# 204. Moving Closure, Length Contraction, and Michelson-Morley This appendix derives longitudinal contraction from the structure of a moving self-sustained closure. The contraction is not inserted as a coordinate rule. It is the geometric deformation required for one bounded transport mode to remain coherent while drifting uniformly through an approximately uniform region. Throughout this appendix, let $k$ denote the local transport speed in that region. The derivation assumes that $k$ is effectively constant over the size of the apparatus during one run. More general backgrounds require path integrals of the same local transport law. ## 204.1 Assumptions and Rest Closure Fix an approximately uniform region in which the local transport speed is the constant $k>0$. Consider one bounded self-sustained mode. In its rest configuration there is no distinguished drift direction. Choose one closure span of length $L_0$ along a chosen axis, and one orthogonal closure span of the same length $L_0$. The quantity $L_0$ is not introduced as an external ruler length. It is the rest span of one internal closure of the mode. One out-and-back closure across such a span has recurrence period $$ T_0=\frac{2L_0}{k}. $$ Now let the whole mode drift uniformly with speed $v$ along the $x$ direction, with $$ 0\le v Let a bounded self-sustained closure drift uniformly through an > approximately uniform region with local transport speed $k$. If the moving > closure remains coherent, then its longitudinal span must be > > $$ > L_\parallel=L_0\sqrt{1-\frac{v^2}{k^2}}. > $$ The conclusion is forced by coherence of the moving closure. It is not an additional convention. ## 204.6 Structural Meaning The contraction has not been imposed as a measurement convention. It has been derived from one structural requirement only: the moving bounded mode must remain one coherent closure. So the meaning of the result is precise: - a mode at rest closes with one recurrence structure - a drifting mode must preserve that closure - preserving that closure forces a longitudinal deformation Length contraction appears here as the geometry required for moving closure, not as an external postulate. ## 204.7 Michelson-Morley Consequence Now consider a Michelson-Morley interferometer built from the same bounded material closures and drifting uniformly through the same approximately uniform region. Let each arm have rest length $L_0$. The arm transverse to the drift keeps that length geometrically, so its round-trip transport time is $$ T_\perp=\frac{2L_0}{\sqrt{k^2-v^2}}. $$ The arm parallel to the drift contracts to $$ L_\parallel=L_0\sqrt{1-\frac{v^2}{k^2}}, $$ so its round-trip transport time is $$ T_\parallel = \frac{L_\parallel}{k-v}+\frac{L_\parallel}{k+v} = \frac{2kL_\parallel}{k^2-v^2}. $$ Substituting the contracted length, $$ T_\parallel = \frac{2kL_0\sqrt{1-v^2/k^2}}{k^2-v^2} = \frac{2L_0}{\sqrt{k^2-v^2}} = T_\perp. $$ Therefore $$ \Delta T = T_\parallel - T_\perp = 0. $$ So a Michelson-Morley device is blind to uniform translational drift through a region with uniform local transport speed $k$. The null result does not arise because nothing moved. It arises because the same transport closure that carries the signal also determines the moving geometry of the device. ## 204.8 Scope This appendix addresses uniform drift in an approximately uniform region. If the surrounding transport conditions vary across the apparatus, then the local speed $k$ must be replaced by the appropriate path-dependent transport speed. The structural point remains the same: transport and geometry must be solved together, because the bounded mode and the signal it carries are governed by the same closure.