NOTE-746: Marconato et al. — All or None: identifiable linear properties of next-token predictors

Contribution

The paper characterizes exactly which next-token predictors of the softmax form induce the same conditional distribution, without the diversity condition and without equal representation dimensions that earlier linear identifiability results (Roeder et al. 2021; Khemakhem et al. 2020) needed. It then defines relational versions of linear properties of language models (relational linearity, linear subspaces, linear probing, linear steering) in terms of the quantities that the characterization fixes. It proves that, under a containment condition, relational linearity holds in every distribution-equivalent model or in none. Parallelism of difference vectors is preserved only inside one subspace. The paper is theory only.

Key insight

Only the part of each representation that enters the logit dot product is fixed by the distribution. Project the embeddings onto M and the unembeddings onto N, and any two equivalent models are related there by an invertible linear map. Off those subspaces they can differ by any distortion at all. A linear property is therefore shared by all equivalent models exactly when it lives in those subspaces. A property that uses the rest of the representation is a fact about one model, not about the distribution.

Assumptions

• Model form. p_{f,g}(x_{t+1} | x_{1:t}) = exp(f(x_{1:t})ᵀ g(x_{t+1})) / Z(x_{1:t}), with f : Seq(A) → ℝᵈ and g : A → ℝᵈ nonparametric (Eq. 1). Appendix A shows a decoder-only transformer’s last-token state with its unembedding and bias has this form.
• Exact equality of distributions. Every result is about p_{f,g} = p_{f̃,g̃} on all of Seq(A) and every next token, not approximate equality.
• A shared vocabulary A. Different vocabularies are left as a hypothesis (§3.4).
• Trained models (§3.4). The step to trained models assumes both reach the global maximum of the likelihood and both can represent the data distribution p_D exactly. The authors note that this fails in practice. The softmax gives positive mass to every sequence, while p_D gives zero to many, so the model class is “inherently misspecified” there. Optimisation issues such as local minima are set aside (footnote 2).
• For Theorem 14: the property is stated on a subspace Γ ⊆ N, and its image Γ_q := Im(A_qᵀ P_Γ) lies in M.

Key results

Notation. F := SIm(f) and G := SIm(g₀) are the spans of the embeddings and the pivoted unembeddings, where g₀(y) := g(y) − g(y₀) for a fixed pivot token y₀. M := Im(P_F P_G) and N := ker(P_F P_G)^⊥ (Eq. 7). dim(M) is called the model’s effective complexity.

• Lemma 2. dim(M) = dim(N) = dim(F) − dim(F ∩ G^⊥). M ⊆ F and N ⊆ G. f(x)ᵀ g₀(y) = (P_M f(x))ᵀ P_N g₀(y) (Eq. 8). So components outside M and N do not contribute to any logit difference.
• Definition 3 (extended linear equivalence, ∼EL). (f, g) ∼EL (f̃, g̃) when dim(M) = dim(M̃) and there are full-rank M, N ∈ ℝ^{d×d̃}, invertible from M̃ to M and from Ñ to N, with Mᵀ N = P_M̃ P_Ñ and P_M f(x) = M P_M̃ f̃(x), P_N g₀(y) = N P_Ñ g̃₀(y) for all x and y (Eqs. 9–10). Appendix B.3 proves it is an equivalence relation.
• Theorem 5 (identifiability). For all (f, g), (f̃, g̃) ∈ Θ, of any dimensions d and d̃: p_{f,g} = p_{f̃,g̃} ⇔ (f, g) ∼EL (f̃, g̃). Hence p_{f,g} = p_{f̃,g̃} ⇒ dim(M) = dim(M̃) (Eq. 13), so effective complexity is a property of the distribution. The proof (B.6) pivots out the normaliser, solves for M with pseudo-inverses on spanning sets of tokens and sequences, and bounds ranks to show rank(M) = rank(N) = dim(M).
• Corollary 6 (Roeder et al., adapted). If d = d̃ and (f, g) satisfies diversity, SIm(f) = SIm(g₀) = ℝᵈ, then f(x) = M f̃(x) and g₀(y) = M^{−ᵀ} g̃₀(y) for an invertible M (Eqs. 14–15). The proof relaxes Roeder et al.’s condition SIm(f₀) = ℝᵈ on pivoted embeddings to SIm(f) = ℝᵈ, and needs it of one model only.
• Counterexample without diversity (B.5). With g(y₀) = (1, 0), g(y₁) = (1, 1), g(y₂) = (1, −1), so that G = span(e₂), the embedding f̃(x) = (f₁(x) + 0.2 cos(40 a₁/π), f₂(x)) gives the same distribution and is not a linear transform of f.
• Lemma 8 (parallelism is a distributional statement). g(y₁) − g(y₀) and g(y₃) − g(y₂) are parallel in N, that is P_N γ = β P_N γ′ with β ≠ 0 (Definition 7), if and only if log p(y₀ | s)/p(y₁ | s) = β · log p(y₂ | s)/p(y₃ | s) for every s ∈ Seq(A) (Eq. 16).
• Definition 9 (relational linearity of a query q on Γ ⊆ SIm(g₀)). There are A_q and a_q with P_Γ f(s ⌢ q) = P_Γ A_q f(s) + a_q for every context s (Eq. 17). Propositions 11 and 13 derive the relational linear subspace (Definition 10) and relational linear probing (Definition 12, softmax(W f(s) + b)ᵢ = p(yᵢ | s ⌢ q; Y_P)) from it. Proposition 19 (C.1) gives relational linear steering under a non-nesting condition on the Γ_q of several queries.
• Theorem 14 (all or none). If (f, g) ∼EL (f̃, g̃), f linearly represents q on Γ ⊆ N and Γ_q ⊆ M, then f̃ linearly represents q on Γ̃ ⊆ Ñ, with Γ̃ = Im(N⁺ P_Γ). By Propositions 11 and 13 the same holds for linear subspaces and linear probing. Under the diversity condition M = N = ℝᵈ, so the containment holds automatically.
• When Γ_q ⊄ M (D.1). The components of f in F ∖ M “can be arbitrarily chosen”, for example as squares of f̃’s coordinates. An equivalent model then has a non-linear dependence of f̃(s ⌢ q) on f̃(s) whenever P_{Γ_q}(P_F − P_M) ≠ 0. So without the condition the property can hold in one model and fail in an equivalent one.
• Theorem 15 (parallelism). For ∼EL-equivalent models, γ and γ′ are parallel within N if and only if γ̃ and γ̃′ are parallel in Ñ. Parallelism in the whole space is not preserved.
• Appendix E. A paraphrase q₂ of q₁ (proportional log-odds over paired replies, Definition 21) forces P_{Γ₁} f(s ⌢ q₁) = β O P_{Γ₂} f(s ⌢ q₂) for an invertible O (Proposition 22), under Γᵢ ⊆ SIm(f)_{qᵢ}. A tautology, a query whose replies ignore the context, is trivially relationally linear: P_G f(s ⌢ q) = P_G a_q (Proposition 24).

Claims

id	claim	strength	support
C1	Softmax next-token predictors induce the same distribution iff they are ∼EL-equivalent, at any dimensions and without diversity	strong	Theorem 5, proof in B.6
C2	Effective complexity dim(M) is fixed by the distribution	strong	Eq. 13, from Theorem 5
C3	Without diversity, equivalent models need not be linearly related	strong	counterexample, B.5
C4	Parallelism in N is equivalent to proportional log-odds over all contexts, and only parallelism in N is shared	strong	Lemma 8; Theorem 15
C5	Relational linearity, linear subspaces and linear probing hold in all or none of the equivalent models when Γ ⊆ N and Γ_q ⊆ M	strong	Theorem 14; Propositions 11, 13
C6	Without Γ_q ⊆ M the property can fail in an equivalent model	moderate	construction in D.1, shown up to proportionality, not as a worked model
C7	The parallelism that PCA shows in trained models is not explained by identifiability, so training biases select within the equivalence class	weak	informal argument, §6, no measurement

Concepts

• Distribution-equivalent models. Two predictors with p_{f,g} = p_{f̃,g̃} on all sequences and tokens.
• Effective complexity. dim(M) = dim(N), the dimension of the part of the representation that enters the logits. It is at most d.
• Diversity condition. SIm(f) = SIm(g₀) = ℝᵈ: one model’s embeddings and pivoted unembeddings span the whole space (Definition 1).
• Context–query–reply string. x = s ⌢ q ⌢ y, with s the context (“subject”), q the query (“relation”) and y one reply token (“object”), for example s = “Rome”, q = “is the capital of”, y = “Italy”.
• Relational linearity. As in Definition 9 above. The information in f(s ⌢ q) that bears on replies in Γ is an affine function of f(s). It refines Paccanaro and Hinton’s linear relational embedding and Hernandez et al.’s relation decoding, and uses only the quantities that Theorem 5 fixes.

Connections

The identifiability result generalizes Roeder et al. (2021) and Khemakhem et al. (2020), whose pivoting argument it follows, and Lachapelle et al. (2023) for the corollary. Its taxonomy of linear properties is Park, Choe and Veitch’s (LIT-304), recast relationally. Park et al. introduce latent binary concepts, while this paper is “agnostic to assumptions on the data-generating process” and posits no latent variables. Jiang et al. (2024) explain linearity from latent variables, which this paper declines to do. The authors’ later work with Nielsen and Dittadi asks when approximate equality of distributions forces representational similarity (LIT-253, held here unread). Picozzi (LIT-936) cites this paper as the theoretical characterisation of identifiable properties of next-token predictors. §6 compares the shared-property hypothesis to the principle of covariance in physics, which says laws should be coordinate-free.

Bearing on the record

• QUESTION-027. The paper is a clean, proved instance of a signature identified when the full latent model is not. Theorem 5 is the characterisation of the observational equivalence classes, and Theorem 14 names properties that are constant on them. But it is the case that QUESTION-027’s “What is already known” calls answered. The observation is the whole conditional distribution on every sequence in Seq(A), so there are no unobserved probes. Lemma 8 makes this explicit: parallelism in N is equivalent to proportional log-odds over all contexts, a property of the observed law itself. Nothing in the paper concerns decisions outside the observed probes, interventions or transport, where QUESTION-027 puts its content. LIT-950’s status note says the paper “proves which linear properties hold in every next-token predictor that induces the same distribution”. It does so only under Γ ⊆ N and Γ_q ⊆ M, and it gives sufficient conditions, not a characterisation of all shared properties. The LIT body’s “That is the question’s pattern” holds only for the unrestricted form.
• CLAIM-148. The paper refines the gauge that CLAIM-148 describes. For a bilinear softmax model, the gauge is an invertible linear map only under diversity and equal dimensions (Corollary 6). Without diversity it is an invertible linear map on M and N, plus arbitrary, even non-linear, distortion off them (B.5, D.1). CLAIM-148’s point stands, and gains a case: a property of one fitted model that depends on components outside M is not a property of the data.
• CLAIM-209. The paper supports only the unrestricted second sentence, which CLAIM-209’s “Where its content is” already grants. It gives nothing for the restricted clause.
• CLAIM-193 and CLAIM-142. Relational linear probing (Definition 12) is shared only for replies whose difference vectors lie in N. That is a concrete case of “what the probes cannot see they cannot separate”: a probe trained on one model reads structure that an equivalent model need not carry, unless the structure lies in N.
• THEORY-175 and the cross-model agreement it records. §6 bears on how to read agreement between trained models. The parallelism that PCA displays is not shared across the equivalence class. So where trained models agree on it, the agreement comes from inductive biases of training or architecture, not from the distribution. Agreement between independently trained models is therefore not by itself evidence that a structure is identified.
• Curation of 2026-10-14. The triage table calls this work “the identifiability half of the relational-signature question”. That overstates it. It is the identifiability half of the unrestricted form only.
• No instruction for machine-learning practice. An anthology topic on representation learning could hold it, as the anthology-candidate flag says. It is held here for QUESTION-027 under ADR-013.

Limitations

• There are no experiments, and no check that any trained model satisfies Γ ⊆ N or Γ_q ⊆ M. Whether the relational properties reported by Hernandez et al. and others fall under Theorem 14 is not tested.
• Exact distributional equality only. The authors grant that trained models are misspecified and are not global maximisers, so the bridge to real models (§3.4) is conditional. Approximate equivalence is left to later work.
• One vocabulary only.
• Footnote 5 says that if Γ ⊄ N, relational linearity “would be trivially satisfied” because γᵀ f(x) = 0. The reasoning fits Γ ⊆ N^⊥, not every Γ ⊄ N, so the footnote is loosely stated. Theorem 14 is not affected.
• The D.1 construction is shown only up to proportionality (”∝”), not as a worked pair of models.
• §6’s explanation of PCA-visible parallelism by inductive bias is a hypothesis. The authors offer it as a framework for future empirical work.

Open questions

• Do trained language models satisfy Γ ⊆ N and Γ_q ⊆ M for the relations in which linearity has been reported? A measurement of M, N and Γ_q in released models would settle it.
• What survives under approximate equivalence, for models with small but non-zero divergence (the question LIT-253 takes up)?
• Which selection within the ∼EL class does training make, and is the selected subset the same across architectures? A result there would turn §6’s hypothesis into a claim.
• For QUESTION-027: are there properties that are not functions of the observed law, yet are shared by every model that a stated training procedure produces? Those would be signatures identified by the procedure rather than by the data.