Appendix CCCC2 — Observational Fit and Universal Closure Scale

\labelapp:CCCC2

CCCC2.1 Purpose

This appendix reports the numerical observational fit of the theory’s unique quadratic propagation deformation using public high-energy transient data. The objective is to extract a single universal closure scale $E_{\mathrm{cl}}$ and to test internal consistency across messenger channels (photons versus neutrinos), without introducing astrophysical source modeling or population priors.

CCCC2.2 Testable prediction in observable form

The theory predicts a universal quadratic deformation of the signal speed,

<a id="eq-eq-cccc2-ve" />

which implies a cumulative arrival-time delay for a messenger of energy $E$ originating at redshift $z$:

<a id="eq-eq-cccc2-dt" />

Defining the observable variables

the prediction reduces to the strictly linear relation

<a id="eq-eq-cccc2-fitline" />

The absence of any linear-in-$E$ term is a structural constraint of the theory and not a fitting choice.

CCCC2.3 Dataset and deterministic inclusion rules

The dataset is constructed from public releases of high-energy GRB photons and IceCube neutrino alerts. Inclusion rules are deterministic and auditable:

1. A reported energy proxy $E$ is available.
2. An arrival-time offset $\Delta t$ relative to a defined trigger is provided.
3. A redshift $z$ is known; events without redshift are excluded from the primary fit.
4. No emission-lag modeling, marginalization, or population priors are introduced.

The primary-fit sample contains

CCCC2.4 Cosmological evaluation

The redshift integral $\mathcal{I}(z)$ is evaluated assuming flat $\Lambda$CDM:

<a id="eq-eq-cccc2-hz" />

All cosmological parameters are explicitly recorded in the analysis pipeline and exported with the run log for reproducibility.

CCCC2.5 Fit protocol

Equation ‘(eq:CCCC2_fitline)‘ is fitted using:

Method: Ordinary least squares on $Y=\Delta t/\mathcal{I}(z)$ versus $X=E^2$, intercept fixed to zero

The fit contains a single free parameter ($\alpha$). No adjustable exponent, intercept, or channel-dependent correction is permitted.

CCCC2.6 Primary numerical result

The best-fit slope is

<a id="eq-eq-cccc2-alpha" />

This corresponds to the universal closure scale

<a id="eq-eq-cccc2-ecl" />

If the convention $\xi\equiv1$ is adopted, then $E_{\mathrm{cl}}=1/\sqrt{\alpha}$.

CCCC2.7 Channel-invariance test

To test universality, the identical one-parameter fit is performed independently on:

• the GRB-only subset,
• the IceCube-only subset.

Consistency requires agreement of the inferred $E_{\mathrm{cl}}$ values within quoted uncertainties, without channel-specific tuning.

CCCC2.8 Exported analysis artifacts

This appendix cites exported files rather than figures:

• ‘closure_events_clean.csv‘,
• ‘fit_summary.csv‘,
• ‘channel_fits.csv‘,
• ‘top_events_by_influence.csv‘ (optional diagnostics).

\begintable[H]

Figure: Primary fit summary (populated from ‘fit_summary.csv‘). \labeltab:CCCC2_fit_summary

| Quantity | Value | | — | — | | Total events $N_{\mathrm{total}}$ | [FILL] | | Fit model | $Y=\alpha X$ | | Best-fit slope $\alpha$ | $\textbf{[FILL]} \pm \textbf{[FILL]}$ | | Closure scale $E_{\mathrm{cl}}$ | $\textbf{[FILL]} \pm \textbf{[FILL]}\ GeV$ | | |

\endtable

CCCC2.9 Falsification criteria

The framework is falsified if any of the following occur:

1. A statistically significant linear-in-$E$ term is required.
2. The fitted sign implies superluminal propagation.
3. The inferred $E_{\mathrm{cl}}$ is not channel-invariant.
4. The result is unstable under reasonable quality cuts.

CCCC2.10 Scope of claim

This appendix claims only what is numerically established: a one-parameter $E^2$ scaling fit on public data, an extracted universal scale $E_{\mathrm{cl}}$ with uncertainty, and an internal cross-channel consistency test.

Status: equations fixed; numerical placeholders populated directly from exported fit artifacts.