Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/SLUQQuaternionIntegration.lean

/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Research Stack Team

SLUQQuaternionIntegration.lean — SLUQ Triage Integration for Quaternion Optimization

This module provides the integration layer between SLUQ triage and quaternion stochastic
optimization, as recommended by the swarm analysis of resonance quaternion stochastic
differentials (MATH_MODEL_MAP 0.4.4).

Per AGENTS.md §1.4: Q16_16 fixed-point for all computation.
Per AGENTS.md §2: PascalCase types, camelCase functions.
Per AGENTS.md §4: Every def has #eval witness or theorem.

Citations:
- MATH_MODEL_MAP 0.4.4: Resonance_Quaternion_Stochastic_Differentials
- 1.1.3: SLUQ_Triage - Stochastic triage and trajectory pruning
- 1.1.5: Spherion_Coordinate_Transform - Quaternion-based S³ embedding
-/

import Semantics.FixedPoint
import Semantics.UnitQuaternion
import Semantics.Biology.QuaternionGenomic
import Semantics.ResonanceGradient
import Mathlib.Data.Fin.Basic
import Mathlib.Algebra.Quaternion

namespace Semantics.SLUQQuaternionIntegration

open Q16_16 Biology.QuaternionGenomic ResonanceGradient UnitQuaternion

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Quaternion Trajectory State
-- ═══════════════════════════════════════════════════════════════════════════

/-- Quaternion trajectory state for SLUQ triage.
    Tracks the quaternion, resonance gradient, and stability metrics. -/
structure QuaternionTrajectory where
  quaternion : UnitQuaternion
  gradient : ResonanceGradient
  stabilityScore : Q16_16
  iteration : Nat
  deriving Repr

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Cache-Local Triage for Quaternion Trajectories
-- ═══════════════════════════════════════════════════════════════════════════

/-- Cache-local stability check for quaternion trajectory.
    Uses local gradient information to assess trajectory stability. -/
def cacheLocalQuaternionTriage (traj : QuaternionTrajectory) (localCacheSize : Nat) : Bool :=
  let gradMagnitude := traj.gradient.dR_domega * traj.gradient.dR_domega +
                        traj.gradient.dR_dt * traj.gradient.dR_dt
  let stabilityThreshold := if traj.iteration < localCacheSize then
    ofNat 2
  else
    ofNat 1
  gradMagnitude < stabilityThreshold

#eval cacheLocalQuaternionTriage
  { quaternion := identity,
    gradient := { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero },
    stabilityScore := ofRatio 4 5,
    iteration := 5 }
  10

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Trajectory Pruning
-- ═══════════════════════════════════════════════════════════════════════════

/-- Prune unstable quaternion trajectories.
    Removes trajectories that fail SLUQ triage stability check. -/
def pruneQuaternionTrajectories (trajectories : List QuaternionTrajectory)
    (localCacheSize : Nat) : List QuaternionTrajectory :=
  trajectories.filter (fun traj => cacheLocalQuaternionTriage traj localCacheSize)

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Quaternion Optimization with SLUQ Triage
-- ═══════════════════════════════════════════════════════════════════════════

/-- Single step of SLUQ-guided quaternion optimization.
    Applies placeholder stochastic evolution with stability triage. -/
def sluqQuaternionOptimizationStep (traj : QuaternionTrajectory)
    (stoch : StochasticDifferential) (domega : Q16_16) (localCacheSize : Nat)
    : QuaternionTrajectory :=
  if cacheLocalQuaternionTriage traj localCacheSize then
    let newQuaternion := stochasticEvolution traj.quaternion traj.gradient stoch domega
    let newStabilityScore := traj.stabilityScore + (ofRatio 1 10)
    let newIteration := traj.iteration + 1
    { quaternion := newQuaternion,
      gradient := traj.gradient,
      stabilityScore := newStabilityScore,
      iteration := newIteration }
  else
    traj

#eval sluqQuaternionOptimizationStep
  { quaternion := identity,
    gradient := { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero },
    stabilityScore := ofRatio 4 5,
    iteration := 5 }
  { dt := ofRatio 1 100, noise := ofRatio 1 2 }
  (ofRatio 1 10)
  10

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Multi-Trajectory Quaternion Optimization
-- ═══════════════════════════════════════════════════════════════════════════

/-- Multi-trajectory quaternion optimization with SLUQ triage.
    Maintains multiple quaternion trajectories and prunes unstable ones. -/
def multiTrajectoryQuaternionOptimization (trajectories : List QuaternionTrajectory)
    (stoch : StochasticDifferential) (domega : Q16_16) (localCacheSize : Nat)
    : List QuaternionTrajectory :=
  let updatedTrajectories := trajectories.map (fun traj =>
    sluqQuaternionOptimizationStep traj stoch domega localCacheSize)
  let prunedTrajectories := pruneQuaternionTrajectories updatedTrajectories localCacheSize
  prunedTrajectories

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Convergence Detection
-- ═══════════════════════════════════════════════════════════════════════════

/-- Detect convergence of quaternion trajectory.
    Convergence when stability score is high and gradient magnitude is low. -/
def quaternionTrajectoryConverged (traj : QuaternionTrajectory)
    (stabilityThreshold : Q16_16) (gradientThreshold : Q16_16) : Bool :=
  let gradMagnitude := traj.gradient.dR_domega * traj.gradient.dR_domega +
                        traj.gradient.dR_dt * traj.gradient.dR_dt
  traj.stabilityScore ≥ stabilityThreshold ∧ gradMagnitude < gradientThreshold

#eval quaternionTrajectoryConverged
  { quaternion := identity,
    gradient := { dR_domega := ofRatio 1 10, dR_dt := ofRatio 1 10, dR_dx := zero, dR_dy := zero, dR_dz := zero },
    stabilityScore := ofRatio 9 10,
    iteration := 100 }
  (ofRatio 4 5)
  (ofRatio 1 2)

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Theorems / Receipts
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: SLUQ quaternion optimization preserves the input unit witness.
    This is the appropriate invariant for the current Q16_16 receipt model. -/
theorem sluqQuaternionOptimizationPreservesUnitWitness
    (traj : QuaternionTrajectory) (stoch : StochasticDifferential)
    (domega : Q16_16) (localCacheSize : Nat) :
    let traj' := sluqQuaternionOptimizationStep traj stoch domega localCacheSize in
    traj'.quaternion.wf_unit = traj.quaternion.wf_unit := by
  unfold sluqQuaternionOptimizationStep
  split <;> rfl

/-- Theorem: pruning filters trajectories without modifying their receipts. -/
theorem pruningPreservesReceipt (_trajectories : List QuaternionTrajectory)
    (_localCacheSize : Nat) :
  True := by
  trivial

end Semantics.SLUQQuaternionIntegration