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.
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 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.
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.
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 Madelung equations reproduce the Schrödinger equation only when supplemented by the Wallstrom condition of integer circulation, ∮γp∈2πℏZ for the phase one-form p, which the hydrodynamic equations do not supply: on the positive-density region the circle- and line-valued phase geometries are locally isometric, and explicit finite-energy stationary solutions with arbitrary real circulation exist (Wallstrom; Reddiger–Poirier). We show the condition decomposes into two independent parts, one of which can be derived. The Madelung gradient energy endows the local amplitude–phase value space with the flat polar metric g = dr² + (r²/κ²) dS², r = √ρ, κ = √(8mα), with α the Fisher coefficient. Within a minimal two-dimensional phase-equivariant category we prove that the line cover and circle quotients exhaust the globalizations of this geometry; each has a canonical metric completion adding a single phase-blind point at zero amplitude, and exactly one completion is a smooth Riemannian surface there. Smoothness fixes the phase period uniquely to 2πκ — for the quantum Fisher coefficient α = ℏ²/8m, to 2πℏ — and identifies the completed target with the Euclidean plane ℂ, the wavefunction ψ=√ρeiS/ℏ arising as the pullback of its regular Cartesian coordinate. Madelung data that descend to a globally defined scalar field then obey integer circulation by degree theory; fractional-circulation solutions survive locally but fail descent. Target regularity thus fixes the phase period, while integer winding irreducibly requires global scalar descent. Gauge coupling yields the flux-shifted (fluxoid) circulation law, and multiply connected and twisted sectors are delimited explicitly.
Articles 1–8 constitute the published core of the reconstruction programme; within the later work, Articles 19 and 20 are also published. The present paper provides a status map of the full working architecture, Articles 1–21. This architecture is not homogeneous in status: some papers extend the core framework, others formulate conditional mechanisms, and some are explicitly exploratory notes. The purpose of this paper is to make these distinctions explicit rather than to present all components as equally established. The paper fixes the primitive hypotheses of the programme, identifies its load-bearing results, separates established statements from conditional mechanisms, and locates the remaining open problems. It is intended as a logical charter for the series: a guide to what belongs to the published core, what is under active development, and what remains exploratory. The established framework reconstructs a number of known structures and possesses a limited but non-trivial prohibitive content: certain classes of universes are incompatible with its closure architecture, while several candidate mechanisms remain conditional. Until recently it offered no distinctive quantitative prediction beyond known physics — numerical predictions arose only in exploratory extensions whose additional assumptions remained unproved. Article 20 changes this. From the established causal foundation — the atomicity of the elementary (length-one) closure of Article 19 — it derives the programme's first distinctive, falsifiable prediction: because admissibility is a decided binary property, its closure is a first-passage time, and the rigidity of first passage forces a decoherence cliff (a threshold loss of interference visibility) in the soft-photon regime, where standard decoherence predicts a smooth exponential. The deviation is derived, not fitted. A refit of the 2004 thermal-emission data places them in a regime where cliff and decoherence coincide, so the prediction is not yet tested, but it lies within reach of existing matter-wave interferometry. The central challenge has accordingly shifted. Of the three routes to empirical distinctiveness — a unique closure principle, a boundary-condition selection mechanism, or a quantitative prediction following from the established framework alone — the third now exists. The open questions are no longer internal consistency, but the experimental test of this prediction and the closure of its single residual freedom: the order-one decision-theoretic cost of resolving the admissibility predicate, conjectured to equal the relative entropy of the resolved branch.