Information Church Turing Ledger Follows 8tick

The ledger's eight-tick step

The Church-Turing thesis states that any function computable by an effective procedure can be computed by a Turing machine. Equivalent formulations say that all reasonable models of computation, such as lambda calculus, recursive functions, and register machines, have equal power. The thesis is not a theorem because the notion of an effective procedure is not formally defined. The halting problem, the busy beaver function, and Kolmogorov complexity are classic examples of uncomputable functions that any adequate model must exclude.

In Recognition Science, the framework models computation as updates to a ledger, a discrete record of events. The declaration ledger_follows_8tick states that each update to this ledger corresponds to one eight-tick cycle, where phase accumulation determines the next state. The framework's library, a machine-checked collection of formal theorems, contains this declaration, but it is marked as a sketch. The declaration concludes in a tautology, meaning it does not yet establish a real result. The intended claim is that the eight-tick structure provides a universal gate set, allowing the ledger to simulate any Turing machine and thus any physical process.

This sketch is part of a larger target: deriving the Church-Turing thesis from Recognition Science principles. The framework also intends to show that the ledger is computationally universal, that physics is computable in principle, and that no hypercomputation is possible. These are targets, not achievements. The library includes a falsifier structure that would be violated if hypercomputation were found or if the ledger were not universal. The declaration ledger_follows_8tick does not prove that the eight-tick cycle is real or that the ledger can actually simulate any computation; it only records the intention to establish such a claim.

/-- The ledger update follows 8-tick phases. -/
theorem ledger_follows_8tick :
    -- Each update corresponds to one 8-tick cycle
    -- Phase accumulation determines the next state
    True := trivial

/-- The ledger update follows 8-tick phases. -/
theorem ledger_follows_8tick :
    -- Each update corresponds to one 8-tick cycle
    -- Phase accumulation determines the next state
    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 declaration does not prove that the eight-tick cycle is real or that the ledger can simulate any computation. The Church-Turing thesis is not derived from Recognition Science principles in this module. No hypercomputation is not established as impossible by this declaration.

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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