False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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False: unrestricted diagonalization respects any resource bound

Statement

An unrestricted universal diagonalization automatically stays within whatever resource bound is claimed for its simulated machines.

Facts & Assumptions

Given: the claimed constant bound $f (n) = 1$ .

[F1]

The fixed clocked simulator first decodes the complete self-delimiting pair and rejects malformed encodings before running the simulated machine (An efficient universal simulation with a clock, A fixed effective binary encoding of deterministic one-tape Turing machines).

Refutation

technique · direct

1.1givenF1construct

Fix a machine $M$ that halts after one transition without reading its input, and consider the valid pairs $⟨ M, x ⟩$ as $∣ x ∣$ grows. The simulated computation respects the bound $f = 1$ , but [F1] requires the specified simulator to read the complete pair before simulation. Reading that unbounded input takes at least linear time in its length.

2.1step 1.1algebra∎

Hence this unrestricted diagonalizer is not an $O (1)$ -time computation, despite every selected run of $M$ taking one step. A candidate's resource bound alone therefore does not bound decoding and universal-simulation overhead; a separate clock and a proved gap are needed.

Depends on

Used by

Overhead breaks an unrestricted diagonalization claim Counterexample

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity, §3.1 (standard reference, not scraped)