Joint Realizability of Quantum Probability Histories An Overview of the Brown Theorems Series

Matthew Rick Brown Brown

This paper solves the joint-realizability problem exactly for generic three-state (N=3) quantum probability histories. It proves that a generic 3\times3 unistochastic matrix has exactly two gauge-inequivalent unitary completions, shows that the reachable endpoint set is genuinely two-dimensional, and eliminates the hidden connecting phases to obtain a finite necessary-and-sufficient algebraic criterion for determining whether three probability matrices A,B,C can arise from one consistent unitary composition. The resulting decision procedure reduces the problem to at most four completion branches and degree-\leq8 polynomial equations, and its floating-point implementation was validated on 500 constructed-realizable and 500 unrelated triples with no errors in the reported tests. The theorem applies to the generic stratum; several nongeneric boundary and singular cases remain for a fully exhaustive N=3 decision procedure.

Luigi Accardi

Giuseppe Nisticò

Matthew Rick Brown Brown

The Brown Rules develops a framework for deciding whether a sequence of probability transitions can arise from one consistent underlying unitary history. The central idea is that individual probability tables can each be valid on their own while still being incompatible when required to describe successive stages of the same process. The paper identifies probability-only tests that can rule out incompatible histories, shows how hidden phase information constrains composition, and demonstrates that in important cases the set of compatible outcomes occupies only a restricted portion of the possible probability space. It also extends the framework to longer histories and outlines computational approaches for testing compatibility. The results apply specifically to finite-dimensional unitary models and provide consistency tests within that setting rather than proposing a modification of quantum mechanics.

Giuseppe Nisticò

Matthew Rick Brown Brown

This paper investigates whether locally consistent quantum probability histories can always be combined into a single globally consistent unitary history. It demonstrates, through an explicit and independently verified counterexample in the simplest nontrivial setting, that every three-time subset of a multi-time history can be individually realizable even though no single sequence of unitary transformations can reproduce the complete collection of transition probabilities. The paper identifies shared connecting phases as the mechanism behind this local-to-global obstruction, develops a general parameter-counting condition for when global realization may be possible, and relates the result to broader quantum compatibility and marginal problems as well as the independently developed Quantum Composition Paradox.