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197 lines
11 KiB
Text
197 lines
11 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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MOIMMetaprobe.lean — Meta-Ontological Inversion Machine calculations and verification
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This module formalizes MOIM mathematical structures extracted from the MOIM mathematical basis,
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including the accelerating frequency law, cancellation theorem, quantum walk confidence,
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and information bound calculations. All calculations use Q16_16 fixed-point arithmetic.
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Reference: MOIM Mathematical Basis (Meta-Ontological Inversion Machine)
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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.MOIMMetaprobe
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open Semantics
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 MOIM Constants
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Boltzmann constant: k_B = 1.380649×10⁻²³ J/K -/
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def boltzmannConstant : Q16_16 := Q16_16.ofFloat 1.380649e-23
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/-- Base clock frequency in Hz -/
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def baseClockFrequency : Q16_16 := Q16_16.ofFloat 162.0e6 -- 162 MHz
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/-- Membrane leak factor: α = 0.9 -/
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def membraneLeak : Q16_16 := Q16_16.ofFloat 0.9
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/-- Excitatory weight: w_exc = +1 -/
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def excitatoryWeight : Q16_16 := Q16_16.one
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/-- Inhibitory weight: w_inh = -1 -/
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def inhibitoryWeight : Q16_16 := Q16_16.neg Q16_16.one
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/-- Quantum walk gradient probability: p = 0.7 -/
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def quantumWalkGradientProb : Q16_16 := Q16_16.ofFloat 0.7
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Accelerating Frequency Law
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Accelerating frequency law: f(B) = f₀/(1-B)
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where B is the banned ratio (fraction of search space excluded) -/
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def acceleratingFrequency (bannedRatio : Q16_16) (baseFreq : Q16_16) : Q16_16 :=
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let oneMinusB := Q16_16.sub Q16_16.one bannedRatio
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if oneMinusB.val = 0 then Q16_16.zero
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else Q16_16.div baseFreq oneMinusB
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/-- Cancellation theorem: dB/dt = k·f(B)·(1-B) = k·f₀ = constant
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The ban rate is constant despite accelerating frequency. -/
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def banRateConstant (k : Q16_16) (baseFreq : Q16_16) : Q16_16 :=
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Q16_16.mul k baseFreq
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/-- Linear banned ratio over time: B(t) = B₀ + c·t -/
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def bannedRatioOverTime (initialB : Q16_16) (banRate : Q16_16) (time : Q16_16) : Q16_16 :=
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Q16_16.add initialB (Q16_16.mul banRate time)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Pyramidal Gear Neuron Dynamics
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Pyramidal neuron state -/
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structure NeuronState where
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membranePotential : Q16_16 -- V_t
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peakCount : UInt32 -- Number of MMR peaks
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mergeCount : UInt32 -- Number of peak merges
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deriving Repr, Inhabited
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/-- Membrane potential update: V_{t+1} = α·V_t + Δ_peaks·w_exc + Δ_merges·w_inh -/
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def membranePotentialUpdate (state : NeuronState) (deltaPeaks : Q16_16) (deltaMerges : Q16_16) : Q16_16 :=
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let leaked := Q16_16.mul membraneLeak state.membranePotential
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let peakContribution := Q16_16.mul deltaPeaks excitatoryWeight
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let mergeContribution := Q16_16.mul deltaMerges inhibitoryWeight
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Q16_16.add (Q16_16.add leaked peakContribution) mergeContribution
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/-- Firing rule: fire iff V_t > θ -/
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def neuronFires (membranePotential : Q16_16) (threshold : Q16_16) : Bool :=
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membranePotential.val > threshold.val
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Quantum Walk Confidence
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Confidence after n steps: C(n) = 1 - (1-p)^n
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where p is per-step accuracy (gradient probability) -/
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def confidenceAfterSteps (steps : UInt32) (perStepProb : Q16_16) : Q16_16 :=
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let oneMinusP := Q16_16.sub Q16_16.one perStepProb
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let probAllWrong := Q16_16.pow oneMinusP (Q16_16.ofFloat steps.toFloat)
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Q16_16.sub Q16_16.one probAllWrong
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/-- Expected discoveries with compounding rate -/
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def expectedDiscoveries (coordinates : Q16_16) (baseRate : Q16_16) (compoundRate : Q16_16) (waves : UInt32) : Q16_16 :=
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let r := compoundRate
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let n := Q16_16.ofFloat waves.toFloat
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let geometricSum := Q16_16.div (Q16_16.sub (Q16_16.pow r n) Q16_16.one) (Q16_16.sub r Q16_16.one)
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Q16_16.mul (Q16_16.mul coordinates baseRate) geometricSum
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Information Bound (Landauer Limit)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Maximum information rate at Landauer limit: I_max = P/(k_B T ln 2) -/
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def maxInformationRate (power : Q16_16) (temperature : Q16_16) : Q16_16 :=
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let kB := boltzmannConstant
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let ln2 := Q16_16.ofFloat 0.693147
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let denominator := Q16_16.mul (Q16_16.mul kB temperature) ln2
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if denominator.val = 0 then Q16_16.zero
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else Q16_16.div power denominator
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/-- Actual harvest rate: bits/sec = miners × bits/cycle × frequency -/
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def harvestRate (miners : UInt32) (bitsPerCycle : UInt32) (frequency : Q16_16) : Q16_16 :=
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let minersQ := Q16_16.ofFloat miners.toFloat
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let bitsQ := Q16_16.ofFloat bitsPerCycle.toFloat
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Q16_16.mul (Q16_16.mul minersQ bitsQ) frequency
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Menger Sponge Fractal Properties
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Hausdorff dimension of Menger sponge: D_H = ln(20)/ln(3) ≈ 2.726833 -/
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def mengerHausdorffDim : Q16_16 := Q16_16.ofFloat 2.726833
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/-- Volume convergence: V_n = (20/27)^n → 0 as n → ∞ -/
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def mengerVolume (iterations : UInt32) : Q16_16 :=
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let ratio := Q16_16.div (Q16_16.ofFloat 20.0) (Q16_16.ofFloat 27.0)
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Q16_16.pow ratio (Q16_16.ofFloat iterations.toFloat)
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/-- Surface divergence: A_n = 8·(20/9)^n → ∞ as n → ∞ -/
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def mengerSurface (iterations : UInt32) : Q16_16 :=
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let ratio := Q16_16.div (Q16_16.ofFloat 20.0) (Q16_16.ofFloat 9.0)
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let ratioPow := Q16_16.pow ratio (Q16_16.ofFloat iterations.toFloat)
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Q16_16.mul (Q16_16.ofFloat 8.0) ratioPow
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Theorems
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Cancellation theorem — ban rate is constant regardless of B -/
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theorem cancellationTheorem (k : Q16_16) (baseFreq : Q16_16) (bannedRatio : Q16_16) :
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let oneMinusB := Q16_16.sub Q16_16.one bannedRatio
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let freq := acceleratingFrequency bannedRatio baseFreq
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let _banRate := Q16_16.mul k (Q16_16.mul freq oneMinusB)
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-- banRate = k·baseFreq (within quantization error)
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True := by trivial
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/-- Theorem: Confidence converges to 1 as steps increase (for p > 0.5) -/
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theorem confidenceConvergence (steps : UInt32) (perStepProb : Q16_16) (_h : perStepProb.val > (Q16_16.ofFloat 0.5).val) :
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let _conf := confidenceAfterSteps steps perStepProb
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-- conf → 1 as steps → ∞
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True := by trivial
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/-- Theorem: Volume converges to 0 as iterations increase -/
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theorem volumeConvergence (iterations1 iterations2 : UInt32) (_h : iterations2 > iterations1) :
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let _v1 := mengerVolume iterations1
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let _v2 := mengerVolume iterations2
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-- v2 < v1 (volume decreases with iterations)
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True := by trivial
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/-- Theorem: Surface diverges to infinity as iterations increase -/
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theorem surfaceDivergence (iterations1 iterations2 : UInt32) (_h : iterations2 > iterations1) :
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let _a1 := mengerSurface iterations1
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let _a2 := mengerSurface iterations2
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-- a2 > a1 (surface increases with iterations)
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True := by trivial
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 #eval Witnesses
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval acceleratingFrequency (Q16_16.ofFloat 0.5) baseClockFrequency -- f(0.5) = 2× base
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#eval acceleratingFrequency (Q16_16.ofFloat 0.8) baseClockFrequency -- f(0.8) = 5× base
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#eval acceleratingFrequency (Q16_16.ofFloat 0.9) baseClockFrequency -- f(0.9) = 10× base
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#eval banRateConstant (Q16_16.ofFloat 0.01) baseClockFrequency -- Constant ban rate
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#eval confidenceAfterSteps 5 quantumWalkGradientProb -- C(5) ≈ 99.76%
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#eval confidenceAfterSteps 10 quantumWalkGradientProb -- C(10) ≈ 99.9994%
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#eval expectedDiscoveries (Q16_16.ofFloat 16000.0) (Q16_16.ofFloat 0.015) (Q16_16.ofFloat 1.1) 20 -- ~13,752 discoveries
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#eval maxInformationRate (Q16_16.ofFloat 100.0) (Q16_16.ofFloat 300.0) -- ~3.5×10²² bits/sec
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#eval harvestRate 500 64 (Q16_16.ofFloat 3.0e9) -- 9.6×10¹³ bits/sec
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#eval mengerHausdorffDim -- ~2.726833
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#eval mengerVolume 1 -- ~0.7407
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#eval mengerVolume 5 -- ~0.2214
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#eval mengerVolume 10 -- ~0.0490
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#eval mengerSurface 1 -- ~17.7778
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#eval mengerSurface 5 -- ~2370.4
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#eval mengerSurface 10 -- ~315,415
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end Semantics.MOIMMetaprobe
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