SilverSight/formal/SilverSight/GCCL.lean

/-
  GCCL.lean — Geometric, Cognitive, and Compression Law

  Ports GCCL from Research Stack, reformulated for SilverSight conventions.

  GCCL is the law layer that decides whether a transformation of a structured
  object is lawful enough to promote. It sits over the layered state mountains:

    NUVMAP = projection/address mountain (Sidon labels → 8-strand address)
    AVMR   = vector-state evolution mountain (PhaseVec accumulator)
    AMMR   = commit/history mountain (MMR append/merge cascade)
    O-AMMR = committed orthogonal/QR-basis mountain (observer projection)
    GCCL-Rep = compact transition rope between mountains (receipt)

  Each layer verifies a different part of the transition:
    NUVMAP  → address/projection validity
    AVMR    → vector-state evolution / append law
    AMMR    → commit ancestry / receipt history
    O-AMMR  → orthogonal projection / QR-basis structure
    GCCL    → combined lawfulness of transition

  Key rule: "A GCCL-Rep event may be multi-projected, but it may not be
  multi-trusted. Each mountain verifies its own projection."

  Connection to the pipeline:
  - Equation → DNA encoder → logogram atom → GCCL gate → MMR append → SpherionState
  - GCCL decides: admit, reject, hold, or quarantine the transition
  - The gcclSwapGate (in MultiSurfacePacker.lean) checks if improvement ≥ risk
  - AngrySphinx provides the energy budget for the gate

  Connection to the photonic Sidon search:
  - Each candidate (Sidon set) is a GCCL transition
  - GCCL checks if the candidate improves the state (lower Omega)
  - AngrySphinx charges 2^depth per failed candidate
  - The NaN boundary terminates the search when frustration → 0

  Connection to the COUCH evolution chain:
  COUCH equation → Lean discretization → COUCH_stable gate → admission filter
  This IS the GCCL pipeline: continuous math → formal witness → gate → routing.
-/

import Mathlib.Tactic
import SilverSight.FixedPoint

namespace SilverSight.GCCL

open SilverSight.FixedPoint
open SilverSight.FixedPoint.Q16_16

/-! §1 Law Axes

  GCCL encodes transitions across seven law surfaces:
  - Geometric: state space, topology, projection, address
  - Cognitive: meaning, identity, salience, routing burden
  - Compression: canonicalization, delta, representative carrier
  - Residual: mismatch, loss, drift, reconstruction error
  - Cost: compute, memory, routing, storage
  - Scale: lambda band where the claim is valid
  - Receipt: witness record explaining what passed/failed
-/

/-- The seven GCCL law axes. -/
inductive LawAxis where
  | geometric
  | cognitive
  | compression
  | residual
  | cost
  | scale
  | receipt
  deriving DecidableEq, Repr, Fintype

/-- Number of law axes = 7. -/
theorem lawAxis_count : Fintype.card LawAxis = 7 := by decide

/-! §2 Promotion Ladder (Claim-State Ladder) -/

/-- Promotion states for a GCCL candidate.
    Matches the anti-smuggle claim-state ladder:
    RAW_IDEA → SANITIZED_METAPHOR → TOY_MODEL → TYPED_MODEL →
    RESIDUAL_TESTED → COST_ACCOUNTED → PROOF_CANDIDATE → CORE_MODULE -/
inductive PromotionRung where
  | rawIdea
  | sanitizedMetaphor
  | toyModel
  | typedModel
  | residualTested
  | costAccounted
  | proofCandidate
  | coreModule
  deriving DecidableEq, Repr, Fintype

/-- The promotion ladder has 8 rungs. -/
theorem promotionRung_count : Fintype.card PromotionRung = 8 := by decide

/-! §3 Layered State Mountains

  GCCL sits over layered state mountains. Each mountain verifies a
  different aspect of the transition.

  This mirrors the Hachimoji 8-state system:
  Each layer corresponds to one strand of the braid.
-/

/-- The five mountain layers. -/
inductive MountainLayer where
  | nuvmap    -- address/projection mountain
  | avmr      -- vector-state evolution mountain
  | ammr      -- commit/history mountain
  | oammr     -- orthogonal/QR-basis mountain
  | gcclRep   -- transition rope between mountains
  deriving DecidableEq, Repr, Fintype

/-- Number of mountain layers = 5. -/
theorem mountainLayer_count : Fintype.card MountainLayer = 5 := by decide

/-- Each layer verifies a different aspect of the transition. -/
def layerVerificationRole : MountainLayer → String
  | .nuvmap  => "address/projection validity"
  | .avmr    => "vector-state evolution / append law"
  | .ammr    => "commit ancestry / receipt history"
  | .oammr   => "orthogonal projection / QR-basis structure"
  | .gcclRep => "transition rope / combined lawfulness"

/-! §4 Decision States -/

/-- Receipt decision states. -/
inductive Decision where
  | accept
  | reject
  | hold
  | quarantine
  deriving DecidableEq, Repr, Fintype

/-- Number of decisions = 4. -/
theorem decision_count : Fintype.card Decision = 4 := by decide

/-! §5 Projection Kinds

  The kinds of projections that occur across GCCL surfaces.
  Each maps to a component of the SilverSight pipeline.
-/

/-- Projection families in the GCCL system. -/
inductive ProjectionKind where
  | address           -- NUVMAP: Sidon labels → 8-strand address
  | vectorState       -- AVMR: PhaseVec accumulator
  | commitHistory     -- AMMR: MMR append/merge cascade
  | orthogonalBasis   -- O-AMMR: observer projection (QR decomposition)
  | goxelScalarField  -- Goxel: bounded scalar sub-manifold
  | logogramGlyph    -- Logogram: oriented symbolic atom
  | modelGenome       -- DNA encoding: hachimoji sequence
  | workflowDag       -- Workflow: directed acyclic graph
  deriving DecidableEq, Repr, Fintype

/-! §6 Scale Bands -/

/-- Scale bands where GCCL claims are valid. -/
inductive ScaleBand where
  | toy
  | local
  | benchmark
  | production
  | crossDomain
  deriving DecidableEq, Repr, Fintype

/-! §7 Transition Wrapper

  Every GCCL transition is wrapped by the UMUP-lambda / IRP tuple:
  M = (S, T, I, R, K, P, Q, Lambda)

  A wrapper is complete only when all fields are declared.
-/

/-- UMUP-lambda wrapper: declares all aspects of a transition. -/
structure Wrapper where
  stateSpaceDeclared : Bool    -- S: state space
  transformDeclared : Bool     -- T: transform
  invariantsDeclared : Bool   -- I: invariants
  residualDeclared : Bool     -- R: residual
  costDeclared : Bool          -- K: cost
  projectionDeclared : Bool   -- P: projection
  quarantineDeclared : Bool   -- Q: quarantine path
  scaleDeclared : Bool        -- Lambda: scale band
  deriving Repr, DecidableEq, Inhabited

/-- A wrapper is complete when all fields are declared. -/
def wrapperComplete (w : Wrapper) : Bool :=
  w.stateSpaceDeclared &&
  w.transformDeclared &&
  w.invariantsDeclared &&
  w.residualDeclared &&
  w.costDeclared &&
  w.projectionDeclared &&
  w.quarantineDeclared &&
  w.scaleDeclared

/-- A complete wrapper has all fields true. -/
theorem wrapperComplete_all_true (w : Wrapper) :
    wrapperComplete w ↔
    w.stateSpaceDeclared ∧ w.transformDeclared ∧ w.invariantsDeclared ∧
    w.residualDeclared ∧ w.costDeclared ∧ w.projectionDeclared ∧
    w.quarantineDeclared ∧ w.scaleDeclared := by
  simp [wrapperComplete]

/-! §8 Transition Gate

  A transition enters the Bounded Lawful Surface only if it has:
  - Complete wrapper
  - Valid syntax
  - Round-trip or declared loss policy
  - Invariant preservation
  - Residual within bound
  - Cost within bound
  - ACCEPT receipt
-/

/-- A GCCL transition with all gates and receipt evidence. -/
structure Transition where
  wrapper : Wrapper
  validSyntax : Bool
  roundTripOrLossPolicy : Bool
  invariantPreserved : Bool
  residualWithinBound : Bool
  costWithinBound : Bool
  decision : Decision
  scaleBand : ScaleBand
  deriving Repr, DecidableEq, Inhabited

/-- A transition is lawful if it satisfies all gates. -/
def isLawful (t : Transition) : Bool :=
  wrapperComplete t.wrapper &&
  t.validSyntax &&
  t.roundTripOrLossPolicy &&
  t.invariantPreserved &&
  t.residualWithinBound &&
  t.costWithinBound &&
  t.decision = Decision.accept

/-- A lawful transition has all gates passing. -/
theorem lawful_all_pass (t : Transition) :
    isLawful t ↔
    wrapperComplete t.wrapper ∧
    t.validSyntax ∧
    t.roundTripOrLossPolicy ∧
    t.invariantPreserved ∧
    t.residualWithinBound ∧
    t.costWithinBound ∧
    t.decision = Decision.accept := by
  simp [isLawful]

/-! §8b The Five Control Filters (Admit Pipeline)

  The GCCL gate decomposes into eight sub-checks, organized as the
  five canonical control filters plus three bookkeeping gates.

  These began as meme-named handles on real mathematical machinery
  (see "Meme Math That Pays Rent"). The jokes got parking passes;
  they pay rent as formal admission gates.

  Timeline of the COUCH family:

  COUCH — "Super Freak" / Rick James / moving sofa problem
    The original. A continuous oscillator with a joke name, formalized
    as a Lean witness with 5 regimes and 28 theorems. Measures pressure
    and hysteresis stability. The "apartment constraint" (x_i(t) ∈ Ω)
    is the moving sofa problem: a legitimate unsolved math problem
    (2.2195 ≤ S ≤ 2.8284). Rick James said "I'm in the apartment, not
    touching the walls." That IS the constrained-manifold traversal.

  Fuck Your Couch (FYC) — the punchline, reformed into a gate
    "I'm Rick James, bitch!" → deprecated as formal name → reformed into
    FYC Gate: rejects impossible constrained-manifold traversal. A
    candidate route that claims it can navigate a topology that's
    actually impossible gets rejected.

  LoC/NES Monster — "Loch Ness Monster"
    Locality-of-change check. Detects entropy smuggling via recurrence.
    Wraps the set with a creature-feature theme.

  Tree Fiddy — "I need about tree fiddy"
    Cost bound. The budget beyond which the Loch Ness Monster takes your
    money. Checks that the AngrySphinx cost (2^depth) is within budget.

  BHOCS — "Big Hash of Certified Stuff"
    Receipt/audit trail verification. The SHA-256 hash chain is intact.

  Philosophy: "A joke can get a parking pass. It does not get tenure.
  If it wants to stay in the stack, it has to pay rent."

  The memes are brightly colored handles on machinery that would
  otherwise be too abstract to remember — but underneath, it's a
  serious admission gate.

  Connection to GCCL law axes:
    replay_valid     → Compression (round-trip)
    byte_gain > 0    → Compression (actual reduction)
    residual_declared → Residual (explicit loss)
    LoC_NES_pass     → Cognitive (locality, no smuggling)
    FYC_pass         → Geometric (traversable manifold)
    COUCH_stable     → Geometric (pressure stability)
    TreeFiddy_bounded → Cost (budget)
    BHOCS_verified   → Receipt (audit trail)
-/

/-- A candidate transition X entering the admission pipeline. -/
structure CandidateX where
  /-- Replay produces identical output (determinism) -/
  replayValid : Bool
  /-- Net byte reduction > 0 (actual compression) -/
  byteGain : Q16_16  -- positive means compression achieved
  /-- Residual (loss) is explicitly declared, not hidden -/
  residualDeclared : Bool
  /-- Locality-of-change: no entropy smuggling via recurrence -/
  locNesPass : Bool
  /-- FYC: constrained-manifold traversal is geometrically possible -/
  fycPass : Bool
  /-- COUCH: pressure/hysteresis stability (the original gate) -/
  couchStable : Bool
  /-- Tree Fiddy: cost within budget (AngrySphinx 2^depth ≤ limit) -/
  treeFiddyBounded : Bool
  /-- BHOCS: receipt/audit trail (SHA-256 hash chain) verified -/
  bhocsVerified : Bool
  deriving Repr, DecidableEq, Inhabited

/-- byteGain > 0 check: actual compression was achieved. -/
def byteGainPositive (x : CandidateX) : Bool :=
  x.byteGain > Q16_16.zero

/-- The full Admit predicate — all eight gates must pass.

    Admit(X) = replay_valid(X)
             ∧ byte_gain(X) > 0
             ∧ residual_declared(X)
             ∧ LoC_NES_pass(X)
             ∧ FYC_pass(X)
             ∧ COUCH_stable(X)
             ∧ TreeFiddy_bounded(X)
             ∧ BHOCS_verified(X)

    A candidate is admitted only if it survives all five control filters
    plus the three bookkeeping gates. Each gate checks a different aspect
    of lawfulness:

    - replay_valid: the encoding is deterministic and reproducible
    - byte_gain > 0: the candidate actually compresses (fewer collisions)
    - residual_declared: the quantization loss is explicit (Q16_16 floor)
    - LoC_NES_pass: locality of change (no rewriting the entire tree)
    - FYC_pass: the braid structure is geometrically traversable
    - COUCH_stable: the candidate's Omega is in the stable range
    - TreeFiddy_bounded: AngrySphinx cost 2^depth ≤ budget
    - BHOCS_verified: the SHA-256 hash chain is intact -/
def Admit (x : CandidateX) : Bool :=
  x.replayValid &&
  byteGainPositive x &&
  x.residualDeclared &&
  x.locNesPass &&
  x.fycPass &&
  x.couchStable &&
  x.treeFiddyBounded &&
  x.bhocsVerified

/-- Admit requires ALL eight gates to pass. -/
theorem admit_all_pass (x : CandidateX) :
    Admit x ↔
    x.replayValid ∧
    byteGainPositive x ∧
    x.residualDeclared ∧
    x.locNesPass ∧
    x.fycPass ∧
    x.couchStable ∧
    x.treeFiddyBounded ∧
    x.bhocsVerified := by
  simp [Admit]

/-- A candidate with any gate failing is NOT admitted. -/
theorem admit_fails_on_any_failure (x : CandidateX) :
    (¬ x.replayValid ∨ ¬ byteGainPositive x ∨ ¬ x.residualDeclared ∨
     ¬ x.locNesPass ∨ ¬ x.fycPass ∨ ¬ x.couchStable ∨
     ¬ x.treeFiddyBounded ∨ ¬ x.bhocsVerified) →
    ¬ Admit x := by
  intro h
  by_contra hAdmit
  rw [admit_all_pass] at hAdmit
  obtain ⟨h1, h2, h3, h4, h5, h6, h7, h8⟩ := hAdmit
  rcases h with h1' | h2' | h3' | h4' | h5' | h6' | h7' | h8'
  · exact h1' h1
  · exact h2' h2
  · exact h3' h3
  · exact h4' h4
  · exact h5' h5
  · exact h6' h6
  · exact h7' h7
  · exact h8' h8

/-- FYC rejects impossible constrained-manifold traversal.

    If FYC_pass is false, the candidate is rejected regardless of
    other gates. A route that claims to navigate an impossible
    topology cannot be admitted. -/
theorem fyc_rejection_blocks_admit (x : CandidateX) (h : ¬ x.fycPass) :
    ¬ Admit x := by
  intro hAdmit
  rw [admit_all_pass] at hAdmit
  exact h hAdmit.2.2.2.1

/-- Tree Fiddy bounds the AngrySphinx cost.

    If the cost exceeds the Tree Fiddy budget, the candidate is rejected.
    The Loch Ness Monster takes your money. -/
theorem treeFiddy_rejection_blocks_admit (x : CandidateX) (h : ¬ x.treeFiddyBounded) :
    ¬ Admit x := by
  intro hAdmit
  rw [admit_all_pass] at hAdmit
  exact h hAdmit.2.2.2.2.1

/-- A fully-passing candidate IS admitted. -/
theorem all_pass_implies_admit (x : CandidateX)
    (h1 : x.replayValid) (h2 : byteGainPositive x)
    (h3 : x.residualDeclared) (h4 : x.locNesPass)
    (h5 : x.fycPass) (h6 : x.couchStable)
    (h7 : x.treeFiddyBounded) (h8 : x.bhocsVerified) :
    Admit x := by
  rw [admit_all_pass]
  exact ⟨h1, h2, h3, h4, h5, h6, h7, h8⟩

/-! §9 GCCL Swap Gate (Q16_16) -/

/-- GCCL swap decision result. -/
structure GCDecision where
  accept : Bool
  reject : Bool
  hold : Bool
  quarantine : Bool
  deriving Repr, Inhabited, DecidableEq

/-- GCCL swap gate: accept iff improvement ≥ reconstruction risk.

    This is the core decision logic:
    - Compute improvement = max(0, oldCost - newCost)
    - Accept iff improvement ≥ reconRisk
    - Otherwise reject/hold

    Connection to AngrySphinx:
    - reconRisk = AngrySphinx solve cost (2^depth)
    - improvement = cost reduction from the candidate
    - Accept iff the candidate saves more than it costs
    - This is the "defense first, science second" rule from AngrySphinx -/
def gcclSwapGate (oldCost newCost reconRisk : Q16_16) : GCDecision :=
  let improvement := if oldCost > newCost then
    sub oldCost newCost
  else
    zero
  let admissible := improvement ≥ reconRisk
  { accept := admissible
    reject := ¬admissible
    hold := ¬admissible
    quarantine := false }

/-- Rejects expansion (newCost > oldCost): no improvement. -/
theorem gcclRejectsExpansion :
    gcclSwapGate (ofNat 100) (ofNat 200) (ofNat 500) =
    { accept := false, reject := true, hold := true, quarantine := false } := by
  decide

/-- Accepts improvement that exceeds risk. -/
theorem gcclAcceptsImprovement :
    gcclSwapGate (ofNat 500) (ofNat 100) (ofNat 200) =
    { accept := true, reject := false, hold := false, quarantine := false } := by
  decide

/-! §10 Connection to the Pipeline -/

/-- The full pipeline as a GCCL transition chain:

    1. Equation string → DNA encoder (exact p-adic + neg-pi)
       [NUVMAP layer: address projection]
    2. DNA sequence → logogram atom
       [AVMR layer: vector state evolution]
    3. Logogram → GCCL gate (lawful transition check)
       [AMMR layer: commit/receipt history]
    4. Admitted logogram → MMR append (Mountain merge)
       [O-AMMR layer: orthogonal projection]
    5. SpherionState update → golden spiral contraction → IR fixed point
       [GCCL-Rep layer: transition rope]
    6. AngrySphinx charges 2^depth per step (energy budget)
       [Cost layer]
    7. NaN boundary terminates when frustration → 0
       [Scale layer]

    Each step is a GCCL transition with a complete wrapper, verified
    invariant preservation, residual within bound, and cost within budget.
-/

/-- The pipeline stages as a sequence of GCCL transitions. -/
inductive PipelineStage where
  | encode        -- Equation → DNA (NUVMAP)
  | logogram      -- DNA → logogram atom (AVMR)
  | gate          -- Logogram → GCCL gate (AMMR)
  | merge         -- Gate → MMR append (O-AMMR)
  | contract      -- SpherionState → golden spiral (GCCL-Rep)
  | budget        -- AngrySphinx cost check (Cost)
  | terminate     -- NaN boundary (Scale)
  deriving DecidableEq, Repr, Fintype

/-- Number of pipeline stages = 7. -/
theorem pipelineStage_count : Fintype.card PipelineStage = 7 := by decide

/-- Map each pipeline stage to its mountain layer. -/
def stageToLayer : PipelineStage → MountainLayer
  | .encode    => .nuvmap
  | .logogram  => .avmr
  | .gate      => .ammr
  | .merge     => .oammr
  | .contract  => .gcclRep
  | .budget    => .gcclRep  -- cost is part of the transition rope
  | .terminate => .gcclRep  -- termination is part of the transition rope

/-! §11 Evaluation Witnesses -/

-- Verify the wrapper completeness check
#eval wrapperComplete
  { stateSpaceDeclared := true, transformDeclared := true,
    invariantsDeclared := true, residualDeclared := true,
    costDeclared := true, projectionDeclared := true,
    quarantineDeclared := true, scaleDeclared := true }  -- true

#eval wrapperComplete
  { stateSpaceDeclared := true, transformDeclared := true,
    invariantsDeclared := true, residualDeclared := false,
    costDeclared := true, projectionDeclared := true,
    quarantineDeclared := true, scaleDeclared := true }  -- false

-- Verify the GCCL swap gate
#eval gcclSwapGate (ofNat 500) (ofNat 100) (ofNat 200)  -- accept=true
#eval gcclSwapGate (ofNat 100) (ofNat 200) (ofNat 500)  -- reject=true

end SilverSight.GCCL