Partition numbers satisfy striking congruences along three arithmetic progressions. I want the notation, the mechanism, and the failure mode visible at the same time.
The mathematical object
A modular form \(f\) on the upper half-plane transforms predictably under fractional linear maps and has a Fourier expansion in \(q=e^{2\pi i\tau}\). Its coefficients often encode arithmetic.
I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).
One explicit computation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Why the identity matters
The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.
Where it can fail
Weight, level, character, and cusp conditions are part of the definition. A formal \(q\)-series is not automatically a modular form.
This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.