Weight Concentration and Category Heterogeneity in Precision Higgs Combinations: A Robustness Bound from H → μ⁺μ⁻

Weight Concentration and Category Heterogeneity in Precision Higgs Combinations: A Robustness Bound from H → μ⁺μ⁻ Precision Higgs measurements combine many production-sensitive categories under one signal strength. This paper asks how much unmodelled category-level heterogeneity can change such a combination, and what can and cannot be checked from published summaries alone. The key quantity is the weight concentration factor f = 1/sqrt(n_eff), Kish's effective sample size of the inverse-variance weights. For independent category deviations of scale sigma_delta, the combined uncertainty grows by sqrt(1 + R^2), with R = mu sigma_delta f / sigma_comb. The same R, together with the number and distribution of the weights, sets the expected excess of Cochran's Q, so it governs both the size of the effect and whether a homogeneity test could see it. Applied to the ATLAS Run-3 H -> mu+ mu- measurement (PRL 135, 231802):- six-group reconstruction: mu = 1.44 +- 0.555, f = 0.507 (n_eff = 3.88), Q = 4.00;- a 2% broadening needs additional independent deviations of about 15-29%, or an additional common shift of 7.5-7.7%;- additional heterogeneity of a few percent is negligible at current precision;- the published summaries do not fix the correlations between the groups (a 13-dimensional family of Gaussian covariances reproduces them), so the homogeneity p-value cannot be rebuilt from summaries alone. Monte Carlo validation, a comparison of category-aware estimators, a worked example, a decision guide, threshold lookup tables, an illustrative HL-LHC scaling and the propagation to kappa_mu are included. All computed results are produced by the accompanying script. VERSION NOTES (version 3) Version 3 is a full rebuild; the title changes to describe the content. Appendix B lists the principal corrections relative to versions 1 and 2:- thresholds re-anchored to the measured Run-3 precision (the conclusion that few-percent heterogeneity matters now is reversed);- corrected weight-concentration values (0.581/0.582, not 0.596/0.589);- the correlated-deviation formula and Table II (the spread rises with rho);- the category-aware estimator, its limits and its bias;- the kappa_mu propagation (not amplified);- numerical and figure-reference items;- the posted code, which did not reproduce the paper;- attribution: Kish's effective sample size; Cochran's Q, I^2 and DerSimonian-Laird; Ball et al. (2010) for the multiplicative-normalization objective and its bias; Valassi and Chierici (2014) on unknown correlations. Versions 1 and 2 had the same scientific content; version 2 removed editorial notes. Code: concept DOI 10.5281/zenodo.20599199 (all versions; version 3 accompanies this paper). Version 1 of the code, 10.5281/zenodo.20599200, is the historical script described in Appendix B. RESEARCH-PROGRAM PROVENANCE This study arose within Kate Lynn Blatt's Energy-Consciousness Observer Framework (ECOF) research program. Foundation reference: Blatt, K. L. (2025). The Energy Consciousness Observer Framework: Foundations of Observer-Coupled Physics -- Toward a Unified Field for General Relativity's Mass-Energy Corrections. https://doi.org/10.5281/zenodo.16944712 The reference records the program's foundation and provenance; the present statistical analysis is self-contained and does not assume an observer-coupled modification of the Standard Model. PERSISTENT LINKS Paper, all versions: https://doi.org/10.5281/zenodo.20434179Code, all versions: https://doi.org/10.5281/zenodo.20599199ECOF foundation, all versions / latest: https://doi.org/10.5281/zenodo.16944711Paper version 3: https://doi.org/10.5281/zenodo.23094240Code version 3: https://doi.org/10.5281/zenodo.23095059 Related studies in the ECOF research program. CP-even information flow · CP-odd information flow · Graph transfer and ultraviolet suppression · Higgs weight concentration (Higgs code).