Information Church Turing Eight Tick Universal Gates

A gate set from eight ticks

The Church-Turing thesis states that anything effectively computable can be computed by a Turing machine. It is a thesis, not a theorem, because "effectively computable" has no formal definition. The thesis is widely accepted because many different models of computation, from lambda calculus to register machines, turn out to be equivalent in power. A Turing machine itself is a simple abstract device with a tape, a read-write head, and a set of states; a universal Turing machine can simulate any other Turing machine.

In Recognition Science, the framework models computation as a sequence of updates to a ledger, a discrete record of events. The framework's account intends to derive the Church-Turing thesis from ledger universality: the ledger can simulate any physical process, any computation is a sequence of ledger updates, and an eight-tick cycle provides a universal gate set. The eight-tick cycle is a repeating sequence of eight recognition events that the framework derives from its cost function. The declaration eight_tick_universal_gates records the intent that the Hadamard gate H and the T gate, a π/4 rotation, generate all single-qubit unitaries, and that adding CNOT gives full universality. Other declarations in the same module record similar intentions: the ledger can simulate any Turing machine, each ledger update follows one eight-tick cycle, and physics is computable in principle with no hypercomputation.

None of these declarations is a result. Every theorem in the module concludes in True or another tautology, which means the statements are placeholders, not proofs. The module is explicitly labeled a sketch. The declaration eight_tick_universal_gates therefore establishes nothing about gates, computation, or the Church-Turing thesis. It records a target: derive the Church-Turing thesis from ledger universality. The intended upgrade path is to replace each sketch with a target_X : Prop declaration that states the real claim, and then prove it.

What the framework does not claim is as important as what it intends. It does not claim that the Church-Turing thesis is proved, that the eight-tick gate set is established, or that the ledger is computationally universal. The module also lists falsifiers: if hypercomputation is found, if the Church-Turing thesis is violated, or if the ledger is not universal, then the account is falsified. The framework's predictions include no hypercomputation, bounded by a time scale τ₀, and that computation has a cost J. These are predictions, not theorems.

The reader can now see the difference between a target and a proof. The machine-checked library of formal theorems contains a sketch module that names a goal, not a result. The honest summary is that the Church-Turing thesis from ledger universality remains open, and the eight-tick gate set is a plan, not an established fact.

/-- **THEOREM**: 8-tick phases give universal quantum gates.

    The Solovay-Kitaev theorem: {H, T} can approximate any unitary
    to accuracy ε with O(log^c(1/ε)) gates. -/
theorem eight_tick_universal_gates :
    -- H and T generate all single-qubit unitaries
    -- Add CNOT for full universality
    True := trivial

/-- **THEOREM**: 8-tick phases give universal quantum gates.

    The Solovay-Kitaev theorem: {H, T} can approximate any unitary
    to accuracy ε with O(log^c(1/ε)) gates. -/
theorem eight_tick_universal_gates :
    -- H and T generate all single-qubit unitaries
    -- Add CNOT for full universality
    True := trivial

/-- **THEOREM**: 8-tick phases give universal quantum gates.

    The Solovay-Kitaev theorem: {H, T} can approximate any unitary
    to accuracy ε with O(log^c(1/ε)) gates. -/
theorem eight_tick_universal_gates :
    -- H and T generate all single-qubit unitaries
    -- Add CNOT for full universality
    True := trivial

What this page does not claim

The Church-Turing thesis is not proved by the framework. The eight-tick gate set is not established as universal. The ledger is not shown to be computationally universal.

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/ChurchTuring.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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