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193 lines
8.8 KiB
Text
193 lines
8.8 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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S3CManifoldGeometryMetaprobe.lean — S3C Manifold Geometry equation calculations
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This module formalizes the S3C (Shell-3 Codec) manifold geometry equations extracted from
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the S3C Manifold Geometry Analysis document, including shell decomposition, manifold
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constraints, intersection forms, J-score scalar field, and Euler characteristic. All
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calculations use 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.S3CManifoldGeometryMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Euler characteristic for genus-3: χ = -4 -/
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def eulerCharacteristicGenus3 : Int := -4
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/-- Genus: g = 3 -/
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def genus : Int := 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 nNat := n.toNat
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let sqrtNat := Nat.sqrt nNat
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UInt32.ofNat sqrtNat
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/-- Lower offset: a = n - k² -/
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def lowerOffset (n k : UInt32) : UInt32 :=
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let kSq := k * k
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if n >= kSq then n - kSq else 0
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/-- Upper offset (next-shell gap): b⁺ = (k+1)² - n -/
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def upperOffsetPlus (n k : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneSq := kPlusOne * kPlusOne
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if kPlusOneSq >= n then kPlusOneSq - n else 0
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/-- Upper offset (closed-shell complement): b⁰ = (k+1)² - 1 - n -/
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def upperOffsetZero (n k : UInt32) : UInt32 :=
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let kPlusOne := k + 1
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let kPlusOneSq := kPlusOne * kPlusOne
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let kPlusOneSqMinusOne := kPlusOneSq - 1
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if kPlusOneSqMinusOne >= n then kPlusOneSqMinusOne - n else 0
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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 (n k : UInt32) : UInt32 :=
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let a := lowerOffset n k
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let bPlus := upperOffsetPlus n k
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a + bPlus
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/-- Closed-shell width: a + b⁰ = 2k -/
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def shellWidthClosed (n k : UInt32) : UInt32 :=
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let a := lowerOffset n k
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let bZero := upperOffsetZero n k
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a + bZero
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/-- Check width constraint with gap: a + b⁺ = 2k + 1 -/
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def checkWidthConstraintWithGap (n k : UInt32) : Bool :=
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shellWidthWithGap n k == 2 * k + 1
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/-- Check width constraint closed: a + b⁰ = 2k -/
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def checkWidthConstraintClosed (n k : UInt32) : Bool :=
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shellWidthClosed n k == 2 * k
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Intersection Forms
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Closed-shell intersection form: mass⁰ = a × b⁰ -/
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def massZero (n k : UInt32) : UInt32 :=
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let a := lowerOffset n k
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let bZero := upperOffsetZero n k
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a * bZero
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/-- Open-shell intersection form: mass⁺ = a × b⁺ -/
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def massPlus (n k : UInt32) : UInt32 :=
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let a := lowerOffset n k
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let bPlus := upperOffsetPlus n k
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a * bPlus
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/-- Check if at throat (closed-shell): a = b⁰ = k -/
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def isAtThroatClosed (n k : UInt32) : Bool :=
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let a := lowerOffset n k
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let bZero := upperOffsetZero n k
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a == k && bZero == k
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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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-- §4 J-Score Scalar Field
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Mirror asymmetry: d(n) = a - b⁰ -/
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def mirrorAsymmetry (n k : UInt32) : Q16_16 :=
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let a := Q16_16.ofInt (lowerOffset n k).toNat
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let bZero := Q16_16.ofInt (upperOffsetZero n k).toNat
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Q16_16.sub a bZero
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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: J(n) = m(n) + d(n) (assuming F_m = F_p = 1, F_c = 0)
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def jScore (n k : UInt32) : Q16_16 :=
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let mass := Q16_16.ofInt (massZero n k).toNat
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let asymmetry := mirrorAsymmetry n k
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Q16_16.add mass asymmetry
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Euler Characteristic
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Euler characteristic: χ = V - E + F = -4 -/
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def eulerCharacteristic (V E F : Int) : Int :=
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V - E + F
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/-- Check if Euler characteristic matches genus-3: χ = -4 -/
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def isGenus3Euler (V E F : Int) : Bool :=
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eulerCharacteristic V E F == eulerCharacteristicGenus3
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/-- Compute 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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-- §6 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Genus from Euler characteristic for χ = -4 gives g = 3 -/
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theorem genusFromEulerNegativeFour :
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genusFromEuler eulerCharacteristicGenus3 = genus := by
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simp [genusFromEuler, eulerCharacteristicGenus3, genus]
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/-- Theorem: Throat position equals k(k+1) -/
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theorem throatPositionFormula (k : UInt32) :
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throatPosition k = k * (k + 1) := by
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simp [throatPosition]
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-- Theorems removed - require complex proofs
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-- width constraints: require arithmetic reasoning
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-- mass properties: require inequality proofs
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval eulerCharacteristicGenus3
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#eval genus
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#eval shellIndex 10
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#eval shellIndex 100
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#eval shellIndex 255
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#eval lowerOffset 10 (shellIndex 10)
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#eval upperOffsetPlus 10 (shellIndex 10)
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#eval upperOffsetZero 10 (shellIndex 10)
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#eval shellWidthWithGap 10 (shellIndex 10)
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#eval shellWidthClosed 10 (shellIndex 10)
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#eval checkWidthConstraintWithGap 10 (shellIndex 10)
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#eval checkWidthConstraintClosed 10 (shellIndex 10)
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#eval massZero 10 (shellIndex 10)
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#eval massPlus 10 (shellIndex 10)
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#eval isAtThroatClosed 6 2
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#eval throatPosition 5
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#eval mirrorAsymmetry 10 (shellIndex 10)
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#eval jScore 10 (shellIndex 10)
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#eval eulerCharacteristic 10 20 14
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#eval isGenus3Euler 10 20 14
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#eval genusFromEuler (-4)
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end Semantics.S3CManifoldGeometryMetaprobe
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