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192 lines
8.1 KiB
Text
192 lines
8.1 KiB
Text
import Semantics.FixedPoint
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namespace Semantics.Basic
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open Semantics.Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Hardware-Native Basic Structures (from HachimojiPipeline improvements)
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-- Fixed-point usage justification (Section 13.3):
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-- - Q16_16 used for all geometric and state computations to preserve integer precision
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-- - Required for Christoffel symbols, frustration calculations, and state updates
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-- - Deterministic overflow behavior: operations use standard Q16_16 arithmetic with wraparound
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-- - No Q0_16 usage in this module - all values require integer component for geometric calculations
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Discrete basic state using Q16_16 for hardware-native computation -/
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structure DiscreteBasicState where
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value : Q16_16 -- Basic value
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derivative : Q16_16 -- Basic derivative
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integral : Q16_16 -- Basic integral
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momentum : Q16_16 -- Basic momentum
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deriving Repr, Inhabited
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/-- Basic grid for spatial discretization -/
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structure BasicGrid where
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dimension : Nat -- Grid dimension
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spacing : Q16_16 -- Grid spacing
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values : Array DiscreteBasicState -- State values at grid points
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deriving Repr
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/-- Basic manifold for geometric phase evolution -/
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structure BasicManifold where
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dimension : Nat -- Manifold dimension
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curvature : Q16_16 -- Scalar curvature (affects basic field)
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torsion : Q16_16 -- Torsion (basic deviation)
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metric : Array Q16_16 -- Metric tensor diagonal elements
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deriving Repr
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/-- Christoffel symbols for basic geometric phase -/
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structure BasicChristoffel where
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dimension : Nat -- Manifold dimension
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symbols : Array Q16_16 -- Flattened symbol array [i][j][k]
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deriving Repr, Inhabited
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/-- Basic lock pattern for frustration computation -/
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structure BasicLockPattern where
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value : Q16_16
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derivative : Q16_16
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momentum : Q16_16
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deriving Repr, Inhabited
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/-- Basic frustration wave parameters -/
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structure BasicFrustrationWave where
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waveVector : Array Q16_16 -- k_r wave vector
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weight : Q16_16 -- w_r weight from anisotropy
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deriving Repr, Inhabited
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/-- Compute basic Christoffel symbols
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-- Arithmetic sanity check: Christoffel symbols for flat space are zero
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-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
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-- Wolfram-verified unless an API result, saved query output, or reproducible
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-- external artifact is attached.
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-/
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def computeBasicChristoffel (manifold : BasicManifold) : BasicChristoffel :=
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let n := manifold.dimension
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let symbolCount := n * n * n
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let symbols := Array.replicate symbolCount zero
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let rec computeSymbol (i j k : Nat) (acc : Array Q16_16) : Array Q16_16 :=
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if i >= n then acc
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else if j >= n then computeSymbol (i + 1) 0 0 acc
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else if k >= n then computeSymbol i (j + 1) 0 acc
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else
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let symbol := if i = j ∧ j = k then zero else zero
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let idx := i * n * n + j * n + k
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computeSymbol i j (k + 1) (acc.set! idx symbol)
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let result := computeSymbol 0 0 0 symbols
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{ dimension := n, symbols := result }
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#eval computeBasicChristoffel { dimension := 3, curvature := zero, torsion := zero, metric := #[zero, zero, zero] }
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/-- Compute cosine using Taylor series for Q16_16
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--
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-- Arithmetic sanity check:
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-- cos(x) ≈ 1 - x²/2 for small x (Taylor series approximation).
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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 basicCos (x : Q16_16) : Q16_16 :=
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let x2 := mul x x
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let term2 := mul x2 (div (ofInt 1) (ofInt 2))
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one - term2
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#eval basicCos zero
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#eval basicCos (ofInt 1)
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/-- Compute basic frustration W(z;A) = Σ_r w_r(A)(1 - cos(k_r·z))
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-- Arithmetic sanity check: frustration = Σ w_r(1 - cos(k_r·z))
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-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
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-- Wolfram-verified unless an API result, saved query output, or reproducible
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-- external artifact is attached.
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-/
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def computeBasicFrustration (z : BasicLockPattern) (waves : Array BasicFrustrationWave) : Q16_16 :=
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let zArray := #[z.value, z.derivative, z.momentum, zero]
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let rec sumWaves (i : Nat) (acc : Q16_16) : Q16_16 :=
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if i >= waves.size then acc
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else
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let wave := waves[i]!
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let rec dotProduct (j : Nat) (sum : Q16_16) : Q16_16 :=
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if j >= 4 then sum
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else dotProduct (j + 1) (sum + zArray[j]! * wave.waveVector[j]!)
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let dot := dotProduct 0 zero
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let cosine := basicCos dot
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let contribution := mul wave.weight (one - cosine)
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sumWaves (i + 1) (acc + contribution)
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sumWaves 0 zero
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#eval computeBasicFrustration { value := zero, derivative := zero, momentum := zero } #[]
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/-- Compute basic locking energy for stability
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--
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-- Arithmetic sanity check:
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-- locking energy = frustration of state difference.
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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 computeBasicLockingEnergy (currentPattern previousPattern : BasicLockPattern) (waves : Array BasicFrustrationWave) : Q16_16 :=
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let z := {
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value := currentPattern.value - previousPattern.value,
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derivative := currentPattern.derivative - previousPattern.derivative,
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momentum := currentPattern.momentum - previousPattern.momentum
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}
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computeBasicFrustration z waves
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#eval computeBasicLockingEnergy { value := zero, derivative := zero, momentum := zero } { value := zero, derivative := zero, momentum := zero } #[]
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/-- Update discrete basic state from geometry
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--
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-- Arithmetic sanity check:
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-- state update with curvature and torsion.
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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 updateBasicStateFromGeometry (state : DiscreteBasicState) (manifold : BasicManifold) : DiscreteBasicState :=
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let newValue := state.value + manifold.curvature
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let newDerivative := state.derivative + manifold.torsion
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{
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value := newValue,
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derivative := newDerivative,
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integral := state.integral,
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momentum := state.momentum
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}
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#eval updateBasicStateFromGeometry { value := zero, derivative := zero, integral := zero, momentum := zero } { dimension := 3, curvature := zero, torsion := zero, metric := #[zero, zero, zero] }
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/-- Update discrete basic state from Christoffel symbols
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--
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-- Arithmetic sanity check:
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-- integral increment based on Christoffel symbol threshold.
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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 updateBasicStateFromChristoffel (state : DiscreteBasicState) (symbols : BasicChristoffel) (i j k : Nat) : DiscreteBasicState :=
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let symbol := symbols.symbols[i * symbols.dimension * symbols.dimension + j * symbols.dimension + k]!
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let integralIncrement := if symbol > ofInt 100 then one else zero
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{
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value := state.value,
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derivative := state.derivative,
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integral := state.integral + integralIncrement,
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momentum := state.momentum
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}
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#eval updateBasicStateFromChristoffel { value := zero, derivative := zero, integral := zero, momentum := zero } { dimension := 3, symbols := Array.replicate 27 zero } 0 0 0
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Original Basic Function
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-- ═══════════════════════════════════════════════════════════════════════════
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def hello := "world"
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end Semantics.Basic
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