Appendix CCCC3 — Deterministic Ranking and Empirical Concentration

\labelapp:CCCC3

CCCC3.1 Purpose

This appendix introduces and evaluates a deterministic ranking statistic constructed solely from directly reported IceCube alert observables. The objective is to demonstrate that the public alert set exhibits strong internal concentration when events are ranked by joint energy and localization precision, without invoking probabilistic association, catalog matching, or astrophysical modeling.

CCCC3.2 Definition of the ranking statistic

For each IceCube alert $i$, we define the dimensionless ranking statistic

<a id="eq-eq-cccc3-k" />

where:

• $E_i$ is the reported deposited-energy proxy in TeV,
• $\sigma_{\theta,i}$ is the angular localization scale in degrees.

In the primary analysis, $\sigma_{\theta,i}$ is taken from the published $50\%$ containment radius:

No weights, priors, or tuning parameters are introduced. The statistic is fully deterministic.

CCCC3.3 Dataset and filtering

The ranking is applied to the public IceCube Gold and Bronze alert archive, after filtering to events with finite reported localization. The resulting sample contains

events.

No temporal coincidence, spatial association, or external catalog information is used.

CCCC3.4 Empirical concentration

The ranked distribution of $\mathcal{K}_i$ exhibits strong concentration. The top-ranked event is

The next-highest ranked events fall significantly below this maximum, producing a clear separation between the leading event and the remainder of the alert population.

CCCC3.5 Interpretation

The observed concentration is a purely data-internal consequence of combining two independently reported observables: energy and angular precision. It does not depend on:

• source catalogs,
• likelihood models,
• background simulations,
• or probabilistic significance assignments.

As such, the ranking exposes intrinsic structure in the alert set that is otherwise obscured when events are treated as statistically equivalent.

CCCC3.6 Robustness checks

The dominance of the top-ranked event persists under:

1. replacement of Error50 by Error90 (with rescaled normalization),
2. restriction to Gold-only or Bronze-only subsets,
3. removal of the single highest-energy event followed by re-ranking.

In all cases, the distribution remains sharply non-uniform.

CCCC3.7 Exported artifacts

This appendix is supported by deterministic outputs generated by the pipeline:

• ‘closure_events.csv‘,
• ‘top_events_by_KILLER.csv‘,
• ‘ranking_summary.txt‘.

These files fully reproduce the ranking without manual intervention.

CCCC3.8 Falsification logic

The ranking framework would be invalidated if:

1. the $\mathcal{K}_i$ distribution were statistically uniform,
2. the top-ranked event were unstable under minimal data-quality cuts,
3. the ordering depended on undocumented or tunable parameters.

None of these conditions are observed.

CCCC3.9 Scope of claim

This appendix claims only that:

• a deterministic ranking can be defined from public alert data,
• the IceCube alert set exhibits strong empirical concentration,
• one event dominates the joint energy–localization metric.

No claim is made regarding astrophysical origin or source association.

Status: equations fixed; numerical values populated directly from exported ranking artifacts.