Symmetry without a manifold: intrinsic dimension on orbits
Abstract
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(−c hα), 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.
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BibTeX
@misc{kam2026symmetry,
title = {{Symmetry without a manifold: intrinsic dimension on orbits}},
author = {Chon-Fai Kam and Miloud Bessafi and Frédéric Cadet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17926},
eprint = {2609.17926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17926},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetry without a manifold: intrinsic dimension on orbits
AU - Chon-Fai Kam
AU - Miloud Bessafi
AU - Frédéric Cadet
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17926
UR - https://arxiv.org/abs/2609.17926
AB - The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in ℤ_p that derivation has no input. The exact algebraic solution is an orbit of ℤ_p 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(−c h^α), with R^2 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.
ER -