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.
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.
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.
Action principles, phase accumulation and quantized closure appear across classical mechanics, quantum theory, semiclassical dynamics, superfluidity and geometric phase phenomena. Their coexistence is familiar; their structural relationship is usually taken as part of the established formalism. Here we reconstruct that relationship in the opposite direction. Starting from persistent relational dynamics, we show how a real two-component rotation supplies a phase coordinate without assuming a primitive complex number, how a relational canonical one-form can connect phase increments to action-like increments, and how stationary phase and global phase consistency impose two distinct selection conditions. The local condition yields stationary action, δA_R = 0. The global condition yields integral symplectic periods. Under the explicit convention F_R = −iΩ_R/κ_A for a U(1) connection curvature, global compatibility requires [Ω_R/(2πκ_A)] ∈ H²(M_R,Z), equivalently ∫_Σ Ω_R = 2πnκ_A on closed two-cycles. This separates the origin of discreteness from the origin of the dimensionful conversion scale. Topology fixes integer classes; it does not determine κ_A. We therefore identify a precise remaining problem: what selects a unique, universal and non-zero κ⋆ across coherently composable sectors? The framework makes this question falsifiable by requiring cross-scale closure, causal ancestry, intervention consistency and independence from observer compression. It also distinguishes a universal phase–action conversion scale from scale-dependent effective couplings. The result is a sharpened bridge from relational phase to stationary dynamics and quantized closure, with a mathematically explicit boundary between what is derived, what is inherited from established geometry, and what remains to be selected dynamically. The revision formalizes an Action-Scale Factorization Theorem together with Exact-Action and Curvature-Compatibility no-go results, thereby separating the existence of quantized closed-path sectors from the still-open physical selection of a universal non-zero scale such as ħ.
We investigate the relationship between phase interference associated with action, the relativistic dynamics of proper time, and the structural origin of action quantization. We start from the general principle of interferential compatibility of action introduced in previous works, in which the dynamic phase of a system is given by ϕ=S/ℏ and phase coincidences between distinct evolutions occur only when ΔS=2πnℏ. We show that when this formalism is specialized to the relativistic action of a free particle S=−mc2τ, the phase evolution naturally implies an internal frequency ωC=mc2/ℏ, corresponding to the Compton clock. When two systems evolve with distinct proper times, their Compton phases diverge, producing a relativistic phase beating. We demonstrate that the interferential coherence condition Δϕ=2πn defines a discrete sequence of synchronization instants and directly implies the relation mc2Δτ=nh, showing that discrete action increments emerge as a geometric consequence of the relativistic periodicity of the phase. We also show that the used phase coincides exactly with the rest phase of the Dirac equation solutions, establishing the connection with the relativistic dynamics of fields. The beating between Compton clocks defines an operational discretization of the action that can be interpreted as a discrete sum, whose continuous form is recovered in the limit of almost perfect synchronization between reference frames. We also discuss the relationship of this formalism with the notion of the internal clock introduced by de Broglie and with the interferential structure of relativistic matter interferometry. In this framework, Planck's constant emerges as the geometric invariant associated with the first closure of the relativistic beating between Compton clocks. Quantization can then be interpreted as a coherence condition between relativistic phases, establishing a direct bridge between quantum interference, proper time relativistic dynamics, and the variational principle of stationary action.
Why do coherent causal alternatives contribute to a sum over histories, and why are detector outcomes described by quadratic measures? This article addresses these questions separately within Recursive Causal Closure (RCC). On finite directed acyclic networks, recorded causal restitution, additive admission and a specified linear local-response rule give an exact nilpotent resolvent. Its finite expansion enumerates all admissible directed routes, without separately postulating a sum over paths. Exact elimination of internal events preserves the boundary response and is associative under successive coarse-grainings. When a persistent phase is supplied by the companion temporal analysis, each coherent route acquires an action-dependent complex factor. A separate positive bilateral detector comparison yields a strongly positive quantum measure with no irreducible third-order interference. As a continuum benchmark, the real causal-set chain construction reproduces the massive retarded Klein–Gordon kernel in 1+1 dimensions in expectation, with a controlled variance. A universal continuum path measure, the microscopic quantum generator, Born normalisation and the identification of the action scale with Planck's constant are not yet derived.
Modern quantum theory treats time as an external parameter, whereas relativity embeds all events within a static spacetime manifold. Neither framework explains the physical reality of the “Now” or the mechanism by which causal progression unfolds. Chronon Field Theory (CFT) introduces an intrinsic causal alignment field Φμ(x) whose self-organization generates both geometry and temporality. Quantization arises from the discrete holonomy of the causal connection, while measurement corresponds to boundary-induced synchronization of causal phases. The characteristic coherence speed vcoh=J/χ=c defines the invariant rate at which causal order advances—an intrinsic causal rate rather than the motion of matter or signals. The geometric action quantum hbargeom, derived from the CFT parameters (J, λ, χ), provides the natural origin of Planck’s constant. Within this framework, quantum discreteness, relativistic invariance, and temporal becoming emerge as complementary manifestations of a single causal–geometric field dynamics.
De Broglie’s “phase harmony” suggests that quantization can be understood as a requirement of global phase consistency after completing the fundamental cycles of a bound orbit. We show that the Einstein–Brillouin–Keller (EBK) quantization rules for planar Coulomb orbits can arise without introducing a matter wave, from a long-lived localized internal clock whose phase advances in proper time at a fixed rest-frame angular frequency. In the slow–fast regime, special-relativistic time dilation makes the clock accumulate a small phase lag relative to laboratory time. After subtracting the trivial rapid ticking at the carrier frequency, the remaining drift defines two independent loop phases on the Kepler torus: one around the azimuthal cycle and one around the radial cycle (the torus generators). To leading order these generator phases are proportional to the corresponding Kepler actions, with a single action scale set by the reduced mass, the speed of light, and the clock frequency. Requiring the clock factor to return to itself after each generator loop—its holonomy, meaning the accumulated loop phase—yields closure conditions identical in form to EBK quantization, including the universal turning-point and Coulomb-origin phase slips known as the Maslov and Langer shifts. Because the Coulomb energy depends only on the principal action given by the sum of the radial and azimuthal actions, these closure conditions reproduce the familiar hydrogenic inverse-square bound-state ladder with the expected reduced-mass scaling. We also show how closure can carry an energetic bias in a frequency-local sealed-clock regime: if electric-type dipole radiation is cancelled exactly at the carrier frequency but nearby detuned sidebands radiate, then a holonomy mismatch shifts spectral weight into detuned lines and increases near-carrier leakage quadratically near closure. The identification of the action scale with Planck’s constant is treated as an external calibration.