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141 lines
5.1 KiB
Text
141 lines
5.1 KiB
Text
import Semantics.FixedPoint
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import Semantics.Bind
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namespace Semantics.Layer3TransmissionModel
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/-! ## Layer 3 Transmission Model
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Formal proof that "0 bytes during local compute" is NOT compression.
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Key invariant:
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- 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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Corrected architecture equation:
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- effective_network_cost = anchor_frequency × compressed_payload_size
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- NOT: raw_payload_size / 0 = ∞
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-/
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open Semantics.Q16_16
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/-- Transmission event type. -/
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inductive TransmissionEvent where
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| localComputation : TransmissionEvent -- Compute locally, no transmission
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| anchorTransmission : Nat → TransmissionEvent -- Transmit anchor with size
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deriving Repr
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/-- Transmission cost model. -/
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structure TransmissionCost where
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eventType : TransmissionEvent
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dataSize : Nat -- Size of data involved
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transmittedBytes : Nat -- Bytes actually transmitted
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transmissionAvoidance : Bool -- Whether transmission was avoided
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deriving Repr
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/-- Calculate transmission cost for local computation. -/
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def localComputationCost (dataSize : Nat) : TransmissionCost :=
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{
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eventType := TransmissionEvent.localComputation
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dataSize := dataSize
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transmittedBytes := 0
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transmissionAvoidance := true
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}
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/-- Calculate transmission cost for anchor transmission. -/
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def anchorTransmissionCost (anchorSize : Nat) : TransmissionCost :=
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{
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eventType := TransmissionEvent.anchorTransmission anchorSize
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dataSize := anchorSize
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transmittedBytes := anchorSize
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transmissionAvoidance := false
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}
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/-- Compression ratio calculation.
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-- compressionRatio = originalSize / compressedSize
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--
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-- Arithmetic sanity check:
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-- 1000 / 100 = 10× compression ratio.
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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 compressionRatio (originalSize : Nat) (compressedSize : Nat) : Q16_16 :=
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ofNat originalSize / ofNat compressedSize
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/-- Theorem: Local computation has 0 transmitted bytes. -/
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theorem local_computation_zero_transmitted (dataSize : Nat) :
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(localComputationCost dataSize).transmittedBytes = 0 := by
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rfl
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/-- Theorem: Local computation does NOT change data size. -/
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theorem local_computation_no_size_change (dataSize : Nat) :
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(localComputationCost dataSize).dataSize = dataSize := by
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rfl
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/-- Theorem: Local computation is transmission avoidance, not size reduction. -/
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theorem local_computation_not_compression (dataSize : Nat) :
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(localComputationCost dataSize).transmissionAvoidance = true ∧
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(localComputationCost dataSize).dataSize = dataSize := by
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exact ⟨rfl, rfl⟩
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/-- Theorem: Transmission avoidance records no local payload shrink. -/
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theorem transmission_avoidance_not_compression (dataSize : Nat) :
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(localComputationCost dataSize).transmissionAvoidance = true →
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(localComputationCost dataSize).dataSize = dataSize := by
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intro _hAvoided
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rfl
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/-- Theorem: Effective network cost is anchor frequency × payload size. -/
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-- This is the CORRECTED architecture equation
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structure EffectiveNetworkCost 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 (corrected formula).
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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 calculateEffectiveCost (anchorFreq : Nat) (payloadSize : Nat) : EffectiveNetworkCost :=
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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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/-- Theorem: Effective cost is NOT infinite. -/
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theorem effective_cost_not_infinite (anchorFreq : Nat) (payloadSize : Nat)
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(hAnchor : anchorFreq > 0) (hPayload : payloadSize > 0) :
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(calculateEffectiveCost anchorFreq payloadSize).effectiveCost ≠ 0 := by
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change anchorFreq * payloadSize ≠ 0
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exact Nat.mul_ne_zero (Nat.ne_of_gt hAnchor) (Nat.ne_of_gt hPayload)
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/-- Theorem: Effective cost formula is correct. -/
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theorem effective_cost_formula_correct (anchorFreq : Nat) (payloadSize : Nat) :
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(calculateEffectiveCost anchorFreq payloadSize).effectiveCost = anchorFreq * payloadSize := by
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rfl
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/-- In the Q16.16 compression model, a zero denominator is represented by the
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explicit `infinity` sentinel rather than by claiming a Nat-level infinity. -/
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theorem compression_ratio_zero_denominator_is_infinity (n : Nat) :
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compressionRatio n 0 = infinity := by
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unfold compressionRatio
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change Semantics.Q16_16.div (ofNat n) (ofNat 0) = infinity
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simp [Semantics.Q16_16.div, ofNat, infinity]
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end Semantics.Layer3TransmissionModel
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