Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
How statement and proof provenance work

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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The space-diagonal machine halts

Statement

Let s(n)≥1 be space constructible and suppose n=O(s(n)). The all-tapes space diagonalizer can be made total in O(s(n)) space by stopping a simulation after its configuration bound.

Facts & Assumptions

Given: the stated constructible cap and a simulated deterministic machine.

Proof

technique · direct
1.1

Compute s(n) in binary, enforce that cap, and run the universal simulation with a binary counter through the at most 2cs(n) configurations allowed by the cap. The simulation uses O(s(n)) space because n=O(s(n)).

givenconstruct
2.1

A nonhalting deterministic run repeats a configuration before the counter expires, while a halting run is seen directly. Declare a fixed answer at expiry. The counter has O(s(n)) bits, so it and the simulator remain within O(s(n)) space.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

3 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