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 $S^2$ 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, $j=\rho\nabla S/m$. 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 $\psi=\sqrt{\rho}\,e^{iS/\hbar}$, 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 $\hbar$.
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.
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 relationn=qmM,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.
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 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.
The Schrödinger Equation from Capacity Geometry The Schrödinger equation is usually either taken as a starting postulate or recovered by adding Fisher information or an exact-uncertainty principle as an independent quantum input. This paper proves a narrower but different result: within a declared fixed-background local action class, the nonrelativistic Schrödinger equation is forced downstream of an independently established geometric scalar rather than by postulating the Fisher term at the quantum layer. More precisely, within the declared local phase-independent quadratic-gradient density class, the Fisher-corrected Madelung action is the unique action family compatible with the admissible-class constraints, and its Euler–Lagrange equations are exactly the Schrödinger equation. The scaffold comes from two companion theorem lanes in The Spectral Geometry of Coherence programme, where Part 1A establishes the canonical logarithmic capacity scalar and the uniqueness of the quadratic base tension in the declared geometric class. And Part 7A proves that the continuous translation-invariant quadratic sector is phase blind, so a phase-sensitive evolution law cannot come from that sector alone. The present paper then shows that continuity-current compatibility fixes the phase sector, the quadratic-gradient and scaling assumptions force the Fisher-form density correction up to a constant, and the bridge to the imported scalar fixes the canonical member of that family. In that precise sense, the logical order is different at the Schrödinger layer: the geometric base comes first, and the Schrödinger action appears downstream of it. The claim is exact within its lane and no broader. It is a fixed-background, single-particle, local theorem, stated away from nodal sets. It does not claim a many-body theory, measurement theory, relativistic completion, global nodal reconstruction, or backreaction of the quantum sector on the background geometry. For programme context, readers can trace the scaffold through The Spectral Geometry of Coherence (SGOC), Volume Two, Parts 1–7, 10.5281/zenodo.18462286, together with Part 1A https://doi.org/10.5281/zenodo.19113748, Part 6A, 10.5281/zenodo.19113871, and Part 7A, 10.5281/zenodo.19113876.
We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics