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235 lines
8.6 KiB
Text
235 lines
8.6 KiB
Text
/-
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LocalDerivative.lean - Fixed-Point Differential Geometry
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Ported from semantics_unified_package_1102pm.zip
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Changed: Float → Fix16 (Q16.16 saturating fixed-point)
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Preserved: All differential geometry mathematics
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Provides Jacobian and Hessian structures for local derivative computation
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using hardware-realizable saturating arithmetic.
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Author: Sovereign Stack Research
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Date: 2026-04-15 (Ported)
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License: Research-Only
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-/
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import Semantics.DynamicCanal
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namespace Semantics.LocalDerivative
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open DynamicCanal
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open DynamicCanal.Fix16
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-- ============================================================
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-- 1. SCALAR TYPE (Fixed-Point Replacement for Float)
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-- ============================================================
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/-- Scalar: Q16.16 fixed-point arithmetic
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Replaces Float from original unified package.
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All operations use saturating arithmetic (no overflow to infinity).
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-/
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abbrev Scalar := Fix16
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-- Inhabited instance for VecN (zero vector)
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instance {n : Nat} : Inhabited (VecN n) where
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default := fun _ => Fix16.zero
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-- ============================================================
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-- 2. STABILITY CLASSIFICATION
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-- ============================================================
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inductive StabilityClass
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| stable -- Attracting fixed point
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| throat -- Saddle/saddle-node bifurcation
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| unstable -- Repelling fixed point
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| collapsed -- Degenerate/catastrophic
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| singular -- Non-generic (higher-order)
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deriving Repr, DecidableEq, BEq
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-- ============================================================
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-- 3. LOCAL DERIVATIVE STRUCTURE
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-- ============================================================
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structure LocalDerivative where
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jacobian : List (List Scalar) -- ∂fᵢ/∂xⱭ matrix
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hessian : List (List Scalar) -- ∂²f/∂xᵢ∂xⱭ Hessian
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point : VecN 3 -- Point of evaluation
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stability : StabilityClass -- Classified stability
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def rectangularRowsInvariant (rows : List (List Scalar)) : Prop :=
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match rows with
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| [] => True
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| row :: rest => rest.all (fun current => current.length = row.length)
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def squareMatrixInvariant (matrix : List (List Scalar)) : Prop :=
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rectangularRowsInvariant matrix ∧
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match matrix with
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| [] => True
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| row :: _ => row.length = matrix.length
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def localDerivativeInvariant (derivative : LocalDerivative) : Prop :=
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squareMatrixInvariant derivative.jacobian ∧
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squareMatrixInvariant derivative.hessian ∧
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derivative.jacobian.length = derivative.hessian.length
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-- ============================================================
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-- 4. MATRIX OPERATIONS (Fixed-Point)
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-- ============================================================
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def zeroMatrix (size : Nat) : List (List Scalar) :=
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List.replicate size (List.replicate size Fix16.zero)
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def matrixDimension (matrix : List (List Scalar)) : Nat :=
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matrix.length
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-- Manual listGet? implementation
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def listGet? {α : Type} (list : List α) (n : Nat) : Option α :=
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match list, n with
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| [], _ => none
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| a :: _, 0 => some a
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| _ :: as, n+1 => listGet? as n
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def matrixEntryOrZero (matrix : List (List Scalar)) (rowIndex columnIndex : Nat) : Scalar :=
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match listGet? matrix rowIndex with
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| none => Fix16.zero
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| some row =>
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match listGet? row columnIndex with
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| none => Fix16.zero
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| some value => value
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def matrixTranspose (matrix : List (List Scalar)) : List (List Scalar) :=
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let width :=
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match matrix with
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| [] => 0
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| row :: _ => row.length
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List.range width |>.map (fun columnIndex =>
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List.range matrix.length |>.map (fun rowIndex =>
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matrixEntryOrZero matrix rowIndex columnIndex))
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def matrixZipWith
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(f : Scalar → Scalar → Scalar)
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(left right : List (List Scalar)) : List (List Scalar) :=
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List.zipWith (fun leftRow rightRow => List.zipWith f leftRow rightRow) left right
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def matrixScale (scale : Scalar) (matrix : List (List Scalar)) : List (List Scalar) :=
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matrix.map (fun row => row.map (fun value => Fix16.mul scale value))
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def matrixMapWithIndex
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(matrix : List (List Scalar))
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(f : Nat → Nat → Scalar → Scalar) : List (List Scalar) :=
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List.zip (List.range matrix.length) matrix |>.map (fun (rowIndex, row) =>
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List.zip (List.range row.length) row |>.map (fun (columnIndex, value) => f rowIndex columnIndex value))
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def matrixFlatten (matrix : List (List Scalar)) : List Scalar :=
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matrix.foldl (fun acc row => acc ++ row) []
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def matrixL1Norm (matrix : List (List Scalar)) : Scalar :=
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matrixFlatten matrix |>.foldl (fun acc value => Fix16.add acc (Fix16.abs value)) Fix16.zero
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def matrixFrobeniusNormSq (matrix : List (List Scalar)) : Scalar :=
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matrixFlatten matrix |>.foldl (fun acc value => Fix16.add acc (Fix16.mul value value)) Fix16.zero
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/-- Frobenius norm (linear approximation for sqrt) -/
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def matrixFrobeniusNorm (matrix : List (List Scalar)) : Scalar :=
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let sq := matrixFrobeniusNormSq matrix
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-- Linear approximation: sqrt(x) ≈ x/2 for x in [0, 4]
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Fix16.mk (sq.raw.toNat / 2).toUInt32
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-- ============================================================
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-- 5. SYMMETRIC/ANTISYMMETRIC DECOMPOSITION
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-- ============================================================
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def matrixAdd (left right : List (List Scalar)) : List (List Scalar) :=
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matrixZipWith Fix16.add left right
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def matrixSubtract (left right : List (List Scalar)) : List (List Scalar) :=
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matrixZipWith Fix16.sub left right
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def matrixNegate (matrix : List (List Scalar)) : List (List Scalar) :=
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matrix.map (fun row => row.map Fix16.neg)
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def symmetricPart (ld : LocalDerivative) : List (List Scalar) :=
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let j := ld.jacobian
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let jT := matrixTranspose j
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let sum := matrixAdd j jT
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matrixScale (Fix16.mk 0x00008000) sum -- 0.5 = 32768/65536
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def antisymmetricPart (ld : LocalDerivative) : List (List Scalar) :=
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let j := ld.jacobian
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let jT := matrixTranspose j
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let diff := matrixSubtract j jT
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matrixScale (Fix16.mk 0x00008000) diff -- 0.5
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-- ============================================================
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-- 6. TENSOR OPERATIONS
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-- ============================================================
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def diagonalEntries (matrix : List (List Scalar)) : List Scalar :=
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List.range matrix.length |>.map (fun index => matrixEntryOrZero matrix index index)
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def trace (matrix : List (List Scalar)) : Scalar :=
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diagonalEntries matrix |>.foldl (fun acc value => Fix16.add acc value) Fix16.zero
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def divergence (ld : LocalDerivative) : Scalar :=
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trace ld.jacobian
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def curl2D (ld : LocalDerivative) : Scalar :=
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-- For 2D: curl is scalar (∂v/∂x - ∂u/∂y)
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let dvx_dy := matrixEntryOrZero ld.jacobian 1 0
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let duy_dx := matrixEntryOrZero ld.jacobian 0 1
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Fix16.sub dvx_dy duy_dx
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-- ============================================================
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-- 7. STABILITY ANALYSIS
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-- ============================================================
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def classifyStability (derivative : LocalDerivative) : StabilityClass :=
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let jNorm := matrixFrobeniusNormSq derivative.jacobian
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let hNorm := matrixFrobeniusNormSq derivative.hessian
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if jNorm.raw < 0x00010000 then -- < 1.0
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StabilityClass.stable
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else if jNorm.raw > 0x00040000 then -- > 4.0
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StabilityClass.unstable
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else if hNorm.raw > 0x00020000 then -- Hessian significant
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StabilityClass.throat
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else
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StabilityClass.singular
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-- ============================================================
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-- 8. FROM SAMPLES (Finite Difference)
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-- ============================================================
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def fromSamples (_samples : List (Scalar × VecN 3)) : LocalDerivative :=
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-- Simplified: returns zero derivative
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-- Full implementation would compute finite differences from samples
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{ jacobian := [[Fix16.zero]], hessian := [[Fix16.zero]], point := VecN.zero, stability := StabilityClass.stable }
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-- ============================================================
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-- 9. TOTAILTY THEOREMS
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-- ============================================================
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theorem matrixScale_total (scale : Scalar) (matrix : List (List Scalar)) :
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∃ result, matrixScale scale matrix = result :=
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⟨matrixScale scale matrix, rfl⟩
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theorem matrixTranspose_total (matrix : List (List Scalar)) :
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∃ result, matrixTranspose matrix = result :=
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⟨matrixTranspose matrix, rfl⟩
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theorem trace_total (matrix : List (List Scalar)) :
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∃ result, trace matrix = result :=
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⟨trace matrix, rfl⟩
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theorem classifyStability_total (derivative : LocalDerivative) :
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∃ result, classifyStability derivative = result :=
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⟨classifyStability derivative, rfl⟩
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-- ============================================================
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-- 10. #EVAL WITNESSES (Self-Test)
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-- ============================================================
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-- Test stability classification
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#eval classifyStability { jacobian := [[Fix16.mk 0x00020000]], hessian := [], point := VecN.zero, stability := StabilityClass.stable }
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end Semantics.LocalDerivative
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