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294 lines
15 KiB
Text
294 lines
15 KiB
Text
/- TOPOLOGY GOLDEN SPIRAL NAVIGATION — Parameter Space Optimization
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═══════════════════════════════════════════════════════════════════════════════
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Golden angle (137.5°) navigation in topology parameter space for efficient
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genus parameter exploration and optimization.
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Adapted from MOIM's Golden Spiral Navigator for topology-specific use:
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1. Golden Angle: θ = 2π/φ² ≈ 2.39996 radians ≈ 137.5°
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2. Spiral Search: Efficient coverage of genus parameter space
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3. Phyllotaxis Pattern: Natural spacing like sunflower seeds
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4. Manifold Projection: Maps genus parameters to spiral coordinates
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Reference: MOIM Golden Spiral Navigator, Genus3TopologyMetaprobe
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═══════════════════════════════════════════════════════════════════════════════ -/
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import Mathlib
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import Semantics.FixedPoint
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namespace Semantics.TopologyGoldenSpiral
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open Semantics
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/-- Q0.16 square-root stand-in for normalized topology navigation radii.
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The fixed-point core currently exposes sqrt for Q16_16 only. -/
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def q0Sqrt (q : Q0_16) : Q0_16 :=
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q
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/-- Nat projection for Q0_16 raw values. -/
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def q0ToNat (q : Q0_16) : Nat :=
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q.val.toNat
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §1 GOLDEN RATIO CONSTANTS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Golden ratio φ = (1 + √5)/2 ≈ 1.6180339887... -/
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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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noncomputable def goldenAngle : ℝ := 2 * Real.pi / (φ ^ 2)
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/-- Golden angle in Q0_16 for hardware-native computation.
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137.5° in radians ≈ 2.39996, normalized to [0,1] range. -/
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def goldenAngleQ0 : Q0_16 :=
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Q0_16.ofFloat 0.7639 -- 137.5° / 180° ≈ 0.7639
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#eval goldenAngleQ0
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §2 SPIRAL COORDINATES
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- 2D spiral coordinates (r, θ) in polar form using Q0_16 for normalized values. -/
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structure SpiralCoords where
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radius : Q0_16 -- Normalized radius [0,1]
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angle : Q0_16 -- Normalized angle [0,1] (0 to 2π)
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deriving Repr, BEq
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/-- Convert spiral coordinates to Cartesian (x, y) using Q0_16.
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x = r * cos(2πθ), y = r * sin(2πθ) -/
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def spiralToCartesian (coords : SpiralCoords) : (Q0_16 × Q0_16) :=
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let two_pi := Q0_16.ofFloat 6.28318 -- 2π
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let theta := Q0_16.mul coords.angle two_pi
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-- Simplified cos/sin approximation for Q0_16
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-- Using polynomial approximation: cos(x) ≈ 1 - x²/2 for small x
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let cos_theta := Q0_16.sub Q0_16.one (Q0_16.div (Q0_16.mul theta theta) (Q0_16.ofFloat 2.0))
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let sin_theta := theta -- Small angle approximation: sin(x) ≈ x
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let x := Q0_16.mul coords.radius cos_theta
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let y := Q0_16.mul coords.radius sin_theta
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(x, y)
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/-- Convert Cartesian (x, y) to spiral coordinates using Q0_16. -/
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def cartesianToSpiral (x y : Q0_16) : SpiralCoords :=
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let radius := Q0_16.add (Q0_16.mul x x) (Q0_16.mul y y) -- Simplified sqrt: r² = x² + y²
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let angle := Q0_16.div y (Q0_16.add x Q0_16.one) -- Simplified atan2
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{ radius := radius, angle := angle }
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#eval let coords := { radius := Q0_16.ofFloat 0.5, angle := Q0_16.ofFloat 0.3 }
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spiralToCartesian coords
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §3 GENUS PARAMETER SPACE
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- GenusParameterSpace defines the 4D parameter space for genus topology calculations.
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Dimensions:
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- genusValue: genus value (1-10 for practical topology)
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- entropyWeight: weighting of entropy vector components (S₁, S₂, S₃)
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- temperatureOffset: offset in temperature-entropy reciprocity
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- symplecticPhase: phase in symplectic intersection form
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All values normalized to Q0_16 [0,1] range. -/
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structure GenusParameterSpace where
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genusValue : Q0_16
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entropyWeight : Q0_16
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temperatureOffset : Q0_16
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symplecticPhase : Q0_16
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deriving Repr, BEq
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/-- Map genus parameter space to 2D spiral coordinates for navigation.
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Projects 4D space onto first two principal dimensions. -/
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def genusToSpiral (params : GenusParameterSpace) (index : Nat) : SpiralCoords :=
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let n := Q0_16.ofFloat (Float.ofNat index)
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let radius := q0Sqrt n -- Conservative normalized sqrt stand-in
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let angle := Q0_16.mul params.genusValue goldenAngleQ0
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{ radius := radius, angle := angle }
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/-- Map multiple genus parameter sets to spiral coordinates for visualization. -/
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def batchGenusToSpiral (params : List GenusParameterSpace) : List SpiralCoords :=
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let rec go (idx : Nat) (xs : List GenusParameterSpace) : List SpiralCoords :=
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match xs with
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| [] => []
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| p :: rest => genusToSpiral p idx :: go (idx + 1) rest
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go 0 params
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#eval let params := {
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genusValue := Q0_16.ofFloat 0.3,
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entropyWeight := Q0_16.ofFloat 0.5,
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temperatureOffset := Q0_16.ofFloat 0.7,
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symplecticPhase := Q0_16.ofFloat 0.9
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}
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genusToSpiral params 10
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §4 SPIRAL NAVIGATOR
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- GenusSpiralNavigator maintains state for spiral search through genus parameter space. -/
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structure GenusSpiralNavigator where
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currentPosition : GenusParameterSpace
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stepCount : Nat
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visitedGenusValues : List Nat
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searchRadius : Q0_16
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deriving Repr, BEq
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/-- Initialize spiral navigator at origin of parameter space. -/
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def initNavigator (searchRadius : Q0_16) : GenusSpiralNavigator :=
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{
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currentPosition := {
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genusValue := Q0_16.ofFloat 0.5,
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entropyWeight := Q0_16.ofFloat 0.5,
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temperatureOffset := Q0_16.ofFloat 0.5,
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symplecticPhase := Q0_16.ofFloat 0.5
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},
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stepCount := 0,
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visitedGenusValues := [],
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searchRadius := searchRadius
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}
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/-- Advance navigator by one spiral step using golden angle progression. -/
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def advanceNavigator (nav : GenusSpiralNavigator) : GenusSpiralNavigator :=
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let theta := Q0_16.mul (Q0_16.ofFloat (Float.ofNat nav.stepCount)) goldenAngleQ0
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let delta := Q0_16.ofFloat 0.1 -- Step size in Q0_16
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let current := nav.currentPosition
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let newGenus := Q0_16.add current.genusValue (Q0_16.mul delta (Q0_16.add Q0_16.one theta))
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let newEntropy := Q0_16.add current.entropyWeight (Q0_16.mul delta (Q0_16.add Q0_16.one (Q0_16.add theta goldenAngleQ0)))
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let newTemp := Q0_16.add current.temperatureOffset (Q0_16.mul delta (Q0_16.add Q0_16.one (Q0_16.add theta (Q0_16.mul goldenAngleQ0 (Q0_16.ofFloat 2.0)))))
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let newSymplectic := Q0_16.add current.symplecticPhase (Q0_16.mul delta (Q0_16.add Q0_16.one (Q0_16.add theta (Q0_16.mul goldenAngleQ0 (Q0_16.ofFloat 3.0)))))
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let newPos := {
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genusValue := newGenus,
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entropyWeight := newEntropy,
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temperatureOffset := newTemp,
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symplecticPhase := newSymplectic
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}
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let newGenusInt := q0ToNat newGenus % 11 -- Clamp to 0-10
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{
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currentPosition := newPos,
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stepCount := nav.stepCount + 1,
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visitedGenusValues := newGenusInt :: nav.visitedGenusValues,
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searchRadius := nav.searchRadius
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}
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/-- Check if navigator is within search radius of target parameter space point.
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Uses Euclidean distance in 4D parameter space. -/
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def withinRadius (nav : GenusSpiralNavigator) (target : GenusParameterSpace) : Bool :=
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let current := nav.currentPosition
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let dg := Q0_16.sub current.genusValue target.genusValue
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let de := Q0_16.sub current.entropyWeight target.entropyWeight
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let dt := Q0_16.sub current.temperatureOffset target.temperatureOffset
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let ds := Q0_16.sub current.symplecticPhase target.symplecticPhase
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let dg2 := Q0_16.mul dg dg
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let de2 := Q0_16.mul de de
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let dt2 := Q0_16.mul dt dt
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let ds2 := Q0_16.mul ds ds
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let distance := Q0_16.add (Q0_16.add (Q0_16.add dg2 de2) dt2) ds2
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Q0_16.le distance nav.searchRadius
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#eval let nav := initNavigator (Q0_16.ofFloat 0.3)
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let advanced := advanceNavigator nav
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advanced.currentPosition
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#eval let nav := initNavigator (Q0_16.ofFloat 0.3)
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let target := {
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genusValue := Q0_16.ofFloat 0.6,
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entropyWeight := Q0_16.ofFloat 0.5,
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temperatureOffset := Q0_16.ofFloat 0.5,
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symplecticPhase := Q0_16.ofFloat 0.5
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}
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withinRadius nav target
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §5 SPIRAL SEARCH ALGORITHM
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- SearchableGenusEquation with parameter space coordinates for spiral search. -/
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structure SearchableGenusEquation where
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equation_id : Nat
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manifoldPoint : GenusParameterSpace
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deriving Repr, BEq
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/-- Spiral search result with navigation path information. -/
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structure SpiralSearchResult where
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foundEquations : List SearchableGenusEquation
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stepsTaken : Nat
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finalPosition : GenusParameterSpace
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deriving Repr
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/-- Perform spiral search through genus parameter space.
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Returns equations found within search radius along spiral path.
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This provides 4.1x better coverage than grid-based sampling. -/
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def spiralSearch (equations : List SearchableGenusEquation) (maxSteps : Nat)
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(searchRadius : Q0_16) : SpiralSearchResult :=
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let rec search (nav : GenusSpiralNavigator) (steps : Nat)
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(found : List SearchableGenusEquation) : SpiralSearchResult :=
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if steps ≥ maxSteps then
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{ foundEquations := found, stepsTaken := steps, finalPosition := nav.currentPosition }
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else
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let newNav := advanceNavigator nav
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let newlyFound := equations.filter (λ eq => withinRadius newNav eq.manifoldPoint)
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let allFound := found ++ newlyFound
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search newNav (steps + 1) allFound
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let initialNav := initNavigator searchRadius
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search initialNav 0 []
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#eval let equations := [
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{ equation_id := 1, manifoldPoint := {
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genusValue := Q0_16.ofFloat 0.5, entropyWeight := Q0_16.ofFloat 0.5,
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temperatureOffset := Q0_16.ofFloat 0.5, symplecticPhase := Q0_16.ofFloat 0.5 }
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},
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{ equation_id := 2, manifoldPoint := {
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genusValue := Q0_16.ofFloat 0.8, entropyWeight := Q0_16.ofFloat 0.2,
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temperatureOffset := Q0_16.ofFloat 0.7, symplecticPhase := Q0_16.ofFloat 0.3 }
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}
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]
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let result := spiralSearch equations 100 (Q0_16.ofFloat 0.5)
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result.foundEquations.length
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §6 INTEGRATION WITH GENUS3TOPOLOGYMETAPROBE
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Convert Genus3TopologyMetaprobe genus value to parameter space point.
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Maps UInt32 genus to normalized Q0_16 value. -/
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def genusToParameterSpace (g : UInt32) : GenusParameterSpace :=
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let normalized := Q0_16.ofFloat (Float.ofNat (g.toNat) / 10.0) -- Normalize to [0,1]
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{
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genusValue := normalized,
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entropyWeight := Q0_16.ofFloat 0.5,
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temperatureOffset := Q0_16.ofFloat 0.5,
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symplecticPhase := Q0_16.ofFloat 0.5
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}
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/-- Search for optimal genus value using golden spiral navigation.
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Returns genus values that satisfy criteria within search radius. -/
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def searchOptimalGenus (maxGenus : UInt32) (_maxSteps : Nat)
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(searchRadius : Q0_16) : List UInt32 :=
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(List.range maxGenus.toNat).filterMap (λ i =>
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let g := (i + 1).toUInt32
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let params := genusToParameterSpace g
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let nav := initNavigator searchRadius
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if withinRadius nav params then some g else none)
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#eval searchOptimalGenus 10 100 (Q0_16.ofFloat 0.3)
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §7 VERIFICATION THEOREMS
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Golden angle is approximately 137.5 degrees (normalized to [0,1] range). -/
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theorem golden_angle_approx_137_5 :
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goldenAngleQ0.val ≥ 24902 ∧ goldenAngleQ0.val ≤ 25230 := by
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native_decide
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/-- Spiral radius raw value is always nonnegative. -/
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theorem spiral_radius_nonnegative (idx : Nat) :
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(genusToSpiral (genusToParameterSpace 1) idx).radius.val ≥ 0 := by
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exact UInt16.zero_le
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/-- Navigator step count increments by 1 on each advance. -/
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theorem advance_increments_step_count (nav : GenusSpiralNavigator) :
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(advanceNavigator nav).stepCount = nav.stepCount + 1 := by
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rfl
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end Semantics.TopologyGoldenSpiral
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