Elastic Time: Adaptive Temporal Framework

• Elastic Time is a concept where time is treated as a flexible, non-uniform parameter, enabling adaptive alignment, sampling, and modulation across various technical domains.
• It extends classical inner-product structures by introducing elastic recurrences and dynamic programming techniques, thereby improving efficiency and robustness in sequence and time series analysis.
• Applications span diverse fields including neural network bottleneck optimization, reinforcement learning with variable control durations, wave modulation in elastic media, and even cosmological modeling.

Elastic Time denotes a family of technical constructions in which time is not treated as a rigid, uniformly sampled index. In the literature, it includes time-elastic inner products on non-uniformly sampled multivariate time series of varying lengths, dynamic-frame-rate bottlenecks for neural audio autoencoders, elastic time steps in off-policy actor-critic control, and elastic media whose constitutive parameters are explicitly modulated in time (Marteau, 2011, Bralios et al., 25 Jun 2026, Wang et al., 2024, Santini et al., 2022). This suggests a shared theme—replacing fixed temporal correspondence by alignment, adaptive sampling, learned duration, or temporal modulation—while the mathematical object called “elastic time” remains domain-specific.

1. Foundational inner-product geometry

The foundational formulation appears in the construction of time-elastic inner products for the set $\mathcal U^*$ of all finite, non-zero value, non-uniformly sampled time series of arbitrary lengths. A sequence $A\in\mathcal U^*$ is written as $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$, with strictly increasing times $t_{a(i)}\in T\subset\mathbb R$. Two operations are defined: scalar multiplication,

$$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$

and a time-wise merge addition $\oplus$ that merges two sequences on their time stamps, sums co-located points in the spatial vector space $S$, and drops points that sum to $0_S$. On this structure, a general Time Elastic Product is introduced recursively: $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$ with initialization $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$ (Marteau, 2011).

The central existence theorem states that this recursive form is an inner product if and only if $A\in\mathcal U^*$0, $A\in\mathcal U^*$1 for all $A\in\mathcal U^*$2, $A\in\mathcal U^*$3 is itself an inner product on $A\in\mathcal U^*$4, and $A\in\mathcal U^*$5. The 2012 discrete formulation presents the same choice as the unique one that makes the elastic product a bona-fide inner product on $A\in\mathcal U^*$6, with $A\in\mathcal U^*$7 symmetric and strictly positive on $A\in\mathcal U^*$8 (Marteau, 2011, Marteau et al., 2012). A common misconception is that any DTW-like recursion can be made into an inner product; these results show that the admissible recursion is sharply constrained.

The Euclidean inner product appears as a limit case. For uniformly sampled series of fixed length $A\in\mathcal U^*$9, with $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$0, standard Euclidean $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$1, and a time-weight such as $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$2 or $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$3, the time-elastic inner product converges to $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$4 as $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$5 (Marteau, 2011, Marteau et al., 2012). For finite $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$6, nearby time points are softly aligned rather than rigidly matched. The same framework also allows multiple embedded elastic dimensions by nesting elastic inner products.

2. Computational realizations for sequence and time-series mining

Once an elastic inner product exists, standard Hilbert-space constructions become available on variable-length and non-uniform sequences. The induced norm and metric are

$(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$7

so elastic comparison no longer has to be expressed only as a dynamic-programming distance (Marteau, 2011). The same paper demonstrates Gram–Schmidt orthogonalization in a Time Elastic Vector Space: with $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$8 and $(a(1),t_{a(1)})\dots(a(p),t_{a(p)})$9, $t_{a(i)}\in T\subset\mathbb R$0, a spike basis becomes a two-lobed pair and a standard sine–cosine Fourier basis of length $t_{a(i)}\in T\subset\mathbb R$1 becomes slightly deformed in amplitude and frequency. These examples are not merely visual curiosities; they show that classical orthogonal bases change shape when alignment is admitted as part of the geometry.

The 2012 discrete framework makes the computational implications explicit. A direct dynamic-programming evaluation of $t_{a(i)}\in T\subset\mathbb R$2 has $t_{a(i)}\in T\subset\mathbb R$3 time and $t_{a(i)}\in T\subset\mathbb R$4 memory, reducible to $t_{a(i)}\in T\subset\mathbb R$5 memory with a two-row implementation. When all series share the same uniform time grid $t_{a(i)}\in T\subset\mathbb R$6, each database series $t_{a(i)}\in T\subset\mathbb R$7 can be embedded as $t_{a(i)}\in T\subset\mathbb R$8, where the elastic matrix has entries $t_{a(i)}\in T\subset\mathbb R$9. Building the index for $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$0 series costs $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$1 offline, after which each lookup $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$2 costs $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$3 (Marteau et al., 2012). This indexed regime is central to the practical claim that elastic geometry can preserve some of DTW’s robustness while approaching Euclidean online cost.

The same line of work also extends beyond numeric time series. A Gaussian kernel built from $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$4 systematically improves over the Euclidean-distance Gaussian SVM for moderate $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$5 on 20 UCR-like datasets, and an “elastic cosine” obtained by setting $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$6 and $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$7 reduces for $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$8 to the classical TF or TF-IDF cosine while respecting word order for $$\lambda\otimes A=((\lambda\cdot a(1),t_{a(1)}),\dots,(\lambda\cdot a(p),t_{a(p)})),$$9 (Marteau, 2011). On 20 UCR benchmark sets, $\oplus$0 1-NN beats $\oplus$1 1-NN on $\oplus$2 of datasets, and $\oplus$3 beats $\oplus$4 on $\oplus$5; on PCBC SCOP95 and CATH95 subsets, $\oplus$6 with moderate elasticity $\oplus$7 achieves mean AUC $\oplus$8, outperforming BLAST $\oplus$9 and the LA-kernel $S$0, and remaining close to Smith–Waterman $S$1 and Needleman–Wunsch $S$2 (Marteau et al., 2012).

3. Multivariate elastic measures and large-scale learning

A complementary strand of work generalizes elastic learning directly under Dynamic Time Warping. Elastic functions are defined by embedding a time series into a matrix space along a warping path and evaluating an ordinary function $S$3 on that embedding. Because the resulting objective is a pointwise minimum or maximum over a finite family of smooth functions, it is piecewise smooth and admits a Clarke subdifferential. This permits generalized-gradient training of elastic perceptron, elastic logistic regression, elastic margin perceptron, and elastic least-squares classifiers. On UCR two-class problems, elastic linear classifiers are reported as on-par with 1-NN(DTW) using all prototypes and substantially better than 1-NN with one prototype per class, while test-time classification costs $S$4 per series rather than the prototype-dependent cost of nearest-neighbor DTW (Jain, 2015).

For multivariate data, seven elastic measures—Derivative DTW, Weighted DTW, Weighted Derivative DTW, Longest Common Subsequence, Edit Distance with Real Penalty, Move–Split–Merge, and Time Warp Edit—are extended through two strategies. The independent strategy applies a univariate measure to each dimension and combines the results with an $S$5 norm, while the dependent strategy replaces scalar pointwise cost by a multivariate cost inside a single dynamic-programming recursion. On 23 fixed-length multivariate datasets from the UEA archive, every measure except $S$6 achieves the highest accuracy on at least one dataset; $S$7 has the best average rank, $S$8 the worst; and either the dependent or the independent version can dominate across all measures, depending on the dataset (Shifaz et al., 2021). A common misconception is that cross-dimensional coupling is always beneficial. The empirical study states the opposite more precisely: high-dimensional data tend to favor the independent strategy, whereas low-dimensional datasets with strong cross-dimensional correlations can favor the dependent strategy.

Scalability motivates yet another formulation: Elastic Product Quantization for Time Series. PQDTW partitions a series into $S$9 equal-length sub-sequences, learns $0_S$0 separate codebooks by $0_S$1k-means under DTW, and approximates elastic distance by precomputed DTW tables between centroids. Because all distance calls reduce to $0_S$2 table look-ups, the query stage is asymmetric and fast; a MODWT-based pre-alignment step moves split positions within a small backward tail window and then re-interpolates each segment to common length. On 48 UCR datasets, PQDTW is on average $0_S$3 faster than exact DTW under the optimal window with compression $0_S$4–$0_S$5, and in clustering it is $0_S$6 faster than cDTW and $0_S$7 faster than full DTW; the pre-alignment step consistently reduces approximation error by $0_S$8–$0_S$9 with negligible extra time $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$0 (Robberechts et al., 2022).

4. Neural architectures and adaptive temporal bottlenecks

Time elasticity has also been internalized into neural architectures. The time elastic neural network (teNN) introduces a differentiable alignment module for multivariate time-series classification. Given an input $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$1 and a learned reference $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$2, each cell $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$3 on an $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$4 grid uses a local matching kernel

$$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$5

where $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$6 is an attention matrix and $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$7 is an activation matrix. Two dynamic-programming-style recurrences produce a final similarity score, and training minimizes a regularized categorical cross-entropy with $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$8 penalties on $$\langle A_1^p,B_1^q\rangle_{tep} = \alpha\langle A_1^{p-1},B_1^q\rangle_{tep} + \beta\langle A_1^{p-1},B_1^{q-1}\rangle_{tep} + f(a(p),b(q))\,g(t_{a(p)},t_{b(q)}) + \alpha\langle A_1^p,B_1^{q-1}\rangle_{tep},$$9 and $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$0, encouraging sparsity in attention and in the alignment corridor (Marteau, 2024). On 30 multivariate datasets from the UEA archive, teNN with one reference per class achieves an average rank of $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$1, comparable to LCEM, MLSTM-FCN, and Random Forest. During training, the learned $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$2 matrices are typically $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$3–$$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$4 zero and $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$5 is $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$6–$$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$7 zero, so the architecture effectively learns a dropout-like pruning of its own alignment grid.

A different use of Elastic Time appears in neural audio coding. Here the objective is not alignment between two series but adaptive latent frame rate. Starting from a pretrained autoencoder $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$8, Elastic Time inserts a Re-Bottleneck module $$\langle\Omega,\Omega\rangle_{tep}=\langle A,\Omega\rangle_{tep}=\langle\Omega,A\rangle_{tep}=\xi$$9 around the latent sequence $A\in\mathcal U^*$00, and trains a lightweight causal GRU-based predictor $A\in\mathcal U^*$01 of $A\in\mathcal U^*$02 M parameters to decide which latent frames can be skipped and later reconstructed. The chunk procedure returns a monotonic boundary array $A\in\mathcal U^*$03, retaining only $A\in\mathcal U^*$04 anchors; dechunk rolls out the predictor inside each retained segment. Boundary selection can be performed by a greedy $A\in\mathcal U^*$05 algorithm or an exact dynamic program with complexity $A\in\mathcal U^*$06 (Bralios et al., 25 Jun 2026).

The empirical claim is deployment-time rate control: a user specifies a kept fraction $A\in\mathcal U^*$07, and the same model can be run at different temporal resolutions without retraining. Using Stable Audio Open VAE as the base autoencoder, a ConvNeXt-V2 Re-Bottleneck, and datasets spanning music, sound effects, Chinese vocal music, and speech, Elastic Time variants achieve lower mel-spectrogram distance and Fréchet Audio Distance than adaptive baselines at matched latent rates $A\in\mathcal U^*$08 Hz–$A\in\mathcal U^*$09 Hz; the greedy solver nearly matches exact DP; and a single-rate model trained at $A\in\mathcal U^*$10 surpasses fixed downsampling at $A\in\mathcal U^*$11 (Bralios et al., 25 Jun 2026). This is a distinct sense of elastic time: not temporal warping between inputs, but learned temporal budget allocation within a latent representation.

5. Elastic time steps in reinforcement learning

In reinforcement learning, elastic time is operationalized as a variable control duration. In Soft Elastic Actor-Critic, each action includes both a control $A\in\mathcal U^*$12 and a duration $A\in\mathcal U^*$13,

$A\in\mathcal U^*$14

with $A\in\mathcal U^*$15 clipped to $A\in\mathcal U^*$16. The critic therefore estimates $A\in\mathcal U^*$17, and the scalarized reward combines task reward, computational cost, and wall-clock time cost: $A\in\mathcal U^*$18 The proposed algorithm, Multi-Objective Soft Elastic Actor-Critic (MOSEAC), keeps the usual SAC machinery—double Q, delayed targets, entropy regularization—but treats $A\in\mathcal U^*$19 as part of the action space (Wang et al., 2024).

The theoretical analysis states that the Bellman operator remains a $A\in\mathcal U^*$20-contraction and that, under standard stochastic approximation conditions, the coupled actor-critic updates converge almost surely to a local Nash-equilibrium of the soft-Q and actor objectives. The practical objective is to select the lowest viable control frequency rather than hold the controller at a fixed update rate. In Trackmania 2023 and two Newtonian-kinematics mazes, MOSEAC converges in $A\in\mathcal U^*$21 M steps on the difficult maze, uses $A\in\mathcal U^*$22 fewer steps than SAC-$A\in\mathcal U^*$23 Hz over $A\in\mathcal U^*$24 trials, completes tasks $A\in\mathcal U^*$25 faster than the best fixed-rate SAC, and in Trackmania achieves average lap time $A\in\mathcal U^*$26 s versus SAC-$A\in\mathcal U^*$27 Hz $A\in\mathcal U^*$28 s while reducing average energy by $A\in\mathcal U^*$29 (Wang et al., 2024). The learned policy increases control rate in sharp turns and reduces it on straights, illustrating an adaptive-frequency interpretation of elastic time.

6. Time-modulated elastic media and wave control

In elastic-wave physics, elastic time denotes explicit temporal modulation of material parameters. For one-dimensional elastic or acoustic waves in a lossy medium with periodically modulated stiffness $A\in\mathcal U^*$30, viscosity $A\in\mathcal U^*$31, and density $A\in\mathcal U^*$32, the wave amplitude can be analyzed by factoring out viscous decay and studying the Floquet exponent $A\in\mathcal U^*$33. The exact loss-compensation condition is

$A\in\mathcal U^*$34

which identifies the regime in which amplification due to time-dependent properties offsets material dissipation (Torrent et al., 2017). The paper states that below the modulation threshold, $A\in\mathcal U^*$35 and losses cannot be compensated; above it, sufficiently strong temporal modulation can support constant-amplitude propagation. A frequent overstatement is that time modulation automatically yields amplification. The exact condition shows that amplification, loss compensation, and instability are distinct regimes.

The same temporal control logic appears in lattices with slowly modulated stiffness. In a 1D spring–mass chain with time-dependent stiffness $A\in\mathcal U^*$36, the adiabatic no-back-scattering condition is

$A\in\mathcal U^*$37

which reduces to an explicit bound on the ramp speed $A\in\mathcal U^*$38. Under this condition, an incoming Bloch wave is frequency converted from $A\in\mathcal U^*$39 to $A\in\mathcal U^*$40 without undesired reflections. In a 2D lattice with $A\in\mathcal U^*$41 fixed and $A\in\mathcal U^*$42 modulated, the group velocity component $A\in\mathcal U^*$43 becomes time dependent, so the propagation angle

$A\in\mathcal U^*$44

bends smoothly in time, yielding scattering-free steering in the adiabatic regime (Santini et al., 2022).

A more abrupt variant uses time interfaces generated by suddenly switching a traveling spatiotemporal stiffness modulation on and off. For a rod with

$A\in\mathcal U^*$45

during a finite temporal slab, continuity of $A\in\mathcal U^*$46 and $A\in\mathcal U^*$47 at the activation and deactivation times produces a global scattering matrix that redistributes an incident mode into Floquet sidebands. The paper distinguishes subsonic and supersonic modulation: subsonic modulation induces nonreciprocal energy reversal and frequency bandgaps, while supersonic modulation leads to nonreciprocal energy amplification and wavenumber bandgaps (Ye et al., 27 Apr 2025). The practical consequence is a route to one-way elastic filters, amplifiers, and frequency converters.

7. Space-time elasticity, scattering delay, and transient response

A more speculative but mathematically explicit use of elastic time appears in continuum models of space-time. In a 5-D embedding framework, the metric of the natural 4-D manifold is written as

$A\in\mathcal U^*$48

so the strain tensor is the non-trivial part of the metric. The action combines the Ricci scalar with an elastic potential

$A\in\mathcal U^*$49

and variation yields generalized Einstein equations with an elastic contribution $A\in\mathcal U^*$50. In this framework, time is treated on the same footing as space, its deformation is encoded in temporal strain components such as $A\in\mathcal U^*$51, and the modified Friedmann equation contains a term that can drive late-time acceleration without a separate dark-energy fluid (0911.3362). This use of elastic time is conceptually distinct from sequence alignment: it is a cosmological constitutive model, not a data-analytic geometry.

In scattering theory, temporal elasticity appears as delay. For low-energy elastic $A\in\mathcal U^*$52-C$A\in\mathcal U^*$53 scattering, the Eisenbud–Wigner–Smith time delay for partial wave $A\in\mathcal U^*$54 is

$A\in\mathcal U^*$55

with $A\in\mathcal U^*$56 extracted from partial-wave phase shifts. Using DFT and Annular Square Well models, the study reports giant attosecond time delays at shape resonances; for example, in the ASW model $A\in\mathcal U^*$57 as, and in the DFT model $A\in\mathcal U^*$58 as (R. et al., 2024). A related Dirac-bubble model for fullerene cages $A\in\mathcal U^*$59 reports peak delays $A\in\mathcal U^*$60–$A\in\mathcal U^*$61 atomic units, corresponding to $A\in\mathcal U^*$62 attoseconds, with resonances attributed to temporary trapping in quasi-bound states (Amusia et al., 2019).

At the scale of amorphous solids, time-dependent elastic response is studied through a local shear transformation that mimics an elementary plastic event. Molecular dynamics simulations show that the long-time averaged displacement field matches the Eshelby inclusion solution despite fluctuations of order one across realizations, and that the propagation of the elastic signal exhibits a crossover from a propagative transmission in weakly damped dynamics to a diffusive transmission for strong damping (Puosi et al., 2014). In the overdamped regime, the full time-dependent response agrees with the solution of a diffusion equation for the displacement field in an elastic medium. Across these examples—cosmology, scattering, and glassy response—elastic time no longer means temporal alignment, but rather a constitutive or dynamical relation between temporal evolution and elastic structure.