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130 lines
4.8 KiB
Text
130 lines
4.8 KiB
Text
import Mathlib.Data.Fintype.Card
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import Mathlib.Data.Fin.Basic
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/-!
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# Human Neural Compression Verification
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This module records the hard finite-state obstruction behind any claimed
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lossless compression of arbitrary neural snapshots: a smaller finite code space
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cannot injectively represent a larger finite source space.
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The result is intentionally modest. It does not rule out useful neural
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compression; it rules out universal lossless compression unless the model
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restricts the admissible state space, permits bounded loss, or uses an explicit
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probabilistic guarantee.
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-/
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namespace Semantics.HumanNeuralCompressionVerification
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/-! ## Published budget constants -/
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/-- Nominal uncompressed human neural snapshot size in gigabytes: 1 PB. -/
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def uncompressedStateGb : Nat := 1000000
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/-- Conservative practical target size in gigabytes. -/
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def targetMaxGb : Nat := 800
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/-- Optimistic practical target size in gigabytes. -/
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def targetMinGb : Nat := 300
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/-- Minimum nominal ratio needed to map 1 PB into 800 GB. -/
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def minimumCompressionRatio : Nat := uncompressedStateGb / targetMaxGb
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/-- Idealized ratio needed to map 1 PB into 300 GB. -/
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def idealCompressionRatio : Nat := uncompressedStateGb / targetMinGb
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/-- Sparse active-neuron estimate used in the exploratory budget model. -/
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def activeRatioPercent : Nat := 15
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/-- Effective uncompressed budget after the active-neuron estimate. -/
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def effectiveUncompressedGb : Nat :=
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(uncompressedStateGb * activeRatioPercent) / 100
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/-- Sparsity-adjusted minimum ratio for an 800 GB target. -/
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def effectiveMinimumRatio : Nat := effectiveUncompressedGb / targetMaxGb
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theorem minimumCompressionRatio_eq : minimumCompressionRatio = 1250 := by
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native_decide
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theorem idealCompressionRatio_eq : idealCompressionRatio = 3333 := by
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native_decide
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theorem effectiveUncompressedGb_eq : effectiveUncompressedGb = 150000 := by
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native_decide
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theorem effectiveMinimumRatio_eq : effectiveMinimumRatio = 187 := by
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native_decide
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/-! ## Byte-budget facts -/
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/-- 1 PB in decimal bytes. -/
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def uncompressedBytes : Nat := 1000000000000000
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/-- 800 GB in decimal bytes. -/
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def compressedBytes : Nat := 800000000000
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/-- The proposed compressed budget is strictly smaller than the source budget. -/
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theorem byte_budget_strictly_smaller : compressedBytes < uncompressedBytes := by
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native_decide
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/-! ## Finite lossless compression obstruction -/
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/--
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A witness for a lossless compression scheme from `sourceCodes` possible source
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states into `compressedCodes` possible code words.
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Injectivity is the key property: two source states may not share a code word if
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decoding is required to be universally lossless.
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-/
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structure LosslessCompressionWitness (sourceCodes compressedCodes : Nat) where
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encode : Fin sourceCodes → Fin compressedCodes
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injective : Function.Injective encode
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/-- Any injective finite encoding requires at least as many code words as inputs. -/
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theorem lossless_witness_requires_capacity
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{sourceCodes compressedCodes : Nat}
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(w : LosslessCompressionWitness sourceCodes compressedCodes) :
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sourceCodes ≤ compressedCodes := by
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simpa using Fintype.card_le_of_injective w.encode w.injective
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/--
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Pigeonhole form: there is no injective encoding from a larger finite type into a
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smaller finite type.
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-/
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theorem no_injective_compression_to_smaller_fintype
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{α β : Type} [Fintype α] [Fintype β]
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(h : Fintype.card β < Fintype.card α) :
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¬ ∃ encode : α → β, Function.Injective encode := by
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rintro ⟨encode, hInjective⟩
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exact Nat.not_le_of_gt h (Fintype.card_le_of_injective encode hInjective)
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/-- Concrete finite-code version of the same obstruction. -/
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theorem no_lossless_universal_compression
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{sourceCodes compressedCodes : Nat}
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(h : compressedCodes < sourceCodes) :
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¬ ∃ encode : Fin sourceCodes → Fin compressedCodes, Function.Injective encode := by
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exact no_injective_compression_to_smaller_fintype
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(α := Fin sourceCodes) (β := Fin compressedCodes) (by simpa using h)
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/--
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Witness form: a claimed universal lossless compressor is impossible whenever the
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compressed code space has lower cardinality than the source space.
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-/
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theorem arbitrary_lossless_compression_impossible
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{sourceCodes compressedCodes : Nat}
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(h : compressedCodes < sourceCodes) :
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¬ Nonempty (LosslessCompressionWitness sourceCodes compressedCodes) := by
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rintro ⟨w⟩
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exact Nat.not_le_of_gt h (lossless_witness_requires_capacity w)
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/--
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Research gate for the 1 PB → 800 GB target: because the byte budget is smaller,
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any successful rigorous model must discharge one of the missing assumptions:
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restricted admissible source states, bounded lossy reconstruction error, or an
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explicit stochastic confidence theorem.
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-/
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theorem onePbTo800Gb_needs_extra_model_structure :
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compressedBytes < uncompressedBytes :=
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byte_budget_strictly_smaller
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end Semantics.HumanNeuralCompressionVerification
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