CALC-111

Feasibility of sensing structural displacement from the reference tone's frequency

The sensing idea raised at G3 on CLM-0903, tested against held figures: velocity and displacement resolution, what the reference carries in the baseline architecture, the projection from reference path to line of sight, and integration drift. Producer-computed 2026-09-17; the producer is barred from verifying its own work, and every headline figure was reproduced independently before this landed (recorded at the end). The answer is that the idea does not work as stated, for three independent reasons, and a narrower use survives.

Provenance. The project's working file ref_doppler_sensing.py, 11,127 bytes, sha256 BD6DAA233DAD45FE, 145 lines, no dependencies. Inputs: CALC-109 bands; CLM-0902's 2.914 mm tolerance and 2.485 GHz reference; RP-1076 (SRC-107) pp.14, 57, 75, 98, 224, 232 read first-hand; SRC-451 Vol III p.34 for the 500 m half-span and 40 m depth.

1. Resolution - frequency counting fails; phase tracking works, and phase tracking is already a displacement measurement. At 2.485 GHz, lambda is 120.7 mm and 1 mm/s is 8.3 mHz. The slowest structural case from CALC-109 - mode 1 at 0.088 Hz and 2.5 mm - moves at 1.4 mm/s, 11 mHz. Counting frequency over one mode period gives 1/T resolution: 91 to 667 mHz, which misses that by 8 to 58x. Tracking the reference phase instead, at RP-1076 p.75's carrier-loop jitter of under 0.1 degree rms, resolves 34 micrometres of path and 4 to 32 micrometres per second - the whole band with margin. But s = (lambda / 2 pi) phi: a tracked phase IS the displacement. Integrating its derivative adds noise and removes nothing. The idea, correctly implemented, collapses into continuous reference-phase tracking.

2. Architecture - in the 1981 baseline there is no Doppler at the element to read, by design. RP-1076 p.98, transcribed from the render: the master slave returnable timing system phase-locks a returned sample so the phase at the element "is driven to, and maintained at, the same phase as the reference signal at point A, even if the path length varies." The loop drives the phase rate at the element to zero; the path-length rate exists inside the phase control centre's correction signal, not at the element. Breadboard residual 4.2 degrees rms (p.224) = 1.41 mm; budget 5 degrees per level over four levels (p.14 Fig III-25), 10 degrees total (p.57) = 3.35 mm, against CLM-0902's 2.914 mm. This is the "distribution scheme that measures and corrects its own path lengths" that CLM-0902's limits said would change the picture - it is the 1981 baseline, held, and uncited on that point. SRC-001's breadboard has no such loop, so a one-metre array would see the Doppler; CASSIOPeiA's distribution is proprietary and unknown either way.

3. Geometry - the reference path is not the line of sight, and the projection is zero or a strain. (i) An in-plane free-space master: normal displacement changes the path only at second order, cos(psi) = 0; raising the master 40 m gives 0.08 at the edge, and order-unity sensitivity needs a stand-off of the array's own radius. (ii) A cable or fibre routed through the structure carries temperature - RP-1076 Fig IV-11, IVPO fibre, 1.3e-11 deg/Hz/m/degC read off the plot, 5.4 mm per degC over a 500 m run at 2.485 GHz, crossing CLM-0902's tolerance at 0.54 degC differential (which is why MSRTS exists) - and cap strain: a bending mode of amplitude A and wavenumber k puts (h/2) k^2 A of path on a cable at the cap and zero on the neutral axis. Mode 1 at 2.5 mm gives 0.6 mm over 500 m; mode 5 at 141 mm gives 420 mm. Bending reaches a cap-routed cable as integrated strain, not displacement, and inverting it to line of sight needs the structural model the idea was meant to supply. (iii) The only reference path with full line-of-sight sensitivity is the pilot from Earth, and its phase is what the conjugator already tracks.

4. Drift - separable from the modes, not from the bow. With phase tracking, white phase noise does not accumulate. Master-oscillator frequency error is common to every element and is piston (CLM-0903). Differential thermal drift of two fibre runs at ASSUMED 0.1 to 1 degC per hour is 0.5 to 5.4 mm per hour, crossing the 2.9 mm tolerance in 30 minutes to 5 hours. Structural velocities are millimetres per second and above at 0.09 Hz and above: two decades apart, separable by a high-pass - at the price of losing the quasi-static thermal bow, which is exactly what CALC-109's amplitude band is.

Could measured displacement replace CALC-110's assumed amplitude? Not directly. What survives: tracking the reference path's phase at the phase control centre gives the high-passed strain signature of the modes, and that constrains the mode mix - the spectral exponent p that moves CALC-110's answer by 2x - if the cable routing and bending kinematics are known. A narrower and real use, and it needs the same structural model as approach 3. The reference tree already reaches every power module (101,552 in RP-1076 p.98), so the sensing density is the tree's.

Limits. (1) The fibre coefficient is read off a plotted curve at 60 MHz and 1060 m and scaled linearly to 2.485 GHz and 500 m. (2) Cable routing at the cap is ASSUMED for the strain figure; on the neutral axis it is zero. (3) Drift rates are ASSUMED. (4) CASSIOPeiA's own reference distribution is not disclosed; the architectural finding is about the RP-1076 baseline. (5) Mode 5 and 10 velocities use the full thermal amplitude as a bound; CALC-110's p >= 1 says higher modes carry far less, so mode 1's rows are the realistic ones.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-17, without the script: lambda 120.7 mm; 1 mm/s as 8.3 mHz; mode 1 at 0.088 Hz and 2.5 mm as 1.38 mm/s and 11.5 mHz; 1/T over 1.5 to 11 s as 91 to 667 mHz; 0.1, 4.2 and 10 degrees of lambda as 34 micrometres, 1.41 mm and 3.35 mm; mode-1 cap strain over 500 m as 0.6 mm. All match. The script runs clean at its declared digest.

POSTSCRIPT 2026-09-18. Rockwell's 1981 reference distribution is free-space from an illuminator at D/2 to 3D above the aperture (CLM-0999), which gives cos(psi) of 0.71 to 0.99 at the aperture edge rather than the 0 to 0.08 computed here for an in-plane master; the geometric objection in section 3(i) is weak for that design. The architectural objection in section 2 stands: Rockwell's phase bridge and servo normalise each subarray's phase to a flat aperture, removing the deformation before it could be read, as MSRTS does.

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