Cyclic Equivariance Contracts

Claim boundary proved finite equivariance executable checks AI quality separate no continuous-rotation claim

Equivariance says that transforming the input transforms the output in the same way. The AI literature uses this idea in group convolutions and steerable networks, including Group Equivariant Convolutional Networks, Distill’s equivariance overview, and E(2)-equivariant steerable CNNs.

Circle Calculus keeps the first contract finite:

shift(s, x)[i] = x[i - s]
f is equivariant when f(shift(s, x)) = shift(s, f(x))

Lean Surface

import Circle.Applications.CyclicEquivariance
import Circle.Applications.Public

The theorem ids are CC-T0149 through CC-T0160. They cover the finite cyclic shift action, closure of cyclic-equivariant maps, circulant-layer equivariance, sum-pooling invariance, reflection involution, reflection/shift interaction, and minimal dihedral identity facts.

Python Example

from circle_math.core import (
    circulant_equivariance_report,
    cyclic_sum_invariance_report,
    dihedral_transform,
)

print(dihedral_transform([10, 20, 30, 40], shift=1, reflected=True))

report = circulant_equivariance_report(
    [2, 0, 1, 0],
    [[1, 2, 0, -1], [0, 3, 1, 2]],
)
print(report.passed)
print(report.max_abs_delta)

pooling = cyclic_sum_invariance_report([[1, 2, 3, 4]])
print(pooling.passed)

These are finite structural checks. The Python examples are executable references, not proof artifacts and not model-quality claims. They do not claim robustness, data efficiency, or continuous rotation equivariance.