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.
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.
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.
Record-Conditioned Reduction and the Decoherence Cliff This paper proposes a falsifiable reconstruction-based account of quantum measurement. Its central claim is that the loss of interference is not merely a smooth Markovian decay, but a first-passage phenomenon: coherence persists until a reconstructive admissibility threshold is crossed. Purpose Building on the causal-segment structure of Article 19, the paper interprets measurement as the moment when a coherent reconstruction can no longer remain admissible because the record has accumulated enough information to distinguish the branches. Main Idea A superposition remains reconstructively admissible while the branch information is below a threshold. Once the accumulated record crosses that boundary, the coherent segment closes and the system undergoes record-conditioned reduction. The Decoherence Cliff The visibility of interference is identified with the survival probability of a first-passage process. This predicts a characteristic “cliff”: a sharp loss of visibility around a critical number of elementary causal steps. The cliff location is governed by: Nc = Iadm / Îmut where Îmut is the per-step mutual Fisher information and Iadm is the total admissibility threshold. Empirical Content The model predicts a threshold law rather than a simple exponential decay. Its distinctive signature is a sharp transition whose location scales with the coherence window, together with a Fano-factor structure different from ordinary Markovian decoherence. Relation to Decision Theory The admissibility threshold is later identified with the branch entropy H(p), giving Iadm = ln 2 for a balanced superposition. This turns the cliff location into a parameter-free prediction in the ideal pure-branch case. Status The paper is a preprint-level proposal in the foundations of quantum mechanics. Its strongest content is the falsifiable replacement of smooth decoherence by a first-passage admissibility threshold. Its remaining burden is to establish the strong impossibility result against signed-increment or hidden Markovian reformulations. Conclusion This paper reframes measurement as the crossing of a reconstructive admissibility boundary. If correct, reduction is not an added postulate but the observable cliff produced when a coherent causal segment ceases to be globally 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.
Any finite-resolution description of a physical system incurs two costs: an information cost, quantifying microscopic complexity via relative entropy, and an ordering cost, quantifying geometric or structural organization. The Entropy–Order Balance (EOB) framework postulates that realized physical laws correspond to Lyapunov-stable equilibria of the total cost functional. Part I showed that the gravitational sector — Einstein's field equations and the cosmological constant — emerges as the stable ordering response. This paper shows that quantum mechanics emerges as the stable information response. Starting from the same five operational axioms applied to resolution windows with smooth Riemannian structure, supplemented by Lyapunov stability and two standard GPT-to-quantum hypotheses (reversible subgroup, local tomography/purification), the paper derives: State-space convexity from variational stability — a result that standard quantum reconstruction programs (Hardy 2001, Chiribella–D'Ariano–Perinotti 2011) assume as a postulate Hilbert space structure and completely positive trace-preserving (CPTP) dynamics Born rule probabilities from decision-theoretic stability over KL divergence Measurement as a barrier-dominance phenomenon: inter-sector transitions in the EOB functional are exponentially suppressed in apparatus size, producing effective state reduction without a collapse postulate A universal decoherence timescale τ_dec = d_S/c depending only on system dimension and the same constant c that determines Λ = cH₀² in Part I Pointer basis selection via Holevo quantity maximization Conditional extensions requiring additional AQFT inputs (Bisognano–Wichmann property, Haag–Kastler net structure) yield an area-law entropy exponent d = 2 and identification of the EOB balance scale with the Planck length. Part I (companion paper) derives Einstein's field equations and the cosmological constant from the same axiomatic framework.
Quantum mechanics represents physical states through complex amplitudes, state vectors, and density operators, yet the ontological status of these objects remains interpretation-dependent. This paper develops a closure-based formulation in which a quantum state is interpreted as a representation of contextually partial relational closure: the state possesses definite mathematical and relational structure while remaining unresolved relative to one or more observational contexts. Within this language, decoherence differentiates closure channels and redistributes locally accessible coherence into relational structure; environmental redundancy stabilizes records; and classical facts emerge as approximately fixed, effectively commuting, redundantly recorded structures. Measurement is separated into three problems—context, weight, and outcome—so that decoherence, Born weighting, and unique outcome selection are not conflated. The paper then distinguishes two stronger research programs from the exact reformulation. First, it asks whether Hilbert-space quantum mechanics can be reconstructed from primitive closure, incompatibility, composition, and consistency principles. Second, it formulates a falsifiable closure-history extension in which physically relevant information might exist beyond the present density operator. Quantum Partial Closure Theory is therefore presented as a relational interpretation and information calculus, a reconstruction program, and a prospective experimental framework rather than as an established replacement for quantum mechanics. Keywords: quantum foundations; closure; wavefunction ontology; coherence; entanglement; decoherence; measurement; classical emergence; quantum information; Hilbert-space reconstruction The wavefunction question Quantum theory is operationally clear about what a wavefunction does. For a normalized pure-state vector ψ∈H and a projective measurement with alternatives {Π_i}, the Born weights are p_i=Tr(ρ_ψ Π_i ), ρ_ψ=ψψ^†. For a rank-one outcome Π_i=e_i e_i^†, this reduces to p_i=|e_i^† ψ|^2. Yet the mathematical clarity of the formalism does not by itself settle what the state vector is ontologically. Interpretations of quantum theory differ over whether the state should be regarded as physical, epistemic, relational, branch-relative, dispositional, or otherwise. The present paper proposes a specific interpretive and reconstructive language: partial relational closure. The central claim is not that the wavefunction is literally a new material substance called closure. Rather, ▭(ψ "is interpreted as a Hilbert-space representation of coherent partial relational closure" .) The word partial is crucial. It does not mean that a state is a scalar fraction of being real or complete. It means that information and constraint are distributed across local, coherent, and relational sectors, and that closure is always specified relative to a context. A Bell state, for example, can be locally indefinite while being relationally exact. A decohered state can be closed with respect to one effective macroscopic context while retaining a larger quantum embedding. A selective outcome can be locally definite without implying that every larger-scale relation in the universe is globally closed. This formulation is motivated by three observations. First, projective dephasing is naturally closure-like because it is idempotent. Second, modern quantum information already supplies exact measures for local coherence, correlations, and record redundancy. Third, the quantum-to-classical transition is naturally relational: local coherence can disappear from a subsystem while information migrates into system-environment correlations. The paper has three epistemic layers. Layer A uses established quantum mathematics. Layer B derives exact information identities and interprets them in closure language. Layer C states reconstruction conjectures and an explicitly speculative beyond-QM closure-history hypothesis. These layers are kept separate throughout. A quantum system is not structureless before measurement. Its amplitudes, phases, conservation relations, and correlations are tightly organized. What remains unresolved is how that structure will become locally definite relative to a chosen measurement context. The partial-closure view therefore distinguishes three broad regimes: Coherent partial closure. Alternatives remain phase-related and can interfere. Contextual differentiation. Interaction or dephasing suppresses coherence between alternatives relative to a context without necessarily choosing one. Selective local closure. A particular outcome is conditioned or realized relative to the measurement relation. In this language, wave-like behavior corresponds to spatially or otherwise contextually unresolved coherent closure, while particle-like detection corresponds to localized closure. Entanglement is closure stored in relations rather than in isolated subsystems. Decoherence is not simply destruction: in a globally pure description it can be a transfer of local coherence into relational entanglement. Classical objectivity appears when relevant records become stable and redundantly accessible. The framework does not assign consciousness a privileged dynamical role. An observer is a partial closure process for constraining possibilities. Explainer video Watch on YouTube (Roots of Reality): https://www.youtube.com/watch?v=UYx_omk3h3A The video, its thumbnail and description are included in this record.