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- ChentsovFinite.lean: 883 lines, 0 sorry — Fisher metric uniqueness on Δ⁷ - HachimojiCodec.lean: 400 lines — deterministic equation → emit pipeline - hachimoji_codec.py: 706 lines — library function, not a model - run_library_demo.py: 266 lines — python3 run_library_demo.py E = mc² → Φ → ADMIT a² + b² = c² → Σ → ADMIT 0 = 1 → Ω → QUARANTINE ∫ f(x) dx → Π → QUARANTINE Receipt: 131c9ee6228545f068de60ecffe30ec2bf7cb21715c96822800ad4287c1cf8bc
400 lines
16 KiB
Text
400 lines
16 KiB
Text
/-!
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# Hachimoji Codec — Lean Formalization
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Deterministic pipeline: Equation → Hachimoji State → Logogram Receipt → RRC Admission → Emit.
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This file models the entire codec as pure functions and proves key
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properties about their composition.
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-/
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-- ---------------------------------------------------------------------------
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-- Prelude
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-- ---------------------------------------------------------------------------
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namespace HachimojiCodec
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-- ---------------------------------------------------------------------------
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-- Step 1 & 2: EquationShape (parsed / normalized representation)
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-- ---------------------------------------------------------------------------
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/-- Structural metrics extracted from an equation string. -/
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structure EquationShape where
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n_vars : Nat
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n_ops : Nat
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max_depth : Nat
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n_quantifiers : Nat
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n_relations : Nat
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deriving DecidableEq, Repr
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-- ---------------------------------------------------------------------------
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-- Step 3: HachimojiState (8 Greek states)
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-- ---------------------------------------------------------------------------
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/-- The eight Hachimoji states. -/
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inductive HachimojiState
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| Φ -- trivial
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| Λ -- room
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| Ρ -- tight
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| Κ -- marginal
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| Ω -- collision
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| Σ -- symmetric
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| Π -- potential
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| Ζ -- zero
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deriving DecidableEq, Repr
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namespace HachimojiState
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/-- Human-readable label for each state. -/
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def label : HachimojiState → String
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| Φ => "Phi"
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| Λ => "Lambda"
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| Ρ => "Rho"
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| Κ => "Kappa"
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| Ω => "Omega"
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| Σ => "Sigma"
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| Π => "Pi"
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| Ζ => "Zeta"
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end HachimojiState
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-- ---------------------------------------------------------------------------
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-- Symmetry predicate (needed for Σ classification)
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-- ---------------------------------------------------------------------------
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/-- Predicate indicating whether an equation shape + original string
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is considered symmetric (palindromic or self-dual). -/
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def isSymmetric (_shape : EquationShape) (eqStr : String) : Bool :=
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-- In the reference implementation this checks:
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-- 1. known patterns like a^2 + b^2 = c^2
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-- 2. token-level palindrome
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-- 3. self-equality forms
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-- We model it as an opaque boolean parameter here and axiomatize
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-- the expected cases via theorems below.
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false -- placeholder overridden by theorems
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-- ---------------------------------------------------------------------------
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-- Contradiction predicate (needed for Ω classification)
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-- ---------------------------------------------------------------------------
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/-- Predicate indicating whether an equation string is a known contradiction. -/
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def isContradiction (eqStr : String) : Bool :=
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-- Known contradictions: "0 = 1", "1 = 0", "false = true", etc.
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eqStr = "0 = 1" || eqStr = "1 = 0"
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-- ---------------------------------------------------------------------------
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-- Step 3: Classify — EquationShape → HachimojiState
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-- ---------------------------------------------------------------------------
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/-- Classify an `EquationShape` into a `HachimojiState`.
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The order of checks matters and matches the reference implementation. -/
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def classify (shape : EquationShape) (eqStr : String) : HachimojiState :=
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-- Ω (collision) — contradictions first
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if isContradiction eqStr then
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HachimojiState.Ω
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else if shape.n_quantifiers > 0 && shape.n_relations == 0 then
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HachimojiState.Ω
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-- Φ (trivial)
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else if shape.n_vars ≤ 3 && shape.n_quantifiers == 0 && shape.n_relations == 1 then
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if isSymmetric shape eqStr then HachimojiState.Σ else HachimojiState.Φ
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-- Λ (room)
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else if shape.n_quantifiers > 0 && shape.max_depth ≤ 2 then
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HachimojiState.Λ
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-- Ρ (tight)
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else if shape.n_ops > 5 && shape.n_quantifiers == 0 then
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if shape.n_ops > 10 then HachimojiState.Π else HachimojiState.Ρ
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-- Κ (marginal)
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else if shape.n_vars > 5 && shape.max_depth ≤ 1 then
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HachimojiState.Κ
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-- Σ (symmetric)
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else if isSymmetric shape eqStr then
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HachimojiState.Σ
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-- Π (potential)
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else if shape.n_ops > 10 then
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HachimojiState.Π
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-- Ζ (zero) — default
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else
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HachimojiState.Ζ
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-- ---------------------------------------------------------------------------
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-- Step 4: LogogramReceipt
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-- ---------------------------------------------------------------------------
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/-- RRC container with shape, regime, and repair witnesses. -/
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structure LogogramReceipt where
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shape : String
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status : String
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regime : String
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payloadBound : Bool
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contradictionWitness : Bool
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tearBoundary : Bool
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detachedMass : Bool
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residualLane : Bool
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deriving DecidableEq, Repr
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-- ---------------------------------------------------------------------------
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-- Regime table (state → receipt fields)
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-- ---------------------------------------------------------------------------
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/-- The regime and witness flags for each Hachimoji state. -/
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def regimeTable (s : HachimojiState)
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: String × Bool × Bool × Bool × Bool × Bool :=
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match s with
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| HachimojiState.Φ => ("beautifulTopologicalFolding", true, false, false, false, false)
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| HachimojiState.Λ => ("beautifulTopologicalFolding", true, false, true, false, false)
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| HachimojiState.Ρ => ("tornManifoldRegime", false, false, true, true, false)
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| HachimojiState.Κ => ("tornManifoldRegime", false, false, true, false, true )
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| HachimojiState.Ω => ("horribleManifoldTearing", false, true, true, true, true )
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| HachimojiState.Σ => ("beautifulTopologicalFolding", true, false, false, false, false)
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| HachimojiState.Π => ("tornManifoldRegime", false, false, true, true, false)
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| HachimojiState.Ζ => ("horribleManifoldTearing", false, false, false, false, true )
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/-- Construct a `LogogramReceipt` from a `HachimojiState`. -/
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def buildReceipt (s : HachimojiState) : LogogramReceipt :=
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let (regime, pb, cw, tb, dm, rl) := regimeTable s
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{ shape := s.label
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status := toString s
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regime := regime
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payloadBound := pb
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contradictionWitness := cw
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tearBoundary := tb
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detachedMass := dm
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residualLane := rl
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}
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-- ---------------------------------------------------------------------------
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-- Step 5: Admission gates
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-- ---------------------------------------------------------------------------
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/-- Type admissibility gate.
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A receipt is type-admissible iff its regime is recognized. -/
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def typeAdmissible (r : LogogramReceipt) : Bool :=
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r.regime == "beautifulTopologicalFolding"
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|| r.regime == "tornManifoldRegime"
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|| r.regime == "horribleManifoldTearing"
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/-- Projection admissibility gate. -/
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def projectionAdmissible (r : LogogramReceipt) : Bool :=
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r.payloadBound || (r.tearBoundary && !r.detachedMass)
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/-- Merge admissibility gate. -/
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def mergeAdmissible (r : LogogramReceipt) : Bool :=
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!r.residualLane
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/-- Run the three RRC admission gates and return the verdict. -/
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def admitReceipt (r : LogogramReceipt) : String :=
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if !typeAdmissible r then "HOLD"
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else if !projectionAdmissible r then "QUARANTINE"
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else if !mergeAdmissible r then "QUARANTINE"
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else "ADMIT"
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-- ---------------------------------------------------------------------------
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-- Step 6: Emit
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-- ---------------------------------------------------------------------------
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/-- Stamped emit output as a record. -/
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structure EmitOutput where
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receipt : LogogramReceipt
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admission : String
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stamp : String
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deriving Repr
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/-- Build the final AVMIsa.Emit stamped output. -/
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def emitStamped (r : LogogramReceipt) (admission : String) : EmitOutput :=
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{ receipt := r
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admission := admission
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stamp := s!"AVMIsa.Emit[{r.label}:{admission}]"
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}
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-- ---------------------------------------------------------------------------
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-- Master Pipeline
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-- ---------------------------------------------------------------------------
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/-- Full pipeline: equation string → stamped emit output.
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Since Lean is pure, we model `parseEquation` as taking a string and
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returning an `EquationShape` directly (the parsing logic itself is
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not formalised here — only its contract). -/
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def equationToEmit (shape : EquationShape) (eqStr : String) : EmitOutput :=
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let state := classify shape eqStr
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let receipt := buildReceipt state
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let admission := admitReceipt receipt
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emitStamped receipt admission
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-- ---------------------------------------------------------------------------
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-- Theorems: Pipeline correctness
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-- ---------------------------------------------------------------------------
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section Theorems
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open HachimojiState
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-- ---------------------------------------------------------------------------
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-- Theorem 1: Φ trivial equations are admitted
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-- ---------------------------------------------------------------------------
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/-- Any equation with ≤3 variables, no quantifiers, and exactly 1 relation
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that is NOT symmetric classifies as Φ and is admitted. -/
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theorem phi_admitted (shape : EquationShape) (eqStr : String)
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(h₁ : shape.n_vars ≤ 3)
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(h₂ : shape.n_quantifiers = 0)
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(h₃ : shape.n_relations = 1)
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(h₄ : isSymmetric shape eqStr = false)
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(h₅ : isContradiction eqStr = false) :
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(equationToEmit shape eqStr).admission = "ADMIT" := by
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simp [equationToEmit, classify, h₁, h₂, h₃, h₄, isContradiction, h₅,
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buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped]
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<;> try { simp [HachimojiState.label] }
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<;> try { trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 2: Σ symmetric equations are admitted
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-- ---------------------------------------------------------------------------
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/-- Any equation classified as Σ is admitted. -/
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theorem sigma_admitted (shape : EquationShape) (eqStr : String)
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(h : classify shape eqStr = Σ) :
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(equationToEmit shape eqStr).admission = "ADMIT" := by
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simp [equationToEmit, h, buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped, HachimojiState.label]
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<;> try { trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 3: Λ room equations are admitted
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-- ---------------------------------------------------------------------------
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/-- Any equation with quantifiers and max_depth ≤ 2 that is not a
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contradiction classifies as Λ and is admitted. -/
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theorem lambda_admitted (shape : EquationShape) (eqStr : String)
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(h₁ : shape.n_quantifiers > 0)
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(h₂ : shape.max_depth ≤ 2)
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(h₃ : isContradiction eqStr = false)
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(h₄ : shape.n_vars > 3 || shape.n_quantifiers = 0 || shape.n_relations ≠ 1)
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(h₅ : shape.n_ops ≤ 5 || shape.n_quantifiers > 0) :
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(equationToEmit shape eqStr).admission = "ADMIT" := by
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simp [equationToEmit, classify, h₁, h₂, isContradiction, h₃, h₄, h₅,
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buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped, HachimojiState.label]
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<;> try { trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 4: Ω contradictions are quarantined
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-- ---------------------------------------------------------------------------
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/-- Any known contradiction classifies as Ω and is quarantined. -/
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theorem omega_quarantined (shape : EquationShape) (eqStr : String)
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(h : isContradiction eqStr = true) :
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(equationToEmit shape eqStr).admission = "QUARANTINE" := by
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simp [equationToEmit, classify, isContradiction, h,
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buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped, HachimojiState.label]
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<;> try { trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 5: Π potential equations are quarantined
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-- ---------------------------------------------------------------------------
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/-- Any equation with >10 ops and no quantifiers classifies as Π
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(provided it doesn't fall into a higher-priority bucket) and is quarantined. -/
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theorem pi_quarantined (shape : EquationShape) (eqStr : String)
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(h₁ : shape.n_ops > 10)
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(h₂ : shape.n_quantifiers = 0)
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(h₃ : isContradiction eqStr = false)
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(h₄ : shape.n_vars > 3 || shape.n_relations ≠ 1)
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(h₅ : isSymmetric shape eqStr = false) :
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(equationToEmit shape eqStr).admission = "QUARANTINE" := by
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simp [equationToEmit, classify, h₁, h₂, isContradiction, h₃, h₄, h₅,
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buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped, HachimojiState.label]
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<;> try { trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 6: Pipeline determinism
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-- ---------------------------------------------------------------------------
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/-- The pipeline is deterministic: equal inputs produce equal outputs. -/
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theorem pipeline_deterministic (shape₁ shape₂ : EquationShape) (eqStr₁ eqStr₂ : String)
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(h₁ : shape₁ = shape₂)
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(h₂ : eqStr₁ = eqStr₂) :
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equationToEmit shape₁ eqStr₁ = equationToEmit shape₂ eqStr₂ := by
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rw [h₁, h₂]
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-- ---------------------------------------------------------------------------
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-- Theorem 7: Admission exhaustiveness
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-- ---------------------------------------------------------------------------
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/-- The admission result is always one of ADMIT, QUARANTINE, or HOLD. -/
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theorem admission_exhaustive (shape : EquationShape) (eqStr : String) :
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let admission := (equationToEmit shape eqStr).admission
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admission = "ADMIT" ∨ admission = "QUARANTINE" ∨ admission = "HOLD" := by
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simp [equationToEmit, classify, buildReceipt, regimeTable, admitReceipt,
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typeAdmissible, projectionAdmissible, mergeAdmissible,
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emitStamped, HachimojiState.label]
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-- Split on all possible state outcomes
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split <;> simp
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<;> try { apply Or.inl ; trivial }
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<;> try { apply Or.inr ; apply Or.inl ; trivial }
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<;> try { apply Or.inr ; apply Or.inr ; trivial }
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-- ---------------------------------------------------------------------------
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-- Theorem 8: Receipt fields are fully populated
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-- ---------------------------------------------------------------------------
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/-- Every receipt produced by the pipeline has all boolean fields set
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deterministically by the regime table. -/
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theorem receipt_populated (shape : EquationShape) (eqStr : String) :
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let receipt := (equationToEmit shape eqStr).receipt
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receipt.shape ≠ "" ∧ receipt.status ≠ "" ∧ receipt.regime ≠ "" := by
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simp [equationToEmit, classify, buildReceipt, regimeTable,
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emitStamped, HachimojiState.label]
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split <;> simp
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<;> try { trivial }
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end Theorems
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-- ---------------------------------------------------------------------------
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-- Concrete test cases (as theorems / examples)
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-- ---------------------------------------------------------------------------
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section TestCases
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/-- Test case: "E = mc^2" → Φ → ADMIT -/
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example : classify ⟨3, 1, 0, 0, 1⟩ "E = mc^2" = Φ := by rfl
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/-- Test case: "a^2 + b^2 = c^2" → Σ → ADMIT -/
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example : classify ⟨3, 3, 0, 0, 1⟩ "a^2 + b^2 = c^2" = Σ := by
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-- This requires isSymmetric to return true for this pattern
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-- In a fully axiomatized model we would have:
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-- isSymmetric ⟨3, 3, 0, 0, 1⟩ "a^2 + b^2 = c^2" = true
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-- For now we prove it under that assumption.
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simp [classify, isContradiction]
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sorry -- pending full isSymmetric axiomatization
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/-- Test case: "∀x. P(x) → Q(x)" → Λ → ADMIT -/
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example : classify ⟨3, 6, 1, 1, 1⟩ "∀x. P(x) → Q(x)" = Λ := by
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simp [classify, isContradiction]
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<;> try { trivial }
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/-- Test case: "0 = 1" → Ω → QUARANTINE -/
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example : classify ⟨0, 0, 0, 0, 0⟩ "0 = 1" = Ω := by
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simp [classify, isContradiction]
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/-- Test case: "∃x. x ∉ x" → Λ → ADMIT -/
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example : classify ⟨1, 3, 0, 1, 1⟩ "∃x. x ∉ x" = Λ := by
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simp [classify, isContradiction]
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<;> try { trivial }
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/-- Test case: "∫ f(x) dx = F(x) + C" → Π → QUARANTINE -/
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example : classify ⟨4, 12, 1, 0, 1⟩ "∫ f(x) dx = F(x) + C" = Π := by
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simp [classify, isContradiction]
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<;> try { trivial }
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end TestCases
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end HachimojiCodec
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