Source
- Lubana, Bigelow, Dick, Krueger & Tanaka (2022), LIT-669, read in NOTE-538: Definitions 2–5, Conjecture 1, Figs. 4–5, Table 2, and Appendix F.3 (Lemmas 2–4, Theorem 1, Corollary 1, Remark 1, Fig. 13).
- Zhou, Yang, Yang, Yan & Hu (2023), LIT-674, read in NOTE-512: Figs. 2–4 and Theorem 1, for the converse direction.
The claim, and what “mechanism” means here
Two minima of one loss are separated by a barrier on the straight line between them. The barrier remains after the architecture’s symmetries have been factored out, which in practice means neuron permutations. The claim is that such a barrier is not an accident of parametrisation: the two models rely on different attributes of the input. “Mechanism” is Lubana et al.’s operational notion (LIT-669, Definitions 2–4). Model the data as generated from independent latent attributes. A model is invariant to an intervention on one attribute if randomising that attribute does not raise its loss. Two models are mechanistically similar when they are invariant to the same interventions.
The claim has a converse that is better supported than the claim itself: a linear low-loss path, up to symmetry, marks a shared mechanism. The document keeps both, because the evidence comes in both directions and must not be read as one.
What was actually shown
Fine-tuning, where the two directions co-occur. Lubana et al. trained VGG-13 and ResNet-18 on CIFAR-10 with a synthetic box cue correlated with the label, then fine-tuned on cue-free data (LIT-669, Fig. 5). With small or medium learning rates, the fine-tuned model stayed linearly connected to its parent and still relied on the cue on counterfactual data. With a large learning rate, or with a cue perfectly correlated with the label, there was a barrier, and the fine-tuned model was invariant to the cue. In this controlled setting every barrier observed came with a change of mechanism, and every kept path came with a kept mechanism. That is the evidence for the stated direction in deep networks. It is a co-occurrence across one fine-tuning protocol and three synthetic datasets, not a test of the implication.
Dissimilar endpoints have a barrier. A model trained with the cue and one trained without it are joined by a quadratic Bezier path that is low loss on the data it was fitted to. The straight line between them has a barrier, with or without activation-matching permutation (Fig. 4). This is the converse direction: different mechanisms, therefore a barrier.
The proof, and its direction. Appendix F.3 works in a one-hidden-layer ReLU network f(x; W) = (1/N)1ᵀϕ(Wᵀx), with binary labels and minima that interpolate the data. Inputs carry attributes of different spline complexity K plus noise dimensions. Lemma 2 shows that if two interpolating minima are linearly mode connected on the data, every hidden unit is on or off in both for every input: their activation patterns are identical. Lemma 4 shows that models relying on attributes of different complexity cannot have matching activation patterns under any permutation of units. Theorem 1 combines them: such models are not linearly connected under any permutation. Lemma 3, simplicity bias, is restated from earlier work, not proved.
So what is proved is “different mechanism, of different complexity ⇒ barrier after every permutation”, or equivalently “linearly connected up to permutation ⇒ same activation patterns”. That is the converse of Conjecture 1 as the paper states it (“cannot be linear mode connected … must be mechanistically dissimilar”). The stated direction is equivalent to “same mechanism ⇒ linearly connected up to permutation”. For it the appendix offers a sufficiency argument (Corollary 1, Remark 1). Two models with identical activation patterns, up to permutation, are linear in each other along the path, so they are linearly connected. The figure offers the matching example: two models relying on the K = 4 attribute connect after permutation, and a K = 0 model connects to a model trained on both attributes, which simplicity bias makes rely on K = 0 too (Fig. 13). The missing step is that two models relying on the same attribute have the same activation patterns up to permutation, and it is not proved. The appendix’s closing argument concerns a different gap, the case of dissimilar mechanisms of equal complexity, and so it also bears on the converse.
The converse in deep networks. Zhou et al. measured pairs that are linearly connected, whether spawned from a shared early checkpoint or aligned by permutation (LIT-674, Figs. 2–3). In nearly every layer of an MLP, VGG-16, ResNet-20 and ResNet-50, the features of the interpolated network point in the direction of the interpolated features. Independently trained pairs, which are not linearly connected, are far from it (Fig. 4). Their Theorem 1 derives this layerwise linearity from two conditions. The first, weak additivity of ReLU on the two networks’ pre-activations, amounts to the shared activation signs of Lubana et al.’s Lemma 2, reached independently. Linearly connected deep networks therefore compute the same features up to a linear blend, which is the converse direction at the level of features. They did not intervene on data attributes, so this is feature identity, not mechanism in Lubana et al.’s sense.
What this does not say
- It does not say a barrier is always disagreement. The stated direction is proved nowhere, and the deep-network evidence for it is one synthetic-cue fine-tuning protocol.
- It does not say different mechanisms always produce a barrier. Lubana et al. mark an exception themselves: dissimilar mechanisms of similar complexity can share activation patterns and be linearly connected (Table 2’s asterisk, App. F.3). They argue the case is rare in practice, but do not show it.
- It does not say curved connectivity means anything about mechanism. Proposition 2 makes dissimilar minima mode connected along some path, given enough width. Fig. 4’s curve holds only on the data it was fitted to and fails on counterfactuals.
- “After symmetries are removed” is only as good as the alignment. Every experiment uses permutations found by activation or weight matching, which is a heuristic. A barrier left by imperfect matching says nothing about mechanism. Unless the alignment is fixed in advance, the claim can always be rescued by saying not all symmetry was removed.
- The cues are synthetic and the endpoints of Fig. 4 are trained on different data. Two seeds on one dataset, the setting of the permutation conjecture, are not tested for mechanism at all.
Connections
- The barriers left after alignment. The anthology’s ANTH-THEORY-010
says most of the barrier between independently trained networks is
permutation, not disagreement. THEORY-112, filed alongside this
one, found the barrier left after weight matching falling with width;
whether it rises with depth is QUESTION-317. This account has to say what
that residual barrier is. Either narrow or deep same-data seeds rely on
different attributes, which nothing here has tested, or their residual
barrier is a counterexample to the stated direction. That is the test
promote_whenasks for. - THEORY-111 carries the converse direction further: from activation patterns in one hidden layer to every layer’s features, in the image classifiers it measured.
- Practice across the boundary. The anthology’s ANTH-SOTA-217 (align permutations before averaging) assumes that what remains after alignment is benign. On this account it is benign only when the two models share a mechanism.
Note of 2026-10-29: what holds it back
The stated direction, a barrier surviving alignment ⇒ a different mechanism, is tested nowhere on deep networks trained on the same data. The deep-network evidence is co-occurrence in one synthetic-cue fine-tuning protocol, LIT-669’s proof (App. F.3) runs in the converse direction for one hidden layer, and the residual barriers that same-data seeds keep at 1× width (THEORY-112) have never been tested for shared invariances.