Eight outcomes already produced through ProofAtlas
The same collaborative process has produced checked theorem strengthenings, an accepted finite counterexample, a Lean-checked geometric lower-bound improvement, and unverified manuscripts. Each result remains labeled with its exact evidence status.
Lech Mazur · developed with AI agents through ProofAtlas · 2026
01 · Number theory · Collatz dynamicsAccepted in ProofAtlas · Lean checked
A positive fraction reaches 1 in logarithmic Collatz time
The paper gives explicit c > 0 and X₀ such that, for every natural X ≥ X₀, at least cX positive integers n < X reach 1 within 10.46 ln(n) ordinary Collatz steps.
Release creditLech MazurVersionmanuscript v2.1 of September 6, 2026
Starting frontierEarlier bounds: descent or sublinear counts→AI-developed checked resultA positive fraction reaches 1 within 10.46 ln(n) steps
Why it matters
Reaching 1 with a positive lower-density guarantee
Combines positive lower natural density of reaching 1 with an ordinary-step logarithmic clock and fixed explicit constants. The guaranteed fraction is extremely small and the cutoff extremely large; no practical range is claimed.
How it came together
Explicit constants, manuscript, and pinned Lean source
Manuscript v2.1, explicit constants, public Lean declarations, original infographic and theorem-specific source are connected for inspection. The linked artifact retains the exact build transcript, clean source provenance and public-root axiom checks.
Scope: The full Collatz conjecture remains open. This is a positive lower-density bound, not density-one convergence, an optimal time constant, or a historical-priority claim. The 10.48 corollary uses the same constants.
Starting frontierTao: logarithmic-density almost-boundedness→AI-developed checked resultNatural density + an explicit logarithmic clock
Why it matters
From logarithmic density to ordinary counting
Strengthens the coverage of Tao's landmark theorem and adds one explicit raw-step clock.
How it came together
Phase gaps, density transport, and the raw-time lift
A long multi-agent proof program joined the phase-gap argument, density transport, raw-time lift, paper, and Lean formalization into one inspectable result.
Scope: This permits a density-zero exceptional set and does not prove the full Collatz conjecture or arrival at 1.
Pushes the quantitative predecessor-set lower bound substantially beyond the cited 0.84 benchmark.
How it came together
Exact certificates and adaptive growth
Analytic proof structure, large exact finite certificates, independent computations, three theorem forms, and Lean evidence were bound into one reproducible package.
Scope: This does not prove the Collatz conjecture, give an explicit cutoff, establish optimality, or remove the disclosed native-assisted computation boundary.
A manuscript and complete Lean package claim that, for a nonzero complex polynomial of degree at least two whose zeros all lie in the closed unit disk, every zero lies within distance 1 of a critical point.
Release creditLech MazurSelected review record1 reviewVersionpublic version of 5 August 2026
A complete proof package for a classical conjecture
Makes a complete formal proof candidate for a famous complex-analysis conjecture available for detailed inspection.
How it came together
Manuscript, formal theorem, and open review questions
The manuscript, formal theorem, source, computational boundary, and open review questions are presented together without treating submission as acceptance.
Scope: Independent statement and publication review remains open; the Lean development does not verify the supplemental Python line by line.
05 · Graph theory · counterexampleAccepted in ProofAtlas · Lean-checked counterexample
Jackson Hamilton-decomposition counterexample
An explicit regular orientation of K6,6 on 12 vertices has no Hamilton decomposition, giving a Lean-checked counterexample to the unrestricted conjecture.
Replaces the unrestricted conjectural picture with one exact finite obstruction while leaving sufficiently large-order results untouched.
How it came together
Parity obstruction, enumeration, and Lean
The explicit tournament, parity obstruction, independent enumeration, formal theorem, and source are connected in one evidence dossier.
Scope: Granet already described the flipped-C4 construction and opposite-pair structure; specialist novelty review of the class-size-three obstruction remains open.
06 · Combinatorial game theory · Lean-checked counterexampleUnverified manuscript · adversarial audit attached
Rectangle-reachable Berlekamp counterexample
A manuscript and scoped Lean artifact give a rectangle-reachable 28-cell Domineering position left after 30 moves on an 11 × 8 board, with exact temperature 33/16 > 2.
Release creditLech MazurSelected review record1 reviewVersionpublic version of 14 August 2026 with scoped Lean artifact
Starting frontierBerlekamp's conjectured temperature ceiling 2→AI-developed unverified manuscriptLean-checked rectangle-reachable witness at 33/16
Why it matters
A finite witness above the proposed temperature ceiling
The explicit rectangle-reachable witness moves the finite frontier above 2; how high Domineering temperatures can be and whether an infinite hotter family exists remain open.
How it came together
Rectangle replay, exact game value, and the checked Lean boundary
The August 14 paper, exact 30-move replay, checked Lean statement, finite certificate, public source bundle, and visual explanation are connected at one stable reader route.
Scope: Lean checks the exact existential theorem through the development's narrow HasValueTemperature interface. General thermograph invariance, unrelated reproduction, specialist review, and historical priority remain outside this release.
07 · Graph theory · candidate paper + checked Lean endpointLean formalization checked · acceptance review open
Paper title: “A formalized proof of Bondy's minimum-degree longest-cycle conjecture”
The August 16 paper presents a candidate proof that Bondy's degree threshold forces every path outside a longest cycle to have fewer than k vertices. Paper-to-formal-statement alignment is under review. Lean checks only the exact displayed endpoint Bondy.bondy_longest_cycle.
Release creditLech MazurVersionrevised public version of August 16, 2026
Starting frontierBondy's 1980 open conjecture→AI-developed Lean-checked candidateAugust 16 paper + audited 204-module Lean release
Why it matters
A complete checked endpoint for a longstanding cycle conjecture
Connects the paper's candidate claim to one exact kernel-checked endpoint and its immutable source release while their alignment remains under review.
How it came together
Written proof, 204-module Lean cone, and the sharpness family
The new paper, exact formal statement, 204-module source cone, axiom and no-sorry audit, public source, and retained boundary-example infographic are connected at one reader route.
Scope: The Lean package is not a ProofAtlas accepted result, unrelated replication, specialist peer review, or a historical-priority determination; review of the paper-to-formal-statement correspondence remains open.
A mixed-area lower bound for Moser's convex worm problem
The September 2 paper proves M₊, M± > 0.23743658226923856768, while four Lean theorem pairs check stronger pointwise bounds for direct and reflection-allowed universal covers.
Release creditLech MazurVersionpublic version of September 2, 2026
A quantified improvement while the optimal cover remains open
Raises the known lower bound and closes about 18.1% of the interval from the 2013 lower bound to the cited 2026 preprint upper bound, without claiming the exact optimum.
How it came together
Mixed-area witnesses, exact calibration, and four Lean endpoints
Mixed-area witnesses, exact calibration, an adversarial manuscript audit, the 28-page paper, four pointwise Lean theorem pairs, and immutable source are connected in one evidence-bound result.
Scope: This is a lower-bound improvement, not the optimal area or a solution of Moser's convex worm problem. The Lean endpoints are pointwise cover inequalities; the paper separately takes infima, and independent acceptance review remains open.
