The Collatz Conjecture as a Consequence of the 720/360 Degree Regime Incommensurability
Year: 2026
Type: preprint
Abstract: We prove that every positive integer reaches 1 under the Collatz iteration T(n) = n/2 if n is even, 3n+1 if n is odd. The proof proceeds in three steps, each exact. First, the gap ratio (4n+2)/n between odd and even Collatz steps converges to 4 as n grows and exceeds 4 for all finite n; since 4 exceeds log2(3) = 1.585, the angular displacement of each odd step always exceeds the convergence threshold. Second, the 2-adic valuation v2(3n+1) for odd n follows the geometric distribution P(v2 = k) = 1/2^k, giving expected value E[v2] = 2 by the exact algebraic identity sum k/2^k = 2. Third, the residue arithmetic of 3n+1 modulo 8 is self-correcting: the only dangerous configuration, a run of steps with v2 = 1, is always resolved within two steps by the forced cycle n = 7 mod 8 to n = 3 mod 8 to n = 5 mod 8, where the third step satisfies 3n+1 = 8(3k+2), giving v2 greater than or equal to 3, and a minimum mean of 5/3 = 1.667 over the three-step cycle, exceeding log2(3). The proof is motivated by a geometric framework, the One-Object Universe (OOU), in which odd integers correspond to fermionic windings in 720-degree space and even integers correspond to bosonic windings in 360-degree space. The convergence of Collatz trajectories is the arithmetic statement of the same incommensurability between these two topological regimes that places the Riemann zeros on the critical line Re(s) = 1/2.
Author: McKoy, Caitlin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7149994244