Unified Causal Field Theory: A Proof of Geometric Subsumption and Extension of Causal Inference Methods Into Unified Framework

This document presents a self-contained, field-theoretic framework in a generalized $(n\text{D}+\text{T})$ spacetime to formally unify a wide range of causal inference methods. I prove that established methods are geometric projections of this underlying reality. A key result is the formalization of Hypertime, a `derivative dimension` that models the interaction between spatial concepts and temporal dynamics. Within this causal tensor spacetime, I propose a new `Born Rule` for the emergence of causation from the interaction of vector-like states. This geometric-dynamical paradigm offers a novel mechanism for the detection of unobserved variable bias using operator non-commutativity and provides a mathematical origin for memory effects via the sinusoidal structure of fractional derivatives. The framework extends beyond traditional causal inference to encompass Bayesian statistics (belief updates as field dynamics), decision-making processes (softmax-to-argmax transitions as causal collapse), machine learning (parameter evolution as causal geodesics), and multi-agent systems (strategic interactions as causal field evolution) as manifestations of causal field theory. The empirical success of existing low-dimensional causal inference methods provides a compelling validation for the mathematical necessity and superior performance of this higher-dimensional spacetime framework. This proof serves as both a standalone treatise and the foundational pillar for a broader monograph on Unified Cognitive Field Theory.

Keywords:
causal inference, field theory, Lorentzian geometry, derivative dimension, Born rule, hypertime, memory, geometric subsumption, qualitative primacy, hidden variables, Fubini's theorem, Bayesian inference, decision theory, machine learning, tensor calculus, manifolds, mathematical proof