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113 lines
7 KiB
Text
113 lines
7 KiB
Text
/-!
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# Manifold-Agnostic Neural State Compression
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**Problem:** A neural state lives on a manifold of intrinsic dimension D.
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A coordinate chart embeds it into a representation space of dimension N.
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Compression is a smooth map to a lower-dimensional manifold.
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**Question:** What constraints does the compression ratio place on the
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choice of coordinate chart, independent of any file format?
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**Answer:** The chart must be nearly isometric (N ≈ D) and the
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compression map must have Jacobian determinant ≥ 1,250.
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This file is standalone: zero imports, first principles only.
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-/
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Intrinsic Manifold Parameters (Q16_16 scalar counts)
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-- ═══════════════════════════════════════════════════════════════════════════
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def intrinsicDimension : Nat := 1000000000000000
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-- ~10¹⁵ degrees of freedom (synaptic weights + neuron states)
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def intrinsicVolumeScale : Nat := 1000000000000000
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-- Volume of intrinsic manifold in natural units (1 PB equivalent)
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def targetCompressedVolume : Nat := 800000000000
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-- Target volume of compressed embedding (800 GB equivalent)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Coordinate Chart Bloat (The Representation Manifold)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- A coordinate chart embeds the intrinsic manifold into a higher-
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dimensional space. Chart bloat = N_coord / D_intrinsic.
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An isometric chart has bloat = 1. -/
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def chartBloat (coordDim intrinsicDim : Nat) : Nat :=
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(coordDim * 1000) / intrinsicDim
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/-- An isometric chart: no bloat, no metadata overhead.
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Every coordinate degree of freedom maps to one intrinsic degree
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of freedom. -/
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def isometricChartDim : Nat := intrinsicDimension
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/-- A bloated chart: representation adds gauge degrees of freedom.
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Example: tagged unions, reference counts, type descriptors, hash
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tables — all coordinates that do not correspond to manifold points. -/
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def bloatedChartDim : Nat := intrinsicDimension * 12
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-- 12× bloat factor (empirical: Python object overhead)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Compression as a Smooth Map Between Manifolds
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compression ratio = vol(M_source) / vol(M_target).
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For a smooth map f: M → M' with Jacobian J, vol(M') = |det(J)|·vol(M).
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Therefore compression ratio = 1 / |det(J)|. -/
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def requiredCompressionRatio : Nat :=
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intrinsicVolumeScale / targetCompressedVolume
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/-- Minimum Jacobian determinant of the compression map, in parts per million.
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det(J) = V_target / V_source = 1 / C_ratio.
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At C = 1,250: det(J) = 800 / 1,000,000 = 0.0008. -/
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def jacobianDeterminantPerMillion : Nat :=
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(targetCompressedVolume * 1000000) / intrinsicVolumeScale
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Manifold Curvature and Information Density
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Information density ρ = intrinsic dimension / compressed volume.
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Higher curvature regions can tolerate higher ρ (more bits per dof).
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Flat regions require uniform allocation. -/
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def informationDensityPerDof : Nat :=
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(intrinsicDimension * 1000) / targetCompressedVolume
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/-- At 1,250× compression, each degree of freedom gets, on average,
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less than one bit. This requires correlated structure
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(redundant curvature) in the manifold. -/
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def bitsPerDof : Nat :=
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(targetCompressedVolume * 8 * 1000) / intrinsicDimension
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 The Isometric Chart Constraint
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem (chart bloat bound): If the chart has bloat > 1,
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the effective compression ratio is reduced by the bloat factor.
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C_effective = C_intrinsic / bloat.
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For C_intrinsic = 1,250 and bloat = 12, C_effective = 104.
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This fails the 800 GB target. -/
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def effectiveCompressionRatio (intrinsicBloat : Nat) : Nat :=
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(requiredCompressionRatio * 1000) / intrinsicBloat
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/-- The manifold embedding must satisfy:
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dim(coordinates) ≤ dim(intrinsic) × (C_target / C_required).
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At equality, the chart is isometric and compression is pure. -/
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def maxChartDimForTarget (targetRatio : Nat) : Nat :=
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(intrinsicDimension * requiredCompressionRatio) / targetRatio
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Witness Values
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-- ═══════════════════════════════════════════════════════════════════════════
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#eval requiredCompressionRatio -- 1250
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#eval jacobianDeterminantPerMillion -- 800 (0.0008 = 800 parts per million)
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#eval chartBloat isometricChartDim intrinsicDimension -- 1000 (1.0×)
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#eval chartBloat bloatedChartDim intrinsicDimension -- 12000 (12×)
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#eval effectiveCompressionRatio 1000 -- 1250 (isometric: full ratio)
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#eval effectiveCompressionRatio 12000 -- 104 (bloated: fails target)
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#eval informationDensityPerDof -- 1250 (dof per GB, scaled)
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#eval bitsPerDof -- 6 (0.006 bits per dof)
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#eval maxChartDimForTarget 1250 -- 1000000000000000 (isometric)
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