Appendix MMM — Lorentz Invariance from Closure

MMM.1 Statement of the Result

This appendix proves that Lorentz invariance emerges as a necessary consequence of closure dynamics, given the simultaneous existence of:

• a minimal quantized closure unit (L_\min) (see Appendix OOO, § OOO.5: this is a candidate closure-quantization scale; it is not identified with $\hbar$ here, since that identification is disputed and, per Appendix PPP, fails calibration by 38–112 orders of magnitude),
• a maximal admissible propagation speed (c_\mathrmeff).

No spacetime geometry, metric structure, or relativistic postulate is assumed.

MMM.2 Kinematic Admissibility Constraints

Consider any physical excitation capable of sustaining closure. Such an excitation must satisfy simultaneously:

These inequalities define the full admissible kinematic domain for physical processes.

Any transformation between observers must preserve this domain in order to maintain closure.

MMM.3 Failure of Galilean Transformations

Under a Galilean velocity transformation:

there exists no invariant upper bound on (|\mathbfv'|).

Hence, even if:

one can always choose (\mathbfu) such that:

By Appendix LLL, this implies closure collapse. Therefore, Galilean kinematics is existentially inadmissible.

MMM.4 Requirement of Speed-Bound Invariance

Any admissible transformation between reference frames must preserve the maximal propagation speed:

This condition uniquely fixes the transformation group to those preserving a universal speed bound.

MMM.5 Role of the Quantum Closure Scale

The existence of a minimal closure unit:

(not identified with $\hbar$; see Appendix OOO, § OOO.5) implies that phase-space volume elements cannot be arbitrarily compressed.

Any admissible transformation must therefore preserve:

• causal ordering (Appendix LLL),
• minimal action content (Appendix OOO).

This excludes non-linear or dissipative re-scalings of time or space.

MMM.6 Emergence of Lorentz Transformations

The unique class of transformations that:

• preserve a universal speed bound (c_\mathrmeff),
• preserve minimal action (\hbar),
• preserve closure admissibility,

are the Lorentz transformations with invariant speed (c_\mathrmeff).

No alternative transformation group satisfies all three constraints.

MMM.7 Interpretation

Lorentz invariance does not encode spacetime symmetry. It encodes the algebra of existence.

It arises because:

• faster-than-(c_\mathrmeff) motion is existentially forbidden,
• sub-(\hbar) closure is dynamically impossible,
• physical observers must agree on what can exist.

\paragraph*Concluding Statement