Appendix CCCC — Structural Bounds and No-Go Constraints

CCCC.1 Purpose and Logical Position

This appendix derives model-independent bounds and no-go constraints that follow solely from the structural framework established in Appendices AAAA and BBBB.

No observational data, phenomenological fitting, or astrophysical assumptions are introduced. All results are consequences of: (i) structural stability, (ii) admissible invariant construction, and (iii) long-time dynamical consistency.

The role of this appendix is to delimit what is possible at all within the theory.

CCCC.2 Admissible Observable Class

Let $\mathcal{O}[\Psi]$ denote any admissible observable as defined in Appendix BBBB, satisfying:

1. Determinism (no stochastic inputs),
2. Long-time averaging invariance,
3. Stability under admissible perturbations,
4. Coarse-graining invariance.

Such observables may depend on a finite set of history-averaged structural quantities, e.g.

but not on microscopic fluctuations.

CCCC.3 General Upper Bound (Stability Constraint)

For any admissible observable $\mathcal{O}$, structural stability implies the existence of a finite bound:

<a id="eq-eq-upper-bound" />

where $p \le 2$ and $C_{\mathcal{O}}$ is a theory-internal constant.

Reason.

Higher powers $p>2$ amplify microscopic perturbations and violate the perturbative stability condition of Appendix BBBB. Linear ($p=1$) and quadratic ($p=2$) dependence exhaust the admissible scaling class.

CCCC.4 Lower Bound and Non-Triviality

Non-trivial observables must satisfy:

<a id="eq-eq-lower-bound" />

otherwise they fail to distinguish dynamically stable sectors from noise-dominated histories.

This excludes observables that vanish under averaging or collapse under coarse-graining.

CCCC.5 No-Go Theorem I: Probabilistic Observables

Statement.

No observable that depends explicitly on probabilistic weights, likelihoods, or posterior assignments is admissible.

Proof (sketch).

Probabilistic constructs introduce ensemble-dependence that violates additivity and compositional consistency (Appendix BBBB). Therefore, any observable $\mathcal{O}_{\mathrm{prob}}$ is structurally inadmissible.

CCCC.6 No-Go Theorem II: ML-Dependent Functionals

Statement.

Any observable whose definition depends on machine-learned parameters or training data is structurally forbidden.

Reason.

Such observables: (i) lack invariance under re-sampling, (ii) depend on external priors, and (iii) fail reproducibility under history truncation.

CCCC.7 Universality of Quadratic Scaling

Combining the upper bound (‘§eq:upper_bound‘) with the uniqueness result of Appendix AAAA, we obtain the universal structural form:

<a id="eq-eq-quadratic-universality" />

Any admissible observable that saturates the structural bounds must reduce to this quadratic class up to normalization.

CCCC.8 Consequences for Phenomenology

These bounds imply:

• No arbitrarily sharp event ranking is possible.
• No super-quadratic energy dominance can occur.
• All physically meaningful rankings must collapse to a single structural family.

This explains the empirical concentration observed in Appendix RRR without invoking selection bias.

CCCC.9 Summary

This appendix completes the theoretical constraint layer of the framework.