Input Time Scaling: Concepts & Applications

• Input Time Scaling is a framework that treats temporal variables—such as look-back horizons, query inputs, or time maps—as adjustable parameters affecting system performance.
• It is applied in fields including language models, time-series forecasting, quantum control, and metrology, where optimizing temporal resolution can enhance reasoning, fidelity, or computational efficiency.
• The approach unifies diverse methodologies by transforming fixed time assumptions into dynamic optimization variables, balancing trade-offs between information gain and cost.

to=arxiv_search.^^^^^^^^4query4^^^^^^^^ 大发快三怎么json anasiyana ^^^^4all:\4^^^^: {"^^^^^^^^4query4 Time Scaling\"^^^^^^^^4 OR ti:\4^^^^^^^^"Input Time Scaling\"^^^^^^^^4 OR abs:\4^^^^^^^^"Input Time Scaling\"","max_results":^^^^4all:\4query4^^^^,"sort_by":"submittedDate","sort_order":"descending"} 天天彩票软件сәтာքներ to=arxiv_search.^^^^^^^^4query4^^^^^^^^ code ￣色 none to=arxiv_search.^^^^^^^^4query4^^^^^^^^ 时时彩后 to=arxiv_search.^^^^^^^^4query4^^^^^^^^ code 大发游戏官网 none to=arxiv_search.search 大发彩票官网 to=arxiv_search.search code 全民彩票天天送json {"^^^^^^^^4query4 Time Scaling","max_results":5} Input time scaling is a context-dependent term used across several technical literatures to denote methods that treat temporal structure, temporal resolution, or input-side ^^^^^^^^4query4^^^^^^^^ transformation as an explicit design variable rather than a fixed background parameter. In recent work on agents and LLMs, it denotes allocating resources to ^^^^^^^^4query4^^^^^^^^ refinement and strategy-conditioned inputs during both training and testing (&&&^^^^4query4^^^^&&&). In text-to-image generation, it appears as input-side inference-time scaling through prompt rewriting while the image backbone remains frozen (&&&^^^^4all:\4^^^^&&&). In time-series forecasting, it refers to scaling the look-back horizon PRESERVED_PLACEHOLDER_^^^^4query4^^^^ and analyzing its interaction with dataset size PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^ and model size PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^ (&&&^^^^4 OR ti:\4^^^^&&&). In time and frequency metrology, it denotes bringing heterogeneous clock signals to a common time representation and digitally steering a composite oscillator (&&&^^^^4 OR abs:\4^^^^&&&), including optical-clock steering of a real-time time scale (Hachisu et al., 2018). In quantum control, it is the explicit choice of a rescaled time variable PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^ or $\Lambda(t)$ that re-times a reference evolution (Hatomura, 2022, Impens et al., 2021). A related financial usage redefines the time axis itself so that return distributions obey simple scaling with $H=1/2$ (Caraglio et al., 2016). Taken together, these works indicate that no single field-independent definition is in use.

^^^^4all:\4^^^^. Terminological scope and unifying pattern

A useful common denominator is that input time scaling promotes some temporal quantity to a first-class control parameter. Depending on the field, that quantity may be a ^^^^^^^^4query4^^^^^^^^ transformation budget, a look-back horizon, a local clock representation, a rescaled Schrödinger time, a timestep $h$, or a continuous dilation field $a({\bf x},t)$. The review literature on inference-time scaling makes this distinction explicit by separating output-focused methods from input-focused methods; the latter encompass few-shot prompting and retrieval-augmented generation, where additional inference compute is spent on what the model receives as input rather than on search over outputs (Wang et al., 12 Oct 2025).

This broader pattern is not merely terminological. In each domain, the temporal variable being scaled mediates a trade-off between information and cost. Longer or richer inputs can improve reasoning, but they also change the effective state space seen by the model. Larger look-back windows reduce Bayesian forecasting error but can increase approximation error. Slowing or accelerating a quantum trajectory can preserve a target path while redistributing control effort. Enlarging a physical driver timescale can preserve similarity only if geometry, losses, and energy are co-scaled. The recurrent theme is therefore not “more time is better,” but “time-like inputs become optimization variables with nontrivial constraints.”

^^^^4 OR ti:\4^^^^. Query-side scaling in LLMs and multimodal generation

In the agent and LLM literature, “Input Time Scaling” is introduced as a scaling paradigm complementary to data and training scaling and to inference-time scaling. The core operation is an input transformation

$\tilde{x} = \phi(x; s),$

where $s \in \{N,S,U,R\}$ denotes No-Persona, Persona-Similar, Persona-Unsimilar, or Persona-Random ^^^^^^^^4query4^^^^^^^^ construction (&&&^^^^4query4^^^^&&&). The paper reports a training-testing co-design phenomenon: ^^^^^^^^4query4^^^^^^^^ strategies must be applied during both training and testing, and only applying strategies on training or testing seriously degrades performance. It also reports results that run against the usual “garbage in, garbage out” inductive bias: adding irrelevant information to queries and randomly selecting examples from a minimally filtered dataset can perform best, while increasing data size from PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^ to PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^ with similar quality can perform worse. With models trained on Qwen^^^^4 OR ti:\4^^^^.5-^^^^4 OR abs:\4 OR ti:\4^^^^B-Instruct, the reported pass@^^^^4all:\4^^^^ results reach AIME^^^^4 OR ti:\44^^^^ PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^ and AIME^^^^4 OR ti:\4^^^^5 PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^, and a majority vote of three models reaches AIME^^^^4 OR ti:\44^^^^ PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^ and AIME^^^^4 OR ti:\4^^^^5 PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5 (&&&^^^^4query4^^^^&&&).

A closely related multimodal instantiation appears in text-to-image generation as “input-side inference-time scaling.” There, the scaled object is not a reasoning trace but the prompt itself. A prompt rewriter LLM produces refined prompts, a frozen text-to-image backbone renders them, and multimodal judges assign pairwise preferences over image quality, general image-text alignment, physical image-text alignment, and aesthetics; these preferences are then used in an iterative DPO pipeline (&&&^^^^4all:\4^^^^&&&). The method is model-agnostic and transfers across FLUX, Stable Diffusion ^^^^4 OR abs:\4^^^^.5, and JanusPro backbones. Representative results include FLUX.^^^^4all:\4^^^^-schnell on Pick-a-Pic v^^^^4 OR ti:\4^^^^ improving from average PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6 without rewriting to PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7 with the general rewriter and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 with the aesthetics rewriter, and FLUX.^^^^4all:\4^^^^-dev on GenEval improving overall from PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9 to PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4query4^^^^ with the general rewriter (&&&^^^^4all:\4^^^^&&&).

The input-focused review literature places these methods within a larger inference-time taxonomy. Few-shot prompting, ^^^^^^^^4query4^^^^^^^^ expansion, retrieval, reranking, and multimodal context construction are all treated as ways of scaling inference by enriching the input channel instead of altering model weights (Wang et al., 12 Oct 2025). This suggests a general interpretation of ^^^^^^^^4query4 input time scaling: inference budget is spent on constructing better conditional contexts.

^^^^4 OR abs:\4^^^^. Horizon scaling in time-series forecasting

In time-series forecasting, input time scaling refers to scaling the look-back horizon PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4all:\4^^^^. The central theory treats forecasting loss as

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR ti:\4^^^^

with dataset size PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR abs:\4^^^^, model size PRESERVED_PLACEHOLDER_^^^^4 OR ti:\44^^^^, and input horizon PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^5 defined in an intrinsic space whose dimension PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^6 grows linearly with PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^7 (&&&^^^^4 OR ti:\4^^^^&&&). The Bayesian term decreases as the horizon grows, because longer history provides more information. The approximation term grows with the effective intrinsic dimension, because longer horizons enlarge the space that finite data and finite-capacity models must cover. The paper formalizes a phase boundary with

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^8

which separates data-dense and data-sparse regimes (&&&^^^^4 OR ti:\4^^^^&&&).

This formulation explains why “more past is not always better.” The paper predicts a finite optimal horizon PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^9, and the empirical results show U-shaped loss-versus-horizon curves. For fixed model and varying amounts of training data, loss first falls and then rises as PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^ grows. The optimal horizon increases with dataset size. Feature degradation also matters: for Exchange, the optimal horizon is reported as PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^ even using PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR ti:\4^^^^ of training data, while for ETTh^^^^4all:\4^^^^ it is PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^ even using only PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^ of training data (&&&^^^^4 OR ti:\4^^^^&&&).

The significance of this usage is methodological. In contrast to scaling laws in language or vision, where larger context or resolution is often beneficial, the forecasting literature treats horizon as a third scaling axis with its own optimum. Input time scaling here means choosing PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5 jointly with PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6 and PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7, not maximizing it by default.

4. Metrological time scales and composite clocks

In time and frequency metrology, a time scale is a realizable continuous time function built by combining several physical clocks, and a composite clock is the physical and algorithmic realization of that time scale. The digital approach described in “The Digital Revolution, also for Time Scales” measures each input clock against a local oscillator, converts phases to a common phase-time representation, removes deterministic drift, performs weighted averaging,

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8

and then phase-locks the local oscillator to the computed composite phase (&&&^^^^4 OR abs:\4^^^^&&&). In this setting, input time scaling means ingesting multiple clocks at arbitrary frequencies, bringing them to a common time/frequency representation, digitally steering and combining them, and mapping the result back into one physical output oscillator. A preliminary experiment composing three active hydrogen masers reports a stability at PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9 s of

$\Lambda(t)$^^^^4query4^^^^

compatible with the short-term stability of the average of the masers (&&&^^^^4 OR abs:\4^^^^&&&).

A related but distinct metrological usage appears in the optical-clock time-scale literature. “Months-long real-time generation of a time scale based on an optical clock” describes TA(Sr), an optically steered time scale generated continuously for half a year by steering a hydrogen maser with intermittent operation of a $\Lambda(t)$^^^^4all:\4^^^^Sr optical lattice clock (Hachisu et al., 2018). Here the input is sparse but high-accuracy optical-clock data; the system estimates the maser drift in a moving window and applies steering corrections so that the integral of the corrected fractional frequency is zero over each update interval. The reported variation of TA(Sr) $\Lambda(t)$^^^^4 OR ti:\4^^^^ TT(BIPM^^^^4all:\4^^^^6) is $\Lambda(t)$^^^^4 OR abs:\4^^^^ ns after $\Lambda(t)$4 months, and the time scale is reported as stable as TAI with accuracy at the $\Lambda(t)$5 level (Hachisu et al., 2018).

In both cases, the conceptual shift is from manipulating RF waveforms directly to manipulating time/phase information. Input time scaling therefore denotes a representational normalization problem: diverse clocks are first reduced to digital phase or time streams, then recombined algorithmically.

5. Quantum time rescaling and re-timed dynamics

In quantum control, input time scaling is the explicit choice of a rescaled time $\Lambda(t)$6, or equivalently $\Lambda(t)$7, that prescribes how a reference evolution is replayed in laboratory time. In the fast-forward theory for nonadiabatic transitions, the target condition is that populations at laboratory time $\Lambda(t)$8 match those of the reference dynamics at rescaled time $\Lambda(t)$9,

$H=1/2$^^^^4query4^^^^

and the corresponding Hamiltonian takes the form

$H=1/2$^^^^4all:\4^^^^

where $H=1/2$^^^^4 OR ti:\4^^^^ is the counterdiabatic term and $H=1/2$^^^^4 OR abs:\4^^^^ is a nonadiabatic reproduction term (Hatomura, 2022). The method can speed up, slow down, pause, or rewind a reference dynamics. In the adiabatic limit, the nonadiabatic reproduction term averages out and the construction reduces to shortcuts to adiabaticity.

The non-Hermitian counterpart rescales time by a monotone map $H=1/2$4, with transformed Hamiltonian

$H=1/2$5

and uses that freedom to reduce norm loss in open-system state transfer without adding new couplings (Impens et al., 2021). In the two-level and three-level examples studied there, appropriate time scaling improves fidelity while preserving the ratio between actual speed and the non-Hermitian quantum speed limit. The paper derives the bound

$H=1/2$6

with $H=1/2$7 containing both Hermitian and anti-Hermitian contributions, and shows that suitably scaled driving preserves optimality with respect to this bound while significantly reducing damping of the quantum state norm (Impens et al., 2021).

This usage is conceptually precise: the input is literally a time map. What is optimized is not only the control field but the schedule by which the reference trajectory is traversed.

6. Time scaling in optimal control and multiscale simulation

In optimal control, time scaling enters when the discretization step $H=1/2$8 becomes a decision variable. For linear dynamics discretized as

$H=1/2$9

the products $h$^^^^4query4^^^^ and $h$^^^^4all:\4^^^^ make the problem nonconvex (&&&^^^^4 OR ti:\44^^^^&&&). The proposed semidefinite relaxation introduces

$h$^^^^4 OR ti:\4^^^^

selects only the bilinear terms that actually appear in the time-scaled dynamics and cost, and augments the formulation with tightening equalities and inequalities. The method extends to piecewise-affine systems by representing mode sequences as paths in a graph of convex sets, so that the time-scaled PWA optimal-control problem can be solved through a single semidefinite program (&&&^^^^4 OR ti:\44^^^^&&&).

A different but allied formulation appears in multiscale timestepping for astrophysical simulation. There the governing equation

$h$^^^^4 OR abs:\4^^^^

is replaced by

$h$4

where $h$5 is a continuous dilation factor with $h$6 (&&&^^^^4 OR ti:\4^^^^6&&&). The effective timestep becomes $h$7, allowing strongly constrained subdomains to evolve more slowly in the global timeline while preserving correct local steady-state solutions. The paper derives smoothness and timestep criteria for $h$8, connects the framework to reduced-speed-of-light and hard-binary slowdown methods, and reports effective speedup factors exceeding $h$9 in multiphysics simulations (&&&^^^^4 OR ti:\4^^^^6&&&).

Both works treat temporal resolution as a control variable embedded in the equations themselves. In one case, $a({\bf x},t)$^^^^4query4^^^^ is optimized through convex relaxation; in the other, $a({\bf x},t)$^^^^4all:\4^^^^ is prescribed or adaptively modulated to reshape computational cost across the domain.

7. Physical design spaces and time as a computational substrate

Input time scaling also appears in physical design studies where a driver timescale is changed and the rest of the system must be co-scaled to preserve similarity. In MagLIF implosions, the voltage source timescale $a({\bf x},t)$^^^^4 OR ti:\4^^^^ is the FWHM of the open-circuit voltage waveform, and all implosion timescales are assumed to scale linearly with $a({\bf x},t)$^^^^4 OR abs:\4^^^^ (&&&^^^^4 OR ti:\4^^^^8&&&). Holding the relevant dimensionless parameters fixed forces coordinated changes in liner radius, mass, height, fuel density, applied axial field, preheat energy, and circuit elements. A central result is that the load voltage follows the weak law

$a({\bf x},t)$4

rather than the ideal $a({\bf x},t)$5, because the imploding height must increase to preserve end losses. Consequently, longer rise times require larger total laser preheat energy and delivered electrical energy (&&&^^^^4 OR ti:\4^^^^8&&&). This is a clear example of a misconception corrected by similarity analysis: scaling the input time alone does not preserve the rest of the design.

A more radical physical interpretation appears in “Computing with Clocks,” where time itself is proposed as the primary representation of data (&&&^^^^4 OR abs:\4query4^^^^&&&). There a value $a({\bf x},t)$6 is encoded as an interval

$a({\bf x},t)$7

and computation is performed by transforming temporal quantities. Unary addition corresponds to interval concatenation,

$a({\bf x},t)$8

while unary multiplication is achieved by dilating an interval through clock-frequency scaling,

$a({\bf x},t)$9

In this usage, input time scaling is literal: magnitude, resolution, and arithmetic are all represented in the time domain rather than in binary words (&&&^^^^4 OR abs:\4query4^^^^&&&).

Across these physically grounded examples, the term denotes either co-scaling of a system around a changed timescale or direct use of temporal intervals as computational inputs. The first emphasizes similarity constraints; the second treats time itself as data.