{"paper":{"title":"Wave-particle duality as an uncertainty relation for the average confidence width","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"Shengjun Wu","submitted_at":"2026-06-30T10:19:35Z","abstract_excerpt":"We introduce the average confidence width $\\Delta_a x=\\int_0^1 \\Delta_c x (\\theta_x) d \\theta_x$: the confidence width $\\Delta_c x(\\theta_x)$ -- the smallest position interval carrying a fraction $\\theta_x$ of the probability -- averaged over all levels. It is the first moment of the decreasing rearrangement of $|\\psi|^2$, an $L^1$ mean-absolute-deviation measure of localization, so the product $\\Delta_{a} x\\,\\Delta_{a} p$ is dilation invariant and obeys $\\Delta_{a} x\\,\\Delta_{a} p\\ge c\\,\\hbar$. Reading $1/\\Delta_{a} x$ as a particle character and $1/\\Delta_{a} p$ as a wave character, this low"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31443","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.31443/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}