CALC-084

power density at altitude relative to ground

Supports CLM-0865. A calculation is required: the claim's stated value is arithmetic performed here. Independently reproduced by a cold-start verifier that derived the geometry itself and did not see the producer's implementation (VER-1088).

Model. Beam width taken between the stated aperture and the stated ground spot; power density scales as the inverse square of local beam width. The convergence sign must be checked per geometry - a diverging model carried into a converging one is how the producer first got LEO wrong.

Cross-model agreement at the GEO F-region peak:

point-source diverging      1.0170
linear beam-width           1.0153
Gaussian-beam diffraction   1.0216   <- the one the producer had not tested
amplitude taper             cancels in a far-field ratio
                            ----------------------------------
registrable figure          about 2.2 per cent

Applicability conditions, which are the reason the LEO row differs in kind and not only in value:

                  aperture  freq       range      2D^2/lambda   ratio    far-field
GEO sub-satellite   1 km    2.45 GHz   35,800 km   16,345 km    2.190x   HOLDS
GEO slant range     1 km    2.45 GHz   37,126 km   16,345 km    2.271x   HOLDS
LEO                 375 m   10   GHz      480 km    9,375 km    0.051x   INAPPLICABLE

CORRECTED 2026-09-06. The line this replaces read "GEO 2D^2/lambda = 16,345 km, range 35,800 km -> 2.27x Rayleigh", and those three numbers do not go together. Each was individually right and the PAIRING was wrong: 35,800 / 16,345 = 2.190, while 2.271 requires 37,125.5 km - the SLANT range to a mid-latitude ground station, not the sub-satellite altitude the line stated. Both rows are now given, because which range is meant depends on where the ground station is, and both clear the threshold comfortably. Nothing downstream moves: CLM-0865's comparability conclusion rests on the far-field condition holding, and it holds either way. This was a traceability defect, not a numerical one.

Aperture and frequency columns are added because the LEO row could not be reproduced without them. Its 9,375 km depends on 10 GHz, which appeared NOWHERE in the line beside it; evaluated at 2.45 GHz the same aperture gives 2,298 km and a ratio of 0.209, four times larger. A row whose result silently depends on a parameter it does not print is not reproducible, whatever its provenance. (The 9,375 km figure itself uses lambda = 0.03 m exactly; from c = 299,792,458 m/s it is 9,381 km. Both are defensible roundings and the ratio is 0.051 either way.)

This is the SECOND range-mixing defect in this stream and it ran the OPPOSITE way to the first - earlier a figure was computed at the slant range while the source used the sub-satellite altitude. The recurring fault is not the arithmetic but an UNSTATED REFERENCE FRAME, the same family as the ground-plane versus beam-normal ambiguity recorded against CLM-0867. Found by recomputing from scratch rather than by re-reading, which is why it was found at all.

Reproduction criterion: the ratios, on any hardware. This is arithmetic, not a timed run.

ACCEPT NOTHING THAT DOES NOT REPRODUCE 1.0153 for GEO and 0.0816 for LEO at 300 km. A reconstruction giving 7.111 for LEO has applied a diverging model to a converging geometry - the producer's original error, which the verifier deliberately reproduced to confirm the diagnosis.

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