The Unified Hybrid Synthesis: Algebraic QFT, Operator Gravity, and the Born Rule Functor.

Boyu Yang, James Fullwood

Antony Valentini

Patrick Devlin

The four axioms of Modular Entropic Gravity (MEG) implicitly contain the Doplicher–Haag–Roberts (DHR) superselection structure of algebraic quantum field theory. This paper establishes the result through a four-link chain: Axiom 1 (modular Hamiltonians of local regions) commits the theory to the Haag–Kastler framework; Axiom 3 (Bisognano–Wichmann and the PLES functional) provides a mass gap with exponential clustering; Axiom 2 (JLMS) restricts physical states to a class satisfying the Buchholz–Fredenhagen exponential localisation criterion, as proved for an explicitly defined admissible state class with bounded modular energy; and the DHR theorem then implies superselection sectors labelled by representations of a compact group, with block decomposition of regional operator algebras and block-diagonal admissible states. The paper demonstrates that this result has consequences beyond the gauge theory application for which it was first identified in the MEG programme (Zenodo DOI: 10.5281/zenodo.19635025). Specifically, it completes the Born rule dissolution established in the companion paper on convex field representations (Zenodo DOI: 10.5281/zenodo.19600153). That paper showed that within a single superselection sector, the Born rule is not a separate measurement postulate but a structural feature of the convex entropy field representation. What it could not address was the origin of the sector structure itself. The DHR superselection structure answers three questions left open by the Born rule paper, all from the axioms alone. First, why do sectors exist? — because the DHR theorem reconstructs them from the Haag–Kastler framework with mass gap and exponential localisation. Second, why are cross-sector superpositions forbidden? — because JLMS-admissible states are block-diagonal, a theorem of the DHR framework rather than a postulate. Third, why are cross-sector branches distinguishable? — because states in distinct sectors carry different quantum numbers under the DHR group, forcing their entropy field configurations to differ. The synthesis gives the Born rule and superselection structure from the MEG axioms: the sector decomposition from DHR (this paper), the Born rule within each sector from the convex dissolution (companion paper), and the absence of cross-sector interference from the block-diagonal constraint. No independent probability postulate is required.x

François David

Matthew Eltgroth

This paper develops a structural framework connecting quantum mechanics, category theory, and arithmetic geometry through the language of non-semisimple categories. Within the Algorithmic Motives (AM) and Ramified Arithmetic Coordinates (RAC) frameworks, core phenomena of quantum mechanics—including superposition, measurement, the Born rule, complementarity, and entanglement—are interpreted as consequences of categorical structures such as extension groups, global dimension, and exact functors. Superposition is identified with non-split extensions classified by Ext¹, while measurement is modeled as a functor to a semisimple category, forcing the annihilation of extension classes and yielding wavefunction collapse as an algebraic necessity. The Born rule is derived as the unique probability assignment compatible with the output category’s semisimple structure. Complementarity arises from the algebraic invariance of squared amplitudes under phase transformations. The framework further introduces a geometric interpretation of entanglement via covering spaces and deck transformations, and provides a categorical explanation of the quantum eraser phenomenon. These constructions are extended to arithmetic geometry through a dictionary linking Betti and étale realizations to wave and particle descriptions. A central component is the Ramified Arithmetic Coordinates (RAC) framework, which interprets the critical point of L-functions as a ramification point analogous to a coordinate singularity in general relativity. The modified trace is proposed as a coordinate system resolving this singularity. Within this structure, the Generalized Riemann Hypothesis is reformulated as a faithfulness condition for a modified comparison functor, with precise separation between proved finite-level results and conjectural infinite-level extensions. All statements are explicitly classified as Theorem, Structural Correspondence, or Conjecture, ensuring clarity between established results and proposed extensions. The paper is intended as a research exposition introducing a unified algebraic framework rather than a claim of equivalence between physical and arithmetic systems.

Danny Westwood

We present a first-principles derivation of the Born rule using a curvature-filtered gravitational decoherence model combined with standard environmental decoherence.The derivation focuses on filtered density matrices and geometric projection operators to explain how outcome probabilities emerge as curvature-weighted classicalizations of quantum states. The curvature filter, motivated by renormalization group flow, regulates when vacuum fluctuations become gravitationally classical. We show that the Born rule emerges as a special case of a normalized curvature-weighted density matrix and as the limit of curvature-saturated operator projections. We briefly supplement this with a path-integral formulation and discuss competing theories. This approach offers a novel, observer-independent foundation for quantum measurement.