CALC-110

Modal-basis interpolation residual against CALC-106's closed form

Approach 2 of synthesis/structural-phase-modelling.md. Measure how far the single-correlation-length closed form of CALC-106 sits from the residual of a field built from the first 5 to 10 free-free beam modes, then the sensitivity of the permitted detector spacing to amplitude and spectral shape. Producer-computed 2026-09-17; the producer is barred from verifying its own work, and the headline ratio was reproduced with independent code before this landed (recorded at the end).

Provenance. The project's working file modal_interp_residual.py, 10,872 bytes, sha256 EF9D590178AD888E, 192 lines, numpy only. Runs with no arguments and prints every table below. Inputs: CALC-109's mode family and Lc estimates; CALC-106 eq.(7) and its k-table at the 0.1 dB budget; CLM-0954's amplitude choice (thermal, not assembly).

Model. s(x) = sum a_n phi_n(x), n = 1..M, phi_n the free-free beam modes on [0, L] with unit rms, computed in a form that avoids the cosh-minus-sinh cancellation above n = 4. a_n = A n^-p scaled to total rms sigma_s; p is swept over 0, 1 and 2 because nothing held says how thermal deformation distributes across modes. Modes at different frequencies have random relative sign at any instant (f_n / f_1 = (beta_n / beta_1)^2, CALC-109), so the field is an ensemble over 64 random-sign draws. Detectors at spacing D, 8 grid offsets, linear interpolation, residual at every cell midpoint - CALC-106's worst point - rms over cells, draws and offsets. Against it: eq.(7) with a Gaussian rho of Lc fitted from the ensemble autocorrelation's small-lag curvature, and eq.(7) with the exact ensemble rho.

Geometry enters only as L - the CASSIOPeiA helix's 1 to 1.5 km extent (SRC-088 p.19 via CLM-0942) treated as a one-dimensional beam. The modes are the free-free beam family, not the helix's own: no stiffness data for that geometry exists in the register (CLM-0907, CALC-109).

RESULT 1 - THE FIRST RESULT THE BRIEF ASKED FOR. CALC-106's closed form OVERSTATES the residual of a modal field by a factor of 1.2 to 3 across the design range - the conservative direction. Across 12 cases (L 1 and 1.5 km, M 5 and 10, p 0, 1, 2) at D/Lc from 0.1 to 2.35, the ratio modal residual / Gaussian closed form is 0.33 to 0.84: closest to 1 for a flat spectrum with ten modes (0.82-0.84 at p = 1, M = 10), furthest for a steep one (0.33-0.45 at p = 2). Beyond D/Lc of about 2.5 the ratio crosses 1 for flat spectra - but that is past CALC-106's own crossover where interpolation is already worse than nothing, so it is outside the design range. L drops out once spacing is in units of Lc: the 1 km and 1.5 km rows are identical to three figures.

Why, in one sentence: a field of M modes has no content at scales below about L/M, while a Gaussian-correlated random field with the same fitted Lc has a smooth tail of short-scale content, and linear-interpolation error is set by short-scale curvature. Half the gap is the Gaussian shape and half is non-stationarity: eq.(7) with the field's own exact autocorrelation sits at 0.5-0.97 of the modal answer - closer, still over. CLM-0912's two stated assumptions each cost about the same.

RESULT 2 - permitted spacing at the 2.955 mm budget, L = 1 km. Modal D_max exceeds CALC-106's k x Lc by 1.1 to 1.7x:

RESULT 3 - amplitude sensitivity at fixed spacing, L = 1 km, M = 10, p = 1. Linear in sigma_s, as it must be. At the 18 m bay spacing the budget is exceeded only above about 400 mm rms; at the 10 m subarray spacing, only above about 1.3 m. At 100 m spacing it is exceeded above about 15 mm.

Against CALC-109's thermal band (2.5 to 141 mm for a graphite primary): at the design point of 2.5-7 mm, modal D_max is 400-500 m; at the top of the band, 30-60 m. Detector spacing at subarray or bay scale is comfortable across the entire thermal band under every spectral assumption swept. Only the 1 m static assembly error would drive spacing below 20 m on higher modes, and that is excluded from the amplitude choice per CLM-0954.

Does the answer depend on details only a finite-element model gives? Partly, and the part that does not is the part that matters. Absolute D_max moves by about 2x between p = 1 and p = 2, and p is exactly what only a structural model yields. But the two conclusions the question turns on are robust across the whole sweep: the closed form is a safe bound, never less than 1.1x conservative in the design range, and subarray-scale spacing meets the budget for thermal deformation. Approach 3 is not recommended for this question; only if someone needs the absolute spacing to better than a factor of 2.

A FINDING FROM THE FIRST, WRONG RUN, WHICH BELONGS IN CLM-0912's LIMITS. The first autocorrelation estimator divided every lag by the global variance. Free-free mode shapes carry four times the mean variance at the free ends, so a lag-k product over [0, L - k dx] drops an end region of high variance and comes out low by about 3 k dx / L - a linear fall in rho at small lag, which eq.(7) read as short-scale roughness and turned into a 1 mm residual at D = 7 m for a 10 mm field. The field had no such roughness; the estimator did. The fix is per-lag normalisation. eq.(7) is exquisitely sensitive to 1 - rho at small lag, so anyone who later feeds MEASURED deformation data into CLM-0912 - finite-length and non-stationary by construction - will manufacture a residual unless the autocorrelation is normalised per lag.

Limits. (1) Spectral exponent p is assumed and swept; it moves absolute D_max by 2x and is the one input a structural model would settle. (2) M = 5 and 10 only; more modes at flat p push the ratio toward 1. (3) One-dimensional; a plate has two-dimensional modes and a helix has coupled ones. (4) The Gaussian Lc fit uses lags where rho > 0.8; a different fit window moves Lc by tens of per cent but the modal residual does not depend on it. (5) Ensemble of 64 sign draws; ratios stable to about 2 per cent across seeds. (6) The autocorrelation must be normalised per lag - see above.

INDEPENDENTLY REPRODUCED ON LANDING, 2026-09-17, WITH SEPARATE CODE written without reading the script: five free-free modes on a 1 km beam, p = 1, 10 mm rms, 64 random-sign draws, per-lag-normalised autocorrelation, Gaussian Lc fitted where rho > 0.8, linear interpolation at eight offsets, residual at cell midpoints. Ratios of modal residual to closed form: 0.61 at D/Lc = 0.5, 0.68 at 1.0, 0.83 at 2.0 - inside the declared 0.33-0.84 band and in the conservative direction throughout. The script runs clean and its digest matches the one declared. The direction of Result 1 is therefore confirmed by code the producer did not write; its size at other p and M is the script's alone.

POSTSCRIPT 2026-09-18. The assumed thermal amplitude band of 2.5 to 141 mm is superseded by a stated requirement band: Rockwell's peak out-of-plane deflection of 12 to 48 cm and +/-18 cm reference-aperture flatness (CLM-0997). The 100 mm row of Result 3 is therefore the operative one; subarray-scale spacing still meets the 0.1 dB budget at 480 mm but with about a third of the margin reported above, and the "2.5 to 7 mm design point" language is retired. The mode periods here (1.5 to 11 s) remain antenna-structure periods; the satellite's 0.0012 Hz mode has a 14-minute period two decades away.

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