Appendix CCCC4 — Deterministic Multi-Messenger Closure Protocol

\labelapp:CCCC4

CCCC4.1 Purpose

This appendix specifies a fully deterministic protocol for testing multi-messenger closure between high-energy neutrino alerts and high-energy photon transients. The goal is not source identification, but to test whether a single, theory-fixed deformation (quadratic propagation) produces internally consistent timing across messengers without tuning.

CCCC4.2 Inputs (public, auditable)

The protocol consumes only publicly released quantities:

• IceCube alert time $t_\nu$, direction $(\alpha_\nu,\delta_\nu)$, energy proxy $E_\nu$, and localization scale $\sigma_{\theta,\nu}$.
• GRB trigger time $t_\gamma$, direction $(\alpha_\gamma,\delta_\gamma)$, photon energy proxy $E_\gamma$, and redshift $z$ when available.

No catalog priors, likelihood weights, or background simulations are used.

CCCC4.3 Deterministic coincidence rule

A neutrino–photon pair $(\nu,\gamma)$ is declared eligible if and only if:

1. Temporal gate:

with \Delta t_\max fixed a priori ([FILL]). 2. Angular gate:

where $\kappa$ is fixed and documented; no optimization is permitted. 3. Energy availability: both events provide an energy proxy.

All thresholds are constants recorded in the run log.

CCCC4.4 Closure observable

For each eligible pair we compute the theory-implied normalized delay:

with $\mathcal{I}(z)$ defined in Appendix CCCC2. Closure requires that all pairs lie on the same linear relation $Y=\alpha X$ with a single slope $\alpha$ shared across messengers.

CCCC4.5 No-tuning guarantees

The protocol enforces:

• no adjustment of $\Delta t_{\max}$ or angular factors after inspection,
• no per-event weights or significance scores,
• no energy-dependent windowing,
• no marginalization over emission lags.

Any deviation invalidates the test.

CCCC4.6 Output artifacts

The pipeline deterministically exports:

• ‘eligible_pairs.csv‘ (all passing pairs),
• ‘closure_fit.csv‘ (global $\alpha$, uncertainty),
• ‘pair_residuals.csv‘ ($Y-\alpha X$ diagnostics),
• ‘run_constants.json‘ (all thresholds and cosmology).

CCCC4.7 Summary statistics

For the documented run:

with a null-hypothesis surrogate probability

computed only from time scrambling under fixed windows.

CCCC4.8 Falsification logic (non-negotiable)

The framework is ruled out if any of the following occur:

1. Different messengers require different slopes $\alpha$.
2. A linear-in-$E$ term improves the fit.
3. The sign of the delay is inconsistent across pairs.
4. Results depend on undocumented threshold choices.

CCCC4.9 Scope of claim

This appendix claims only that a deterministic, auditable multi-messenger closure test can be executed and falsified. It makes no claim of source identification or discovery significance.

Status: protocol fixed; numerical placeholders to be filled directly from exported artifacts.