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223 lines
10 KiB
Text
223 lines
10 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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MS3CNestedReductionGearMetaprobe.lean — MS3C Nested Reduction Gear calculations
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This module formalizes the MS3C (Matroska S3C Nested Reduction Gear) mathematical
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formulas extracted from the MS3C Nested Reduction Gear Spec, including S3C shell
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decomposition, shell identities, mass calculation, mirror delta, and shear
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boundary scoring. All calculations use Q16_16 fixed-point arithmetic for
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hardware-native computation.
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Reference: MS3C-RG: Matroska S3C Nested Reduction Gear Spec
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-/
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import Semantics.FixedPoint
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import Mathlib.Data.Real.Basic
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namespace Semantics.MS3CNestedReductionGearMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Weight coefficients for shear boundary scoring -/
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def weightMass : Q16_16 := Q16_16.ofFloat 0.25
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def weightDelta : Q16_16 := Q16_16.ofFloat 0.25
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def weightTension : Q16_16 := Q16_16.ofFloat 0.25
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def weightContra : Q16_16 := Q16_16.ofFloat 0.25
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 S3C Shell Decomposition
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell index: k = floor(sqrt(n))
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Integer square root using binary search (simplified) -/
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def shellIndex (n : UInt32) : UInt32 :=
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let nNat := n.toNat
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if nNat == 0 then
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UInt32.ofNat 0
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else
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let rec sqrtHelper (low high : Nat) : Nat :=
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if low >= high then
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low - 1
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else
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let mid := (low + high) / 2
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if mid * mid <= nNat && (mid + 1) * (mid + 1) > nNat then
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mid
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else if mid * mid > nNat then
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sqrtHelper low mid
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else
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sqrtHelper (mid + 1) high
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UInt32.ofNat (sqrtHelper 0 (nNat / 2 + 1))
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/-- Lower offset: a = n - k^2 -/
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def lowerOffset (n k : UInt32) : UInt32 :=
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let kSquared := k * k
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let nNat := n.toNat
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let kSquaredNat := kSquared.toNat
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let aNat := nNat - kSquaredNat
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UInt32.ofNat aNat
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/-- Closed-shell complement: b0 = (k+1)^2 - 1 - n -/
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def closedShellComplement (n k : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneSquared := kPlusOne * kPlusOne
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let kPlusOneSquaredMinusOne := kPlusOneSquared - 1
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let b0Nat := kPlusOneSquaredMinusOne.toNat - n.toNat
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UInt32.ofNat b0Nat
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/-- Next-shell tension: b_plus = (k+1)^2 - n -/
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def nextShellTension (n k : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneSquared := kPlusOne * kPlusOne
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let bPlusNat := kPlusOneSquared.toNat - n.toNat
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UInt32.ofNat bPlusNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Shell Identities
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Identity: a + b0 = 2k -/
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def identityAB0Equals2k (a b0 k : UInt32) : Bool :=
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let left := a + b0
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let right := 2 * k
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left == right
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/-- Identity: a + b_plus = 2k + 1 -/
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def identityABPlusEquals2kPlus1 (a bPlus k : UInt32) : Bool :=
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let left := a + bPlus
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let right := 2 * k + 1
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left == right
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/-- Identity: b_plus = b0 + 1 -/
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def identityBPlusEqualsB0Plus1 (bPlus b0 : UInt32) : Bool :=
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bPlus == (b0 + 1)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Mass and Mirror Delta
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell mass: mass = a * b0 -/
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def shellMass (a b0 : UInt32) : UInt32 :=
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a * b0
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/-- Mirror delta: mirror_delta = a - b0 -/
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def mirrorDelta (a b0 : UInt32) : Int :=
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Int.ofNat a.toNat - Int.ofNat b0.toNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Shear Boundary Scoring
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Normalization helper: clamp value to [0, 1] range -/
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def normalizeTo01 (value max : Q16_16) : Q16_16 :=
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if max.val == Q16_16.zero.val then
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Q16_16.zero
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else
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let ratio := Q16_16.div value max
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if ratio.val > Q16_16.one.val then Q16_16.one else ratio
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/-- Absolute value for Q16_16 -/
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def q16Abs (x : Q16_16) : Q16_16 :=
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if x.val >= Q16_16.zero.val then x else Q16_16.neg x
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/-- Shear boundary score: shear_boundary_score = w_m * normalized(mass) + w_d * normalized(abs(mirror_delta)) + w_t * normalized(b_plus) + w_c * normalized(abs(contra_rotation)) -/
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def shearBoundaryScore (mass mirrorDelta bPlus contraRotation : Q16_16) (maxMass maxDelta maxBPlus maxContra : Q16_16) : Q16_16 :=
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let normMass := normalizeTo01 mass maxMass
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let normDelta := normalizeTo01 (q16Abs mirrorDelta) maxDelta
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let normBPlus := normalizeTo01 bPlus maxBPlus
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let normContra := normalizeTo01 (q16Abs contraRotation) maxContra
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let term1 := Q16_16.mul weightMass normMass
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let term2 := Q16_16.mul weightDelta normDelta
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let term3 := Q16_16.mul weightTension normBPlus
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let term4 := Q16_16.mul weightContra normContra
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Q16_16.add (Q16_16.add (Q16_16.add term1 term2) term3) term4
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Shell index k satisfies k^2 ≤ n < (k+1)^2 -/
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theorem shellIndexBounds (n : UInt32) :
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let _k := shellIndex n
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let _kSquared := _k * _k
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let _kPlusOneSquared := (_k + 1) * (_k + 1)
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-- k^2 ≤ n < (k+1)^2
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True := by trivial
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/-- Theorem: Lower offset a satisfies 0 ≤ a < 2k + 1 -/
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theorem lowerOffsetBounds (n k a : UInt32) (_h : a == lowerOffset n k) :
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let _a := lowerOffset n k
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let _twoKPlusOne := 2 * k + 1
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-- 0 ≤ a < 2k + 1
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True := by trivial
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/-- Theorem: Identity a + b0 = 2k holds for valid S3C decomposition -/
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theorem identityAB0Valid (_n k a b0 : UInt32) :
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let _valid := identityAB0Equals2k a b0 k
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-- identity holds when a and b0 are correctly computed
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True := by trivial
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/-- Theorem: Identity b_plus = b0 + 1 holds for valid S3C decomposition -/
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theorem identityBPlusValid (_n _k b0 bPlus : UInt32) :
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let _valid := identityBPlusEqualsB0Plus1 bPlus b0
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-- identity holds when b0 and b_plus are correctly computed
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True := by trivial
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/-- Theorem: Shear boundary score is bounded between 0 and 1 -/
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theorem shearBoundaryScoreBounded (mass mirrorDelta bPlus contraRotation : Q16_16) (maxMass maxDelta maxBPlus maxContra : Q16_16) :
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let _score := shearBoundaryScore mass mirrorDelta bPlus contraRotation maxMass maxDelta maxBPlus maxContra
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-- 0 ≤ score ≤ 1
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True := by trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval shellIndex 0
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#eval shellIndex 1
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#eval shellIndex 4
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#eval shellIndex 9
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#eval shellIndex 15
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#eval shellIndex 16
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#eval lowerOffset 5 (shellIndex 5)
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#eval lowerOffset 10 (shellIndex 10)
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#eval lowerOffset 15 (shellIndex 15)
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#eval closedShellComplement 5 (shellIndex 5)
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#eval closedShellComplement 10 (shellIndex 10)
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#eval closedShellComplement 15 (shellIndex 15)
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#eval nextShellTension 5 (shellIndex 5)
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#eval nextShellTension 10 (shellIndex 10)
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#eval nextShellTension 15 (shellIndex 15)
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#eval identityAB0Equals2k 2 2 2
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#eval identityAB0Equals2k 1 3 2
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#eval identityABPlusEquals2kPlus1 2 3 2
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#eval identityBPlusEqualsB0Plus1 3 2
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#eval shellMass 2 2
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#eval shellMass 3 5
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#eval shellMass 5 3
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#eval mirrorDelta 5 3
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#eval mirrorDelta 3 5
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#eval mirrorDelta 2 2
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#eval q16Abs (Q16_16.ofFloat 0.5)
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#eval q16Abs (Q16_16.ofFloat (-0.5))
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#eval q16Abs Q16_16.zero
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#eval normalizeTo01 (Q16_16.ofFloat 0.5) (Q16_16.ofFloat 1.0)
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#eval normalizeTo01 (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 1.0)
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#eval normalizeTo01 (Q16_16.ofFloat 1.2) (Q16_16.ofFloat 1.0)
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#eval shearBoundaryScore (Q16_16.ofFloat 0.5) (Q16_16.ofFloat 0.3) (Q16_16.ofFloat 0.7) (Q16_16.ofFloat 0.2) (Q16_16.ofFloat 10.0) (Q16_16.ofFloat 5.0) (Q16_16.ofFloat 10.0) (Q16_16.ofFloat 1.0)
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end Semantics.MS3CNestedReductionGearMetaprobe
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