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

1.3 Identifiability and aliasing

Single channel. With one channel, E(t) is periodic; many t minimize it (aliases). No unique solution.

Two channels. If periods are commensurate or nearly so (e.g., 1 and 7 days), strong alias valleys remain. A single snapshot can land in a wrong valley at moderate noise.

Three or more channels. Diversity collapses many aliases, but under real noise and short windows a wrong valley can still win.

Takeaway. Aliasing is a feature of periodic inference, not a bug. We do not “tune” the physics; we make the inverse numerically stronger so the correct valley wins (see 1.6).

1.4 Confidence from curvature (local sharpness, v1.1)

At the minimum tau_hat, estimate numeric curvature and map to [0,1) with a bounded, alignment-friendly squash:

curv      = ( E(t+h) - 2*E(t) + E(t-h) ) / (h*h)   # small h in days
curv_norm = curv / max( E(t), eps_E )              # scale-invariant normalization
conf      = tanh( c_conf * curv_norm )             # c_conf > 0, bounded in [0,1)
  • Interpretation. Sharper bowls (higher curv_norm) imply higher local certainty.
  • Parameters. eps_E prevents division by zero; c_conf controls gain.

1.5 Gentle reliability gating (alpha_i)

Design. Weight each channel by a smooth factor that reflects phase “readiness” (e.g., proximity to a cusp). Keep it small and never re-fit physics.

z_i in [0,1]                    # higher is better (e.g., near special angles)
alpha_i = 1 + k_z * (1 - z_i)   # k_z small, e.g., 0.15

One concrete z_i (piecewise-linear, ASCII).
Let d_i = distance (deg) from phi_i_obs to the nearest target angle set.
For rasi cusps, the target set is { 0, 30, 60, ..., 330 }.
Define half-width H (e.g., 15 deg):

z_i = max( 0 , 1 - d_i / H )
  • If a channel is far from any target, its alpha_i rises only slightly, tempering its influence without brittle thresholds.

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