Maximal specieslike clusters with a fixed real founding window
Alexander, Specieslike clusters based on identical ancestor points (2026), Section 6, asks for alternative constraints to common ancestry in maximal-cluster existence. Replace that condition by a fixed real founding duration: every internal founder of S is born no later than the earliest member birthdate plus Delta. On natural-number organism identifiers with real chronological birthdates, finite strict earlier sublevels and finite child sets, does every organism belong to a maximal connected, ancestry-convex, reflecting IAP set satisfying this same window, for each Delta>=0? Maximality is within the specified fixed-window class, not among unrestricted specieslike sets.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Infinite labelled graphs; mathematical ancestry
- Posed by
- Samuel A. Alexander, Specieslike clusters based on identical ancestor points (2026), Section 6; a proposed fixed-real-window replacement constraint.
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-25
- Model
- OpenAI Codex (GPT-6 family)
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Significance
- 6 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Yes. Every organism belongs to an inclusion-maximal member of the fixed-window class. The proof constructs an actual seed, proves reflection under appropriate intersections, proves IAP and the real window survive the required directed unions, and applies constrained maximal extension. Relabelling preserves original real birthdates, including ties, and the duration shared by every candidate and competitor. No finite-root premise is needed. This supplies a checked replacement-constraint theorem responding to the source direction, not merely a conditional result assuming the needed seed. The eight-module proof chain has 50 selected endpoints. It does not yield uniqueness, a partition, an effective classifier or empirical species validation. Maximality keeps connectivity, convexity, IAP, reflection and the same founding window. Biological plausibility, priority and whether the constraint matches the author's intended qualitative distinction remain for review.
What the AI did
A human collaborator selected the research directions and requested source correspondence, explanations, proof checking and submission. Codex agents developed the concrete mathematical construction or deduction, Lean proofs and exposition. Other agents in the project reviewed the arguments and source correspondence. These are internal project checks, not independent human expert review. Model-led discovery of the submitted scoped result is claimed; independent human verification and worldwide priority are not claimed.
Verification
Checked here on 30 September 2026 to the extent the artefacts allow. Lean 4.33.1 with pinned Mathlib; eight modules contribute 50 selected endpoints for the founder-window chain, permitting only propext, Classical.choice and Quot.sound, and the Verify workflow is green on the correct repository. Not rebuilt here and no independent audit of the informal-to-formal correspondence, so Lean-checked. The source URL was corrected for the repository's trailing dash. Partial, as submitted, and the submitter's own scope statement is kept: the whole class is fixed before maximality is asserted, so this is maximality within the fixed-window class and not unrestricted maximal specieslikeness.
Sources
- Lean proofPinned public research note, Lean source and verification receiptsActual original-graph maximal-window theoremReal window and union propertiesActual seed construction
- Problem recordSource maximal-cluster question, section 6
- OtherFrozen founder proof packetSource/question provenance ledgerIntegrated publication coverageSuccessful integrated hosted proof audit
Submitted by ZestyDingo473 on