CALC-037

The sampling plateau that excludes sampling as the cause of a 0.36 dB offset

> THE 0.359 dB RESIDUAL THIS RECORD SUPPORTS IS WITHDRAWN, 2026-09-05. DO NOT QUOTE IT. > The plateau result below stands: refining the observation spacing 77-fold moves the answer > 0.011 dB, so SAMPLING is genuinely excluded as a cause. What does not stand is the residual it > was measuring. The element lattice was underspecified, and once specified it turned out to be > OFF-CENTRE through an endpoint defect: with d = lambda/2 = 0.061182134 m and a = 0.5 m, > 2a/d = 16.3446 is not an integer, so np.arange(-a, a+1e-12, d) gives 17 points from -0.500000 > to +0.478914, centred at -0.010543 m rather than zero. That grid holds exactly 209 elements > inside the radius; a CENTRED grid of the same pitch holds 213. Re-running both codes on both > lattices: off-centre 209 gives a residual of +0.359 dB, centred 213 gives -0.255 dB. > The residual reverses sign, so it is not a property of the tool. Confirmed independently. > The POSITION agreement, better than 0.03 per cent, is unaffected and still holds. > See CLM-0860, which is PARTIAL and carries the full withdrawal.

Supports CLM-0860. This is the evidence that the offset is not an artefact of how the pattern was sampled, which was the first explanation offered for it and is the one this calculation refutes.

Configuration. LyceanEM 0.1.131 (SRC-321, pinned commit 4069f809ab812a0ac5915f153318dcc0ded7786a), run locally on an NVIDIA RTX 4070. Uniform circular aperture, 1 m diameter, 8.172 wavelengths at 2.45 GHz, 209 elements at half-wavelength spacing, uniform illumination. Observation points on an arc at 1000 m swept across 8 to 16 degrees. Exact continuous-aperture reference: first sidelobe -17.5704 dB at 11.5388 degrees.

A 77-fold refinement in observation-point spacing changes the answer by 0.011 dB. The value plateaus. Sampling is excluded as the cause.

Decomposition of the +0.403 dB, and the label defect that had to be corrected first.

LyceanEM                                      11.5600 deg   -17.167  dB
independent direct array sum, same elements   11.5570 deg   -17.526  dB
continuous-aperture Airy, exact               11.5388 deg   -17.5704 dB

discretisation (discrete minus continuous)                  +0.044 dB
UNEXPLAINED RESIDUAL (LyceanEM minus discrete)              +0.359 dB   <-- WITHDRAWN

The original acceptance test compared LyceanEM's discrete 209-element array against the continuous Airy pattern. Those are different objects, so part of the apparent error was the comparison rather than the tool. That is METHOD rule VF1's failure mode exactly, and correcting it is what isolated the residual: discretisation accounts for 11 per cent of the offset and the remaining +0.359 dB does not go away on this lattice. It does not survive a CENTRED lattice, where it becomes -0.255 dB - see the withdrawal notice at the head of this record.

Position agreement between LyceanEM and the independent sum: 0.026 per cent. The tool places the structure correctly and reports one feature's level too high.

The obvious explanation is excluded on sign. Obliquity and Green's-function terms present in LyceanEM's Rayleigh-Sommerfeld kernel and absent from a pure array-factor sum would DEPRESS the sidelobe relative to the peak, not lift it. The candidate that suggests itself first therefore has the wrong sign, and the cause remains unidentified. Recorded as an open, quantified residual with a named excluded candidate, which tells the next person where not to look.

Scope. ONE configuration: one aperture size, uniform illumination, one frequency, one range. The offset must NOT be assumed constant or general, and has not been tested elsewhere.

Origin: produced by the EM simulation pass, sections 18.1 to 18.6. Scripts emsim/lyceanem-local/sampling_convergence.py and emsim/lyceanem-local/run_validation.py. Producer-computed and UNVERIFIED: the producer may not verify its own work, so this needs an independent pass. Sampling was offered as the cause, tested, and recorded its refutation rather than its story.

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