Spending the Same Budget a Thousand Times: Phase Error Accumulation Over Repeated Transits

A vehicle that transits repeatedly does not get a fresh phase error budget on each journey; it spends from the same account. This paper derives how per-pass errors accumulate. Independent random errors of sigma_1 per pass grow as sigma_1 sqrt(N) over N passes, while a systematic bias b grows as bN. Against the 0.0667 rad budget, random errors alone permit 177 passes at sigma_1 = 0.005 rad, 44 passes at 0.01 rad, and 11 passes at 0.02 rad. A bias of 0.001 rad per pass exhausts the budget in 66 passes acting alone, and a representative combination of 0.01 rad random error with 0.0002 rad bias reaches the budget at 43 passes. Bias, growing linearly, always overtakes random error eventually: long service lives are a bias-control problem first and a noise problem second. Computed; not measured. SIMULATED, NOT MEASURED.