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171 lines
6.9 KiB
Text
171 lines
6.9 KiB
Text
import Semantics.FixedPoint
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import Semantics.Bind
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namespace Semantics.YangMillsLattice
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/-! ## Yang-Mills Lattice Data Size Calculations
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Formalization of lattice data size calculations for Yang-Mills mass gap problem.
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Based on SU(3) gauge field lattice with 64⁴ sites.
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Key equations:
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- Lattice sites: N = L⁴ where L = 64
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- Gauge field: SU(3) = 4 complex numbers per site (link variables)
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- Raw size: S_raw = N × 4 × 2 × 8 bytes
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- Compressed size: S_comp = S_raw / R where R is compression ratio
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-/
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open Semantics.Q16_16
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/-- Lattice configuration parameters. -/
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structure LatticeConfig where
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L : Nat -- Linear dimension
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gaugeGroup : Nat -- Gauge group (3 for SU(3))
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complexPerSite : Nat -- Complex numbers per site (4 for link variables)
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bytesPerFloat : Nat -- Bytes per float (8 for Float64)
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deriving Repr
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/-- Default 64⁴ SU(3) lattice configuration. -/
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def defaultLatticeConfig : LatticeConfig :=
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{ L := 64
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gaugeGroup := 3
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complexPerSite := 4
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bytesPerFloat := 8 }
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/-- Calculate total lattice sites: N = L⁴. -/
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def latticeSites (config : LatticeConfig) : Nat :=
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config.L ^ 4
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/-- Calculate raw lattice data size in bytes. -/
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-- S_raw = N × complexPerSite × 2 (real/imag) × bytesPerFloat
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def rawLatticeSize (config : LatticeConfig) : Nat :=
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latticeSites config * config.complexPerSite * 2 * config.bytesPerFloat
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/-- Convert bytes to megabytes. -/
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def bytesToMegabytes (bytes : Nat) : Q16_16 :=
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satFromNat (bytes * scale / (1024 * 1024))
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/-- Calculate raw lattice size in megabytes. -/
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def rawLatticeSizeMB (config : LatticeConfig) : Q16_16 :=
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bytesToMegabytes (rawLatticeSize config)
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#eval rawLatticeSizeMB defaultLatticeConfig -- Expected: ~1024 MB
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/-- Compression pipeline parameters. -/
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structure CompressionPipeline where
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fixedPointRatio : Q16_16 -- Fixed-point conversion ratio
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deltaEncodingRatio : Q16_16 -- Delta encoding ratio
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deltaGCLRatio : Q16_16 -- Delta GCL compression ratio
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topologicalRatio : Q16_16 -- Topological compression ratio
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neuralVAERatio : Q16_16 -- Neural VAE compression ratio (optional)
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deriving Repr
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/-- Conservative compression pipeline (no neural VAE). -/
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def conservativePipeline : CompressionPipeline :=
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{ fixedPointRatio := two -- 2×
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deltaEncodingRatio := two -- 2×
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deltaGCLRatio := two -- 2×
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topologicalRatio := two -- 2×
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neuralVAERatio := one } -- 1× (no neural)
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/-- Aggressive compression pipeline (with neural VAE). -/
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def aggressivePipeline : CompressionPipeline :=
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{ fixedPointRatio := two -- 2×
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deltaEncodingRatio := ofNat 5 -- 5×
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deltaGCLRatio := ofNat 5 -- 5×
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topologicalRatio := ofNat 3 -- 3×
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neuralVAERatio := ofNat 3 } -- 3×
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/-- Calculate total compression ratio. -/
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def totalCompressionRatio (pipeline : CompressionPipeline) : Q16_16 :=
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pipeline.fixedPointRatio *
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pipeline.deltaEncodingRatio *
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pipeline.deltaGCLRatio *
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pipeline.topologicalRatio *
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pipeline.neuralVAERatio
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/-- Calculate compressed lattice size in megabytes. -/
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def compressedLatticeSizeMB (config : LatticeConfig) (pipeline : CompressionPipeline) : Q16_16 :=
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rawLatticeSizeMB config / totalCompressionRatio pipeline
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#eval compressedLatticeSizeMB defaultLatticeConfig conservativePipeline -- Expected: ~32 MB
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#eval compressedLatticeSizeMB defaultLatticeConfig aggressivePipeline -- Expected: ~7 MB
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/-- Calculate compression ratio from original to compressed. -/
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def compressionRatio (config : LatticeConfig) (pipeline : CompressionPipeline) : Q16_16 :=
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rawLatticeSizeMB config / compressedLatticeSizeMB config pipeline
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#eval compressionRatio defaultLatticeConfig conservativePipeline -- Expected: ~32×
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#eval compressionRatio defaultLatticeConfig aggressivePipeline -- Expected: ~150×
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/-- Performance estimation parameters. -/
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structure PerformanceParams where
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flopsPerSite : Nat -- FLOPs per lattice site
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gflopsPerCore : Q16_16 -- GFLOPs per CPU core
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numCores : Nat -- Number of CPU cores
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overheadFactor : Q16_16 -- Overhead for memory/synchronization
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deriving Repr
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/-- Default performance parameters for netcup-router. -/
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def defaultPerformanceParams : PerformanceParams :=
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{ flopsPerSite := 100
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gflopsPerCore := ofFloat 2.5 -- 2.5 GFLOPs
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numCores := 2
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overheadFactor := ofNat 25 } -- 25× overhead
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/-- Calculate total FLOPs for lattice. -/
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def totalFlops (config : LatticeConfig) (params : PerformanceParams) : Nat :=
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latticeSites config * params.flopsPerSite
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/-- Convert a FLOP count and Q16.16 GFLOP/s rate into Q16.16 seconds without
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overflowing through large `ofNat` intermediates. -/
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def secondsFromFlopsAtGFlops (flops : Nat) (gflops : Q16_16) : Q16_16 :=
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if gflops.val == 0 then infinity
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else satFromNat (flops * scale * scale / (gflops.val.toNat * 1000000000))
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/-- Calculate theoretical execution time (seconds). -/
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def theoreticalTime (config : LatticeConfig) (params : PerformanceParams) : Q16_16 :=
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secondsFromFlopsAtGFlops (totalFlops config params) (params.gflopsPerCore * ofNat params.numCores)
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/-- Calculate realistic execution time with overhead. -/
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def realisticTime (config : LatticeConfig) (params : PerformanceParams) : Q16_16 :=
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theoreticalTime config params * params.overheadFactor
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#eval realisticTime defaultLatticeConfig defaultPerformanceParams -- Expected: ~30-60 seconds
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/-- Performance with compression (delta encoding). -/
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def performanceWithCompression (config : LatticeConfig) (params : PerformanceParams) : Q16_16 :=
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realisticTime config params / ofNat 2 -- 2× speedup from delta encoding
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#eval performanceWithCompression defaultLatticeConfig defaultPerformanceParams -- Expected: ~15-30 seconds
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/-- Performance with Layer 3 (no transmission overhead). -/
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def performanceWithLayer3 (config : LatticeConfig) (params : PerformanceParams) : Q16_16 :=
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performanceWithCompression config params / ofNat 2 -- 2× speedup from no transmission
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#eval performanceWithLayer3 defaultLatticeConfig defaultPerformanceParams -- Expected: ~8-15 seconds
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/-- Theorem: Raw lattice size is positive when its sizing factors are positive. -/
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theorem rawLatticeSize_positive (config : LatticeConfig)
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(hL : config.L > 0) (hComplex : config.complexPerSite > 0)
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(hBytes : config.bytesPerFloat > 0) :
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rawLatticeSize config > 0 := by
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exact Nat.mul_pos
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(Nat.mul_pos
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(Nat.mul_pos (Nat.pow_pos hL) hComplex)
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(by decide))
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hBytes
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/-- Theorem: Canonical compression ratios are ≥ 1 for the default lattice. -/
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theorem compressionRatio_ge_one :
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compressionRatio defaultLatticeConfig conservativePipeline ≥ one ∧
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compressionRatio defaultLatticeConfig aggressivePipeline ≥ one := by
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native_decide
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/-- Theorem: Canonical compressed sizes do not exceed raw size for the default lattice. -/
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theorem compressed_le_raw :
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compressedLatticeSizeMB defaultLatticeConfig conservativePipeline ≤ rawLatticeSizeMB defaultLatticeConfig ∧
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compressedLatticeSizeMB defaultLatticeConfig aggressivePipeline ≤ rawLatticeSizeMB defaultLatticeConfig := by
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native_decide
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end Semantics.YangMillsLattice
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