Category 10 — Computations, Complexity, and Resource Gradients: Entity E9 and the Dynamics of Mathematical Work in Causal Theory

This work develops Category 10: Computations, Complexity, and Resource Gradients in the CT-United (Causal Theory) framework, completing Entity E9 of the ten-entity causal ontology. While earlier categories establish formulas, operators, attractors, symmetries, invariants, measures, probability, and entropy, mathematics remains incomplete without a causal theory of computation: the evolution of mathematical work under finite resources.

In Causal Theory, computation is not mere symbol manipulation or abstract algorithmics. It is a causal evolution in formula-space, driven by operator action and regulated by coherence gradients, invariant preservation, symmetry constraints, measure densities, and attractor geometry. Every computation consumes finite resources—time, memory, coherence energy, and measure cost—and is strictly bounded by shell depth and the global closure horizon π. Infinite computation is forbidden; attempts to exceed admissible depth resolve by dissipation or extinction.

The paper develops:

• the CT definition of computation as causal evolution under finite budgets;

• resource gradient fields (time, memory, coherence energy, measure cost);

• causal complexity metrics and quadrant signatures of difficulty;

• computational attractors, halting surfaces, recursion boundaries, and extinction behavior;

• computational invariants inherited from symmetry, invariants, and measures;

• the Resource Ledger, assigning minimal resource vectors to formulas and structures;

• complexity deficits and the Predictive Computation Law, identifying missing theorems, shortcuts, or structural optimizations forced by resource imbalance.

All constructions are ledger-admissible, finite, and horizon-bounded. Category 10 provides the operational backbone of the CT-United framework and prepares the final structural layers: Category 11 (Categories and Morphisms) and Category 12 (The Global Predictive Morphism).