Hidden Equivalences in Sheaf Neural Networks: The Share Credited to Matrix-Valued Transport Is Not Intrinsic

Sheaf neural networks replace scalar edge weights with learned restriction maps, and their gains are routinely credited to this matrix-valued transport. We show that the share of the gain so credited is not intrinsic to a task or an architecture. The algebra its restriction maps generate constrains the propagation operator of a sheaf layer: by the Wedderburn decomposition it fixes the block structure and which blocks act irreducibly, so scalar maps give a real edge-weighted operator, one-angle rotations a complex-weighted (magnetic) one, and of our six rungs only the dense one acts irreducibly. It does not fix the realizable family, since restricted families can generate the same algebra. On this chain of algebras we build an attribution protocol: an exactly nested six-level ladder audited in function space, a depth-matched no-graph baseline, and a necessity diagnostic that can refuse to answer. The share credited to matrix-valued transport is then 53% and 11% on two heterophilous tasks at two layers and 13% and 17% at four, and it moves with width (46% to 73%), with the operator (11% to 35%) and with the order of credit (13% to 94%). The released operator moves the absolute sheaf-exclusive gain by less than $0.01$ at two layers (a point estimate, not an equivalence test) but reverses the sign of the four-layer depth effect on Minesweeper ($-0.034$ against $+0.035$), four fifths of which a $2\times 2$ experiment traces to the normalization alone, acting through the scalar gate as a spectral bound we prove predicts. What the scalar control is allowed to do matters as much as what the sheaf adds: in a preregistered test, a learned degree-3 spectral response on those rungs absorbs 92% of the gain credited to matrix-valued transport on one task, while on the other the interaction runs the other way and is not resolved under a matched training budget. Two later controls on four-layer Minesweeper reduce the gain credited to matrix-valued transport by 98% with learned one-hop filtering and by 71% with fixed three-hop filtering. The larger reduction therefore requires no wider receptive field. On an independently published directed sheaf architecture the capacity effect is $-29.6$ points unconditionally, positive conditional on convergence, and not significant once an identity-containing parameterization removes the collapse. Claims about what matrix-valued transport contributes should therefore state the operator, depth and order of credit under which they were measured.