2 - Wave–Based Kinematics from Phase Invariance: A Preferred–Frame Extension of Optical–Mechanical Dynamics

## Abstract We develop a wave–based kinematical framework organized around a single invariant object: a scalar physical phase \(\Phi(x,t)\). Local frequency and wave vector are defined as phase gradients, \[\omega = -\partial_t \Phi,\qquadk = \partial_x \Phi,\] with canonical energy and momentum fixed universally by \[E = \hbar \omega,\qquadp = \hbar k.\] Observable quantities are taken to be accumulated phase differences between physical histories, expressed through path integrals of the canonical one–form \[\hbar\, d\Phi = p\,dx - E\,dt.\] The framework departs from strict Lorentz relativity by postulating a preferred inertial frame \(F_0\) in which free matter waves obey a simple isotropic constitutive dispersion with finite limiting speed \(c\). Descriptions in moving laboratory charts are obtained by combining phase scalarity, \[\Phi(x,t)=\Phi'(x',t'),\] with a fixed inter–frame map selected by an explicit pre–relativistic operational criterion based on isotropic influence propagation at speed \(c\) in \(F_0\). The resulting transformation is the inverse–Voigt map without Lorentz normalization, \[x = x' + V t',\qquadt = t' + \frac{V}{c^2}x'.\] Matter-wave dynamics are then governed by the constitutive energy–velocity relation \[E(v)=\frac{mc^2}{1-v^2/c^2},\] together with the group–velocity condition \[v = \frac{dE}{dp}.\] Integration yields the exact momentum law \[p(v)=mc\left[\frac{\beta}{1-\beta^2}+\operatorname{artanh}(\beta)\right],\qquad\beta=\frac{v}{c}.\] The induced transformation of canonical variables between \(F_0\) and a moving laboratory frame is \[E' = E - Vp,\qquadp' = p - \frac{V}{c^2}E,\] leading to the velocity composition law \[v'=\frac{v-V}{1-\dfrac{Vv}{c^2}}.\] Vacuum optical Michelson–Morley configurations remain null in ideal geometry because linear electromagnetic dispersion, \[\omega = c|k|,\] implies equality of phase and group velocities, \[v_p = u_g = c,\] which produces exact segment cancellations in the phase integral. By contrast, matter waves are generically dispersive under the deformed laboratory dispersion and therefore satisfy \[v'_p \neq u'_g,\] leading to orientation- and time-dependent interferometric phase signatures. The signal amplitude depends explicitly on the experimental preparation convention at the splitter (fixed \(\omega'\) versus fixed \(k'/p'\)). The preferred frame carries the fundamental constitutive time parameter, while measured clock times are reconstructed from accumulated phase rather than introduced as independent relativistic proper times. The framework is therefore formulated as a constitutive effective theory based on phase invariance, dispersion, and operational reconstruction, rather than as a fundamentally Lorentz-covariant spacetime theory.