From 0b936c8e3960e75a8a05535599ee5221db1a0d96 Mon Sep 17 00:00:00 2001 From: Brandon Schneider Date: Tue, 19 May 2026 14:29:41 +0000 Subject: [PATCH] Semantics: modularize quaternion, resolve circular dependencies, and eliminate float operations --- 0-Core-Formalism/lean/Semantics/Biology.lean | 3 + .../Semantics/Biology/QuaternionGenomic.lean | 258 +----------------- .../Semantics/Semantics/Quantization.lean | 3 +- .../Semantics/ResonanceGradient.lean | 49 ++-- .../Semantics/SLUQQuaternionIntegration.lean | 30 +- .../Semantics/Semantics/UnitQuaternion.lean | 193 +++++++++++++ 6 files changed, 249 insertions(+), 287 deletions(-) create mode 100644 0-Core-Formalism/lean/Semantics/Biology.lean create mode 100644 0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean diff --git a/0-Core-Formalism/lean/Semantics/Biology.lean b/0-Core-Formalism/lean/Semantics/Biology.lean new file mode 100644 index 00000000..66e60aaa --- /dev/null +++ b/0-Core-Formalism/lean/Semantics/Biology.lean @@ -0,0 +1,3 @@ +import Semantics.Biology.BioRxivFormalization +import Semantics.Biology.QuaternionGenomic +import Semantics.Biology.RGFlowBioinformatics diff --git a/0-Core-Formalism/lean/Semantics/Semantics/Biology/QuaternionGenomic.lean b/0-Core-Formalism/lean/Semantics/Semantics/Biology/QuaternionGenomic.lean index ca27c9ca..7f5bdbf9 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/Biology/QuaternionGenomic.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/Biology/QuaternionGenomic.lean @@ -24,6 +24,7 @@ import Semantics.FixedPoint import Semantics.SLUG3 import Semantics.GenomicCompression import Semantics.ResonanceGradient +import Semantics.UnitQuaternion import Mathlib.Data.Fin.Basic import Mathlib.Algebra.Quaternion @@ -31,238 +32,6 @@ namespace Semantics.QuaternionGenomic open Q16_16 SLUG3 GenomicCompression --- ═══════════════════════════════════════════════════════════════════════════ --- §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 - --- ═══════════════════════════════════════════════════════════════════════════ --- §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 := - -- Placeholder: polynomial approximation or LUT - -- Full implementation: Chebyshev polynomial or CORDIC - q16_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 := - -- Placeholder: Taylor series approximation - 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 := - -- Placeholder: linear approximation - -- Full implementation: use precomputed LUT or polynomial - q16_one - x -- Approximation: arccos(x) ≈ 1 - x (for small angles) - -namespace UnitQuaternion - -/-- Identity quaternion (1, 0, 0, 0) - neutral element -/ -def identity : UnitQuaternion := - { w := one - x := zero - y := zero - z := zero - wf_unit := by simp [one, zero] } - -/-- 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 - -- Note: wf_unit proof omitted for computational use - -- In production: verify norm preservation - { w := w', x := x', y := y', z := z', - wf_unit := by trivial } - -/-- 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 := - -- arccos approximation via lookup table (hardware-efficient) - -- Full implementation would use cordic or polynomial approximation - let d := dot a b - -- Map [-1, 1] to [0, π] using piecewise linear approx - if d.val ≥ 0x00010000 then -- d ≥ 1.0 - zero -- distance = 0 (identical) - else if d.val ≤ 0xFFFF0000 then -- d ≤ -1.0 - ofUInt32 0x0003243F -- π ≈ 3.14159 in Q16_16 - else - -- Linear interpolation: distance ≈ arccos(d) - -- Simplified: use precomputed LUT for arccos values - q16_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 := negQ q.x, y := negQ q.y, z := negQ q.z, - wf_unit := by simp [one, q.wf_unit] } - -/-- 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 - -- Represent v as pure quaternion [0, vx, vy, vz] - let p := { w := zero, x := vx, y := vy, z := vz, wf_unit := by trivial } - -- Compute q · p · q⁻¹ - let rotated := (q.mul p).mul q.inv - -- Extract vector part - (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. - Standard robotics/computer graphics convention. --- --- Arithmetic sanity check: --- quaternion from axis-angle formula. --- --- 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 fromAxisAngle (axis : Q16_16 × Q16_16 × Q16_16) (angle : Q16_16) : UnitQuaternion := - let (ux, uy, uz) := axis - -- Arithmetic sanity check: - -- cos(θ/2) for quaternion rotation. - let cosHalf := Q16_16.cos (angle / ofInt 2) - -- Arithmetic sanity check: - -- sin(θ/2) for quaternion rotation. - let sinHalf := Q16_16.sin (angle / ofInt 2) - -- Arithmetic sanity check: - -- √(x² + y² + z²) for vector normalization. - 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 := by trivial } - -/-- 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. - Formula: slerp(q1, q2, t) = sin((1-t)·Ω)/sin(Ω) · q1 + sin(t·Ω)/sin(Ω) · q2 - where Ω = arccos(q1 · q2) and t ∈ [0, 1]. - Used for smooth DNA backbone interpolation between nucleotide states. -/ -def slerp (a b : UnitQuaternion) (t : Q16_16) : UnitQuaternion := - let dotAB := a.dot b - -- Ensure we take the shortest path (flip sign if dot < 0) - let (b', dotAB') := if dotAB.val < 0x00008000 then - ({ b with w := negQ b.w, x := negQ b.x, y := negQ b.y, z := negQ b.z, - wf_unit := b.wf_unit }, negQ dotAB) - else - (b, dotAB) - - let omega := acos dotAB' -- Angle between quaternions - let sinOmega := sin omega - - if sinOmega.val < 0x00000100 then -- Quaternions nearly parallel - -- Use linear interpolation (LERP) to avoid division by near-zero - 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 := by trivial } - else - -- Full SLERP - 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 := by trivial } - -/-- Convert unit quaternion to 3×3 rotation matrix (row-major). - Matrix entries derived from Hamilton product algebra. - Used for rendering DNA backbone in 3D visualization. -/ -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 - - -- First row - let m00 := one - two * (y * y + z * z) - let m01 := two * (x * y - z * w) - let m02 := two * (x * z + y * w) - - -- Second row - let m10 := two * (x * y + z * w) - let m11 := one - two * (x * x + z * z) - let m12 := two * (y * z - x * w) - - -- Third row - 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. - Two quaternions are "incompatible" if their product has negative scalar part. - This corresponds to right-hand vs left-hand chirality mismatch. -/ -def chiralIncompatible (a b : UnitQuaternion) : Bool := - let product := mul a b - product.w.val < 0x00008000 -- scalar part < 0 (negative) - -/-- Ternary classification from quaternion dot product (SLUG-3 gate). - Maps dot product threshold to ternary state: - - dot ≥ threshold: high (compatible) - - |dot| < threshold: mid (uncertain) - - dot ≤ -threshold: low (incompatible/collapse) - -/ -def toTernary (a b : UnitQuaternion) (threshold : Q16_16) : Ternary := - let d := dot a b - if d ≥ threshold then - Ternary.high - else if d ≤ negQ threshold then - Ternary.low - else - Ternary.mid - -end UnitQuaternion - -- ═══════════════════════════════════════════════════════════════════════════ -- §2 Nucleotide-to-Quaternion Mapping (Prime-Addressed) -- ═══════════════════════════════════════════════════════════════════════════ @@ -427,10 +196,7 @@ def primeIndexedQuaternion (primeIdx : Nat) (hPrime : Nat.Prime primeIdx) : Unit let axisZ := ofNat (primeIdx + 6) -- Normalize axis - let norm := Float.sqrt (axisX.toFloat * axisX.toFloat + - axisY.toFloat * axisY.toFloat + - axisZ.toFloat * axisZ.toFloat) - let n := ofFloat norm + let n := Q16_16.sqrt (axisX * axisX + axisY * axisY + axisZ * axisZ) { w := cos angle, -- Approximation: use lookup x := sin angle * axisX / n, @@ -516,9 +282,9 @@ def stochasticEvolution (q : UnitQuaternion) (grad : ResonanceGradient.Resonance #eval stochasticEvolution identity - { 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) + { 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) /-- Resonance-tuned quaternion rotation. @@ -531,7 +297,7 @@ def resonanceTunedRotation (q : UnitQuaternion) (axis : Q16_16 × Q16_16 × Q16_ let (ax, ay, az) := axis -- Gradient magnitude determines rotation angle let gradMagnitude := grad.dR_domega * grad.dR_domega + grad.dR_dt * grad.dR_dt - let optimalAngle := gradMagnitude * (toQ16_16 0.5) -- Simplified scaling + let optimalAngle := gradMagnitude * (ofRatio 1 2) -- Simplified scaling -- Apply rotation with optimal angle fromAxisAngle axis optimalAngle @@ -539,7 +305,7 @@ def resonanceTunedRotation (q : UnitQuaternion) (axis : Q16_16 × Q16_16 × Q16_ #eval resonanceTunedRotation identity (one, zero, zero) - { 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 } + { dR_domega := ofRatio 1 2, dR_dt := ofRatio 3 10, dR_dx := zero, dR_dy := zero, dR_dz := zero } -- Expected: rotation around x-axis with angle proportional to gradient magnitude /-- Stochastic quaternion optimization. @@ -552,16 +318,16 @@ def stochasticQuaternionOptimization (q : UnitQuaternion) (grad : ResonanceGradi -- Check stability via SLUQ triage if ResonanceGradient.sluqQuaternionTriage q grad stabilityThreshold then -- Stable: apply stochastic evolution - stochasticEvolution q grad stoch (toQ16_16 0.1) + stochasticEvolution q grad stoch (ofRatio 1 10) else -- Unstable: skip update (prune trajectory) q #eval stochasticQuaternionOptimization identity - { 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 1.0) + { 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 } + one -- Expected: unchanged quaternion (stable check passes, but evolution is placeholder) /-- Theorem: Stochastic quaternion evolution preserves unit norm. @@ -574,7 +340,7 @@ theorem stochasticEvolutionPreservesUnitNorm q'.w * q'.w + q'.x * q'.x + q'.y * q'.y + q'.z * q'.z = one := by -- Proof: Quaternion exponential map preserves unit norm -- Stochastic increment is applied via rotation, which preserves norm - trivial + sorry /-- Theorem: Resonance-tuned rotation preserves unit norm. Axis-angle representation always produces unit quaternions. -/ diff --git a/0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean b/0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean index 2569874f..c6f1b5c8 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/Quantization.lean @@ -14,7 +14,6 @@ This module formalizes: import Semantics.FixedPoint import Mathlib.Data.Fin.Basic -import Mathlib.Algebra.Order.Interval.Set namespace Semantics.Quantization @@ -113,7 +112,7 @@ def activationScale (η ε : Q16_16) : Q16_16 := /-- Clip operation for activations -/ def clipActivation (x scale : Q16_16) (ε : Q16_16) : Q16_16 := let Qb_val := ofNat (2 ^ Q_b - 1) - let lower := negQ Qb_val + ε + let lower := neg Qb_val + ε let upper := Qb_val - ε let scaled := x * scale if scaled < lower then lower diff --git a/0-Core-Formalism/lean/Semantics/Semantics/ResonanceGradient.lean b/0-Core-Formalism/lean/Semantics/Semantics/ResonanceGradient.lean index 15eb8a45..bb04f9e0 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/ResonanceGradient.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/ResonanceGradient.lean @@ -23,13 +23,13 @@ Citations: -/ import Semantics.FixedPoint -import Semantics.Biology.QuaternionGenomic +import Semantics.UnitQuaternion import Mathlib.Data.Fin.Basic import Mathlib.Algebra.Quaternion namespace Semantics.ResonanceGradient -open Q16_16 Biology.QuaternionGenomic +open Q16_16 -- ═══════════════════════════════════════════════════════════════════════════ -- §1 Resonance Amplitude Type @@ -77,9 +77,9 @@ def resonanceDifferential (grad : ResonanceGradient) (domega : Q16_16) (dt : Q16 grad.dR_domega * domega + grad.dR_dt * dt #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 diff --git a/0-Core-Formalism/lean/Semantics/Semantics/SLUQQuaternionIntegration.lean b/0-Core-Formalism/lean/Semantics/Semantics/SLUQQuaternionIntegration.lean index 890cdede..087cf087 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/SLUQQuaternionIntegration.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/SLUQQuaternionIntegration.lean @@ -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. -/ diff --git a/0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean b/0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean new file mode 100644 index 00000000..1ae04841 --- /dev/null +++ b/0-Core-Formalism/lean/Semantics/Semantics/UnitQuaternion.lean @@ -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