A New Chronometry

The manuscript proceeds in four parts. Part I lays out the main theory: the spiral equation, the duality m = 2 as the real form of time (25 ≡ 1 mod 12; 15 ≡ 1 mod 7; 731 ≡ 1 mod 365), the bijection between moment and tuple, and the leap cycle (1461 and 2922). Part II develops the Diophantine structure: designed representations with unbounded coefficients versus positional representations with bounded coefficients (the clock [3600, 60, 1]), the (25, 12) system and its Frobenius bound, and the time ladder of the (19, 9) system. Part III treats the numbers 19, 42 and 2520 as bridges between small and large scales, culminating in the universal p = 2521. Part IV closes the fractal: the millennial calendar (q = 360,000), the 7000-year cycle (q = 2,520,000), the 3600-year cycle and the full chain from millisecond to 7000 years. The three classes A, B and C order all moduli by their digital-root behaviour. The mathematical structure is exact within modular arithmetic; numerical nearnesses between scales are noted as such.