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130 lines
5.4 KiB
Text
130 lines
5.4 KiB
Text
import Semantics.FixedPoint
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import Semantics.Bind
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namespace Semantics.YangMillsCompressionBounds
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/-! ## Yang-Mills Compression Bounds
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Formalization of compression bounds separating:
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- Lossless compression (exact reconstruction)
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- Lossy compression (acceptable precision loss)
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- Precision-reduced compression (fixed-point conversion)
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- Transmission avoidance (Layer 3 local computation)
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Key invariant: compressionRatio ≠ transmissionAvoidance
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Layer 3 can reduce when data is transmitted. It does not make the data smaller by itself.
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-/
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open Semantics.Q16_16
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/-- Compression type classification. -/
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inductive CompressionType where
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| lossless : CompressionType -- Exact reconstruction
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| lossy : CompressionType -- Acceptable precision loss
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| precisionReduced : CompressionType -- Fixed-point conversion
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| transmissionAvoidance : CompressionType -- Layer 3 local computation
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deriving Repr, DecidableEq
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/-- Compression bounds for each type. -/
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structure CompressionBounds where
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compressionType : CompressionType
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minRatio : Q16_16 -- Minimum compression ratio (≥ 1)
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maxRatio : Q16_16 -- Maximum compression ratio (theoretical limit)
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requiresBenchmark : Bool -- Requires empirical benchmark
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justification : String
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deriving Repr
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/-- Lossless compression bounds (Delta GCL). -/
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-- Based on actual achievements: 92% on structured data, 99.9% on metadata
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-- Field data less compressible: 2-5× realistic, 10-20× theoretical
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def losslessBounds : CompressionBounds :=
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{ compressionType := CompressionType.lossless
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minRatio := two -- 2× (conservative)
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maxRatio := ofNat 20 -- 20× (theoretical upper bound for field data)
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requiresBenchmark := true
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justification := "Based on Delta GCL achievements (92% on structured data). Field data less compressible than metadata. Requires empirical zstd/ndzip comparison." }
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/-- Lossy compression bounds (neural VAE). -/
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-- Optional second stage with acceptable precision loss
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def lossyBounds : CompressionBounds :=
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{ compressionType := CompressionType.lossy
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minRatio := two -- 2× (conservative)
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maxRatio := ofNat 20 -- 20× (depends on acceptable precision loss)
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requiresBenchmark := true
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justification := "Neural VAE compression with acceptable precision loss. Requires benchmark against precision requirements." }
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/-- Precision-reduced compression bounds (fixed-point). -/
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-- Float64 → Q16_16: 2× reduction in bit width
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def precisionReducedBounds : CompressionBounds :=
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{ compressionType := CompressionType.precisionReduced
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minRatio := two -- 2× (Float64 → Q16_16)
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maxRatio := two -- 2× (fixed, no variation)
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requiresBenchmark := false
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justification := "Fixed-point conversion: Float64 → Q16_16 is exactly 2× bit width reduction. No benchmark required." }
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/-- Transmission avoidance bounds (Layer 3). -/
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-- Layer 3 does NOT compress data - it avoids transmission during local computation
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-- This is NOT a compression ratio, it's a transmission reduction factor
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def transmissionAvoidanceBounds : CompressionBounds :=
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{ compressionType := CompressionType.transmissionAvoidance
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minRatio := one -- 1× (no compression)
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maxRatio := one -- 1× (no compression)
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requiresBenchmark := false
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justification := "Layer 3 local computation avoids transmission but does NOT compress data. transmissionAvoidance ≠ compressionRatio." }
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/-- Calculate effective network cost. -/
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-- effective_network_cost = anchor_frequency × compressed_payload_size
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structure NetworkCost where
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anchorFrequency : Nat -- Number of anchors per unit time
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compressedPayloadSize : Nat -- Size after compression
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effectiveCost : Nat -- Total network cost
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deriving Repr
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/-- Calculate effective network cost.
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--
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-- Arithmetic sanity check:
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-- effective_network_cost = anchor_frequency × compressed_payload_size.
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--
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-- Example:
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-- 10 anchors × 1 MiB = 10 MiB.
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--
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-- Provenance note:
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-- This is not a compression ratio. It is a scheduling/transmission-cost model.
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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 effectiveNetworkCost (anchorFreq : Nat) (payloadSize : Nat) : NetworkCost :=
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{
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anchorFrequency := anchorFreq
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compressedPayloadSize := payloadSize
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effectiveCost := anchorFreq * payloadSize
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}
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#eval effectiveNetworkCost 10 1000000 -- 10 anchors × 1MB = 10MB effective cost
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/-- Theorem: Lossless compression ratio ≥ 1. -/
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theorem lossless_ratio_ge_one : losslessBounds.minRatio ≥ one := by
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native_decide
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/-- Theorem: Precision reduction is exactly 2×. -/
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theorem precision_reduction_exact_two : precisionReducedBounds.minRatio = two ∧ precisionReducedBounds.maxRatio = two := by
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native_decide
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/-- Theorem: Transmission avoidance does NOT compress data. -/
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theorem transmission_avoidance_no_compression : transmissionAvoidanceBounds.minRatio = one ∧ transmissionAvoidanceBounds.maxRatio = one := by
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native_decide
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/-- Theorem: compressionRatio ≠ transmissionAvoidance. -/
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theorem compression_not_transmission_avoidance :
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losslessBounds.compressionType ≠ CompressionType.transmissionAvoidance := by
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native_decide
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/-- Theorem: Effective network cost is anchor frequency × payload size. -/
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theorem effective_cost_formula (anchorFreq : Nat) (payloadSize : Nat) :
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(effectiveNetworkCost anchorFreq payloadSize).effectiveCost = anchorFreq * payloadSize := by
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rfl
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end Semantics.YangMillsCompressionBounds
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