Statement

The Minkowski Representation Hypothesis (MRH) claims that a layer’s activation space $X$ satisfies $X=\bigoplus_i P_i$, the Minkowski sum of disjoint tile polytopes $P_i=\mathrm{conv}(A_{T_i})$ over a partitioned archetype dictionary, and that any activation is a block-sparse convex combination $x=\sum_{i\in S} z_i A_{T_i}$ with $z_i\in\Delta^{|T_i|}$ and $|S|\ll m$ (only a few tiles active at once).

This node is the hypothesis (a falsifiable general claim about how a whole layer is organized). The geometric object it postulates — the Minkowski sum of polytopes itself — lives at Minkowski Sum of Tile Polytopes (Minkowski Representation Hypothesis) (type: geometric-object). Keeping the two separate follows the map’s rule that a hypothesis must not carry the same type as the mathematical object it asserts.

Why it is a hypothesis, not a shape

Finding one activation inside a vector sum of two polytopes is consistent with MRH but does not establish it; MRH is universally quantified over the whole layer and predicts (i) curved geodesics along a token k-NN graph, (ii) convex/archetypal coding matching SAE reconstruction with few archetypes, and (iii) spontaneous block-diagonal co-activation structure. Each is a separate falsifiable consequence.

Key evidence

Fel, Wang, Lepori, Kowal, Lee, Balestriero, Joseph, Lubana, Konkle, Ba & Wattenberg (ICLR 2026, arXiv:2510.08638) report all three signatures on DINOv2-B with a 32k-atom SAE dictionary, while stating the evidence is compatible, not conclusive (“multiple mechanisms can mimic the same surface phenomena”) and running no causal steering test. See dino-activation-space-is-consistent-with-a-minkowski-sum-of-tile-polytopes.