Connectivity of Binary Transition-Count Fibers Under Interior Symbol Swaps

We study the connectivity of finite binary words under swaps of two unequal strictly interior symbols that preserve the directed adjacent-pair count vector (M_2=(c₀₀,c₀₁,c₁₀,c₁₁)). We establish a constructive characterization of the connected components: two binary words of the same length belong to the same component if and only if they have identical directed adjacent-pair counts and identical first and last symbols. The proof uses the alternating run decomposition of binary words. Local transfers between neighboring runs of the same symbol are implemented by single admissible interior swaps. A deterministic normalization procedure moves surplus run lengths toward designated target runs while decreasing a nonnegative integer-valued transport potential. Every word in a fixed transition-count and endpoint fiber therefore reaches a unique canonical representative. Consequently, the number of connected components for binary words of length (n≥ 2) is [ C(n)=n(n-1)+2. ] We supplement the mathematical argument with reproducible finite computational audits, including exhaustive word enumeration, independent swap-graph component calculations, boundary-case tests, constructive path verification, and checks of the normalization potential. The result is situated within the combinatorics of words, directed transition-count statistics, Eulerian-trail representations, and the theory of Markov bases. Related connectivity questions have previously been studied in algebraic statistics; accordingly, no claim of absolute priority or comprehensive novelty is made. The accompanying Python implementation uses only the standard library and can be executed offline, including on Android using Pydroid 3. Status: Mathematical preprint; not externally peer-reviewed. Mathematical verification and originality statement The manuscript presents a constructive, length-independent mathematical argument supported by reproducible finite computational experiments. The computational audit includes direct verification of local run transfers, exhaustive enumeration of binary words for selected lengths, independent construction of admissible-swap graphs, verification of connected components, and testing of canonical normalization procedures. Finite computational verification is distinguished from the general mathematical proof. The accompanying code is not a formal proof assistant. Related research on Markov bases, transition-count fibers, and homogeneous Markov chain models is discussed in the manuscript. The author does not claim that the general subject of fiber connectivity is new, nor that a comprehensive priority search has established the novelty of the particular theorem or move set. This is a publicly released mathematical preprint and has not undergone external peer review. Declaration of AI assistance and author responsibility OpenAI ChatGPT was used substantially in mathematical exploration, programming, computational certificate design, literature scouting, proof organization, and manuscript drafting. The named author, Ivo Vajc, is responsible for reviewing and validating the mathematical arguments, checking the released source files and computational results, verifying bibliographic references and claims of originality, and making the final decision regarding submission and publication. The use of AI assistance does not constitute independent mathematical verification, external peer review, or certification of originality.