This paper clarifies the role of the quantum Fisher coupling in reconstructive dynamics. Article 5 showed that the reconstructive action reproduces the Madelung equations, and hence the Schrödinger equation, when the Fisher term carries the coupling λ = ℏ2/8m. The present article argues that the fundamental object to be derived is not ℏ or m separately, but their natural combination: κ = ℏ2/m. Three results are established. First, κ is the unique coupling of dimension energy times length squared compatible with the Fisher information term in the Madelung Hamiltonian. Second, κ/2 appears as the scale of the biharmonic flow obtained by linearising the Fisher gradient flow around a flat reference state. Third, the decomposition κ = ℏ2/m requires additional geometric input: m as the inertial coefficient of Wasserstein transport, and ℏ as the prequantisation constant associated with a non-trivial symplectic loop. The paper does not derive κ from hypotheses H1–H5. Rather, it establishes that κ is the natural single object to be derived and identifies the geometric conditions under which its decomposition into ℏ and m may become accessible.
This paper clarifies the role of the quantum Fisher coupling in reconstructive dynamics. Article 5 showed that the reconstructive action reproduces the Madelung equations, and hence the Schrödinger equation, when the Fisher term carries the coupling λ = ℏ2/8m. The present article argues that the fundamental object to be derived is not ℏ or m separately, but their natural combination: κ = ℏ2/m. Three results are established. First, κ is the unique coupling of dimension energy times length squared compatible with the Fisher information term in the Madelung Hamiltonian. Second, κ/2 appears as the scale of the biharmonic flow obtained by linearising the Fisher gradient flow around a flat reference state. Third, the decomposition κ = ℏ2/m requires additional geometric input: m as the inertial coefficient of Wasserstein transport, and ℏ as the prequantisation constant associated with a non-trivial symplectic loop. The paper does not derive κ from hypotheses H1–H5. Rather, it establishes that κ is the natural single object to be derived and identifies the geometric conditions under which its decomposition into ℏ and m may become accessible.
Uniqueness of the Fisher–Madelung Coupling in the Minimal Kähler Reconstruction This article addresses a key structural question left open by Articles 5 and 6: is the quantum Fisher coupling merely chosen to match the Schrödinger equation, or is it uniquely forced by the reconstruction framework? Purpose Articles 5 and 6 showed that the reconstructive action reproduces the Madelung–Schrödinger dynamics when the Fisher coupling is λ = ℏ² / 8m, equivalently κ = ℏ² / m. Article 11 strengthens this result by showing that this coupling is the unique one compatible with the minimal Fisher–Wasserstein–Madelung–Kähler structure. Main Result The paper proves that four structural constraints determine the coupling uniquely: Fisher compatibility: the Fisher term must have the correct energy dimension, forcing λ to be proportional to ℏ²/m. Wasserstein compatibility: the Fisher term must combine consistently with the mass-scaled Wasserstein kinetic term. Kähler compatibility: the Madelung phase space must admit a compatible Kähler structure. This fixes the numerical factor exactly: λ = ℏ² / 8m. Global constancy: Markov monotonicity excludes spatially varying couplings, so λ must be a global constant. Interpretation The article does not derive the numerical value of κ = ℏ²/m from first principles. Rather, it shows that once such a coupling exists, its form and normalisation are not arbitrary. Any alternative coupling would violate at least one of the Fisher, Wasserstein, Kähler, or Markov-monotonicity constraints. Why It Matters The result clarifies why the Schrödinger form appears in the reconstruction programme: not as an imposed quantum rule, but as the unique regular Kähler completion of the Fisher–Wasserstein dynamics. The Fisher–Bohm identity and the factor 1/8 arise from the same algebraic source, namely the identity I(ρ) = 4 ∫ |∇√ρ|² dx. Conclusion Article 11 establishes the necessity of the Fisher–Madelung coupling within the minimal reconstruction framework. It does not solve the remaining problem of deriving the value of κ from the geometry of the reconstruction space, but it shows that the Schrödinger-compatible coupling is the only one structurally admissible.
The Reconstruction Theorem: Minimal Kähler Completion of Fisher–Rao Dynamics This article serves as a synthesis paper for the reconstruction programme. Rather than introducing new results, it identifies the programme's deductive core and compresses the results of Articles 1–11 into a single logical structure. Purpose The goal is to determine the minimal set of assumptions required to recover Madelung–Schrödinger dynamics from information-geometric principles. The paper distinguishes clearly between established results, conditional results, and open problems. The Five Hypotheses The reconstruction programme rests on five assumptions: H1: Reparametrisation invariance. H2: Markov monotonicity. H3: Cohesion (finite Fisher volume). H4: Compact reconstruction domain. H5: Variational selection by Fisher minimisation. H1 and H2 uniquely determine the Fisher–Rao geometry. H3 and H4 provide regularity. H5 remains a working variational principle whose justification is discussed elsewhere. Main Results Theorem A (Minimal Kähler Completion). The unique minimal admissible completion of the Fisher–Rao reconstruction space is the cotangent bundle T*P(M) equipped with its canonical symplectic structure and Kähler geometry. Through the Madelung map, this completion is isometric to the projective Hilbert space CP(L²(M)) endowed with the Fubini–Study metric. Theorem B (Coupling Uniqueness). The unique coupling compatible with Fisher geometry, Wasserstein dynamics, Kähler structure, and Markov monotonicity is λ = ℏ²/(8m). Corollary. Under the standard Madelung conditions (positive density and irrotational flow), the resulting dynamics is equivalent to the Schrödinger equation. The Deductive Kernel The article condenses the programme into a single chain: Fisher–Rao geometry + Wasserstein dynamics + canonical symplectic structure + minimal Kähler completion → Madelung–Schrödinger dynamics. Within this framework, the quantum structure appears as the unique admissible completion of the reconstruction dynamics rather than as an independent postulate. Three Explicit No-Go Results The existence of an action constant is forced by the symplectic completion, but its numerical value is not derived. Physical time is not obtained directly from H1–H5 and remains associated with the causal reconstruction layer. Gauge sectors are not derived from the present framework. Conclusion Article 12 functions as the logical charter of the reconstruction programme. Its main contribution is to identify the minimal deductive pathway leading from the Fisher–Rao reconstruction hypotheses to quantum dynamics, while explicitly separating established results from open problems.
This paper extends the causal reconstruction programme from static optimal states to finite-action dynamical histories. A reconstructive history is modelled as a curve of probability measures in the mass-scaled Wasserstein space (P(M), W2,m), and admissible histories are selected by a reconstructive action combining a Wasserstein kinetic term, a Fisher information term, and a causal constraint. The resulting Euler–Lagrange equations form a coupled system: a continuity equation for the evolving density and a generalized Hamilton–Jacobi equation for the velocity potential. The central result is the Fisher–Bohm identity. For any sufficiently regular positive density ρ, the functional derivative of Fisher information satisfies δI / δρ(x) = -4 Δ√ρ / √ρ, so that the Bohm–Madelung quantum potential is exactly Q(x) = (ℏ2 / 8m) · δI / δρ(x). Thus the quantum potential is the Fisher information gradient of the probability density, scaled by ℏ2/8m. When the Fisher coupling in the reconstructive action is set to λ = ℏ2/8m, the Euler–Lagrange equations coincide with the Madelung equations and are therefore equivalent to the Schrödinger equation, under the standard assumptions of positive density and irrotational flow. This establishes a structural correspondence between reconstructive dynamics and non-relativistic quantum mechanics. The result is not presented as a derivation of quantum mechanics: the coupling λ = ℏ2/8m is matched through the Madelung Hamiltonian rather than derived from the reconstruction hypotheses. The geometric origin of this coupling remains an open problem.
A Solvable Annulus Model for Reconstructive Dynamics This paper provides the first fully solvable model of the reconstruction programme. It studies the flat cylinder C = [0,L] × S¹, obtained after the logarithmic change of variable ρ = ln(T), and shows that the central quantities of the programme can be computed explicitly. Purpose While the previous articles established the general theoretical framework, the goal here is to demonstrate that the programme is not merely conceptual but genuinely calculable within a simple and analytically tractable model. Main Results Optimal reconstruction state. Under Neumann boundary conditions, the preferred state is exactly the Jeffreys reference measure: ν* = μF. The computation shows that non-trivial reconstruction states require additional sector-specific boundary data, supplied by the Liouville sector. Dirac spectrum. The Dirac spectrum of the flat cylinder is obtained in closed form and confirms the existence of the zero mode previously identified in the programme. Fisher gradient flow. The linearised dynamics is biharmonic, and every non-trivial mode decays exponentially. The uniform state emerges as the unique stable equilibrium. Quantum potential. The Fisher–Bohm identity is verified explicitly for Gaussian states, confirming the connection between information geometry and the quantum potential. Quantum Fisher coupling κ = ℏ²/m. The flat geometry alone does not determine κ. An additional energy scale is required, highlighting one of the programme’s remaining open problems. Bohr–Sommerfeld condition. For the trivial reconstruction state, the velocity potential vanishes and ℏ remains undetermined. Determining ℏ therefore requires a non-trivial sector, expected to arise from the Liouville specification. What the Model Demonstrates The flat-cylinder model serves as an analytical laboratory for the reconstruction programme. It clearly separates: what can already be computed exactly; what follows directly from hypotheses H1–H5; where additional sector-specific assumptions enter; which open problems remain unresolved. Conclusion The paper demonstrates that the reconstruction programme possesses a fully solvable instance. Several general results obtained in earlier articles are verified explicitly, while the remaining open questions are sharply identified. In particular, the model clarifies why non-trivial reconstruction states, the determination of ℏ, and the derivation of the quantum coupling κ require additional structure beyond the flat-cylinder geometry itself.
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
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.