OSTOE — open-source theory of everything, claims registry

Established The Prime Tread Wheel — counting numbers on a growing wheel; spokes are rationals, channels carry at most two primes, and the gear net is exactly the sieve

id: prime-tread-wheel · filed: 2026-10-07 · depends: method-mechanism-first · falsifier: any counterexample from probes/verify.py — a spoke carrying three primes, a layer whose prime-seat count differs from φ(m), or the gear net missing or inventing a single prime

Mechanism

[S] One rule: layer m of the wheel holds the m numbers T(m−1)+1 .. T(m), where T(m) = m(m+1)/2 is the m-th triangular number, and tick k of layer m sits at angle 2π·k/m. The wheel grows by one tick per layer while the tick never changes size — this is Floyd's triangle rolled onto polar coordinates, and no other structure is added.

What follows, at this resolution:

• [F] A spoke is the set of positions whose reduced angle fraction a/b is fixed. Every rational in (0,1] owns exactly one spoke; irrationals own none.
• [F] Adjacent layers misalign by the fractional part of the running count — "ratio as slip". A spoke a/b re-opens every b layers, exactly when b divides the layer index.
• [F] (Multiplexer reading, §8 of the paper) placing n on layer m computes n mod m spatially — each lap is a complete residue system, and the frame length equals the lap index, so frame growth is the enumeration of moduli. Caveat recorded in the paper: on even laps the origin sits a half-turn off (T(m−1) ≡ m/2 mod m), so "slot" means steps since the lap's own origin.

What was tried (the established components)

• [E] Channel capacity is two. The visit formula x = g(gb² − b + 2a)/2 (visit g of spoke a/b) is exact, and for g ≥ 3 the visit number divides the occupant (g odd → g|x; g even → (g/2)|x with cofactor > 1). Proven, and verified for every visit to layer 5000: no spoke ever carries a third prime.
• [E] Every layer m ≥ 3 has exactly φ(m) prime seats — fresh ticks with gcd(k,m)=1. Verified; on layers m ≡ 2 (mod 4) the fresh ticks are all even and the seats swap to second-visit ticks, and the count still lands on φ(m) exactly.
• [E] The gear net is the sieve of Eratosthenes, restated. A gear p fires exactly when cos(2πx/p) > cos(π/p), which for integers is precisely p|x. Planting primes as their own weights and switching them on at p² is the sieve's square-root boundary. The "net" therefore catches every prime below the largest gear squared and dilutes past it — it is a residue-class detector, not a trained approximation. The wheel shows the sieve's boundary clearly; it does not move it.
• Verification: probes/verify.py passes clean (Python + numpy, ~6 s); independent hand-checks of the algebra agreed with the code.

Downstream

The registry's other wheel entry, gap-prime-between-triangulars, depends on this one — the gap is about primes between layers, which presupposes the layer/slot structure established here. If any [E] above is ever falsified, the gap entry must be re-filed, not merely re-dated.

Live: the instrument (the §8 multiplexer panel — pick a channel a/b, scrub laps, watch the frame open and close over it), problem wheels, and the repo with the paper and all probes.