CALC-106

Detector spacing against structural correlation length

REGISTERED 2026-09-14. The closed form two live records already lean on - CLM-0908's accuracy cap and CALC-104's whole premise - which until now existed only in a working file. Provenance, all three hashes re-measured before this row was written and all three matching the values the draft declared:

A NAME IN THE SOURCE DOCUMENT DOES NOT RESOLVE. The model file section 1.6 credits the numerical check to interp_check.py. No such file is held. The check is cassiopeia-interp-residual.py above, confirmed by opening it: NTRIAL = 200_000, an exact function, an asymptotic function, and the design-relation solver at the 0.1 dB budget. The name is stale, not the work. Recorded so the next reader does not go looking for a file that was never acquired under that name.

Purpose. To turn "how far apart may the phase detectors sit" into a relation that can be reasoned about, given that the patent states no spacing and no structural model is held.

Method. Model the line-of-sight displacement s(x) as a stationary random field of rms sigma_s and autocorrelation rho(delta). Detectors at spacing D, linear interpolation between two of them, residual evaluated at the cell midpoint - the worst point inside a cell, so the result is conservative within the cell. Then

    E[eps^2] / k_t^2  =  sigma_s^2 ( 3/2  -  2 rho(D/2)  +  rho(D)/2 )              (7)

Both limits are the right shape, and the second is the interesting one. As D -> 0 the residual vanishes. As D >> Lc, (7) tends to 1.5 sigma_s^2 - worse than applying no correction at all, which would leave sigma_s^2, because uncorrelated neighbours contribute their own variance. Interpolating from detectors spaced further apart than the structure's correlation length actively harms the beam.

It reduces to CALC-085 as required. Where the residual is uncorrelated between elements, eps_n is a random phase error, sigma_phi = k_t sigma_e, and that drops straight into G/G0 = exp(-sigma_phi^2) and the Ruze table.

Verification, and it is two independent checks not one. The author ran 200,000 trials per point against the exact 3x3 covariance. This row re-evaluated the closed form independently, in a separate script, before accepting any of it - not re-running the author's code, evaluating equation (7) from the written form. Gaussian rho(delta) = exp(-(delta/Lc)^2) throughout.

Design relation at 2.45 GHz against CALC-085's 0.1 dB budget, which caps the residual at 2.955 mm rms. Solved from the exact form (7) by bisection, re-solved here:

THE 1 m ROW IS CORRECTED HERE AND THE CORRECTION RUNS IN THE UNSAFE DIRECTION. The draft tabulates 0.083 Lc. The exact solution is 0.0827 Lc, and 0.083 puts the residual at 2.979 mm against a 2.955 mm budget - 0.8 per cent over. The other three rows round down and stay inside the budget; this one rounds up and steps outside it. It is a small number and the direction is what matters: a rounded-up spacing licenses detectors further apart than the budget allows. This row publishes 0.082 Lc. Note what this is an instance of - the record's own limits warn that solving from the quartic overstates the permitted spacing by 34 per cent in the stiffest case, an error in exactly this direction; the rounding does the same thing two orders of magnitude smaller, in the row the quartic warning is about.

THE SMALL-SPACING QUARTIC MUST NOT BE USED AS THE DESIGN RELATION. The asymptote sigma_e -> (sqrt(3)/4) sigma_s (D/Lc)^2 is good to about 1 per cent only below D/Lc = 0.3; at D/Lc = 4 it overstates the residual by a factor of 33. It is recorded rather than dropped because the next reader will re-derive it - it is the natural small-parameter expansion - and needs to know where it stops being usable and which way the error runs.

What this does not establish. The result is a RATIO, and that is the honest limit of it. It says detectors must sit closer than a fraction of the structure's correlation length, and that the fraction falls as the square root of how far the structure moves. Turning it into a spacing in metres needs modal shapes and amplitudes for a 2 km deployable structure, and no held source gives them - see CLM-0907. A real deployable's modes are neither stationary nor single-scale, so this is a relation for reasoning about the trade, not a prediction for a specific structure. The gap it exposes is better defined than the gap the thread began with, which is the useful part.

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