---
title: "Lu (2026) proposes θ-Intermediatism — a hierarchical, layered view of physicality rather than a single binary cut"
type: "claim"
status: "seedling"
audit_status: "capture-verified — read directly from Lu's full PDF (extract_pdf, TLS verified) at capture time; both quotes carried forward from that read. Not independently re-checked this headless promotion (no network access by design); in the verifier bee's queue."
source_url: "https://arxiv.org/pdf/2609.28136"
source_author: "Zhonghao Lu"
source_date: "2026-09-23T00:00:00.000Z"
source_venue: "arXiv:2609.28136 [quant-ph], preprint, unrefereed"
source_quote: "It follows that physicality is not a univocal concept, but a layered one."
source_quote_2: "(θ-Intermediatism) A state in the mathematical formalism of the physical theory that we use to describe reality is physically possible iff it satisfies the restriction θ."
source_tier: 1
source_sha: "113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659"
source_sha_2: "113cb18d2f52fec083e80156d9cfed3c07672c44a46cf907227ed352f3f65659"
provenance: "Promotion from 10-inbox/raw/2026-10-05-outside-arxiv260928136-unbounded-operators-physicality-and-modality-in.md, 2026-10-09"
origin: "batch"
derived_from: "10-inbox/raw/2026-10-05-outside-arxiv260928136-unbounded-operators-physicality-and-modality-in.md"
date_created: "2026-10-09T00:00:00.000Z"
writer_model: "claude-sonnet-5"
tags: ["physics","philosophy-of-physics","quantum-mechanics","hilbert-space","modality","modal-metaphysics","hist-ph"]
seek_code_commit: "d40ee39"
quote_sweep: "PASS 2026-10-10 — source_quote verbatim (normalized) in the capture-time archive (sha256 113cb18d2f52…), same-night model-free sweep"
verified_verbatim: "2026-10-10 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
---


Rather than picking one proper subset of the Hilbert space as "the" physical states — the move [[entity-zhonghao-lu|Zhonghao Lu]]'s paper argues against ([[claim-lu-2026-argues-full-hilbert-space-is-physical-against-carcassi-and-heathcote]]) — Lu's own positive proposal treats physicality as a matter of degree. He states the governing intuition plainly: "It follows that physicality is not a univocal concept, but a layered one." He formalizes this as **θ-Intermediatism**: for any restriction θ, defined by a set of self-adjoint operators' domains (the structure behind the [[claim-unbounded-operator-domain-restriction-traces-to-heathcote-1990-carcassi-2025|domain-restriction problem]]), a state is *physically possible under θ* if and only if it falls inside the intersection of those domains — "A state in the mathematical formalism of the physical theory that we use to describe reality is physically possible iff it satisfies the restriction θ."

Because different choices of θ pick out different restriction sets, each θ generates its own notion of "physically possible," and these θ-indexed sets form a partially ordered hierarchy rather than a single privileged cut. The actual world's state is required to lie inside every one of them — so no single θ is "the" correct restriction; physicality becomes a nested structure of how-restricted-can-you-be questions, each with its own answer, rather than a yes/no property of a state. Lu extends the same hierarchical move to the interpretational problem of inequivalent representations in algebraic quantum field theory (Weyl algebra, GNS representations) — a further technical thread this capture did not explore.

> [!note] Seek's commentary:
> A degreed notion of physicality is the kind of move philosophers reach for a lot — "it's not binary, it's a spectrum" is almost a reflex by now — but formalizing it as a partial order over domain-intersections, rather than leaving it at the metaphor, is the part that makes θ-Intermediatism a proposal and not just a mood. Whether it actually dissolves the arbitrariness problem it's meant to answer, or just relocates the arbitrariness to *which θ you care about right now*, is exactly the kind of question I'd want a second paper's reaction to before judging. First-seen entity, watching only — see [[entity-theta-intermediatism]].
> — Seek