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

/-
  Field Equation System Integration
  Extending Holy Diver/ENE with Σ-selector, MMR, Pentagonal Squares
  
  This module integrates the mathematical concepts from the FAMM/DP/IUTT
  conversation into the Lean formalization, including:
  - Unified field equation system
  - Σ-selector (nexus operator)
  - Merkle Mountain Range (MMR) with self-feeding
  - Pentagonal square computational cells
  - Near-miss tension function (Fermat sieve)
  - Web stabilization constraints
  - Soft/hard collapse mechanisms
-/

import Mathlib.Data.Nat.Basic
import Mathlib.Data.List.Basic
import Mathlib.Data.Fin.Basic
import Semantics.RealityContractMassNumber

namespace HolyDiver
namespace FieldSystem

/-! ## Core field enumerations -/

/-- Collapse mode: soft (preserve residual) or hard (absolute zero). -/
inductive CollapseMode where
  | soft   -- Preserve minimum residual signal
  | hard   -- Absolute zero
  deriving Repr, DecidableEq

/-- The four primary field types in the unified equation. -/
inductive FieldType where
  | famm      -- FAMM geometric transformation
  | iutt      -- IUTT quantum path-splitting
  | center    -- Center mathematical models
  | dp        -- Dynamic programming optimization
  deriving Repr, DecidableEq

/-- Field state representation using natural-number weights. -/
structure FieldState where
  fieldType : FieldType
  value     : Nat
  weight    : Nat
  deriving Repr

/-! ## Unified field equation system -/

/-- 
The unified field equation:
  Ψ(t) = (1/4)[F(t) ⊗ Φ(t) ⊗ C(t) ⊗ D(t)]
  
This represents the coupled composite field at time t.
-/
structure UnifiedState where
  famm   : FieldState
  iutt   : FieldState
  center : FieldState
  dp     : FieldState
  deriving Repr

/-- Compute the composite value of the unified state. -/
def UnifiedState.composite (s : UnifiedState) : Nat :=
  (s.famm.value + s.iutt.value + s.center.value + s.dp.value) / 4

/-- Compute the weighted composite (includes field weights). -/
def UnifiedState.weightedComposite (s : UnifiedState) : Nat :=
  (s.famm.value * s.famm.weight +
   s.iutt.value * s.iutt.weight +
   s.center.value * s.center.weight +
   s.dp.value * s.dp.weight) / 4

/-! ## Σ-selector (nexus operator) -/

/-- 
The Σ-selector (nexus operator) evaluates and selects the best
cross-field continuation from candidate field states.
-/
structure SigmaSelector where
  scoringFunction : Nat → Nat → Nat → Nat → Nat
  threshold      : Nat

/-- Score a candidate field configuration. -/
def SigmaSelector.score
    (σ : SigmaSelector)
    (f i c d : Nat) : Nat :=
  σ.scoringFunction f i c d

/-! ## Pentagonal squares -/

/--
A pentagonal square is a computational cell with four corners and a fifth
nexus center. The center value sigma must balance the corners.
Simplified version using Nat fields for easier integration.
-/
structure PentagonalSquare where
  famm   : Nat  -- FAMM geometric transformation value
  iutt   : Nat  -- IUTT quantum path-splitting value
  center : Nat  -- Center mathematical models value
  dp     : Nat  -- Dynamic programming optimization value
  sigma  : Nat  -- 5th center nexus value
  deriving Repr

/-- Compute the pentagonal closure (sum of corners plus center). -/
def PentagonalSquare.closure (p : PentagonalSquare) : Nat :=
  p.famm + p.iutt + p.center + p.dp + p.sigma

/-- Check if the pentagonal square is balanced (center equals average of corners). -/
def PentagonalSquare.isBalanced (p : PentagonalSquare) : Bool :=
  let cornerSum := p.famm + p.iutt + p.center + p.dp
  p.sigma * 4 == cornerSum

/-! ## Morphic cores for identity-through-transform -/

/-- 
Morphic core: stable identity handle for when an idea changes representation across fields.
Preserves identity through transformations while allowing field representation changes.
-/
structure MorphicCore where
  coreId         : Nat  -- Unique identifier
  signature      : Nat  -- Cryptographic signature
  invariantMass  : Nat  -- Mass preserved through transforms
  transformCount : Nat  -- Number of transformations
  stable         : Bool -- Core stability status
  deriving Repr

/-- Check if a morphic core is stable (Rule 56). -/
def MorphicCore.isStable (m : MorphicCore) : Bool :=
  m.stable

/-! ## History indirection for long-term stability -/

/-- 
History reference: typed indirection to MMR history.
This provides semantic metadata about the history root without
embedding the full history structure in each record.
-/
structure HistoryRef where
  root       : Nat  -- MMR root hash
  generation : Nat  -- Registry generation number
  verified   : Bool -- Cryptographic verification status
  deriving Repr

/-! ## Enhanced candidate structures -/

/-- 
Enhanced candidate: extends the base Candidate with field equation state,
tension score, typed history reference, and morphic core for identity continuity.
-/
structure EnhancedCandidate where
  base       : ENE.Candidate
  core       : MorphicCore  -- Identity-through-transform handle
  fieldState : PentagonalSquare
  tension    : Nat  -- Near-miss tension score
  historyRef : HistoryRef  -- Typed history indirection
  deriving Repr

/-- 
Rule 56 — Morphic Core Invariance:
A transformed candidate may be treated as the same forest object only when its morphic core remains stable.
-/
def morphicCoreInvariant (cand : EnhancedCandidate) : Bool :=
  cand.core.stable

/-! ## Merkle Mountain Range (MMR) -/

/-- A Merkle Mountain Range node. -/
structure MMRNode where
  value : Nat
  hash  : Nat
  deriving Repr

/-- Simple hash function for demonstration (in practice, use cryptographic hash). -/
def mmrHash (value : Nat) : Nat :=
  value * 31 + 17

/-- Create an MMR node from a value. -/
def createMMRNode (value : Nat) : MMRNode :=
  { value := value, hash := mmrHash value }

/-- 
A Merkle Mountain Range: append-only history structure.
  Each mountain is a perfect binary tree of height h.
-/
structure MMRMountain where
  height : Nat
  nodes  : List MMRNode
  deriving Repr

/-- Compute the root hash of a mountain. -/
def MMRMountain.root (m : MMRMountain) : Nat :=
  match m.nodes with
  | [] => 0
  | [n] => n.hash
  | _ =>
    -- Simplified: XOR all hashes (in practice, proper Merkle tree)
    m.nodes.foldl (fun acc n => Nat.xor acc n.hash) 0

/-- 
An MMR with self-feeding: the root of the previous state
  becomes part of the next selection criterion.
-/
structure SelfFeedingMMR where
  mountains : List MMRMountain
  currentRoot : Nat
  deriving Repr

/-- Append a new value to the MMR and update the root. -/
def SelfFeedingMMR.append (mmr : SelfFeedingMMR) (value : Nat) : SelfFeedingMMR :=
  let newNode := createMMRNode value
  let newMountain := { height := 1, nodes := [newNode] }
  let newMountains := newMountain :: mmr.mountains
  let newRoot := mmrHash (mmr.currentRoot + newNode.hash)
  { mountains := newMountains, currentRoot := newRoot }

/-! ## Near-miss tension function (Fermat sieve) -/

/-- 
The near-miss error function:
  ε(P) = |(x^n + y^n)^(1/n) - z|
  
For Lean-core compatibility, we use a simplified version
  that measures distance from a target.
-/
structure NearMissPoint where
  x : Nat
  y : Nat
  z : Nat
  n : Nat
  deriving Repr

/-- Compute the near-miss error (simplified for Lean-core). -/
def NearMissPoint.epsilon (p : NearMissPoint) : Nat :=
  -- Simplified: |x + y - z| (avoiding fractional exponents)
  if p.x + p.y > p.z then p.x + p.y - p.z else p.z - (p.x + p.y)

/-- 
The tension function:
  T(P) = |ε(P) - μ| + 1/(|ε(P) - μ| + δ)
  
where μ is the average error and δ prevents division by zero.
-/
structure TensionFunction where
  delta : Nat  -- Safety value to prevent division by zero
  deriving Repr

/-- Compute the average error over a list of near-miss points. -/
def averageError (points : List NearMissPoint) : Nat :=
  match points with
  | [] => 0
  | _ =>
    let totalError := points.foldl (fun acc p => acc + p.epsilon) 0
    totalError / points.length

/-- Compute the tension score for a point given the average error. -/
def TensionFunction.tension
    (tf : TensionFunction)
    (point : NearMissPoint)
    (avgError : Nat) : Nat :=
  let diff := if point.epsilon > avgError then point.epsilon - avgError else avgError - point.epsilon
  let denom := diff + tf.delta
  diff + (if denom == 0 then 0 else 1000 / denom)  -- Simplified division

/-- 
The Fermat Near-Miss Sieve: classifies candidates based on
  their tension score.
-/
inductive SieveClassification where
  | genuine      -- Truly valid
  | suspicious   -- Near-miss, high tension
  | invalid      -- Obviously wrong
  deriving Repr, DecidableEq

/-- Classify a point using the tension function. -/
def TensionFunction.classify
    (tf : TensionFunction)
    (point : NearMissPoint)
    (avgError : Nat)
    (lowThreshold : Nat)
    (highThreshold : Nat) : SieveClassification :=
  let tension := tf.tension point avgError
  if tension < lowThreshold then
    SieveClassification.genuine
  else if tension > highThreshold then
    SieveClassification.invalid
  else
    SieveClassification.suspicious

/-! ## Collapse mechanisms -/

/-- Collapse a value based on threshold and mode. -/
def collapseValue
    (mode : CollapseMode)
    (threshold : Nat)
    (epsilon : Nat)
    (value : Nat) : Nat :=
  if value < threshold then
    match mode with
    | CollapseMode.soft => epsilon
    | CollapseMode.hard => 0
  else
    value

/-! ## Web stabilization constraints -/

/-- A web constraint connecting two field states. -/
structure WebConstraint where
  source      : Nat  -- Index of source field
  target      : Nat  -- Index of target field
  strength    : Nat  -- Constraint strength
  deriving Repr

/-- Web stabilization system. -/
structure WebSystem where
  constraints : List WebConstraint
  deriving Repr

/-- Helper to safely get nth element from list with default. -/
def getNthDefault : List Nat → Nat → Nat → Nat
  | [], _, default => default
  | x :: _, 0, _ => x
  | _ :: xs, n, default => getNthDefault xs (n - 1) default

/-- Apply web constraints to stabilize a field state. -/
def WebSystem.stabilize
    (ws : WebSystem)
    (state : List Nat) : List Nat :=
  ws.constraints.foldl
    (fun acc c =>
      if c.source < state.length ∧ c.target < state.length then
        let sourceVal := getNthDefault state c.source 0
        let targetVal := getNthDefault state c.target 0
        let stabilized := (sourceVal * c.strength + targetVal) / (c.strength + 1)
        acc.modify c.target (fun _ => stabilized)
      else
        acc)
    state

/-! ## Integrated field cell with full architecture -/

/-- 
The complete integrated field cell:
  - Pentagonal square base
  - MMR history commitment
  - Web stabilization
  - Σ-selector with tension-aware scoring
-/
structure IntegratedFieldCell where
  pentagon : PentagonalSquare
  mmr      : SelfFeedingMMR
  webs     : WebSystem
  tension  : TensionFunction
  deriving Repr

/-- 
The full update step for an integrated field cell:
  1. Apply web stabilization
  2. Compute tension score
  3. Update MMR with new state
  4. Σ-selector uses MMR root for next decision
-/
def IntegratedFieldCell.update
    (cell : IntegratedFieldCell) : IntegratedFieldCell :=
  -- Step 1: Apply web stabilization
  let currentState := [cell.pentagon.famm, cell.pentagon.iutt,
                      cell.pentagon.center, cell.pentagon.dp]
  let stabilized := cell.webs.stabilize currentState
  
  -- Step 2: Update pentagonal square (using Nat fields directly)
  let newPentagon :=
    { famm   := getNthDefault stabilized 0 0,
      iutt   := getNthDefault stabilized 1 0,
      center := getNthDefault stabilized 2 0,
      dp     := getNthDefault stabilized 3 0,
      sigma := cell.pentagon.sigma }
  
  -- Step 3: Update MMR with new composite value
  let composite := newPentagon.closure
  let newMMR := cell.mmr.append composite
  
  -- Step 4: Return updated cell
  { pentagon := newPentagon,
    mmr      := newMMR,
    webs     := cell.webs,
    tension  := cell.tension }

/-! ## Auto-mapping structure -/

/-- 
Auto-mapping entry: maps a concept from the JSON conversation
  to its Lean formalization counterpart.
-/
structure AutoMapping where
  jsonConcept    : String
  leanStructure  : String
  description    : String
  confidence     : Nat  -- 0-100 confidence score
  deriving Repr

/-- The auto-mapping registry. -/
def autoMappingRegistry : List AutoMapping :=
  [
    { jsonConcept   := "Σ-selector",
      leanStructure := "SigmaSelector",
      description  := "Nexus operator that evaluates and selects best cross-field continuation",
      confidence    := 95 },
    { jsonConcept   := "MMR (Merkle Mountain Range)",
      leanStructure := "SelfFeedingMMR",
      description  := "Append-only history commitment that feeds back into selector",
      confidence    := 90 },
    { jsonConcept   := "Pentagonal square",
      leanStructure := "PentagonalSquare",
      description  := "4-corner computational cell with 5th center/nexus constraint",
      confidence    := 95 },
    { jsonConcept   := "F(t) - FAMM",
      leanStructure := "FieldType.famm",
      description  := "FAMM geometric transformation field",
      confidence    := 100 },
    { jsonConcept   := "Φ(t) - IUTT",
      leanStructure := "FieldType.iutt",
      description  := "IUTT quantum path-splitting field",
      confidence    := 100 },
    { jsonConcept   := "C(t) - Center models",
      leanStructure := "FieldType.center",
      description  := "Center mathematical models field",
      confidence    := 100 },
    { jsonConcept   := "D(t) - Dynamic programming",
      leanStructure := "FieldType.dp",
      description  := "Dynamic programming optimization field",
      confidence    := 100 },
    { jsonConcept   := "Unified equation Ψ(t)",
      leanStructure := "UnifiedState",
      description  := "Coupled composite field state",
      confidence    := 95 },
    { jsonConcept   := "Tension function T(P)",
      leanStructure := "TensionFunction",
      description  := "Near-miss detection and tension scoring",
      confidence    := 90 },
    { jsonConcept   := "Fermat Near-Miss Sieve",
      leanStructure := "SieveClassification",
      description  := "Classification of candidates as genuine/suspicious/invalid",
      confidence    := 85 },
    { jsonConcept   := "Soft/Hard collapse",
      leanStructure := "CollapseMode",
      description  := "Collapse mode for field values",
      confidence    := 95 },
    { jsonConcept   := "Web stabilization",
      leanStructure := "WebSystem",
      description  := "Constraint edges that stabilize field geometry",
      confidence    := 90 },
    { jsonConcept   := "Integrated field cell",
      leanStructure := "IntegratedFieldCell",
      description  := "Complete cell with pentagonal base, MMR, webs, and tension",
      confidence    := 85 }
  ]

/-- Lookup a mapping by JSON concept name. -/
def lookupMapping (concept : String) : Option AutoMapping :=
  autoMappingRegistry.find? (fun m => m.jsonConcept == concept)

/-- Lookup a mapping by Lean structure name. -/
def lookupLeanMapping (structName : String) : Option AutoMapping :=
  autoMappingRegistry.find? (fun m => m.leanStructure == structName)

end FieldSystem
end HolyDiver