Spectral closure certification in ALEPH data. BKT–94–A02–B01

Spectral closure certification in ALEPH data. BKT-94-A02-B01 A family of relational estimators and definitions of closure and non-closure of physical structures Author: Robert KupskiResearch program PJM USC GTWSSF GTCWArticle identifier: BKT-94-A02-B01 Abstract This work develops mathematical and operator-based certification of spectral response using relational estimators from the PJM USC GTWSSF GTCW research program. The analysis concerns the inclusive non-strange vector channel in ALEPH hadronic τ-decay data and a separate application to the exclusive Belle two-pion channel. Closure and non-closure are defined relative to a specified operator, probe subspace and admissible measure class. Four published ALEPH moments, together with normalization and full covariance, constrain a positive measure describing the decay-distribution shape. For the exact central moments, positive definiteness of the Hankel matrix H₂ requires every compatible measure to have at least three support points. In the canonical realization H(y) = √y with a constant probe, the closure defect is D = Var_p(1/√y), where y = s/mτ². On the support [0.0238, 1], its sharp nominal range is approximately [0.20363958, 1.60918725]. The extremal measures are unique Gauss–Radau realizations. Accounting for a Gaussian moment-uncertainty region at the nominal 95% level and rounding of the published central values, standard deviations and correlations yields the outer band 0.1948387786 ≤ D ≤ 1.7095360557. Its statistical interpretation is conditional on the adopted error model and covariance class; sharpness of this band's upper bound has not been established. Analyses of 79 ALEPH histogram bins and 62 Belle bins provide lower bounds on spectral-reduction error. In their respective normalized coordinates, these are W₁ ≥ 0.03110788 for an ALEPH reduction to at most three measure atoms and W₁ ≥ 0.02427224 for a Belle reduction to at most two atoms. These are separate conditional certificates. For the exact central moments and stipulated support [0.0238, 1], removing the kinematic filter yields a certified global minimum of the dimensionless rescaled current functional at q = 1, I_min ≈ 4.79572393614, while its supremum remains infinite under the moment constraints alone. The framework combines positivity, moment problems, operator reductions, resolvent readouts and uncertainty control. It assesses spectral representation order, approximation limits and model distinguishability beyond the calibration set. It provides tools for spectral reconstruction, dispersive analyses and comparisons with QCD under matched observable definitions. The results are theorems and reproducible calculations within the stated measure and operator class; they neither identify the full QCD Hamiltonian nor establish a departure from the Standard Model. Introduction A finite set of spectral data does not, in general, uniquely determine a physical representation of the system under study. Different positive measures may reproduce the same moments and remain compatible with their uncertainties while yielding different responses for other observables. The resulting problem is to determine which spectral properties hold across the entire class of representations consistent with the data, what minimum representation order is required and how much error accompanies reduction. This article formulates that problem as spectral closure certification. Within the PJM USC GTWSSF GTCW research program, relational estimators quantify the compatibility of a structure with a specified set of conditions [M5, M6, M7, M8, M9]. In the present application, these conditions receive a precise operator formulation. Closure means that the designated probe subspace reduces the specified operator. In the adopted definition, the defect measures the component of the inverse operator A = H⁻¹ acting outside the probe subspace. The magnitude of the defect depends on the operator, probe, normalization and reference class. This work gives these concepts a concrete computational realization; its proofs do not require the hypothesis of an overarching informational layer as an additional premise. The principal empirical inputs are the published moments of the ALEPH non-strange vector channel and their full covariance matrix [Z1]. The analyzed measure describes the normalized decay-distribution shape and includes the kinematic filter. Relating it to an unfiltered current spectral function requires a separate transformation. Positivity of the measure and its moment matrices permits bounds valid for all admissible representations. In this formulation, the central moments require at least three support points, while the range of the closure defect is obtained by global extremization. Covariance and the precision of the printed data define a separate uncertainty level for these conclusions. The analysis also covers the full ALEPH histogram and the exclusive τ⁻ → π⁻π⁰ντ channel measured by Belle [Z2]. The distribution of mass within each histogram bin remains part of the representation ambiguity. This permits lower bounds on reduction error in Wasserstein distance and total variation distance without identifying the histogram with an arbitrarily chosen interpolation. The results explain why fitting a few moments can ensure accuracy for selected integrals while remaining insufficient to reproduce the full distribution. A separate analysis of the kinematic filter identifies information lost at its zeros and explains why a finite upper bound on the functional after filter removal requires additional control of the unfiltered measure, such as a bound on its total mass. The article connects these results with Krylov-subspace analysis, semidefinite representations, resolvent readouts and identifiability tests. Observables used to determine parameters serve a calibration role, while readouts withheld from that stage can test the selected realization. Applications include controlling positive reduced models, reconstructing spectra from Euclidean correlators, and comparisons with dispersive integrals and QCD sum rules [Z3, Z17]. Physical comparisons require matched definitions of the current, integral kernel, normalization and corrections. The principal outcome is a quantitative account of the information limits of the data: properties enforced by observations, admissible representation ambiguity and conditions for distinguishing models.