Rushing Time Functions: A Cross-Disciplinary Overview

• Rushing Time Functions are constructs that reparameterize and constrain time measurements across various domains by altering clock mechanisms or descent lengths.
• In musical performance analysis, they quantify microtiming residuals relative to a metrical grid, distinguishing between rushing (negative residuals) and dragging (positive residuals).
• In abstract settings, such functions underpin causality in spacetime, clock-based penalization in stochastic processes, and runtime estimations in computational proof theory.

Searching arXiv for the cited papers to ground the article. arxiv_search(query="(Gordon et al., 2019)", max_results=5, sort_by="relevance") arxiv_search(query="(Gordon et al., 2019)", max_results=5, sort_by="relevance") “Rushing time functions” does not denote a single standardized object across the arXiv literature. The expression, or closely related constructions explicitly interpreted that way, appears in several technically distinct settings: as microtiming residuals that quantify “rushing” and “dragging” relative to a metrical grid in drum performance; as continuous functions on abstract spacetimes satisfying a metric-dominating monotonicity inequality; as families of random times (“clocks”) used to take long-time limits in Lévy-process penalization; and as runtime measures induced by ordinally or combinatorially indexed witnessing flows in proof theory (Gordon et al., 2019, Minguzzi, 16 Aug 2025, Takeda et al., 2022, Tabatabai, 2024). Related nearby usages also occur in numerical studies of metastable bacterial respiration and in work on reducing execution time in R, but those uses are terminologically looser than the formal notions above (Kundu et al., 2023, Uyttendaele, 2015).

1. Taxonomy of the term

The main usages can be organized as follows.

Domain	Object called or interpreted as a rushing time function	Core role
Drum microtiming	Residuals $\delta t_i = t_i - (t_0 + iT)$	Ahead/behind-grid timing
Abstract spacetime	Rushing function $f$ with $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$	Encodes order and distance
Lévy penalization	Random clocks such as $e_q$, $T_a$, $T_{a,b}$, $\eta^a_u$	Sends observation time to infinity
Witnessing flows	Length of ordinal or $k$-flow descent	Bounds feasible computation time

In the drumming literature, the operative object is a time series of residuals relative to a nominal beat period. In the spacetime literature, the object is a continuous real-valued function that dominates proper time along causal pairs. In Lévy-process penalization, the relevant objects are random times along which limits are taken. In witnessing-flow analysis, the relevant object is a runtime induced by the length of a descending sequence of feasible proof-theoretic transformations. This suggests that the common semantic core is not a single definition but a family of constructions in which “time” is reparameterized, constrained, or accelerated in a domain-specific way.

2. Microtiming residuals in performance analysis

In the musical setting, the relevant quantities are defined relative to a metrical grid. For hi-hat onset times $t_i$, inter-onset intervals are $\tau_i = t_{i+1}-t_i$, and the microtiming residuals are

$f$0

The sign convention is explicit: $f$1 means rushing, while $f$2 means dragging. For the “Tom Sawyer” task analyzed at $f$3 bpm, the quarter-note period is $f$4 and the nominal sixteenth-note interval is $f$5; complementary fluctuation measures are the mean-centered IOI deviations $f$6 and the onset amplitudes $f$7 (Gordon et al., 2019).

The empirical analysis used $f$8 hi-hat samples from $f$9 participants, including experienced drummers and non-drummers. Onset detection used a 5th-order Butterworth low-pass with cutoff $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$0–$x \le y \Rightarrow f(x)+d(x,y)\le f(y)$1 Hz and Matlab’s findpeaks() within a prediction window centered on $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$2 with $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$3 ms tolerance; erroneous intervals with $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$4 deviating beyond $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$5 ms were removed. Detrended fluctuation analysis (DFA) was then applied to the integrated, mean-centered timing or amplitude series. With

$x \le y \Rightarrow f(x)+d(x,y)\le f(y)$6

window size $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$7, linear detrending order $x \le y \Rightarrow f(x)+d(x,y)\le f(y)$8, and

$x \le y \Rightarrow f(x)+d(x,y)\le f(y)$9

the exponent $e_q$0 was interpreted conventionally: $e_q$1 for short-range anti-correlations, $e_q$2 for white noise, and $e_q$3 for long-range positive correlations. For stationary fractional Gaussian noise, the spectral relation is $e_q$4.

The principal result is negative with respect to universality. A previous Porcaro analysis had reported a clear crossover from short-range anti-correlations to long-range positive correlations, but in the larger “Tom Sawyer” dataset positive crossovers were rare: $e_q$5 of IOI series and $e_q$6 of amplitude series. Most series were statistically close to white noise: $e_q$7 of IOI series and $e_q$8 of amplitude series were approximately linear in $e_q$9 versus $T_a$0, typically with $T_a$1. Negative crossovers occurred in $T_a$2 of IOI series and $T_a$3 of amplitude series, and were interpreted cautiously as likely DFA artifacts or nonstationarities. No systematic differences in $T_a$4 or crossover prevalence were found between experienced drummers and non-drummers, or between single-handed and double-handed techniques within this task (Gordon et al., 2019).

The same study also provides a methodological warning that is central to this usage. DFA is described as prone to bias and spurious crossovers, especially for anti-correlated signals. Strong anti-correlations with $T_a$5 often produce curvature in $T_a$6 versus $T_a$7 and unstable exponent estimates; missing data, window-range choice, detrending order, and subjective crossover placement can all distort inference. In this literature, therefore, a “rushing time function” is not merely a residual series but an object whose interpretation depends strongly on robust fluctuation analysis.

3. Rushing functions in abstract spacetime theory

In the spacetime setting, the terminology is formal and axiomatic. A spacetime may be defined as a quadruple

$T_a$8

where $T_a$9 is a closed preordered space and $T_{a,b}$0 is $T_{a,b}$1-upper semi-continuous, vanishes off the preorder, and satisfies the reverse triangle inequality

$T_{a,b}$2

An equivalent time-separation form uses

$T_{a,b}$3

with $T_{a,b}$4 and $T_{a,b}$5 (Minguzzi, 16 Aug 2025).

Against this background, a function $T_{a,b}$6 is rushing if

$T_{a,b}$7

equivalently

$T_{a,b}$8

A time function is a continuous utility that is strictly isotone, and a rushing time function is both rushing and a time function. The intended intuition is explicit: such an $T_{a,b}$9 “runs faster than proper time.” Along any nontrivial causal pair $\eta^a_u$0 one therefore has

$\eta^a_u$1

The central technical device is the product trick. One passes from $\eta^a_u$2 to

$\eta^a_u$3

equipped with product topology and preorder

$\eta^a_u$4

equivalently $\eta^a_u$5. A translationally invariant function

$\eta^a_u$6

is continuous and isotone on $\eta^a_u$7 if and only if $\eta^a_u$8 is a continuous rushing function on $\eta^a_u$9. This reduces “metricity” to causality in one higher dimension and unifies proper time with time functions.

The representation theorems are the distinguishing feature of this usage. For locally compact, $k$0-compact, stable spacetimes, the family $k$1 of continuous rushing functions reconstructs topology, order, and Lorentzian distance:

$k$2

Under second-countability, one may restrict to the family $k$3 of rushing time functions without losing reconstruction. In this sense, spacetime can be represented as a family of rushing time functions, and “time fully characterizes spacetime” (Minguzzi, 16 Aug 2025).

The paper’s Minkowski example makes the idea concrete. In $k$4 with coordinates $k$5, the linear functions

$k$6

are strictly isotone and rushing, and they recover the Lorentzian distance through an infimum formula. Here the term is not metaphorical: it names a function class with a precise order-metric inequality and a full reconstruction theorem.

4. Random clocks in Lévy-process penalization

For one-dimensional Lévy processes, the pertinent objects are families of random times used to define long-time penalization limits. The process is observed on càdlàg path space with canonical filtration, and the local time at $k$7, denoted $k$8, is assumed well defined under the regularity hypotheses stated in the paper. The inverse local time is

$k$9

The penalization scheme studies weighted expectations of the form

$t_i$0

where $t_i$1 is not deterministic but chosen from a parametrized family of random clocks (Takeda et al., 2022).

The paper considers four such clocks: the exponential clock $t_i$2 with $t_i$3, the hitting time clock $t_i$4 with $t_i$5, the two-point hitting time clock $t_i$6 with $t_i$7 under a balance condition, and the inverse local time clock $t_i$8 with either $t_i$9 or $\tau_i = t_{i+1}-t_i$0. These families are explicitly interpreted as “rushing time functions” in the sense that they push the observation horizon to infinity in different probabilistic ways: exponential scaling, hitting distant levels, hitting one of two distant levels, or accumulating local time.

The limit law is characterized by a non-negative martingale built from the invariant function

$\tau_i = t_{i+1}-t_i$1

and

$\tau_i = t_{i+1}-t_i$2

The process killed upon hitting zero has $\tau_i = t_{i+1}-t_i$3 as an invariant function, and the limiting penalized law is the corresponding Doob $\tau_i = t_{i+1}-t_i$4-transform. The choice of clock determines $\tau_i = t_{i+1}-t_i$5: the exponential clock yields $\tau_i = t_{i+1}-t_i$6, one-sided hitting clocks yield $\tau_i = t_{i+1}-t_i$7, the two-point clock yields $\tau_i = t_{i+1}-t_i$8 depending on the asymptotic balance, and inverse local time clocks with $\tau_i = t_{i+1}-t_i$9 again yield $f$00.

A central distinction is between recurrent finite-variance and infinite-variance regimes. When $f$01, different clocks produce genuinely different limit martingales and limit laws; the paper states that “the limit law varies according to the chosen clock when $f$02.” When $f$03, one has $f$04 and $f$05, so all four clock families lead to the same limit. In the transient regime, the penalization becomes clock-independent again, governed by a single martingale. In this literature, therefore, a rushing time function is not a deterministic function on state space but a clocking mechanism that selects a particular asymptotic excursion bias.

5. Rushing time in witnessing flows and bounded arithmetic

In proof theory and bounded arithmetic, the expression is used for a runtime induced by witnessing mechanisms rather than for a temporal observable or a geometric function. The paper develops ordinal flows and $f$06-flows, each consisting of feasible, PV- or PV$f$07-provable stepwise transformations whose length acts as a time measure (Tabatabai, 2024).

For a ptime ordinal representation

$f$08

an $f$09-flow from $f$10 to $f$11 is a pair $f$12 satisfying

$f$13

$f$14

and

$f$15

The associated ordinal local search program $f$16 starts from $f$17, applies a modifier $f$18 while strictly decreasing an ordinal measure through

$f$19

and reads a witness at level $f$20 via $f$21.

Here the rushing time function is the total runtime induced by this descent. If each modification step is polynomial-time and the number of steps is bounded by the length of the descending $f$22-chain, then

$f$23

where $f$24 measures the number of decreases to reach $f$25. The controlling principle is well-foundedness: termination is guaranteed by transfinite induction or, equivalently at the $f$26 level, by the absence of primitive recursive descending sequences below $f$27.

The paper’s main characterization theorem states

$f$28

and

$f$29

For bounded arithmetic, a $f$30-flow of length $f$31 is defined by

$f$32

and

$f$33

This yields explicit runtime bounds:

$f$34

Thus the “rushing” is the proof-extracted descent through ordinal height or bounded iteration length. Unlike the geometric and stochastic usages, the object is explicitly computational.

6. Related usages, limits of unification, and common confusions

Two additional papers use nearby but looser language. In the Fairen–Velarde model of bacterial respiration, the reported phenomenon is “a fast time scale (after leaving the zone of limit cycle) in rushing towards the stable fixed point.” The model exhibits two time scales: a slow metastable lifetime near the stable limit cycle and a faster late-time decay toward the stable fixed point. The analysis uses semilogarithmic plots of peak–dip amplitudes, where straight lines indicate exponential decay and define $f$35 and $f$36, with $f$37 observed; decreasing the nutrient parameter $f$38 reduces the gross residence time near the limit cycle (Kundu et al., 2023). This is a dynamical-timescale usage rather than a formal function class.

In “How to speed up R code,” the subject is the reduction of total execution time rather than an object called a rushing time function. The paper discusses measurement with proc.time() and system.time(), profiling via Rprof, vectorization, preallocation, compiled C code via .C, shared-memory parallelism, and cluster submission. Reported examples include a loop-to-apply change from $f$39 to $f$40, a kernel-smoothing rewrite from $f$41 to $f$42, and a pseudo-observation rewrite from $f$43 to $f$44 (Uyttendaele, 2015). This is best understood as a runtime-optimization context, not as a definition of the encyclopedia topic.

A recurrent misconception would be to treat all these usages as instances of one mathematical object. The sources do not support that. In performance analysis, the object is a residual time series $f$45; in spacetime theory, it is a continuous rushing function satisfying a causal inequality; in Lévy penalization, it is a family of random clocks; in witnessing flows, it is a runtime function derived from descent length. A plausible implication is that “rushing time function” functions more as a cross-domain label for accelerated, dominant, or asymptotically directing temporal structures than as a universally fixed term.

Another misconception would be to identify “rushing” with a universal signature or with a single correlation law. The drumming study explicitly rejects universality of the Porcaro-style crossover and stresses that most timing and amplitude fluctuations in its larger dataset are consistent with white noise, while the Lévy-process study shows that asymptotic behavior can depend sharply on the chosen clock when $f$46 (Gordon et al., 2019, Takeda et al., 2022).

Across these literatures, the most robust commonality is structural rather than definitional: a rushing time function is a device that constrains how time-like progression is measured. Whether that progression is musical, causal, stochastic, or computational depends entirely on the ambient theory.