Superposition

Source. Toy Models of Superposition (Elhage et al., 2022).

Description. A ReLU network trained to reconstruct sparse features stores more of them than it has dimensions, giving each feature a non-orthogonal direction and absorbing the interference those directions create; the linear model differing only in activation function shows no superposition at any sparsity. Where the tradeoff falls is mapped over an importance $×$ sparsity grid against closed-form losses for each candidate weight configuration, and the geometries the trained models settle into are uniform polytopes solving a generalized Thomson problem. Evaluated on: toy autoencoders on synthetic features whose sparsity, importance and correlation are set by hand, across four architectures and sizes from $(2,1)$ to 400 features in 30 dimensions, plus an absolute-value model that computes through a compressed layer rather than storing. Description mode: representational.

Readings of the claim

Superposition. Readings of the claim, from the version the authors state to the version the field cites.

Reading	Verdict	Missing
Primary. A ReLU network on sparse features represents more features than it has dimensions, in geometries set by sparsity and importance	Mechanistically Supported	A converse arm, with some property shown intact while superposition is removed (I6); correction for the lowest-loss selection the reported geometries are drawn from (M7); an input distribution outside the synthetic family (E2)
The reported geometries are properties of the optimum rather than of the optimizer	Underdetermined	The $m=2$ case is reported as much harder for gradient descent and its solutions are selected by loss (I7, M7); an invited comment in the same article shows the global minimum can have a smaller basin of attraction than nearby local minima, and the paper leaves it open (V3)
The phase diagram is fixed by sparsity and importance alone	Disconfirmed	Nothing — tested and failed: the paper holds the activation function fixed across every one of its own experiments, and a replication carried in the same article finds the phase diagrams look quite different under other activation functions (M3, M6)
Language models represent features in superposition	Insufficient	Every measurement is on a toy model, and the paper labels its real-model section validation by consistency with existing reports rather than measurement taken here (E4); no natural-data distribution appears anywhere, and Open Questions asks whether real importance and sparsity curves can be estimated at all (E2)

Verdict

Superposition. Verdict: Mechanistically Supported. The claim stops at the rung marked Blocked; rungs below it are Reached and rungs above it are blocked by that rung. Criteria in bold are absent or adverse at that rung; only those the rung requires can block it.

Tier	Requires	Missing
Proposed	construct (C1–C2)	Reached.
Causally Suggestive	measurement (M2); internal (I1)	Reached.
Mechanistically Supported	internal (I2, I4); external (E1)	Reached.
Triangulated	construct (C3–C4); internal (I5–I7); external (E2, E4)	Blocked. I6 double dissociation — Importance and sparsity are crossed, but on one outcome; no converse arm is run
Validated	construct (C5–C6); measurement (M1–M6); internal (I3, I10–I12); external (E2–E6); interpretive (V1–V5)	Blocked at Triangulated. C6 complementation validity — Ground truth is available by construction and the test is not run; I10 rescue reversibility — Adversarial training reduces superposition; nothing is corrupted and then restored; I12 offset coupling — Offset never run; the learning-dynamics section is limited by the authors’ own account

36-criterion audit

Superposition. Full 36-criterion audit. Status: C = Confirmed, PC = Partially confirmed, U = Untested, I = Inconclusive, D = Disconfirmed, N/A = Not applicable.

ID	Criterion	Status	Evidence
Construct Validity
C1	Falsifiability	C	Point prediction against a linear control, plus a two-parameter phase diagram
C2	Structural plausibility	C	Mechanism derived in closed form, then matched against the trained weights
C3	Convergent validity	C	Theory and experiment agree, and three outside groups replicated before publication
C4	Discriminant validity	PC	Three-way discrimination against planted features; PCA separated as a limiting case
C5	Nomological validity	C	Formal links to compressed sensing, the Thomson problem and distributed codes
C6	Complementation validity	U	Ground truth is available by construction and the test is not run
Measurement Validity
M1	Reliability	PC	Repetition run and reported; aggregation is selection by loss, with no dispersion
M2	Baseline separation	C	Linear control cannot superpose, runs at every sparsity, and separates completely
M3	Stability	PC	Sparsity, importance, correlation and size all swept; activation function held fixed
M4	Calibration	PC	Two purpose-built measures with stated readings and no external calibration
M5	Sensitivity	C	Planted ground-truth features give a known positive at every point
M6	Invariance	PC	Holds across sizes and data structures; one activation function and one loss
M7	Selection correction	U	Best-of-1000 and worst-discarded selection documented, with no correction applied
Internal Validity
I1	Necessity	C	Output nonlinearity, sparsity and negative bias each shown to be required
I2	Sufficiency	C	Built from the hypothesized ingredients alone, and superposition appears
I3	Minimality	PC	Load-bearing parts named and the smallest case solved; data assumptions untested
I4	Specificity	N/A	One task and one loss, and the paper says cross-superposition loss comparison fails
I5	Rival mechanism exclusion	PC	PCA named as a limiting case; incidental polysemanticity untested (Lecomte et al., 2023)
I6	Double dissociation	U	Importance and sparsity are crossed, but on one outcome; no converse arm is run
I7	Confound control	PC	Data confounds set by construction; optimization left as an uncontrolled one
I8	Confounding sensitivity	U	Unattempted; nothing bounds an unmeasured confounder in the synthetic setup
I9	Epistatic interaction	C	Feature interactions are the subject once uniformity is dropped, and each is measured
I10	Rescue reversibility	U	Adversarial training reduces superposition; nothing is corrupted and then restored
I11	Onset coupling	PC	Onset tracked across training; feature dimensionality jumps as the loss drops
I12	Offset coupling	U	Offset never run; the learning-dynamics section is limited by the authors’ own account
External Validity
E1	Intervention reach	PC	Four intervention forms including two that remove superposition, cost reported
E2	Prompt generalization	PC	Input distribution swept along every axis the theory names, all synthetic
E3	Cross-task generalization	PC	Two tasks, with computation carrying the weight; representativeness left open
E4	Cross-model recurrence	PC	Four architectures inside the toy family; real-model support is consistency only
E5	Graded response	C	Dose-response is the core object: sparsity swept, response monotone and located
E6	Novel prediction	C	Two predictions that could have failed: polytope geometry and adversarial fragility
Interpretive Validity
V1	Level declaration	C	Levels separated before any experiment; the chosen definition flagged as circular
V2	Level-evidence match	C	Claims stay at the toy-model level, and every step up from it is marked
V3	Alternative level	PC	PCA examined as a limiting case; the optimization alternative raised and left open
V4	Unlicensed labeling	C	Metaphors are physical and cashed out; agentive verbs appear but stay scare-quoted
V5	Scope declaration	C	Scope declared in the title, in Key Results, in the Discussion and in Open Questions
Total: 15 Confirmed, 14 Partially confirmed, 6 Untested, 1 Not applicable
Verdict: Mechanistically Supported

Exploratory Lens Analysis

An exploratory reading of this claim through the framework’s five lenses, written for this site and not part of the paper, is at Superposition — exploratory lens analysis.