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

This article completes the bridge between the causal layer and the quantum layer of the reconstruction programme: it proves, under three explicit regularity assumptions, that the Madelung–Schrödinger equation is the continuum limit of discrete causal chains. Status markers: E established, C conditional, O open. The route from elementary causal succession to quantum dynamics factors through two arrows: a discrete causal chain → the continuous reconstructive action → the Madelung–Schrödinger dynamics. The second arrow (action → Madelung) was already established as an Euler–Lagrange theorem in Finite-Action Reconstructive Histories and the Fisher Origin of the Quantum Potential. This paper proves the first — that discrete causal chains converge, in a mathematically controlled way, to the continuous reconstructive action. The key identification is that the reconstructive cost between consecutive states is precisely a discrete Lagrangian in the sense of variational-integrator theory (Marsden–West), so a causal chain is a discrete variational system whose critical chains satisfy discrete Euler–Lagrange equations expressing momentum continuity at each node. E The discrete action then Γ-converges to the continuous reconstructive action as the action quantum tends to zero at fixed total duration, and the discrete solutions converge to solutions of the Madelung system; the discrete symplectic form converges to the canonical Kähler form of the reconstruction phase space (The Reconstruction Theorem: Minimal Kähler Completion of Fisher–Rao Dynamics), so the Kähler structure is preserved in the limit. C Combined with that Madelung result, this completes a fully variational chain from elementary causal succession (Causal Reconstruction: Elementary Succession and the Planck Energy) to the Schrödinger equation. A substantive contribution is the regularity analysis. In the density variable the Fisher information diverges as the density approaches zero; this is shown to be a coordinate artefact, not a genuine obstruction. In the amplitude variable u = √ρ the Fisher information is exactly a quadratic Dirichlet energy, 4∥∇u∥², and in the wavefunction variable ψ the reconstructive action is smooth on the natural Hilbert space H¹(M, ℂ) with no lower bound on the density — so the C² regularity required by assumption (A1) holds in those charts, and the quantum potential appears as the first variation. E The continuum limit reduces to three explicit conditions: tightness of minimising sequences (C1, needed because the state space is not locally compact); a small-step regime keeping each transition inside the injectivity radius (C2); and non-degeneracy of the second variation, excluding conjugate points (C3). Condition C2 holds identically in the limit E; C1 and C3 are the operative analytical hypotheses, expected to hold in the small-step regime but not yet established for the programme’s action O. The same non-degeneracy condition C3 plays a double role across the causal tier: here it secures the regularity of the cost, while in Causal Succession by Segments: Tunnel Effect, Superposition, and Measurement in the Reconstructive Order it makes the covering relation single-valued — and its failure (conjugate points, multiple equal-length geodesics) is exactly the multivaluation that that article identifies with quantum superposition. Conceptually, the causal-chain picture is thereby promoted from a heuristic interpretation to a variational approximation theorem: a quantum trajectory is not a fundamental continuous object but the continuum limit of a chain of elementary reconstructive transitions, each carrying one action quantum. Status. E the discrete-Lagrangian identification and the discrete Euler–Lagrange / momentum-conservation equations; the resolution of the apparent Fisher singularity as a coordinate artefact and the resulting C² regularity in the amplitude and wavefunction charts; the automatic small-step condition C2; the preservation of the Kähler structure · C the Γ-convergence and the convergence of discrete solutions to the Madelung system, at the level of the standard variational-integrator theory and conditional on the stated assumptions · O the two operative regularity hypotheses C1 (tightness) and C3 (non-degeneracy), expected in the small-step regime but not yet proven for the reconstructive action.