Value chain · CLM-0917
In plain terms
Sensors placed too far apart do not merely help less. Past a certain spacing they make the beam worse than doing nothing, because each is reporting a bend that has nothing to do with the antenna it is being used to correct. That point is about two and a third times the distance over which the structure's bending stays similar to itself - not, as first drafted, one times it.
The claim as recorded
Interpolating pilot phase between sparse detectors stops helping, and then starts hurting, at a computable spacing. Using CLM-0912's closed form, the residual variance equals the uncorrected variance at a detector spacing of 2.3515 structural correlation lengths, and rises to a limit of 1.5 times the uncorrected variance as spacing grows further. Beyond about 2.35 correlation lengths, a linear interpolation between detectors leaves the beam worse than applying no phase correction at all, because samples uncorrelated with the element they are correcting contribute their own variance instead of cancelling the element's. Below that threshold interpolation still helps, and helps strongly: at one correlation length the residual is 0.1263 of the uncorrected variance.
Evidence behind this: T0-derivation. The limiting behaviour of CLM-0912's closed form, with the crossover solved numerically and confirmed by Monte Carlo. Both the crossover and the table were re-solved independently on registration.
Scope and limits
A property of one interpolation scheme, not a law about sparse sampling. A scheme that weighted its samples by measured correlation, or fell back to the design phase where correlation is low, would degrade toward the uncorrected variance rather than past it - it would lose the benefit without incurring the penalty. SRC-088 states no interpolation scheme at all, which is why the naive case is the one worth recording. The midpoint is the worst point within a cell, so both the threshold and the asymptote are conservative for an element nearer a detector. The crossover figure inherits every assumption of CLM-0912's closed form, including a stationary field with a single Gaussian correlation length, which a real deployable's modes are not. No structural model is held - see CLM-0907 - so 2.3515 Lc cannot be converted to a spacing in metres.
What it assumes
Linear interpolation between two detectors, evaluated at the cell midpoint, and that "no correction" means applying the design-geometry phase and therefore carrying the full sigma_s^2.
Where it comes from
This is the project's own working, shown in full:
Closed form from CLM-0912. Crossover solved by bisection at g(t) = 1, giving 2.3515; asymptote 1.5000. Monte Carlo over 200,000 draws from the exact three-point covariance gives 0.9961 at the crossover against 1.0000, and 1.4553 at D/Lc = 4 against 1.4634. RE-SOLVED INDEPENDENTLY ON REGISTRATION, 2026-09-14, from the closed form as written rather than by re-running the author's script: crossover 2.3515 to four figures, and 0.0106, 0.1263, 0.4131, 0.7734, 1.2893, 1.4634, 1.5000 at D/Lc of 0.5, 1.0, 1.5, 2.0, 3.0, 4.0 and 10. The table is in synthesis/calcs.md under the CALC-107 heading, as a further step of the same relation, not as a separate calculation.
Basis: dimensionless (residual variance as a multiple of deformation variance; spacing as a multiple of correlation length)
Related records
- Answers question
- QST-013 not published in this release
Cite this
energyaperture.com/c/clm-0917
CLM-0917, Energy Aperture, as of 2026-09-14.