CALC-103

Frequency-offset residual

Feeds CLM-0904. REGISTERED 2026-09-14, HAVING BEEN DRAFTED AND NEVER ENTERED. CLM-0904 cited it by a working label that matched no register grammar, so every reference check passed it and the record read as though an independent calculation stood behind its power-beam branch. It did exist - in a draft file with full arithmetic and a script - and it had simply never been registered.

Purpose. What a pilot-to-power frequency offset costs a retrodirective array, using the offset of a built system.

Relation. A heterodyne conjugator receiving at f_p and transmitting at f_t applies -k_p.L_n where focusing requires -k_t.L_n, leaving eps_n = (k_t - k_p).L_n = dk.L_n. With the design geometry pre-compensated, eps_n = dk.s_n.

Method, per CALC-101's rule. The offset is taken from SRC-001's stated frequencies, and BOTH OF ITS FREQUENCY PLANS GIVE 20 MHz - design 2.496 minus 2.476, as-measured 2.495 minus 2.475 - so no plan-crossing is possible here. This calculation uses the DESIGN plan throughout and every constant traces to it: mixing plans across steps is what produced the 9 MHz defect in CALC-101.

Inputs. f_p = 2.476 GHz, f_t = 2.496 GHz, df = 20 MHz (SRC-001). dk = 2.pi.df/c = 0.41917 rad/m. k_t = 2.pi.f_t/c = 52.312 rad/m. Budget sigma_phi = 0.15175 rad (CALC-085, 0.1 dB).

Results. - fractional offset df/f_t = 0.008013 - error without conjugation, per metre of line-of-sight shift: 52.312 rad - error with offset conjugation, per metre: 0.41917 rad - improvement factor `f_t/df` = 124.8 - tolerable rms displacement from this term alone: 0.15175/0.41917 = 0.3620 m

Quadrature combination. The interpolation and frequency-offset residuals are independent: sigma_phi^2 = [k_t.(s - I[s])]^2 + [dk.s]^2. At 20 MHz a structure moving 300 mm rms spends 0.12575 of the 0.15175 rad budget on the offset term, leaving 0.08494 rad - 1.624 mm of path error for the interpolation residual and everything else.

Provenance. cassiopeia-freq-offset.py, sha256 692fc06a8b7f4945, standard library only, run 2026-09-13. Every figure above was recomputed independently on 2026-09-14 before registration and all reproduce exactly, and the script hash was verified against the file rather than taken from the draft.

What this does not establish. Dispersive contributions from the ionosphere and the feed network, a separate mechanism handled in the literature with multi-tone pilots. It is a single-term budget: it says what the offset costs, not what the whole error budget is.

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