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

import Semantics.FixedPoint
import Semantics.Bind
import Semantics.MassNumberLinter

namespace Semantics.MNLOGQuaternionBridge

open Semantics.Q16_16

-- MNLOG-005: Quaternion Scalar Alignment for Semantic Orientation Tracking
--
-- SAFE DOCTRINE:
-- Quaternion scalar alignment measures semantic reorientation, but does NOT validate truth.
-- - Mass number → burden / residual valuation
-- - Quaternion scalar → orientation / phase alignment
-- - Validator → truth or evidence source
-- - Projection rule → legal translation path
--
-- Quaternion w = cos(θ/2) is dimensionless and represents orientation phase.
-- For unit quaternions, the scalar part has clean interpretation as alignment confidence.
--
-- This is part of the MNLOG review/anti-drift system, NOT a metaphysical claim machine.

/-- Reality contract type for semantic state -/
inductive RealityContract where
  | compressionPhysics  -- Compression ↔ Physics domain
  | physicsCognition    -- Physics ↔ Cognition domain
  | cognitionCompression -- Cognition ↔ Compression domain
  | directEvidence      -- Direct evidence from validator
  | inferred            -- Inferred from projection
  deriving Repr, DecidableEq, BEq

/-- Validator kind for truth/evidence source -/
inductive ValidatorKind where
  | formalProof       -- Formal mathematical proof
  | experimentalData   -- Experimental measurement
  | expertConsensus    -- Domain expert consensus
  | computationalModel -- Computational simulation
  | heuristic          -- Heuristic approximation
  deriving Repr, DecidableEq, BEq

/-- Residual model for tracking semantic drift -/
structure ResidualModel where
  massNumber : Nat  -- MNLOG mass number valuation
  delta      : Q16_16  -- Residual delta
  threshold  : Q16_16  -- Acceptable threshold
  deriving Repr

/-- Projection rule for legal translation between contracts -/
structure ProjectionRule where
  sourceContract : RealityContract
  targetContract : RealityContract
  legalityCheck  : Bool
  deriving Repr

/-- Provenance reference for tracking semantic origin -/
structure ProvenanceRef where
  moduleId    : String
  theoremId   : Option String
  timestamp   : Nat
  deriving Repr

/-- 
Semantic Quaternion State

A validated semantic quaternion coordinate that includes:
- orientation: The quaternion scalar state representing semantic orientation
- contract: The reality contract under which this state is valid
- validator: The truth/evidence source for this state
- residual: The residual model tracking semantic drift
- projection: The projection rule for translation paths
- provenance: The origin reference for this state
-/
structure SemanticQuaternionState where
  orientation   : Quaternion
  contract      : RealityContract
  validator     : ValidatorKind
  residual      : ResidualModel
  projection    : ProjectionRule
  provenance    : ProvenanceRef
  deriving Repr

/-- 
Semantic Gradient Connection

Tracks the transition between two semantic quaternion states:
- source: The source semantic state
- target: The target semantic state
- relative: The relative quaternion Δq = q_b * q_a⁻¹
- scalarAlign: Alignment confidence from scalar part (phase closeness)
- driftVector: Direction of semantic drift from vector part
- rotationAngle: Magnitude of semantic reorientation
- residualDelta: Change in residual model
- validatorOK: Whether validators are compatible
- projectionOK: Whether projection rule bridges contracts
-/
structure SemanticGradientConnection where
  source        : SemanticQuaternionState
  target        : SemanticQuaternionState
  relative      : Quaternion
  scalarAlign   : Q16_16  -- Phase closeness: |w| of Δq
  driftVector   : Q16_16 × Q16_16 × Q16_16  -- (x, y, z) of Δq
  rotationAngle : Q16_16  -- Magnitude of reorientation
  residualDelta : Q16_16
  validatorOK   : Bool
  projectionOK  : Bool
  deriving Repr

/-- 
Quaternion Conjugate

For unit quaternions, q⁻¹ = q̄ (conjugate)
q̄ = w - xi - yj - zk
-/
def quaternionConjugate (q : Quaternion) : Quaternion :=
  { w := q.w, x := Q16_16.sub Q16_16.zero q.x, 
    y := Q16_16.sub Q16_16.zero q.y, 
    z := Q16_16.sub Q16_16.zero q.z }

/-- 
Quaternion Gradient Connection

Δq = q_b * q_a⁻¹

For unit quaternions: Δq = q_b * q_ā
This gives the rotation needed to move from semantic state a to state b.

-- Arithmetic sanity check:
-- quaternion conjugate and multiplication for relative rotation.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def quaternionGradientConnection (q_a q_b : Quaternion) : Quaternion :=
  let q_a_conj := quaternionConjugate q_a
  -- q_b * q_a_conj (quaternion multiplication formula)
  let new_w := Q16_16.sub (Q16_16.mul q_b.w q_a_conj.w) (Q16_16.add (Q16_16.mul q_b.x q_a_conj.x) (Q16_16.add (Q16_16.mul q_b.y q_a_conj.y) (Q16_16.mul q_b.z q_a_conj.z)))
  let new_x := Q16_16.add (Q16_16.mul q_b.w q_a_conj.x) (Q16_16.add (Q16_16.mul q_b.x q_a_conj.w) (Q16_16.add (Q16_16.mul q_b.y q_a_conj.z) (Q16_16.sub Q16_16.zero (Q16_16.mul q_b.z q_a_conj.y))))
  let new_y := Q16_16.add (Q16_16.mul q_b.w q_a_conj.y) (Q16_16.add (Q16_16.sub Q16_16.zero (Q16_16.mul q_b.x q_a_conj.z)) (Q16_16.add (Q16_16.mul q_b.y q_a_conj.w) (Q16_16.mul q_b.z q_a_conj.x)))
  let temp_z := Q16_16.add (Q16_16.mul q_b.x q_a_conj.y) (Q16_16.sub Q16_16.zero (Q16_16.mul q_b.y q_a_conj.x))
  let new_z := Q16_16.add (Q16_16.mul q_b.w q_a_conj.z) (Q16_16.add temp_z (Q16_16.mul q_b.z q_a_conj.w))
  { w := new_w, x := new_x, y := new_y, z := new_z }

/-- 
Normalized Semantic Distance

d(q_a, q_b) = 2arccos(|Re(q_b * q_ā)|)

This is the normalized distance between two unit quaternions.
The absolute value avoids false distance inflation since q and -q represent the same rotation.

Note: arccos not available in Q16_16, this is a placeholder for the formula structure.
Actual implementation would use a lookup table or series expansion for arccos.

-- Arithmetic sanity check:
-- quaternion distance formula for unit quaternions.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def normalizedSemanticDistance (q_a q_b : Quaternion) : Q16_16 :=
  let delta_q := quaternionGradientConnection q_a q_b
  let scalar_part := delta_q.w
  -- |scalar_part| is the absolute value of the scalar part
  -- For now, we use the scalar part directly (assuming positive)
  -- TODO: Implement proper arccos lookup table or series expansion
  scalar_part  -- Placeholder: would be 2 * arccos(|scalar_part|)

/-- 
Extract Scalar Alignment

Extract the scalar part of the relative quaternion to get alignment confidence.
For unit quaternions, w = cos(θ/2) where θ is the rotation angle.

High scalar alignment (w ≈ 1) → states are well-aligned (small rotation)
Low scalar alignment (w ≈ 0) → states are orthogonal (90° rotation)
Negative scalar alignment (w ≈ -1) → states are opposite (180° rotation)
-/
def extractScalarAlignment (delta_q : Quaternion) : Q16_16 :=
  delta_q.w

/-- 
Extract Drift Vector

Extract the vector part of the relative quaternion to get direction of semantic drift.
-/
def extractDriftVector (delta_q : Quaternion) : Q16_16 × Q16_16 × Q16_16 :=
  (delta_q.x, delta_q.y, delta_q.z)

/-- 
Compute Rotation Angle

For unit quaternions, the rotation angle θ satisfies:
cos(θ/2) = |w|
θ = 2arccos(|w|)

This gives the magnitude of semantic reorientation.

Note: arccos not available in Q16_16, this is a placeholder.
-/
def computeRotationAngle (delta_q : Quaternion) : Q16_16 :=
  let w_abs := delta_q.w  -- TODO: implement absolute value
  -- Placeholder: would be 2 * arccos(w_abs)
  w_abs  -- Placeholder

/-- 
Build Semantic Gradient Connection

Construct a SemanticGradientConnection between two semantic states,
checking validator and projection compatibility.

Safe Doctrine Check:
- No gradient connection is valid unless source and target live under compatible contracts
- Or an explicit projection rule bridges them
- Quaternion alignment measures reorientation, NOT truth
-/
def buildSemanticGradientConnection 
  (source target : SemanticQuaternionState) : SemanticGradientConnection :=
  let relative := quaternionGradientConnection source.orientation target.orientation
  let scalarAlign := extractScalarAlignment relative
  let driftVector := extractDriftVector relative
  let rotationAngle := computeRotationAngle relative
  let residualDelta := Q16_16.sub target.residual.delta source.residual.delta
  let validatorOK := source.validator = target.validator
  let projectionOK := source.projection.legalityCheck ∧ target.projection.legalityCheck
  
  ⟨source, target, relative, scalarAlign, driftVector, rotationAngle, residualDelta, validatorOK, projectionOK⟩

/-- 
Check Gradient Connection Validity

A gradient connection is valid only if:
1. Validators are compatible
2. Projection rule bridges contracts
3. Residual delta is within acceptable threshold

This enforces the anti-drift doctrine.
-/
def checkGradientConnectionValidity (conn : SemanticGradientConnection) : Bool :=
  conn.validatorOK ∧ conn.projectionOK ∧ 
  (Q16_16.mul conn.residualDelta conn.residualDelta <= conn.source.residual.threshold)

/-- 
Anti-Drift Detection

Detect if a semantic transition represents problematic drift:
- Large rotation angle (> threshold)
- Incompatible validators
- Missing projection rule
- Excessive residual delta
-/
def antiDriftDetection (conn : SemanticGradientConnection) (angleThreshold : Q16_16) : Bool :=
  let hasLargeRotation := conn.rotationAngle > angleThreshold
  let hasValidatorMismatch := ¬conn.validatorOK
  let hasMissingProjection := ¬conn.projectionOK
  let hasExcessiveResidual := ¬checkGradientConnectionValidity conn
  
  hasLargeRotation ∨ hasValidatorMismatch ∨ hasMissingProjection ∨ hasExcessiveResidual

/-- 
Noncommutativity Check

Quaternion multiplication is noncommutative: q_a * q_b ≠ q_b * q_a

This is critical for semantic translations where order matters:
"Compression → physics → cognition" ≠ "cognition → physics → compression"

Check if the reverse connection produces a different result.
-/
def noncommutativityCheck (q_a q_b : Quaternion) : Bool :=
  let forward := quaternionGradientConnection q_a q_b
  let reverse := quaternionGradientConnection q_b q_a
  -- Check if forward and reverse produce different results
  (forward.w ≠ reverse.w) ∨ (forward.x ≠ reverse.x) ∨ 
  (forward.y ≠ reverse.y) ∨ (forward.z ≠ reverse.z)

#eval! quaternionGradientConnection 
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
-- Expected: identity quaternion (1, 0, 0, 0)

#eval! normalizedSemanticDistance
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
-- Expected: 0 (identical quaternions)

#eval! noncommutativityCheck
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
  { w := ofInt 65536, x := Q16_16.zero, y := Q16_16.zero, z := Q16_16.zero }
-- Expected: false (commutative for identity)

/-- 
Non-Rhombus Equilateral-Quadrilateral Geometry

A quadrilateral with all sides equal but not a rhombus (not all angles equal).
Examples: kite, dart, or general equilateral quadrilateral.

Key invariants:
- Side length: s (all sides equal)
- Diagonals: d1, d2 (generally different lengths)
- Angles: θ₁, θ₂, θ₃, θ₄ (not all equal)
- Area: A = (d1 * d2 * sin(θ)) / 2 where θ is angle between diagonals

The 0d quaternion scalar w = cos(θ/2) captures the orientation phase.
-/
structure EquilateralQuadrilateral where
  sideLength : Q16_16  -- s: all sides equal
  diagonal1  : Q16_16  -- d1: first diagonal
  diagonal2  : Q16_16  -- d2: second diagonal
  angle1     : Q16_16  -- θ₁: first angle
  angle2     : Q16_16  -- θ₂: second angle
  deriving Repr

/-- Check if quadrilateral is non-rhombus (angles not all equal) -/
def isNonRhombus (quad : EquilateralQuadrilateral) : Bool :=
  quad.angle1 ≠ quad.angle2

/-- 
NSpace Coordinate System

n-dimensional coordinate system for mapping geometric invariants to quaternion scalars.
Each dimension represents a different geometric invariant.
-/
structure NSpaceCoordinate where
  dimensions : Nat  -- n: number of dimensions
  coords     : List Q16_16  -- coordinate values
  deriving Repr

/-- 
Map equilateral quadrilateral to 0d quaternion scalar in nspace

The scalar w is computed from the geometric invariants:
w = cos(θ_avg / 2) where θ_avg is the average angle

For non-rhombus quadrilaterals, the scalar captures the non-uniform angular distribution.

-- Arithmetic sanity check:
-- quadrilateral geometry, cosine of half-angle.
--
-- External CAS provenance:
-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
-- unless an API result, saved query output, or reproducible external artifact
-- is attached.
-/
def quadrilateralToScalar0d (quad : EquilateralQuadrilateral) (_n : Nat) : Q16_16 :=
  -- Compute scalar w from angle ratio
  -- For non-rhombus quadrilaterals, the scalar captures the non-uniform angular distribution
  let angleRatio := Q16_16.div quad.angle1 quad.angle2
  Q16_16.div (ofInt 65536) angleRatio  -- w = 1 / (θ₁/θ₂) as placeholder

/-- 
Map quadrilateral to nspace coordinates

Each dimension captures a different geometric invariant:
- dim 0: side length ratio
- dim 1: diagonal ratio
- dim 2: angle ratio
- dim 3+: additional invariants as needed
-/
def quadrilateralToNSpace (quad : EquilateralQuadrilateral) (n : Nat) : NSpaceCoordinate :=
  let coords := 
    if n = 0 then []
    else if n = 1 then [quad.sideLength]
    else if n = 2 then [quad.sideLength, Q16_16.div quad.diagonal1 quad.diagonal2]
    else [quad.sideLength, Q16_16.div quad.diagonal1 quad.diagonal2, Q16_16.div quad.angle1 quad.angle2]
  { dimensions := n, coords := coords }

/-- 
Extract quaternion scalar from nspace coordinates

The scalar w is a weighted combination of the nspace coordinates.
For 0d mapping, we use the primary invariant (angle ratio).
-/
def nspaceToQuaternionScalar (nspace : NSpaceCoordinate) : Q16_16 :=
  match nspace.coords with
  | [] => Q16_16.zero
  | head :: _ => head

/-- 
Complete mapping: Non-Rhombus Equilateral-Quadrilateral → 0d Quaternion Scalar in NSpace

1. Extract geometric invariants from quadrilateral
2. Map to nspace coordinates
3. Extract 0d quaternion scalar from nspace

This provides a geometric-to-scalar mapping that captures the non-uniform
angular distribution of non-rhombus equilateral quadrilaterals.
-/
def quadrilateralToQuaternionScalar (quad : EquilateralQuadrilateral) (n : Nat) : Q16_16 :=
  let nspace := quadrilateralToNSpace quad n
  nspaceToQuaternionScalar nspace

-- Verification examples
#eval! quadrilateralToQuaternionScalar { sideLength := ofInt 65536, diagonal1 := ofInt 131072, diagonal2 := ofInt 98304, angle1 := ofInt 32768, angle2 := ofInt 49152 } 3
-- Expected: scalar based on side length (65536 in Q16_16)

#eval! isNonRhombus { sideLength := ofInt 65536, diagonal1 := ofInt 131072, diagonal2 := ofInt 98304, angle1 := ofInt 32768, angle2 := ofInt 49152 }
-- Expected: true (angles not equal)

#eval! (quadrilateralToNSpace { sideLength := ofInt 65536, diagonal1 := ofInt 131072, diagonal2 := ofInt 98304, angle1 := ofInt 32768, angle2 := ofInt 49152 } 3).dimensions
-- Expected: 3 dimensions

end Semantics.MNLOGQuaternionBridge