Infinite paths in the composite-restricted visible lattice
Overview
The graph joins coprime integer pairs greater than one by unit steps, with at least one composite coordinate at every vertex. The theorem gives an infinite path and, for almost every ratio in (4/3,5/3), a ray with that limiting coordinate ratio. Earlier local determinant and finite-path observations by rafalwrona and Duncan are credited. The proof supplies the additional rough-composite bands, safe supporting columns, simultaneous direction estimates and connections across scales needed for the global theorem.
Original public questions
Erdős Problem 1212: an infinite path in the composite-restricted visible lattice
Erdős Problem 1212 — An infinite unit-step path through coprime integer pairs greater than one, with at least one composite coordinate at every vertex.
rafalwrona's mixed-modulus determinant observation — Earlier public contribution; exact scope and chronology stated in the attribution correction.
Duncan's drift and finite-path observations — Earlier public contribution; exact scope and chronology stated in the attribution correction.
Original abstract (English)
We consider the nearest-neighbour graph on pairs of integers greater than one that are coprime and have at least one composite coordinate. We prove that this graph contains an infinite simple path, answering Erdős Problem 1212 affirmatively. More precisely, for almost every slope in a fixed interval away from the diagonal, there is a ray with that limiting coordinate ratio. The proof obtains pairwise coprime rough composite rows in short bands, supplies safe composite supporting columns using a polynomial Jacobsthal bound, and forces a small integer interpolation polynomial to vanish at a point of every bad crossing. A summable simultaneous estimate for exceptional directions and planar crossing duality connect the resulting crossings across scales.
Prior mechanism and attribution. This revision credits rafalwrona's mixed-modulus determinant observation of 23 August 2026 and Duncan's drift and finite-path observations of 21 July 2026. The former shares the local arithmetic mechanism used in the interpolation step. These observations do not supply the global infinite-path construction. The present proof additionally provides rough composite bands, safe supports, simultaneous exceptional-direction estimates and connections across scales. The mathematical statements and proofs are unchanged.
Mathematical review
The principal conclusions have a public kernel-checked proof package and reviewed correspondence with the paper. This is distinct from external peer review.
Review standard