How much can unpublished correlations among the 14 short-range coefficients move a limit on a mesh signal?

A short sensitivity study continuing three earlier notes (https://doi.org/10.5281/zenodo.24063799, https://doi.org/10.5281/zenodo.24063701, https://doi.org/10.5281/zenodo.24097646). Paper 1 limited the amplitude kappa of a cubic-mesh signal by fitting the 14 coefficients of the combined HUST/IU short-range gravity analysis with independent errors, because their correlations are not published. This note asks how far that assumption can move the limit and which correlations matter. For any correlation matrix the fitted uncertainty on kappa cannot grow by more than sqrt(14) = 3.7 times (a bound reached only for perfectly correlated coefficients) and has no lower bound. Random correlation matrices with mean identity raise the information on average (Jensen's inequality), so their median limits, 0.96 to 0.50 times the uncorrelated one, show how far the limit can move, not that the true limit is probably tighter; the strongest ensemble is nearly singular. A single strongly correlated pair weakens the limit in 19 to 30 per cent of draws, by at most 30 to 35 per cent, while several pairs tighten it, so the direction of the change depends on the structure. The limit is most sensitive to the correlations among the five best-measured coefficients (XXXZ, XXYZ, XYYZ, XYZZ, YYYZ; 76 per cent of the squared sensitivity). Within the structures studied other than nearly singular matrices a nearest-neighbour mesh at 86 micron stays below 0.6 sigma; nearly singular covariances could make it arbitrarily significant, so the real covariance is needed, and the note recommends requesting it. It uses only published numbers and assumes correlation structures rather than measuring them. Produced with AI assistance (see the disclosure in the PDF). Scripts and saved output (MIT licence) are included in paper5_scripts.zip, and a one-page illustrated guide in paper5_diagram.html.