Hilbert space fundamentalism (HSF) claims that all physical structure — the tensor product structure (TPS) identifying subsystems, locality, and ultimately space — should emerge from the bare triple (H,H,∣ψ⟩)(\mathcal{H},\mathsf{H},|\psi\rangle) of Hilbert space, Hamiltonian, and state vector. A recent exchange — Cotler–Penington–Ranard's spectrum-based locality theorem, Stoica's no-go results, and Soulas–Franzmann–Di Biagio's invariant-based construction of a unique TPS, followed by Stoica's 2026 Comment on it — has narrowed but not settled the question of what survives of this program. The Comment leaves three linked open questions: (1) whether any construction depending only on unitary-invariant data of the triple, without hand-specified invariant values, can yield a non-unique but tightly constrained family of TPSs that supports empirical predictions while remaining compatible with the observed time-dependence of entanglement — or whether a no-go theorem extends to such weakened uniqueness; (2) the delimitation, for concrete theories including quantum field theory and quantum gravity candidates, of the physically relevant subgroup of U(H)\mathrm{U}(\mathcal{H}) relative to which relational uniqueness should be formulated; and (3) the scope of a law-oriented HSF, i.e., which lawful features are decodable from the spectrum or other invariant data alone and which are irreducibly underdetermined.

Background

A quantum theory is standardly specified by more than a state vector and a Hamiltonian: besides QT1 (a Hilbert space H\mathcal{H}, a Hamiltonian H\mathsf{H}, and a unit vector ∣ψ⟩|\psi\rangle evolving unitarily), one also postulates QT2 — a tensor product structure (TPS), i.e., a decomposition H≃H1⊗⋯⊗Hn\mathcal{H}\simeq\mathcal{H}_1\otimes\cdots\otimes\mathcal{H}_n identifying subsystems or spatial regions — and QT3, an assignment of physical meaning to operators (which are positions, momenta, or local observable algebras). Hilbert space fundamentalism (HSF) is the thesis that QT2 and QT3 need not be postulated: all physical structure should emerge uniquely from the triple (H,H,∣ψ⟩)(\mathcal{H},\mathsf{H},|\psi\rangle) alone, a position popularized as "Mad-Dog Everettianism" by Carroll and Singh. The field of quantum mereology studies which decompositions of a Hilbert space into subsystems are preferred; since a TPS is equivalently fixed by its subalgebras of observables (Zanardi, Lidar, and Lloyd), the question is what, if anything, fixes those algebras without QT3-type input.

Three results define the frontier. Cotler, Penington, and Ranard argued that the Hamiltonian's spectrum alone almost always determines the local degrees of freedom up to an equivalence notion they defined. Stoica proved no-go results in the opposite direction: (i) selecting a physically unique TPS requires fixing a number of continuous parameters that grows exponentially with the number of subsystems; and (ii) Theorem 2 of his 2022 paper: an emergent structure constructed invariantly from the triple, essentially unique (unique up to physically unobservable differences), and "physically relevant" — meaning it stands in an observable, time-dependent relation with ∣ψ(t)⟩|\psi(t)\rangle — leads to a contradiction, so invariance, uniqueness, and compatibility with observed change cannot coexist. Soulas, Franzmann, and Di Biagio responded by distinguishing absolute uniqueness (uniqueness of a labelled TPS) from relational uniqueness (uniqueness of a pair of structures up to a global unitary on H\mathcal{H}), arguing that only the relational notion is physically meaningful, and proving that (H,∣ψ⟩)(\mathsf{H},|\psi\rangle) plus a fixed list of entanglement-entropy invariants — the von Neumann entropies (the entropy S(ρ)=−tr(ρln⁡ρ)S(\rho)=-\mathrm{tr}(\rho\ln\rho) of a reduced density matrix) of the subsystem states of R(H)∣ψ⟩R(\mathsf{H})|\psi\rangle for all polynomials RR — determines a unique TPS under genericity assumptions, reviving the hope that HSF is tenable.

Stoica's 2026 Comment on that paper contests both the physical content and the dynamical compatibility of this construction. First, fixing the invariant list amounts to selecting an orbit by hand — analogous to claiming that the electron's mass "emerges" by fixing the Casimir invariants (the mass and spin labels of irreducible Poincaré-group representations) — so the construction can accommodate any desired TPS and constrains nothing. Second, under time evolution the construction faces a trilemma: either its infinitely many invariants depend on time in a strictly choreographed way, or they are fixed at one absolute moment of time (both options break the invariance the construction needs), or entanglement between subsystems can never change (contradicting observation). The Comment concludes that the construction illustrates rather than refutes the no-go results. It then asks, without answering, whether giving up uniqueness helps — whether a physically relevant structure could emerge unique only up to a weaker equivalence relation. It further argues that "relational philosophy" should not quotient by the full unitary group U(H)\mathrm{U}(\mathcal{H}): the symmetry group of quantum theory is much smaller (spacetime isometries and gauge groups), just as the Euclidean group sits strictly inside the affine group, but the correct subgroup for concrete theories is not identified. Finally, it proposes a weakened, law-oriented HSF (HSF-law) — two triples depict the same physical law if and only if they are isomorphic — noting that Cotler et al.'s result shows some law-level structure (the local form of the Hamiltonian) is decodable from the spectrum, while Stoica's clock-smearing results (infinitely many physically different worlds share the translation-generator Hamiltonian −iℏ ∂/∂τ-i\hbar\,\partial/\partial\tau) show the spectrum alone is often insufficient even for the law.

Problem Statement

Determine what, if anything, of the program of deriving preferred physical structure from the bare quantum triple survives the no-go results, in three linked parts:

(1) The central existence question. Construct, or rule out, a procedure that (i) is invariant — it depends on the triple (H,H,∣ψ⟩)(\mathcal{H},\mathsf{H},|\psi\rangle) only through unitary-invariant data, with no hand-specified invariant values such as the entropy list sR,ks_{R,k} of Soulas et al. — and (ii) yields a non-unique but tightly constrained family of TPSs, narrow enough to support empirical predictions, retrodictions, or explanations and varying with H\mathsf{H} and ∣ψ⟩|\psi\rangle, while (iii) remaining compatible with the empirical fact that subsystem entanglement changes in time. A negative answer would be a no-go theorem extending Theorem 2 of Stoica (2022) from essentially unique emergent structures to such weakened-uniqueness families; a positive answer would be an explicit construction together with a demonstration that it excludes TPSs and predicts or explains something not fixed by hand.

(2) The symmetry-group delimitation that makes "relational uniqueness" well-defined. Identify, for explicitly named concrete theories, including at least one interacting quantum field theory and at least one quantum-gravity candidate, the physically relevant subgroup of U(H)\mathrm{U}(\mathcal{H}) relative to which uniqueness should be formulated (as the Euclidean group sits inside the affine group), with a justification tied to the theory's empirical symmetries, and reformulate the uniqueness question relative to that subgroup.

(3) The scope of law-oriented HSF. Characterize which lawful features — interaction structure, locality, causal order, the number and dimensions of subsystems — can be decoded from the spectrum of H\mathsf{H} or other invariant data of the triple alone, and which are irreducibly underdetermined (as the clock-smearing degeneracies suggest); in particular, classify each of these four features as decodable or as underdetermined.

Parts (2) and (3) sharpen part (1) — part (2) fixes the equivalence relation relative to which uniqueness is weakened, and part (3) fixes what remains recoverable if no ontic structure can — but the three parts are independently answerable and each is a mandatory deliverable: a positive construction or a no-go theorem settling part (1) does not by itself settle parts (2) and (3), and a result settling (2) or (3) does not by itself settle part (1). A complete answer addresses all three under the single objective of deciding the viability of a weakened, relational Hilbert-space fundamentalism.

References

  1. Ovidiu Cristinel Stoica (2026). Comment on "On the emergence of preferred structures in quantum theory" by Soulas, Franzmann, and Di Biagio. arXiv:2603.07674, DOI 10.48550/arXiv.2603.07674. https://arxiv.org/abs/2603.07674
  2. Antoine Soulas, Guilherme Franzmann, Andrea Di Biagio (2025). On the emergence of preferred structures in quantum theory. arXiv:2512.07468, DOI 10.48550/arXiv.2512.07468. https://arxiv.org/abs/2512.07468
  3. Ovidiu Cristinel Stoica (2022). 3D-Space and the preferred basis cannot uniquely emerge from the quantum structure. Advances in Theoretical and Mathematical Physics 26(10), 3895-3962. DOI 10.4310/ATMP.2022.v26.n10.a12, arXiv:2102.08620. https://arxiv.org/abs/2102.08620
  4. Ovidiu Cristinel Stoica (2024). Does the Hamiltonian determine the tensor product structure and the 3d space? arXiv:2401.01793, DOI 10.48550/arXiv.2401.01793. https://arxiv.org/abs/2401.01793
  5. Jordan S. Cotler, Geoffrey R. Penington, Daniel H. Ranard (2019). Locality from the Spectrum. Communications in Mathematical Physics 368(3), 1267–1296. DOI 10.1007/s00220-019-03376-w, arXiv:1702.06142. https://arxiv.org/abs/1702.06142
  6. Sean M. Carroll, Ashmeet Singh (2019). Mad-Dog Everettianism: Quantum Mechanics at Its Most Minimal. In: What is Fundamental? (Frontiers Collection, Springer). DOI 10.1007/978-3-030-11301-8_10, arXiv:1801.08132. https://arxiv.org/abs/1801.08132
  7. Sean M. Carroll (2022). Reality as a Vector in Hilbert Space. In: Quantum Mechanics and Fundamentality (Synthese Library, Springer). DOI 10.1007/978-3-030-99642-0_15, arXiv:2103.09780. https://arxiv.org/abs/2103.09780
  8. Paolo Zanardi, Daniel A. Lidar, Seth Lloyd (2004). Quantum tensor product structures are observable induced. Physical Review Letters 92(6), 060402. DOI 10.1103/PhysRevLett.92.060402, arXiv:quant-ph/0308043. https://arxiv.org/abs/quant-ph/0308043
  9. Paolo Zanardi, Emanuel Dallas, Faidon Andreadakis, Seth Lloyd (2024). Operational Quantum Mereology and Minimal Scrambling. Quantum 8, 1406. DOI 10.22331/q-2024-07-11-1406, arXiv:2212.14340. https://arxiv.org/abs/2212.14340
  10. Nicolas Loizeau, Dries Sels (2025). Quantum mereology and subsystems from the spectrum. Foundations of Physics 55(1), 3. DOI 10.1007/s10701-024-00813-2, arXiv:2409.01391. https://arxiv.org/abs/2409.01391
  11. Arsalan Adil, Manuel S. Rudolph, Andrew Arrasmith, Zoë Holmes, Andreas Albrecht, Andrew S. Sornborger (2026). Search for classical subsystems in quantum worlds. Physical Review D 113(10), 103535. DOI 10.1103/vj4x-96fk, arXiv:2403.10895. https://arxiv.org/abs/2403.10895
  12. Antoine Soulas (2025). Disentangling tensor product structures. International Journal of Quantum Information 23(7), 2550027. DOI 10.1142/S0219749925500273, arXiv:2506.21173. https://arxiv.org/abs/2506.21173
  13. Varun Kushwaha, Nicolas Loizeau, Alexei Grinbaum, Oliver Friedrich (2026). Causality from the spectrum: Emergence of causal order from process-matrix mereology. arXiv:2607.15345, DOI 10.48550/arXiv.2607.15345. https://arxiv.org/abs/2607.15345
  14. Ye-Ming Meng, Zhe-Yu Shi (2026). O-Sensing: Operator Sensing for Interaction Geometry and Symmetries. arXiv:2603.03826, DOI 10.48550/arXiv.2603.03826. https://arxiv.org/abs/2603.03826
  15. Ovidiu Cristinel Stoica (2026). No change in Hilbert space fundamentalism. arXiv:2602.20331, DOI 10.48550/arXiv.2602.20331. https://arxiv.org/abs/2602.20331
  16. Ovidiu Cristinel Stoica (2022). The Problem of Irreversible Change in Quantum Mechanics. arXiv:2204.02270, DOI 10.48550/arXiv.2204.02270. https://arxiv.org/abs/2204.02270
  17. Ovidiu Cristinel Stoica (2024). Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations. Physica Scripta 99(8), 085102. DOI 10.1088/1402-4896/ad59d2, arXiv:2204.01426. https://arxiv.org/abs/2204.01426
  18. Ovidiu Cristinel Stoica (2026). The preferred-time problem in the conditional interpretation of time. arXiv:2608.16732, DOI 10.48550/arXiv.2608.16732. https://arxiv.org/abs/2608.16732
  19. Augustin Vanrietvelde, Octave Mestoudjian, Pablo Arrighi (2025). Partitions in quantum theory. arXiv:2506.22218, DOI 10.48550/arXiv.2506.22218. https://arxiv.org/abs/2506.22218
  20. Philipp A. Hoehn, Isha Kotecha, Fabio M. Mele (2023). Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics. arXiv:2308.09131, DOI 10.48550/arXiv.2308.09131. https://arxiv.org/abs/2308.09131