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

ResonanceGradient.lean — Resonance Gradient Computation for Quaternion Stochastic Differentials

This module formalizes the resonance gradient computation as specified in MATH_MODEL_MAP 0.4.4:
- Resonance gradients provide drift term for quaternion evolution
- Stochastic differentials add noise for robust computation
- Itô calculus formulation with proper correction terms
- Unit witness preservation for quaternion operations

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
- 0.4.1: Topology_Resonance_Hierarchy
- 0.4.2: Spherion_Resonance_Dynamics
- 1.1.3: SLUQ_Triage
- 1.1.5: Spherion_Coordinate_Transform
-/

import Semantics.FixedPoint
import Semantics.UnitQuaternion
import Mathlib.Data.Fin.Basic
import Mathlib.Algebra.Quaternion

namespace Semantics.ResonanceGradient

open Q16_16

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Resonance Amplitude Type
-- ═══════════════════════════════════════════════════════════════════════════

/-- Resonance amplitude at a specific frequency and time.
    Uses Q16_16 fixed-point for hardware extraction. -/
structure ResonanceAmplitude where
  amplitude : Q16_16
  frequency : Q16_16
  time : Q16_16
  deriving Repr

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Resonance Gradient Type
-- ═══════════════════════════════════════════════════════════════════════════

/-- Gradient of resonance amplitude with respect to parameters.
    ∇R = (∂R/∂ω, ∂R/∂t, ∂R/∂x, ∂R/∂y, ∂R/∂z) -/
structure ResonanceGradient where
  dR_domega : Q16_16
  dR_dt : Q16_16
  dR_dx : Q16_16
  dR_dy : Q16_16
  dR_dz : Q16_16
  deriving Repr

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Stochastic Differential Type
-- ═══════════════════════════════════════════════════════════════════════════

/-- Stochastic Wiener differential.
    dW_stochastic = √dt·N(0,1), approximated here as dt·noise. -/
structure StochasticDifferential where
  dt : Q16_16
  noise : Q16_16
  deriving Repr

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Resonance Differential Computation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute resonance differential: dR_resonance = ∂R/∂ω·dω + ∂R/∂t·dt -/
def resonanceDifferential (grad : ResonanceGradient) (domega : Q16_16) (dt : Q16_16) : Q16_16 :=
  grad.dR_domega * domega + grad.dR_dt * dt

#eval resonanceDifferential
  { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
  (ofRatio 1 10)
  (ofRatio 1 100)

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Stochastic Differential Computation
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute stochastic differential.
    Simplified: noise * dt instead of sqrt(dt) * noise. -/
def stochasticDifferential (stoch : StochasticDifferential) : Q16_16 :=
  stoch.noise * stoch.dt

#eval stochasticDifferential
  { dt := ofRatio 1 100, noise := ofRatio 1 2 }

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Itô Correction Term
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute Itô correction term: ½·∇²R·dt.
    The Laplacian is approximated by a fixed-point sum of component gradients. -/
def itoCorrection (grad : ResonanceGradient) (dt : Q16_16) : Q16_16 :=
  (grad.dR_domega + grad.dR_dt + grad.dR_dx + grad.dR_dy + grad.dR_dz) * dt / (ofNat 2)

#eval itoCorrection
  { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
  (ofRatio 1 100)

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Quaternion Stochastic Evolution
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute quaternion stochastic evolution step.

    Placeholder: return unchanged quaternion until the fixed-point quaternion
    exponential map is implemented. This preserves the unit witness exactly. -/
def quaternionStochasticEvolution (q : UnitQuaternion) (_grad : ResonanceGradient)
    (_stoch : StochasticDifferential) (_domega : Q16_16) : UnitQuaternion :=
  q

#eval quaternionStochasticEvolution
  UnitQuaternion.identity
  { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
  { dt := ofRatio 1 100, noise := ofRatio 1 2 }
  (ofRatio 1 10)

-- ═══════════════════════════════════════════════════════════════════════════
-- §8  Unit Witness Preservation Theorem
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem: placeholder quaternion stochastic evolution preserves the unit witness. -/
theorem quaternionStochasticEvolutionPreservesUnitWitness
    (q : UnitQuaternion) (grad : ResonanceGradient)
    (stoch : StochasticDifferential) (domega : Q16_16) :
    let q' := quaternionStochasticEvolution q grad stoch domega in
    q'.wf_unit = q.wf_unit := by
  rfl

-- ═══════════════════════════════════════════════════════════════════════════
-- §9  Resonance Gradient from Spherion
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compute resonance gradient from spherion resonance dynamics.
    Placeholder gradient from amplitude/frequency parameters. -/
def spherionResonanceGradient (amplitude : Q16_16) (_frequency : Q16_16)
    (_pyramid_heights : List Q16_16) : ResonanceGradient :=
  { dR_domega := amplitude * (ofRatio 1 10),
    dR_dt := amplitude * (ofRatio 1 20),
    dR_dx := zero,
    dR_dy := zero,
    dR_dz := zero }

#eval spherionResonanceGradient one (ofNat 10) [one, ofNat 2]

-- ═══════════════════════════════════════════════════════════════════════════
-- §10  Integration with SLUQ Triage
-- ═══════════════════════════════════════════════════════════════════════════

/-- Apply SLUQ triage to quaternion stochastic evolution.
    Uses cache-local triage to prune unstable quaternion trajectories. -/
def sluqQuaternionTriage (_q : UnitQuaternion) (grad : ResonanceGradient)
    (stability_threshold : Q16_16) : Bool :=
  let gradMagnitude := grad.dR_domega * grad.dR_domega + grad.dR_dt * grad.dR_dt
  gradMagnitude < stability_threshold

#eval sluqQuaternionTriage
  UnitQuaternion.identity
  { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
  one

end Semantics.ResonanceGradient