CALC-101

Timing reference accuracy and reference-network path tolerance

Purpose. Establish what accuracy a distributed timing reference needs, and what path-length accuracy the reference distribution network needs.

Inputs. CORRECTED 2026-09-14: THIS LINE PREVIOUSLY CROSSED SRC-001's TWO FREQUENCY PLANS AND THAT IS WHERE THE 9 MHz CAME FROM. It took the pilot from the design plan and the reference from the as-measured plan, and 2.485 - 2.476 = 9 MHz exactly. The source carries two self-consistent plans and states neither as 9 MHz.

Both plans give f_IF = 10 MHz and a 20 MHz offset, and SRC-001 states the IF directly twice - "in the 10 MHz range". "9 MHz" occurs nowhere in the paper. Figures below use the AS-MEASURED plan and say so where it matters. CONSTANTS NORMALISED TO THE AS-MEASURED PLAN, 2026-09-14, AND THE PREVIOUS PAIRING DID NOT HOLD. k_t = 52.291 rad/m (2.495 GHz) and k_ref = 52.082 rad/m (2.485 GHz), both recomputed as 2.pi.f/c. The superseded values were 52.312, which is the DESIGN beam at 2.496 GHz in a section declaring the as-measured plan, and 52.070, which implies 2.48444 GHz and is neither plan's reference. A REPAIR THAT ATTACHED A FREQUENCY LABEL TO A NUMBER IT DID NOT COME FROM IS THE ORIGINAL DEFECT OF THIS CALCULATION, COMMITTED BY ITS OWN FIX. Nothing downstream moves: both plans agree at the stated precision and no claim cites either wavenumber. Budget sigma_phi = 0.15175 rad (CALC-085, 0.1 dB), independent of the frequency plan.

Method, and it is the rule this calculation exists to carry forward: THE INTERMEDIATE FREQUENCY IS TAKEN AS THE SOURCE STATES IT, NOT DERIVED BY SUBTRACTING TWO FREQUENCIES. SRC-001 gives the IF in words - "down converted to a lower frequency IF signal (in the 10 MHz range)" - and labels a 10 MHz VCXO on Figure 9. SUBTRACTING IS WHAT PRODUCED THE ERROR: two self-consistent plans sit one megahertz apart on the same printed page, the figure is what the eye goes to and the text is what gets quoted, so a pilot from one and a reference from the other differ by 9 MHz - a number present in neither convention and in no part of the paper. TWO READERS IN SEPARATE SESSIONS, WITH NO CONTACT, MADE THAT IDENTICAL SUBTRACTION FROM THAT PAGE. That is a property of the page, not of the readers. Where a source states a derived quantity directly, use the statement and treat the arithmetic as a cross-check that must agree - not the other way round.

Step 1, what is measured. A detector reports the pilot's phase relative to the local reference. d(psi)/dt = 2.pi.f_IF + k.v. A differential timing error dt between detectors produces a phase error d(psi)/dt . dt. Common timing offset is piston and costs nothing.

Step 2, structural branch. dt < sigma_phi / (k_t . 2.pi.f_m.a). Parametric, since no modal model exists:

Step 3, beat branch. dt < sigma_phi / (2.pi.f_IF). At 10 MHz: 0.15175/(2.pi.10e6) = 2.415 ns. At 1 MHz: 24.15 ns. At zero IF: the term vanishes.

Step 4, scale against the structure. Light crosses 2 km in 2000/c = 6.671 us. That is 2,762 times the 10 MHz requirement, and 692 times smaller than the 1 Hz / 0.1 m structural requirement. So a broadcast tick is harmless against structural motion and inadequate against a megahertz beat unless its propagation delay is compensated.

Step 5, residual after static delay compensation. Delay error from displacement along the tick path is s/c, costing 2.pi.f_IF.s/c against a useful sensitivity k_p.s. Ratio f_IF/f_p = 10e6/2.475e9 = 0.004040, a degradation factor of 247.5 (as-measured pilot). The superseded 275 OVERSTATED the penalty, so unlike Step 3 this correction ran in the conservative direction, not the lenient one.

Step 6, reference network path tolerance. A heterodyne element transmits phi_LO,n - k_p.L_n, so phi_LO,n reaches the far field with coefficient 1. Tolerable path error = sigma_phi / k_ref = 0.15175/52.082 = 2.914 mm, against 2.955 mm for element position from CALC-085.

Provenance. cassiopeia-timing-reference.py, sha256 65ae978d6399885d, 4,854 bytes. Standard library only. Run 2026-09-13, Python 3.14.0.

What this does not establish. Any real structure's modal behaviour, so step 2 is parametric and no value from it may be quoted as a requirement. Nor whether any distribution scheme meets step 6.

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