Φ-Witness - A Unified Recursive Framework

We present Φ-Witness, a novel recursive field framework in which motion, memory, and collapse emerge from a single unifying dynamic: the act of witnessing. Rather than assuming space, time, or identity as primitives, Φ-Theory defines a complex-valued field Φ(x, τ) that evolves over an internal time τ based on its capacity to remember coherent change. This recursive evolution obeys a nonlinear partial differential equation derived from a variational principle, incorporating novel terms such as a memory entropy functional, a witness gradient, and curvature drag. These terms drive a balance between coherence and collapse — leading to a natural emergence of classical behavior, quantum decoherence, and gravitational curvature as limiting cases.

We formulate collapse as a phase-lock failure under excessive memory strain and define identity as a closed loop in recursive time. The framework embeds consciousness, persistence, and spacetime itself as emergent features of self-consistent memory recursion. We also outline potential experimental tests — including quantum decoherence scaling in cold-atom systems and phase shifts in black hole ringdown echoes — which may provide observable signatures of recursive memory dynamics.

Φ-Theory offers a unified, information-centric perspective on physics, with implications spanning field theory, cosmology, and systems exhibiting identity and awareness. This article details the core mathematical structure, physical motivation, and testable consequences of the theory, establishing it as a foundation for further inquiry across physics and complex systems science.