Computational Wave Field Theory

Abstract

We propose a framework for thinking about computation in the way physicists think about quantum mechanics: as the dynamics of a wave-like field, with events occurring probabilistically according to the squared amplitude of that field at each location. The proposal is not that computation is quantum mechanics, nor that physical quantum mechanics gives rise to computation. Computation is treated as a domain in its own right (formal systems, cellular automata, neural networks, brains, distributed systems), and the claim is that this domain admits a wave-mechanical description with structure parallel to, but distinct from, the wave mechanics of physical particles. The framework introduces a substrate-relative quantum of action hbar_c playing the role analogous to Planck's h, and a substrate-relative speed bound c_c analogous to the speed of light; both depend on the substrate and are calibrated across a five-substrate pilot catalog. The book develops this proposal from postulates to executed experiments, and its results split cleanly rather than confirming one grand claim. A computational analog of gravity emerges from spatial gradients of a measured gain field: horizons, clock freezing, and an area law on a holographic code class are realized, including a linearised computational Einstein equation, while the thermodynamic route to full Einstein dynamics fails generically across dissipative substrates (kinematics yes, Einstein dynamics no). The optimal belief state of an embedded observer gains a real compression advantage from wave-style amplitudes, but no epistemic restriction over a definite computational ground can force complex interference; the missing phase is traced instead to self-reference, where two theorems show that a continuous reversible realization of self-negation forces the imaginary unit, and that the geometric phase of quantum mechanics is the holonomy of that structure, with the Maxwell action recovered as the unique self-consistency cost. What remains is a single physical wager: that nature derives its complex phase this way. Every claim carries an explicit status label separating rigorous theorems, bridge identifications, computed toy results, and speculation; negative results are reported in full, and named falsifiable bets (including a concrete prediction for cellular automaton Rule 54) invite refutation. All experiments are seeded, CPU-scale, and reproducible from the accompanying code and records.