A coarse helix-geometry null accounts for the apparent membrane-centering of transmembrane coordination-number maxima

We asked whether the depth at which a transmembrane (TM) helix bundle's local Cα coordination number is maximal -- a candidate proxy for a functional mechanical pivot -- sits closer to the bilayer midplane in G protein-coupled receptors (GPCRs) than in other polytopic membrane protein folds, and, more generally, whether this coordination-number maximum clusters near the membrane center across polytopic membrane proteins regardless of fold. Across 96 experimentally determined structures (40 GPCRs, Class A/B/C/F; 56 non-GPCR structures spanning five architectures), 72.9% had a Cα coordination-number maximum within ±6 Å of the bilayer midplane, and 88.5% within ±9 Å -- more than expected under a uniform distribution across the physically plausible TM window. To test whether this reflects a residue-scale packing preference or a coarser property of TM helix arrangement, we built a straight-rod surrogate: for each real TM segment, we fit a straight line through its Cα coordinates and generated idealized points evenly spaced along that line, spanning the segment's true extent, then recomputed the coordination-number profile. Across four point-density choices (each segment's own residue count, or fixed 20/30/40 points), the surrogate reproduces the aggregate centering closely (74.7-77.9% within ±6 Å, n=95 of 96) and correlates with the real coordination-number maximum's depth (Spearman rho=0.47-0.58). Structure-level agreement is moderate rather than exact: 17 of 95 paired structures disagree on within-±6-Å classification, unevenly across fold families -- most notably VGIC_6TM (observed 35.7% vs. surrogate 21.4% within ±6 Å). We conclude that a coarse geometric property -- each TM helix's true 3D position and extent, independent of its detailed packing -- accounts for most of the aggregate centering pattern, and that this pattern alone should not be interpreted as evidence of a residue-scale packing preference or lipid-pressure-driven mechanism. The narrower, GPCR-specific comparison that motivated this study is separately not robust at the family level: the estimated contrast (non-GPCR minus GPCR family-mean |depth|) shrinks from 7.6 Å under the least-restricted comparison to 1.8 Å (family-level p=0.14) once restricted to genuinely transmembrane residues and TM-TM contacts, and a contact-source decomposition indicates part of the unrestricted estimate reflects overall protein architecture rather than TM-helix-bundle packing specifically. Note on this version: this record supersedes an earlier version reporting a stronger, GPCR-specific and general-centering claim. Further critical review and a coarse-geometry surrogate analysis showed that most of the aggregate membrane-centering pattern is attributable to bundle geometry rather than a residue-scale packing preference, and that the original GPCR-specific comparison is not robust at the family level once restricted to genuinely transmembrane residues and contacts. This version reports that narrower, more cautious conclusion. The companion analysis code is archived at Zenodo, DOI 10.5281/zenodo.21533799.