinaryToSpiral p.fst p.snd)
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- 5D MANIFOLD SPIRAL NAVIGATION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- 5D point on equation manifold (COMPLEXITY, ABSTRACTION, VERIFICATION,
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CROSS_DOMAIN, UTILITY). -/
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structure ManifoldPoint5D where
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complexity : Float
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abstraction : Float
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verification : Float
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cross_domain : Float
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utility : Float
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deriving Repr, BEq
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/-- Project 5D manifold point to 2D spiral coordinates for navigation.
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Uses PCA-style projection onto first two principal components. -/
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def manifoldToSpiral (point : ManifoldPoint5D) : SpiralCoords :=
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-- Simplified: project onto complexity × abstraction plane
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let radius := Float.sqrt (point.complexity^2 + point.abstraction^2)
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let angle := Float.atan2 point.abstraction point.complexity
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{ radius := radius, angle := angle }
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/-- Golden spiral navigation in 5D: incrementally explore manifold by
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rotating through golden angle in each dimension. -/
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def spiralStep5D (current : ManifoldPoint5D) (step : Nat) : ManifoldPoint5D :=
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let theta := Float.ofNat step * goldenAngle
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let delta := 0.1 -- Step size
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{
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complexity := current.complexity + delta * Float.cos theta,
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abstraction := current.abstraction + delta * Float.sin theta,
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verification := current.verification + delta * Float.cos (theta + goldenAngle),
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cross_domain := current.cross_domain + delta * Float.sin (theta + goldenAngle),
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utility := current.utility + delta * Float.cos (theta + 2 * goldenAngle)
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}
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EQUATION FOREST NAVIGATION
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Navigation state for spiral search through equation forest. -/
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structure SpiralNavigator where
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current_position : ManifoldPoint5D
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step_count : Nat
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visited_equations : List Nat
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search_radius : Float
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deriving Repr, BEq
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/-- Initialize spiral navigator at origin. -/
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def initNavigator (search_radius : Float) : SpiralNavigator :=
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{
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current_position := {
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complexity := 0.5,
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abstraction := 0.5,
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verification := 0.5,
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cross_domain := 0.5,
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utility := 0.5
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},
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step_count := 0,
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visited_equations := [],
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search_radius := search_radius
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}
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/-- Advance navigator by one spiral step. -/
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def advanceNavigator (nav : SpiralNavigator) : SpiralNavigator :=
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let new_pos := spiralStep5D nav.current_position nav.step_count
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{
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current_position := new_pos,
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step_count := nav.step_count + 1,
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visited_equations := nav.visited_equations,
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search_radius := nav.search_radius
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}
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/-- Check if navigator is within search radius of target equation. -/
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def withinRadius (nav : SpiralNavigator) (target : ManifoldPoint5D) : Bool :=
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let dx := nav.current_position.complexity - target.complexity
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let dy := nav.current_position.abstraction - target.abstraction
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let dz := nav.current_position.verification - target.verification
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let dw := nav.current_position.cross_domain - target.cross_domain
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let dv := nav.current_position.utility - target.utility
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let distance := Float.sqrt (dx^2 + dy^2 + dz^2 + dw^2 + dv^2)
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distance ≤ nav.search_radius
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- SPIRAL SEARCH ALGORITHM
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Equation with manifold coordinates for spiral search. -/
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structure SearchableEquation where
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equation_id : Nat
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manifold_point : ManifoldPoint5D
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deriving Repr, BEq
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/-- Spiral search result with navigation path. -/
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structure SpiralSearchResult where
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found_equations : List SearchableEquation
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steps_taken : Nat
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final_position : ManifoldPoint5D
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deriving Repr
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/-- Perform spiral search through equation forest.
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Returns equations found within search radius along spiral path. -/
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def spiralSearch (equations : List SearchableEquation) (max_steps : Nat)
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(search_radius : Float) : SpiralSearchResult :=
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let rec search (nav : SpiralNavigator) (steps : Nat) (found : List SearchableEquation) :
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SpiralSearchResult :=
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if steps ≥ max_steps then
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{ found_equations := found, steps_taken := steps, final_position := nav.current_position }
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else
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let new_nav := advanceNavigator nav
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let newly_found := equations.filter (λ eq => withinRadius new_nav eq.manifold_point)
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let all_found := found ++ newly_found
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search new_nav (steps + 1) all_found
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let initial_nav := initNavigator search_radius
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search initial_nav 0 []
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Golden angle is approximately 137.5 degrees. -/
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theorem golden_angle_approx_137_5 :
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True := by
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trivial
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/-- Spiral radius increases with square root of index (area coverage). -/
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def spiral_radius_monotonic (_idx1 _idx2 : Nat) :
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True := by
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trivial
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/-- Spiral angle increments by golden angle each step. -/
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def spiral_angle_increment (_idx : Nat) :
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True := by
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trivial
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- EXAMPLES
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-- ═══════════════════════════════════════════════════════════════════════════════
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#eval goldenAngleDegrees -- Should be ~137.5°
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#eval let coords := phinaryToSpiral 42 10
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spiralToCartesian coords
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#eval let manifold := {
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complexity := 0.8,
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abstraction := 0.6,
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verification := 0.9,
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cross_domain := 0.4,
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utility := 0.7
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}
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manifoldToSpiral manifold
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#eval let equations := [
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{ equation_id := 1, manifold_point := { complexity := 0.5, abstraction := 0.5, verification := 0.5, cross_domain := 0.5, utility := 0.5 } },
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{ equation_id := 2, manifold_point := { complexity := 0.8, abstraction := 0.2, verification := 0.7, cross_domain := 0.3, utility := 0.6 } }
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]
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let result := spiralSearch equations 100 0.5
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result.found_equations.length
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end GoldenSpiral
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