Foundational Closure of Game Theoretic and Proof Resolution Symmetry

We introduce a formal framework for resolution symmetry in cyclic, turn-based systems, establishing precise conditions under which terminal claims such as completion, victory, or finality are logically self-referentially closed. For a system of x ≥ 2 distinct agents acting in fixed cyclic order, we prove that a terminal condition is valid if and only if it occurs after a number of moves congruent to 0 ≡ mod(x). This yields the Resolution Symmetry Closure Theorem, which characterizes closure as a parity condition on causal opportunity rather than as a property of the terminal state itself. We further show that the minimal unit of causal symmetry in such systems is 1/x, defining a fundamental symmetry increment that governs partial and complete resolution. This leads to a normalization principle for infinite, non-convergent alternating operator processes: rather than possessing a classical terminal value, such systems admit a symmetric normalization constant equal to 1/x, representing the barycentric center of their operator cycle. In the two-operator case, this recovers the normalization 1/2, providing a unified structural explanation for closure in two-player games, the light-switch paradox, and the symmetric interpretation of the series 1 - 1 + 1 - 1 + ....Finally, we establish that in any cyclic operator system satisfying resolution symmetry, the normalized terminal state of an infinite process is uniquely determined by its initiating state, resolving apparent indeterminacy as a consequence of omitted parity completion rather than intrinsic ambiguity. Together, these results reframe game-theoretic termination, alternating series normalization, and paradoxical infinite processes as manifestations of a single principle: causal symmetry must be fully traversed for a claim of finality to be well-defined.