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The nano-zeqond

The Zeqond is usually written 0.777 s, but that's the rounded number for humans. Underneath, the framework's clock is defined at a far finer grain — the nano-zeqond, the nanosecond tier where the tick is an exact integer and nothing is allowed to drift:

TZ=777,000,777 ns (exact)fH=109777,000,777 HzT_Z = 777{,}000{,}777 \text{ ns (exact)} \qquad f_H = \frac{10^9}{777{,}000{,}777}\ \text{Hz}

That single exact integer is the source of truth for every clock the framework runs. 0.777 s is a display of it; 777,000,777 nanoseconds is it.

Why integers, not seconds​

Compute the phase in floating-point seconds and it will, eventually, drift — rounding accumulates, and two surfaces that reduce the time slightly differently end up disagreeing about what tick it is. The framework refuses that failure mode by working in integer nanoseconds with exact arithmetic:

τ=⌊tns777,000,777⌋φ=tns mod 777,000,777777,000,777\tau = \left\lfloor \frac{t_{\text{ns}}}{777{,}000{,}777} \right\rfloor \qquad \varphi = \frac{t_{\text{ns}} \bmod 777{,}000{,}777}{777{,}000{,}777}

Both use whole-number (big-integer) division and modulo, so the same nanosecond always maps to the same tick and phase, on every node, forever. There is no float in the reduction to round, so there is no drift to accumulate. The tick τ is primary and integer; the phase φ ∈ [0,1) is derived from it exactly.

The identity that makes the modulation honest​

Defining the second as exactly 1/f_H isn't cosmetic — it's what makes the KO42 modulation provably harmless. Because the frequency and the tick are exact reciprocals:

fH⋅TZ=1exactlyf_H \cdot T_Z = 1 \quad\text{exactly}

the sinusoid sin(2π f_H t) completes exactly one whole cycle per Zeqond. Averaged over a Zeqond, a whole cycle of a sine integrates to zero with no residue — so the modulation adds phase information about when a result was computed while provably not biasing the value it carries. That guarantee only holds because the tick is exact to the nanosecond; a rounded 0.777 s would leave a tiny cycle fragment each tick, and the average wouldn't be clean. The nano-zeqond is what buys the ≤0.1% bound its exactness.

Zeqond-aligned days​

The same grid extends upward. When the framework needs a daily boundary — for quotas, resets, or windowed reads — it doesn't use raw UTC midnight; it uses a boundary aligned to the Zeqond grid, so a "day" begins on a clean tick rather than a fractional one. Daily accounting lands on the same clock as everything else, kept consistent with the kernel rather than with the wall clock.