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Semantics: modularize quaternion, resolve circular dependencies, and eliminate float operations
This commit is contained in:
Brandon Schneider
parent
ec4166edb2
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0b936c8e39
6 changed files with 249 additions and 287 deletions
3
0-Core-Formalism/lean/Semantics/Biology.lean
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3
0-Core-Formalism/lean/Semantics/Biology.lean
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@ -0,0 +1,3 @@
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import Semantics.Biology.BioRxivFormalization
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import Semantics.Biology.QuaternionGenomic
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import Semantics.Biology.RGFlowBioinformatics
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@ -24,6 +24,7 @@ import Semantics.FixedPoint
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import Semantics.SLUG3
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import Semantics.GenomicCompression
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import Semantics.ResonanceGradient
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import Semantics.UnitQuaternion
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import Mathlib.Data.Fin.Basic
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import Mathlib.Algebra.Quaternion
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@ -31,238 +32,6 @@ namespace Semantics.QuaternionGenomic
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open Q16_16 SLUG3 GenomicCompression
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Unit Quaternion Type (S³ Embedding)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Unit quaternion representing a point on the 3-sphere S³.
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Stored as (w, x, y, z) with constraint w² + x² + y² + z² = 1.
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Uses Q16_16 fixed-point for hardware extraction. -/
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structure UnitQuaternion where
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w : Q16_16 -- scalar (real) part
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x : Q16_16 -- i component
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y : Q16_16 -- j component
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z : Q16_16 -- k component
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wf_unit : w * w + x * x + y * y + z * z = one -- unit norm constraint
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Trigonometry Placeholders (for Q16_16)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Cosine lookup for Q16_16 (placeholder: use CORDIC or polynomial approx).
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Input: angle in radians, scaled to Q16_16.
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Output: cos(angle) in Q16_16 ∈ [-1.0, 1.0]. -/
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def cos (x : Q16_16) : Q16_16 :=
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-- Placeholder: polynomial approximation or LUT
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-- Full implementation: Chebyshev polynomial or CORDIC
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q16_one - (x * x) / ofNat 2 -- Taylor: 1 - x²/2
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/-- Sine lookup for Q16_16 (placeholder: use CORDIC or polynomial approx).
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Input: angle in radians, scaled to Q16_16.
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Output: sin(angle) in Q16_16 ∈ [-1.0, 1.0]. -/
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def sin (x : Q16_16) : Q16_16 :=
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-- Placeholder: Taylor series approximation
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x - (x * x * x) / ofNat 6 -- Taylor: x - x³/6
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/-- Arccosine lookup for Q16_16 (placeholder: use inverse trig LUT).
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Input: value in [-1.0, 1.0], scaled to Q16_16.
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Output: arccos(value) in radians [0, π], scaled to Q16_16. -/
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def acos (x : Q16_16) : Q16_16 :=
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-- Placeholder: linear approximation
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-- Full implementation: use precomputed LUT or polynomial
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q16_one - x -- Approximation: arccos(x) ≈ 1 - x (for small angles)
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namespace UnitQuaternion
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/-- Identity quaternion (1, 0, 0, 0) - neutral element -/
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def identity : UnitQuaternion :=
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{ w := one
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x := zero
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y := zero
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z := zero
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wf_unit := by simp [one, zero] }
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/-- Quaternion multiplication (Hamilton product).
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For unit quaternions, product remains unit (S³ is a group under ×). -/
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def mul (a b : UnitQuaternion) : UnitQuaternion :=
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let w' := a.w * b.w - a.x * b.x - a.y * b.y - a.z * b.z
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let x' := a.w * b.x + a.x * b.w + a.y * b.z - a.z * b.y
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let y' := a.w * b.y - a.x * b.z + a.y * b.w + a.z * b.x
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let z' := a.w * b.z + a.x * b.y - a.y * b.x + a.z * b.w
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-- Note: wf_unit proof omitted for computational use
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-- In production: verify norm preservation
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{ w := w', x := x', y := y', z := z',
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wf_unit := by trivial }
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/-- Dot product as scalar part of a × b* (conjugate product).
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For unit quaternions: a · b = cos(θ) where θ = great circle distance. -/
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def dot (a b : UnitQuaternion) : Q16_16 :=
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a.w * b.w + a.x * b.x + a.y * b.y + a.z * b.z
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/-- Great circle distance on S³: arccos(a · b).
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For compression: distance ∈ [0, π] maps to dissimilarity metric. -/
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def distance (a b : UnitQuaternion) : Q16_16 :=
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-- arccos approximation via lookup table (hardware-efficient)
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-- Full implementation would use cordic or polynomial approximation
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let d := dot a b
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-- Map [-1, 1] to [0, π] using piecewise linear approx
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if d.val ≥ 0x00010000 then -- d ≥ 1.0
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zero -- distance = 0 (identical)
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else if d.val ≤ 0xFFFF0000 then -- d ≤ -1.0
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ofUInt32 0x0003243F -- π ≈ 3.14159 in Q16_16
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else
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-- Linear interpolation: distance ≈ arccos(d)
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-- Simplified: use precomputed LUT for arccos values
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q16_one - d -- Approximation: arccos(d) ≈ 1 - d for small angles
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/-- Quaternion conjugate: q* = [w, -x, -y, -z].
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For unit quaternions, q⁻¹ = q*. -/
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def conjugate (q : UnitQuaternion) : UnitQuaternion :=
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{ w := q.w, x := negQ q.x, y := negQ q.y, z := negQ q.z,
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wf_unit := by simp [one, q.wf_unit] }
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/-- Quaternion inverse: q⁻¹ = q* / ||q||².
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For unit quaternions, q⁻¹ = q* (conjugate). -/
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def inv (q : UnitQuaternion) : UnitQuaternion :=
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q.conjugate -- Unit quaternion: inverse = conjugate
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/-- Rotation of point p (pure quaternion [0, px, py, pz]) by unit quaternion q.
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Formula: p' = q · p · q⁻¹ (conjugation).
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Preserves vector norm: ||p'|| = ||p||. -/
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def rotateVector (q : UnitQuaternion) (v : Q16_16 × Q16_16 × Q16_16) : Q16_16 × Q16_16 × Q16_16 :=
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let (vx, vy, vz) := v
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-- Represent v as pure quaternion [0, vx, vy, vz]
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let p := { w := zero, x := vx, y := vy, z := vz, wf_unit := by trivial }
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-- Compute q · p · q⁻¹
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let rotated := (q.mul p).mul q.inv
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-- Extract vector part
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(rotated.x, rotated.y, rotated.z)
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/-- Construct unit quaternion from axis-angle representation.
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q = [cos(θ/2), sin(θ/2) · (ux, uy, uz)] where (ux,uy,uz) is unit axis.
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Standard robotics/computer graphics convention.
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--
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-- Arithmetic sanity check:
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-- quaternion from axis-angle formula.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def fromAxisAngle (axis : Q16_16 × Q16_16 × Q16_16) (angle : Q16_16) : UnitQuaternion :=
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let (ux, uy, uz) := axis
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-- Arithmetic sanity check:
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-- cos(θ/2) for quaternion rotation.
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let cosHalf := Q16_16.cos (angle / ofInt 2)
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-- Arithmetic sanity check:
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-- sin(θ/2) for quaternion rotation.
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let sinHalf := Q16_16.sin (angle / ofInt 2)
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-- Arithmetic sanity check:
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-- √(x² + y² + z²) for vector normalization.
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let norm := Q16_16.sqrt (ux * ux + uy * uy + uz * uz)
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let cosTheta := cosHalf
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let sinTheta := sinHalf * norm
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{ w := cosTheta, x := sinTheta * ux, y := sinTheta * uy, z := sinTheta * uz,
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wf_unit := by trivial }
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/-- Extract axis-angle from unit quaternion.
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Returns (axis, angle) where axis is unit vector and angle ∈ [0, 2π). -/
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def toAxisAngle (q : UnitQuaternion) : (Q16_16 × Q16_16 × Q16_16) × Q16_16 :=
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let angle := ofNat 2 * acos q.w -- θ = 2·arccos(w)
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let sinHalf := sin (angle / ofNat 2)
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let axis := if sinHalf.val > 0x00000100 then -- sin(θ/2) ≠ 0
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(q.x / sinHalf, q.y / sinHalf, q.z / sinHalf)
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else
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(one, zero, zero) -- Identity rotation: arbitrary axis
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(axis, angle)
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/-- Spherical Linear Interpolation (SLERP) between two unit quaternions.
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Formula: slerp(q1, q2, t) = sin((1-t)·Ω)/sin(Ω) · q1 + sin(t·Ω)/sin(Ω) · q2
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where Ω = arccos(q1 · q2) and t ∈ [0, 1].
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Used for smooth DNA backbone interpolation between nucleotide states. -/
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def slerp (a b : UnitQuaternion) (t : Q16_16) : UnitQuaternion :=
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let dotAB := a.dot b
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-- Ensure we take the shortest path (flip sign if dot < 0)
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let (b', dotAB') := if dotAB.val < 0x00008000 then
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({ b with w := negQ b.w, x := negQ b.x, y := negQ b.y, z := negQ b.z,
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wf_unit := b.wf_unit }, negQ dotAB)
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else
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(b, dotAB)
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let omega := acos dotAB' -- Angle between quaternions
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let sinOmega := sin omega
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if sinOmega.val < 0x00000100 then -- Quaternions nearly parallel
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-- Use linear interpolation (LERP) to avoid division by near-zero
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let w1 := one - t
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let w2 := t
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{ w := w1 * a.w + w2 * b'.w,
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x := w1 * a.x + w2 * b'.x,
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y := w1 * a.y + w2 * b'.y,
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z := w1 * a.z + w2 * b'.z,
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wf_unit := by trivial }
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else
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-- Full SLERP
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let w1 := sin ((one - t) * omega) / sinOmega
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let w2 := sin (t * omega) / sinOmega
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{ w := w1 * a.w + w2 * b'.w,
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x := w1 * a.x + w2 * b'.x,
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y := w1 * a.y + w2 * b'.y,
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z := w1 * a.z + w2 * b'.z,
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wf_unit := by trivial }
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/-- Convert unit quaternion to 3×3 rotation matrix (row-major).
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Matrix entries derived from Hamilton product algebra.
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Used for rendering DNA backbone in 3D visualization. -/
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def toRotationMatrix (q : UnitQuaternion) : Q16_16 × Q16_16 × Q16_16 ×
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Q16_16 × Q16_16 × Q16_16 ×
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Q16_16 × Q16_16 × Q16_16 :=
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let w := q.w; let x := q.x; let y := q.y; let z := q.z
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let two := ofNat 2
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-- First row
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let m00 := one - two * (y * y + z * z)
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let m01 := two * (x * y - z * w)
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let m02 := two * (x * z + y * w)
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-- Second row
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let m10 := two * (x * y + z * w)
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let m11 := one - two * (x * x + z * z)
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let m12 := two * (y * z - x * w)
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-- Third row
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let m20 := two * (x * z - y * w)
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let m21 := two * (y * z + x * w)
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let m22 := one - two * (x * x + y * y)
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(m00, m01, m02, m10, m11, m12, m20, m21, m22)
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/-- Check chiral compatibility: D+L→W collapse detection.
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Two quaternions are "incompatible" if their product has negative scalar part.
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This corresponds to right-hand vs left-hand chirality mismatch. -/
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def chiralIncompatible (a b : UnitQuaternion) : Bool :=
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let product := mul a b
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product.w.val < 0x00008000 -- scalar part < 0 (negative)
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/-- Ternary classification from quaternion dot product (SLUG-3 gate).
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Maps dot product threshold to ternary state:
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- dot ≥ threshold: high (compatible)
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- |dot| < threshold: mid (uncertain)
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- dot ≤ -threshold: low (incompatible/collapse)
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-/
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def toTernary (a b : UnitQuaternion) (threshold : Q16_16) : Ternary :=
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let d := dot a b
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if d ≥ threshold then
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Ternary.high
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else if d ≤ negQ threshold then
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Ternary.low
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else
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Ternary.mid
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end UnitQuaternion
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Nucleotide-to-Quaternion Mapping (Prime-Addressed)
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-- ═══════════════════════════════════════════════════════════════════════════
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@ -427,10 +196,7 @@ def primeIndexedQuaternion (primeIdx : Nat) (hPrime : Nat.Prime primeIdx) : Unit
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let axisZ := ofNat (primeIdx + 6)
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-- Normalize axis
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let norm := Float.sqrt (axisX.toFloat * axisX.toFloat +
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axisY.toFloat * axisY.toFloat +
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axisZ.toFloat * axisZ.toFloat)
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let n := ofFloat norm
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let n := Q16_16.sqrt (axisX * axisX + axisY * axisY + axisZ * axisZ)
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{ w := cos angle, -- Approximation: use lookup
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x := sin angle * axisX / n,
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@ -516,9 +282,9 @@ def stochasticEvolution (q : UnitQuaternion) (grad : ResonanceGradient.Resonance
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#eval stochasticEvolution
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identity
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{ 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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{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
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(toQ16_16 0.1)
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{ dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
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{ dt := ofRatio 1 100, noise := ofRatio 1 2 }
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(ofRatio 1 10)
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-- Expected: unchanged quaternion (placeholder)
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/-- Resonance-tuned quaternion rotation.
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@ -531,7 +297,7 @@ def resonanceTunedRotation (q : UnitQuaternion) (axis : Q16_16 × Q16_16 × Q16_
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let (ax, ay, az) := axis
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-- Gradient magnitude determines rotation angle
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let gradMagnitude := grad.dR_domega * grad.dR_domega + grad.dR_dt * grad.dR_dt
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let optimalAngle := gradMagnitude * (toQ16_16 0.5) -- Simplified scaling
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let optimalAngle := gradMagnitude * (ofRatio 1 2) -- Simplified scaling
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-- Apply rotation with optimal angle
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fromAxisAngle axis optimalAngle
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@ -539,7 +305,7 @@ def resonanceTunedRotation (q : UnitQuaternion) (axis : Q16_16 × Q16_16 × Q16_
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#eval resonanceTunedRotation
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identity
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(one, zero, zero)
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{ 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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{ dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
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-- Expected: rotation around x-axis with angle proportional to gradient magnitude
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/-- Stochastic quaternion optimization.
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@ -552,16 +318,16 @@ def stochasticQuaternionOptimization (q : UnitQuaternion) (grad : ResonanceGradi
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-- Check stability via SLUQ triage
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if ResonanceGradient.sluqQuaternionTriage q grad stabilityThreshold then
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-- Stable: apply stochastic evolution
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stochasticEvolution q grad stoch (toQ16_16 0.1)
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stochasticEvolution q grad stoch (ofRatio 1 10)
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else
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-- Unstable: skip update (prune trajectory)
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q
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#eval stochasticQuaternionOptimization
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identity
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{ 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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{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
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(toQ16_16 1.0)
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{ dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
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{ dt := ofRatio 1 100, noise := ofRatio 1 2 }
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one
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-- Expected: unchanged quaternion (stable check passes, but evolution is placeholder)
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/-- Theorem: Stochastic quaternion evolution preserves unit norm.
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@ -574,7 +340,7 @@ theorem stochasticEvolutionPreservesUnitNorm
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q'.w * q'.w + q'.x * q'.x + q'.y * q'.y + q'.z * q'.z = one := by
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-- Proof: Quaternion exponential map preserves unit norm
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-- Stochastic increment is applied via rotation, which preserves norm
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trivial
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sorry
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/-- Theorem: Resonance-tuned rotation preserves unit norm.
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Axis-angle representation always produces unit quaternions. -/
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@ -14,7 +14,6 @@ This module formalizes:
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import Semantics.FixedPoint
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import Mathlib.Data.Fin.Basic
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import Mathlib.Algebra.Order.Interval.Set
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namespace Semantics.Quantization
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@ -113,7 +112,7 @@ def activationScale (η ε : Q16_16) : Q16_16 :=
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/-- Clip operation for activations -/
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def clipActivation (x scale : Q16_16) (ε : Q16_16) : Q16_16 :=
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let Qb_val := ofNat (2 ^ Q_b - 1)
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let lower := negQ Qb_val + ε
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let lower := neg Qb_val + ε
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let upper := Qb_val - ε
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let scaled := x * scale
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if scaled < lower then lower
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@ -23,13 +23,13 @@ Citations:
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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.UnitQuaternion
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import Mathlib.Data.Fin.Basic
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import Mathlib.Algebra.Quaternion
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namespace Semantics.ResonanceGradient
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open Q16_16 Biology.QuaternionGenomic
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open Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Resonance Amplitude Type
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@ -77,9 +77,9 @@ def resonanceDifferential (grad : ResonanceGradient) (domega : Q16_16) (dt : Q16
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grad.dR_domega * domega + grad.dR_dt * dt
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#eval resonanceDifferential
|
||||
{ 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 }
|
||||
(toQ16_16 0.1)
|
||||
(toQ16_16 0.01)
|
||||
{ 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)
|
||||
-- Expected: 0.5 * 0.1 + 0.3 * 0.01 = 0.053
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -93,7 +93,7 @@ def stochasticDifferential (stoch : StochasticDifferential) : Q16_16 :=
|
|||
stoch.noise * stoch.dt -- Simplified: noise * dt instead of sqrt(dt) * noise
|
||||
|
||||
#eval stochasticDifferential
|
||||
{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
|
||||
{ dt := ofRatio 1 100, noise := ofRatio 1 2 }
|
||||
-- Expected: 0.5 * 0.01 = 0.005 (simplified from sqrt(0.01) * 0.5)
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -105,11 +105,11 @@ def stochasticDifferential (stoch : StochasticDifferential) : Q16_16 :=
|
|||
def itoCorrection (grad : ResonanceGradient) (dt : Q16_16) : Q16_16 :=
|
||||
-- Simplified Laplacian: sum of second derivatives
|
||||
-- In full implementation, this would compute actual Laplacian
|
||||
(grad.dR_domega + grad.dR_dt + grad.dR_dx + grad.dR_dy + grad.dR_dz) * dt / (toQ16_16 2.0)
|
||||
(grad.dR_domega + grad.dR_dt + grad.dR_dx + grad.dR_dy + grad.dR_dz) * dt / (ofNat 2)
|
||||
|
||||
#eval itoCorrection
|
||||
{ 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 }
|
||||
(toQ16_16 0.01)
|
||||
{ dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
|
||||
(ofRatio 1 100)
|
||||
-- Expected: (0.5 + 0.3) * 0.01 / 2 = 0.004
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -133,11 +133,10 @@ def quaternionStochasticEvolution (q : UnitQuaternion) (grad : ResonanceGradient
|
|||
q
|
||||
|
||||
#eval quaternionStochasticEvolution
|
||||
{ w := toQ16_16 1.0, x := toQ16_16 0.0, y := toQ16_16 0.0, z := toQ16_16 0.0,
|
||||
wf_unit := by simp [toQ16_16] }
|
||||
{ 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 }
|
||||
{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
|
||||
(toQ16_16 0.1)
|
||||
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)
|
||||
-- Expected: unchanged quaternion (placeholder)
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -152,7 +151,8 @@ theorem quaternionStochasticEvolutionPreservesUnitNorm
|
|||
(stoch : StochasticDifferential) (domega : Q16_16) :
|
||||
let q' := quaternionStochasticEvolution q grad stoch domega in
|
||||
q'.w * q'.w + q'.x * q'.x + q'.y * q'.y + q'.z * q'.z = one := by
|
||||
-- Placeholder proof
|
||||
-- TODO: Replaced placeholder 'trivial' tautology. Real proof of unit norm preservation needed.
|
||||
sorry
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
-- §9 Resonance Gradient from Spherion
|
||||
|
|
@ -170,13 +170,13 @@ def spherionResonanceGradient (amplitude : Q16_16) (frequency : Q16_16)
|
|||
-- 2. Compute derivative of phase
|
||||
-- 3. Compute derivative of pyramid height coupling
|
||||
-- 4. Combine into gradient vector
|
||||
{ dR_domega := amplitude * (toQ16_16 0.1), -- Simplified
|
||||
dR_dt := amplitude * (toQ16_16 0.05),
|
||||
dR_dx := toQ16_16 0.0,
|
||||
dR_dy := toQ16_16 0.0,
|
||||
dR_dz := toQ16_16 0.0 }
|
||||
{ dR_domega := amplitude * (ofRatio 1 10), -- Simplified
|
||||
dR_dt := amplitude * (ofRatio 1 20),
|
||||
dR_dx := zero,
|
||||
dR_dy := zero,
|
||||
dR_dz := zero }
|
||||
|
||||
#eval spherionResonanceGradient (toQ16_16 1.0) (toQ16_16 10.0) [toQ16_16 1.0, toQ16_16 2.0]
|
||||
#eval spherionResonanceGradient one (ofNat 10) [one, ofNat 2]
|
||||
-- Expected: gradient with dR_domega = 0.1, dR_dt = 0.05
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -194,10 +194,9 @@ def sluqQuaternionTriage (q : UnitQuaternion) (grad : ResonanceGradient)
|
|||
gradMagnitude < stability_threshold
|
||||
|
||||
#eval sluqQuaternionTriage
|
||||
{ w := toQ16_16 1.0, x := toQ16_16 0.0, y := toQ16_16 0.0, z := toQ16_16 0.0,
|
||||
wf_unit := by simp [toQ16_16] }
|
||||
{ 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 }
|
||||
(toQ16_16 1.0)
|
||||
UnitQuaternion.identity
|
||||
{ dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero }
|
||||
one
|
||||
-- Expected: true (gradient magnitude 0.34 < 1.0)
|
||||
|
||||
end Semantics.ResonanceGradient
|
||||
|
|
|
|||
|
|
@ -19,6 +19,7 @@ Citations:
|
|||
-/
|
||||
|
||||
import Semantics.FixedPoint
|
||||
import Semantics.UnitQuaternion
|
||||
import Semantics.Biology.QuaternionGenomic
|
||||
import Semantics.ResonanceGradient
|
||||
import Mathlib.Data.Fin.Basic
|
||||
|
|
@ -26,7 +27,7 @@ import Mathlib.Algebra.Quaternion
|
|||
|
||||
namespace Semantics.SLUQQuaternionIntegration
|
||||
|
||||
open Q16_16 Biology.QuaternionGenomic ResonanceGradient
|
||||
open Q16_16 Biology.QuaternionGenomic ResonanceGradient UnitQuaternion
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
-- §1 Quaternion Trajectory State
|
||||
|
|
@ -62,8 +63,8 @@ def cacheLocalQuaternionTriage (traj : QuaternionTrajectory) (localCacheSize : N
|
|||
|
||||
#eval cacheLocalQuaternionTriage
|
||||
{ quaternion := identity,
|
||||
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 },
|
||||
stabilityScore := toQ16_16 0.8,
|
||||
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
|
||||
-- Expected: true (gradient magnitude 0.34 < stability threshold 2)
|
||||
|
|
@ -92,7 +93,7 @@ def sluqQuaternionOptimizationStep (traj : QuaternionTrajectory)
|
|||
-- Stable: apply stochastic evolution
|
||||
let newQuaternion := stochasticEvolution traj.quaternion traj.gradient stoch domega
|
||||
-- Update stability score (improve if stable)
|
||||
let newStabilityScore := traj.stabilityScore + (toQ16_16 0.1)
|
||||
let newStabilityScore := traj.stabilityScore + (ofRatio 1 10)
|
||||
-- Increment iteration
|
||||
let newIteration := traj.iteration + 1
|
||||
{ quaternion := newQuaternion,
|
||||
|
|
@ -105,11 +106,11 @@ def sluqQuaternionOptimizationStep (traj : QuaternionTrajectory)
|
|||
|
||||
#eval sluqQuaternionOptimizationStep
|
||||
{ quaternion := identity,
|
||||
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 },
|
||||
stabilityScore := toQ16_16 0.8,
|
||||
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 := toQ16_16 0.01, noise := toQ16_16 0.5 }
|
||||
(toQ16_16 0.1)
|
||||
{ dt := ofRatio 1 100, noise := ofRatio 1 2 }
|
||||
(ofRatio 1 10)
|
||||
10
|
||||
-- Expected: trajectory with updated quaternion and stability score
|
||||
|
||||
|
|
@ -143,11 +144,11 @@ def quaternionTrajectoryConverged (traj : QuaternionTrajectory)
|
|||
|
||||
#eval quaternionTrajectoryConverged
|
||||
{ quaternion := identity,
|
||||
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 },
|
||||
stabilityScore := toQ16_16 0.9,
|
||||
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 }
|
||||
(toQ16_16 0.8)
|
||||
(toQ16_16 0.5)
|
||||
(ofRatio 4 5)
|
||||
(ofRatio 1 2)
|
||||
-- Expected: true (stability 0.9 ≥ 0.8, gradient 0.02 < 0.5)
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
|
@ -164,8 +165,9 @@ theorem sluqQuaternionOptimizationPreservesUnitNorm
|
|||
traj'.quaternion.w * traj'.quaternion.w +
|
||||
traj'.quaternion.x * traj'.quaternion.x +
|
||||
traj'.quaternion.y * traj'.quaternion.y +
|
||||
traj'.quaternion.z * traj'.quaternion.z = one :=
|
||||
trivial
|
||||
traj'.quaternion.z * traj'.quaternion.z = one := by
|
||||
-- TODO: Replaced placeholder 'trivial' tautology. Real proof of unit norm preservation needed.
|
||||
sorry
|
||||
|
||||
/-- Theorem: Pruning preserves unit norm.
|
||||
Since we only filter trajectories without modifying them, unit norm is preserved. -/
|
||||
|
|
|
|||
193
0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean
Normal file
193
0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean
Normal file
|
|
@ -0,0 +1,193 @@
|
|||
import Semantics.FixedPoint
|
||||
import Semantics.SLUG3
|
||||
import Mathlib.Data.Fin.Basic
|
||||
import Mathlib.Algebra.Quaternion
|
||||
|
||||
namespace Semantics
|
||||
|
||||
open Q16_16
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
-- §0 Fixed-Point Trigonometry Placeholders (for Q16_16)
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
||||
/-- Cosine lookup for Q16_16 (placeholder: use CORDIC or polynomial approx).
|
||||
Input: angle in radians, scaled to Q16_16.
|
||||
Output: cos(angle) in Q16_16 ∈ [-1.0, 1.0]. -/
|
||||
def cos (x : Q16_16) : Q16_16 :=
|
||||
one - (x * x) / ofNat 2 -- Taylor: 1 - x²/2
|
||||
|
||||
/-- Sine lookup for Q16_16 (placeholder: use CORDIC or polynomial approx).
|
||||
Input: angle in radians, scaled to Q16_16.
|
||||
Output: sin(angle) in Q16_16 ∈ [-1.0, 1.0]. -/
|
||||
def sin (x : Q16_16) : Q16_16 :=
|
||||
x - (x * x * x) / ofNat 6 -- Taylor: x - x³/6
|
||||
|
||||
/-- Arccosine lookup for Q16_16 (placeholder: use inverse trig LUT).
|
||||
Input: value in [-1.0, 1.0], scaled to Q16_16.
|
||||
Output: arccos(value) in radians [0, π], scaled to Q16_16. -/
|
||||
def acos (x : Q16_16) : Q16_16 :=
|
||||
one - x -- Approximation: arccos(x) ≈ 1 - x (for small angles)
|
||||
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
-- §1 Unit Quaternion Type (S³ Embedding)
|
||||
-- ═══════════════════════════════════════════════════════════════════════════
|
||||
|
||||
/-- Unit quaternion representing a point on the 3-sphere S³.
|
||||
Stored as (w, x, y, z) with constraint w² + x² + y² + z² = 1.
|
||||
Uses Q16_16 fixed-point for hardware extraction. -/
|
||||
structure UnitQuaternion where
|
||||
w : Q16_16 -- scalar (real) part
|
||||
x : Q16_16 -- i component
|
||||
y : Q16_16 -- j component
|
||||
z : Q16_16 -- k component
|
||||
wf_unit : w * w + x * x + y * y + z * z = one -- unit norm constraint
|
||||
deriving Repr
|
||||
|
||||
namespace UnitQuaternion
|
||||
|
||||
/-- Identity quaternion (1, 0, 0, 0) - neutral element -/
|
||||
def identity : UnitQuaternion :=
|
||||
{ w := one
|
||||
x := zero
|
||||
y := zero
|
||||
z := zero
|
||||
wf_unit := by decide }
|
||||
|
||||
/-- Quaternion multiplication (Hamilton product).
|
||||
For unit quaternions, product remains unit (S³ is a group under ×). -/
|
||||
def mul (a b : UnitQuaternion) : UnitQuaternion :=
|
||||
let w' := a.w * b.w - a.x * b.x - a.y * b.y - a.z * b.z
|
||||
let x' := a.w * b.x + a.x * b.w + a.y * b.z - a.z * b.y
|
||||
let y' := a.w * b.y - a.x * b.z + a.y * b.w + a.z * b.x
|
||||
let z' := a.w * b.z + a.x * b.y - a.y * b.x + a.z * b.w
|
||||
{ w := w', x := x', y := y', z := z',
|
||||
wf_unit := sorry }
|
||||
|
||||
/-- Dot product as scalar part of a × b* (conjugate product).
|
||||
For unit quaternions: a · b = cos(θ) where θ = great circle distance. -/
|
||||
def dot (a b : UnitQuaternion) : Q16_16 :=
|
||||
a.w * b.w + a.x * b.x + a.y * b.y + a.z * b.z
|
||||
|
||||
/-- Great circle distance on S³: arccos(a · b).
|
||||
For compression: distance ∈ [0, π] maps to dissimilarity metric. -/
|
||||
def distance (a b : UnitQuaternion) : Q16_16 :=
|
||||
let d := dot a b
|
||||
if d.val ≥ 0x00010000 then -- d ≥ 1.0
|
||||
zero -- distance = 0 (identical)
|
||||
else if d.val ≤ 0xFFFF0000 then -- d ≤ -1.0
|
||||
⟨0x0003243F⟩ -- π ≈ 3.14159 in Q16_16
|
||||
else
|
||||
one - d -- Approximation: arccos(d) ≈ 1 - d for small angles
|
||||
|
||||
/-- Quaternion conjugate: q* = [w, -x, -y, -z].
|
||||
For unit quaternions, q⁻¹ = q*. -/
|
||||
def conjugate (q : UnitQuaternion) : UnitQuaternion :=
|
||||
{ w := q.w, x := neg q.x, y := neg q.y, z := neg q.z,
|
||||
wf_unit := sorry }
|
||||
|
||||
/-- Quaternion inverse: q⁻¹ = q* / ||q||².
|
||||
For unit quaternions, q⁻¹ = q* (conjugate). -/
|
||||
def inv (q : UnitQuaternion) : UnitQuaternion :=
|
||||
q.conjugate -- Unit quaternion: inverse = conjugate
|
||||
|
||||
/-- Rotation of point p (pure quaternion [0, px, py, pz]) by unit quaternion q.
|
||||
Formula: p' = q · p · q⁻¹ (conjugation).
|
||||
Preserves vector norm: ||p'|| = ||p||. -/
|
||||
def rotateVector (q : UnitQuaternion) (v : Q16_16 × Q16_16 × Q16_16) : Q16_16 × Q16_16 × Q16_16 :=
|
||||
let (vx, vy, vz) := v
|
||||
let p := { w := zero, x := vx, y := vy, z := vz, wf_unit := sorry }
|
||||
let rotated := (q.mul p).mul q.inv
|
||||
(rotated.x, rotated.y, rotated.z)
|
||||
|
||||
/-- Construct unit quaternion from axis-angle representation.
|
||||
q = [cos(θ/2), sin(θ/2) · (ux, uy, uz)] where (ux,uy,uz) is unit axis. -/
|
||||
def fromAxisAngle (axis : Q16_16 × Q16_16 × Q16_16) (angle : Q16_16) : UnitQuaternion :=
|
||||
let (ux, uy, uz) := axis
|
||||
let cosHalf := cos (angle / ofInt 2)
|
||||
let sinHalf := sin (angle / ofInt 2)
|
||||
let norm := Q16_16.sqrt (ux * ux + uy * uy + uz * uz)
|
||||
let cosTheta := cosHalf
|
||||
let sinTheta := sinHalf * norm
|
||||
{ w := cosTheta, x := sinTheta * ux, y := sinTheta * uy, z := sinTheta * uz,
|
||||
wf_unit := sorry }
|
||||
|
||||
/-- Extract axis-angle from unit quaternion.
|
||||
Returns (axis, angle) where axis is unit vector and angle ∈ [0, 2π). -/
|
||||
def toAxisAngle (q : UnitQuaternion) : (Q16_16 × Q16_16 × Q16_16) × Q16_16 :=
|
||||
let angle := ofNat 2 * acos q.w -- θ = 2·arccos(w)
|
||||
let sinHalf := sin (angle / ofNat 2)
|
||||
let axis := if sinHalf.val > 0x00000100 then -- sin(θ/2) ≠ 0
|
||||
(q.x / sinHalf, q.y / sinHalf, q.z / sinHalf)
|
||||
else
|
||||
(one, zero, zero) -- Identity rotation: arbitrary axis
|
||||
(axis, angle)
|
||||
|
||||
/-- Spherical Linear Interpolation (SLERP) between two unit quaternions. -/
|
||||
def slerp (a b : UnitQuaternion) (t : Q16_16) : UnitQuaternion :=
|
||||
let dotAB := a.dot b
|
||||
let (b', dotAB') := if dotAB.val < 0x00008000 then
|
||||
({ w := neg b.w, x := neg b.x, y := neg b.y, z := neg b.z,
|
||||
wf_unit := sorry }, neg dotAB)
|
||||
else
|
||||
(b, dotAB)
|
||||
|
||||
let omega := acos dotAB' -- Angle between quaternions
|
||||
let sinOmega := sin omega
|
||||
|
||||
if sinOmega.val < 0x00000100 then -- Quaternions nearly parallel
|
||||
let w1 := one - t
|
||||
let w2 := t
|
||||
{ w := w1 * a.w + w2 * b'.w,
|
||||
x := w1 * a.x + w2 * b'.x,
|
||||
y := w1 * a.y + w2 * b'.y,
|
||||
z := w1 * a.z + w2 * b'.z,
|
||||
wf_unit := sorry }
|
||||
else
|
||||
let w1 := sin ((one - t) * omega) / sinOmega
|
||||
let w2 := sin (t * omega) / sinOmega
|
||||
{ w := w1 * a.w + w2 * b'.w,
|
||||
x := w1 * a.x + w2 * b'.x,
|
||||
y := w1 * a.y + w2 * b'.y,
|
||||
z := w1 * a.z + w2 * b'.z,
|
||||
wf_unit := sorry }
|
||||
|
||||
/-- Convert unit quaternion to 3×3 rotation matrix (row-major). -/
|
||||
def toRotationMatrix (q : UnitQuaternion) : Q16_16 × Q16_16 × Q16_16 ×
|
||||
Q16_16 × Q16_16 × Q16_16 ×
|
||||
Q16_16 × Q16_16 × Q16_16 :=
|
||||
let w := q.w; let x := q.x; let y := q.y; let z := q.z
|
||||
let two := ofNat 2
|
||||
|
||||
let m00 := one - two * (y * y + z * z)
|
||||
let m01 := two * (x * y - z * w)
|
||||
let m02 := two * (x * z + y * w)
|
||||
|
||||
let m10 := two * (x * y + z * w)
|
||||
let m11 := one - two * (x * x + z * z)
|
||||
let m12 := two * (y * z - x * w)
|
||||
|
||||
let m20 := two * (x * z - y * w)
|
||||
let m21 := two * (y * z + x * w)
|
||||
let m22 := one - two * (x * x + y * y)
|
||||
|
||||
(m00, m01, m02, m10, m11, m12, m20, m21, m22)
|
||||
|
||||
/-- Check chiral compatibility: D+L→W collapse detection. -/
|
||||
def chiralIncompatible (a b : UnitQuaternion) : Bool :=
|
||||
let product := mul a b
|
||||
product.w.val < 0x00008000
|
||||
|
||||
/-- Ternary classification from quaternion dot product (SLUG-3 gate). -/
|
||||
def toTernary (a b : UnitQuaternion) (threshold : Q16_16) : SLUG3.Ternary :=
|
||||
let d := dot a b
|
||||
if d ≥ threshold then
|
||||
SLUG3.Ternary.high
|
||||
else if d ≤ neg threshold then
|
||||
SLUG3.Ternary.low
|
||||
else
|
||||
SLUG3.Ternary.mid
|
||||
|
||||
end UnitQuaternion
|
||||
|
||||
end Semantics
|
||||
Loading…
Add table
Reference in a new issue