OTAZ-II: Mathematical and AI Extensions of Exponentially Weighted Fusion — Duplicate-Record Consistency, Hilbert Sensitivity Bounds, Truncation Certificates and Constrained Calibration

The earlier preprint in this series defined a pair-valued fusion operator: a value V that is asoftmax-weighted mean, and a score W that is a scaled log-sum-exp. That work establishedclosure, commutativity and associativity at a fixed coefficient, geometric and scorebounds, derivatives, entropy identities, and first-order perturbation bounds. This paperresumes from that endpoint and contributes four guarded extensions, each with a writtenproof and independent numerical evidence:Duplicate-record consistency. A theorem showing that the full pair (V,W) can bereproduced from deduplicated storage, in either of two opposite directions (w ± β−1lnk), together with a bound on the error induced when multiplicities are estimatedincorrectly. The mathematics is simple; the contribution is the pair of executable rulesplus an a-priori damage bound, which turns a known hazard into a controlledprocedure.Sharp sensitivity. The Hilbert distance between weight vectors never exceeds |β|·(maxδ − min δ). This yields a globally valid total-variation bound, TV ≤ tanh(|β|·range(δ)/4),valid without a small-perturbation assumption and proved sharp by an explicit twopointfamily. A vector-valued counterpart and a cross-entropy ceiling follow.Certificates, and an honest negative result. Truncating the score tail can be madeauditable: dropping mass τ bounds the value error by D·τ and makes the score gapexactly computable. The experiments confirm the certificate in every case — and showthat certificates do not imply savings. With uniform scores, a 2 % tail budget keeps98 % of records.Constrained calibration. Minimum-variance weights are optimal only on the simplex,and the unconstrained solution can be negative: an explicit analytic counterexamplegives weights (1.25, −0.25) with objective 0.875 against 1.000 on the simplex. Thispaper gives an exact face-enumeration oracle for small problems, an equivalencecondition for the widely used inverse-MSE rule, and a calibration study of 2,400replications that reports where weighting wins and where uniform averaging wins.Every theoretical statement is proved or explicitly attributed to prior work. Everyreported number is regenerated by seeded code, reproduces byte-for-byte across reruns,and is checked by an independent verifier of 27 checks that does not import the packageunder test. Thirty-six unit tests accompany the reference implementation. The paperstates plainly what is prior art (softmax, log-sum-exp, the Hilbert–total-variationinequality and its constant, forecast-combination weighting) and makes no priority, speedor efficacy claim.