Take the triangular numbers — the cumulative sums of natural numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45 … Reduce each to its digital root (sum digits until single digit). The result is a period-9 cycle: [1, 3, 6, 1, 6, 3, 1, 9, 9]. This cycle repeats exactly every 9 terms. It is called the Triangular Digital Root Sequence (TDRS).
Digital roots of T(1) through T(9): [1, 3, 6, 10→1, 15→6, 21→3, 28→1, 36→9, 45→9]
When the cycle is reordered to its palindromic form, it reads the same forward and backward. The center value is 1. The sequence is symmetric around unity. The 9s anchor both ends.
The sum of the full cycle is 1+3+6+1+6+3+1+9+9 = 39, which reduces to 3. The sum of the palindromic form is identical. This is not a coincidence — it is the structure.
The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … — when each term is reduced to its digital root, produces a cycle of exactly 24 terms before repeating. This is called the Pisano period for digital roots.
The cycle repeats exactly every 24 terms.
108 reduces to 9. The cycle sums to 9.
Positions 12 and 24 — the midpoint and the end.
Figurate numbers describe how points arrange into regular geometric shapes. They are the bridge between arithmetic and geometry — the same number expressed as both a count and a form.
The 13-month natural calendar divides the solar year into 13 months of exactly 28 days each, plus one day out of time. 13 × 28 = 364. The triangular number T(13) = 91. And 91 × 4 = 364. The calendar year is four times the 13th triangular number.
28 is the 7th triangular number: T(7) = 28. Its digital root is 1. Each 28-day month completes a full triangular cycle. The TDRS period-9 cycle maps cleanly onto the 28-day structure: three complete TDRS cycles (27 days) plus one — the anchor day that begins the next cycle.
The Triplet-Window Lattice is a 13×13 grid structure that emerged from the intersection of the TDRS, the Fibonacci 24-cycle, and the figurate number sequences. Each cell in the lattice corresponds to a specific numerical relationship between day-of-year position, life path number, and triangular digital root. The lattice revealed itself on March 10 through direct pattern recognition — not constructed, observed.
The Fibonacci 24-cycle divides naturally into three windows of 8. Each window of 8 terms sums to 36 (digital root 9). Three windows × 8 terms = 24. The 13-month calendar contains 52 weeks — 52 = 2 × 26 = 2 × 2 × 13. The Fibonacci 24-cycle and the 13-month structure share the same base-9 digital root architecture.
| Sequence | Key Number | Calendar Connection |
|---|---|---|
| TDRS | Period 9 | 3 cycles of 9 = 27 days; +1 anchor = 28-day month |
| Fibonacci 24-cycle | 24 terms, sum 108 | 24 = 13 + 11; 108 ÷ 13 = 8.3… (Fibonacci ratio) |
| Triangular T(n) | T(13) = 91 | 91 × 4 = 364 = 13 × 28 (full calendar year) |
| Tetrahedral Te(n) | Te(7) = 84 | 84 = 3 × 28 (three natural months) |
| Square S(n) | S(13) = 169 | 169 = 13² — the Triplet-Window Lattice grid |
The academic foundation for these sequences is documented in the Coherosance Hexalogy — six papers available on Zenodo.