Appendix BBBB — Structural Invariants and Observable Projections

BBBB.1 Role of this appendix in the logical chain

This appendix serves a precise and limited purpose: to establish the mapping between the structural measure on histories derived in Appendix AAAA and the operational invariants that can be computed, ranked, and tested in numerical and observational data.

No new dynamics are introduced here. No numerical results are reported. Instead, this appendix defines the minimal set of quantities that

1. are invariant under admissible perturbations,
2. descend unambiguously from the structural measure,
3. admit deterministic extraction from finite data,
4. and are suitable for falsifiable validation.

This appendix therefore mediates between:

BBBB.2 From history-space measure to event-level invariants

Appendix AAAA establishes a unique quadratic structural weighting

defined over equivalence classes of admissible histories.

In practice, finite datasets do not provide access to entire histories. Instead, they provide events or snapshots that act as compressed representatives of underlying dynamical trajectories.

The guiding principle is therefore:

\beginquote An admissible observable must act as a monotonic proxy for the structural weight of the histories that generated it. \endquote

This requirement sharply restricts the class of acceptable observables.

BBBB.3 Structural admissibility criteria for observables

An observable $\mathcal{O}$ is considered structurally admissible if it satisfies:

1. Determinism. $\mathcal{O}$ must be computable from the data without stochastic modeling, training procedures, or prior-dependent inference.
2. Monotonicity. Higher values of $\mathcal{O}$ must correspond to greater structural dominance of the underlying history sector.
3. Stability. Small perturbations in the data induce only subleading variations in $\mathcal{O}$.
4. Scale compression. Extremal values should be enhanced without allowing any single factor to diverge or dominate pathologically.

These criteria eliminate probability assignments, likelihood ratios, and machine-learned classifiers as structurally inadmissible for foundational testing.

BBBB.4 Canonical structural proxies

Two classes of proxies arise naturally from the structural measure:

(A) Amplitude proxies.

Quantities that increase monotonically with the strength of the dynamical response, e.g. energy deposition, rotation amplitude, or integrated flux. Such quantities act as first-order surrogates for $\langle |\mathbf{L}| \rangle$.

(B) Precision or coherence proxies.

Quantities that quantify the concentration or coherence of the dynamical outcome, such as angular localization uncertainty or spatial dispersion. These suppress structurally noisy or diffuse histories.

Structurally meaningful observables must combine both aspects.

BBBB.5 Dimensionless invariant construction

Given an amplitude proxy $A$ and a coherence proxy $C$, the simplest structurally admissible invariant takes the form

where $f$ and $g$ are monotonic, dimensionless mappings chosen to compress dynamic range while preserving ordering.

Logarithmic compression of amplitude and linear or sublinear penalization of dispersion are structurally preferred, as they

1. suppress extreme outliers,
2. stabilize rankings,
3. and preserve additivity under coarse graining.

This generic form underlies all operational invariants introduced in subsequent appendices.

BBBB.6 Relation to numerical validation (Appendix RRR)

Appendix RRR instantiates the abstract invariant $\mathcal{I}$ using concrete proxies extracted from NPZ-resolved dynamics. The unified scoring functional defined there is not arbitrary; it is a direct realization of the admissible structure defined here.

Specifically:

• flatness metrics act as coherence proxies,
• amplitude metrics act as structural strength proxies,
• lensing normalization enters only through compressed factors,
• and final rankings emerge deterministically.

Thus Appendix RRR should be read not as an empirical guess, but as the first controlled execution of the structural dictionary formalized in this appendix.

BBBB.7 Forward compatibility with observational data

The same admissibility logic applies to event-based observational datasets. When applied to discrete astrophysical events, the invariant $\mathcal{I}$ becomes an event-ranking statistic that orders observations by their intrinsic structural informativeness, independent of astrophysical source modeling.

This forward compatibility is exploited explicitly in later appendices concerned with real detector data.

BBBB.8 Summary

This appendix establishes the following:

1. The structural measure of Appendix AAAA does not remain abstract; it enforces strict constraints on admissible observables.
2. Only deterministic, monotonic, and stable combinations of amplitude and coherence are structurally valid.
3. All numerical and observational tests in this work instantiate this same invariant logic.

Appendix BBBB therefore completes the conceptual bridge between structural theory and empirical falsification.