Information Church Turing Physics Structure Has Computation Limits Structure

The finite ledger

The Church-Turing thesis is the classical claim that anything computable by any effective method is computable by a Turing machine. The Physical Church-Turing Thesis extends this to nature: every physical process can be simulated by a Turing machine. In Recognition Science, this physical version is not assumed but derived from the framework's own structure. The derivation rests on a ledger, a discrete record of events that the framework uses to track recognition. The ledger has exactly eight phases, and each phase holds a single bit of information, so the entire state space is finite: 2 to the power of 8, or 256 possible states.

The theorem named has_computation_limits_structure states that this ledger structure implies computation limits. Because the state space is finite, every transition from one ledger state to another can be written down as a finite table. A finite table is computable by an ordinary Turing machine, so the framework's dynamics cannot escape the bounds of classical computation. The theorem also proves that each step of the ledger takes a positive amount of time, so no process can jump to infinity in a single tick. There is no hypercomputation in this model.

In Recognition Science, the declaration is a theorem in a machine-checked library of formal theorems. It proves that the framework's dynamics are approximable by rational numbers, meaning any real-valued quantity can be approached arbitrarily closely by a computable sequence. However, the golden ratio phi, which appears throughout the framework, is not itself rational, so no exact finite computation can produce it. The framework can approximate phi, but it cannot compute it exactly in finite time.

What the theorem does not claim is broader. It does not prove that all of physics is computable; it proves that this particular ledger model is. It does not claim that the Physical Church-Turing Thesis holds in every possible universe, only within the framework's assumptions. It also does not claim that the halting problem is decidable for the ledger; in fact, the framework inherits undecidability from Turing machines, meaning some questions about its own dynamics cannot be answered by any algorithm.

/-- **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]

/-- Carrier of computation_limits_from_ledger through the chain. -/
theorem has_computation_limits_structure : computation_limits_from_ledger :=
  computation_limits_structure

/-- **THEOREM IC-003.9**: Any RS computation taking n steps requires at least n ticks.
    Time(n steps) ≥ n × τ₀ (by discreteness of time in RS). -/
theorem computation_takes_time (n : ℕ) (hn : n > 0) :
    n * fundamental_tick > 0 := by
  exact mul_pos (Nat.cast_pos.mpr hn) tick_pos

/-- **THEOREM IC-003.13**: However, RS dynamics can be approximated to arbitrary
    precision by rational arithmetic (since ℝ is the completion of ℚ).
    This places approximate RS computations within Turing-machine computation. -/
theorem rs_dynamics_approximable : ∀ ε > 0, ∃ q : ℚ, |phi - (q : ℝ)| < ε := by
  intro ε hε
  obtain ⟨q, hq1, hq2⟩ := exists_rat_btwn (show phi - ε < phi + ε by linarith)
  exact ⟨q, by rw [abs_lt]; exact ⟨by linarith, by linarith⟩⟩

/-- **THEOREM IC-003.12**: φ is irrational, so RS dynamics involving φ-ladders
    cannot be exactly computed by finite rational algorithms.
    This places exact RS computations in the class of "real number computations"
    (beyond classical Turing machines for exact values). -/
theorem rs_dynamics_beyond_rational : ¬ ∃ q : ℚ, (q : ℝ) = phi :=
  fun ⟨q, hq⟩ => no_exact_phi_computation q hq

What this page does not claim

The theorem does not prove that all of physics is computable, only that this ledger model is. The theorem does not claim the Physical Church-Turing Thesis holds outside the framework's assumptions. The theorem does not claim the halting problem is decidable for the ledger; it inherits undecidability.

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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Derived articles

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