Appendix V: The Emergent Causal Cone — Causality Without Spacetime Geometry

V.1 Purpose and Claim

Relativity encodes causality through a geometric primitive: the light cone of a spacetime metric. In the present framework, no such primitive is assumed. There is:

• no metric $g_{\mu\nu}$,
• no null condition,
• no geodesic structure,
• no a priori notion of a causal cone.

Nevertheless, the validated dynamics enforce a strict causal ordering constraint: coherent influence can propagate only within a finite, horizon-invariant cone. This appendix defines that cone operationally and proves its emergence from dissipation, coherence, and horizon robustness.

Central Result.

V.2 Operational Definition of Causal Influence

In this work, “causal influence” is not defined by geometry; it is defined by reproducibility and phase coherence under the same validation standards used to classify inertial phases.

Definition V.1 (Causal influence).

Let $\delta\Psi_{\mathcal{A}}$ be a localized perturbation introduced in region $\mathcal{A}$ at time $t_0$. We say $\mathcal{A}$ causally influences region $\mathcal{B}$ at time $t>t_0$ if the following holds:

1. A statistically reproducible response $\Delta \mathcal{O}_{\mathcal{B}}(t)$ occurs in $\mathcal{B}$, for a fixed diagnostic observable $\mathcal{O}$ used in phase classification.
2. The response is phase-coherent in the sense that it survives:

• repeat trials (seed variation),
• grid refinement,
• horizon extension $T\mapsto kT$ with $k\in\{1,2,4\}$.

If any of these tests fail, the apparent influence is classified as transient and is not counted as causal transmission.

V.3 The Finite Maximum Propagation Speed

Appendix R established that coherent influence admits a finite maximum speed $c_{\mathrm{eff}}$, enforced by coherence lifetime and dissipation. We restate the operative bound here:

No additional assumptions are required: this bound follows from the incompatibility of coherence transmission with dissipation beyond the coherence timescale.

V.4 Definition of the Emergent Causal Cone

We now define the causal cone purely as a dynamical reachability set.

Let $d(\mathcal{A},\mathcal{B})$ denote the Euclidean distance between regions in the simulation domain (or, more generally, the metric distance on the computational manifold used to evaluate spatial gradients and fluxes in the governing PDE).

Definition V.2 (Causal cone).

For a perturbation injected at $(\mathcal{A},t_0)$, define the causal accessibility set at time $t$:

The boundary

is the emergent causal cone surface. It is not postulated; it is the maximal domain within which coherent influence can be validated.

V.5 Theorem: Cone Emergence as a Robustness Constraint

Theorem V.1 (Emergent causal cone).

In any dissipative system supporting horizon-robust inertial organization, the set of regions that can be coherently influenced by a localized perturbation at time $t_0$ is contained in the cone $\mathfrak{C}(\mathcal{A},t_0;t)$ defined by $c_{\mathrm{eff}}$.

Proof (robustness-based).

Assume for contradiction that coherent influence is observed at $(\mathcal{B},t)$ with

Then the implied effective propagation speed

satisfies $v_{\mathrm{eff}} > c_{\mathrm{eff}}$. But Appendix R shows that any influence propagating faster than $c_{\mathrm{eff}}$ cannot retain phase coherence under horizon extension and therefore fails the robustness criteria used to certify persistent dynamics. This contradicts the hypothesis that the response in $\mathcal{B}$ is horizon-robust and coherent. Hence coherent influence is restricted to the cone. $\square$

V.6 Cone Invariance Across Admissible Frames

Appendix S established that the admissible transformation group is exactly the set of transformations preserving $c_{\mathrm{eff}}$. Therefore the causal cone defined by $c_{\mathrm{eff}}$ is invariant under all admissible frames.

Corollary V.1 (Frame invariance of the cone).

If $\mathcal{T}\in\mathcal{G}_{\mathrm{adm}}$ and $\mathcal{T}$ preserves $c_{\mathrm{eff}}$, then $\mathcal{T}$ maps cone boundaries to cone boundaries:

Thus, causal structure is not geometric input; it is a dynamical invariant of the admissible description class.

V.7 Inside, On, and Outside the Cone

The cone partitions dynamical effects into three regimes:

1. \textbfInside the cone ($d < c_{\mathrm{eff}}\Delta t$). Coherent influence can propagate; perturbations may produce reproducible phase responses.
2. \textbfOn the cone ($d = c_{\mathrm{eff}}\Delta t$). This is the saturation boundary. The only excitations that can persist on the boundary are coherence-saturated modes.
3. \textbfOutside the cone ($d > c_{\mathrm{eff}}\Delta t$). Any observed correlation is necessarily transient or non-coherent and fails horizon robustness. No causal transmission exists in the operational sense of Definition V.1.

V.8 Relation to “Light” as Saturation

Appendix T identified “light” with coherence-saturation propagation:

Therefore, the cone boundary is precisely the dynamical locus of lightlike propagation. This reproduces the logical role of null cones without introducing null geometry:

V.9 Why the Cone is Fundamental in This Framework

In standard theories, the cone is fundamental and forces are placed inside it. Here, the logical order is reversed:

Thus, causality is not a spacetime axiom. It is the stability envelope of coherent dynamical organization.

V.10 Summary