Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27
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 ZFC inequalities of Cichoń's diagram

Statement

In ZFC, the ten cardinals add⁡(N), cov⁡(N), non⁡(N), cof⁡(N), add⁡(M), cov⁡(M), non⁡(M), cof⁡(M), b, and d satisfy the standard Cichoń-diagram inequalities: the elementary ideal arrows, the null-to-meagre arrows, the cross-ideal arrows, the eventual-domination arrows stated below, and all their transitive consequences. No independence or completeness assertion is part of this theorem.

Facts & Assumptions

Given: The null and meagre ideals and eventual-domination cardinals in ZFC.

[F1]

For I=N,M, add⁡(I)≤cov⁡(I), add⁡(I)≤non⁡(I), cov⁡(I)≤cof⁡(I), and non⁡(I)≤cof⁡(I). (Elementary bounds on ideal cardinal invariants)

[F2]

add⁡(N)≤add⁡(M) and cof⁡(M)≤cof⁡(N). (Null-to-meagre Tukey inequalities)

[F3]

cov⁡(N)≤non⁡(M), cov⁡(M)≤non⁡(N), add⁡(M)≤b≤non⁡(M), and cov⁡(M)≤d≤cof⁡(M). (Cross-ideal and bounding inequalities in Cichoń's diagram)

[F4]

b≤d; their definitions and the ideal minima are evaluated in ZFC, where AC supplies cardinal comparison. (Basic bounding and dominating relations, Eventual domination and the numbers b and d, The Axiom of Choice)

Proof

technique · assemble proved generating inequalities
1.1

Draw the ten named cardinals as nodes. For each of N and M, insert the four elementary arrows of [F1]. Insert the two null-to-meagre arrows of [F2], the six cross and bounding arrows of [F3], and b≤d from [F4]. Every inserted arrow is an inequality already proved under the same ZFC conventions.

F1F2F3F4
2.1

If a≤b and b≤c are among these arrows, ordinal/cardinal order transitivity gives a≤c. Repeated application yields exactly the transitive consequences asserted in the Statement. The statement makes no claim that an omitted arrow is independent of ZFC or that a diagram drawing captures every possible relation. AC is used in the cited supplier proofs and in regarding the ten minima as comparable cardinals; no additional selection occurs in this assembly. ∎

step 1.1F4

Depends on

Used by

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Dependency tree · two levels

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Sources