A GENERALIZED COLLATZ MULTI-CYCLE TERMINATION EXAMPLE: THE 3x+5 MAP AND ITS SOURCE DECOMPOSITION

Abstract. We study the Collatz-type mapT3,5(n) = n/2,n even,3n +5, n odd.We formalize a conditional convergence strategy based on: (i) a reduction to sources 3u(with u odd) via an anti-9 lifting, (ii) a component-wise well-founded descent on the sourceparameter u produced by dyadic leaf certificates modulo Mt = 9 · 2L+6t (with L = 3), and (iii)a terminating certificate forest with six roots U0 = {1,3,5,17,41,57}. Under these hypotheses,every trajectory reaches one of the six empirically known positive terminal cycles (seed values1, 5,19,23,187,347)