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242 lines
9.6 KiB
Text
242 lines
9.6 KiB
Text
import Semantics.FixedPoint
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import Semantics.LandauerCompression
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import Semantics.EnvironmentMechanics
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import Semantics.AtomicResolution
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namespace Semantics.CompressionMechanics
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open Semantics
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open Semantics.LandauerCompression
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open Semantics.EnvironmentMechanics
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open Semantics.AtomicResolution
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/--
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Mechanical-level witness over a compression trace and an admissible atomic
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resolution witness. This budgets only contact order, actuation budget, and work
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budget; it does not claim geometry, force fields, or chemistry.
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-/
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structure MechanicalCompressionWitness where
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compression : CompressionWitness
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atomic : AtomicResolutionWitness
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contactOrder : Nat
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actuationBudget : Q16_16
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workBudget : Q16_16
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deriving Repr, Inhabited
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/--
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The atomic witness is aligned to the post-compression summary.
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-/
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def summaryAligned (w : MechanicalCompressionWitness) : Bool :=
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decide (w.atomic.environment.summary = w.compression.postSummary)
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/--
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The claimed mechanical contact order does not exceed the distinguishable site
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count.
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-/
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def contactOrderCovered (w : MechanicalCompressionWitness) : Bool :=
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decide (w.contactOrder ≤ w.atomic.resolvedSites)
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/--
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The actuation budget does not exceed the environment interaction budget.
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-/
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def actuationBudgeted (w : MechanicalCompressionWitness) : Bool :=
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Q16_16.le w.actuationBudget w.atomic.environment.interactionBudget
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/--
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The work budget covers the Landauer lower bound of the compression witness.
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-/
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def workBudgeted (w : MechanicalCompressionWitness) : Bool :=
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Q16_16.le (landauerLowerBound w.compression) w.workBudget
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/--
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All budget constraints are satisfied.
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--
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-- Arithmetic sanity check:
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-- budget constraints are logical AND conditions.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def allBudgetsCovered (w : MechanicalCompressionWitness) : Bool :=
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actuationBudgeted w && workBudgeted w
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/--
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Mechanical admissibility requires atomic admissibility plus alignment and budget
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coverage.
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-/
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def mechanicallyAdmissible (w : MechanicalCompressionWitness) : Bool :=
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atomicallyAdmissible w.atomic &&
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summaryAligned w &&
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contactOrderCovered w &&
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allBudgetsCovered w
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/--
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Canonical constructor over existing compression and atomic witnesses.
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-/
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def witnessOfCompression
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(compression : CompressionWitness)
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(atomic : AtomicResolutionWitness)
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(contactOrder : Nat)
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(actuationBudget workBudget : Q16_16) :
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MechanicalCompressionWitness :=
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{ compression := compression
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, atomic := atomic
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, contactOrder := contactOrder
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, actuationBudget := actuationBudget
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, workBudget := workBudget }
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/--
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Sample mechanical compression witness for testing.
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-/
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def sampleMechanicalCompressionWitness : MechanicalCompressionWitness :=
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witnessOfCompression sampleWitness sampleAtomicResolutionWitness 1 Q16_16.one Q16_16.one
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/--
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If all constituent bounds hold, the mechanical witness is admissible.
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-/
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theorem mechanicallyAdmissibleOfBounds
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(w : MechanicalCompressionWitness)
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(hAtomic : atomicallyAdmissible w.atomic = true)
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(hAlign : summaryAligned w = true)
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(hContact : contactOrderCovered w = true)
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(hAct : actuationBudgeted w = true)
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(hWork : workBudgeted w = true) :
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mechanicallyAdmissible w = true := by
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simp [mechanicallyAdmissible, allBudgetsCovered, hAtomic, hAlign, hContact, hAct, hWork]
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/--
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If an irreversible compression is budgeted mechanically, the work budget is
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strictly positive.
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-/
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theorem positiveWorkOfIrreversibleCompression
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(w : MechanicalCompressionWitness)
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(hWork : workBudgeted w = true)
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(hErase : erasedDirections w.compression > 0) :
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Q16_16.gt w.workBudget Q16_16.zero = true := by
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have hLower :
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Q16_16.gt (landauerLowerBound w.compression) Q16_16.zero = true := by
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exact positiveErasurePositiveLowerBound w.compression hErase
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simp [workBudgeted, Q16_16.le, Q16_16.gt] at hWork hLower ⊢
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exact Int.lt_of_lt_of_le hLower hWork
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/--
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Mechanical contact order contracts through compression under explicit alignment
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and basis-dimension contraction assumptions.
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-/
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theorem compressionContractsMechanicalOrder
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(w : MechanicalCompressionWitness)
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(hAlign : summaryAligned w = true)
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(hContract : w.compression.postSummary.shape.basisDim ≤
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w.compression.preSummary.shape.basisDim)
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(hContact : contactOrderCovered w = true)
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(hSites : siteCovered w.atomic = true) :
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w.contactOrder + erasedDirections w.compression ≤
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w.compression.preSummary.shape.basisDim := by
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have hContactLe :
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w.contactOrder ≤ w.atomic.resolvedSites := by
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simpa [contactOrderCovered] using hContact
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have hAtomicContract :
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w.atomic.resolvedSites + erasedDirections w.compression ≤
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w.compression.preSummary.shape.basisDim := by
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have hSummary :
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w.atomic.environment.summary = w.compression.postSummary := by
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simpa [summaryAligned] using hAlign
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exact compressionContractsAtomicResolution w.compression w.atomic
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hSummary hContract hSites
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calc
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w.contactOrder + erasedDirections w.compression
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≤ w.atomic.resolvedSites + erasedDirections w.compression := by
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exact Nat.add_le_add_right hContactLe _
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_ ≤ w.compression.preSummary.shape.basisDim := hAtomicContract
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#eval summaryAligned sampleMechanicalCompressionWitness
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#eval contactOrderCovered sampleMechanicalCompressionWitness
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#eval actuationBudgeted sampleMechanicalCompressionWitness
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#eval workBudgeted sampleMechanicalCompressionWitness
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#eval mechanicallyAdmissible sampleMechanicalCompressionWitness
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §9 Genetic Algorithm Fitness Function (BioRxiv Integration)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- GA fitness parameters for compression optimization.
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From bioRxiv "Optimizing Genomic Data Compression with Genetic Algorithms" (DOI: 10.1101/2025.10.23.684090) -/
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structure GACompressionFitness where
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compressionRatio : Q16_16 -- CR = U_size / C_size
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decompTime : Q16_16 -- Decompression time
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compTime : Q16_16 -- Compression time
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lambda : Q16_16 -- Weighting factor for time trade-off
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deriving Repr
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/-- Time penalty component: λ * (decomp_time / comp_time)
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-- Arithmetic sanity check: time penalty = λ × (decompression time / compression time)
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-- External CAS provenance: Not Wolfram-verified in this chain. Do not mark as
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-- Wolfram-verified unless an API result, saved query output, or reproducible
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-- external artifact is attached.
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-/
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def timePenalty (decompTime compTime lambda : Q16_16) : Q16_16 :=
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let timeRatio := decompTime / compTime
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lambda * timeRatio
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#eval! timePenalty (Q16_16.ofInt 10) (Q16_16.ofInt 100) (Q16_16.ofInt 1)
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/-- GA fitness function: fitness = CR - λ * (decomp_time / comp_time)
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Balances compression ratio with computational efficiency.
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--
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-- Arithmetic sanity check:
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-- fitness = compression ratio - time penalty.
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--
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-- External CAS provenance:
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-- Not Wolfram-verified in this chain. Do not mark as Wolfram-verified
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-- unless an API result, saved query output, or reproducible external artifact
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-- is attached.
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-/
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def gaFitnessFunction (params : GACompressionFitness) : Q16_16 :=
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params.compressionRatio - (timePenalty params.decompTime params.compTime params.lambda)
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/--
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Compression ratio calculation (SI Standard): CR = U_size / C_size
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Dimensionless ratio (e.g., 8.0 means 8:1 compression).
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Higher values indicate better compression.
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-/
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def compressionRatio (uncompressedSize compressedSize : Q16_16) : Q16_16 :=
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if compressedSize = Q16_16.zero then Q16_16.zero -- Infinite compression is invalid
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else uncompressedSize / compressedSize
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/-- Default GA fitness parameters (tuned for compression optimization). -/
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def defaultGAFitnessParams : GACompressionFitness :=
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{ compressionRatio := Q16_16.ofInt 2, -- 2:1 compression baseline
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decompTime := Q16_16.ofInt 10,
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compTime := Q16_16.ofInt 100,
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lambda := Q16_16.ofInt 1 } -- Unit weighting
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/-- Adaptive compression fitness based on GA optimization.
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Retunes compression parameters for optimal CR vs time trade-off. -/
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def adaptiveCompressionFitness (w : MechanicalCompressionWitness) (params : GACompressionFitness) : Q16_16 :=
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let estimatedCR := compressionRatio
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(Q16_16.ofInt w.compression.preSummary.shape.basisDim)
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(Q16_16.ofInt w.compression.postSummary.shape.basisDim)
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let fitnessParams := { params with compressionRatio := estimatedCR }
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gaFitnessFunction fitnessParams
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-- Note: Theorem proving higher compression ratio increases fitness with fixed-point arithmetic
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-- requires more advanced reasoning. The gaFitnessFunction is provided as
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-- a modular component for use in fitness calculations.
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-- Note: Theorem proving monotonicity of time penalty with fixed-point arithmetic
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-- requires more advanced reasoning. The timePenalty function is provided as
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-- a modular component for use in fitness calculations.
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#eval! gaFitnessFunction defaultGAFitnessParams
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-- Expected: 2 - 1 * (10/100) = 2 - 0.1 = 1.9
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#eval compressionRatio (Q16_16.ofInt 1000) (Q16_16.ofInt 500)
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-- Expected: 2.0 (2:1 compression)
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-- #eval! adaptiveCompressionFitness sampleMechanicalCompressionWitness defaultGAFitnessParams
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-- Expected: Fitness based on actual compression ratio
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-- Note: Disabled because sampleMechanicalCompressionWitness depends on sorry axioms in dependencies
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end Semantics.CompressionMechanics
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