OSTOE — open-source theory of everything, claims registry

Open gap GAP — is there always a prime between consecutive triangular numbers?

id: gap-prime-between-triangulars · filed: 2026-10-07 · depends: prime-tread-wheel · falsifier: a layer of the wheel containing no prime — this would falsify OEIS A066888 and instantiate a counterexample to Sierpiński's Hypothesis H1

[G] Every layer m of the Prime Tread Wheel holds the m consecutive integers T(m−1)+1 .. T(m). The wheel's geometry (see prime-tread-wheel) makes each layer a complete residue system and routes every number to a slot — but routing is free; deciding which occupant is prime is the sieve, and this entry is the one place the wheel currently leans on an unproved statement about the answer.

Conjecture (A066888 / Sierpiński's Hypothesis H1 at this instance): at least one prime lies in every interval (T(m−1), T(m)).

Checked computationally to layer 10,000 in this program (sieve of Eratosthenes against the wheel's own placement). Externally, A066888 is tabulated and the conjecture appears to be open in the literature; Visser (2025) surveys the hypothesis family. Necessary at the present resolution; mechanism not yet decompressed — no proof from the wheel's own structure is known, and the wheel's theorems (channel capacity two, φ(m) seats, the gear net) all survive whether or not this gap closes, because none of them asserts anything about a prime's existence in a layer, only about where primes sit if they land.

Not load-bearing for any other entry in this registry; load-bearing for any future wheel claim of the form "every layer sees a prime" or "the wheel generates primes by construction."