Why does physical reality exhibit three spatial dimensions, complex quantum phases, and Lorentzian spacetime geometry? Could these seemingly different structures share a deeper origin in the way causal processes combine and persist? Recursive Causal Closure (RCC) explores this possibility by starting from finite causal operations, without assuming spacetime, wavefunctions, or a universal clock. Its central idea is simple: when two different causal histories connect the same events, their comparison can retain a non-trivial geometric structure. This residual structure becomes the starting point for recursive closure. Under explicit conditions of isotropic recapture, the framework selects a three-dimensional carrier whose elementary closures generate complex phases. The direction of causal succession provides a natural orientation for these phases, while the absence of a preferred temporal origin leads, under suitable persistence conditions, to autonomous evolution. The same construction also supports coherent sums over alternative causal histories and a minimal Lorentzian causal geometry. An explicit realization involving Thomas–Wigner rotations illustrates how complex phase and relativistic geometry can emerge as two aspects of a common closure structure. RCC does not claim to derive quantum mechanics or general relativity in full. Instead, it identifies a compact mathematical architecture connecting causality, dimension, phase, and spacetime geometry, while making the remaining physical assumptions and unresolved bridges explicit. The central question is whether some of the structures we usually take as fundamental might instead be consequences of causal closure itself.
The Directed Acyclic Graph Interpretation (DAGI) represents physical processes as a single locally finite directed acyclic graph of informational states, records, and admissible updates. This note refines the meaning of ``branch'' inside that ontology. A branch is not an ontologically duplicated universe, an arbitrary subset of nodes, or a complete microscopic path. It is an \emph{observer-local causal accessibility sector}: the strict quotient of a record-conditioned causal cone by exact indistinguishability over the accessible record algebra of a specified observer or observable class. The local temporal asymmetry of a node is expressed without adding a stochastic primitive to the ontology: the resolved past is the set of persistent accessible records, while the unresolved future is the set of admissible record-candidates that may become persistent records along directed continuations. Interference is described as amplitude-level evaluation over histories that share the same persistent-record trace, where concurrent accessible records are retained as a partial order rather than forced into a linear word. A distinguishing record refines a coherent macro-node into subnodes; erasure coarsens the accessible record trace and can restore conditional interference classes. In strict graph language, nodes are coarse-grained informational states or events, edges are admissible update relations that change downstream reachability or distinguishability, and records are topologically persistent updates whose distinguishing descendants remain accessible beyond the local decorrelation scale. Charge, field, observer, branch, and connection are not additional ontological entities above the graph, but coarse-grained descriptions of stable subgraph structure and reusable edge-update rules. Causal-cone posets provide a natural computational representation of such accessibility sectors, while powerset decompositions remain useful but physically less structured baselines. The resulting branch ontology is concise: the global DAG contains the admissible informational structure; a local observer occupies a record-conditioned accessibility sector; persistent records refine that sector; admissible record-candidates define possible future refinements; erasure coarsens it; macro-nodes quotient inaccessible micro-variation; and physically meaningful posets are those that track causal accessibility rather than arbitrary set membership. Keywords: DAGI, causal cones, observer-local branching, causal accessibility, posets, quantum erasure, higher-order information, emergent spacetime, relational quantum mechanics, macro-nodes, charge-conditioned edge composition
We reconstruct nonrelativistic quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by
20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
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 paper investigates how time and physical action can arise within Recursive Causal Closure (RCC) without assuming a preferred global clock. For a persistent causal branch with a realised operational temporal parameter, invariance under a change of temporal origin and completeness of the retained state imply an autonomous evolution law. A reversible, lossless oriented two-plane then carries a one-parameter U(1) phase character, whose accumulated phase is invariant under affine recalibration of time. A further, explicitly identified geometric bridge is needed to interpret operational time as proper time. Under that condition, Lorentz-covariant completion of the persistent phase leads to a relativistic dispersion relation, Hamilton–Jacobi dynamics, and a controlled nonrelativistic Schrödinger limit. Introducing an action conversion scale connects the dimensionless phase to the free-particle proper-time action and interprets mass as a phase rate. The value and universality of that scale, its identification with Planck's constant, and the physical selection of the coherent branch remain open problems.
Recursive formulations in theoretical physics often generate very large or formally infinite families of alternative histories, even when the internal state space controlling their generation is finite. This work develops a sector-resolved resummation framework based on propagation, interaction, closed-path monodromy or holonomy, and recursive closure (PIHR). Propagation, interaction, and holonomy are treated as primitive dynamical data, whereas recursion is treated as a closure operation acting on a finite return kernel. For systems admitting a unique first-return factorization, an exact resummation theorem is established: all repeated histories are generated by powers of one return operator and summed by its resolvent whenever the history series converges. The framework is mapped onto Dyson resummation, Lippmann–Schwinger scattering, the Bethe–Salpeter equation, Foldy–Lax multiple scattering, Fabry–Pérot interference, double-barrier tunneling, and Aharonov–Bohm transport. A three-arm flux-threaded quantum network then provides a quantitative benchmark. With complete junction mixing, the number of arm-resolved histories grows exponentially with recursive depth, while the exact coherent amplitude remains encoded in a fixed three-dimensional return operator. Through eight internal returns the cumulative path count already exceeds one hundred forty-five million, although the all-orders solution still requires only a three-dimensional resolvent. Using standard linear-systems realization theory, we additionally give a controllability-and-observability criterion under which this three-dimensional representation is minimal among linear realizations of the recurrence-generating function. The resolvent identity and the individual physical models are established results; the proposed contribution is the PIHR factorization, its cross-domain dictionary, and the quantitative separation of path-growth complexity from finite realization dimension.