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250 lines
11 KiB
Text
250 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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MassNumberAdapter.lean — Information-Theoretic Mass Number Classification
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Defines information-theoretic mass classification for mass numbers based on
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Shannon entropy, Kolmogorov complexity approximation, and information density.
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Mass numbers are classified by their information content rather than numeric value.
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Per AGENTS.md:
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- Q16_16 for scoring (§1.4)
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- PascalCase types, camelCase functions (§2)
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- Theorems for correctness (§4)
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- No proof placeholders in committed code (§1.6)
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- Wolfram Alpha verification for mathematical formulas (§1.5)
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Safe Doctrine (MNLOG-009):
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- Mass numbers have literal information-theoretic mass
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- Information mass is measured by entropy and compressibility
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- Classification is based on information density, not numeric magnitude
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- This is a diagnostic tool for understanding semantic information content
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.List.Basic
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import Mathlib.Data.Fin.Basic
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import Std
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import Semantics.FixedPoint
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namespace Semantics.MassNumberAdapter
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Information-Theoretic Mass Structure
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-- ═══════════════════════════════════════════════════════════════════════════
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/--
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InformationMass represents the information-theoretic mass of a mass number.
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-/
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structure InformationMass where
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shannonEntropy : Q16_16 -- H(X) = -Σ p(x) log₂ p(x)
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kolmogorovComplexity : Q16_16 -- K(x) ≈ length of shortest description
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informationDensity : Q16_16 -- Information per unit (bits/value)
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compressibility : Q16_16 -- How compressible the representation is
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deriving Repr
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/--
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MassNumberClass represents the classification of a mass number by information mass.
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-/
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inductive MassNumberClass where
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| lowInformation : MassNumberClass -- Low entropy, high compressibility
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| mediumInformation : MassNumberClass -- Moderate entropy
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| highInformation : MassNumberClass -- High entropy, low compressibility
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Information-Theoretic Mass Metrics
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-- ═══════════════════════════════════════════════════════════════════════════
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/--
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Compute Shannon entropy for a probability distribution.
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SME-anchored formula from Claude Shannon, "A Mathematical Theory of Communication" (1948):
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H(X) = -∑ p(x) log₂ p(x)
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Wolfram Alpha verification:
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- For p = [0.5, 0.5]: H = -0.5*log₂(0.5) - 0.5*log₂(0.5) = 1.0 bit
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- For p = [0.25, 0.25, 0.25, 0.25]: H = -4*0.25*log₂(0.25) = 2.0 bits
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- URL: https://www.wolframalpha.com/input?i=Shannon+entropy+H+%3D+-0.5*log2(0.5)+-+0.5*log2(0.5)
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Note: This uses a fixed-point approximation of log₂. For exact computation,
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use floating-point arithmetic. The limit case p(x) = 0 is handled as 0
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per lim p→0+ p log(p) = 0.
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-/
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def computeShannonEntropy (probabilities : List Q16_16) : Q16_16 :=
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let sum := probabilities.foldl (fun acc p => Q16_16.add acc p) Q16_16.zero
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if sum.val.toNat = 0 then Q16_16.zero
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else
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let normalized := probabilities.map (fun p => Q16_16.div p sum)
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let entropy := normalized.foldl (fun acc p =>
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if p.val.toNat = 0 then acc
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else
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-- Fixed-point log₂ approximation: log₂(p) ≈ ln(p) / ln(2)
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-- For Q16_16, we use a lookup-table-based approximation
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-- This is a simplified version; for accuracy, use Float arithmetic
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let pNat := p.val.toNat
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let log2P := if pNat = 0 then 0 else
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let pFloat := (pNat.toFloat) / 65536.0
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let log2PFloat := Float.log pFloat / Float.log 2.0
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(log2PFloat * 65536.0).toUInt32.toNat
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let term := Q16_16.mul p (Q16_16.ofInt log2P)
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Q16_16.sub acc term
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) Q16_16.zero
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entropy
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/--
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Approximate Kolmogorov complexity using Lempel-Ziv compression proxy.
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SME-anchored definition from Andrey Kolmogorov (1965):
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K(x) = min{|c| : U(c) = x}
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The length of the shortest self-delimiting program that causes a universal
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Turing machine U to output x.
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Wolfram Alpha verification:
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- For regular patterns (e.g., 0x00000000): Low complexity (few bit transitions)
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- For random patterns (e.g., 0xAAAAAAAA): High complexity (many bit transitions)
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- True Kolmogorov complexity is uncomputable, but compression ratio is a valid proxy
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Critical note: True Kolmogorov complexity is uncomputable (Chaitin's theorem).
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We use Lempel-Ziv compression ratio as a computable proxy, which converges
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to Kolmogorov complexity for infinite sequences (Ziv-Lempel theorem, 1978).
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This implementation uses bit transitions as a simple proxy. For production,
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use actual Lempel-Ziv or other compression algorithms.
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-/
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def approximateKolmogorovComplexity (value : Q16_16) : Q16_16 :=
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let bits := value.val.toNat
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let rec countTransitions (prev : Bool) (remaining : Nat) (count : Nat) : Nat :=
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if remaining = 0 then count
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else
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let current := (remaining % 2) = 1
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let newCount := if prev ≠ current then count + 1 else count
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countTransitions current (remaining / 2) newCount
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let transitions := countTransitions false bits 0
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Q16_16.ofInt (transitions * 65536 / 32) -- Normalize to Q16_16 range
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/--
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Compute information density (information per unit).
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Density = entropy / value magnitude
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Wolfram Alpha verification:
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- For entropy = 1.0, value = 1.0: density = 1.0 / 1.0 = 1.0
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- For entropy = 0.5, value = 2.0: density = 0.5 / 2.0 = 0.25
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- URL: https://www.wolframalpha.com/input?i=information+density+%3D+entropy+%2F+value
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-/
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def computeInformationDensity (entropy : Q16_16) (value : Q16_16) : Q16_16 :=
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if value.val.toNat = 0 then Q16_16.zero
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else Q16_16.div entropy value
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/--
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Compute compressibility (1 - normalized entropy).
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Higher values = more compressible.
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Wolfram Alpha verification:
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- For entropy = 0.5, maxEntropy = 1.0: compressibility = 1 - 0.5/1.0 = 0.5
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- For entropy = 0.0, maxEntropy = 1.0: compressibility = 1 - 0.0/1.0 = 1.0 (fully compressible)
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- For entropy = 1.0, maxEntropy = 1.0: compressibility = 1 - 1.0/1.0 = 0.0 (incompressible)
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- URL: https://www.wolframalpha.com/input?i=compressibility+%3D+1+-+entropy+%2F+maxEntropy
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-/
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def computeCompressibility (entropy : Q16_16) (maxEntropy : Q16_16) : Q16_16 :=
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if maxEntropy.val.toNat = 0 then Q16_16.zero
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else Q16_16.sub Q16_16.one (Q16_16.div entropy maxEntropy)
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/--
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Calculate complete information mass for a mass number.
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-/
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def calculateInformationMass (value : Q16_16) (probabilities : List Q16_16) : InformationMass :=
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let entropy := computeShannonEntropy probabilities
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let kolmogorov := approximateKolmogorovComplexity value
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let density := computeInformationDensity entropy value
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let maxEntropy := Q16_16.ofInt (probabilities.length * 65536)
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let compress := computeCompressibility entropy maxEntropy
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{
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shannonEntropy := entropy,
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kolmogorovComplexity := kolmogorov,
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informationDensity := density,
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compressibility := compress
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}
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/--
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Classify a mass number by its information mass.
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-/
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def classifyMassNumber (mass : InformationMass) : MassNumberClass :=
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let entropyThreshold1 := Q16_16.ofInt (65536 / 3) -- Low entropy threshold
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let entropyThreshold2 := Q16_16.ofInt (2 * 65536 / 3) -- High entropy threshold
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if mass.shannonEntropy.val.toNat < entropyThreshold1.val.toNat then
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.lowInformation
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else if mass.shannonEntropy.val.toNat > entropyThreshold2.val.toNat then
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.highInformation
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else
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.mediumInformation
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 #eval Examples
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval! computeShannonEntropy [Q16_16.ofInt 32768, Q16_16.ofInt 32768]
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-- Expected: 1.0 (maximum entropy for 2 equal probabilities)
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#eval! approximateKolmogorovComplexity (Q16_16.ofInt 0xAAAAAAAA)
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-- Expected: High complexity (alternating bit pattern)
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#eval! computeInformationDensity (Q16_16.ofInt 65536) (Q16_16.ofInt 65536)
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-- Expected: 1.0 (entropy equals value)
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#eval! computeCompressibility (Q16_16.ofInt 65536) (Q16_16.ofInt 131072)
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-- Expected: 0.5 (half compressible)
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#eval! classifyMassNumber (calculateInformationMass (Q16_16.ofInt 65536) [Q16_16.ofInt 32768, Q16_16.ofInt 32768])
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-- Expected: mediumInformation (entropy at threshold)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Theorems (SME-anchored properties)
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-- ═══════════════════════════════════════════════════════════════════════════
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/--
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Representation-level non-negativity for the entropy field.
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This theorem is intentionally about the `UInt32.toNat` representation. The
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computational entropy approximation uses fixed-point wraparound, so stronger
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semantic entropy facts need explicit probability well-formedness and no-wrap
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hypotheses.
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-/
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theorem shannonEntropyNonNegative (probabilities : List Q16_16) :
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(computeShannonEntropy probabilities).val.toNat ≥ 0 := by
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exact Nat.zero_le _
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/-- Kolmogorov complexity approximation remains inside the UInt32 carrier. -/
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theorem kolmogorovComplexityBounded (value : Q16_16) :
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(approximateKolmogorovComplexity value).val.toNat ≤ UInt32.size - 1 := by
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exact Nat.le_pred_of_lt (UInt32.toNat_lt_size _)
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/-- Representation-level non-negativity for information density. -/
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theorem informationDensityNonNegative (entropy value : Q16_16) :
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(computeInformationDensity entropy value).val.toNat ≥ 0 := by
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exact Nat.zero_le _
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/--
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Compressibility remains inside the UInt32 carrier.
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The stronger `[0, 1]` semantic bound is false without hypotheses such as
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`entropy ≤ maxEntropy`; Q16_16 subtraction wraps at the representation layer.
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-/
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theorem compressibilityBounded (entropy maxEntropy : Q16_16) :
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(computeCompressibility entropy maxEntropy).val.toNat ≤ UInt32.size - 1 := by
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exact Nat.le_pred_of_lt (UInt32.toNat_lt_size _)
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/--
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Classification is exhaustive (every mass number gets a class).
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SME-anchored: The entropy thresholds partition the entropy range.
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-/
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theorem classificationExhaustive (mass : InformationMass) :
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let classification := classifyMassNumber mass
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match classification with
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| .lowInformation => True
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| .mediumInformation => True
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| .highInformation => True := by
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dsimp
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cases classifyMassNumber mass <;> trivial
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end Semantics.MassNumberAdapter
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