OTAZ-II: Mathematical and AI Extensions of Exponentially Weighted Fusion — Duplicate-Record Consistency, Hilbert Sensitivity Bounds, Truncation Certificates and Constrained Calibration
Abstract
The earlier preprint in this series defined a pair-valued fusion operator: a value V that is a
softmax-weighted mean, and a score W that is a scaled log-sum-exp. That work established
closure, commutativity and associativity at a fixed coefficient, geometric and score
bounds, derivatives, entropy identities, and first-order perturbation bounds. This paper
resumes from that endpoint and contributes four guarded extensions, each with a written
proof and independent numerical evidence:
Duplicate-record consistency. A theorem showing that the full pair (V,W) can be
reproduced from deduplicated storage, in either of two opposite directions (w ± β−1ln
k), together with a bound on the error induced when multiplicities are estimated
incorrectly. The mathematics is simple; the contribution is the pair of executable rules
plus an a-priori damage bound, which turns a known hazard into a controlled
procedure.
Sharp sensitivity. The Hilbert distance between weight vectors never exceeds |β|·(max
δ − 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 twopoint
family. A vector-valued counterpart and a cross-entropy ceiling follow.
Certificates, and an honest negative result. Truncating the score tail can be made
auditable: dropping mass τ bounds the value error by D·τ and makes the score gap
exactly computable. The experiments confirm the certificate in every case — and show
that certificates do not imply savings. With uniform scores, a 2 % tail budget keeps
98 % of records.
Constrained calibration. Minimum-variance weights are optimal only on the simplex,
and the unconstrained solution can be negative: an explicit analytic counterexample
gives weights (1.25, −0.25) with objective 0.875 against 1.000 on the simplex. This
paper gives an exact face-enumeration oracle for small problems, an equivalence
condition for the widely used inverse-MSE rule, and a calibration study of 2,400
replications that reports where weighting wins and where uniform averaging wins.
Every theoretical statement is proved or explicitly attributed to prior work. Every
reported 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 package
under test. Thirty-six unit tests accompany the reference implementation. The paper
states plainly what is prior art (softmax, log-sum-exp, the Hilbert–total-variation
inequality and its constant, forecast-combination weighting) and makes no priority, speed
or efficacy claim.
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BibTeX
@misc{aldawdeh2026otazii,
title = {{OTAZ-II: Mathematical and AI Extensions of Exponentially Weighted Fusion — Duplicate-Record Consistency, Hilbert Sensitivity Bounds, Truncation Certificates and Constrained Calibration}},
author = {Al-Mu'taz Billah Salah Aldawdeh},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23678745},
url = {https://zenodo.org/records/23678745},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - OTAZ-II: Mathematical and AI Extensions of Exponentially Weighted Fusion — Duplicate-Record Consistency, Hilbert Sensitivity Bounds, Truncation Certificates and Constrained Calibration
AU - Al-Mu'taz Billah Salah Aldawdeh
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23678745
UR - https://zenodo.org/records/23678745
AB - The earlier preprint in this series defined a pair-valued fusion operator: a value V that is a softmax-weighted mean, and a score W that is a scaled log-sum-exp. That work established closure, commutativity and associativity at a fixed coefficient, geometric and score bounds, derivatives, entropy identities, and first-order perturbation bounds. This paper resumes from that endpoint and contributes four guarded extensions, each with a written proof and independent numerical evidence: Duplicate-record consistency. A theorem showing that the full pair (V,W) can be reproduced from deduplicated storage, in either of two opposite directions (w ± β−1ln k), together with a bound on the error induced when multiplicities are estimated incorrectly. The mathematics is simple; the contribution is the pair of executable rules plus an a-priori damage bound, which turns a known hazard into a controlled procedure. Sharp sensitivity. The Hilbert distance between weight vectors never exceeds |β|·(max δ − 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 twopoint family. A vector-valued counterpart and a cross-entropy ceiling follow. Certificates, and an honest negative result. Truncating the score tail can be made auditable: dropping mass τ bounds the value error by D·τ and makes the score gap exactly computable. The experiments confirm the certificate in every case — and show that certificates do not imply savings. With uniform scores, a 2 % tail budget keeps 98 % of records. Constrained calibration. Minimum-variance weights are optimal only on the simplex, and the unconstrained solution can be negative: an explicit analytic counterexample gives weights (1.25, −0.25) with objective 0.875 against 1.000 on the simplex. This paper gives an exact face-enumeration oracle for small problems, an equivalence condition for the widely used inverse-MSE rule, and a calibration study of 2,400 replications that reports where weighting wins and where uniform averaging wins. Every theoretical statement is proved or explicitly attributed to prior work. Every reported 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 package under test. Thirty-six unit tests accompany the reference implementation. The paper states plainly what is prior art (softmax, log-sum-exp, the Hilbert–total-variation inequality and its constant, forecast-combination weighting) and makes no priority, speed or efficacy claim.
ER -