HexaClock — Technical Specification and Theoretical Foundations Exact phase arithmetic in Z/BZ and discrete KAM resonance analysis (Resonance Web)

HexaClock is a number representation system and a set of algebraic operators built on cyclic groups Z mod BZ, where B is an adaptive multiple of 60. It provides exact integer arithmetic for a strictly delimited class of operations: (1) rational fractions with controlled denominators (Phase in Z mod BZ), (2) rotational geometry (Phase), (3) Dirac spinor algebra in standard representations (closure over {+1, -1, +i, -i, 0}), and (4) selected constructions from Clifford algebras via Bott periodicity. HexaClock is not a universal replacement for float/double. It is a deliberately constrained tool which, within its applicability range, achieves zero numerical error for phase operations while operating on integer types. The Magnitude layer (values) is fixed-point and may introduce truncation error under division if the scale does not cover the denominators of results (not only inputs). This subtlety is critical and confirmed by reference tests. Research extension (KAM / resonances): KAM theory studies the persistence of quasi-periodic invariant tori in Hamiltonian systems close to integrable ones, where resonances and "small denominators" are the main practical difficulty. In numerical computations, resonance detection and resonance proximity estimates are often affected by floating-point error and limited reproducibility across platforms. We introduce the HexaClock Resonance Web formalism W_B_K(rho_B) for torus rotations/maps, where: (i) distance to resonance is computed exactly using integer arithmetic via a symmetric remainder modulo B, (ii) small denominators of the form exp(2pii*k·rho) - 1 admit strict bounding bands, and (iii) the number of resonant hyperplanes on the grid has a closed form based on gcd combinatorics. The document includes numerical experiments (resonance web and the standard map) and plots. DSP clarification: when we say "exact twiddle factors" for N divides B, we mean phase exactness (indices in Z mod BZ and phase multiplication as addition modulo). A full DFT/FFT for general data requires values cos(2pik/N) and sin(2pik/N); for many N these are irrational, so they fall into the Magnitude layer or require symbolic machinery (e.g., ring extensions, NTT-like approaches, or cyclotomic rings).