What was already known
Lorieau stated the factor-two two-size iterative-rounding claim as Conjecture 3.1.1 in a 2024 thesis and proved a factor-two guarantee for a different greedy rule. The unrestricted Gasoline conjecture remains open.
AI involvement
OpenAI Codex is credited with the suffix-LP formula, rounding invariant, sharpness family and Lean formalization under human-directed objectives and verification requests.
The project credits Codex with the suffix-LP formula, rounding invariant, sharpness family and Lean formalization. A human set objectives and requested verification. The central credited mathematics supports AI-led involvement.
Importance
Proves the two-size conjecture and identifies the exact worst-case ratio for the stated tie rule.
Limitations
The result uses doubly stochastic row and column sums for the residual LP, resolving an ambiguity between the source thesis text and one printed formulation. The unrestricted one-dimensional and multidimensional cases remain open. The separate O(n) implementation complexity is not part of this record.
Technical verification
The repository reports a clean Lean build of the formal two-size guarantee and sharpness family. It also reports a Nanoda replay for earlier targets; the later assignment-LP bridge is Lean-checked but outside that replay. This is source-reported checking at the pinned revision; AIProofWatch did not rerun it.
Formal checking
The repository reports a clean Lean build of the formal guarantee and sharpness family, with a separate Nanoda replay for the earlier formal targets; the later LP bridge is Lean-checked but not covered by that Nanoda replay.
- Checker
- Lean kernel, as reported by MathIsEvenEasier
- Scope
- For positive-integer two-size Gasoline instances with deliveries in {1,K}, the position-first iterative-rounding algorithm under the doubly stochastic assignment-LP interpretation has capacity at most OPT+K−2. For K≥3, the original-index tie rule has matching instances with OPT=K and returned capacity 2K−2, giving worst-case ratio 2−2/K.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The sources describe human direction and presentation review but do not document a named person explaining the central proof. Few is the default from this evidence gap, not a finding that people cannot understand it.
Correction and publication history
Primary theorem, prior baseline, AI attribution and documented proof scope reviewed for non-notifying local inclusion.
Read this published revision