Null-Directional Kinematics (NDK) begins with a statistical ensemble of invariant-speed propagation directions rather than massive particle trajectories as primitive. Previous work showed that a small set of compositional axioms uniquely selects the Bhattacharyya coefficient as a statistical clock affinity and that its binary reversal sector reproduces the Lorentz clock factor. A companion closure result further showed that this statistical clock law, together with the inertial and null-direction structural inputs of NDK, closes the kinematic sector to the proper orthochronous Lorentz group. A subsequent construction connected the local geometry of the same statistical structure to nonrelativistic Schrödinger dynamics. The present paper asks a stronger question. Can the clock sector itself constrain a relativistic quantum dynamics? It is shown that the invariant proper occupation associated with the null-direction current is exactly the extensive form of the same Bhattacharyya overlap that defines the NDK clock. Within the local, derivative-free class of Lorentz-scalar transport actions that are extensive in occupation, this proper occupation fixes the free clock action up to the normalization of the inertial mass. Its exact quadratic auxiliary representation introduces a positive scalar density $q$ that classically coincides with the proper occupation. The clock action then fixes the phase block of the local $(q,S)$ geometry, while compatibility with the inherited density-phase two-form uniquely fixes the reciprocal Fisher density block within the stated local symmetry class. The natural complex coordinate $\Psi=\sqrt q\,e^{iS/\kappa}$ and the linear Klein-Gordon equation are not assumed separately. Within that class, they follow from the inherited density–phase geometry. The quantum correction enters through the relation\[n=q\frac{M}{m},\]where $M$ is the local invariant mass parameter, relating proper occupation to amplitude density while leaving the kinematic NDK clock ratio unchanged. Its nonrelativistic limit recovers the previously obtained Madelung-Fisher structure, while its first relativistic correction has the standard scalar mass-velocity coefficient. With standard external electromagnetic minimal coupling, the theory also reproduces the exact Klein-Gordon Coulomb spectrum. The Klein-Gordon equation, Fisher geometry, and auxiliary-field methods are established results and are not claimed as new. The proposed foundations result is the chain connecting a uniquely selected null-direction clock overlap to a relativistic scalar quantum dynamics. In this construction, the relativistic clock does more than share a statistical representation with quantum mechanics. It helps determine the form of the relativistic quantum transport theory.
Null-Directional Kinematics (NDK) begins with a statistical ensemble of invariant-speed propagation directions rather than massive particle trajectories as primitive. Previous work showed that a small set of compositional axioms uniquely selects the Bhattacharyya coefficient as a statistical clock affinity and that its binary reversal sector reproduces the Lorentz clock factor. A companion closure result further showed that this statistical clock law, together with the inertial and null-direction structural inputs of NDK, closes the kinematic sector to the proper orthochronous Lorentz group. A subsequent construction connected the local geometry of the same statistical structure to nonrelativistic Schrödinger dynamics. The present paper asks a stronger question. Can the clock sector itself constrain a relativistic quantum dynamics? It is shown that the invariant proper occupation associated with the null-direction current is exactly the extensive form of the same Bhattacharyya overlap that defines the NDK clock. Within the local, derivative-free class of Lorentz-scalar transport actions that are extensive in occupation, this proper occupation fixes the free clock action up to the normalization of the inertial mass. Its exact quadratic auxiliary representation introduces a positive scalar density q that classically coincides with the proper occupation. The clock action then fixes the phase block of the local (q,S) geometry, while compatibility with the inherited density-phase two-form uniquely fixes the reciprocal Fisher density block within the stated local symmetry class. The natural complex coordinate Ψ=qeiS/κ and the linear Klein-Gordon equation are not assumed separately. Within that class, they follow from the inherited density–phase geometry. The quantum correction enters through the relation[n=q\frac{M}{m},]where M is the local invariant mass parameter, relating proper occupation to amplitude density while leaving the kinematic NDK clock ratio unchanged. Its nonrelativistic limit recovers the previously obtained Madelung-Fisher structure, while its first relativistic correction has the standard scalar mass-velocity coefficient. With standard external electromagnetic minimal coupling, the theory also reproduces the exact Klein-Gordon Coulomb spectrum. The Klein-Gordon equation, Fisher geometry, and auxiliary-field methods are established results and are not claimed as new. The proposed foundations result is the chain connecting a uniquely selected null-direction clock overlap to a relativistic scalar quantum dynamics. In this construction, the relativistic clock does more than share a statistical representation with quantum mechanics. It helps determine the form of the relativistic quantum transport theory.
A companion letter established that four simple axioms on statistical comparison uniquely select the Bhattacharyya coefficient as a clock functional, and that its binary reversal sector yields the Lorentz time-dilation factor without presupposing a spacetime metric. The present paper asks whether that uniqueness structure can be extended from kinematics to dynamics. Five linked results are established. First, the continuum parity-reversal extension of the clock functional on the local celestial sphere S2 yields, in the weak-anisotropy dipole sector, a locally Minkowskian quadratic norm. Second, on the resulting spatial slices, a local action written in current form and constrained by probability conservation forces the transport current to be potential, j=ρ∇S/m. Third, an intrinsic second-order density geometry, taken as the local continuum descendant of the unique monotone Fisher-Rao metric identified by Čencov, fixes the density-gradient term uniquely up to overall scale as the Fisher information. Fourth, the resulting Euler-Lagrange system is exactly the Madelung form of quantum mechanics. Finally, by the standard substitution ψ=ρeiS/ℏ, the Schrödinger equation follows. The central result is not merely that Fisher information reproduces the Schrödinger equation, but that the Lorentz clock factor and the Schrödinger equation arise from sibling uniqueness structures on the same probability geometry: Bhattacharyya affinity and Fisher-Rao metric. The only undetermined dimensional constants are c and ℏ.
In previous work, the master action of the null-directional kinematics (NDK) framework was constructed from Fisher-information transport on the sphere bundle over spacetime, yielding a real amplitude field ψ(k^;x) with a quadratic kinetic sector inherited from the information geometry of probability densities. This paper derives the emergent Schrödinger dynamics of that framework in a controlled low-energy limit. The derivation proceeds in four steps. First, a controlled angular reduction justified by the spectral gap of the sphere-bundle mode tower truncates the full field to a spacetime envelope R(x) times a frozen angular profile. Second, the resulting reduced action is shown to be a real Fisher-density theory in spacetime. Third, the requirement that this density evolve locally and reversibly forces a canonically conjugate transport potential S(x,t) via a continuity constraint. Fourth, the resulting (ρ,S) system is shown to be exactly the Madelung form of a Schrödinger equation. The complex field is therefore not grafted onto the real theory but emerges as the compact packaging of a conjugate pair forced by reversible transport. Independent arguments from the dipole-current sector and the isotropic rest-mode sector pin the effective mass and the Fisher coefficient in terms of master-action couplings, yielding an emergent quantum scale of the form ℏeff2∼24βΛ∗/c2, with the precise numerical coefficient fixed up to order-unity convention-dependent factors. Stress-energy matching of the canonically normalized positive-frequency envelope fixes the density-normalization bridge at n∗=1. A systematic analysis then shows that all internal one-particle consistency conditions—free-particle dispersion, probability current, Galilean covariance, and circulation quantization—close without contradiction but do not generate an additional equation fixing the numerical value of ℏeff. The program therefore derives the structure of quantum mechanics from the NDK directional substrate, while the absolute quantum action scale remains open and is isolated to a single product of master-action couplings.
This paper studies normalized probability states on pairs of parity-related null channels. The total weights of the two channels define an imbalance β=w+−w−, while the Bhattacharyya affinity between a state and its parity reverse defines a dimensionless clock functional χ. For an arbitrary internal channel coordinate, the exact factorizationχ=1−β2A(R+,R−)is established, where R± are the normalized internal profiles of the two channels and A(R+,R−) is their Bhattacharyya affinity. Consequently,χ≤1−β2,with equality, away from the completely polarized endpoints, if and only if the two channels have the same normalized internal profile and differ only in their total weights. After identifying β=v/c, the saturation value is the special-relativistic proper-time factor. Rapidity becomes one half of the channel log-odds ratio, and reciprocal exponential reweighting furnishes a representation of collinear Lorentz boosts with additive rapidity and Einstein velocity composition. The analysis further shows that the ordinary directional first moment is profile-dependent, whereas the Lorentz clock factor is universal across the full common-profile saturation manifold. Rotational covariance constrains but does not uniquely determine the shared profile; maximum entropy supplies a canonical exponential family. Off saturation, the clock deficit is exactly controlled by the Hellinger separation of the two internal profiles and is quadratic for small profile mismatch. As an operational illustration, a pure ℓ=3 directional moment is coupled to the gauge-invariant electric-octupole curvatureEijk=D⟨iDjEk⟩.In a linear-response completion, this produces a one-sided clock shift quadratic in Eijk while leaving normalization and the directional first moment unchanged at leading order. The magnitude depends on an undetermined matter--directional Wilson coefficient, so the construction defines a controlled null test rather than a parameter-free forecast. The construction therefore distinguishes the empirically established Lorentz sector from a broader null-directional state space without, by itself, asserting that the additional states are physically realized.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
This work develops a phase-based formulation of relativistic wave dynamics, demonstrating that the Klein--Gordon equation emerges naturally from the foundational assumption of intrinsic phase periodicity in material fields. Mapping the phase directly onto the classical action, we postulate that localized excitations possess an invariant rest-frame oscillation governed by a proper frequency ω0. This physical condition establishes an operational mass-frequency relation, m=ℏω0/c2, without requiring rest energy as an independent, axiomatic input. We show that the Klein--Gordon equation arises as the minimal local, linear, Lorentz-invariant field equation compatible with this internal phase structure. Within this framework, mass acts as an intrinsic frequency scale governing wave propagation, and relativistic kinematics is fully recovered as a structural consequence of phase coherence. This approach provides a unified wave-mechanical interpretation where particle dynamics maps onto the group velocity of dispersive wave packets, offering an intuitive account of free propagation, dispersion, and tunneling across potential barriers.
This paper asks what must be added to the one normalized relational interval and selected finite-dimensional carriers inherited from the first two Complementarity-First Time papers before those carriers function as quantum clocks, support endpoint comparisons, and share a universal matter coupling. The analysis first separates four parameter types: raw relation magnitude ρ, normalized relational interval ℓ, supplied local proper interval τ, and a shared comparison parameter s. In the declared order-unit placement class, the raw interval and raw generator compensators cancel exactly, leaving phase evolution per ℓ. For any one chosen parameter, a selected state-readout statistic is minimally phase-readable exactly when its grouped Bohr-frequency expansion has nonzero weight in at least one nonzero mode. A declared endpoint observable factors into signal transport and local clock calibration after the signal law, path, observers, and transitions are supplied. Complete labeled gap transfer is affine on each connected spectral graph. On a supplied common semiclassical path, a complete mass-energy operator is sufficient for common internal spectral scaling at the retained weak-field order; it does not derive universal trajectories or a unique matter action. An explicitly typed worldline-reparameterization-covariant, complement-even scalar model gives species-dependent clock rates, proving that the inherited one-time geometry and complement parity do not force universality. A two-level commuting perturbation has an internal affine blind spot, whereas an addressed three-level sector supports an anchor-free spectral-shape contrast. The prospective multiclock architecture therefore has rank three, with independent vector z = (Δ_ST, Δ_01, N_3)^T. It remains at STOP-SOURCE. No second physical time dimension, universal matter action, unique signal action, executed experiment, numerical redshift prediction, absolute second, numerical ℏ, objective chance law, total-carrier completeness, or thermodynamic arrow is claimed.