Independently trained neural networks have no shared neuron-index reference frame, so comparing them requires accounting for coordinate freedom. Neural Collapse sharpens this problem: networks converge toward a shared, low-dimensional geometry, raising the question of whether trajectory-specific functional variation remains distinguishable after convergence. We distinguish three claims - detectability, transplantability, and causal persistence - and address the first. Using five independently trained networks reconstructing Neural Collapse on MNIST, we apply a verified affine-correct alignment mapping donor heads into recipient coordinates. Donor-specific functional fingerprints remain distinguishable after recipient-level baseline correction: all 20 ordered donor-recipient pairs are correctly identified, with an exact permutation p=0.0083, robust to a leakage audit. These findings establish detectability under the test used here, but not transplantability or causal persistence. The study shows how alignment, ambiguity diagnostics, and leakage control combine to test cross-network variation in a controlled setting; whether this generalizes beyond it is open.
While neural collapse (NC) predicts that a $K$-class-balanced classifier should organize terminal representations as a $(K-1)$-dimensional simplex equiangular tight frame (ETF), modular addition consistently enters a different regime: networks compress to a two-dimensional cyclic geometry in which both classifier weights and token embeddings lie on circles. We refine the explanation of this phenomenon in three directions. First, we formalize a layerwise non-uniform training mechanism: downstream classifier weights are driven by dense cross-entropy gradients into a rank-2 equiangular configuration before upstream embeddings fully reorganize, and once this classifier plane forms, backpropagated feature gradients constrain embedding motion to the same plane while weight decay suppresses orthogonal components. Second, after this subspace locking, the induced in-plane dynamics admit an entropy-regularized transport interpretation on $S^1$; combined with modular-addition labels, this reduces embedding formation to phase alignment, whose minimizers are single-frequency characters of $\mathbb{Z}/P\mathbb{Z}$ and hence equal-angle points on a circle. Third, we quantify why this solution prevails over NC: a simplex ETF gains only an $O(1)$ advantage in cross-entropy, whereas the cyclic rank-2 solution enjoys a $Θ(K)$ advantage under Schatten or weight-decay surrogates, yielding a critical threshold $λ_{\mathrm{crit}} = Θ(1/K)$. Our results explain both why classifier weights move first and why embeddings subsequently align with them, showing that grokking on modular arithmetic is governed not by maximal separation alone but by a task-structured trade-off between separation, symmetry, and complexity.