A companion to Learn, using the named TLF seed throughout.
Products and modular labels
An ordinary N × N multiplication table has entries r · c for r,c = 1,…,N. Its largest entry is N²; extending the table indefinitely gives unbounded products. A modular grid instead retains a remainder and displays it using a declared label convention.
Write [x]N for the nonnegative remainder 0,…,N−1. One common positive representative convention writes remainder 0 as N and leaves other remainders unchanged. The published TLF display uses a different convention: add 1 to every remainder, so 0 becomes label 1 and N−1 becomes label N. Either is a representation choice; neither creates extra residue coverage.
Offsets and phase in TLF
With integer N ≥ 2 and r,c,k in 1,…,N, the current definition is:
wrapN(x,N) = ((x−1) mod N) + 1
A_TLF(r,c,k,N) = (((r−1) · wrapN(c−1,N) + (k−1)) mod N) + 1
= [ (r−1)(c−1) + (k−1) ]_N + 1
The second form follows because wrapN(c−1,N) is congruent to c−1 modulo N. The coordinate offsets r−1 and c−1 matter: TLF is not obtained just by reducing the ordinary r · c table. Phase k adds the same offset k−1 before reduction. It changes labels cyclically. Ordinary multiplication tables can also be animated or transformed; here phase has this specific mathematical definition.
A worked N = 9 comparison
At r = 4, c = 3, the ordinary product is 12, with remainder 3 modulo 9. For TLF at k = 1, the offset product is 3 · 2 = 6, displayed as label 7. At k = 2 it becomes remainder 7, displayed as label 8. The table applies those same declared conventions at selected columns; its values are direct substitutions.
| c | Product 4c | Product remainder (0–8) | TLF label, k = 1 (1–9) | TLF label, k = 2 (1–9) |
|---|---|---|---|---|
| 1 | 4 | 4 | 1 | 2 |
| 2 | 8 | 8 | 4 | 5 |
| 3 | 12 | 3 | 7 | 8 |
| 4 | 16 | 7 | 1 | 2 |
| 9 | 36 | 0 | 7 | 8 |
Exactly when a row or column visits every label
A sequence [a · t + b]N has least positive period N/gcd(N,a). Indeed, a shift h repeats it precisely when N divides a · h; dividing by gcd(N,a) gives the stated minimum. It visits every residue once over N positions exactly when gcd(N,a) = 1. Adding b or changing the display labels does not change that result.
Thus TLF row r has horizontal period N/gcd(N,r−1), and column c has vertical period N/gcd(N,c−1). Each is a permutation of all N labels exactly when the corresponding gcd is 1. For N = 9, rows and columns with index in {2,3,5,6,8,9} meet that condition. Index 1 has period 1; indices 4 and 7 have period 3. In the worked row, k = 1 cycles through 1,4,7 and k = 2 through 2,5,8.
Arithmetic periods versus visual motifs
Exact modular repetition does not by itself prove visible rectangular windows. The Chinese Remainder Theorem describes residues through pairwise coprime prime-power components; a visual claim also needs its phase, orientation, projection or color rule, and layout specified. Historical window observations are retained in the archive note.
The Family defines two seeds and eight named parameter choices, including reflections. They need not yield eight distinct grids at every size: positive and negative signs coincide modulo 2. See Research for evidence and limits beyond this elementary arithmetic.