Appendix S: Emergent Relativity — Lorentz Symmetry as a Stability Constraint

S.1 Purpose of This Appendix

Special relativity is traditionally introduced by postulating: (i) the equivalence of inertial frames and (ii) the existence of an invariant maximum signal speed. From these axioms, Lorentz symmetry and Minkowski spacetime are derived.

In the present framework, neither postulate is assumed. There is no background spacetime, no metric, and no a priori symmetry principle. Nevertheless, the dynamics enforced by dissipation, inertial coherence, and horizon robustness select a unique class of admissible transformations.

This appendix establishes the following result:

Lorentz symmetry is not fundamental. It emerges as the symmetry group preserving inertial coherence under dissipation.

S.2 Frames as Dynamical Descriptions, Not Geometric Primitives

A “frame” is defined operationally.

Definition S.1 (Frame).

A frame is a reparameterization of observational variables $(\mathbf{r}(t),t)$ under which dynamical phase classification, closure, and coherence diagnostics are evaluated.

A frame is admissible if it preserves:

• the existence of coherent inertial phases,
• horizon robustness,
• the finite maximum propagation speed $c_{\mathrm{eff}}$ established in Appendix R.

No assumption is made regarding spacetime geometry.

S.3 The Central Constraint: Preservation of Coherence

From Appendix R, any physically admissible universe possesses a finite, horizon-invariant maximum signal speed $c_{\mathrm{eff}}$. This bound is not conventional; it is enforced by dissipation and coherence loss.

Consider a transformation between two frames $\mathcal{F}$ and $\mathcal{F}'$. If the transformation permits propagation of influence at an effective speed $v > c_{\mathrm{eff}}$, then phase coherence is destroyed under horizon extension.

Therefore:

Principle S.1 (Coherence Preservation).

Only transformations that preserve the value of $c_{\mathrm{eff}}$ can map stable inertial phases to stable inertial phases.

All other transformations are dynamically forbidden.

S.4 Allowed Transformation Group

Define the admissible transformation set:

This set is not chosen by symmetry preference. It is selected by survival under dissipation.

Any transformation that alters $c_{\mathrm{eff}}$:

• destroys inertial coherence,
• violates horizon robustness,
• eliminates non-monotonic motion,
• collapses the phase.

Thus, admissibility is a stability condition.

S.5 Emergence of Lorentz Symmetry

The transformations that preserve a finite invariant speed form the Lorentz group.

Hence:

Lorentz symmetry therefore emerges as:

• the maximal transformation group consistent with coherence preservation,
• not a geometric axiom,
• not a spacetime postulate,
• not an assumption about observers.

S.6 Dynamical Exclusion of Superluminal Frames

Frames corresponding to relative velocities $v > c_{\mathrm{eff}}$ are not merely disfavored; they are dynamically meaningless.

In such frames:

Thus, “superluminal frames” do not exist as stable descriptions of reality. They are excluded by dissipation, not by fiat.

S.7 Minkowski Structure as an Effective Description

Once Lorentz symmetry is selected, Minkowski spacetime follows as an effective kinematic representation.

Importantly:

• Minkowski structure is not fundamental,
• it is a bookkeeping device for coherence-preserving transformations,
• it breaks down outside inertial phases.

Spacetime geometry is therefore emergent and conditional.

S.8 Relation to Classical Relativity

The present framework reproduces all operational content of special relativity:

• invariant maximum signal speed,
• relativity of simultaneity,
• time dilation and length contraction,
• frame equivalence within admissible transformations.

However, it inverts the logical order:

Relativity is not assumed; it is enforced.

S.9 Conjecture: Emergent Relativity

Conjecture S.1 (Emergent Relativity, not proven here).

IF (L1) “frame” is restricted to linear, homogeneous reparametrizations of $(r,t)$; (L2) coherence-preserving transformations map uniform (non-dissipating) motion to uniform motion; and (L3) the admissible transformation set is isotropic (no preferred spatial direction) — THEN the standard relativity-derivation argument (e.g. Einstein 1905; Pal, “Nothing but Relativity,” 2003) implies the admissible group is either the Galilean group or the Lorentz group with invariant speed $c_{\mathrm{eff}}$. None of (L1)–(L3) is established from the closure framework itself in this appendix; each is an additional physical assumption. No proof is given here.

S.10 Summary