Discrete-to-Continuum Limit of Causal Chains in the Kähler Reconstruction

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.