Constraint-Algebra Basis Hypothesis

structures / Hypotheses / Constraint-Algebra Basis Hypothesis

CLAIMhypothesisintermediatehow it's classified →

When a task's validity conditions are governed by an explicit algebraic/combinatorial constraint structure rather than by the surface units a human observer would naturally decompose the state into, a trained model's linearly-decodable world-model features align with the constraint structure's own units, not the surface units — e.g. a Sudoku-solving transformer's linear features are organized by row/column/box membership, not by individual cell, because presence-in-a-substructure, not cell identity per se, is what determines placement validity.

Replicationcomputed from the corpus — never hand-assigned

1 paper1 architecture class1 domain1 model family

Filled = two or more values reported by papers that share no author — replication. Outlined = two or more values, but all from a single study — breadth, not replication. Grey = a single value. Derived from paper authorship and each model's architecture class, domain and family; it updates itself when a paper is added.

Statement

Let a task’s state space have a natural surface decomposition into units $u_1,\ldots,u_n$ (e.g. individual grid cells) that a human observer would use to describe the state, and a separate, coarser or differently-shaped constraint algebra — a collection of subsets or groupings $S_1,\ldots,S_m$ of those surface units (e.g. rows, columns, boxes) over which the task’s validity rules actually operate. The hypothesis claims that when these two decompositions genuinely differ (the rules act on the $S_j$ ‘s, not directly and only on individual $u_i$ ‘s), a model trained end-to-end on the task will build linearly-decodable features organized around the constraint algebra’s units $S_j$ , not the surface units $u_i$ — even though the surface units are what the task is visually/notationally presented in terms of, and even though probes fit directly at the surface-unit granularity may fail to achieve clean linear separability.

Intuition

Earlier world-model findings (OthelloGPT’s per-cell board occupancy, maze-transformers’ per-cell wall connectivity) all happened to have surface unit and constraint unit coincide: in Othello, whether a cell is a legal move genuinely depends on that cell and its neighbors directly, so “one feature per cell” was both the natural surface description and the right computational unit. Sudoku breaks this coincidence on purpose: a cell’s own identity is almost irrelevant to whether a digit belongs there — what matters is which digits already occupy the row, column, and box it belongs to. This hypothesis says a trained model “notices” this asymmetry and organizes its internal geometry around the quantity that actually does the causal work (substructure membership), not the quantity a human would default to describing the board with (cell identity).

Properties

A counter-example to a tacit “surface unit = representational unit” assumption, not merely another confirming instance of it. Every prior world-model finding in this corpus (othello-relative-frame-direction, maze-single-token-linear-decoding) happens to have surface and algebraic units coincide, so this is the first paper in the cluster where the two genuinely diverge and the representational unit tracks the algebraic one.
Diagnostic signature: graded correlation among same-level probe directions. If probes fit at the surface-unit granularity (e.g. per-cell candidate-set probes) are not independent — their weight vectors show higher cosine similarity for surface units that share more of the constraint algebra’s structure (e.g. two cells in the same row-and-box align more than two cells sharing only a row) — that graded alignment pattern is evidence the “true” linear basis sits one level up, at the constraint-algebra units themselves, not at the surface units the probes were fit to.
Falsifiable via a two-granularity probe comparison. Fit probes at both the surface-unit and the algebraic-unit granularity for the same underlying state; the hypothesis predicts the algebraic-unit probes reach strictly higher decoding accuracy (in the originating paper: 100% exact-match vs. 80% top-1 for cell-identity probes) and that this same algebraic decomposition also explains the surface-level probes’ own correlation structure, rather than the two probe families simply measuring unrelated things.
Requires causal, not just correlational, confirmation — the same bar as every other structure claim in this map. Decodability of the algebra-aligned directions is not by itself sufficient; the hypothesis is only supported once those directions are shown to be load-bearing via intervention (patching or ablation), exactly the causality gap Linear Representation Hypothesis itself flags for any candidate direction.
Scope is explicitly domain-conditional, not universal. The hypothesis only makes a distinguishing prediction for tasks whose validity or difficulty genuinely factors through a constraint algebra distinct from the surface decomposition; for a task where the two coincide (Othello’s per-cell legality), it collapses to, and makes no different prediction from, the naive surface-unit assumption — a narrower, more falsifiable scope than a fully general claim about all world models.

Exercises

Base

In Sudoku, why does “digit $d$ is absent from row $r$ , column $c$ , and box $k$ ” determine “digit $d$ is a valid candidate for cell $(r,c)$ ,” while the converse implication does not hold?

Solution

A digit is a valid candidate for a cell exactly when it doesn’t already violate any of the three constraints (row, column, box) that cell participates in — so absence from all three substructures is precisely the validity condition, giving the forward implication. The converse fails because a digit being a valid candidate for a cell (not yet excluded) does not mean it is the digit actually placed there — the cell might still be empty, or another valid candidate might end up chosen instead — so “candidate-valid” is strictly weaker than “this is the filled value,” even though both are downstream of the same substructure-absence fact.

Why is 100% exact-match accuracy for 243 substructure probes stronger evidence for the hypothesis than 100% accuracy for a single substructure probe would be?

Solution

A single substructure probe achieving perfect accuracy would only show that one particular (substructure, digit) fact is linearly decodable — consistent with the ordinary Linear Representation Hypothesis claim that individual human-interpretable facts tend to be linear, without saying anything about representational granularity. Showing all 243 probes (every substructure crossed with every digit) succeed simultaneously, while the alternative per-cell framing plateaus below ceiling, is what specifically supports the comparative claim — that the algebraic unit, not the surface unit, is the representation’s true organizing basis — rather than just reconfirming that some linear structure exists somewhere.

Middle

A follow-up study probes a different constraint-satisfaction puzzle where every constraint happens to involve only a single surface cell (no genuine multi-cell substructures). Would confirming per-cell linear decodability in that setting count as evidence for or against this hypothesis?

Solution

Neither strongly for nor against — per the hypothesis’s domain-conditional scope, when surface units and algebraic units coincide (each constraint acts on exactly one cell), the hypothesis makes the same prediction as the naive surface-unit assumption, so confirming per-cell decodability in that setting is consistent with the hypothesis but doesn’t distinguish it from the null “surface unit = representational unit” default — it’s simply outside the regime where the hypothesis has any distinguishing content, the same way Hypotheses‘s own discussion of vacuous confirmation applies here.

The originating paper finds that cell-candidate probe vectors for cells sharing two substructures (e.g. a row and a box) align at cosine similarity $\approx 0.65$ , while cells sharing one substructure align at $\approx 0.33$ (roughly double). Propose a simple linear model of how a cell’s candidate-probe direction is built from substructure-level directions that would produce approximately this doubling pattern.

Solution

A natural model: each cell’s candidate-probe direction for digit $d$ is (approximately) a sum of the three substructure-absence directions it depends on, $w_{(r,c),d} \approx w_{\mathsf{Row}_r,d} + w_{\mathsf{Col}_c,d} + w_{\mathsf{Box}_k,d}$ , with the three substructure directions roughly orthogonal to each other and of similar norm. Two cells sharing exactly one substructure (say the same row, different column and box) then share exactly one of the three summed terms, giving a cosine similarity proportional to (one shared term)/(three terms each) $\approx 1/3\approx0.33$ ; two cells sharing two substructures (e.g. same row and same box, different column) share two of three terms, giving cosine similarity $\approx 2/3\approx0.65$ — reproducing the observed roughly-doubling pattern directly from how many of the three additive substructure components the two cells’ directions have in common.

Pro

Design a causal-validation experiment (using either Activation patching or Causal interventions (steering)) that would distinguish “the algebraic-unit directions are genuinely load-bearing for the model’s own computation” from “the algebraic-unit directions are merely a more efficient encoding of information the model actually still processes cell-by-cell downstream.”

Solution

Intervene at the algebraic-unit direction only — e.g. patch or ablate just the $(\mathsf{Row}_r, d)$ -direction component of the residual stream (leaving all cell-specific residual content otherwise untouched) so that the model’s representation now falsely indicates $d$ is present in row $r$ — and check whether the model’s predictions change consistently for every cell in that row (not just one). If the model still correctly avoids placing $d$ in some cells of that row despite the row-level direction being corrupted, this would suggest the model retains an independent, cell-specific fallback computation and the algebraic-unit direction is not the sole causal pathway; if instead the intervention uniformly and simultaneously corrupts predictions across all affected cells in the row, this is strong evidence the algebraic-unit direction is genuinely the shared causal bottleneck the model routes all of those cells’ predictions through — exactly the pattern the originating paper reports (Section 3.3): a single substructure-direction patch changes the top-1 prediction in 99% of cases across many different affected cells.

Found in (1 observation · 1 family)

Sudoku Transformer

Transformers Linearly Represent Highly Structured World Models (2026)measured

A Sudoku transformer builds substructure-level linear constraint directions

Details

Kniazev & Fijalkow train an 8-layer transformer on Sudoku solving traces and probe its residual stream at the clues-end token [kniazev-fijalkow-2026] Per-cell digit probes plateau at 80% (imperfect linear separability), but per-cell candidate probes reach 1.0 exact-match in mid-layers [kniazev-fijalkow-2026] Candidate probe directions align by shared substructure (cosine ~0.33 for one shared row/column/box, ~0.65 for two), consistent with each being a linear combination of substructure-level directions [kniazev-fijalkow-2026] Substructure "digit present in row/column/box" probes reach perfect accuracy, and patching only their direction components flips the top-1 prediction in 99% of cases [kniazev-fijalkow-2026] Mid-layer heads specialize by substructure, and a final-MLP set of monosemantic "naked-single" neurons commits the answer [kniazev-fijalkow-2026]

models: Sudoku Transformer (8 layers, 8 heads, d_model=576) · method: Linear probing, Activation patching, Causal interventions (steering), Direct Logit Attribution (DLA), Logit Lens