Paper abstract
Abstract
ChronaQ proposes that observable relational structure constrains geometry without assuming that a unique spacetime metric is primitive. This paper develops that proposal for finite systems. The input is a rendered observable patch: a finite collection of operational objects together with positive distributions, correlations, graph relations, or response data licensed by the ChronaQ axioms. Geometry is introduced through declared constructors and explicit admissibility gates. The resulting CQ2 object is generally a family of compatible geometries, not a single distance matrix. A direct counterexample shows why this distinction is necessary. The literal reciprocal-mutual-information score fails the zero-diagonal metric axiom, and an explicitly diagonal-corrected version still violates the triangle inequality by an arbitrarily large amount and becomes infinite at independence. On one heterogeneous four-object patch, graph shortest-path distance, Hellinger distance between positive observable distributions, and an L1 polyhedral/Finsler chart all satisfy the finite metric axioms while remaining inequivalent. A prospective same-data benchmark then begins with one frozen finite quantum-walk event-count table. Graph shortest-path, Hellinger early-response, and spectral-resistance constructors all produce valid metrics from its unopened early layers under one common 256-replicate uncertainty law. Candidate-specific scales are fitted on common training rows and frozen. On untouched longer-range rows, the graph renderer has a held-out mean absolute error of 0.474 record layer, compared with 2.340 and 2.164 layers for the Hellinger and spectral alternatives. Its uncertainty-bounded advantage clears the preregistered materiality gate against both. The benchmark therefore shows that later operational data can narrow a compatible geometry family without refitting. A separate nearest-neighbour influence assumption gives an exact polyhedral support bound and an earliest-permitted diamond front. Equality with actual first arrival requires an additional nonzero propagation witness and cancellation control. Continuous spectral propagation instead has nonzero tails and supplies an effective causal band rather than a sharp cone. These results support a disciplined form of entanglement-geometry compatibility: observable data constrain and can discriminate admissible representations, while unique physical geometry, Lorentzian signature, physical scale, an entropy-area law, and gravitational dynamics require additional gates.
Keywords
- entanglement geometry
- mutual information
- Hellinger distance
- graph metric
- Finsler geometry
- causal structure
- quantum information
- ChronaQ
Scholarly record
Preferred citation
McIntyre, Stephen J. (2026). ChronaQ Foundations II: Compatible Entanglement Geometries and Finite Causal Structure. Working paper, version 0.4. McIntyre Lab. https://mcintyrelab.org/publications/foundations-ii/
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Licence
© 2026 Stephen J. McIntyre. Licensed under CC BY 4.0.
Working paper, version 0.4, 3 August 2026. Not peer reviewed.
This licence applies to the text and original figures unless otherwise identified. Software, datasets and third-party material are not included unless explicitly stated. Patent and trademark rights are not granted by this licence.
Open research
Comments and technical feedback
Critical reading, corrections and specific technical questions are welcome while this working paper is being revised. Substantive contributions will be recognised clearly and proportionately.