Information Church Turing Physics Structure Church Turing Physics Structure

The computable ledger

The Church-Turing thesis states that anything computable by any mechanical procedure is computable by a Turing machine. The Physical Church-Turing Thesis asks the same question about nature itself: can every physical process be simulated, to arbitrary precision, by a Turing machine? For most of physics this remains a philosophical question, but within Recognition Science (RS) it is a proved theorem. The framework's machine-checked library of formal theorems contains a declaration, church_turing_physics_structure, that establishes the answer: in RS, physics cannot outrun a Turing machine.

The proof rests on a discrete ledger: a record of events that changes in fixed steps. In RS, each ledger entry is a ratio of two quantities, but the dynamics are governed by an eight-tick cycle on a phase space of exactly eight states. The library proves the phase space is finite, and that a single transition from one ledger state to the next is a function from a finite set to itself. Any such function is computable, because it can be written as a finite lookup table. The theorem eight_tick_step_computable states this directly: every step from one phase to another has a finite table that describes it. Since the entire dynamics is made of finitely many such steps, the whole process is Turing-computable.

The declaration also encodes a rate limit. A companion theorem, computation_takes_time, proves that any computation of positive length takes positive time: the ledger cannot jump to infinity in a single tick. This is what rules out hypercomputation, any process that could solve a problem no Turing machine can. The framework's answer is not that physics is easy to simulate, only that it is not beyond computation. The halting problem for RS dynamics inherits undecidability from ordinary Turing machines, so the framework does not claim to predict everything, only that whatever happens is computable in principle.

What the declaration does not claim is just as important. It does not prove that the Physical Church-Turing Thesis holds in general physics; it proves it for the RS model of dynamics. The framework models physics as a ledger, and the theorem applies to that model. It also does not claim that the golden ratio phi, which appears throughout RS, is exactly representable in the ledger. A separate theorem, rs_dynamics_beyond_rational, proves that no rational number equals phi, and another shows phi can be approximated arbitrarily well by rationals. The ledger is discrete, but the quantities it tracks need not be rational; they are only approximated at each tick.

In plain terms, the declaration says: if you accept RS's discrete ledger picture of physics, then physics is computable. It is a structural result, not a practical one. It does not say how long a simulation takes, only that a finite description exists. It leaves open whether the physical Church-Turing thesis holds outside RS, and it does not identify which physical processes are efficiently simulable. The value of the theorem is that it turns a philosophical question into a precise, checkable statement inside a formal system.

/-- **THEOREM IC-003.5**: The Church-Turing physics thesis holds.
    Physical processes in RS are computable because:
    - The phase space is finite (8 phases)
    - Transitions are computable functions on finite types
    - The tick rate is bounded by 1/τ₀
    This is formalized through the irrationality constraint: even though φ is
    irrational, the DYNAMICS (which phase sequences occur) are computable. -/
theorem church_turing_physics_structure : church_turing_physics_from_ledger :=
  has_computation_limits_structure

/-- **THEOREM IC-003.1**: The 8-tick phase space has exactly 8 elements. -/
theorem phase_space_finite : Fintype.card Phase = 8 := by
  simp [Phase]

/-- **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-003.11**: The 8-tick step function is computable (it's a function
    on a finite phase space, hence encodable as a lookup table). -/
theorem eight_tick_step_computable (step : Phase → Phase) :
    ∃ (table : Finset (Phase × Phase)),
      ∀ p : Phase, ∃ p' : Phase, (p, p') ∈ table ∧ step p = p' :=
  finite_function_is_computable (α := Phase) (β := Phase) step

/-- **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-003.12**: 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

/-- **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⟩⟩

What this page does not claim

The Physical Church-Turing Thesis is proved for all of physics, only for the RS ledger model. The golden ratio phi is exactly representable in the discrete ledger. The declaration provides an efficient simulation algorithm, only a proof of computability in principle.

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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