The Causal Prediction Engine for Category 1: Coherence Deficits, Missing Causes, and Required Theorems (Pre-Horizon Engine, π-Bounded)

This paper activates the Causal Prediction Engine of Causal Theory (CT) for Category 1, which contains the central formulas, conjectures, and structures of mathematics. Building on the causal ontology, operator geometry, ledger framework, and attractor dynamics developed in Parts 1–5, the paper performs a systematic causal diagnosis of twelve foundational mathematical problems.

Each object—ranging from Collatz, Goldbach, and Twin Primes to the Riemann Hypothesis and Birch–Swinnerton–Dyer—is represented as a causal state vector characterized by quadrant composition, minimal Son-unit level, momentum, and attractor type. The Prediction Engine computes the coherence deficit of each object and identifies the precise causal components that are missing: invariants, Lyapunov functionals, flow laws, density theorems, or categorical morphisms.

For every non-zero coherence deficit, the engine outputs a predicted theorem-type: the minimal mathematical statement required to restore causal closure under the global π-horizon constraint. All predicted theorem-types are finite, resource-bounded, and ledger-indexed, with no admissible construction exceeding the compilation limit of the finite table T1–T118.

This paper does not propose speculative conjectures; it provides a causal classification of why major conjectures persist and specifies the structural form that their resolving theorems must take. Category 1 thus establishes the first operational demonstration of predictive mathematics within CT, transforming incompleteness from an open-ended mystery into a finite, structured diagnostic process.