Centrality, Noncentral Selection, and the Scope of Bell–CHSH: A Measure-Theoretic Critique, Sharp Inflation Bounds, Canonical Reweighting Constructions, and an Experimental Protocol

Bell--CHSH is frequently paraphrased as ``local realism is impossible'' after experimental violations of CHSH. The theorem is correct, but the paraphrase often suppresses a structural assumption: \emph{selection centrality} (fair sampling), i.e.\ that the detected sample is not a setting- and hidden-variable--dependent reweighting of the hidden prior. We give a measure-theoretic separation between (i) the \emph{central} sector, where $E_{\mathrm{obs}}=E_{\mathrm{full}}$ and CHSH$\le 2$ holds under measurement independence and locality, and (ii) the \emph{noncentral} sector, where the observed correlations are computed under a reweighted measure $d\nu_{a,b}=w(a,b,\lambda)\,d\rho$. We prove a sharp CHSH inflation bound \[ S_{\mathrm{obs}}\le 2+4\delta, \qquad \delta:=\sup_{a,b}\int |w(a,b,\lambda)-1|\,\rho(d\lambda), \] so reproducing Tsirelson's $2\sqrt2$ by strictly local measurement-independent models via selection requires $\delta\ge (2\sqrt2-2)/4\approx0.2071$. We provide (A) a canonical Radon--Nikod\'ym construction (piecewise constants on sign regions plus a small positive remainder), (B) an explicit local factorized detection-loophole construction that reproduces arbitrary finite correlation tables, and (C) an implementable optics protocol to estimate or upper-bound $\delta$ from time-tag data via window dithering, threshold and spectral sweeps, and auxiliary-tag binning. This paper critiques not the validity of Bell's theorem, but the common overstatement of what CHSH violations imply without an explicit, quantitative centrality audit.