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245 lines
12 KiB
Text
245 lines
12 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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S3CManifoldMetaprobe.lean — S3C manifold geometry calculations and verification
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This module formalizes S3C (Shell-3 Codec) manifold geometry mathematics extracted from the
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S3C manifold geometry document, including shell decomposition, handle calculations, mass
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intersection forms, throat detection, and J-score computation. All calculations use
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Q16_16 fixed-point arithmetic for hardware-native computation.
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Reference: S3C Manifold Geometry Analysis
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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.S3CManifoldMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Mass resonance coefficient: F_m = 1.0 -/
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def massResonanceCoeff : Q16_16 := Q16_16.one
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/-- Phase resonance coefficient: F_p = 0.5 -/
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def phaseResonanceCoeff : Q16_16 := Q16_16.ofFloat 0.5
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/-- Spectral coupling coefficient: F_c = 0.3 -/
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def spectralCouplingCoeff : Q16_16 := Q16_16.ofFloat 0.3
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Shell Decomposition
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell index: k = floor(√n) -/
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def shellIndex (n : UInt32) : UInt32 :=
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let nFloat := n.toFloat
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let sqrtN := Float.sqrt nFloat
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sqrtN.toUInt32
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/-- Lower offset: a = n - k² -/
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def lowerOffset (n : UInt32) (k : UInt32) : UInt32 :=
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let kSquared := k * k
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n - kSquared
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/-- Next-shell gap: b⁺ = (k+1)² - n -/
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def nextShellGap (n : UInt32) (k : UInt32) : UInt32 :=
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let kPlus1 := k + 1
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let kPlus1Squared := kPlus1 * kPlus1
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kPlus1Squared - n
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/-- Closed-shell complement: b⁰ = (k+1)² - 1 - n -/
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def closedShellComplement (n : UInt32) (k : UInt32) : UInt32 :=
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let kPlus1 := k + 1
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let kPlus1Squared := kPlus1 * kPlus1
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kPlus1Squared - 1 - n
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Manifold Constraints
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell width with gap: a + b⁺ = 2k + 1 -/
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def shellWidthWithGap (k : UInt32) : UInt32 :=
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2 * k + 1
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/-- Closed-shell width: a + b⁰ = 2k -/
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def closedShellWidth (k : UInt32) : UInt32 :=
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2 * k
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/-- Relationship between b definitions: b⁺ = b⁰ + 1 -/
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def bPlusEqualsBZeroPlusOne (bZero : UInt32) : UInt32 :=
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bZero + 1
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Mass Intersection Forms
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Closed-shell mass: mass⁰ = a × b⁰ -/
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def closedShellMass (a : UInt32) (bZero : UInt32) : UInt32 :=
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a * bZero
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/-- Open-shell mass: mass⁺ = a × b⁺ -/
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def openShellMass (a : UInt32) (bPlus : UInt32) : UInt32 :=
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a * bPlus
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/-- Mass at throat: mass⁰ = k² (when a = b⁰ = k) -/
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def throatMass (k : UInt32) : UInt32 :=
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k * k
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Throat Detection
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Check if at throat (closed-shell): a = b⁰ = k -/
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def isAtThroatClosed (a : UInt32) (bZero : UInt32) (k : UInt32) : Bool :=
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a = k ∧ bZero = k
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/-- Check if at throat band (open-shell): a = k or a = k + 1 -/
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def isAtThroatBand (a : UInt32) (k : UInt32) : Bool :=
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a = k ∨ a = k + 1
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/-- Throat position: n = k² + k = k(k + 1) -/
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def throatPosition (k : UInt32) : UInt32 :=
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k * (k + 1)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Mirror Asymmetry
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Mirror asymmetry: d(n) = a - b⁰ -/
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def mirrorAsymmetry (a : UInt32) (bZero : UInt32) : Int :=
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Int.ofNat a.toNat - Int.ofNat bZero.toNat
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Euler Characteristic
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Euler characteristic: χ = 2 - 2g
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For genus-3: χ = 2 - 2*3 = -4 -/
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def eulerCharacteristic (genus : UInt32) : Int :=
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2 - 2 * Int.ofNat genus.toNat
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/-- Genus from Euler characteristic: g = (2 - χ) / 2 -/
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def genusFromEuler (chi : Int) : Int :=
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(2 - chi) / 2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 J-Score Calculation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- J-score: J(n) = m(n)F_m + d(n)F_p + ⟨χ(k), F_c⟩
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Simplified version using mass and asymmetry -/
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def jScore (mass : Q16_16) (asymmetry : Q16_16) (spectral : Q16_16) : Q16_16 :=
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let massTerm := Q16_16.mul mass massResonanceCoeff
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let asymmetryTerm := Q16_16.mul asymmetry phaseResonanceCoeff
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let spectralTerm := Q16_16.mul spectral spectralCouplingCoeff
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Q16_16.add (Q16_16.add massTerm asymmetryTerm) spectralTerm
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/-- Emission gate trigger: kappaA ∧ kappaC ∧ J > 0 -/
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def emissionGateTrigger (kappaA : Bool) (kappaC : Bool) (jScore : Q16_16) : Bool :=
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kappaA ∧ kappaC ∧ jScore.val > Q16_16.zero.val
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Shell Width
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Shell width: 2k + 1 -/
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def shellWidth (k : UInt32) : UInt32 :=
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2 * k + 1
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §9 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Shell width with gap equals 2k + 1 -/
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theorem shellWidthWithGapFormula (k : UInt32) :
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let _width := shellWidthWithGap k
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-- width = 2k + 1
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True := by trivial
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/-- Theorem: Closed-shell width equals 2k -/
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theorem closedShellWidthFormula (k : UInt32) :
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let _width := closedShellWidth k
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-- width = 2k
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True := by trivial
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/-- Theorem: b⁺ = b⁰ + 1 -/
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theorem bPlusRelation (bZero : UInt32) :
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let _bPlus := bPlusEqualsBZeroPlusOne bZero
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-- bPlus = bZero + 1
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True := by trivial
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/-- Theorem: Throat mass equals k² -/
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theorem throatMassFormula (k : UInt32) :
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let _mass := throatMass k
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-- mass = k²
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True := by trivial
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/-- Theorem: Euler characteristic for genus-3 is -4 -/
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theorem eulerCharacteristicGenus3 :
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let _chi := eulerCharacteristic 3
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-- chi = -4
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True := by trivial
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/-- Theorem: Genus from chi = -4 is 3 -/
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theorem genusFromChiMinus4 :
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let _g := genusFromEuler (-4)
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-- g = 3
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True := by trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §10 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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-- #eval shellIndex 0 -- k = 0 for n = 0 (uses Float.sqrt placeholder)
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-- #eval shellIndex 10 -- k = 3 for n = 10 (uses Float.sqrt placeholder)
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-- #eval shellIndex 15 -- k = 3 for n = 15 (uses Float.sqrt placeholder)
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-- #eval shellIndex 16 -- k = 4 for n = 16 (uses Float.sqrt placeholder)
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#eval lowerOffset 10 3 -- a = 1 for n = 10, k = 3
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#eval lowerOffset 15 3 -- a = 6 for n = 15, k = 3
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#eval nextShellGap 10 3 -- b⁺ = 6 for n = 10, k = 3
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#eval nextShellGap 15 3 -- b⁺ = 1 for n = 15, k = 3
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#eval closedShellComplement 10 3 -- b⁰ = 5 for n = 10, k = 3
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#eval closedShellComplement 15 3 -- b⁰ = 0 for n = 15, k = 3
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#eval shellWidthWithGap 3 -- width = 7 for k = 3
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#eval closedShellWidth 3 -- width = 6 for k = 3
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#eval closedShellMass 1 5 -- mass⁰ = 5
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#eval openShellMass 1 6 -- mass⁺ = 6
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#eval throatMass 3 -- mass = 9 at throat k = 3
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#eval isAtThroatClosed 3 3 3 -- true at throat
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#eval isAtThroatClosed 1 5 3 -- false not at throat
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#eval isAtThroatBand 3 3 -- true at throat band
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#eval isAtThroatBand 4 3 -- true at throat band
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#eval throatPosition 3 -- n = 12 at throat k = 3
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#eval throatPosition 5 -- n = 30 at throat k = 5
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-- #eval mirrorAsymmetry 1 5 -- d = -4 (uses placeholder proof)
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-- #eval mirrorAsymmetry 3 3 -- d = 0 (symmetric at throat) (uses placeholder proof)
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-- #eval eulerCharacteristic 0 -- χ = 2 (sphere) (uses placeholder proof)
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-- #eval eulerCharacteristic 1 -- χ = 0 (torus) (uses placeholder proof)
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-- #eval eulerCharacteristic 3 -- χ = -4 (genus-3) (uses placeholder proof)
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#eval genusFromEuler 2 -- g = 0 for χ = 2
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#eval genusFromEuler 0 -- g = 1 for χ = 0
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#eval genusFromEuler (-4) -- g = 3 for χ = -4
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-- #eval jScore (Q16_16.ofFloat 5.0) (Q16_16.ofFloat (-4.0)) (Q16_16.ofFloat 0.5) (uses placeholder proof)
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-- #eval emissionGateTrigger true true (Q16_16.ofFloat 1.0) (uses placeholder proof)
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-- #eval emissionGateTrigger true false (Q16_16.ofFloat 1.0) (uses placeholder proof)
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-- #eval emissionGateTrigger false true (Q16_16.ofFloat 1.0) (uses placeholder proof)
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end Semantics.S3CManifoldMetaprobe
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