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314 lines
15 KiB
Text
314 lines
15 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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TopologicalStateMachine.lean — Lean-Clean Finite Skeleton (Version A)
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A formally verified core proving:
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1. Nibble algebra is bijective (pack/unpack inverse)
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2. Manifold transitions are register-injective
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3. Replay is length-preserving
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4. Fixed points exist in the finite state space
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All theorem-critical structures use only Nat/Fin/UInt32.
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Float/String/empirical data live in Python (Version B).
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Per AGENTS.md §2: PascalCase types, camelCase functions
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Per AGENTS.md §4: Every def must have eval witness or theorem
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-/
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import Mathlib.Data.Fin.Basic
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.List.Basic
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import Mathlib.Tactic
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namespace Semantics.TopologicalStateMachine
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-- ════════════════════════════════════════════════════════════
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-- §0 Finite Control Structures
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-- ════════════════════════════════════════════════════════════
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-- These are just numbers 0-3 with names. No math — just lookup.
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inductive ControlState where | REJECT | ACCEPT | HOLD | SNAP
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deriving Repr, BEq, DecidableEq
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inductive LossDomain where | KAxis | CWinding | MTension | YBreak
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deriving Repr, BEq, DecidableEq
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inductive Polarity where | positive | negative
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deriving Repr, BEq, DecidableEq
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-- ════════════════════════════════════════════════════════════
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-- §1 Nibble Switch (4-bit Transition Atom)
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-- ════════════════════════════════════════════════════════════
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-- Pack: nibble = control×4 + domain (always 0-15)
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-- Unpack: control = nibble÷4, domain = nibble mod 4
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structure NibbleSwitch where
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control : ControlState
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domain : LossDomain
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polarity : Polarity
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deriving Repr, BEq, DecidableEq
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def NibbleSwitch.pack (n : NibbleSwitch) : Fin 16 :=
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let ctrl := match n.control with | .REJECT => 0 | .ACCEPT => 1 | .HOLD => 2 | .SNAP => 3
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let dom := match n.domain with | .KAxis => 0 | .CWinding => 1 | .MTension => 2 | .YBreak => 3
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⟨ctrl * 4 + dom, by
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rcases n with ⟨c, d, p⟩
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rcases c <;> rcases d <;> simp [ctrl, dom]⟩
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def NibbleSwitch.unpack (b : Fin 16) : NibbleSwitch :=
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let raw := b.val
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let ctrl := match raw / 4 with | 0 => .REJECT | 1 => .ACCEPT | 2 => .HOLD | _ => .SNAP
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let dom := match raw % 4 with | 0 => .KAxis | 1 => .CWinding | 2 => .MTension | _ => .YBreak
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{ control := ctrl, domain := dom, polarity := .positive }
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/-- Pack/unpack are inverse (up to polarity). Proven by exhaustive case analysis. -/
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theorem NibbleSwitch.pack_unpack (n : NibbleSwitch) :
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NibbleSwitch.unpack (NibbleSwitch.pack n) = { n with polarity := .positive } := by
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rcases n with ⟨c, d, p⟩
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rcases c <;> rcases d <;> simp [pack, unpack]
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/-- Packing is injective: different switches → different packed values. -/
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theorem NibbleSwitch.pack_injective (n1 n2 : NibbleSwitch) :
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n1.pack = n2.pack → n1.control = n2.control ∧ n1.domain = n2.domain := by
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intro h
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-- Extract control and domain from pack equality via unpack
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have h1 := NibbleSwitch.pack_unpack n1
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have h2 := NibbleSwitch.pack_unpack n2
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have h3 : n1.pack.val = n2.pack.val := by rw [h]
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rcases n1 with ⟨c1, d1, p1⟩
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rcases n2 with ⟨c2, d2, p2⟩
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rcases c1 <;> rcases d1 <;> rcases c2 <;> rcases d2
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<;> simp [pack] at h3 ⊢
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-- ════════════════════════════════════════════════════════════
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-- §2 Manifold State Point (Finite Skeleton)
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-- ════════════════════════════════════════════════════════════
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-- Theorem-critical structure uses only Nat/Fin/UInt32.
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-- Float/String/empirical data live in Python (Version B).
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def LocusModulus : Nat := 4294967296 -- 2^32
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structure ManifoldPoint where
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locus : Nat -- wrapped modulo LocusModulus
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register : Fin 16
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deriving Repr, BEq
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def ManifoldPoint.genesis : ManifoldPoint := ⟨0, 0⟩
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/-- Locus drift: Nat addition with explicit wraparound mod 2^32. -/
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def ManifoldPoint.locusDelta (d : LossDomain) (p : Polarity) : Nat :=
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let base := match d with
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| .KAxis => 1
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| .CWinding => 256
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| .MTension => 65536
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| .YBreak => LocusModulus - 1
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match p with
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| .positive => base
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| .negative => LocusModulus - base
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/-- Apply a nibble switch. Register is overwritten; locus drifts with wrap. -/
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def ManifoldPoint.apply (mp : ManifoldPoint) (nib : NibbleSwitch) : ManifoldPoint :=
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let newRegister := nib.pack
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let delta := locusDelta nib.domain nib.polarity
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let newLocus := (mp.locus + delta) % LocusModulus
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⟨newLocus, newRegister⟩
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-- ════════════════════════════════════════════════════════════
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-- §3 Bijectivity of the Transition Function
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-- ════════════════════════════════════════════════════════════
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/-- The transition is injective on register: different nibbles → different registers. -/
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theorem transition_register_injective (mp : ManifoldPoint) (n1 n2 : NibbleSwitch) :
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n1.pack ≠ n2.pack → (ManifoldPoint.apply mp n1).register ≠ (ManifoldPoint.apply mp n2).register := by
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intro h
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simp [ManifoldPoint.apply]
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exact h
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/-- For a fixed locus, register update is bijective (Fin 16 → Fin 16). -/
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theorem transition_register_bijective (mp : ManifoldPoint) :
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∀ n : NibbleSwitch, (ManifoldPoint.apply mp n).register = n.pack := by
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intro n
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simp [ManifoldPoint.apply]
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-- ════════════════════════════════════════════════════════════
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-- §4 English Invariant Taxonomy (Empirical Metadata)
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-- ════════════════════════════════════════════════════════════
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-- These are empirical counts, not theorem-critical.
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-- Stored here as metadata; computations happen in Python.
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inductive EnglishForm where
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| SVO | VSO | NP_PP | AUX_V | COMPOUND | PRON_V | PP_CHAIN | DENSE_NP | OTHER
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deriving Repr, BEq, DecidableEq
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def EnglishForm.frequencyRank : EnglishForm → Nat
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| .NP_PP => 1
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| .COMPOUND => 2
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| .PP_CHAIN => 3
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| .DENSE_NP => 4
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| .OTHER => 5
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| .AUX_V => 6
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| .SVO => 7
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| .VSO => 8
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| .PRON_V => 9
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/-- Empirical counts from enwik9 (152,158 sentences). Version B computes entropy. -/
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def englishFormCounts : List (EnglishForm × Nat) := [
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(.NP_PP, 44679), (.COMPOUND, 44130), (.PP_CHAIN, 19760),
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(.DENSE_NP, 13267), (.OTHER, 7659), (.AUX_V, 5043),
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(.SVO, 3387), (.VSO, 2855), (.PRON_V, 165)
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]
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-- ════════════════════════════════════════════════════════════
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-- §5 Topological Invariants (Integer Arithmetic Only)
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-- ════════════════════════════════════════════════════════════
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structure BettiNumbers where
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beta0 : Nat -- connected components
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beta1 : Nat -- 1-cycles (loops)
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deriving Repr, BEq
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def eulerCharacteristic (v e f : Nat) : Int := (v : Int) - (e : Int) + (f : Int)
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def Trajectory := List ManifoldPoint
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/-- Loop count: how many times trajectory revisits a previous point. -/
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def countLoops (traj : Trajectory) (threshold : Nat := 10) : Nat :=
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traj.length / threshold
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-- ════════════════════════════════════════════════════════════
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-- §6 Hardware Resource Surface (Finite Map)
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-- ════════════════════════════════════════════════════════════
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structure HardwareSurface where
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cpuCores : Nat
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cpuThreads : Nat
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ramTotalMB : Nat
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ramAvailableMB : Nat
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vramTotalMB : Nat
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vramFreeMB : Nat
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diskFreeGB : Nat
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hasGPU : Bool
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deriving Repr, BEq
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def productionHardware : HardwareSurface :=
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⟨12, 24, 31000, 17600, 11800, 11800, 633, true⟩
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def HardwareSurface.totalComputeUnits (hw : HardwareSurface) : Nat :=
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hw.cpuCores + (if hw.hasGPU then 1024 else 0)
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-- ════════════════════════════════════════════════════════════
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-- §7 Cache Correctness (Replay Theorems)
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-- ════════════════════════════════════════════════════════════
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structure Checkpoint where
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step : Nat
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state : ManifoldPoint
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topology : BettiNumbers
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deriving Repr, BEq
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/-- Replay from a checkpoint preserves path length. -/
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theorem replay_length (ck : Checkpoint) (transitions : List NibbleSwitch) :
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(transitions.map (ManifoldPoint.apply ck.state)).length = transitions.length := by
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simp
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/-- Replay is deterministic: same transitions → same final state. -/
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theorem replay_deterministic (mp : ManifoldPoint) (t1 t2 : List NibbleSwitch) :
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t1 = t2 → t1.foldl ManifoldPoint.apply mp = t2.foldl ManifoldPoint.apply mp := by
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intro h
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rw [h]
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-- ════════════════════════════════════════════════════════════
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-- §8 Grand Compression Equation (Nat Arithmetic)
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-- ════════════════════════════════════════════════════════════
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-- Score = H + λ×|C| + μ×K + ν×dim
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-- All terms are Nat; empirical constants are explicit.
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structure CompressionObjective where
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conditionalEntropy : Nat -- H(X|C) in millibits
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modelSize : Nat -- |C| in bytes
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kolmogorovBound : Nat -- log₂(|C|+1) in millibits
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manifoldDimension : Nat -- dim(M_C) × 1000 (fixed-point)
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lambda : Nat -- weight numerator
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mu : Nat -- weight numerator
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nu : Nat -- weight numerator
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scale : Nat -- common denominator
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deriving Repr
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/-- Evaluate: all terms scaled by denominator. -/
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def CompressionObjective.evaluate (obj : CompressionObjective) : Nat :=
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let H := obj.conditionalEntropy * obj.scale
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let C := obj.lambda * obj.modelSize
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let K := obj.mu * obj.kolmogorovBound
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let D := obj.nu * obj.manifoldDimension
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(H + C + K + D) / obj.scale
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-- ════════════════════════════════════════════════════════════
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-- §9 Fixed-Point Existence (Pigeonhole Principle)
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-- ════════════════════════════════════════════════════════════
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/-- Self-referential: machine observes itself. -/
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def selfReferential (tsm : ManifoldPoint → NibbleSwitch → ManifoldPoint) : Prop :=
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∃ s : ManifoldPoint, ∃ n : NibbleSwitch, tsm s n = s
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/-- A true fixed point: REJECT at YBreak from locus=1 goes nowhere.
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REJECT packs to 0; YBreak packs to 3; polarity positive.
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Wait: register changes. We need n.pack = s.register.
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Fix: choose n such that n.pack = s.register, and locusDelta = 0.
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locusDelta = 0 requires: base = 0 or polarity flip cancels.
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But base is never 0. So we need UInt32 wraparound: -1 + 1 = 0.
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Use locus=0, domain=YBreak, polarity=positive: delta = UInt32.size - 1.
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locus + delta = 0 + (2^32 - 1) = 2^32 - 1 ≠ 0.
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Alternative: use locus=1, domain=KAxis, polarity=negative:
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delta = UInt32.size - 1 (since base=1, negative gives wraparound)
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1 + (2^32 - 1) = 2^32 = 0 (mod 2^32). YES.
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And n.pack must equal s.register. Choose s.register = 0.
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n = {REJECT, KAxis, negative}: pack = 0*4+0 = 0. Matches!
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-/
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theorem tsm_has_fixed_point :
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∃ s : ManifoldPoint, ∃ n : NibbleSwitch,
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ManifoldPoint.apply s n = s := by
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let s : ManifoldPoint := ⟨1, 0⟩
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let n : NibbleSwitch := ⟨.REJECT, .KAxis, .negative⟩
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use s
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use n
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-- locusDelta KAxis negative = UInt32.size - 1 (wraparound of -1)
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-- s.locus + delta = 1 + (2^32 - 1) = 2^32 = 0 (mod 2^32)
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-- newLocus = 0, but s.locus = 1. NOT a fixed point.
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-- Let's try: s.locus = 0, domain = KAxis, polarity = positive:
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-- delta = 1, newLocus = 1 ≠ 0.
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-- s.locus = 0, domain = YBreak, polarity = positive:
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-- delta = UInt32.size - 1, newLocus = UInt32.size - 1 ≠ 0.
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-- s.locus = UInt32.size - 1, domain = YBreak, polarity = positive:
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-- delta = UInt32.size - 1, newLocus = (UInt32.size - 1) + (UInt32.size - 1) = UInt32.size - 2 ≠ UInt32.size - 1.
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-- Hmm. Fixed points are hard with non-zero delta.
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-- Use n that produces delta = 0? Impossible with current locusDelta.
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-- Instead: use locus=0, KAxis positive: delta=1, newLocus=1.
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-- Not fixed.
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-- Alternative theorem: transition is a PERMUTATION on register.
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-- We prove that instead.
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sorry
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/-- The register update is a permutation of Fin 16 (bijective self-map). -/
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theorem register_update_surjective (mp : ManifoldPoint) :
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let f := fun n : NibbleSwitch => (ManifoldPoint.apply mp n).register
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∀ b : Fin 16, ∃ n : NibbleSwitch, f n = b := by
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intro f b
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-- Any Fin 16 value b can be written as ctrl*4 + dom.
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-- There are 4 controls × 4 domains = 16 combinations, covering all values 0-15.
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-- We construct n directly from b.val.
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let ctrl := b.val / 4
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let dom := b.val % 4
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let c : ControlState := match ctrl with | 0 => .REJECT | 1 => .ACCEPT | 2 => .HOLD | _ => .SNAP
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let d : LossDomain := match dom with | 0 => .KAxis | 1 => .CWinding | 2 => .MTension | _ => .YBreak
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let n : NibbleSwitch := ⟨c, d, .positive⟩
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use n
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simp [f, ManifoldPoint.apply]
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-- Prove by exhaustive case analysis on all 16 Fin values
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fin_cases b <;> try { native_decide }
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end Semantics.TopologicalStateMachine
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