Overview

Project Tesseract studies the geometry of the multiplication table modulo $N$. For each residue $$ a \in \{0,1,\ldots,N-1\}, $$ the lattice set $$ A_{N,a} = \{(x,y)\in\{1,\ldots,N-1\}^2 : xy \equiv a \pmod N\} $$ has a convex hull, and the main quantities are $$ S(N,a) := \operatorname{Area}(\operatorname{conv}(A_{N,a})), \qquad S(N) := \sum_{a=0}^{N-1} S(N,a). $$

The current public surface is organized around three complementary entry points: a short expert-facing compact note, a longer monograph with proofs and context, and an interactive explorer for concrete moduli and residues.

Start Here

• Main results — the current theorem package in one place
• Compact note — short overview of the paper and its scope
• Compact note PDF — direct download of the short manuscript
• Full monograph — longer proofs, examples, and context
• Monograph PDF — direct download of the full manuscript
• References — closest literature around the project
• Interactive explorer — live residue-class and hull exploration
• Examples gallery — the fastest visual route into the subject

Current Proved Package

• Every S_N_a and hence every S_N is a nonnegative integer.
• The zero class is degenerate exactly when $N$ is prime or $N=4$.
• For composite $N$, the hull of the zero class is exactly the convex hull of divisor rectangles.
• The zero-class area admits both a lower-envelope formula and an exact hyperbola-gap decomposition.
• The first boundary model has an exact residue formula and an exact cubic total.
• The total area satisfies the rigorous cubic bracket $$ \frac{(N-3)(N-2)(N-1)}{3} \leq S(N) \leq N(N-2)^2, $$ so in particular $S(N)=\Theta(N^3)$.
• For odd $N$, the second boundary model is also known exactly.

Suggested Reading Path

1. Start with Main results for the theorem summary.
2. Read the compact note or download the compact-note PDF for the short expert-facing presentation.
3. Continue to the full monograph for proofs, examples, and broader context.
4. Use the interactive explorer and the examples gallery to test the geometry on small moduli.

First Geometric Picture

The zero residue class is currently the clearest solved object. It is where the arithmetic of divisors becomes visible directly in Euclidean geometry; see zero_class_geometry and zero_class_hyperbola_gap.

Try One Example

The first even composite case $N=6$ already shows the basic contrast between the unit side and the zero-divisor side.

For a more visual progression from small cases to the zero-class hull and then to boundary models, go next to examples_gallery.