SSM-Clock—Core objects & Horizon (1.1–1.2)

1.1 Core objects (forward and inverse)

Forward phase model (per channel i).

phi_i(t) = wrap360( b0_i + w_i * t )  # degrees
  • t = time in days on a latent scalar axis.
  • b0_i = frozen initial phase (deg).
  • w_i = frozen angular speed (deg/day).
  • All channels share the same t (the latent “now”).

Inverse clock (estimate time from observed phases).

err_i(t) = angdiff( phi_i_obs , b0_i + w_i * t )  # shortest signed diff in deg
E(t)     = SUM_i [ alpha_i * (err_i(t))^2 ]
tau_hat  = argmin_t E(t)
  • phi_i_obs = measured phase (deg) for channel i.
  • alpha_i >= 1 = gentle reliability weights (see 1.5).
  • E(t) is a scalar; choose the t that minimizes it.
  • The solution lives on a circular horizon (see 1.2).

Angle utilities (ASCII, branch-safe).

wrap360(x):  y = x % 360.0 ; return (y + 360.0) if (y < 0.0) else y
angdiff(a,b): return ((a - b + 180.0) % 360.0) - 180.0

Why squared residuals?

  • Smooth + non-negative. Squared angular distance keeps E(t) smooth and ≥ 0, with a unique bottom inside each valley.
  • Curvature-ready. Absolute errors locate valleys similarly, but squared errors yield a clean curvature for confidence (1.4).

1.2 Horizon selection (T_search) and circular distance

Rule (deterministic).

  • If all periods are integers (days):
T_search = LCM(periods)
  • Else:
T_search = max(periods)

Rationale. The inverse only needs to search one fundamental window that repeats. When any period is non-integer (e.g., 29.5306), the exact joint repeat is astronomically long; using max(periods) is a tight, sufficient window when combined with stacking (1.6).

Circular error (time reported modulo horizon).

err_time_days = min( |tau_hat - tau_true| ,
                     T_search - |tau_hat - tau_true| )

This matches the physics: the same phase consensus repeats every horizon.

Navigation

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Next: SSM-Clock—Identifiability, Confidence & Reliability (1.3–1.5)

Explore further:

https://github.com/OMPSHUNYAYA/Symbolic-Mathematical-Clock