A π-day is a day whose calendar fields, read in order and stripped of separators, spell the digits of π. In the American middle-endian reading, 3/14/15 becomes 31415 — the first five digits. A collision is rarer: a single day on which two or more independent calendars each read π at once.

chrono-pi does not look for these by walking the calendar day by day. Each (calendar, reckoning) pair compiles to a period P and a set of active residues A ⊆ Z/P — the days, modulo one supercycle, on which that calendar reads π. Two calendars collide exactly where their residue classes are simultaneously satisfiable, which the Chinese Remainder Theorem resolves in closed form. The witnesses are then enumerated arithmetically, t = t₀ + m·L, over whatever range you ask for — even ranges millions of years out that no scan could ever afford.

That is the whole trick, and the rest of this blog is about the mathematics that makes it exact: the Kalenderkrockssaten (the collision theorem), the Tågrälssatsen (the witness-class algebra), and the Tibiasatsen (the dynamic layer for calendars that refuse to be periodic).