Compact Note

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The compact note is the short expert-facing paper for the current theorem package on the geometry of the multiplication table modulo $N$. It introduces the residue classes $$ A_{N,a}=\{(x,y)\in\{1,\ldots,N-1\}^2 : xy \equiv a \pmod N\}, $$ their convex-hull areas $$ S(N,a)=\operatorname{Area}(\operatorname{conv}(A_{N,a})), \qquad S(N)=\sum_{a=0}^{N-1} S(N,a), $$ and the exact results that are currently strongest.

What The Compact Note Contains

The note is centered on the proved core:

  1. integrality of every residue-class area $S(N,a)$
  2. the sharp zero-class degeneracy criterion
  3. the exact divisor-rectangle description of $\operatorname{conv}(A_{N,0})$
  4. the lower-envelope and hyperbola-gap formulas for $S(N,0)$
  5. the exact first-boundary model
  6. the rigorous cubic-order theorem $S(N)=\Theta(N^3)$

The emphasis is on exact geometric statements, not on a broad survey of secondary extensions.

Mathematical Point Of View

Each fixed congruence class $A_{N,a}$ is already a modular hyperbola in the standard sense. The note takes a different perspective: it keeps the whole residue family in view, treats the convex-hull areas as global invariants of one multiplication table, and makes the zero-divisor geometry of the zero class completely explicit.

Best Companions

Core Supporting Pages