Appendix R: Emergent Causality and the Existence of a Maximum Signal Speed

R.1 Motivation and Scope

Classical relativity postulates the existence of an invariant maximum signal speed as a foundational axiom. In contrast, the framework developed in this work introduces no such assumption. There is no spacetime metric, no light cone, no Lorentz symmetry, and no a priori causal structure.

Nevertheless, the numerical evidence forces a nontrivial conclusion: there exists a finite, horizon-invariant upper bound on the propagation speed of coherent influence. This appendix establishes that result and shows that causality emerges as a dynamical constraint imposed by dissipation and coherence loss.

R.2 What Is Meant by “Signal Speed”

In the present framework, a “signal” is not defined as a particle or wave excitation propagating on a background geometry. Instead, it is defined operationally:

Definition R.1 (Coherent Influence).

A signal is said to propagate from region $\mathcal{A}$ to region $\mathcal{B}$ if a localized perturbation introduced in $\mathcal{A}$ produces a reproducible, phase-coherent response in $\mathcal{B}$ that survives the full horizon-extension and repeatability tests defining robustness.

This definition is purely dynamical and makes no reference to spacetime structure.

R.3 Empirical Constraints from the Validator

The validated simulations impose three non-negotiable empirical constraints:

1. Dissipation. The system is explicitly non-Hamiltonian and dissipative, characterized by a damping parameter $\gamma>0$. There exists a dissipation timescale $\tau_{\mathrm{decay}}\sim\gamma^{-1}$ beyond which stored inertial memory is erased.
2. Horizon Robustness. All claims of stability and persistence are tested under horizon extension $T \rightarrow kT$ with $k\in\{1,2,4\}$. Any apparent structure that fails under extension is rejected as transient.
3. Coherence Loss Beyond Critical Damping. For $\gamma>\gamma_c$, ensemble-averaged angular momentum decays to zero and all non-monotonic behavior vanishes. In this regime, no persistent influence survives long horizons.

Together, these constraints forbid arbitrarily fast or arbitrarily long-lived propagation of influence.

R.4 Emergence of a Finite Propagation Bound

Let $\ell_{\mathrm{coh}}$ denote the maximum spatial extent over which phase-coherent inertial organization can be maintained before dissipation destroys correlation. Let $\tau_{\mathrm{decay}}\sim\gamma^{-1}$ denote the characteristic coherence lifetime.

Any influence propagating with speed $v$ over a distance $\ell$ requires a time $\tau \sim \ell / v$. Coherence can survive this propagation only if $\tau \lesssim \tau_{\mathrm{decay}}$.

Thus, coherence imposes the inequality:

This leads directly to the existence of a finite maximum propagation speed:

No faster propagation is dynamically admissible, because it would erase the very coherence required for influence to be meaningfully transmitted.

R.5 Horizon Invariance of the Bound

Crucially, the bound $c_{\mathrm{eff}}$ is not an artifact of finite simulation time. The validator explicitly enforces horizon invariance:

Any candidate propagation speed exceeding $c_{\mathrm{eff}}$ produces effects that decay or decorrelate under horizon extension and therefore fails the robustness criterion. Hence, the bound is invariant under time rescaling and constitutes a genuine dynamical limit.

R.6 Independence from Spacetime Axioms

At no point in the derivation of $c_{\mathrm{eff}}$ is spacetime geometry invoked. There is:

• no metric,
• no causal cone postulate,
• no assumption of Lorentz invariance,
• no predefined notion of simultaneity.

The existence of a finite signal speed arises solely from:

1. dissipation,
2. finite coherence lifetime,
3. horizon-robust persistence.

Causality, in this framework, is therefore not a geometric primitive but a stability constraint.

R.7 Replacement of the Relativistic Postulate

Special relativity postulates an invariant maximum speed. Here, that postulate is replaced by a theorem:

Theorem R.1 (Emergent Causality).

In any dissipative system that supports horizon-robust inertial organization, there exists a finite, observer-independent maximum speed of coherent influence $c_{\mathrm{eff}}$. No influence propagating faster than $c_{\mathrm{eff}}$ can retain phase coherence or dynamical relevance.

This theorem supplies the causal backbone traditionally attributed to spacetime structure, but derives it from dynamics alone.

R.8 Interpretation

The emergent speed $c_{\mathrm{eff}}$ plays the operational role of the invariant speed in relativity: it bounds causal influence, enforces temporal ordering, and defines admissible transformations. However, it is not fundamental. It is a saturation velocity determined by the balance between inertial memory storage and dissipation.

In subsequent appendices, this bound will be shown to give rise to effective Lorentz symmetry, massless propagation, and gravitational light bending without introducing new axioms.

R.9 Summary