STATUS: IN PROCESS — PUBLIC DRAFT, NOT EXECUTED
A developing adjudication protocol for Park, Choe & Veitch (ICML 2024) against the twelve fractures. The respondent and claim structure are fixed. The experimental design and thresholds are published for inspection while the protocol remains in process. Open for criticism until the freeze date of 31 September 2026. Nothing here is certified. The experiments have not been run, and this draft may be corrected before it is formally frozen.
This is a protocol, not a certificate. It states what is proposed to be measured, what the respondent's theorems predict, what the fractures predict, and the current candidate thresholds. It is published before execution so that the design can be challenged in public. The protocol is not yet frozen; every revision made before freezing will remain visible in the revision record.
Within this framework, Certification is the terminal step of the chain and is not available before the empirical step. No Certification is claimed here and the word is not used below except to say that it is withheld. The Decision-of-fact is stated. The Attestation is granted where it is earned. The verdict is left open.
Anyone may run this. The protocol does not depend on who executes it. If a critic runs it and the fractures fail, that result is as publishable as the other one, and the publication rule below binds us to say so.
This page is being published before the protocol is complete. Its purpose is to make the intended respondent, claims, measurements and candidate decision rules visible while they can still be criticized. Publication of this draft does not preregister the current numbers and does not convert them into evidence.
Before execution, a dated frozen version will be issued. That version will preserve this draft, enumerate every substantive change, fix the model revision, dataset, seeds, implementation, statistics and thresholds, and mark the point after which the publication rule becomes binding. No result will be collected for adjudication before that freeze.
The freeze is dated. This draft is open for criticism until 31 August 2026. On that date the protocol is frozen as it then stands, the revision record is published, and the publication rule below becomes binding. A protocol that stays open indefinitely is not a protocol; it is a position that can be adjusted after the fact and never has to be. The date exists so that this one cannot become that.
The freeze date binds the design, not the execution. Nothing here commits any party to run the experiments by any date, and execution remains open to anyone at any time. What the date fixes is that the claims, the measurements and the thresholds stop moving.
| Field | Value |
|---|---|
| Result | The Linear Representation Hypothesis and the Geometry of Large Language Models |
| Authors | Kiho Park, Yo Joong Choe, Victor Veitch (University of Chicago) |
| Locator | arXiv:2311.03658v2 · ICML 2024 · PMLR 235:39643–39666 |
| Code | github.com/KihoPark/linear_rep_geometry |
| Model under test | LLaMA-2 7B — the respondent's own testbed, used unchanged |
| Disposition | Explicit ontological claim. The global structure is asserted, and asserted as theorems. |
Most of the field does not argue for the manifold. It stands on it. Sparse autoencoders, steering vectors and linear probes assume global structure and never state the assumption, which makes them difficult to prosecute and impossible to falsify on their own terms. This paper is the exception. It states the assumption cleanly, proves consequences from it, and supplies the machinery that the silent users depend on. It is the load-bearing member that is visible.
It is also the strongest possible respondent for a further reason: two of the three claims below are theorems. A model can retreat to being an approximation. A theorem cannot. If a theorem's prediction fails on the machine, an assumption failed — and the assumption is about the space.
That high-level concepts in a transformer are linearly represented, that the representations found in one place transport to another, and that a single inner product on the representation space renders causally separable concepts orthogonal.
Granted in full, and not contested anywhere in this protocol: concepts have local linear structure. Counterfactual pairs for a concept do align along a common direction more than chance allows, the alignment is demonstrated rather than assumed, and it is useful. Probes work. Steering works at moderate magnitudes. The paper's contribution in formalizing what "linear representation" means — separating the subspace, measurement and intervention senses, and proving they connect — is real work and is granted without reservation.
The falsification does not deny local operability. It denies that local structure glues. Everything under adjudication below is a transport claim or a global claim. Nothing under adjudication below is the existence of a concept direction.
| # | Claim | Where | What it asserts globally | Fracture |
|---|---|---|---|---|
| C1 | Concepts are directions: counterfactual differences lie in a common cone | Definition 2.1 | Nothing. Local. | none — granted |
| C2 | Measurement: the logit is linear in the representation, and the direction used for prediction is the same for all counterfactual pairs of the concept | Theorem 2.2 | One direction transports across all instances of the concept | transport |
| C3 | Intervention: adding c·λ̄W leaves off-target probability constant in c ∈ ℝ and increases target probability, increasing in c ∈ ℝ | Theorem 2.5 | Unbounded, monotone, leak-free transport along the concept direction | F4 |
| C4 | A single causal inner product, M = Cov(γ)−1, makes causally separable concepts orthogonal and unifies the two representation spaces | Theorem 3.4 + a stated choice | One global metric over the whole representation space | F9 / MH-D |
C1 is granted and is not tested. C2, C3 and C4 are the respondents. C3 and C4 are the two that carry experiments.
Two concessions are on the record in the paper, and this protocol does not need to establish them — only to measure what they cost.
| Concession | Where | What it means |
|---|---|---|
| The metric is not identified. Causal orthogonality imposes d(d−1)/2 constraints against d(d−1)/2 + d degrees of freedom, leaving a d-parameter family indexed by a free diagonal D. The paper states it has no principle for picking a unique D, and works with the choice D = I. | §3.2 | "The causal inner product" is one arbitrary member of a d-dimensional family. Every downstream orthogonality result is conditional on a choice with no stated warrant. E2b measures whether the data prefer that choice at all. |
| The metric is estimated over words sampled uniformly from the vocabulary. The paper's own footnote concedes that in natural language there may be non-causal correlations between causally separable concepts. | Assumption 3.3 | The metric is fitted on a distribution that inference does not run on. E2c measures whether the orthogonality survives the distribution the claim is actually made about. |
Against C3, Theorem 2.5. Tests fracture 4.
Setup. LLaMA-2 7B, unchanged. Concept set: the respondent's own 27 concepts. For each concept W, construct λ̄W by the respondent's own method — the unembedding representation mapped through the Riesz isomorphism under M = Cov(γ)−1. For each W, select an off-target concept Z that the respondent treats as causally separable with W. Context set: the respondent's own samples, N = 100 contexts per concept.
Sweep. Define the dimensionless steering magnitude κ = c·‖λ̄W‖ / mean‖λ(x)‖ over the context set. Sweep κ from −8 to +8 in steps of 0.25. The theorem is quantified over all c ∈ ℝ, so the negative arm is part of the claim and is swept.
Measured. ℓW(κ), the target-concept logit under the constrained pair, averaged over contexts and counterfactual pairs; and ℓZ(κ), the same for the off-target concept. Per-point standard error from the 100 contexts.
| Quantity | Theorem 2.5 predicts | Fracture 4 predicts | Threshold |
|---|---|---|---|
| Monotonicity of ℓW | Non-decreasing across the entire swept range, both arms | Monotonicity breaks at a finite κ* | A violation is a decrease exceeding 3 standard errors, sustained over ≥3 consecutive grid points. LRH holds iff no violation for |κ| ≤ 8. Fractures hold iff κ* exists with |κ*| ≤ 8, on ≥50% of the 27 concepts. |
| Constancy of ℓZ | Constant in c, exactly | Off-target leakage appears | LRH holds iff |ℓZ(κ) − ℓZ(0)| < 0.5 logits throughout. Fractures hold iff that bound is exceeded at some |κ| ≤ |κ*| — leakage must arrive at or before the monotonicity break, not after it. |
The second threshold is deliberately strict against us. Everything breaks at large enough magnitude, and a finding that steering degrades after it has already stopped working is worth nothing. The claim is that the leak is present while the steering still appears to be working — that the off-target contamination is not a downstream artifact of a saturated intervention but a property of transport itself.
Against C4, Theorem 3.4. Tests fracture 9 and MH-D. No forward passes. No GPU. Cov(γ) is computed from the unembedding matrix alone — a 32000 × 4096 matrix — and the whole of E2 is linear algebra that runs on a laptop in minutes. This is the cheaper experiment and the sharper one.
Statistic throughout. For each pair (W, Z) that the respondent treats as causally separable, the causal cosine cosM(W,Z) = γ̄W⊤ M γ̄Z / (‖γ̄W‖M‖γ̄Z‖M). Theorem 3.4 says this is zero. Every sub-experiment asks whether that zero is a property of the model or a property of the sample, the choice, or the distribution.
| Procedure | LRH predicts | Fractures predict | Threshold | |
|---|---|---|---|---|
| E2a sample drift | Partition the vocabulary into two disjoint seeded halves V₁, V₂. Compute M₁ = Cov(γ|V₁)−1, M₂ = Cov(γ|V₂)−1. Compare cosM₁ and cosM₂ across all causally separable pairs. Null control: the same statistic on random direction pairs and on a covariance-matched Gaussian surrogate. | The metric is a property of the model. |cos| ≈ 0 under both, and the two agree. | The metric is a property of the sample. The orthogonality verdict itself depends on which half of the vocabulary was looked at. | LRH holds iff |cosM₁ − cosM₂| < 0.05 for ≥90% of pairs and both |cos| < 0.1. Fractures hold iff median |Δcos| ≥ 0.10 and exceeds the Gaussian surrogate drift by ≥2×. |
| E2b the free choice | Their D = I fixes M = Cov(γ)−1. Sweep the whitening exponent: Mp = Cov(γ)−p for p ∈ [0, 1.5], step 0.05. p = 0 is Euclidean, which the paper shows is not causal. Plot mean |cos| over causally separable pairs against p. | The data select their choice. A distinct minimum at p = 1. | The data are indifferent. D = I was a choice with no empirical warrant — which the paper concedes, and which nobody downstream has priced in. | LRH holds iff the minimum lies at p = 1.0 ± 0.1 and mean |cos| rises by ≥50% at both p = 0.8 and p = 1.2. Fractures hold iff mean |cos| varies by <20% across p ∈ [0.7, 1.3], or the minimum lies outside p = 1.0 ± 0.2. |
| E2c wrong distribution | Recompute Cov(γ) weighted by empirical token frequency over a natural-language corpus rather than uniformly over the vocabulary. Compare cosuniform and cosfrequency. | Orthogonality survives the distribution inference actually runs on. | It does not. Assumption 3.3 holds where the metric is fitted and fails where the claim is used. | LRH holds iff |Δcos| < 0.05 for ≥90% of pairs. Fractures hold iff median |Δcos| ≥ 0.10. |
Stated here so that it cannot be quietly dropped later.
| If | Then |
|---|---|
| E1 finds no monotonicity break for |κ| ≤ 8 on more than half the concepts, and off-target leakage stays under 0.5 logits throughout | The operational clause of fracture 4 is not supported at accessible steering magnitudes. This is published as a failure of the prediction, in the same place and at the same size as any other result. |
| E2a shows causal cosines stable across vocabulary halves, E2b shows a sharp minimum at p = 1, and E2c shows the orthogonality surviving frequency reweighting | Fracture 9 is not supported at the unembedding level, and the causal inner product is identified by the data after all. Published as such. |
| Both | The twelve fractures lose their two operational clauses against the strongest explicit statement of the manifold thesis, and the falsification is materially weakened. That is a real outcome and it is on the table. |
Three defences are available to the respondent's position, and each is answered in advance so that it cannot be used to make the result unfalsifiable after the fact.
| Retreat | Answer |
|---|---|
| "Theorem 2.5 holds only for the exact embedding representation of Definition 2.3, and the constructed one is approximate." | Then the theorem has no empirical content, because the exact representation cannot be constructed — the paper says so, which is why it builds the representation through the Riesz map instead. A theorem that holds only for an object nobody can obtain is not available to the steering literature that relies on it. Either horn is a finding, and both are published. |
| "Large steering magnitudes are out of distribution; of course it breaks." | The theorem is quantified over all c ∈ ℝ, not over a distribution of c. And the leakage threshold in E1 is set to require the break at or before the monotonicity failure, precisely to exclude the trivial version of this objection. |
| "D = I is just a convenient normalization." | Then E2b will show the data prefer it. If it does, we say so. |
Proposed now; binding when the protocol is frozen before execution.
The execution record is published in full — every threshold, every deviation from this protocol, and the pass or fail on each — whether it confirms the fractures or defeats them. After the protocol is frozen, no sub-experiment is dropped after the fact and no threshold is revised after the fact. If a deviation from this protocol proves necessary, it is recorded as a deviation and the original is left standing on the page beside it. Only after the execution record exists does the Certification chain complete, and only then does the word Certificate become available.
| Field | Value |
|---|---|
| Status | IN PROCESS — NOT YET FROZEN OR EXECUTED |
| Cost | E2 needs no GPU and runs in minutes. E1 needs one 7B model on one card. |
| Who may run it | Anyone. Supporters, critics, the respondents. The protocol does not depend on the operator, which is the point of publishing it before the result. |
| Freeze date | 31 August 2026. Open for criticism until then; frozen as it stands on that date, with the revision record published. |
| Standing commitment | The frozen version, every later deviation, and the complete execution record remain public whatever the result. The freeze binds the design on the date above. It does not bind execution to any date, and execution stays open to any party. |
Reserved. Nothing has been run. This section will remain empty until the protocol is frozen and an execution record exists.
Whatever this protocol returns, it settles only what is not the case. Suppose it goes against the respondent. Suppose semantic transport is not governed by a persistent global metric, that the direction found in one context does not carry to another, and that the space concepts are supposed to live in cannot be glued from the pieces we can see.
Something still supports inference. The models work. The question the fractures cannot answer, and the reason the falsification was worth driving at all, is the one that remains after the demolition:
If semantic transport is not governed by a persistent global manifold, what mathematical object does remain stable enough to support inference?
That question is not answered here.