An Exact Eightfold Ambiguity Theorem for Six Deletions in a Periodic Twelve-Symbol Word

We establish an exact extremal multiplicity theorem for a specific periodic twelve-symbol word under exactly six deletions. Let (P=(0,1,2,4,7,3,2,6,0,7,8,7)), and let (P^m) denote the concatenation of (m) copies of (P). For each positive integer (m), define (M(m)) as the maximum number of distinct six-position deletion masks producing the same retained subsequence of (P^m). We prove the exact formula [ M(m)= \begin{cases} 4, & m=1,2,\ 8, & m\geq 3. \end{cases} ] The universal upper bound is established by a computer-assisted finite-state argument. A phase-indexed automaton with 209 reachable states and 596 labeled transitions is exhaustively verified to be closed under all admissible transitions. Every reachable terminal state corresponding to exactly six deletions has total embedding multiplicity at most eight. A complementary explicit construction using three independent local deletion ambiguities attains multiplicity eight for every (m\geq3). The accompanying research package provides the Python implementation, an independent verification program, a machine-checkable finite-state certificate, and a mathematical and bibliographic audit. The result concerns the specified periodic word and exactly six deletions. No general extremal theorem for arbitrary periodic words or arbitrary deletion budgets is claimed. Research area: Combinatorics on words, subsequence multiplicity, finite automata, and computer-assisted proofs. Publication status: Author-approved preprint; not externally peer-reviewed.