Small Examples Atlas

Why this page exists

The first worked moduli should be read as complete mini-cases: raw table first, then the residue classes, then the hull overlay, and only after that the area values. This is the most concrete bridge from the elementary arithmetic definition of MTMN to the geometric quantities S_N_a and S_N.

The case $N=5$

Since $5$ is prime, every nonzero index is coprime to $5$, so this is the cleanest first example.

Raw multiplication table modulo $5$, showing only the residue values.

$Residue classes $A_{5,a}$ and their convex hulls. Each panel title records $a$ and $S(5,a)$.$

Multiplication table modulo $5$ with convex hulls of all residue classes overlaid.

The exact values are $$ (S(5,0),S(5,1),S(5,2),S(5,3),S(5,4)) = (0,3,4,4,3), $$ so $$ S(5)=14. $$

The case $N=6$

The first even composite case already shows the new zero-divisor side.

Raw multiplication table modulo $6$, showing the residue values before any geometric grouping.

$Residue classes $A_{6,a}$ and their convex hulls. The zero class is now present and already two-dimensional.$

Multiplication table modulo $6$ with convex hulls of all residue classes overlaid.

The exact values are $$ (S(6,0),S(6,1),S(6,2),S(6,3),S(6,4),S(6,5)) = (2,0,9,8,9,0), $$ so $$ S(6)=28. $$

What the atlas teaches early

prime moduli show the clean permutation-plot regime on the nonzero grid
composite moduli introduce a visible zero class and zero-divisor geometry
the convex-hull areas are already nontrivial integers in very small examples
these pages prepare the later exact stories for first_boundary_model and zero_class_geometry

MTMN — the main object illustrated here
A_N_a — residue classes displayed panel by panel
S_N_a — per-residue areas listed in the examples
S_N — total area sums
zero_class_geometry — the composite-side picture visible at $N=6$

Small Examples Atlas

Why this page exists

The case $N=5$

The case $N=6$

What the atlas teaches early

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