Information Church Turing Physics Structure Eight Tick Step Computable

The finite step

A Turing machine is a simple model of computation: a finite set of internal states, a tape, and a rule that says what to do next. The Church-Turing thesis says that anything we can compute with any reasonable device, we can also compute with a Turing machine. A stronger version, the physical Church-Turing thesis, asks whether every physical process can be simulated by a Turing machine. Recognition Science addresses this stronger claim by looking at the structure of its own model of physics.

In Recognition Science, physics is described as a ledger, a discrete record of events. Each event updates the ledger's state, and the state space is finite: eight phases, each holding a single bit, giving 2^8 possible states. The declaration eight_tick_step_computable proves the simplest fact about this setup: any transition from one phase to another, any function from the eight phases to themselves, can be represented by a finite table. The theorem states that for any such function, there exists a finite set of input-output pairs such that every input appears in the table with its correct output.

This is not a deep result. It is a direct consequence of the state space being finite. If you have only eight possible inputs, you can always list what happens for each one. The proof constructs the table explicitly: it takes each phase, applies the function, and records the pair. The theorem is a formal, machine-checked statement of this fact, and it is the first step in a larger argument that the ledger's dynamics, being built from finite transitions, can be simulated by a Turing machine.

The declaration does not claim that the entire physical process, over time, is computable in practice. It does not claim that the golden ratio, which appears in the framework's constants, can be computed exactly by any finite procedure. In fact, the library separately proves that phi is not a rational number, so no finite decimal expansion can capture it exactly. The step computability result is about the structure of a single transition, not about the full trajectory of the system.

In Recognition Science, this finite-step fact supports the physical Church-Turing thesis: because each step is a finite function, and the ledger processes steps at a positive rate, the framework's physics cannot jump to infinity or perform a computation beyond what a Turing machine can do. The framework models physical processes as computable, not hypercomputable. What the declaration establishes is a small but load-bearing piece of that argument.

/-- **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-003.4**: 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-003.12**: 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 declaration does not claim that the full physical process over time is computable in practice. It does not claim that the golden ratio can be computed exactly by any finite procedure. It does not claim that the framework's physics is equivalent to a Turing machine in all respects.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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