Minimal Finite Hankel Closure in Elastic Proton-Proton Scattering: Structural Invariants, Covariance-Aware Public-Data Benchmarks, and Preregistered LHC Tests

We formulate and test a minimal finite Hankel closure for the dip–bump structure in elasticproton-proton scattering. The scope of the claim is deliberately precise. We do not present amicroscopic derivation from QCD and we do not claim universal exclusion of the full hadronicphenomenology. Rather, we establish a conditional theorem, confront its surrogate realizationwith public data, and state explicit near-term tests. First, assuming positivity, radiality, finitemoments, a Born-dominant forward-plus-first-dip window, self-similar scaling, and minimal finiteHankel closure with one simple node, we prove that the unique lowest-complexity sector is the firstLaguerre deformation, which yields a polynomial-times-Gaussian amplitude. Second, we derivestructural relations for the forward slope, dip scale, forward curvature, and the drift observable O_excl = ∆[B0|t|dip], and we prove non-reducibility against the one-scale geometric class for whichthe corresponding invariants are energy independent. Third, we test the closure on two levels ofpublic-data benchmark. In the restricted internal comparison on 83 differential-cross-section pointsat 2.76 and 13 TeV under a common weighted log-space score and shared cross-energy flow, thetwo-scale copy-time surrogate yields χ²_log = 461.19, AIC = 487.19, and BIC = 518.64, compared with(60493.49,60507.49,60524.43) and (59942.77,59964.77,59991.38) for two one-scale baselines. We thenperform a stronger covariance-aware benchmark in log space, using per-dataset block covariances builtfrom the published statistical errors together with fully correlated systematic blocks, and comparethe copy-time surrogate to the internal one-scale baseline, a canonical Regge-pole-plus-Odderonamplitude, a canonical complex Regge-eikonal baseline, and the fixed Kohara–Ferreira–Kodamaparametrization. In that stronger test the covariance-aware copy-time fit remains the best modelin the benchmark set, with χ²_{cov log} = 1687.73, compared with 5440.69 for the strongest externalbaseline. At fixed 13 TeV, however, the split-sector one-scale surrogate remains competitive inthe dedicated local fit, showing that the main empirical leverage of the closure is intrinsicallymulti-energy rather than a consequence of the 13 TeV line shape alone. We also report a hold-outvalidation at 8 TeV, an explicit continuation to 13.6 and 14 TeV, and a narrow-window robustnessscan showing that the forecast sign pattern is stable under moderate perturbations of the real-sectorcontinuation. Within the explicit axiomatic, statistical, and benchmark choices adopted here, theclosure is therefore mathematically constrained, experimentally discriminating, and favored over theimplemented internal and external baselines.