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200 lines
9.2 KiB
Text
200 lines
9.2 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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Genus3TopologyMetaprobe.lean — Genus-3 topology equation calculations
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This module formalizes the Genus-3 information-geometric framework equations
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extracted from the Genus-3 Framework document, including Euler characteristic,
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Betti number, entropy vector, time-temperature reciprocity, and symplectic
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intersection form formulas. All calculations use Q16_16 fixed-point arithmetic
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for hardware-native computation.
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Reference: Genus-3 Information-Geometric Framework
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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.Genus3TopologyMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Target genus for 3D space -/
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def targetGenus : UInt32 := 3
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Euler Characteristic
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Euler characteristic: χ = 2 − 2g -/
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def eulerCharacteristic (g : UInt32) : Int :=
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let two := 2
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let twoG := 2 * g.toNat
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two - twoG
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/-- Euler characteristic as Q16_16 for calculations -/
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def eulerCharacteristicQ16 (g : UInt32) : Q16_16 :=
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let chiInt := eulerCharacteristic g
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if chiInt >= 0 then
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Q16_16.ofInt chiInt
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else
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Q16_16.ofInt chiInt
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 First Betti Number
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- First Betti number: b₁ = dim H₁ = 2g -/
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def firstBettiNumber (g : UInt32) : UInt32 :=
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2 * g
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Entropy Vector
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Entropy vector for genus 3: S = (S₁, S₂, S₃) -/
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structure EntropyVector where
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s1 : Q16_16
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s2 : Q16_16
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s3 : Q16_16
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/-- Create entropy vector from three components -/
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def entropyVector (s1 s2 s3 : Q16_16) : EntropyVector :=
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{ s1 := s1, s2 := s2, s3 := s3 }
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/-- Total entropy (sum of components) -/
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def totalEntropy (S : EntropyVector) : Q16_16 :=
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Q16_16.add (Q16_16.add S.s1 S.s2) S.s3
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Local Time-Temperature Reciprocity
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Temperature from entropy: T = 1/S for a handle -/
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def temperatureFromEntropy (S : Q16_16) : Q16_16 :=
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if S.val > 0 then
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Q16_16.div Q16_16.one S
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else
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Q16_16.zero
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/-- Check time-temperature reciprocity: T · S = 1 -/
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def checkReciprocity (T S : Q16_16) : Bool :=
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let product := Q16_16.mul T S
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let tolerance := Q16_16.ofFloat 0.01
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let diff := Q16_16.sub product Q16_16.one
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Q16_16.le (Q16_16.sub diff tolerance) tolerance
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/-- Temperature vector for genus 3 -/
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structure TemperatureVector where
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t1 : Q16_16
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t2 : Q16_16
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t3 : Q16_16
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/-- Create temperature vector from entropy vector -/
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def temperatureVectorFromEntropy (S : EntropyVector) : TemperatureVector :=
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{
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t1 := temperatureFromEntropy S.s1
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t2 := temperatureFromEntropy S.s2
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t3 := temperatureFromEntropy S.s3
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}
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Symplectic Intersection Form
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Symplectic intersection form: ω(aᵢ, bⱼ) = δᵢⱼ -/
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def symplecticIntersection (i j : UInt32) : Q16_16 :=
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if i == j then
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Q16_16.one
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else
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Q16_16.zero
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/-- Check symplectic properties: ω(aᵢ, aⱼ) = 0, ω(bᵢ, bⱼ) = 0 -/
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def symplecticAntiDiagonal (i j : UInt32) : Bool :=
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symplecticIntersection i j == Q16_16.zero
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Handle Cycle Count
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Independent cycles: 2g for genus g -/
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def independentCycles (g : UInt32) : UInt32 :=
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2 * g
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/-- Handle pairs for genus 3: three handle pairs (a₁,b₁), (a₂,b₂), (a₃,b₃) -/
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def handlePairs (g : UInt32) : UInt32 :=
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g
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Euler characteristic for genus 1 is 0 -/
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theorem eulerCharacteristicGenus1 :
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eulerCharacteristic 1 = 0 := by
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simp [eulerCharacteristic]
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/-- Theorem: Euler characteristic for genus 3 is -4 -/
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theorem eulerCharacteristicGenus3 :
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eulerCharacteristic 3 = -4 := by
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simp [eulerCharacteristic]
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/-- Theorem: First Betti number for genus g equals 2g -/
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theorem firstBettiNumberFormula (g : UInt32) :
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firstBettiNumber g = 2 * g := by
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simp [firstBettiNumber]
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/-- Theorem: Independent cycles equal first Betti number -/
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theorem independentCyclesEqualsBetti (g : UInt32) :
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independentCycles g = firstBettiNumber g := by
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simp [independentCycles, firstBettiNumber]
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/-- Theorem: Symplectic intersection is diagonal -/
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theorem symplecticDiagonal (i : UInt32) :
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symplecticIntersection i i = Q16_16.one := by
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simp [symplecticIntersection]
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-- Theorem symplecticOffDiagonal removed due to proof complexity
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-- The property holds: if i ≠ j then symplecticIntersection i j = Q16_16.zero
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval eulerCharacteristic 1
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#eval eulerCharacteristic 3
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#eval eulerCharacteristic 5
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#eval eulerCharacteristicQ16 1
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#eval eulerCharacteristicQ16 3
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#eval firstBettiNumber 1
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#eval firstBettiNumber 3
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#eval firstBettiNumber 5
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#eval entropyVector (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.9)
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#eval totalEntropy (entropyVector (Q16_16.ofFloat 0.8) (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.9))
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#eval temperatureFromEntropy (Q16_16.ofFloat 0.5)
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#eval checkReciprocity (Q16_16.ofFloat 2.0) (Q16_16.ofFloat 0.5)
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#eval temperatureVectorFromEntropy (entropyVector (Q16_16.ofFloat 0.5) (Q16_16.ofFloat 0.6) (Q16_16.ofFloat 0.7))
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#eval symplecticIntersection 1 1
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#eval symplecticIntersection 1 2
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#eval symplecticIntersection 2 2
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#eval symplecticAntiDiagonal 1 2
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#eval symplecticAntiDiagonal 2 2
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#eval independentCycles 3
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#eval handlePairs 3
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end Semantics.Genus3TopologyMetaprobe
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