Beyond Bayesian Inference in Environmental Biomonitoring: Possibilistic Geometry Surfaces Actionable Epistemic Structure Invisible to Probabilistic Methods

For 24 of the 69 chemicals measured in U.S. National Health and Nutrition Examination Survey (NHANES) urine biomonitoring with data for both children aged 3–5 and adults aged 66 and older — including di-2-ethylhexyl phthalate (DEHP) and inorganic tin — no single regulatory exposure standard can be simultaneously epistemically grounded for both populations. This finding, which we term severe regulatory incommensurability, cannot be obtained from Bayesian inference or any significance test: it requires a geometric measure of the overlap between population-specific feasibility regions that has no probabilistic analog.We derive this result by applying the Theory of Epistemic Abductive Geometry (TEAG) — a possibilistic, constraint-based inference framework grounded in possibility theory and tropical mathematics — to the complete 179-chemical, 11-demographic-group dataset of Stanfield et al. (2022), the gold-standard Bayesian biomonitoring pipeline. TEAG recovers Bayesian median intake rate estimates with near-perfect agreement (r = 0.9965, RMSE = 0.15 log₁₀), establishing that the two frameworks agree on point estimates while diverging fundamentally on the geometric structure of the inference.The primary findings are: (i) the κ pairwise overlap coefficient is below 0.5 for 24 chemicals, meaning no intake rate achieves simultaneous epistemic feasibility above 50% for both age groups, with child-to-elder fold differences up to 8.6×; (ii) the TEAG admissible epistemic basin is on average 20.3× narrower than the Bayesian 95% credible interval, reflecting the geometric separation of measurement censoring, metabolite ambiguity, and demographic variability rather than false precision; (iii) demographic groups can be ordered by falsification priority — children aged 3–5 rank first at 1.89× mean distance from the committed population estimate; and (iv) 70% of 138 chemicals with longitudinal NHANES data (1999–2016, 9 cohorts) undergo epistemic phase transitions across cohorts, with atrazine mercapturate showing a 1.21 log₁₀ commitment reversal and arsenous acid — a severely incommensurate chemical — undergoing a persistent PCRB status change beginning in 2011–2012. A formal proof establishes that the κ incommensurability coefficient cannot be reproduced from any function of Bayesian posterior summary statistics, even given identical posterior means, variances, and credible interval widths.We call explicitly for population-differentiated reference doses for the 24 severely incommensurate chemicals and propose that κ < 0.5 between children and elderly adults in NHANES biomonitoring data be adopted as a standing geometric criterion for triggering age-stratified regulatory review.