Examples Gallery

These examples are the quickest route from the arithmetic definition to the geometric objects. They are not meant to replace proofs; they are meant to make the proofs easier to read once you get there.

1. Small Moduli: $N=5$ And $N=6$

The first two worked cases already show the main split of the subject.

$N=5$ is the clean prime case: no zero class, full permutation-side geometry on the nonzero grid.
$N=6$ is the first even composite case: the zero class appears and is already two-dimensional.

Residue classes for $N=5$ and their convex hulls.

Residue classes for $N=6$ and their convex hulls. The zero class is now visible.

Best companions:

2. The Zero Class At $N=12$

The modulus $N=12$ is the most informative single worked example in the present project. It is the place where divisor rectangles, lower-envelope geometry, and the hyperbola-gap correction can all be seen in one picture.

The zero class for $N=12$ is enlarged by intermediate divisors, not only by the extreme pair.

The lower envelope through divisor points controls the whole zero-class hull.

The polygonal lower hull lies above the continuous hyperbola $y=N/x$.

Best companions:

3. Boundary Models In One Residue Class

The first boundary model is not an ad hoc truncation. It is the first exact geometric layer forced by the border-by-border construction of the table.

For $N=11$ and $a=3$, the first-boundary parallelogram gives an explicit inner model of the full convex hull.

The four outer-frame points already determine the exact first-boundary parallelogram.

Best companions:

4. Try The Composite Example Live

Use the explorer for $N=12$ to compare residue classes, hulls, and boundary models directly.

Where To Go Next

If you want the strongest theorem package, go to main_results.
If you want the short expert-facing route, go to compact_note.
If you want the longer proof route, go to full_monograph.