C.1 Identifiability via stacking (alias suppression)
Let E_s(t) be the per-snapshot objective using model phases shifted by s * Delta days. Suppose a wrong valley at t = t* + delta fits snapshot s = 0. For non-zero Delta, snapshot s compares observations to model phases at effective model time t + s*Delta. Unless delta is a period-exact alias for all channels and all offsets s*Delta, residuals decorrelate across s, hence:
E_stack(t* + delta) = SUM_s E_s(t* + delta) > SUM_s E_s(t*) = E_stack(t*).
Therefore t* remains the global minimizer of E_stack under mild noise when channels are sufficiently diverse and at least one non-integer offset Delta is used.
Practical corollary (tested): S >= 3 with Delta = 0.5 d suffices for {1, 7, 29.5306, 365.2422}.
C.2 Curvature -> confidence (monotone, bounded)
Estimate curvature at t = tau_hat via a centered second difference and squash to [0,1):
curv = ( E(t+h) - 2*E(t) + E(t-h) ) / (h*h)
curv_norm = curv / max( E(t), eps_E )
conf = tanh( c_conf * curv_norm ) # c_conf > 0
Properties. conf is strictly increasing in curv_norm, satisfies 0 <= conf < 1, and:
conf -> 1ascurv_norm -> +inf(sharp bowls, low estimator variance),conf -> 0ascurv_norm -> 0(flat bowls, high uncertainty).
C.3 Circular error metric correctness
For horizon T_search, define the circular (geodesic) distance:
err_T(a,b) = min( |a - b| , T_search - |a - b| ).
This is the distance on the quotient circle R / (T_search * Z): symmetric, non-negative, = 0 iff a ≡ b (mod T_search), and it respects the triangle inequality on the quotient space. It is therefore the appropriate notion of time error modulo the horizon.
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