Total Area Sum $S(N)$

Definition

$$ S(N) = \sum_{a=0}^{N-1} S(N,a). $$ This is the simplest scalar summary of the full residue-area profile.

What is already known

Every S_N_a is a nonnegative integer, so $S(N)$ is a nonnegative integer.
Exact small values are already visible in the first worked examples and tables for small moduli.
The boundary models first_boundary_model and second_boundary_model give exact cubic lower models.
One always has the rigorous cubic bracket $$ \frac{(N-3)(N-2)(N-1)}{3} \leq S(N) \leq N(N-2)^2. $$ So the exponent $3$ is already forced at the level of order of magnitude.
The series $$ \sum_{N=4}^{\infty} \frac{1}{S(N)} \qquad\text{and}\qquad \sum_{N=4}^{\infty} \frac{N}{S(N)} $$ both converge by comparison with the first-boundary series.

Small exact values

$N$	$S(N)$
4	2
5	14
6	28
7	70
8	108
9	205
10	334

For the residue-by-residue tables behind these totals, see S_N_a and small_examples_atlas.

Growth and arithmetic sensitivity

The current proved global statement is the cubic-order theorem $$ S(N)=\Theta(N^3), $$ coming from the exact first-boundary lower model together with the square-window upper bound. The solved zero class in zero_class_geometry shows directly that zero-divisor geometry contributes in a mathematically rigid way to the total.

Figures

$Comparison of the exact total $S(N)$ with the first-boundary model $S^{(1)}(N)$.$

Overlay of residue-class hulls across several moduli. The total $S(N)$ is the sum of these per-residue areas.

S_N_a — individual residue-class areas
small_examples_atlas — first exact totals in small moduli
first_boundary_model — exact cubic lower model $S^{(1)}(N)$
second_boundary_model — exact odd-$N$ second-layer lower model
zero_class_geometry — solved contribution from the zero residue