Locality and Symmetry as Primary Coordinates: Finer Structure Behind Scalar Labels in Recent Many-Body Results

Abstract

Version 3 (2026-09-26). Version 3 corrects version 2 and narrows its thesis and title, following an independent audit and a second review of the corrected draft. (1) Version 2 relied in part on arXiv:2605.30238 (Surace, Minagawa and Kunjwal) for a reversal of the real-complex hierarchy under indefinite causal order. On 2026-06-05 its authors posted that the claimed separation between real and complex quantum theory "is therefore not established" and that they are revising the affected claims. The "even reality" strand built on it has been removed, and a notice is included. (2) The locality-dimension claim was inverted. Lu, Fu and Liu (arXiv:2605.31441) define the dimension intrinsically from the code, independent of background geometry, so it is not an externally imposed metric, and the text now says so. (3) A statement that one state can carry volume-law entanglement and be prepared by a constant-depth circuit was wrong and has been replaced (arXiv:2605.31448). A garbled description of pseudoentanglement has been rewritten. (4) The local SW-SSB diagnostic of arXiv:2605.28967 is detected, with poly(N) resources, up to a volume scale of O(log N). Version 2 wrongly said it needs only O(log N) resources. (5) Smaller overstatements have been corrected: the frustration-graph wording, "independent confirmation", a claim about universality classes that is not in the source, the ECF condition, the relation between entropies in arXiv:2605.30798, and a "fault-tolerant" qualifier. With these corrections, version 2's thesis that dimensionality, universality and "even reality" are conditional on a choice of locality metric no longer holds. Version 3 states a narrower thesis: several scalar labels of many-body physics are refined by finer locality structure and symmetry resolution. The title changes accordingly, from "Locality and Symmetry as Primary Coordinates: A Structural Reorganization of Quantum Many-Body Hilbert Space" to "Locality and Symmetry as Primary Coordinates: Finer Structure Behind Scalar Labels in Recent Many-Body Results". The file has a neutral name, and the full list of corrections is at the top of the PDF. Version 2 was revised in response to an external structural review and an automated critique pass; its change log is kept as the "Response to Review" appendix in the PDF. A recurring pattern in the recent quant-ph, math-ph, math.QA, and cond-mat.stat-mech preprints, read over a thirty-day window ending 2026-06-02, suggests that several quantities long treated as primary scalar coordinates of many-body Hilbert space are coarse summaries of finer structure. Entanglement magnitude, the existence of a free-fermion language, the classification of quantum phase transitions, the dimension that bounds a quantum code, and eigenstate entropy each appear, in independent recent work, to refine into finer distinctions once one asks two prior questions: what is the locality structure? and which symmetry resolves the count? We collect six specific findings: Gheorghiu's constant-depth pseudoentanglement separation arXiv:2605.31448v1, Lu, Fu and Liu's intrinsic-locality dimension for stabilizer codes arXiv:2605.31441v1, Jindal and Hosur's non-Abelian-ETH entropy correction arXiv:2605.30798v1, Fukai, Pozsgay and Vona's path-product expansion for hidden free fermions arXiv:2605.31453v1, Sinha and collaborators' hidden Ising structure from a generalized Yang-Baxter equation arXiv:2605.30007v1, and Balducci and collaborators' traversable-versus-nontraversable quantum phase transitions arXiv:2605.31472v1. We also use Divi, Lessa and Wang's local strong-to-weak SSB diagnostic arXiv:2605.28967v1 as an internal cross-check. (Version 2 also used a result on the real-complex hierarchy under indefinite causal order that its authors now state is not established; see the Version 3 note.) The thesis we synthesize is a heuristic reading, not a derivation from a shared formal structure: several scalar labels of a quantum many-body system (a code's dimension, the entanglement entropy across a cut, eigenstate entropy, free-fermion solvability, the type of a phase transition) are refined by finer structural data, namely a locality structure (an intrinsic dimension, a frustration graph, a shallow-circuit geometry, a local diagnostic) and a symmetry resolution, and the scalar alone does not determine the structural conclusion. The six results sit in different formalisms; the pattern we identify across them is interpretive. The falsification path: each refinement we cite is operationally testable (a constructed example, a closed-form bound, or a finite-data diagnostic), and the thesis fails if the underlying mechanisms are retracted, or if a counter-example emerges in which the finer locality or symmetry data do not change the structural conclusion. The contribution here is the synthesis, not the underlying results. Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted synthesis from an arXiv preprint corpus, originally drafted 2026-06-02, produced under the direction of Cristian Ruvalcaba, the accountable human author. Not peer-reviewed. Cited arXiv preprints: arXiv:2605.28967v1, arXiv:2605.30007v1, arXiv:2605.30798v1, arXiv:2605.31441v1, arXiv:2605.31448v1, arXiv:2605.31453v1, arXiv:2605.31472v1, arXiv:2605.30238v1 (cited in version 2 only; its authors now state the result used is not established, and it is removed in version 3) AI disclosure. This work was produced with an agentic AI research apparatus operated by Saluca Labs. The apparatus drafted, searched and analysed under direction. Cristian Ruvalcaba is the human author and is accountable for the content. No AI system is listed as an author or contributor, because authorship entails accountability that a model cannot hold; this disclosure is the credit, and it is deliberately the whole of it.