/-!
# Manifold-Agnostic Neural State Compression

**Problem:** A neural state lives on a manifold of intrinsic dimension D.
A coordinate chart embeds it into a representation space of dimension N.
Compression is a smooth map to a lower-dimensional manifold.

**Question:** What constraints does the compression ratio place on the
choice of coordinate chart, independent of any file format?

**Answer:** The chart must be nearly isometric (N ≈ D) and the
compression map must have Jacobian determinant ≥ 1,250.

This file is standalone: zero imports, first principles only.
-/

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Intrinsic Manifold Parameters (Q16_16 scalar counts)
-- ═══════════════════════════════════════════════════════════════════════════

def intrinsicDimension : Nat := 1000000000000000
  -- ~10¹⁵ degrees of freedom (synaptic weights + neuron states)

def intrinsicVolumeScale : Nat := 1000000000000000
  -- Volume of intrinsic manifold in natural units (1 PB equivalent)

def targetCompressedVolume : Nat := 800000000000
  -- Target volume of compressed embedding (800 GB equivalent)

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Coordinate Chart Bloat (The Representation Manifold)
-- ═══════════════════════════════════════════════════════════════════════════

/-- A coordinate chart embeds the intrinsic manifold into a higher-
    dimensional space. Chart bloat = N_coord / D_intrinsic.
    An isometric chart has bloat = 1. -/
def chartBloat (coordDim intrinsicDim : Nat) : Nat :=
  (coordDim * 1000) / intrinsicDim

/-- An isometric chart: no bloat, no metadata overhead.
    Every coordinate degree of freedom maps to one intrinsic degree
    of freedom. -/
def isometricChartDim : Nat := intrinsicDimension

/-- A bloated chart: representation adds gauge degrees of freedom.
    Example: tagged unions, reference counts, type descriptors, hash
    tables — all coordinates that do not correspond to manifold points. -/
def bloatedChartDim : Nat := intrinsicDimension * 12
  -- 12× bloat factor (empirical: Python object overhead)

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Compression as a Smooth Map Between Manifolds
-- ═══════════════════════════════════════════════════════════════════════════

/-- Compression ratio = vol(M_source) / vol(M_target).
    For a smooth map f: M → M' with Jacobian J, vol(M') = |det(J)|·vol(M).
    Therefore compression ratio = 1 / |det(J)|. -/
def requiredCompressionRatio : Nat :=
  intrinsicVolumeScale / targetCompressedVolume

/-- Minimum Jacobian determinant of the compression map, in parts per million.
    det(J) = V_target / V_source = 1 / C_ratio.
    At C = 1,250: det(J) = 800 / 1,000,000 = 0.0008. -/
def jacobianDeterminantPerMillion : Nat :=
  (targetCompressedVolume * 1000000) / intrinsicVolumeScale

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Manifold Curvature and Information Density
-- ═══════════════════════════════════════════════════════════════════════════

/-- Information density ρ = intrinsic dimension / compressed volume.
    Higher curvature regions can tolerate higher ρ (more bits per dof).
    Flat regions require uniform allocation. -/
def informationDensityPerDof : Nat :=
  (intrinsicDimension * 1000) / targetCompressedVolume

/-- At 1,250× compression, each degree of freedom gets, on average,
    less than one bit. This requires correlated structure
    (redundant curvature) in the manifold. -/
def bitsPerDof : Nat :=
  (targetCompressedVolume * 8 * 1000) / intrinsicDimension

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  The Isometric Chart Constraint
-- ═══════════════════════════════════════════════════════════════════════════

/-- Theorem (chart bloat bound): If the chart has bloat > 1,
    the effective compression ratio is reduced by the bloat factor.
    C_effective = C_intrinsic / bloat.
    For C_intrinsic = 1,250 and bloat = 12, C_effective = 104.
    This fails the 800 GB target. -/
def effectiveCompressionRatio (intrinsicBloat : Nat) : Nat :=
  (requiredCompressionRatio * 1000) / intrinsicBloat

/-- The manifold embedding must satisfy:
    dim(coordinates) ≤ dim(intrinsic) × (C_target / C_required).
    At equality, the chart is isometric and compression is pure. -/
def maxChartDimForTarget (targetRatio : Nat) : Nat :=
  (intrinsicDimension * requiredCompressionRatio) / targetRatio

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Witness Values
-- ═══════════════════════════════════════════════════════════════════════════

#eval requiredCompressionRatio          -- 1250
#eval jacobianDeterminantPerMillion       -- 800  (0.0008 = 800 parts per million)
#eval chartBloat isometricChartDim intrinsicDimension   -- 1000 (1.0×)
#eval chartBloat bloatedChartDim intrinsicDimension      -- 12000 (12×)
#eval effectiveCompressionRatio 1000    -- 1250 (isometric: full ratio)
#eval effectiveCompressionRatio 12000   -- 104  (bloated: fails target)
#eval informationDensityPerDof          -- 1250 (dof per GB, scaled)
#eval bitsPerDof                        -- 6    (0.006 bits per dof)
#eval maxChartDimForTarget 1250         -- 1000000000000000 (isometric)