Source
- Carroll (2021), LIT-123, pp. 4–9, as read in NOTE-134.
- Van Rijsbergen (2004), LIT-262: Prologue p. 6, ch. 1 p. 19, ch. 4 p. 57, ch. 6, as read in NOTE-239.
- Coecke, Sadrzadeh & Clark (2010), LIT-273, §§3–5, as read in NOTE-245.
- Kartsaklis, Sadrzadeh, Pulman & Coecke (2014), LIT-272, §§2–4 and 6, as read in NOTE-246.
- For learned representations: THEORY-004.
What was actually shown
The mathematical core is elementary. A finite-dimensional Hilbert space with no further structure is determined by its dimension. The unitary group acts transitively on its orthonormal bases, so no basis is distinguished, and neither is any factorisation H ≅ H_A ⊗ H_B of given dimensions. Only quantities invariant under that group can be read off the space alone: inner products, traces, and the spectra of operators placed on it. A non-degenerate self-adjoint operator does single out a basis, its eigenbasis, up to phases. Its unitarily invariant content, though, is only its eigenvalue list. Coecke, Pavlovic & Vicary (LIT-274, read in NOTE-247) make the basis side exact, for complex spaces. On a finite-dimensional complex Hilbert space, the commutative †-Frobenius algebras are in bijection with the orthogonal bases, the basis being the vectors the comultiplication copies (Thm 5.1). The special ones correspond exactly to the orthonormal bases, with the vectors fixed exactly, phases included (§6). The paper calls this “classical data”; the later categorical literature calls these algebras “classical structures”.
Each source takes one position on where the extra structure comes from.
| Work | What it treats as intrinsic | Where the basis or factorisation comes from |
|---|---|---|
| Carroll, LIT-123 | The spectrum {E_n}: “all the information in the Hamiltonian” (pp. 4–5). Changes of basis have “no physical importance whatsoever” (p. 4). | The dynamics. Among factorisations, the Hamiltonian’s form picks out a local one (Cotler et al., unique when it exists) or a quasi-classical one (Carroll & Singh, numerical examples only) (pp. 7–9). |
| Van Rijsbergen, LIT-262 | Traces and inner products. Every quantity NOTE-239 records the book computing is a trace or inner product, and so invariant under a joint unitary change of basis (the reader’s survey, not the book’s statement). | The observable. Each observable’s eigenbasis is “a particular perspective”, and a change of basis “constitutes a change of point of view” (Prologue p. 6; ch. 1 p. 19). |
| Coecke, Sadrzadeh & Clark, LIT-273 | The ε/η composition. Orthogonal changes of the noun and sentence spaces carry the sentence vector covariantly (checked numerically in NOTE-245). The paper does not state this; proved in the note of 2026-10-29. | Only the self-duality V* ≅ V, which the paper grounds in “a fixed base” (p. 13). The meaning computed does not depend on which base. |
| Kartsaklis et al., LIT-272 | Nothing basis-free beyond the functor’s object map. | A stipulation. W is the space of the 2,000 most frequent context lemmas with that basis fixed, and the Frobenius copying map is the one this basis induces. The paper notes that W* ≅ W “is not natural” (§2). A sentence is a Hadamard product in this basis, and one random rotation moved a sentence–subject cosine from 0.008 to −0.53 (NOTE-246). That the Hadamard composition depends on the basis is proved in the note of 2026-10-29, not stated in the paper. |
| THEORY-004 (representations) | The kernel (Gram matrix). A representation is fixed by it only up to an orthogonal transformation. | Nowhere. That is why the comparison measures (CKA, CCA, Procrustes) are the invariant ones (THEORY-008). |
So the same fork appears in physics, retrieval and compositional semantics. A model either computes only invariants, or it takes on a basis or factorisation from somewhere: a Hamiltonian, an observable, a corpus’s vocabulary. What it can say about “parts” (subsystems, terms, features, meaning components) depends on that choice.
Two consequences follow from the record’s other threads. These are this document’s inferences, not claims in the sources.
- Contextuality needs more than one basis. A single orthonormal basis, equivalently (on a complex space) a single commutative special †-Frobenius algebra, supports one global probability distribution over its outcomes. Kochen–Specker contextuality (THEORY-012) is the failure of the distributions from several incompatible bases to glue into one. Van Rijsbergen’s “contexts are bases” (NOTE-239) is the same statement from the retrieval side. A model with one stipulated basis, such as LIT-272’s, cannot be contextual in this sense, whatever its categorical dress.
- Boolean structure needs a basis. The projections diagonal in one orthonormal basis form a Boolean algebra. The full subspace lattice, which is basis-free and defined by orthogonality alone, is orthomodular and not distributive (NOTE-240, “two polarities of one inner product”). Distributive, set-like concept structure is therefore evidence that a basis has been chosen.
What this does not say
- That basis-dependent models are wrong. On LIT-272’s disambiguation task, the basis-dependent models lead: copy-object at ρ = 0.172, and the elementwise Kron and Multp at 0.168 and 0.163. The additive baseline, whose cosines are rotation-invariant, is at 0.050 (NOTE-246). The claim here is about what such a model has assumed, not about how it performs. A corpus’s context words may be exactly the right basis for word meaning. If so, that is an empirical fact about the corpus, not a consequence of the formalism.
- That the extra structure is arbitrary. Carroll’s whole programme is that dynamics selects the factorisation, and that the selected one is “real, though not fundamental” (LIT-123, pp. 7, 11–12). This document records where the structure comes from. It takes no side on whether it is thereby real.
- That the basis–Frobenius correspondence holds on real spaces. LIT-274’s theorem is for complex Hilbert spaces, and it fails over the reals. On ℝ², the complex numbers, with multiplication scaled by 1/√2, form a commutative special †-Frobenius algebra with no nonzero copyable vector. The reader of NOTE-247 found this, and I checked it numerically when filing. LIT-273, LIT-272 and THEORY-004 all work over the reals. There a basis still induces a Frobenius algebra, which is all LIT-272 uses, but a Frobenius algebra need not come from a basis. So the table’s placements stand, and “basis” and “Frobenius algebra” are one notion only in the complex case.
- Anything about continuous observables, or about infinite-dimensional
spaces beyond discrete bases. Abramsky & Heunen (LIT-275, read in
NOTE-248) show that a unital Frobenius algebra exists in Hilb only in
finite dimension (Lemma 3). Dropping the unit, they prove that on a
complex Hilbert space of any dimension, separable or not, a commutative
special †-Frobenius algebra is induced by an orthonormal basis exactly
when it satisfies Ambrose’s H*-axiom, equivalently when it is semisimple
(Thm 22). So a discrete basis is still extra structure supplied as an
algebra on the space, in any dimension. Whether the H*-axiom is automatic
was left open there (Prop. 23).
- Continuous observables have no eigenbasis and are outside that result (its §6). There, Carroll notes, an algebra of observables must be supplied from the start (Haag, LIT-123 p. 5). That is the setting of the GNS strand (LIT-241, THEORY-004).
- Scope of this document. The table stays finite-dimensional, as its sources are. The real-scalar caveat above holds in every dimension: over ℝ the H*-axiom does not force copyable vectors, since ℂ regarded as a real algebra satisfies it and has none (NOTE-248).
- That Carroll’s selection criteria work. The local-factorisation uniqueness is a cited theorem. The quasi-classical criterion rests on numerical examples in a companion paper, and emergent fields on a hope (NOTE-134).
- That a tensor factorisation is just a basis. A factorisation is coarser than a basis, since many bases respect it, and it carries more structure than a list of basis vectors. It is also what “parts” means in Carroll’s mereology. The table puts them in one column because both are structure beyond the space, not because they are the same structure.
Note of 2026-10-19: the core filed apart
The mathematics in “The mathematical core is elementary” and in the real-scalar caveat is now THEORY-721, Active, with the counterexample written out. What keeps this THEORY Proposed is its synthesis: whether, in each placement in the table, the invariance or dependence was measured or proved in the source rather than inferred by the reader. LIT-273’s covariance and LIT-272’s rotation sensitivity are at present the readers’ numerical checks.
Note of 2026-10-29: the two placements proved
Made by the tenth status review under ADR-047’s ruling on re-derivations, for the two cells the promote_when named.
LIT-273’s ε/η composition is orthogonally covariant. A transitive sentence is s = (ε ⊗ 1_S ⊗ ε)(subj ⊗ ψ ⊗ obj), so in components s_j = Σ ψ_ijk subj_i obj_k, with ψ in N ⊗ S ⊗ N. Change the noun space by an orthogonal O_N and the sentence space by an orthogonal O_S, applied consistently: subj’ = O_N subj, obj’ = O_N obj and ψ’ = (O_N ⊗ O_S ⊗ O_N)ψ. Then s’_j = Σ (O_N)_ia (O_S)_jb (O_N)_kc ψ_abc (O_N)_ia’ subj_a’ (O_N)_kc’ obj_c’. Summing over i and k uses (O_Nᵀ O_N)_aa’ = δ_aa’ and likewise for c, which is ε’s invariance under orthogonal maps, so s’ = O_S s. Cosines between sentence vectors are therefore unchanged. Checked numerically on random tensors; with a non-orthogonal change of the noun space in place of O_N, as a positive control, the covariance fails.
LIT-272’s Frobenius composition depends on the basis. The copying map induced by a basis gives μ(e_i ⊗ e_j) = δ_ij e_i, so μ(x ⊗ y) is the elementwise product x ⊙ y in that basis. On ℝ², take x = (1, 0) and y = (1, 1)/√2. Then x ⊙ y = (1/√2, 0), and its cosine with x is 1. Rotate the basis by 45°: Ox = (1, 1)/√2 and Oy = (0, 1), so Ox ⊙ Oy = (0, 1/√2), whose cosine with Ox is 1/√2. A sentence–subject cosine moves under a change of basis, as NOTE-246 found in one random rotation.
Neither paper states these. They are facts about the constructions the papers use, proved here, and the table now says so.