Sep 15, 2026 · cs.LGJ/K move · Enter open · S save
Chon-Fai Kam, Miloud Bessafi, Frédéric Cadet
Dipartimento di Fisica e Chimica, Universit`a degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy · EnergyLab, Universit´e de La R´eunion, F-97715 Saint-Denis, France · Universit´e Paris Cit´e and Universit´e de La R´eunion, BIGR, INSERM UMR S1134, F-75014 Paris, France · PEACCEL, AI for Biologics, F-75013 Paris, France
The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in
Zp that derivation has no input. The exact algebraic solution is an orbit of
Zp acting by isometries. Transitivity alone makes the ratio statistic underlying the standard dimension estimator a point mass, so the estimator is undefined, and here the two nearest neighbour distances coincide exactly. Breaking the symmetry at scale
ε returns a number, but one that tracks
1/ε with no scale free plateau. We show that the failure is general, since on any finite orbit of a group acting by isometries the estimator reports the resolution at which the set is probed rather than a dimension. What replaces the power law is exponential in hidden width,
L(h)=L∞+Aexp(−chα), with
R2 between 0.982 and 0.995 against 0.857 to 0.906 for a power law admitting the same floor and fitted under the same protocol. Where the data supply is sufficient the rate belongs to the regulariser rather than to the group, since weight decay moves
c by a factor of 47 while group order moves it by 1.10, a residual below seed to seed resolution, for every fixed
α between 0.75 and 2. The critical width falls with group order rather than rising, against capacity counting that assigns a fixed number of neurons to each irreducible representation.