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

/-
Goxel.lean — bounded geometric-volume packets with witness accounting

This module formalizes the current Research Stack "Goxel" surface as a finite,
receipt-bearing admission model.  It intentionally keeps the analytic manifold
language at the boundary and gives the build a discrete witness that can be
checked by native decision.

External mathematical anchor:
  Dongming Merrick Hua, Antoine Song, Stefan Tudose,
  "On Talagrand's Convexity Conjecture", arXiv:2605.10908,
  DOI: 10.48550/arXiv.2605.10908, released 2026-05-11.

Bounded claim:
  The Talagrand field below is a project witness shape for
  dimension-independent convex/probabilistic cover accounting.  It is not a
  formal proof of Talagrand's conjecture.
-/

namespace Semantics.Goxel

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Citation and claim boundary
-- ═══════════════════════════════════════════════════════════════════════════

/-- Minimal citation payload for external mathematical anchors. -/
structure ArticleAnchor where
  title : String
  authors : List String
  released : String
  doi : String
  arxiv : String
  notes : String
  deriving Repr, DecidableEq

/-- Hua-Song-Tudose Talagrand anchor supplied by the project notes. -/
def talagrandConvexityAnchor : ArticleAnchor :=
  { title := "On Talagrand's Convexity Conjecture"
  , authors := ["Dongming Merrick Hua", "Antoine Song", "Stefan Tudose"]
  , released := "2026-05-11"
  , doi := "10.48550/arXiv.2605.10908"
  , arxiv := "2605.10908"
  , notes :=
      "External mathematical anchor for dimension-independent convex covering and geometry-probability translation."
  }

/-- The formal boundary for this module's Talagrand-related definitions. -/
def talagrandClaimBoundary : String :=
  "cover-witness-shape-only-not-a-proof-of-talagrand-convexity"

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Finite Goxel coordinates and scalar fields
-- ═══════════════════════════════════════════════════════════════════════════

/-- Finite coordinate/state vector inside an active ambient n-space. -/
structure GoxelPoint where
  coords : List Int
  deriving Repr, DecidableEq

/-- The active ambient manifold is represented by its dimension and samples. -/
structure AmbientManifold where
  activeDim : Nat
  samples : List GoxelPoint
  deriving Repr, DecidableEq

/-- Scalar ingredients of the Goxel potential at a point.

All fields are nonnegative integer receipts.  Analytic norms and distances are
encoded before entering this gate, keeping this module deterministic and free of
floating point constructors.
-/
structure LocalGoxelState where
  density : Nat
  shearMismatch : Nat
  spectralMismatch : Nat
  packetDistance : Nat
  boundaryPressure : Nat
  residualScar : Nat
  deriving Repr, DecidableEq

/-- Law-axis weights for the complete Goxel potential. -/
structure GoxelWeights where
  densityWeight : Nat
  shearWeight : Nat
  spectralWeight : Nat
  packetWeight : Nat
  boundaryWeight : Nat
  residualWeight : Nat
  deriving Repr, DecidableEq

/-- Unit weights: every field contributes directly. -/
def unitWeights : GoxelWeights :=
  { densityWeight := 1
  , shearWeight := 1
  , spectralWeight := 1
  , packetWeight := 1
  , boundaryWeight := 1
  , residualWeight := 1
  }

/-- Complete finite Goxel potential.

This is the discrete counterpart of
`λρρ + λS‖S-I‖ + λC‖C-UΛUᵀ‖ + λΓ dΓ + λB B + λε ε`.
-/
def goxelPotential (w : GoxelWeights) (s : LocalGoxelState) : Nat :=
  w.densityWeight * s.density
  + w.shearWeight * s.shearMismatch
  + w.spectralWeight * s.spectralMismatch
  + w.packetWeight * s.packetDistance
  + w.boundaryWeight * s.boundaryPressure
  + w.residualWeight * s.residualScar

/-- A sampled scalar field over the ambient manifold. -/
abbrev GoxelField := GoxelPoint → LocalGoxelState

/-- A point is inside the Goxel when its potential is below the iso-threshold. -/
def insideGoxel (w : GoxelWeights) (iso : Nat) (field : GoxelField) (v : GoxelPoint) : Bool :=
  goxelPotential w (field v) <= iso

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Packet, witness, and full Goxel object
-- ═══════════════════════════════════════════════════════════════════════════

/-- Packet identity receipt Γ = γ ⊗ χ ⊗ κ ⊗ τ ⊗ spectral ⊗ θ ⊗ ε. -/
structure PacketIdentity where
  gain : Nat
  chirality : Int
  curvature : Nat
  torsion : Nat
  spectralMode : Nat
  phase : Nat
  scar : Nat
  deriving Repr, DecidableEq

/-- Internal spectral witness `C = UΛUᵀ`, represented by finite mode receipts. -/
structure SpectralWitness where
  basisHash : Nat
  eigenvalueHash : Nat
  correlationCost : Nat
  deriving Repr, DecidableEq

/-- The receipt tuple required for a Goxel admission. -/
structure GoxelWitness where
  fieldWitness : Bool
  shearWitness : Bool
  packetWitness : Bool
  spectralWitness : Bool
  residualWitness : Bool
  coverWitness : Bool
  deriving Repr, DecidableEq

/-- All witness dimensions must pass. -/
def GoxelWitness.valid (w : GoxelWitness) : Bool :=
  w.fieldWitness
    && w.shearWitness
    && w.packetWitness
    && w.spectralWitness
    && w.residualWitness
    && w.coverWitness

/-- Full proof-bearing Goxel object.

`domainSamples` are the finite `Dᵢ` witness, and `field` is the finite
potential source used to test membership in the bounded sublevel domain.
-/
structure Goxel where
  ambient : AmbientManifold
  isoThreshold : Nat
  weights : GoxelWeights
  field : GoxelField
  domainSamples : List GoxelPoint
  localDensityCost : Nat
  shearCost : Nat
  spectral : SpectralWitness
  packet : PacketIdentity
  witness : GoxelWitness
  residualScarCost : Nat
  encodedCost : Nat

/-- Domain predicate `Dᵢ = {v ∈ Mⁿ : Φᵢ(v) ≤ ιᵢ}`. -/
def Goxel.domain (g : Goxel) (v : GoxelPoint) : Bool :=
  insideGoxel g.weights g.isoThreshold g.field v

/-- Boundary predicate `∂Dᵢ = {v ∈ Mⁿ : Φᵢ(v) = ιᵢ}`. -/
def Goxel.boundary (g : Goxel) (v : GoxelPoint) : Bool :=
  goxelPotential g.weights (g.field v) = g.isoThreshold

/-- Finite-volume proxy: samples are bounded by an explicit maximum count. -/
def finiteVolumeWitness (sampleBound : Nat) (g : Goxel) : Bool :=
  g.domainSamples.length <= sampleBound

/-- Nonempty sampled domain witness. -/
def nonemptyDomainWitness (g : Goxel) : Bool :=
  g.domainSamples.any g.domain

/-- Full admissibility gate:
nonempty domain, finite sample volume, bounded residual, and valid witness. -/
def admissibleGoxel (sampleBound residualMax : Nat) (g : Goxel) : Bool :=
  nonemptyDomainWitness g
    && finiteVolumeWitness sampleBound g
    && (g.residualScarCost <= residualMax)
    && g.witness.valid

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Residual, cost, and Talagrand-style cover witness
-- ═══════════════════════════════════════════════════════════════════════════

/-- Residual/scar accounting:
boundary error + spectral mismatch + shear mismatch + packet mismatch. -/
structure ResidualTerms where
  boundaryReconstructionError : Nat
  spectralMismatch : Nat
  shearMismatch : Nat
  packetMismatch : Nat
  deriving Repr, DecidableEq

/-- Complete residual cost εᴳᵢ. -/
def residualCost (r : ResidualTerms) : Nat :=
  r.boundaryReconstructionError + r.spectralMismatch + r.shearMismatch + r.packetMismatch

/-- Encoded burden `L(Gᵢ) = K(Θᵢ) + K(Wᵢ) + K(εᵢ)`. -/
structure GoxelCostTerms where
  generatorCost : Nat
  witnessCost : Nat
  residualRepairCost : Nat
  deriving Repr, DecidableEq

/-- Total compression-native semantic mass / generator burden. -/
def goxelCost (c : GoxelCostTerms) : Nat :=
  c.generatorCost + c.witnessCost + c.residualRepairCost

/-- Talagrand-style dimension-independent cover witness.

`generatorCount ≤ universalBound` is the formal slot for the bounded cover
count; `coverResidual ≤ residualMax` records the residual outside the cover.
-/
structure TalagrandCoverWitness where
  generatorCount : Nat
  universalBound : Nat
  coverResidual : Nat
  residualMax : Nat
  deriving Repr, DecidableEq

/-- A cover is admissible when both the generator count and residual are bounded. -/
def TalagrandCoverWitness.valid (w : TalagrandCoverWitness) : Bool :=
  w.generatorCount <= w.universalBound && w.coverResidual <= w.residualMax

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  Goxel bind / merge
-- ═══════════════════════════════════════════════════════════════════════════

/-- Boundary, packet, spectral, and shear mismatch terms for binding. -/
structure MergeDistance where
  boundaryDistance : Nat
  packetDistance : Nat
  spectralDistance : Nat
  shearDistance : Nat
  deriving Repr, DecidableEq

/-- Weighted merge distance `dₘ(Gₐ,Gᵦ)`. -/
def mergeDistance (w : GoxelWeights) (d : MergeDistance) : Nat :=
  w.boundaryWeight * d.boundaryDistance
  + w.packetWeight * d.packetDistance
  + w.spectralWeight * d.spectralDistance
  + w.shearWeight * d.shearDistance

/-- Two Goxels bind when mismatch plus residuals stay under threshold. -/
def bindAdmissible
    (w : GoxelWeights) (threshold residualA residualB : Nat) (d : MergeDistance) : Bool :=
  mergeDistance w d + residualA + residualB <= threshold

/-- Hard potential composition.  Smooth log-sum-exp blending is kept outside
this finite gate because it is analytic/real-valued. -/
inductive MergeMode where
  | intersection
  | union
  deriving Repr, DecidableEq

/-- Intersection uses `max Φₐ Φᵦ`; union uses `min Φₐ Φᵦ`. -/
def mergePotential (mode : MergeMode) (potentialA potentialB : Nat) : Nat :=
  match mode with
  | .intersection => max potentialA potentialB
  | .union => min potentialA potentialB

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Dynamic evolution
-- ═══════════════════════════════════════════════════════════════════════════

/-- A lawful transition carries both the next Goxel and its admission check. -/
structure GoxelTransition where
  before : Goxel
  after : Goxel
  sampleBound : Nat
  residualMax : Nat

/-- Dynamic Goxel evolution is lawful exactly when the successor is admissible. -/
def GoxelTransition.lawful (t : GoxelTransition) : Bool :=
  admissibleGoxel t.sampleBound t.residualMax t.after

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Executable witness surface
-- ═══════════════════════════════════════════════════════════════════════════

def originPoint : GoxelPoint := { coords := [0, 0, 0] }

def boundaryPoint : GoxelPoint := { coords := [1, 0, 0] }

def outsidePoint : GoxelPoint := { coords := [9, 9, 9] }

/-- A small deterministic field with inside, boundary, and outside samples. -/
def exampleField : GoxelField := fun v =>
  if v = originPoint then
    { density := 1, shearMismatch := 0, spectralMismatch := 0
    , packetDistance := 0, boundaryPressure := 0, residualScar := 0 }
  else if v = boundaryPoint then
    { density := 1, shearMismatch := 1, spectralMismatch := 1
    , packetDistance := 0, boundaryPressure := 0, residualScar := 0 }
  else
    { density := 9, shearMismatch := 9, spectralMismatch := 9
    , packetDistance := 9, boundaryPressure := 9, residualScar := 9 }

def allWitnessesValid : GoxelWitness :=
  { fieldWitness := true
  , shearWitness := true
  , packetWitness := true
  , spectralWitness := true
  , residualWitness := true
  , coverWitness := true
  }

def exampleGoxel : Goxel :=
  { ambient := { activeDim := 3, samples := [originPoint, boundaryPoint, outsidePoint] }
  , isoThreshold := 3
  , weights := unitWeights
  , field := exampleField
  , domainSamples := [originPoint, boundaryPoint]
  , localDensityCost := 1
  , shearCost := 1
  , spectral := { basisHash := 13, eigenvalueHash := 21, correlationCost := 1 }
  , packet :=
      { gain := 1
      , chirality := 1
      , curvature := 0
      , torsion := 0
      , spectralMode := 21
      , phase := 0
      , scar := 0 }
  , witness := allWitnessesValid
  , residualScarCost := 1
  , encodedCost := goxelCost
      { generatorCost := 5, witnessCost := 6, residualRepairCost := 1 }
  }

/-- The origin is inside the example Goxel. -/
theorem origin_inside_example :
    exampleGoxel.domain originPoint = true := by
  native_decide

/-- The boundary sample lies exactly on the iso-threshold. -/
theorem boundary_is_boundary_example :
    exampleGoxel.boundary boundaryPoint = true := by
  native_decide

/-- The far sample is outside the example Goxel. -/
theorem outside_not_inside_example :
    exampleGoxel.domain outsidePoint = false := by
  native_decide

/-- The example Goxel passes the finite admission gate. -/
theorem example_admissible :
    admissibleGoxel 8 2 exampleGoxel = true := by
  native_decide

/-- Residual accounting is additive over the four scar dimensions. -/
theorem residual_example :
    residualCost
      { boundaryReconstructionError := 1
      , spectralMismatch := 2
      , shearMismatch := 3
      , packetMismatch := 4 } = 10 := by
  native_decide

/-- Compression-native Goxel burden is generator + witness + repair cost. -/
theorem cost_example :
    goxelCost { generatorCost := 5, witnessCost := 6, residualRepairCost := 1 } = 12 := by
  native_decide

/-- Dimension-independent cover witness accepts bounded generator count. -/
theorem talagrand_cover_example :
    (TalagrandCoverWitness.valid
      { generatorCount := 4, universalBound := 8, coverResidual := 1, residualMax := 2 }) = true := by
  native_decide

/-- Merge gate accepts low mismatch plus low residuals. -/
theorem bind_admissible_example :
    bindAdmissible unitWeights 10 1 1
      { boundaryDistance := 2, packetDistance := 1, spectralDistance := 1, shearDistance := 1 } = true := by
  native_decide

#eval! talagrandConvexityAnchor.arxiv
#eval! talagrandClaimBoundary
#eval! goxelPotential unitWeights (exampleField originPoint)
#eval! goxelPotential unitWeights (exampleField boundaryPoint)
#eval! admissibleGoxel 8 2 exampleGoxel
#eval! TalagrandCoverWitness.valid
  { generatorCount := 4, universalBound := 8, coverResidual := 1, residualMax := 2 }

end Semantics.Goxel