Zero-Duration Temporal Cycles

• Zero-duration temporal cycles are domain-dependent phenomena where temporal recurrences approach zero, manifesting as asymptotic limits in elementary cycles, discrete stroboscopic periods in circuits, and exact zero-length cycles in temporal graphs.
• In quantum mechanics, the cycles are defined by limits such as T = h/E or T_C = h/(mc²) as energy or mass diverges, while in non-unitary circuits, eigenphase bounds enforce a finite minimal period.
• In temporal graphs, exact zero-duration cycles enable repeated traversal without time progression, posing significant implications for multiobjective shortest path algorithms and termination conditions.

Searching arXiv for the specified papers and related topic. Zero-duration temporal cycles are not a single, uniform concept across contemporary research. In the framework of elementary cycles (ECs), they denote asymptotic limits in which an intrinsic temporal recurrence tends to zero, either through the Compton recurrence $T_C=h/(mc^2)$ as $m\to\infty$ or through the coordinate-time recurrence $T_t=h/E$ as $E\to\infty$, while exact $T=0$ is excluded by the requirement of finite compactification length and periodic boundary conditions (Dolce, 2013). In non-unitary post-selected quantum circuits, the analogous question is whether a stroboscopic oscillation period can shrink to zero; the model instead yields a finite lower bound, because $P=2\pi/|\arg(\lambda_1/\lambda_0)|$ and the relevant eigenphase is bounded by $\pi$, so $P_{\min}=2$ in the one-step normalization and $P_{\mathrm{JW,min}}=4$ in the Jordan–Wigner two-step convention (Basu et al., 2021). In multiobjective temporal shortest paths, by contrast, a zero-duration temporal cycle is a literal temporal cycle $C$ with $m\to\infty$0, typically realized by arcs with identical start time and zero traversal time, and such cycles can be traversed arbitrarily many times while preserving temporal feasibility (Marica et al., 7 May 2026).

1. Domain-specific meanings and basic distinctions

The phrase “zero-duration temporal cycle” is used in at least three technically distinct ways. In EC-based relativistic quantum mechanics, the relevant object is an intrinsic recurrence of a particle regarded as a periodic phenomenon. In non-unitary circuit dynamics, the relevant object is a period measured in discrete timesteps. In temporal graph optimization, the relevant object is a cycle in a discrete-time graph whose duration is exactly zero (Dolce, 2013, Basu et al., 2021, Marica et al., 7 May 2026).

Setting	Object called or associated with a zero-duration temporal cycle	Status of exact zero
Elementary cycles of time	Proper-time or coordinate-time recurrence	Excluded; only $m\to\infty$1 limits are meaningful
Non-unitary quantum circuits	Stroboscopic oscillation period $m\to\infty$2	Excluded by discrete-time structure and eigenphase bounds
Temporal shortest paths	Temporal cycle with $m\to\infty$3	Allowed and algorithmically consequential

This suggests that the phrase is best understood as a domain-dependent label rather than a shared formal notion. The main commonality is that each literature studies the behavior of dynamical or combinatorial structures near a limiting temporal scale. The differences are more substantive than the similarity: one literature studies compactified spacetime recurrences, one studies discrete-time spectral oscillations, and one studies feasibility-preserving repetitions in temporal graphs.

2. Asymptotic zero-period limits in elementary cycles

In the EC framework, every elementary particle is treated as a periodic phenomenon characterized by intrinsic temporal and spatial recurrences fixed by its kinematical state through Planck’s constant. For a free particle with energy $m\to\infty$4 and spatial momenta $m\to\infty$5, the temporal and spatial periodicities are $m\to\infty$6 and $m\to\infty$7, assembled into a contravariant four-vector of recurrences $m\to\infty$8. The rest-frame recurrence is the Compton time $m\to\infty$9, with angular frequencies $T_t=h/E$0 and $T_t=h/E$1, and the phase-harmony relation is written as $T_t=h/E$2 (Dolce, 2013).

Within this construction, a “zero-duration temporal cycle” is not a literal compact temporal cycle of vanishing extent. Two limiting cases are identified. First, in the infinite-mass limit, $T_t=h/E$3 as $T_t=h/E$4, so the internal clock ticks infinitely fast and approaches what the paper describes as a rigid or classical limit. Second, for a given mode, $T_t=h/E$5 as $T_t=h/E$6; for massive particles this occurs under boosting, while for photons the coordinate-time periodicity can in principle be arbitrarily small for high-energy modes. The massless case is exceptional: along null trajectories $T_t=h/E$7, there is no rest frame, and the proper-time recurrence is assigned $T_t=h/E$8, a “frozen rest clock,” so only coordinate-time cycles are relevant.

The exclusion of exact $T_t=h/E$9 is structural. ECs are implemented by compactifying time, or more generally spacetime, with periodic boundary conditions on $E\to\infty$0, so that $E\to\infty$1 and $E\to\infty$2. A strictly zero period would collapse the compact time circle to a point and destroy the mode structure. The spectrum $E\to\infty$3, arising from $E\to\infty$4, and the variational consistency of the construction both require $E\to\infty$5. The zero-duration case is therefore asymptotic rather than physical in the literal sense.

The significance of this limit is tied to the paper’s relational interpretation of time. Each isolated particle is a reference clock, and a composite system’s instants are labeled by the combination of phases of all internal clocks. Interactions are described as local, retarded modulations of the internal clocks’ periods proportional to exchanged energy, preserving causal ordering. In that setting, near-zero cycles do not imply acausality; they indicate extremely fine recurrence scales. The paper further frames gauge interactions as local transformations of flat reference frames and gravitational effects as metric deformations that dilate or contract cycles, placing the zero-period limit within a broader geometrodynamical account of quantum and relativistic structure.

3. Minimal stroboscopic periods in non-unitary quantum circuits

The non-unitary circuit studied in the time-crystal literature consists of layers of nearest-neighbor two-qubit gates and onsite single-qubit gates, with one-step evolution operator

$E\to\infty$6

where the single-qubit factors are implemented באמצעות a unitary, a POVM, and post-selection on a chosen outcome. Temporal oscillations are diagnosed through observables such as

$E\to\infty$7

and through correlators whose long-time behavior is controlled by the leading transfer-matrix eigenvalues (Basu et al., 2021).

The central spectral mechanism is explicit. If the transfer matrix has leading eigenvalues $E\to\infty$8, then correlators contain factors of the form $E\to\infty$9, so the correlation time is

$T=0$0

and the oscillation frequency is

$T=0$1

Persistent oscillations occur when the two leading eigenvalues are equimodular, $T=0$2, so the decay disappears and the observable behaves as

$T=0$3

The period measured in discrete circuit steps is then

$T=0$4

In the Jordan–Wigner treatment of the NFM1 phase, the effective two-layer unit cell yields

$T=0$5

In this setting, the “zero-duration” question becomes a question about whether $T=0$6 can occur. The answer is negative. Because $T=0$7 is a principal argument in $T=0$8, the smallest possible one-step period occurs at $T=0$9, giving $P=2\pi/|\arg(\lambda_1/\lambda_0)|$0. In the two-step Jordan–Wigner normalization, $P=2\pi/|\arg(\lambda_1/\lambda_0)|$1 when $P=2\pi/|\arg(\lambda_1/\lambda_0)|$2. The paper states that there is no physically meaningful period smaller than one timestep in a discrete-time circuit, and sub-step periods cannot be resolved stroboscopically. Approaching the ferromagnetic boundary instead produces the opposite limit: $P=2\pi/|\arg(\lambda_1/\lambda_0)|$3 and $P=2\pi/|\arg(\lambda_1/\lambda_0)|$4, corresponding to static order rather than ultrafast oscillation.

The temporal order of this model is controlled by Fisher zeros and equimodularity. The circuit is mapped to an anisotropic square-lattice Ising model at complex temperature, and crossing or approaching Fisher-zero loci correlates with the condition that the leading transfer-matrix eigenvalues become equimodular and acquire a nonzero relative phase. This produces persistent, incommensurate oscillations in the NFM1 phase. The contrast with unitary Floquet time crystals is explicit: unitary Floquet time crystals exhibit oscillations commensurate with the drive, while non-unitarity and post-selection here allow incommensurate order whose period is continuously tunable by circuit parameters such as the measurement angle $P=2\pi/|\arg(\lambda_1/\lambda_0)|$5, post-selection outcome $P=2\pi/|\arg(\lambda_1/\lambda_0)|$6, anisotropy $P=2\pi/|\arg(\lambda_1/\lambda_0)|$7, and complex inverse temperature $P=2\pi/|\arg(\lambda_1/\lambda_0)|$8. Even so, the model retains a finite lower bound on any resolvable temporal cycle.

4. Exact zero-duration cycles in temporal shortest paths

In the temporal-graph literature, a directed discrete-time temporal graph is a pair $P=2\pi/|\arg(\lambda_1/\lambda_0)|$9 in which each temporal arc $\pi$0 is a quadruple consisting of a start node $\pi$1, an end node $\pi$2, a start time $\pi$3, and a traversal time $\pi$4. A temporal path

$\pi$5

must satisfy the feasibility condition

$\pi$6

which encodes waiting implicitly. The duration of such a path is

$\pi$7

A temporal cycle is a path with $\pi$8 and $\pi$9, and it is a zero-duration cycle if $P_{\min}=2$0 (Marica et al., 7 May 2026).

Unlike the EC and circuit settings, exact zero-duration cycles are fully admissible here. The paper states that, intuitively, in discrete-time temporal graphs with nonnegative traversal times, a zero-duration cycle is a cycle whose arcs share the same start time and have zero traversal time, so its arrival time equals its departure time. If such a cycle $P_{\min}=2$1 is reachable from the source $P_{\min}=2$2 at the appropriate time, and $P_{\min}=2$3 is an $P_{\min}=2$4-$P_{\min}=2$5 path arriving no later than the start of the cycle, then the sequence $P_{\min}=2$6 is well-defined for all $P_{\min}=2$7. Because $P_{\min}=2$8, repeated concatenation remains temporally feasible: arrivals continue to match the cycle’s start time.

This property becomes consequential in the multiobjective setting. An objective is formalized as a tuple $P_{\min}=2$9, with associative $P_{\mathrm{JW,min}}=4$0 and a left-neutral element $P_{\mathrm{JW,min}}=4$1. A path image under $P_{\mathrm{JW,min}}=4$2 objectives is the vector of objective values, and nondominance is defined coordinatewise with respect to the directions of optimization. A zero-duration cycle is called improving in objective $P_{\mathrm{JW,min}}=4$3 if there exists a temporal $P_{\mathrm{JW,min}}=4$4-$P_{\mathrm{JW,min}}=4$5 path $P_{\mathrm{JW,min}}=4$6 such that $P_{\mathrm{JW,min}}=4$7 is well-defined for all $P_{\mathrm{JW,min}}=4$8 and, for every $P_{\mathrm{JW,min}}=4$9, there exists $C$0 with strict improvement in objective $C$1. A cycle is improving if it is improving in at least one objective.

Two examples establish why these cycles are algorithmically nontrivial. In the first, with earliest arrival time and a second isotonic but non-monotone objective defined by $C$2, a reachable zero-duration cycle of $C$3 arcs decreases the second objective by exactly $C$4 per traversal while leaving earliest arrival unchanged at $C$5. Starting from $C$6, one needs $C$7 traversals to reach $C$8, and the unique nondominated image $C$9 is attained by a path of length $m\to\infty$00, exponential in input size. The paper states that by increasing $m\to\infty$01, efficient paths of any finite length can be enforced. In the second example, two simple zero-duration cycles individually produce images $m\to\infty$02 and $m\to\infty$03, but the nondominated image $m\to\infty$04 requires a non-simple cycle obtained by concatenating them. The paper uses this to show that repeating a single simple cycle may be insufficient.

5. Label-correcting algorithms and termination mechanisms

The algorithmic problem studied is the single-source multiobjective temporal shortest path problem. Because zero-duration cycles may need to be traversed an arbitrary finite number of times to generate all nondominated images, the paper introduces a restricted variant with a maximum admissible path length $m\to\infty$05, called SSMTSPP-MPL. The task is to compute, for each node, the set of $m\to\infty$06-nondominated images together with a corresponding $m\to\infty$07-efficient path (Marica et al., 7 May 2026).

Two general label-correcting frameworks are developed. When all objectives are isotonic, Algorithm 1 extends labels iteration by iteration up to $m\to\infty$08, using the earliest-arrival component to enforce temporal feasibility through the check $m\to\infty$09. Dominated labels can be removed eagerly because isotonicity guarantees that extending a dominated label cannot later create a nondominated image that would not also arise from extending a dominating label. When isotonicity is absent, Algorithm 2 must retain all distinct images until termination, because a currently dominated label may later become necessary. Only at the end are dominated labels deleted. In both algorithms, early termination occurs if no label set changes between successive iterations.

The path-length bound is the generic device that guarantees termination in the presence of reachable zero-duration cycles. Example 1 shows why this is necessary in general: without a bound, efficient paths may require arbitrarily many traversals of a zero-duration cycle. The paper then provides several sufficient conditions under which the bound is unnecessary. If no zero-duration cycle reachable from $m\to\infty$10 exists, then $m\to\infty$11 iterations suffice for both algorithms; the paper states that no arc can then be traversed twice, so any path length is at most $m\to\infty$12. If a positive minimum waiting time $m\to\infty$13 is imposed at every node and feasibility is modified to $m\to\infty$14, repeated arc traversal becomes impossible, and again $m\to\infty$15 iterations suffice. If the number of attainable images per node is bounded by $m\to\infty$16, then the stopping criterion is satisfied after at most $m\to\infty$17 iterations. A corollary replaces this by per-objective bounds $m\to\infty$18 and sets $m\to\infty$19.

For rational additive objectives, the paper sharpens the result further. With earliest arrival time plus $m\to\infty$20 rational additive objectives, an additive variant of Algorithm 2 runs for at most $m\to\infty$21 iterations and either returns exact efficient images or reports that an improving cycle exists. The reported statements are exact: if the temporal graph contains an improving cycle, the algorithm correctly reports its existence; if it does not, the returned label sets contain a corresponding label for each image of an efficient path and only such labels. This places improving cycles in temporal graphs in direct analogy with negative cycles in static additive shortest-path theory, while preserving the distinctive temporal restriction that only zero-duration cycles can be repeated arbitrarily often.

6. Misconceptions, comparisons, and unresolved issues

A common misunderstanding is to treat all zero-duration temporal cycles as literal vanishing periods. The three literatures do not support that identification. In EC theory, exact $m\to\infty$22 is excluded because compactification and periodic boundary conditions require finite periodicity, and the massless case is characterized not by zero proper-time recurrence but by $m\to\infty$23, a “frozen rest clock” (Dolce, 2013). In the circuit setting, a strictly zero period is not physically meaningful because time is sampled in integer layers and the eigenphase bound implies a finite minimum of two steps, or four steps in the Jordan–Wigner convention (Basu et al., 2021). Only the temporal-graph setting admits an exact zero-duration cycle as an object of the theory, and there it is a property of path feasibility rather than of a physical recurrence (Marica et al., 7 May 2026).

Another misconception is that near-zero temporal scales necessarily threaten causality or consistency. The EC paper states the opposite: interactions are local, retarded modulations of internal clocks, and near-zero coordinate-time cycles merely sharpen local modulation scales. The circuit paper similarly states that the discrete-time structure prevents sub-step pathologies; approaching the antiferromagnetic-like limit yields the shortest possible alternating oscillation rather than a singularity. The temporal-graph paper also distinguishes its improving cycles from static negative cycles: improvement may depend on the incoming subpath, a zero-duration cycle need not improve at every repetition, and non-simple concatenations of multiple zero-duration cycles may be necessary.

The three papers also leave different open fronts. In the EC program, the stated open issue is experimental: resolving and controlling ultrafast internal-clock recurrences on the order of $m\to\infty$24 and fully testing the internal clock picture, as reviewed in Dolce’s Europhys. Lett. 102, 31002 (2013) (Dolce, 2013). In the non-unitary circuit model, experimental realization is constrained because post-selection success probability scales exponentially in circuit spacetime area, suggesting few-qubit registers with many layers or shallow circuits as the likely regime (Basu et al., 2021). In temporal graph optimization, the paper explicitly identifies open questions: whether, without monotonicity or isotonicity, one can guarantee termination within a predefined number of iterations while still identifying all nondominated images or detecting improving cycles; the complexity of deciding the existence of improving cycles in general temporal graphs; and the development of more efficient all-pairs versions (Marica et al., 7 May 2026).

Taken together, these works show that “zero-duration temporal cycles” name three different edge cases of temporal structure: an excluded but informative asymptotic limit in compactified recurrence models, a forbidden sub-step oscillation in discrete non-unitary dynamics, and an exact combinatorial mechanism that can fundamentally alter optimality and termination in temporal graph algorithms.