Research-Stack/0-Core-Formalism/lean/Semantics/Semantics/PandigitalSpectralMass.lean

/-
  PandigitalSpectralMass.lean

  Compact "pandigital" representations for eigenvectors and semantic mass.

  Core insight: Just as π = 3.8415926 - 0.7 uses each digit once,
  eigenvectors and mass triples can be encoded with minimal unique components
  and reconstructed via simple operations.

  Three compression strategies:
  1. ContinuedFractionEigenvector - Store convergents, not floats
  2. ZNCompactMass - Pack (Z, N) into single value, derive A = Z + N
  3. SpectralMassFusion - Eigenvectors with semantic mass weights

  Domain: LAYER_D_INVARIANTS (geometric_bind)
  Per AGENTS.md §1.4: Uses Q16_16 for hardware-native computation.
-/

import Mathlib.Data.Nat.Basic
import Mathlib.Data.Fin.Basic
import Semantics.FixedPoint

namespace Semantics.PandigitalSpectralMass

open Semantics.Q16_16
open Semantics.FixedPoint.PandigitalPi

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Continued Fraction Eigenvector Components
-- ═══════════════════════════════════════════════════════════════════════════

/--
Continued fraction convergent for eigenvector component storage.
Instead of storing Q16.16 float, store (numerator, denominator) as Nat pair.
Reconstruct: component = numerator / denominator

Space efficiency:
- Direct Q16.16: 4 bytes per component
- Continued fraction: 2-4 bytes per component (small denominators compress better)
- Example: 355/113 approximates π to 6 digits, stored in ~2 bytes
-/
structure CFConvergent where
  num : Nat  -- numerator
  den : Nat  -- denominator (non-zero)
  deriving Repr, DecidableEq, Inhabited

/-- Reconstruct Q16.16 from continued fraction convergent -/
def cfConvergentToQ16 (cf : CFConvergent) : Q16_16 :=
  if cf.den = 0 then zero
  else ofRatio cf.num cf.den

/-- Optimal continued fraction for golden ratio φ = [1; 1, 1, 1, ...] -/
def phiConvergents : List CFConvergent := [
  ⟨1, 1⟩,    -- 1/1 = 1.0
  ⟨2, 1⟩,    -- 2/1 = 2.0  (actually 1+1/1)
  ⟨3, 2⟩,    -- 3/2 = 1.5
  ⟨5, 3⟩,    -- 5/3 ≈ 1.667
  ⟨8, 5⟩,    -- 8/5 = 1.6
  ⟨13, 8⟩,   -- 13/8 = 1.625
  ⟨21, 13⟩,  -- 21/13 ≈ 1.615
  ⟨34, 21⟩,  -- 34/21 ≈ 1.619
  ⟨55, 34⟩,  -- 55/34 ≈ 1.6176
  ⟨89, 55⟩   -- 89/55 ≈ 1.61818 (6 digits accurate)
]

/-- Optimal continued fraction for π convergents -/
def piConvergents : List CFConvergent := [
  ⟨3, 1⟩,      -- 3/1 = 3.0
  ⟨22, 7⟩,     -- 22/7 ≈ 3.142857 (2 digits)
  ⟨333, 106⟩,  -- 333/106 ≈ 3.141509 (4 digits)
  ⟨355, 113⟩,  -- 355/113 ≈ 3.1415929 (6 digits) ← BEST
  ⟨103993, 33102⟩  -- 9 digits (overkill for Q16.16)
]

/-- Select best convergent for target precision in Q16.16 -/
def selectConvergent (convergents : List CFConvergent) (target : Q16_16) (tolerance : Q16_16) : CFConvergent :=
  match convergents with
  | [] => ⟨0, 1⟩  -- default
  | cf :: rest =>
    let reconstructed := cfConvergentToQ16 cf
    if abs (reconstructed - target) ≤ tolerance then
      cf
    else
      selectConvergent rest target tolerance

-- Verification: 355/113 is within Q16.16 resolution of pandigital pi
#eval cfConvergentToQ16 ⟨355, 113⟩  -- Expected: ~3.14159
#eval abs (cfConvergentToQ16 ⟨355, 113⟩ - PandigitalPi.piPandigital)  -- Expected: small

-- ═══════════════════════════════════════════════════════════════════════════
-- §1.5  Mass Number Type Definitions (Local to avoid otom dependency)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Direction of Z/N imbalance for semantic mass -/
inductive BiasSign where
  | structuredHeavy  -- Z > N: control/witness/archive mass dominates
  | balanced         -- Z = N or within tolerance
  | stressHeavy      -- N > Z: dynamics/residual/drain mass dominates
  deriving Repr, DecidableEq, Inhabited

/-- Operational phase after mass classification -/
inductive MassPhase where
  | grounded
  | driftBalanced
  | structuredDrift
  | stressDrift
  | seismic
  deriving Repr, DecidableEq, Inhabited

/-- Downstream route from collapsed mass field -/
inductive MassRoute where
  | promote
  | standard
  | bhocsCommit
  | fammDrain
  | quarantine
  deriving Repr, DecidableEq, Inhabited

/-- S3C shell address for total mass number A -/
structure S3CShellAddress where
  totalMass : Nat  -- A = Z + N
  shellK    : Nat  -- k = floor(sqrt A)
  shellA    : Nat  -- a = A - k^2
  shellB0   : Nat  -- b0 = (k+1)^2 - 1 - A
  shellBPlus : Nat -- b+ = (k+1)^2 - A
  mass0     : Nat  -- m0 = a * b0
  massPlus  : Nat  -- m+ = a * b+
  deriving Repr, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Compact Z/N Mass Encoding (Pandigital-Style)
-- ═══════════════════════════════════════════════════════════════════════════

/--
Compact encoding of (Z, N) mass pair into single value.

Encoding: compact = Z * 65536 + N (concatenation in Q16.16 space)
Constraint: Z < 65536, N < 65536 (within Q16.16 integer range)
Derivation: A = Z + N (total mass), bias = sign(Z - N)

Space: 4 bytes stores both Z and N (vs 8 bytes separate)
-/
def encodeZNCompact (Z N : Nat) : Q16_16 :=
  let zClamped := min Z 65535
  let nClamped := min N 65535
  ofNat (zClamped * 65536 + nClamped)

/-- Decode compact Z/N encoding -/
def decodeZNCompact (compact : Q16_16) : (Nat × Nat) :=
  let raw := compact.toInt.natAbs
  let Z := raw / 65536
  let N := raw % 65536
  (Z, N)

/-- Verify round-trip encoding -/
theorem znRoundTrip (Z N : Nat) (hZ : Z < 65536) (hN : N < 65536) :
  decodeZNCompact (encodeZNCompact Z N) = (Z, N) := by
  sorry  -- TODO: Complete proof with omega after verifying clamping logic

/-- Derive total mass A from compact encoding -/
def deriveAFromCompact (compact : Q16_16) : Nat :=
  let (Z, N) := decodeZNCompact compact
  Z + N

/-- Derive bias sign from compact encoding -/
def deriveBiasFromCompact (compact : Q16_16) : BiasSign :=
  let (Z, N) := decodeZNCompact compact
  if Z > N then .structuredHeavy
  else if N > Z then .stressHeavy
  else .balanced

-- Example encodings
#eval encodeZNCompact 400 100   -- Structured heavy (Z > N)
#eval deriveAFromCompact (encodeZNCompact 400 100)  -- Expected: 500
#eval deriveBiasFromCompact (encodeZNCompact 400 100)  -- Expected: structuredHeavy

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Spectral-Mass Eigenvector (Pandigital Fusion)
-- ═══════════════════════════════════════════════════════════════════════════

/--
Eigenvector component with semantic mass weighting.

Standard eigenvector: stores n float components (4n bytes)
Pandigital spectral-mass: stores (convergent, mass-weight) pairs
  - convergent: CFConvergent (compact rational approximation)
  - mass-weight: Q16.16 weight (Z/N ratio or total mass influence)

Reconstruction: component_i = (num_i/den_i) * massWeight_i
-/
structure SpectralMassComponent where
  cf : CFConvergent      -- Rational approximation of eigenvector component
  massWeight : Q16_16    -- Semantic mass scaling factor
  phase : Q16_16         -- Phase angle for complex components (optional)
  deriving Repr, Inhabited

/-- Reconstruct full component value -/
def reconstructComponent (smc : SpectralMassComponent) : Q16_16 :=
  let rationalPart := cfConvergentToQ16 smc.cf
  rationalPart * smc.massWeight

/--
Sparse spectral-mass eigenvector: only store non-zero components.
Uses pandigital principle: store (index, component) pairs, reconstruct sparse vector.
-/
structure SparseSpectralEigenvector (n : Nat) where
  dimension : Nat              -- full dimension n
  nonZeroCount : Nat           -- number of stored components
  components : Fin nonZeroCount → SpectralMassComponent  -- compact components
  indices : Fin nonZeroCount → Fin n                   -- positions in full vector
  deriving Repr

/-- Reconstruct full eigenvector component at index i -/
def reconstructEigenvectorComponent {n : Nat} (_v : SparseSpectralEigenvector n) (_i : Fin n) : Q16_16 :=
  -- Search for component at index i (simplified - returns zero)
  -- Full implementation would search indices array and return matching component
  zero

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Pandigital Mass Number Field (Compact Collapsed Field)
-- ═══════════════════════════════════════════════════════════════════════════

/--
Ultra-compact mass number field using pandigital encoding principles.

Standard MassNumberField: stores (Z, N, A, packets, biasSign) separately
Pandigital version: stores single compact value + derived fields

Components:
  - znCompact: Q16.16 encoding of (Z, N) pair
  - shellAddress: S3C shell address (k, a, b0, b+ computed from A)
  - phase: derived from Z/N bias
  - route: derived from phase + thresholds

Space: ~8 bytes vs ~32+ bytes for full MassNumberField
-/
structure PandigitalMassField where
  znCompact : Q16_16      -- Encoded (Z, N) pair
  shellK : Nat          -- k = floor(sqrt(A)) where A = Z + N
  lyapunovResidual : Q16_16  -- Residual from PIST witness
  deriving Repr, Inhabited

/-- Construct from full components (collapse step) -/
def fromFullComponents (Z N : Nat) (lyap : Q16_16) : PandigitalMassField :=
  let compact := encodeZNCompact Z N
  let A := Z + N
  let k := Nat.sqrt A
  { znCompact := compact, shellK := k, lyapunovResidual := lyap }

/-- Reconstruct full S3C shell address -/
def reconstructShellAddress (pmf : PandigitalMassField) : S3CShellAddress :=
  let (Z, N) := decodeZNCompact pmf.znCompact
  let A := Z + N
  let k := pmf.shellK
  let a := A - k * k
  let b0 := (k + 1) * (k + 1) - 1 - A
  let bPlus := (k + 1) * (k + 1) - A
  let m0 := a * b0
  let mPlus := a * bPlus
  { totalMass := A, shellK := k, shellA := a, shellB0 := b0, shellBPlus := bPlus, mass0 := m0, massPlus := mPlus }

/-- Derive mass phase from pandigital encoding -/
def deriveMassPhase (pmf : PandigitalMassField) : MassPhase :=
  let (Z, N) := decodeZNCompact pmf.znCompact
  let A := Z + N
  if pmf.lyapunovResidual > ofNat 50000 then  -- threshold for seismic
    .seismic
  else if Z > N && Z > A / 3 then
    .structuredDrift
  else if N > Z && N > A / 3 then
    .stressDrift
  else
    .driftBalanced

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Verification and Examples
-- ═══════════════════════════════════════════════════════════════════════════

/-- Example: Compact encoding of (Z=400000, N=100000) mass pair -/
def exampleCompact400k : Q16_16 := encodeZNCompact 400000 100000
#eval exampleCompact400k.toInt  -- Will saturate due to >65535 limits

/-- Example: Small mass pair within range -/
def exampleCompactSmall : Q16_16 := encodeZNCompact 400 100
#eval exampleCompactSmall.toInt   -- Expected: 400 * 65536 + 100 = 26214500

-- Verify reconstruction
#eval deriveAFromCompact exampleCompactSmall  -- Expected: 500
#eval deriveBiasFromCompact exampleCompactSmall  -- Expected: structuredHeavy

-- Example: Spectral-mass component using 355/113 π convergent
def examplePiComponent : SpectralMassComponent := {
  cf := ⟨355, 113⟩,
  massWeight := Q16_16.one,  -- unit weight
  phase := zero
}
#eval reconstructComponent examplePiComponent  -- Expected: ~3.14159

-- Example: φ-weighted component (golden ratio mass weighting)
def examplePhiWeightedComponent : SpectralMassComponent := {
  cf := ⟨355, 113⟩,  -- π approximation
  massWeight := ofNat 106039,  -- φ ≈ 1.618 in Q16.16
  phase := zero
}
#eval reconstructComponent examplePhiWeightedComponent  -- Expected: ~5.086

end Semantics.PandigitalSpectralMass

namespace Semantics
export PandigitalSpectralMass (
  CFConvergent cfConvergentToQ16
  piConvergents phiConvergents selectConvergent
  encodeZNCompact decodeZNCompact deriveAFromCompact deriveBiasFromCompact
  SpectralMassComponent reconstructComponent
  SparseSpectralEigenvector
  PandigitalMassField fromFullComponents reconstructShellAddress deriveMassPhase
)
end Semantics