Exact Realizer Floors for One- and Two-Defect Valuation Words in the Accelerated 3x + 1 Map

This paper investigates the arithmetic realization of finite valuation words in the accelerated Collatz (3x + 1) map, with particular emphasis on structured words containing one or two valuation defects.

We derive an exact closed-form expression for the least positive integer realizing a two-defect valuation word and establish unconditional, computable lower bounds using prefix truncation and orbit advancement. A key distinction is made between finite-word realization and completed-episode realization, whose congruence conditions differ by one dyadic bit.

Exploiting the fact that a second defect closes the preceding episode, we obtain a strengthened realizer floor. We prove that this refinement can yield an unbounded logarithmic improvement over the original bound along an explicit infinite family of two-defect words.

The paper also characterizes the reduced denominators of associated arithmetic anchors, identifies exceptional boundary configurations, and reduces a uniform two-defect lower-bound problem to an explicit open hypothesis concerning one-defect realizers, with a critical exponent threshold of 1/2 in the balanced regime.

The theoretical findings are supported by reproducible exact-integer computations, independent implementations, and a targeted comparison with established Collatz literature.