Geometric Complexity Hierarchy and Complexity Classes\\ in Computational Universe:\\ Volume Growth, Curvature, and Computability Boundaries\\ Under Unified Time Scale

In previous systematic studies of ``computational universe'' U_{comp} = (X,T,C,I), we have successively constructed discrete complexity geometry, discrete information geometry, control manifold (M,G) induced by unified time scale, task information manifold (S_Q,g_Q), time--information--complexity joint variational principle, and established equivalence between physical universe category and computational universe category on reversible quantum cellular automaton (QCA) subclass. On the other hand, complexity classes (such as P, NP, BQP) in classical complexity theory are mainly defined through ``upper bound of steps/gates as input size n varies'', lacking systematic correspondence with geometric structures. This paper proposes within computational universe geometric framework a geometric complexity hierarchy theory: through complexity distance d_{comp}, complexity ball volume growth V_{x_0}(T), discrete Ricci curvature, and geodesic structure of control manifold (M,G), we give geometric characterizations for a family of natural complexity classes, and prove several constraint theorems of form ``complexity class \leftrightarrow geometric invariants''. Specifically, we first introduce into computational universe input encoding family \{\iota_n:\Sigma^n\to X\}, viewing decision of language L\subset\Sigma^\ast as reachability problem from input configuration to ``acceptance region'' A\subset X. For each input length n, we use complexity distance to define minimum computation radius T_L(n), and accordingly introduce geometric complexity classes $ GC(poly) = \{ L : T_L(n)\le C n^k \}, \quad GC(exp) = \{ L : T_L(n)\le 2^{O(n)} \}, etc. We prove: under natural ``Turing--QCA--computational universe equivalence'' assumption, GC(poly) is equivalent to traditional P class (differing by at most polynomial rescaling), while GC(exp) covers EXP class. Second, we introduce volume growth and complexity dimension into geometric scaling of complexity classes: for given basepoint x_0 and complexity ball B_T(x_0), volume growth exponent \dim_{comp}(x_0) = \limsup_{T\to\infty} \log V_{x_0(T)}{\log T} is used to characterize ``dimension'' of local computation space. We prove a polynomial dimension constraint theorem: if in some region \dim_{comp}(x_0)\le d, and all related language decision trajectories are confined to that region, then geometric complexity functions T_L(n) of all these languages are at most polynomial, with exponent k controlled by d. Conversely, in regions where negative curvature leads to exponential volume growth, we can construct language families whose geometric complexity functions necessarily reach or approach exponential level, reflecting ``negative curvature \leftrightarrow exponential complexity'' geometric--complexity connection. Third, we introduce geometric complexity horizon: for given basepoint and growth order f(n), define language family decidable within radius f(n), and use discrete Ricci curvature lower bound with volume comparison theorem to prove: in non-negative curvature regions, if volume growth bounded by polynomial upper bound, then there exist no ``geometrically essentially exponentially hard'' languages; while in local strong negative curvature regions, there exist natural language families whose complexity horizons necessarily exceed any given polynomial radius. Finally, we discuss quantum case: in QCA universe, through analysis of control--scattering manifold (M,G)$ and phase interference structure, we give geometric upper bound for BQP class: under premises of unified time scale and appropriate interference regularity, minimum geodesic length on control manifold for languages in BQP still bounded by polynomial upper bound, while volume explosion and negative curvature more affect ``non-interference-exploitable'' classical complexity part. This paper systematically connects traditional complexity classes with geometric invariants (volume growth, curvature, horizons) in computational universe, providing foundation for subsequent geometrization of higher-level concepts such as ``complexity phase transitions'' and ``capability--risk frontiers''.