/- SilverSight Core — The Self-Indexing Dual-Domain Machine ======================================================== This is the MINIMAL core. Nothing else belongs here. The core defines: 1. The 8 Hachimoji states (the alphabet of classification) 2. The receipt format (the interface between core and libraries) 3. The AVM transition function δ : S × I → S' 4. The TIC axiom (TIC is derived, not a driver) 5. The loop invariant (computation generates time) What this file does NOT contain: - Chaos game implementation (library: SearchLib) - Finsler metric computation (library: MetricLib) - QUBO/QAOA optimization (library: OptimizeLib) - Token parsing (library: LexLib) - PVGS bridge (library: QuantumLib) - Chirality classification (library: StructureLib) - TIC counter implementation (library: EventLib) - Receipt validation (library: AuditLib) Library method: core defines the contract. Libraries implement it. No library code in core. No core code depends on libraries. Author: allaunthefox Date: 2026-06-21 License: MIT -/ namespace SilverSight.Core -- ============================================================================= -- §1. THE 8 HACHIMOJI STATES -- ============================================================================= /- The 8 Hachimoji states are the classification alphabet. Every mathematical expression resolves to one of these 8 states. This is the OUTPUT of the core machine. Φ (Phi) — trivial, well-understood, admits immediately Λ (Lambda) — room for exploration, interesting but tractable Ρ (Rho) — tight, requires specific tools or conditions Κ (Kappa) — marginal, edge case, may need special handling Ω (Omega) — collision, contradiction, quarantine Σ (Sigma) — symmetric, balanced, multiple valid interpretations Π (Pi) — potential, promising direction for further work Ζ (Zeta) — zero, undefined, no-information state -/ inductive HachimojiState | Φ | Λ | Ρ | Κ | Ω | Σ | Π | Ζ deriving DecidableEq, Repr, Fintype, BEq def HachimojiState.toString : HachimojiState → String | .Φ => "Φ" | .Λ => "Λ" | .Ρ => "Ρ" | .Κ => "Κ" | .Ω => "Ω" | .Σ => "Σ" | .Π => "Π" | .Ζ => "Ζ" instance : ToString HachimojiState := ⟨HachimojiState.toString⟩ -- ============================================================================= -- §2. THE RECEIPT FORMAT (interface between core and libraries) -- ============================================================================= /- The Receipt is the ONLY interface between the core and libraries. Libraries produce receipts. The core consumes them. Nothing else crosses the boundary. receiptID — unique identifier for this classification expression — the input expression (as a string) finalState — the Hachimoji state assigned ticCount — how many events occurred during classification fuelUsed — how many iterations were expended pathCost — Finsler distance traversed (if available) libraryRefs — which libraries contributed to the classification verified — whether the result has been independently checked -/ structure Receipt where receiptID : String expression : String finalState : HachimojiState ticCount : Nat fuelUsed : Nat pathCost : Option Float libraryRefs : List String verified : Bool deriving Repr, BEq -- Empty receipt (no classification performed) def Receipt.empty : Receipt := { receiptID := "" , expression := "" , finalState := .Ζ , ticCount := 0 , fuelUsed := 0 , pathCost := none , libraryRefs := [] , verified := false } -- Receipt is valid if it has an ID and a non-zero state @[simp] def Receipt.isValid (r : Receipt) : Bool := r.receiptID.length > 0 && r.finalState != .Ζ -- ============================================================================= -- §3. THE AVM TRANSITION FUNCTION δ : S × I → S' -- ============================================================================= /- The AVM (Abstract Virtual Machine) is a stack machine with: - A single stack of HachimojiStates - A fuel counter (prevents infinite loops) - An instruction type Instructions are what libraries produce. The core only executes them. -/ inductive Instruction | Classify (expr : String) -- classify expression, push result | LookupLib (name : String) -- reference a library | Merge (s1 s2 : HachimojiState) -- combine two states | Reflect (fuel : Nat) -- chaos game reflection step | Verify (receipt : Receipt) -- verify a receipt | Halt -- stop execution deriving DecidableEq, Repr def Instruction.toString : Instruction → String | .Classify expr => s!"Classify({expr})" | .LookupLib name => s!"LookupLib({name})" | .Merge s1 s2 => s!"Merge({s1}, {s2})" | .Reflect fuel => s!"Reflect({fuel})" | .Verify r => s!"Verify({r.receiptID})" | .Halt => "Halt" instance : ToString Instruction := ⟨Instruction.toString⟩ -- The AVM state structure AVMState where stack : List HachimojiState -- computation stack fuel : Nat -- remaining fuel tic : Nat -- current TIC count history : List Instruction -- executed instructions deriving Repr def AVMState.initial (fuel : Nat) : AVMState := { stack := [], fuel := fuel, tic := 0, history := [] } -- δ : S × I → S' (state transition function) def δ (s : AVMState) (i : Instruction) : AVMState := if s.fuel = 0 then -- Out of fuel: halt { s with history := s.history ++ [.Halt] } else let s' := { s with fuel := s.fuel - 1, tic := s.tic + 1, history := s.history ++ [i] } match i with | .Classify _expr => { s' with stack := .Φ :: s'.stack } -- default: Φ | .LookupLib _name => s' -- library lookup (no stack change) | .Merge s1 s2 => -- Merge: take the "more informative" state let merged := match (s1, s2) with | (.Ω, _) | (_, .Ω) => .Ω -- contradiction dominates | (.Ζ, s) | (s, .Ζ) => s -- non-zero dominates | (s, _) => s -- first wins (default) { s' with stack := merged :: s'.stack.tailD [] } | .Reflect _fuel => { s' with tic := s'.tic + 1 } -- reflection costs extra | .Verify _receipt => s' -- verification (no stack change) | .Halt => { s' with fuel := 0 } -- force halt -- Run a program (list of instructions) from initial state def runAVM (program : List Instruction) (fuel : Nat) : AVMState := let initial := AVMState.initial fuel program.foldl δ initial -- Extract the top of stack as the final state def AVMState.result (s : AVMState) : HachimojiState := s.stack.headD .Ζ -- ============================================================================= -- §4. THE TIC AXIOM (TIC is derived, not a driver) -- ============================================================================= /- AXIOM: The TIC clock is strictly derived from physical events. It is NEVER the driver of computation. Formally: T_{n+1} = T_n + E(S_n, τ) Where: T_n = TIC count at step n S_n = AVM state at step n τ = metric distance (parameter) E = event detection function (E ≥ 0) CRITICAL: There is NO inverse function: ∄ F such that T_n → S_n The TIC counts events. It does not cause them. -/ -- Event detection: every state transition produces exactly 1 event @[simp] def eventDetect (s s' : AVMState) : Nat := if s != s' then 1 else 0 -- TIC update: derived from events @[simp] def ticUpdate (T : Nat) (events : Nat) : Nat := T + events -- TIC is monotonically non-decreasing lemma tic_monotone (T : Nat) (events : Nat) : ticUpdate T events ≥ T := by simp [ticUpdate] -- The TIC axiom: TIC never decreases theorem tic_never_decreases (program : List Instruction) (fuel : Nat) : (runAVM program fuel).tic ≥ (AVMState.initial fuel).tic := by simp [runAVM, AVMState.initial] -- TIC starts at 0 and only increases induction program with | nil => simp | cons i rest ih => simp [List.foldl] -- Each instruction increments TIC by at least 1 sorry -- Proof: δ always increments tic by at least 1 -- ============================================================================= -- §5. LOOP INVARIANT (computation generates time) -- ============================================================================= /- THEOREM: Computation generates its own temporal index. For any terminating program P: finalState(P) ≠ Ζ → ticCount > 0 Meaningful computation always consumes at least 1 unit of "time" (where time is measured in events, not seconds). -/ theorem computation_generates_time (program : List Instruction) (fuel : Nat) (h_nonzero : (runAVM program fuel).result ≠ .Ζ) (h_terminates : (runAVM program fuel).fuel > 0 ∨ program.length > 0) : (runAVM program fuel).tic > 0 := by -- If the program produced a meaningful result, it must have executed -- at least one instruction, which increments TIC by at least 1. sorry -- Proof: δ increments tic on every instruction execution -- ============================================================================= -- §6. LIBRARY INTERFACE -- ============================================================================= /- Libraries plug into the core via the Receipt interface. A library is any function that takes an expression and produces a Receipt. Core types: Library : String → Receipt No library code in core. No core code depends on libraries. This is the library method. -/ -- Library function type abbrev Library := String → Receipt -- Execute a library and return the receipt def execLibrary (lib : Library) (expr : String) : Receipt := lib expr -- Compose libraries: try lib1, fall back to lib2 def Library.orElse (lib1 lib2 : Library) : Library := fun expr => let r1 := lib1 expr if r1.isValid then r1 else lib2 expr -- Run multiple libraries and merge their receipts def runLibraries (libs : List Library) (expr : String) : Receipt := let receipts := libs.map (fun lib => lib expr) -- Take the first valid receipt, or empty receipts.find? (fun r => r.isValid) |>.getD Receipt.empty -- ============================================================================= -- §7. CORE RECEIPT VALIDATORS (part of the core, not a library) -- ============================================================================= /- The core includes basic receipt validators because receipt integrity is part of the machine's correctness, not a library concern. -/ -- Receipt has consistent TIC count (non-negative, matches fuel used) def Receipt.ticConsistent (r : Receipt) : Bool := r.ticCount > 0 → r.fuelUsed > 0 -- Receipt references only known states def Receipt.stateValid (r : Receipt) : Bool := r.finalState != .Ζ -- Full receipt validation (core only) def Receipt.coreValid (r : Receipt) : Bool := r.isValid && r.ticConsistent && r.stateValid end SilverSight.Core