This page collects papers related to the Euclidean decomposition of the multiplication table.
| Title | Description | Link |
|---|---|---|
| Euclidean Decomposition of the Multiplication Table | Introduces the master array A(n,q,r), recording the joint distribution of quotients and remainders obtained from multiplication table entries. | 10.5281/zenodo.18960358 |
| Divisor-Window Formula for the Euclidean Decomposition of the Multiplication Table | Studies a zero-based divisor-window viewpoint and related slices of the Euclidean decomposition. | 10.5281/zenodo.18991381 |
| The Support of the Euclidean Decomposition of the Multiplication Table | Studies the support indicator of the multiplication table and its square-grid visualizations. | 10.5281/zenodo.19102497 |
| Euclidean Transport Update for the Multiplication Table | Short earlier update note on the passage from A_n to A_{n+1}. Kept here as part of the development history. | 10.5281/zenodo.19121230 |
| Euclidean Transport of Multiplication | Main transport paper. Defines the quotient-remainder array A_d(n,q,r), proves the rank-independent transport map for fixed products, and studies transport under base change. | 10.5281/zenodo.19273836 |
| On the Motion of Numbers | Studies the motion and appearances of fixed integers across growing multiplication tables. | 10.5281/zenodo.19390886 |
| Continuous Euclidean Transport | Records continuous Euclidean transport under a change of positive real base. | 10.5281/zenodo.19899679 |
| Motion and Animation in Multiplication | We record immanent coordinate systems carried by the support of the multiplication tables and distinguish fixed-number Euclidean motion from Zusean frame-to-frame animation. | 10.5281/zenodo.19899679 |
For a fixed integer n, multiplication table entries may be written as
i*j = q*n + r,
where q is the quotient and r is the remainder.
The master array records the number of ways a state occurs:
A(n,q,r) = #{(i,j) : i*j = q*n + r}.
The papers study this object through its projections, marginals, supports, visualizations, and transport behavior.