Information Church Turing Physics Structure Church Turing Implies Limits

Computation limits from the ledger

The Church-Turing thesis is a claim about the power of a Turing machine, the idealized computer that reads and writes one symbol at a time on an infinite tape. The physical Church-Turing thesis extends the idea: it asks whether every physical process, not just every algorithm, can be simulated by a Turing machine. The declaration church_turing_implies_limits in the Recognition Science library is a formal statement about that extension. It says that if the physical Church-Turing thesis holds, then computation has limits, and it proves this as a theorem in the framework's machine-checked library of formal theorems.

The proof rests on a specific model of physical computation. In Recognition Science, the framework models a physical process as a ledger, a discrete record of events. The record is not continuous; it advances in eight distinct phases, called ticks. Each tick updates a finite set of entries, and the whole state of the ledger at any moment is a finite object: a function from the eight phases to a Boolean value, giving 2^8 = 256 possible states. The framework proves that any transition from one such state to another is computable, because any function on a finite set can be written as a finite lookup table. That is the core of the argument: finiteness forces computability.

The declaration itself is a small logical step. It takes the statement church_turing_physics_from_ledger, which asserts that the ledger structure implies the physical Church-Turing thesis, and it derives computation_limits_from_ledger, the statement that computation is bounded. The theorem is a conditional: if the former holds, the latter follows. The library also proves supporting facts: the tick rate is positive, so any computation takes time; the phase space is finite, with exactly 8 phases; and the golden ratio phi, which appears throughout the framework, is not exactly representable as a rational number, though it can be approximated arbitrarily well. These facts together give the limits their concrete shape.

What the declaration does not claim is just as important. It does not prove that the physical Church-Turing thesis is true; it takes that as a hypothesis and shows what follows. It does not claim that the framework's model of physics is the only one, or that the ledger is the true structure of the universe. It does not claim that all physical processes are in fact simulable, only that within this model, any process that is simulable is Turing-computable. The theorem is a statement about the consequences of a premise, not an empirical discovery about the world.

/-- **THEOREM IC-003.6**: Church-Turing physics implies computation limits hold. -/
theorem church_turing_implies_limits (h : church_turing_physics_from_ledger) :
    computation_limits_from_ledger := h

/-- The 8-tick phase space: phases 0 through 7. -/
abbrev Phase := Fin 8

/-- **THEOREM IC-003.4**: The discrete ledger state space is finite (exactly 2^8 = 256). -/
theorem ledger_state_space_finite :
    Fintype.card DiscreteLedgerState = 2 ^ 8 := by
  simp [DiscreteLedgerState, Fintype.card_pi, Fintype.card_fin, Fintype.card_bool]

/-- **THEOREM IC-003.3**: Any ledger transition on the 8-tick phase space is
    a function on a finite type, hence computable by table lookup.
    Since there are only 2^8 = 256 possible discrete ledger states, any
    transition function can be pre-computed as a finite lookup table. -/
theorem discrete_ledger_computable (t : LedgerTransition) :
    ∃ (table : Finset (DiscreteLedgerState × DiscreteLedgerState)),
      ∀ (s : DiscreteLedgerState),
        ∃ (s' : DiscreteLedgerState), (s, s') ∈ table ∧ t s = s' := by
  use Finset.image (fun s => (s, t s)) Finset.univ
  intro s
  exact ⟨t s, Finset.mem_image.mpr ⟨s, Finset.mem_univ s, rfl⟩, rfl⟩

What this page does not claim

The declaration does not prove that the physical Church-Turing thesis is true. It does not claim that the ledger is the true structure of the universe, only a model within the framework. It does not claim that all physical processes are in fact simulable, only that within the model, simulable implies Turing-computable.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/ChurchTuringPhysicsStructure.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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