Value chain · CLM-0912
In plain terms
How far apart the sensors may sit depends on how smoothly the structure bends, not on how big it is. The answer comes out as a fraction of the distance over which the bending stays similar to itself, and that fraction shrinks as the square root of how much the structure moves. Spacing them more than about 2.35 times that distance apart is worse than not correcting at all. To turn any of it into a number of metres somebody has to model how a two-kilometre structure actually flexes, and nobody here has.
The claim as recorded
For a line-of-sight displacement field of rms sigma_s and autocorrelation rho, sampled at detector spacing D and linearly interpolated, the residual variance is sigma_s^2 ( 3/2 - 2 rho(D/2) + rho(D)/2 ), evaluated at the cell midpoint which is the worst point in the cell. Against CALC-085's 0.1 dB budget at 2.45 GHz, which caps the residual at 2.955 mm rms, the permitted detector spacing is 2.30 correlation lengths for 3 mm rms deformation, 0.89 for 10 mm, 0.26 for 100 mm and 0.082 for 1 m. The relation also fixes its own ceiling, and the threshold is NOT one correlation length. The residual rises to a limit of 1.5 sigma_s^2 as D grows, which is worse than applying no correction at all, because uncorrelated neighbours contribute their own variance instead of cancelling the element's - but it does not reach the uncorrected variance until D = 2.3515 Lc, and at D = Lc the residual is 0.1263 sigma_s^2, still removing 87 per cent of the variance. See CLM-0917. The result is a RATIO of detector spacing to structural correlation length, and converting it to a spacing in metres requires modal shapes and amplitudes for a 2 km deployable structure, which no held source gives. That structural model is the gap, and it is a better defined gap than the one this thread began with.
Evidence behind this: T0-derivation with numerical verification. CALC-106 carries the closed form, the 200,000-trial Monte Carlo agreeing with it to 0.2 per cent at four spacings, the worked design table and the re-solve that corrected one row.
Scope and limits
A real deployable's modes are neither stationary nor single-scale, so this is a design relation for reasoning about the trade, not a prediction for a specific structure - which is one of the reasons the numerical model is worth building. The midpoint choice makes it conservative within a cell. THE SMALL-SPACING QUARTIC MUST NOT BE USED AS THE DESIGN RELATION. The asymptote (sqrt(3)/4) sigma_s (D/Lc)^2 is accurate only below about D/Lc = 0.3; at D/Lc = 4 it overstates the residual variance by a factor of 33 (5.7 in rms). Solving the design relation from the quartic rather than the exact form gives 1.508 Lc against the exact 2.301 Lc in the stiffest case tabulated: it UNDERSTATES the permitted spacing by 34 per cent, because it overstates the residual, which is the safe direction - it would license detectors closer than the budget needs, not further apart. It is barred as a design relation for being wrong by a third, not for being unsafe. The figures above are solved from the exact form by bisection. THE 1 m ROW WAS CORRECTED ON REGISTRATION, AND THAT ONE DID RUN IN THE UNSAFE DIRECTION. The draft tabulated 0.083 Lc; the exact solution is 0.0827 and 0.083 puts the residual at 2.979 mm against the 2.955 mm budget, 0.8 per cent outside it. The other three rows round down and stay inside. This record publishes 0.082. It is a small number and the direction is the point: a rounding that licenses a spacing the budget does not cover, in the row where the margin is thinnest. NO STRUCTURAL MODEL IS HELD - see CLM-0907 for the scope of that absence. A TRAP FOR ANYONE FEEDING MEASURED DATA INTO THIS RELATION, FOUND 2026-09-17 (CALC-110). The residual is exquisitely sensitive to 1 - rho at small lag. A finite-length, non-stationary field - which every real deformation record is - yields a spuriously linear fall in rho at small lag if the autocorrelation is normalised by the global variance rather than per lag, and this relation reads that as short-scale roughness and manufactures a residual: 1 mm at 7 m spacing for a 10 mm field that had no such roughness. Normalise the autocorrelation per lag. The same run also established that this closed form OVERSTATES the residual of a modal field by 1.2 to 3x across the design range, so it is a safe bound rather than an estimate (CLM-0955).
What it assumes
A stationary random displacement field with a single correlation length, Gaussian autocorrelation for the worked figures, and linear interpolation between two detectors.
Where it comes from
This is the project's own working, shown in full:
CALC-106. Closed form verified against a 200,000-trial Monte Carlo drawing from the exact three-point covariance: agreement within 0.2 per cent at D/Lc of 0.1, 0.5, 1.0 and 4.0. VERIFIED ON REGISTRATION BY RE-EVALUATION RATHER THAN RE-RUNNING, 2026-09-14. The equation was evaluated from its written form in a separate script, not by executing the author's: 1.8672e-05, 1.0574e-02, 1.2634e-01 and 1.4634e+00 at those four spacings, reproducing the published closed-form column. The design table was re-solved by bisection from the exact form, giving 2.3011, 0.8948, 0.2631 and 0.0827 Lc. Provenance re-measured before use: the working file cassiopeia-conjugation-model.md (sha256 prefix bdbd8ba5de6da995, 18,325 bytes) and the script cassiopeia-interp-residual.py (b92c8407ddc51f64, 4,212 bytes), both matching the hashes the draft declared.
Basis: dimensionless (detector spacing as a multiple of structural correlation length)
Related records
- Answers question
- QST-013 not published in this release
Cite this
energyaperture.com/c/clm-0912
CLM-0912, Energy Aperture, as of 2026-09-13.