The divisors of any positive integer H form a lattice — a mathematical structure where every pair of elements has a greatest common divisor (meet) and least common multiple (join).
For H = 360 = 2^3 x 3^2 x 5, there are 24 divisors. Each one is a valid analysis window that tiles H exactly.
The lattice decomposes as a product of independent chains, one per prime dividing H:
Div(360) ≅ {0,1,2,3} × {0,1,2} × {0,1}
prime 2 prime 3 prime 5
Each prime is an independent scale axis. Moving one step on prime-2 doubles the window. Moving on prime-3 triples it.
When analyzing a signal at multiple scales (window sizes), three properties are desirable:
- Artifact-free. Every window tiles the domain without remainder.
- Perfectly nested. Every smaller window partitions every larger window exactly.
- Closed under refinement. The finest shared scale of any two windows is also in the family.
The divisor set Div(H) is the unique maximal family with all three properties. No other window family achieves this.
Each divisor d of H has a unique coordinate vector — its prime exponents:
360 = 2³ × 3² × 5¹ → (3, 2, 1)
60 = 2² × 3 × 5 → (2, 1, 1)
12 = 2² × 3 → (2, 1, 0)
These coordinates give every scale a fixed address in the lattice. The address doesn't depend on analyst choices — it's determined by arithmetic. Two signals with the same H have the same coordinates, so their features at each scale are directly comparable.
This is the basis for structural invariance in SignalForge.