TID_Part_IX_Computational_Limits_and_Tau_Geometry

Temporal Information Dynamics (TID) — Part IX extends the Scalar Temporal Field Ontology (STFO) into the domain of temporal control theory, unifying classical control, quantum control, channel steering, and metrological optimization under a single τ-geometric framework. Building on Parts I–VIII, this installment constructs: the τ–Control Hamiltonian, derived from the global information metric GAB[τ]G_{AB}[\tau]GAB[τ]; the Temporal Controllability Tensor, governing which information degrees of freedom can be driven by τ-geometry; the τ–Response Operator, linking external controls to disformal temporal curvature; the Universal Control Bound, the analogue of a “geometric speed limit” for how information can be steered under scalar temporal flow; the τ–Quantum Control Equation, describing how coherence, populations, and entanglement respond to temporal curvature and noise kernels; the Metrological Reachability Criterion, determining which states are accessible under τ-geometry, noise, and Hamiltonian structure. This work shows that all classical, thermodynamic, geometric, computational, and quantum control processes are special cases of Temporal Steering, the evolution of information functionals under the universal τ–flow equation introduced in Part VIII. Major contributions: A unified theory of controllability for classical and quantum systems in curved temporal geometry. The first derivation of control capacity limits from scalar temporal curvature. A geometric criterion for stabilizing coherence and entanglement using τ–aligned controls. A complete framework connecting control theory, channel theory, entropy production, and quantum metrology. This Part IX installment completes the control-theoretic pillar of TID and prepares the ground for Part X, which will finalize the full Temporal Information Dynamics program by establishing the fundamental limits of computation, sensing, and predictability in the Scalar Temporal Field Ontology. This document is part of the Temporal Information Dynamics series: Part I — Temporal Depth and Topological Information Part II — Channel Geometry and the Energetics of τ-Information Part III — Information Transport and Temporal Currents Part IV — Temporal Information Thermodynamics and the τ–Second Law Part V — Temporal Complexity and Information Geometry Part VI — Field-Theoretic Information Action and the τ–Hamiltonian Flow Part VII — Quantum Information Dynamics, Decoherence, and Entanglement Flow Part VIII — Global Information Geometry and the Universal τ–Flow Equation Part IX — Temporal Control Theory and the Limits of Information Steering (this work) All research authored by Cale Scott Howe (2025).This work includes the Howe License: Exclusive authorship, no derivatives, no commercial use, no patents, and strict AI-use restrictions.