Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction Previous paper (Causal Reconstruction: Elementary Succession and the Planck Energy) proposed that the Schrödinger equation should arise as the continuum limit of discrete causal chains. Article 18 establishes this result under explicit regularity assumptions, completing the bridge between elementary causal succession and continuous quantum dynamics. Purpose The reconstruction programme contains two distinct transitions: Discrete causal chain → continuum reconstructive action → Madelung–Schrödinger dynamics. The second step was already established in Article 5. The purpose of this paper is to prove the first step: that discrete causal chains converge to the continuous reconstructive action in a mathematically controlled way. Main Idea The reconstructive cost between two states is interpreted as a discrete Lagrangian in the sense of variational integrator theory. A causal chain therefore becomes a discrete variational system whose dynamics is governed by discrete Euler–Lagrange equations. The key observation is that the discrete action SN = Σ C(ρk,ρk+1) plays exactly the role of a discrete Lagrangian approximation of the reconstructive action Srec. Main Results Discrete Lagrangian formulation. The reconstructive cost is shown to define a discrete Lagrangian under suitable regularity conditions. Discrete Euler–Lagrange equations. Critical causal chains satisfy discrete Euler–Lagrange equations expressing momentum continuity across each causal node. Γ-convergence. The discrete action SN Γ-converges to the continuous reconstructive action Srec as the elementary action quantum tends to zero while total duration remains fixed. Convergence of solutions. Solutions of the discrete causal-chain equations converge to solutions of the continuous Madelung equations. Preservation of Kähler structure. The discrete symplectic structure converges to the canonical Kähler symplectic form of the reconstruction phase space. Recovery of Schrödinger dynamics. Combining the present result with Article 5 yields a complete chain from discrete causal succession to the Schrödinger equation. Regularity Analysis A significant contribution of the paper is the analysis of the Fisher information functional in alternative coordinates. In the density variable ρ, Fisher information becomes singular as ρ approaches zero. In the amplitude variable u = √ρ, Fisher information becomes exactly a Dirichlet energy: I(ρ) = 4||∇u||². In the wavefunction variable ψ = √ρ exp(iS/ℏ), the reconstructive action becomes smooth on the natural Hilbert space H¹(M,C). This shows that the apparent singularities of Fisher information are coordinate artefacts rather than genuine obstructions. The Three Operative Conditions The continuum limit ultimately depends on three explicit conditions: C1 — Tightness: minimising sequences remain precompact. C2 — Small-step regime: elementary causal transitions remain inside the injectivity radius. C3 — Non-degeneracy: the second variation of the reconstructive action remains positive, excluding conjugate points and ensuring local uniqueness. Condition C2 becomes automatic in the continuum limit. Conditions C1 and C3 remain the principal analytical assumptions. Conceptual Significance The paper transforms the causal-chain picture from a heuristic interpretation into a variational approximation theorem. Discrete causal succession is no longer merely compatible with quantum dynamics; it converges to it. In this view, a quantum trajectory is not a fundamental continuous object. It is the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Conclusion This paper completes the mathematical bridge between the causal layer and the quantum layer of the reconstruction programme. Under explicit regularity assumptions, discrete causal chains converge to the reconstructive action, the reconstructive action yields the Madelung equations, and the Madelung formulation is equivalent to Schrödinger dynamics. The resulting picture is a fully variational route from elementary causal succession to quantum mechanics.
Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction Previous paper (Causal Reconstruction: Elementary Succession and the Planck Energy) proposed that the Schrödinger equation should arise as the continuum limit of discrete causal chains. Article 18 establishes this result under explicit regularity assumptions, completing the bridge between elementary causal succession and continuous quantum dynamics. Purpose The reconstruction programme contains two distinct transitions: Discrete causal chain → continuum reconstructive action → Madelung–Schrödinger dynamics. The second step was already established in Article 5. The purpose of this paper is to prove the first step: that discrete causal chains converge to the continuous reconstructive action in a mathematically controlled way. Main Idea The reconstructive cost between two states is interpreted as a discrete Lagrangian in the sense of variational integrator theory. A causal chain therefore becomes a discrete variational system whose dynamics is governed by discrete Euler–Lagrange equations. The key observation is that the discrete action SN = Σ C(ρk,ρk+1) plays exactly the role of a discrete Lagrangian approximation of the reconstructive action Srec. Main Results Discrete Lagrangian formulation. The reconstructive cost is shown to define a discrete Lagrangian under suitable regularity conditions. Discrete Euler–Lagrange equations. Critical causal chains satisfy discrete Euler–Lagrange equations expressing momentum continuity across each causal node. Γ-convergence. The discrete action SN Γ-converges to the continuous reconstructive action Srec as the elementary action quantum tends to zero while total duration remains fixed. Convergence of solutions. Solutions of the discrete causal-chain equations converge to solutions of the continuous Madelung equations. Preservation of Kähler structure. The discrete symplectic structure converges to the canonical Kähler symplectic form of the reconstruction phase space. Recovery of Schrödinger dynamics. Combining the present result with Article 5 yields a complete chain from discrete causal succession to the Schrödinger equation. Regularity Analysis A significant contribution of the paper is the analysis of the Fisher information functional in alternative coordinates. In the density variable ρ, Fisher information becomes singular as ρ approaches zero. In the amplitude variable u = √ρ, Fisher information becomes exactly a Dirichlet energy: I(ρ) = 4||∇u||². In the wavefunction variable ψ = √ρ exp(iS/ℏ), the reconstructive action becomes smooth on the natural Hilbert space H¹(M,C). This shows that the apparent singularities of Fisher information are coordinate artefacts rather than genuine obstructions. The Three Operative Conditions The continuum limit ultimately depends on three explicit conditions: C1 — Tightness: minimising sequences remain precompact. C2 — Small-step regime: elementary causal transitions remain inside the injectivity radius. C3 — Non-degeneracy: the second variation of the reconstructive action remains positive, excluding conjugate points and ensuring local uniqueness. Condition C2 becomes automatic in the continuum limit. Conditions C1 and C3 remain the principal analytical assumptions. Conceptual Significance The paper transforms the causal-chain picture from a heuristic interpretation into a variational approximation theorem. Discrete causal succession is no longer merely compatible with quantum dynamics; it converges to it. In this view, a quantum trajectory is not a fundamental continuous object. It is the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Conclusion This paper completes the mathematical bridge between the causal layer and the quantum layer of the reconstruction programme. Under explicit regularity assumptions, discrete causal chains converge to the reconstructive action, the reconstructive action yields the Madelung equations, and the Madelung formulation is equivalent to Schrödinger dynamics. The resulting picture is a fully variational route from elementary causal succession to quantum mechanics.
This article develops the causal core of the reconstruction programme. It completes a chain: Causal Reconstruction: Elementary Succession and the Planck Energy defines elementary causal succession and quantises histories in units of the action quantum (recovering the Planck energy), and Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction proves, by Γ-convergence, that discrete causal chains tend to the reconstructive action — yielding the Madelung–Schrödinger equation. The present article extends elementary succession to coverings of variable length and, on that basis, represents tunnelling, superposition, decoherence and measurement within a single segmental mechanism. Status markers: [E] established, [C] conditional, [O] open. The elementary causal succession of the first of these — unit reconstructive cost together with a strict entropy increase — cannot accommodate quantum transitions in which the entropy condition is violated at intermediate nodes, in particular tunnelling through a barrier V > E. This paper generalises it to causal succession by segments. A segment is a minimal entropic closure: the shortest unit-cost chain along which the relative entropy first recovers; this defines the covering relation, admissible when it precedes decoherence (n·τmin ≤ τD), and the causal order ≺ is its transitive closure. The main result is that ≺ is a strict partial order on effective nodes — not by hypothesis, but because the relative entropy H(·|μF) is a strict order-potential along the covering relation, Fisher information appearing as the elementary closure cost through the de Bruijn identity, the potential being canonical up to scale. [E] Three open problems left by the elementary-succession article are structurally reduced: the minimal closure exists iff the transition is physically admissible (forbidden transitions are exactly those with no closure inside the decoherence window); the terminal node is unique up to Wasserstein isometry, the residual non-uniqueness being the geometric signature of quantum superposition (the covering becomes multivalued); and the decoherence time τD = τmin/Îmut has its form derived from the system–environment mutual Fisher information, its magnitude depending on the coupling. [C] Tunnel effect, superposition, decoherence and measurement are then represented within the segment structure with no further postulate. The semiclassical WKB exponent is recovered as the segmental-closure cost of an admissible segment, the factor 2 in 2∫κ being the Madelung square ρ = (√ρ)² rather than an imported convention. [E] Quantum non-locality is not a failure of transitivity — the order is everywhere transitive — but the atomicity of a covering of length greater than one. Measurement is not an independent axiom: a strong environmental coupling enforces length-one closures and suppresses inter-branch coherence — so what the mechanism fixes is the collapse rate, not the definite-outcome selection, which remains the measurement problem proper (Reconstructive Reduction as an Admissibility Threshold: A Record-Conditioned Collapse Mechanism). The transition probabilities follow a closure law, P(Γ) ≈ exp(−ΔS[Γ]) to exponential order, identified as the maximum-caliber (maximum-entropy-on-paths) assignment; the closure cost carries an unconditional metric (triangle-inequality) structure, and physical closures are its variational minimisers. These results are conditional on a full large-deviation formulation of the reconstruction dynamics. [C] Status. [E] the segment definition and the strict-partial-order theorem via the order-potential; the recovery of elementary succession in the classical limit; the WKB exponent as closure cost; the metric structure · [C] the large-deviation / closure-law formulation, and the form (not the magnitude) of τD · [O] five problems remain explicitly open — a rigorous derivation of τD; the well-foundedness of ≺ (the existence of a causal beginning); the classification of superposition degeneracies (two-branch, n-branch, continuous); global existence and unique factorisation of every admissible history into atomic closures; and the existence and modular stability of reconstructive solitons (persistent objects, candidate self-focusing term: Schrödinger–Newton self-gravity), which would supply the notion of a persistent object and may fix the inertial mass.
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.
Exploratory paper in information geometry and local statistical reconstruction. This work studies the Fisher–Rao geometry induced by local probabilistic reconstruction frameworks and investigates a conditional variational extension. Revision note. In the initial version of the programme, causal structure was treated at the level of continuous reconstructive histories. In the revised architecture, Article 1 is restricted to its proper role: it establishes the local Fisher–Rao geometry of infinitesimal distinguishability between reconstructed states. Causal succession is not assumed at this stage. It is introduced later, in Article 17, where two states are defined as elementarily successive when the infimum of the reconstructive action connecting them equals the action quantum a0. Article 18 then shows, under explicit regularity assumptions, that the continuous histories used in Articles 5–6 arise as the variational continuum limit of such discrete causal chains.
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.
COS-PRI develops the time-evolution and spacetime-limit interface of the Collapsing-Structure (COS) program. The module investigates how discrete operative time evolution, CPTP channel dynamics, GKSL continuous-time limits, conditional trajectory descriptions, and quasiclassical spacetime reconstruction can be organized within a common shell-filament framework. The manuscript formulates COS time evolution through quantum instruments and branch-ignored CPTP channels on the physical sector, carefully distinguishing unconditional physical dynamics from conditional Kraus branches, unravelings, and postselected trajectory descriptions. In the continuous-time scaling regime, the channel dynamics is related to GKSL generators, entropy contraction, mixing diagnostics, and residual-monitoring mechanisms. COS-PRI also records the conditional bridge from discrete shell-filament dynamics to Regge-type and Einstein-Hilbert continuum structures. The classical spacetime limit is treated as a controlled variational and reconstruction program, relying on admissibility, compactness, regularity, coercivity, and scaling assumptions. It is not presented as an unconditional classicization theorem or a global Einstein-attractor result. The module further discusses constraint-residual monitoring, quasiclassical trajectory selection, Γ-convergence interfaces, Regge-to-Einstein reconstruction, and large-deviation-inspired trajectory interpretations. These elements are separated by status: operative constructions, controlled conditional bridges, and open programmatic components. A dedicated causality layer formulates Lieb-Robinson-type influence estimates and finite-speed propagation bounds for locally implementable admissibility filtering and quasi-local GKSL decompositions. These claims apply only within the specified local channel class; global nonlocal projectors or unrestricted microdynamics are not automatically covered. COS-PRI therefore serves as the operative time-evolution, causality-envelope, and spacetime-limit status map of the COS program, linking COS-QD, COS-CL, COS-STAB, COS-CNS, and related modules while maintaining explicit distinction between proved constructions, conditional continuum/locality bridges, heuristic calibration interfaces, and open problems.