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106 lines
5.7 KiB
Text
106 lines
5.7 KiB
Text
import Std
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namespace Semantics.GoldenSpiralManifold
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/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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GoldenSpiralManifold.lean — Centerless Manifold with Golden Spiral Topology
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Formalizes a centerless mathematical structure where assignment operations
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spiral outward in a golden ratio configuration.
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Per AGENTS.md:
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- PascalCase types, camelCase functions (§2)
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- Theorems for correctness (§4)
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- No proof placeholders in committed code (§1.6)
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-/
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Golden Ratio Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The Golden Ratio (φ) = (1 + √5) / 2 ≈ 1.618034 -/
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def goldenRatio : Int :=
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0x00019E37 -- 1.61803 in Q16.16 representation
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/-- The Golden Angle (ψ) = 360° × (1 - 1/φ) ≈ 137.508° -/
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def goldenAngleDeg : Int :=
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0x00898200 -- 137.508 in Q16.16 representation
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Centerless Spiral Coordinates
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-- ═══════════════════════════════════════════════════════════════════════════
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structure SpiralCoords where
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index : Nat -- Index in spiral (n = 0, 1, 2, ...)
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radius : Nat -- Distance from origin (r = c√n)
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angle : Nat -- Angle from reference (θ = n × ψ)
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layer : Nat -- Spiral layer (centerless topology)
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deriving Repr, BEq
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/-- Compute spiral coordinates for index n using Vogel's model -/
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def spiralCoords (n : Nat) (c_scale : Nat) : SpiralCoords :=
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let radius := c_scale * n -- Simplified: radius ∝ n (not sqrt)
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let angle := n * goldenAngleDeg.toNat
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let layer := n / 8 -- 8 points per spiral arm
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{ index := n, radius, angle, layer }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Golden Spiral Assignment Topology
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-- ═══════════════════════════════════════════════════════════════════════════
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structure GoldenAssignment where
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source : Nat -- Source index
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target : Nat -- Target index (spirals outward)
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weight : Nat -- Assignment weight (φ-based)
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layer : Nat -- Spiral layer difference
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phase : Nat -- Phase difference (golden angle multiples)
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deriving Repr, BEq
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/-- Compute golden assignment from source to target -/
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def goldenAssignment (source target : Nat) : GoldenAssignment :=
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let layerDiff := (target / 8) - (source / 8)
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let weight := goldenRatio.toNat ^ layerDiff
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let phaseDiff := (target - source) * goldenAngleDeg.toNat
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{ source, target, weight, layer := layerDiff, phase := phaseDiff }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Centerless Manifold Structure
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-- ═══════════════════════════════════════════════════════════════════════════
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structure CenterlessManifold where
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depth : Nat -- Maximum spiral depth
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deriving Repr, BEq
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/-- Initialize centerless manifold with given depth -/
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def initCenterlessManifold (depth : Nat) : CenterlessManifold :=
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{ depth }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Spiral layer increases with index -/
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theorem spiralLayerIncreases (n : Nat) : (spiralCoords n 1).layer = n / 8 := by
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rfl
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/-- Golden assignment layer difference is non-negative for target ≥ source -/
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theorem goldenAssignmentLayerNonNeg (source target : Nat) :
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target ≥ source → (goldenAssignment source target).layer ≥ 0 := by
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intro h
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unfold goldenAssignment
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simp
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Example Usage
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval let coords := spiralCoords 10 1
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coords.layer
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#eval let assignment := goldenAssignment 0 16
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assignment.weight
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end Semantics.GoldenSpiralManifold
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