Candidate Cycles of the Collatz Conjecture: A Corrected Smooth Framework and the Exact Diophantine Bridge

This revised Part I separates arbitrary formal affine words, positive exponent vectors specifying candidate odd-to-odd blocks, and genuine Collatz-admissible blocks. It corrects the extrema of the affine constant and develops an exact Diophantine bridge: a positive exponent vector closes to a positive Collatz cycle if and only if D = 2^b - 3^a is positive and divides its affine numerator S(p).

The telescoping identity excludes all candidate words with no entry equal to 1, except for the trivial all-2 word. Within the class p_i in {1, 2}, the paper excludes cases containing exactly one, two, or three entries equal to 1. Further results include a largest-gap certificate, exact neighboring-cut and quotient identities, finite congruence windows, and prime-power certificates for selected classes. The paper also shows why a proposed uniform cyclic size estimate cannot settle the general problem.

The remaining arithmetic problems are stated explicitly. This paper does not prove the full Collatz conjecture, does not exclude all nontrivial cycles, and does not address unbounded trajectories. The previously established result for the single basic-fractal cyclic class is retained as an input and is not reproved here.

This description reflects the revised August 2026 PDF attached to this record.