From Jumps to Signatures: a Generative Method for Temporal Point Processes

# From Jumps to Signatures: a Generative Method for Temporal Point Processes
Source: [https://arxiv.org/html/2607.06652](https://arxiv.org/html/2607.06652)
Niels Cariou\-Kotlarek and Vasileios Lampos Centre for Artificial Intelligence Department of Computer Science University College London \{niels\.kotlarek\.23,v\.lampos\}@ucl\.ac\.uk

###### Abstract

Rough path signatures are a universal feature map for continuous paths and, via the expected signature, characterise path distributions\. These guarantees do not directly extend to càdlàg paths of Temporal Point Processes \(TPPs\), limiting the use of signature methods for event sequences\. Furthermore, neural TPP models, including recent generative approaches, optimise per\-event objectives with no global sequence\-level loss, while evaluation of variable\-length event sequences lacks distributional discrepancy measures\. This paper proposes a common pathwise framework for addressing these limitations\. We introduce the interarrival embedding, a stable, injective lift from jump paths to continuous paths of bounded variation, extending signature methods to discrete event sequences\. Our theoretical contributions give rise toSigTPP, the first signature\-based generative model for TPPs, trained using a path\-level loss on complete trajectories\. We further analyse the space of counting paths and derive 3 distributional discrepancies, providing mathematically justified tools for evaluating generative TPP models\. Across synthetic and real\-world datasets,SigTPPachieves the best average rank based on 8 complementary metrics, outperforms or is within a standard error of the strongest baseline in64%64\\%of the dataset\-metric pairs, and according to a relative score, improves against every baseline by at least19%19\\%on average\. Source code and datasets:[github\.com/Code\-Cornelius/sigtpp](https://github.com/Code-Cornelius/sigtpp)

## 1Introduction

Temporal Point Processes \(TPPs\) model discrete events occurring in continuous time\[[24](https://arxiv.org/html/2607.06652#bib.bib23),[12](https://arxiv.org/html/2607.06652#bib.bib19)\]\. They underpin applications across finance\[[3](https://arxiv.org/html/2607.06652#bib.bib22)\], seismology\[[14](https://arxiv.org/html/2607.06652#bib.bib51)\], information diffusion\[[25](https://arxiv.org/html/2607.06652#bib.bib49)\], and health\[[51](https://arxiv.org/html/2607.06652#bib.bib48),[20](https://arxiv.org/html/2607.06652#bib.bib47)\]\. Neural TPPs \(NTPPs\) parametrise the conditional or interarrival density with expressive neural networks\[[16](https://arxiv.org/html/2607.06652#bib.bib6),[38](https://arxiv.org/html/2607.06652#bib.bib21),[43](https://arxiv.org/html/2607.06652#bib.bib62),[56](https://arxiv.org/html/2607.06652#bib.bib26)\], including continuous\-time state\-space architectures\[[5](https://arxiv.org/html/2607.06652#bib.bib54)\], and are often trained via log\-likelihood maximisation\. Yet, likelihood\-based training introduces fundamental constraints\[[49](https://arxiv.org/html/2607.06652#bib.bib15)\]\. It either restricts models to tractable likelihood families\[[48](https://arxiv.org/html/2607.06652#bib.bib24)\]or relies on expensive numerical approximations\[[38](https://arxiv.org/html/2607.06652#bib.bib21),[43](https://arxiv.org/html/2607.06652#bib.bib62)\], limiting both expressiveness and scalability\. This has led to a growing interest in alternative generative paradigms, such as diffusion\-based methods\[[55](https://arxiv.org/html/2607.06652#bib.bib16),[35](https://arxiv.org/html/2607.06652#bib.bib55)\], normalising flows\[[48](https://arxiv.org/html/2607.06652#bib.bib24)\], adversarial training\[[53](https://arxiv.org/html/2607.06652#bib.bib18)\], and reinforcement learning\[[29](https://arxiv.org/html/2607.06652#bib.bib17)\], with frameworks\[[32](https://arxiv.org/html/2607.06652#bib.bib12)\]unifying several of these perspectives\. At the same time, the signature transform from rough path theory\[[37](https://arxiv.org/html/2607.06652#bib.bib13),[19](https://arxiv.org/html/2607.06652#bib.bib7)\]has become central to time series modelling\[[6](https://arxiv.org/html/2607.06652#bib.bib4),[27](https://arxiv.org/html/2607.06652#bib.bib2),[11](https://arxiv.org/html/2607.06652#bib.bib35),[44](https://arxiv.org/html/2607.06652#bib.bib46)\]\. Acting as a universal feature map, it encodes pathwise information into a tensor series up to tree\-like equivalence\[[23](https://arxiv.org/html/2607.06652#bib.bib8),[37](https://arxiv.org/html/2607.06652#bib.bib13)\], while the expected signature becomes a moment generating function on path space\[[7](https://arxiv.org/html/2607.06652#bib.bib3),[8](https://arxiv.org/html/2607.06652#bib.bib5)\], supporting generative models such as the Signature\-Wasserstein GAN\[[40](https://arxiv.org/html/2607.06652#bib.bib14),[30](https://arxiv.org/html/2607.06652#bib.bib10)\]\.

Despite this activity, the current landscape has significant structural gaps\. The prevailing training objectives are local as they decompose into a sum of per\-event losses, each comparing one event to the ground truth conditional on preceding events\. This is also true for recent non\-autoregressive models as they fit reverse denoising transitions\[[33](https://arxiv.org/html/2607.06652#bib.bib56)\], transport fields\[[26](https://arxiv.org/html/2607.06652#bib.bib57)\], or edit rates at sampled intermediate states\[[34](https://arxiv.org/html/2607.06652#bib.bib52)\], rather than directly minimising a single global discrepancy between generated and empirical trajectories\. Consequently, global structural properties, such as autocorrelation, long\-range inter\-event dependencies, and joint timing distributions, receive no direct loss signal\. Furthermore, the evaluation methods for generative TPP models are underdeveloped\[[39](https://arxiv.org/html/2607.06652#bib.bib29),[49](https://arxiv.org/html/2607.06652#bib.bib15)\]\. Negative log\-likelihood \(NLL\) measures local conditional fit and is insensitive to global sequence structure\. Low NLL does not preclude mode collapse, misspecified tails, or distorted inter\-event dependencies\[[31](https://arxiv.org/html/2607.06652#bib.bib11),[32](https://arxiv.org/html/2607.06652#bib.bib12),[39](https://arxiv.org/html/2607.06652#bib.bib29)\]\. 1\-step\-ahead pointwise metrics \(e\.g\. mean absolute error\) share this limitation and additionally favour underdispersed predictions\[[41](https://arxiv.org/html/2607.06652#bib.bib28),[42](https://arxiv.org/html/2607.06652#bib.bib32),[59](https://arxiv.org/html/2607.06652#bib.bib33),[58](https://arxiv.org/html/2607.06652#bib.bib27),[57](https://arxiv.org/html/2607.06652#bib.bib30)\]\. A practical shift towards sequence\-level distributional losses, such as the maximum mean discrepancy \(MMD\) with a Gaussian kernel\[[48](https://arxiv.org/html/2607.06652#bib.bib24),[35](https://arxiv.org/html/2607.06652#bib.bib55)\]and the Wasserstein\-1 distance\[[33](https://arxiv.org/html/2607.06652#bib.bib56),[34](https://arxiv.org/html/2607.06652#bib.bib52)\], has emerged, yet whether these losses define meaningful distances on the space of counting paths has not been theoretically established\. Signatures would be a natural candidate to address the aforementioned shortcomings, as they handle naturally variable\-length sequences\. However, the determinacy and universal nonlinearity that underpin signature\-based generative modelling\[[7](https://arxiv.org/html/2607.06652#bib.bib3),[23](https://arxiv.org/html/2607.06652#bib.bib8),[8](https://arxiv.org/html/2607.06652#bib.bib5)\]are formulated for continuous paths of bounded variation\. TPP sample paths are càdlàg step functions, on which these guarantees fail\.

This paper attempts to address these limitations with the following contributions:

1. 1\.We introduce the interarrival embedding, a stable, injective lift to continuous paths of bounded\-variation, enabling signature methods on jump processes \(Section[3\.1](https://arxiv.org/html/2607.06652#S3.SS1)\)\.
2. 2\.Building on this embedding approach, we proposeSigTPP, a signature\-based generative model for TPPs that replaces per\-event objectives with a global signature loss on complete counting paths \(Section[3\.2](https://arxiv.org/html/2607.06652#S3.SS2)\)\.
3. 3\.We provide 3 theoretically grounded discrepancies on counting\-path distributions, namely the energy \(ℰ\\mathcal\{E\}\), Wasserstein\-1 \(W1\\mathrm\{W\}\_\{1\}\), and signature\-Wasserstein\-1 distances \(Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\), for evaluating generative TPP models \(Section[3\.4](https://arxiv.org/html/2607.06652#S3.SS4)\)\. We also showcase the utility of other diagnostic metrics for NTPP evaluation, including moment and histogram\-based summaries\.
4. 4\.Using synthetic and real\-world datasets, we demonstrate thatSigTPPis a strong generative model for TPPs\. It achieves the best relative score on 7 out of the 8 metrics\.SigTPPachieves the best performance or is within a standard error of the strongest baseline in64%64\\%of all metric\-dataset pairs \(Section[4](https://arxiv.org/html/2607.06652#S4)\)\.

## 2Preliminaries

Temporal Point Processes\[[12](https://arxiv.org/html/2607.06652#bib.bib19)\]\.Fix a finite horizonTmax∈ℝ\>0T\_\{\\max\}\\in\\mathbb\{R\}\_\{\>0\}\. A realisation of a TPP on\[0,Tmax\)\[0,T\_\{\\max\}\)is a finite sequence of strictly increasing event times,0<t1<⋯<tm<Tmax0<t\_\{1\}<\\cdots<t\_\{m\}<T\_\{\\max\}\. To define a metric on event sequences and study its topological properties, we work with the equivalent counting\-path representation\. Let𝒩\\mathcal\{N\}denote the set of nondecreasing, càdlàg functionsη:\[0,Tmax\]→ℕ\\eta\\colon\[0,T\_\{\\max\}\]\\to\\mathbb\{N\}withη0=0\\eta\_\{0\}=0,ηTmax<∞\\eta\_\{T\_\{\\max\}\}<\\infty, unit jumps only, and no jump atTmaxT\_\{\\max\}\.η\\etaand\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}are related byηt=\#​\{k:tk≤t\}\\eta\_\{t\}=\\\#\\\{k:t\_\{k\}\\leq t\\\}\. A TPP is an𝒩\\mathcal\{N\}\-valued random variableNNon a probability space\(Ω,ℱ,ℙ\)\(\\Omega,\\mathcal\{F\},\\mathbb\{P\}\)\. The conditional intensityλ​\(t∣ℋt\):=limΔ​t↓0Δ​t−1​𝔼​\[Nt\+Δ​t−Nt\|ℋt\]\\lambda\(t\\mid\\mathcal\{H\}\_\{t\}\):=\\lim\_\{\\Delta t\\downarrow 0\}\\Delta t^\{\-1\}\\,\\mathbb\{E\}\\bigl\[N\_\{t\+\\Delta t\}\-N\_\{t\}\\,\|\\,\\mathcal\{H\}\_\{t\}\\bigr\], whereℋt:=σ\(Ns:s<t\)\\mathcal\{H\}\_\{t\}:=\\sigma\(N\_\{s\}:s<t\), characterises the instantaneous rate of events given history\.

We adopt the following boundary convention throughout, since the definition ofΦ\\Phi\(Definition[4](https://arxiv.org/html/2607.06652#Thmtheorem4)\) and the stability theorem \(Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)\) rely on the same augmented event grid\.

###### Convention 1\(Boundary Convention\)\.

Forη∈𝒩\\eta\\in\\mathcal\{N\}with event times0<t1<⋯<tm<Tmax0<t\_\{1\}<\\cdots<t\_\{m\}<T\_\{\\max\}, we augment the sequence by settingt0:=0t\_\{0\}:=0andtm\+1:=Tmaxt\_\{m\+1\}:=T\_\{\\max\}, and define the corresponding interarrival timesτ0:=0\\tau\_\{0\}:=0,τk:=tk−tk−1\\tau\_\{k\}:=t\_\{k\}\-t\_\{k\-1\}wherek=1,…,mk=1,\\dots,m, andτm\+1:=Tmax−tm\\tau\_\{m\+1\}:=T\_\{\\max\}\-t\_\{m\}\.

Rough Path Signatures\[[37](https://arxiv.org/html/2607.06652#bib.bib13),[19](https://arxiv.org/html/2607.06652#bib.bib7)\]\.The signature transform provides an injective representation, up to tree\-like equivalence, of continuous paths of finite variation into the tensor algebra,T​\(\(ℝd\)\)T\(\(\\mathbb\{R\}^\{d\}\)\)\. We restrict attention to continuous paths of bounded variation \(as opposed to finitepp\-variation\)\.

###### Definition 2\(Signature\)\.

LetX:\[0,Tmax\]→ℝdX:\[0,T\_\{\\max\}\]\\to\\mathbb\{R\}^\{d\}be a continuous path of bounded variation\. The signature ofXX, denoted byS​\(X\)S\(X\), is defined as the infinite sequence of iterated integrals

S​\(X\)=\(1,S​\(X\)1,S​\(X\)2,…\),whereS​\(X\)k=∫0<u1<⋯<uk<Tmaxd​Xu1⊗⋯⊗d​Xuk\.S\(X\)=\(1,S\(X\)^\{1\},S\(X\)^\{2\},\\dots\),\\quad\\text\{where\}\\quad S\(X\)^\{k\}=\\int\_\{0<u\_\{1\}<\\cdots<u\_\{k\}<T\_\{\\max\}\}\\mathop\{\}\\\!\\mathrm\{d\}X\_\{u\_\{1\}\}\\otimes\\cdots\\otimes\\mathop\{\}\\\!\\mathrm\{d\}X\_\{u\_\{k\}\}\\,\.\(1\)SM​\(X\)S^\{M\}\(X\)denotes the truncated signature ofXXof degreeMM, i\.e\.SM​\(X\)=\(1,S​\(X\)1,…,S​\(X\)M\)\.S^\{M\}\(X\)=\(1,S\(X\)^\{1\},\\dots,S\(X\)^\{M\}\)\.

###### Theorem 3\(Determinacy\[[7](https://arxiv.org/html/2607.06652#bib.bib3)\]\)\.

LetXXandYYtwo continuous random paths of bounded variation\. If𝔼​\[S​\(X\)\]=𝔼​\[S​\(Y\)\]\\mathbb\{E\}\[S\(X\)\]=\\mathbb\{E\}\[S\(Y\)\]and𝔼​\[S​\(X\)\]\\mathbb\{E\}\[S\(X\)\]has infinite radius of convergence, thenX=YX=Yin distribution\.

## 3Signature\-based Generative TPPs \(SigTPP\)

The signature transform is defined on continuous paths of bounded variation, whereas counting paths in𝒩\\mathcal\{N\}are càdlàg step functions\. Our approach to addressing this rests on two components\. First, we introduce a deterministic embeddingΦ\\Phiinto the space𝒞:=C​\(\[0,Tmax\],ℝ\)\\mathcal\{C\}:=C\(\[0,T\_\{\\max\}\],\\mathbb\{R\}\)of continuous functions on\[0,Tmax\]\[0,T\_\{\\max\}\]\.Φ\\Phimaps a discrete event sequence to a continuous, piecewise\-linear path on which the signature transform applies \(Section[3\.1](https://arxiv.org/html/2607.06652#S3.SS1)\)\. Then, we train the proposedSigTPPon the Euclidean distance between the truncated expected signatures of the embedded model\-generated and observed counting paths \(Section[3\.2](https://arxiv.org/html/2607.06652#S3.SS2)\)\. We also provide theoretical guarantees ensuring the signature loss reflects genuine discrepancies between counting\-path distributions \(Section[3\.3](https://arxiv.org/html/2607.06652#S3.SS3)\), and we propose principled evaluation metrics for generative TPPs \(Section[3\.4](https://arxiv.org/html/2607.06652#S3.SS4)\)\.

### 3\.1The Interarrival EmbeddingΦ\\Phi

###### Definition 4\(Interarrival Embedding\)\.

Forη∈𝒩\\eta\\in\\mathcal\{N\}withmmevents, let0=:t0<t1<⋯<tm<tm\+1:=Tmax0=:t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\}<t\_\{m\+1\}:=T\_\{\\max\}be the augmented grid andτk:=tk−tk−1\\tau\_\{k\}:=t\_\{k\}\-t\_\{k\-1\}the interarrival times\. The interarrival embedding is the continuous, bounded\-variation, piecewise\-linear functionx:=Φ​\(η\)∈𝒞x:=\\Phi\(\\eta\)\\in\\mathcal\{C\}satisfying

x​\(tk\)\\displaystyle x\(t\_\{k\}\)=τkfor​k=0,…,m\+1,and\\displaystyle=\\tau\_\{k\}\\quad\\text\{ for \}k=0,\\dots,m\{\+\}1\\,,\\text\{ and\}\(2\)Φ​\(η\)t\\displaystyle\\Phi\(\\eta\)\_\{t\}=τk\+t−tktk\+1−tk​\(τk\+1−τk\),where​t∈\[tk,tk\+1\],k=0,…,m\.\\displaystyle\\;=\\;\\tau\_\{k\}\+\\frac\{t\-t\_\{k\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\\bigl\(\\tau\_\{k\+1\}\-\\tau\_\{k\}\\bigr\)\\,,\\quad\\text\{where \}t\\in\[t\_\{k\},t\_\{k\+1\}\],\\ k=0,\\dots,m\\,\.

![Refer to caption](https://arxiv.org/html/2607.06652v1/x1.png)Figure 1:Illustration of the interarrival embeddingΦ\\Phi\(Definition[4](https://arxiv.org/html/2607.06652#Thmtheorem4)\) for a sampled Poisson process withλ=1\\lambda=1andTmax=7T\_\{\\max\}=7\.Equivalently,Φ​\(η\)\\Phi\(\\eta\)is the unique continuous function that is affine on every interarrival time and satisfiesΦ​\(η\)tk=τk\\Phi\(\\eta\)\_\{t\_\{k\}\}\\;=\\;\\tau\_\{k\},k≤m\+1k\\leq m\+1\. Figure[1](https://arxiv.org/html/2607.06652#S3.F1)illustrates this on a Poisson sample: the counting process \(blue, càdlàg\) records cumulative event counts, the marks \(orange\) record the duration of the most recent interarrival interval, andΦ​\(η\)\\Phi\(\\eta\)\(green dashed line\) is the piecewise\-linear interpolant of this staircase on the augmented event grid\. The embedded path is continuous and of bounded variation, satisfying the assumptions required by the signature transform, see Definition[2](https://arxiv.org/html/2607.06652#Thmtheorem2)\. In practice, we apply the signature transform not toΦ​\(η\)\\Phi\(\\eta\)alone but to the time\-augmented curvet↦\(t,Φ​\(η\)t\)t\\mapsto\(t,\\Phi\(\\eta\)\_\{t\}\)inℝ2\\mathbb\{R\}^\{2\}\. The time augmentation encodes both absolute event times and interarrival durations, the two standard input features of NTPPs\[[16](https://arxiv.org/html/2607.06652#bib.bib6),[32](https://arxiv.org/html/2607.06652#bib.bib12)\], in a single curve\. Theoretically, it removes tree\-like components from the path, so that the signature is injective\[[23](https://arxiv.org/html/2607.06652#bib.bib8)\]\. Note that other interpolation schemes are possible \(e\.g\. step function\), but linear interpolation is the standard choice in signature\-based methods\[[6](https://arxiv.org/html/2607.06652#bib.bib4)\], and it is the simplest scheme yielding continuous paths\. The stability of the mapΦ\\Phiand a constructive inverseΨ:=Φ−1\\Psi:=\\Phi^\{\-1\}are established in Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)\.

### 3\.2Model Optimisation

Our goal is to train a parametric generatorμθ\\mu\_\{\\theta\}whose output distribution faithfully reproduces the unknown data distributionν\\nu, without specifying a parametric intensity model\. A natural loss between distributionsμθ\\mu\_\{\\theta\}andν\\nuis the Wasserstein\-1 distance used in Wasserstein GAN models\[[53](https://arxiv.org/html/2607.06652#bib.bib18)\], expressible via Kantorovich\-Rubinstein duality as

W1​\(μθ,ν\)=sup‖f‖Lip≤1𝔼x∼μθ​\[f​\(x\)\]−𝔼y∼ν​\[f​\(y\)\]\.\\mathrm\{W\}\_\{1\}\(\\mu\_\{\\theta\},\\nu\)=\\sup\_\{\\left\\lVert f\\right\\rVert\_\{\\mathrm\{Lip\}\}\\leq 1\}\\mathbb\{E\}\_\{x\\sim\\mu\_\{\\theta\}\}\\bigl\[f\(x\)\\bigr\]\-\\mathbb\{E\}\_\{y\\sim\\nu\}\\bigl\[f\(y\)\\bigr\]\\,\.\(3\)Approximating the supremum over 1\-Lipschitz functions often requires training a neural network discriminator with gradient penalties or weight clipping\. This adversarial procedure can lead to training instability and convergence issues\[[2](https://arxiv.org/html/2607.06652#bib.bib1)\], especially when modelling TPPs\[[53](https://arxiv.org/html/2607.06652#bib.bib18)\]\. Recent work on generative modelling of continuous paths proposes replacing the adversarial discriminator with a signature\-based metric, denoted bySig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\[[40](https://arxiv.org/html/2607.06652#bib.bib14),[30](https://arxiv.org/html/2607.06652#bib.bib10)\]\. By leveraging the universal nonlinearity signature property, any 1\-Lipschitz test function can be approximated by a linear functional on the signature\. Consequently, the optimisation over the infinite\-dimensional function space reduces to a linear operation on the \(truncated for numerical purposes\) tensor algebraTM​\(ℝd\)T^\{M\}\(\\mathbb\{R\}^\{d\}\)\. Given that TPP sample paths are càdlàg step functions, we first lift them with the interarrival embeddingΦ\\Phiand then apply the signature loss on the embedded path space\. Hence, the loss for optimising the proposed model,SigTPP, is given by:

Sig​\-​W1\(M\)​\(μθ,ν\):=‖𝔼Nθ∼μθ​\[SM​\(Φ​\(Nθ\)\)\]−𝔼N∼ν​\[SM​\(Φ​\(N\)\)\]‖2\.\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}\(\\mu\_\{\\theta\},\\nu\):=\\left\\lVert\\mathbb\{E\}\_\{N\_\{\\theta\}\\sim\\mu\_\{\\theta\}\}\\bigl\[S^\{M\}\(\\Phi\(N\_\{\\theta\}\)\)\\bigr\]\-\\mathbb\{E\}\_\{N\\sim\\nu\}\\bigl\[S^\{M\}\(\\Phi\(N\)\)\\bigr\]\\right\\rVert\_\{2\}\\,\.\(4\)Here∥⋅∥2\\left\\lVert\\cdot\\right\\rVert\_\{2\}denotes theℓ2\\ell^\{2\}norm on the truncated tensor algebraTM​\(ℝ2\)T^\{M\}\(\\mathbb\{R\}^\{2\}\)\. The training procedure is summarised in Figure[2](https://arxiv.org/html/2607.06652#S3.F2)\. The expected signature under the data measureν\\nuis precomputed once on the training set; each gradient step requires only computing signatures underμθ\\mu\_\{\\theta\}\.

N∼μθN\\sim\\mu\_\{\\theta\}![Refer to caption](https://arxiv.org/html/2607.06652v1/x2.png)Φ\\Phi![Refer to caption](https://arxiv.org/html/2607.06652v1/x3.png)SM\\mathrm\{S\}^\{M\}S1S\_\{1\}S2S\_\{2\}SMS\_\{M\}𝔼\\mathbb\{E\}Sig​\-​W1\(M\)​\(μθ,ν\)\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}\(\\mu\_\{\\theta\},\\nu\)TM​\(ℝ2\)T^\{M\}\\\!\\bigl\(\\mathbb\{R\}^\{2\}\\bigr\)∇θ\\nabla\_\{\\theta\}backpropagation throughμθ\\mu\_\{\\theta\}

Figure 2:SigTPP’s optimisation\. A counting path sampled fromμθ\\mu\_\{\\theta\}is embedded withΦ\\Phito a continuous path, mapped to its truncated signature inTM​\(ℝ2\)T^\{M\}\(\\mathbb\{R\}^\{2\}\), and summarised by its expectation under the model\. The loss \(Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\) compares this against the expected signature of the embedded targets’ paths, opposed to the usual per\-event loss\. Loss gradient flows back throughμθ\\mu\_\{\\theta\}\.SigTPPis a 1\-step autoregressive generator that follows the standard NTPP design\[[32](https://arxiv.org/html/2607.06652#bib.bib12)\]\. A single\-layer LSTM encodes the history of tunable hidden size \(HH\) withh0h\_\{0\}\. Each log\-interarrival time is first mapped to a 64\-dimensional learnable trigonometric embedding and processed recurrently to produce the history statehih\_\{i\}\. The decoder is a shallow perceptron and predictslog⁡τi\+1\\log\\tau\_\{i\+1\}from the recurrent statehih\_\{i\}, the current cumulative time and an exogenous noise variableε∼log⁡\(Exp​\(1\)\)\\varepsilon\\sim\\log\(\\mathrm\{Exp\}\(1\)\)\.

### 3\.3Theoretical Guarantees

To analyse the stability of the embeddingΦ\\Phiand to quantify discrepancies between counting\-path distributions, we propose to work with the following metric\.

###### Definition 5\(Counting Path Metric\)\.

We define the pathwise TPP metric between two counting processesη,ξ∈𝒩\\eta,\\xi\\in\\mathcal\{N\}on\[0,Tmax\)\[0,T\_\{\\max\}\)byd𝒩​\(η,ξ\):=∫0Tmax\|ηt−ξt\|​d​td\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\):=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert\\eta\_\{t\}\-\\xi\_\{t\}\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\.

Given that counting paths are càdlàg, distinct event setsη≠ξ\\eta\\neq\\xiyield paths that differ on an interval of positive Lebesgue measure\. Hence,d𝒩​\(η,ξ\)=0d\_\{\\mathcal\{N\}\}\(\\eta,\\xi\)=0impliesη=ξ\\eta=\\xi, andd𝒩d\_\{\\mathcal\{N\}\}is a genuine metric on𝒩\\mathcal\{N\}\(see also Proposition[S3](https://arxiv.org/html/2607.06652#Thmtheorem3a)\)\. As shown in Appendix[A\.3](https://arxiv.org/html/2607.06652#A1.SS3),d𝒩d\_\{\\mathcal\{N\}\}coincides with the Wasserstein\-1 distance between event\-time point measures padded with Dirac masses atTmaxT\_\{\\max\}\. This result puts the optimal\-transport perspective ofXiaoet al\.\[[53](https://arxiv.org/html/2607.06652#bib.bib18)\]on a rigorous footing, yielding a genuine metric on𝒩\\mathcal\{N\}\. The space𝒩\\mathcal\{N\}is not a linear subspace ofL1​\(\[0,Tmax\]\)L^\{1\}\(\[0,T\_\{\\max\}\]\)\. The differenceηt−ξt\\eta\_\{t\}\-\\xi\_\{t\}is signed, scalar multiples of counting paths are not counting paths, and the sum or difference of two counting paths need not have unit jumps\. The metric structure is the natural level at which to study stability\. Consequently, standard tools from linear and functional analysis are not directly available\. Most notably, there is no differentiable structure on which to apply an inverse function theorem, no open mapping theorem to promote a continuous bijection to a bi\-Lipschitz one, and no compact embedding to fall back on since event counts are unbounded\. We therefore work throughout withL1L^\{1\}\-type metrics on both𝒩\\mathcal\{N\}and𝒞\\mathcal\{C\}, which is also natural for signature\-based losses that are computed through integrals\.

Hence, the embedding chain from counting paths to the tensor algebra is given by:

\(𝒩,d𝒩\)→Φ\(𝒞,d1\)→t↦\(t,Φ​\(η\)t\)C​\(\[0,Tmax\],ℝ2\)→SignatureT​\(\(ℝ2\)\),\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)\\xrightarrow\{\\;\\Phi\\;\}\(\\mathcal\{C\},d\_\{1\}\)\\xrightarrow\{\\;\\mathrm\{t\\mapsto\(t,\\Phi\(\\eta\)\_\{t\}\)\}\\;\}C\(\[0,T\_\{\\max\}\],\\mathbb\{R\}^\{2\}\)\\xrightarrow\{\\;\\mathrm\{Signature\}\\;\}T\(\(\\mathbb\{R\}^\{2\}\)\)\\,,\(5\)where𝒩\\mathcal\{N\}is equipped with a metricd𝒩d\_\{\\mathcal\{N\}\}and\(𝒞,d1\)\(\\mathcal\{C\},d\_\{1\}\)denotes the space of continuous paths\[0,Tmax\]→ℝ\[0,T\_\{\\max\}\]\\to\\mathbb\{R\}equipped with theL1L^\{1\}metricd1​\(x,y\):=∫0Tmax\|x​\(t\)−y​\(t\)\|​d​td\_\{1\}\(x,y\):=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert x\(t\)\-y\(t\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\. Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)establishes thatΦ\\Phiis Lipschitz and injective, with a controlled inverse under mild assumptions\. The Lipschitz bound ensures that the signature loss \(Eq\.[4](https://arxiv.org/html/2607.06652#S3.E4)\) inherits regularity from the data distribution\. Closeness in counting\-path space implies closeness of expected signatures\. Injectivity guarantees that no event information is lost in the embedding, which is what allowsSig​\-​W1\(M\)\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}to separate distinct laws on𝒩\\mathcal\{N\}\. Finally, the Hölder bound gives the converse direction under a mild separation assumption that is satisfied by any temporal data recorded at finite precision\.

###### Theorem 6\(Stability of the Embedding; full proofs in Appendix[B](https://arxiv.org/html/2607.06652#A2)\)\.

The embeddingΦ:\(𝒩,d𝒩\)→\(𝒞,d1\)\\Phi\\colon\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)\\to\(\\mathcal\{C\},d\_\{1\}\)satisfies the following properties:

1. 1\.Lipschitz continuity\.For allη,ξ∈𝒩\\eta,\\xi\\in\\mathcal\{N\},d1​\(Φ​\(η\),Φ​\(ξ\)\)≤5⋅Tmax⋅d𝒩​\(η,ξ\)d\_\{1\}\\bigl\(\\Phi\(\\eta\),\\Phi\(\\xi\)\\bigr\)\\leq 5\\cdot T\_\{\\max\}\\cdot d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\. Hence, it is continuous and measurable\(Appendix[B\.1](https://arxiv.org/html/2607.06652#A2.SS1), Theorem[S11](https://arxiv.org/html/2607.06652#Thmtheorem11)\)\. The constant55is not tight but sufficient\.
2. 2\.Bijection\.Φ\\Phiis injective on𝒩\\mathcal\{N\}, hence a bijectionΦ:𝒩→∼Φ​\(𝒩\)\\Phi\\colon\\mathcal\{N\}\\xrightarrow\{\\sim\}\\Phi\(\\mathcal\{N\}\)\. The inverseΨ:=Φ−1\\Psi:=\\Phi^\{\-1\}is recovered constructively by a level\-set recursion\(Appendix[B\.2](https://arxiv.org/html/2607.06652#A2.SS2), Theorem[S16](https://arxiv.org/html/2607.06652#Thmtheorem16)\)\.

Moreover, on counting paths whose interarrival timesτ1,…,τm\+1\\tau\_\{1\},\\dots,\\tau\_\{m\+1\}\(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\) are bounded below by any fixedδ\>0\\delta\>0, the inverseΨ\\Psiis Hölder continuous of exponent1/21/2\(see also Appendix[B\.3](https://arxiv.org/html/2607.06652#A2.SS3), Theorem[S29](https://arxiv.org/html/2607.06652#Thmtheorem29)\)\.

### 3\.4Evaluation Metrics for Counting\-Path Distributions

Evaluating generative TPPs requires discrepancy measures between distributions on𝒩\\mathcal\{N\}\. Current practice borrows tools from continuous\-data settings, e\.g\. MMD with Gaussian kernels\[[48](https://arxiv.org/html/2607.06652#bib.bib24),[35](https://arxiv.org/html/2607.06652#bib.bib55)\]orW1\\mathrm\{W\}\_\{1\}on event times\[[33](https://arxiv.org/html/2607.06652#bib.bib56),[34](https://arxiv.org/html/2607.06652#bib.bib52)\], without verifying that these are valid distances on counting\-path space\. We show that the metricd𝒩d\_\{\\mathcal\{N\}\}gives rise to mathematically valid distributional discrepancies on𝒩\\mathcal\{N\}\.

Throughout this section, uppercaseP,Q∈𝒫1​\(𝒩\)P,Q\\in\\mathcal\{P\}\_\{1\}\(\\mathcal\{N\}\)denote probability distributions over counting paths𝒩\\mathcal\{N\}with finite firstd𝒩d\_\{\\mathcal\{N\}\}\-moment\. The metricd𝒩d\_\{\\mathcal\{N\}\}acts at the path level, while the distribution\-level discrepanciesℰ\\mathcal\{E\},W1\\mathrm\{W\}\_\{1\}, andSig​\-​W1\(M\)\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}introduced in this section act withd𝒩d\_\{\\mathcal\{N\}\}as ground cost\.

Energy distance \(ℰ\\mathcal\{E\}\)\.The energy distance \(ℰ\\mathcal\{E\}\) is a standard metric induced by an underlying distance function\[[21](https://arxiv.org/html/2607.06652#bib.bib38),[50](https://arxiv.org/html/2607.06652#bib.bib36)\], see Appendix[C](https://arxiv.org/html/2607.06652#A3)\. Nonnegativity ofℰ\\mathcal\{E\}requires that\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)be of negative type\[[50](https://arxiv.org/html/2607.06652#bib.bib36)\]\. The mapη↦\(t↦ηt\)\\eta\\mapsto\(t\\mapsto\\eta\_\{t\}\)is an isometry from\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)intoL1​\(\[0,Tmax\]\)L^\{1\}\(\[0,T\_\{\\max\}\]\), which is of negative type, and the property is inherited by subspaces \(Appendix[A\.1](https://arxiv.org/html/2607.06652#A1.SS1)\)\. HenceD:=ℰD:=\\sqrt\{\\mathcal\{E\}\}is a well\-defined discrepancy\. We further show in Appendix[A\.4](https://arxiv.org/html/2607.06652#A1.SS4)thatd𝒩d\_\{\\mathcal\{N\}\}is not of strong negative type, soDDis a pseudometric rather than a metric\. For the purpose of ranking models, this is sufficient\.

Wasserstein\-1 distance \(W1\\mathrm\{W\}\_\{1\}\)\.Since\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)is a metric space, the Wasserstein\-1 distance\[[46](https://arxiv.org/html/2607.06652#bib.bib58),[52](https://arxiv.org/html/2607.06652#bib.bib37)\]is a well\-defined metric on𝒫1​\(𝒩\)\\mathcal\{P\}\_\{1\}\(\\mathcal\{N\}\)\. We recall its definition in Appendix[C](https://arxiv.org/html/2607.06652#A3)\.

Signature discrepancy \(Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\)\.The lossSig​\-​W1\(M\)\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}\(Eq\.[4](https://arxiv.org/html/2607.06652#S3.E4)\) can also be used as an evaluation metric in the continuous setting and we adapt it to TPPs with our embedding\. SinceΦ\\Phiis injective and measurable \(Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)\), distinct lawsP≠QP\\neq Qon𝒩\\mathcal\{N\}induce distinct pushforwardsΦ♯​P≠Φ♯​Q\\Phi\_\{\\sharp\}P\\neq\\Phi\_\{\\sharp\}Q\. If these pushforward laws satisfy the moment condition of Theorem[3](https://arxiv.org/html/2607.06652#Thmtheorem3), then the expected signature determines the law, and hence𝔼η∼P​\[S​\(Φ​\(η\)\)\]≠𝔼ξ∼Q​\[S​\(Φ​\(ξ\)\)\]\\mathbb\{E\}\_\{\\eta\\sim P\}\[S\(\\Phi\(\\eta\)\)\]\\neq\\mathbb\{E\}\_\{\\xi\\sim Q\}\[S\(\\Phi\(\\xi\)\)\]\. Consequently, for any such pairP≠QP\\neq Q, there exists a finite truncation levelMMsuch thatSig​\-​W1\(M\)​\(P,Q\)\>0\.\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}\(P,Q\)\>0\.For each fixedMM,Sig​\-​W1\(M\)\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}^\{\(M\)\}, being anℓ2\\ell^\{2\}distance between expected truncated signatures, is symmetric and satisfies the triangle inequality, and is therefore a pseudometric\.

Pointwise Losses\.As mentioned in the introduction, pointwise losses fail in a quantifiable, easily reproducible way on the very benchmarks the community uses to evaluate generative TPPs\. For an interarrival distributionPPwith finite first moment, MAE is minimised byτ^=median​\(P\)\\hat\{\\tau\}=\\mathrm\{median\}\(P\)and MSE byτ^=𝔼​\[P\]\\hat\{\\tau\}=\\mathbb\{E\}\[P\]\[[21](https://arxiv.org/html/2607.06652#bib.bib38)\]\. Both optima are Dirac measures\. Neither can, in principle, distinguish a calibrated generative model from a deterministic regressor\. See Table[S3](https://arxiv.org/html/2607.06652#A4.T3)for an empirical confirmation\.

## 4Experimental Results

We evaluateSigTPPon controlled synthetic regimes and real\-world tasks\. We assess performance across multiple axes, from path\-level distributional fidelity to moment structure, and examine the reliability of pointwise metrics as evaluation criteria\. We further study the dependence ofSigTPPon the signature truncation degree and its generation cost relative to baselines\.

### 4\.1Datasets, Baselines, and Evaluation Metrics

We use 9 datasets in total, 4 synthetic and 5 real\-world\. Table[S1](https://arxiv.org/html/2607.06652#A4.T1)and Figures[S2](https://arxiv.org/html/2607.06652#A4.F2)\-[S10](https://arxiv.org/html/2607.06652#A4.F10)summarise their basic characteristics\. The synthetic corpus covers the canonical regimes of TPP modelling: Poisson process \(PS\), inhomogeneous Poisson \(IP\), univariate Hawkes process \(H1\), and 3\-dimensional Hawkes process \(H3\)\. The real\-world corpus encompasses datasets from prior work\[[54](https://arxiv.org/html/2607.06652#bib.bib31),[34](https://arxiv.org/html/2607.06652#bib.bib52)\], namely Earthquake \(EQ\), Stack Overflow \(SO\), Taobao \(TB\), Taxi \(TX\), and Yelp \(YLP\)\.

We compareSigTPPagainst 5 baselines spanning the main paradigms for generative TPP modelling\. We focus on autoregressive baselines that have a neural architecture similar toSigTPP\(similar amount of parameters\) to make comparisons fair\. These include the denoising diffusion probabilistic model \(DDPM\) and variational autoencoder \(VAE\)\[[32](https://arxiv.org/html/2607.06652#bib.bib12)\], the Wasserstein GAN \(WGAN\)\[[53](https://arxiv.org/html/2607.06652#bib.bib18)\], and a maximum\-likelihood Gamma fit\[[9](https://arxiv.org/html/2607.06652#bib.bib43),[10](https://arxiv.org/html/2607.06652#bib.bib44)\]as a parametric reference \(Gamma\)\. To isolate the contribution of distributional modelling, we also includeDeter\[[32](https://arxiv.org/html/2607.06652#bib.bib12)\], a deterministic regressor whose predictive law is a Dirac measure and which by construction cannot capture distributional structure\.

We evaluate the goodness\-of\-fit along the following axes: pathwise discrepancies viaℰ\\mathcal\{E\}\(energy distance\) andW1\\mathrm\{W\}\_\{1\}under our proposed metricd𝒩d\_\{\\mathcal\{N\}\}, signature\-based discrepancy viaSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}, pointwise probabilistic accuracy via the continuous ranked probability score \(CRPS\), marginal and temporal structure via histograms of interarrival times \(Llog⁡\(τ\)L\_\{\\log\(\\tau\)\}\) and intensity \(LλL\_\{\\lambda\}\), pairwise correlation discrepancy \(PCD\), and autocorrelation discrepancy \(ACD\) across sequence components\. Lower is better for all reported metrics\. Metric definitions and a more expansive justification for their appropriateness are deferred to Appendix[C](https://arxiv.org/html/2607.06652#A3)\. Results in tables report the mean test performance of each model, with standard errors in parentheses, computed as the sample standard deviation over100100bootstrap replicate estimates\.

For each dataset, hyperparameters are selected on the validation split using a rank\-based aggregate criterion over the validation diagnostics\. The criterion sums within\-dataset ranks across the ranked validation metrics\. During training, checkpoints are selected by the validationLlog⁡\(τ\)L\_\{\\log\(\\tau\)\}\. The aggregate rank criterion is used only for model selection and is not reported as a performance measure\.

Since evaluation metrics can differ substantially in scale across datasets, we report the relative score computed as a 2\-step process \(see also\[[1](https://arxiv.org/html/2607.06652#bib.bib69),[13](https://arxiv.org/html/2607.06652#bib.bib70),[22](https://arxiv.org/html/2607.06652#bib.bib71)\]\)\. First, we compute, for each model, a relative score defined as the ratio between the model’s score and that of a baseline model, hereDeter\. We then aggregate these relative scores across tasks using the geometric mean\[[17](https://arxiv.org/html/2607.06652#bib.bib68)\], which is the appropriate summary statistic for multiplicative relative comparisons, and make the model ordering independent of the choice of the baseline \(see Appendix[C\.4](https://arxiv.org/html/2607.06652#A3.SS4)\)\.

### 4\.2Recovering Known Synthetic Processes

Table 1:Performance comparison ofSigTPPagainst other methods on synthetic TPP tasks \(PS, IP, H1, H3\) using several discrepancy metrics \(see Appendix[C](https://arxiv.org/html/2607.06652#A3)for metric definitions\)\. Values report standard error in parentheses over 100 bootstrap replicates\. Lower is better\(↓\)\(\\downarrow\)\. The best result for each dataset and metric pair isbolded, and the second\-best isunderlined\.We first use synthetic point processes with known data\-generating mechanisms to evaluateSigTPP\. Table[1](https://arxiv.org/html/2607.06652#S4.T1)presents the discrepancy metricsℰ\\mathcal\{E\}andW1\\mathrm\{W\}\_\{1\}underd𝒩d\_\{\\mathcal\{N\}\},Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}, CRPS,Llog⁡\(τ\)L\_\{\\log\(\\tau\)\},LλL\_\{\\lambda\}, ACD, and PCD\. Our first observation is thatSigTPPimproves distributional fidelity on𝒩\\mathcal\{N\}relative to the strongest existing baselines \(see also Section[3\.4](https://arxiv.org/html/2607.06652#S3.SS4)\)\.SigTPPobtains the lowestℰ\\mathcal\{E\}andW1\\mathrm\{W\}\_\{1\}on all datasets \(or 3 out of 4 forSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\)\. The relative score of our model is the lowest underℰ\\mathcal\{E\},W1\\mathrm\{W\}\_\{1\}andSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}, and, compared to the second\-best model \(VAE\),SigTPPimproves the performance by86%86\\%,18%18\\%and39%39\\%, respectively \(see Figure[S11](https://arxiv.org/html/2607.06652#A4.F11)and Table[S6](https://arxiv.org/html/2607.06652#A4.T6)\)\. These gains are consistent withSigTPP’s training objective, which targets the joint temporal law rather than local event\-level behaviour\.

However, it is also striking that whileSigTPPimproves the distributional fit, it underperforms on the CRPS \(by approximately6\.5%6\.5\\%compared to VAE and DDPM\)\. This suggests that optimising a path\-level distributional objective favours global distributional alignment, but does not necessarily improve conditional calibration relative to methods trained directly with conditional losses\. Nevertheless, a number of summary\-statistic discrepancies indicate thatSigTPPremains competitive\.Llog⁡\(τ\),Lλ,ACDL\_\{\\log\(\\tau\)\},L\_\{\\lambda\},\\mathrm\{ACD\}, andPCD\\mathrm\{PCD\}exhibit larger sampling variability, and many methods yield overlapping standard errors\.SigTPPis either the best method or within a standard error of the best baseline in 13 out of 16 metric\-dataset pairs: in3​\(\+1\)/43\(\+1\)/4datasets forLlog⁡\(τ\)L\_\{\\log\(\\tau\)\},2​\(\+1\)/42\(\+1\)/4forLλL\_\{\\lambda\},0​\(\+2\)/40\(\+2\)/4for ACD, and1​\(\+3\)/41\(\+3\)/4for PCD, where the number in parentheses denotes outcomes within a standard error of the best\. When ranking methods separately for each dataset and metric,SigTPPhas the best average rank of2\.02\.0, followed by VAE with an average rank of2\.42\.4\(Table[S5](https://arxiv.org/html/2607.06652#A4.T5)\)\. Figure[3](https://arxiv.org/html/2607.06652#S4.F3)\(a\) showcases this interpretation for H3, a representative challenging synthetic task\. The QQ plot shows thatSigTPPmatches the empirical interarrival\-time distribution, including the upper tail where VAE and DDPM deviate visibly\. The qualitative summaries in Figure[S13](https://arxiv.org/html/2607.06652#A4.F13)further demonstrate thatSigTPPcaptures the overall intensity profile, as well as the correlation and autocorrelation structures\. These visual diagnostics are consistent with the favourableLlog⁡\(τ\)L\_\{\\log\(\\tau\)\}andLλL\_\{\\lambda\}results, since the former reflects interarrival time fit and the latter agreement in the intensity shape\. Overall, across all synthetic metric\-dataset pairs,SigTPPis best in17/3217/32cases and within a standard error of the best in another8/328/32cases, making it competitive in about80%80\\%of them\. This is confirmed by the relative score whereSigTPPhas the best score on 5 out of 8 metrics, the second lowest on 2, and is only outperformed when looking at CRPS\. In pairwise comparisons against each baseline, averaged over datasets and metrics,SigTPPoutperforms WGAN by52%52\\%, VAE by34%34\\%, and DDPM by48%48\\%\.

![Refer to caption](https://arxiv.org/html/2607.06652v1/x4.png)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x5.png)

\(a\)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x6.png)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x7.png)

\(b\)

Figure 3:\(a\) Quantile\-quantile \(QQ\) plots \(generated vs\. reference sample quantiles on x\-axis and y\-axis, respectively\) for all models on synthetic H3\. \(b\) Empirical fit on the real\-world task SO forSigTPP, WGAN, and DDPM\. Top: histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale\. Bottom: empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)from the same sequences\.
### 4\.3Real\-World Tasks

Table 2:Performance comparison ofSigTPPagainst other methods on real\-world TPP tasks \(EQ, SO, TB, TX, YLP\) using several discrepancy metrics \(see Appendix[C](https://arxiv.org/html/2607.06652#A3)for metric definitions\)\. Values report standard error in parentheses over 100 bootstrap replicates\. Lower is better\(↓\)\(\\downarrow\)\. The best result for each dataset and metric pair isbolded, and the second\-best isunderlined\.We now focus our empirical analysis on real\-world tasks under the same discrepancy metrics\. The results are enumerated in Table[2](https://arxiv.org/html/2607.06652#S4.T2)and, again, indicate strong performance forSigTPP\(see Table[S6](https://arxiv.org/html/2607.06652#A4.T6)\)\. It achieves the bestW1\\mathrm\{W\}\_\{1\}andSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}on4/54/5datasets\. This is confirmed by the relative scores, for which it obtains the lowestℰ\\mathcal\{E\},W1\\mathrm\{W\}\_\{1\}, andSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}among all models\. The second\-best model on real\-world tasks is VAE, which is outperformed on geometric average by6%6\\%across all metrics and datasets\. The contrast with the synthetic setting is informative\. Synthetic data is generated by a known mechanism, so distributional recovery is assessed under correct specification\. The real\-world datasets instead combine heterogeneous trajectories, variable observation windows, and latent contextual effects, so the empirical path distribution is only an approximation to a single data\-generating law\. This makes path\-level comparison intrinsically harder and likely explains why the gains ofSigTPPare smaller and less uniform than in the synthetic experiments\. Among the distributional metrics,W1\\mathrm\{W\}\_\{1\}appears to be the most stable across datasets, showing the smallest inter\-model variability and the most consistent advantage forSigTPP\.

The remaining summary\-statistic discrepancies provide a more nuanced comparison\. As on the synthetic datasets,Llog⁡\(τ\)L\_\{\\log\(\\tau\)\},LλL\_\{\\lambda\}, ACD, and PCD exhibit larger sampling variability, and several methods have overlapping standard errors\. Across these metrics,SigTPPis the best method in 7/20 metric\-dataset pairs and lies within a standard error of the best method in an additional 3/20 cases, covering half of all pairs\. Thus, although CRPS consistently favours the conditional baselines, withSigTPPtrailing VAE, WGAN, and DDPM by19%19\\%,11%11\\%, and21%21\\%, respectively, it remains competitive on real\-world data\.

Aggregating across datasets and metrics provides a similar qualitative conclusion\. In pairwise comparisons,SigTPPimproves over WGAN by25%25\\%, VAE by6%6\\%, and DDPM by30%30\\%\. It has the lowest relative score on 5 out of 8 metrics and is second\-best on 2, being outperformed only on CRPS\. Figure[3](https://arxiv.org/html/2607.06652#S4.F3)\(b\) supports this interpretation on SO, a commonly studied task in the literature\. The interarrival times PDF shows thatSigTPPcaptures the overall distribution shape without overweighting the tail as DDPM does\. The intensity profile is also captured accurately\. Appendix Figures[S14](https://arxiv.org/html/2607.06652#A4.F14)\-[S18](https://arxiv.org/html/2607.06652#A4.F18)provide dataset\-level qualitative comparisons of the 6 models and further illustrate these trends\. Average ranks corroborate these observations \(Table[S5](https://arxiv.org/html/2607.06652#A4.T5)\)\.SigTPPachieves the best average rank of2\.252\.25, followed by VAE \(2\.452\.45\)\. Relative scoring reaffirms these outcomes \(see Figure[S12](https://arxiv.org/html/2607.06652#A4.F12)\)\. Overall, the real\-world results mirror the synthetic results, but with smaller margins and greater dataset\-level variability\. The persistence of improvements inW1\\mathrm\{W\}\_\{1\},Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}, pairwise comparisons, and average ranks suggests that the signature\-based objective continues to provide useful distributional alignment under misspecification\. We provide the relative score plot in Figure[4](https://arxiv.org/html/2607.06652#S4.F4)for a comparison across all datasets\.

Finally, Table[S3](https://arxiv.org/html/2607.06652#A4.T3)illustrates why pointwise errors are not suitable primary metrics for generative TPP evaluation\. The deterministic baselineDeteroutputs a single prediction for each event time and is therefore directly optimised for pointwise accuracy rather than distributional fidelity\. Although it performs poorly on most path\-level and distributional metrics, in some cases by several orders of magnitude, it achieves rank1\.01\.0on MAE, outperforming every distributional model on every synthetic dataset\. Even when using the optimal\-predictor variants MAEmedand MSEmean\(see definitions in Appendix[C](https://arxiv.org/html/2607.06652#A3)\),Deterremains highly competitive, with average ranks of2\.42\.4on real\-world tasks and3\.53\.5on synthetic tasks for MAEmed, despite producing no predictive distribution\. These results show that MAE and MSE on event times primarily assess point prediction quality, not generative quality\. Hence, we do not use them as metrics for generative TPP evaluation, and instead recommend path\-level distributional distances\. For forecasting\-focused tasks, longer\-horizon predictive metrics may also be appropriate\.

![Refer to caption](https://arxiv.org/html/2607.06652v1/x8.png)Figure 4:Relative scores with respect toDeter, on all datasets \(see Table[S6](https://arxiv.org/html/2607.06652#A4.T6)for the actual values\)\. The leftmost group shows the average relative score across all metrics, followed by the metric\-wise scores for each metric \(see Appendix[C](https://arxiv.org/html/2607.06652#A3)for metric definitions\)\. A lower score is better\.
### 4\.4Ablation of the Signature Degree

Table 3:Ablation over signature truncation levelMMon TX and SO\. Lower is better\(↓\)\(\\downarrow\)for all metrics\.We isolate in Table[3](https://arxiv.org/html/2607.06652#S4.T3)the effect of the signature truncation orderMMon TX and SO, 2 datasets with different statistical properties and sizes \(see Table[S1](https://arxiv.org/html/2607.06652#A4.T1)\)\. Although the number of signature coordinates grows as𝒪​\(dM\)\\mathcal\{O\}\(d^\{M\}\), increasingMMgives dataset\-dependent gains\. On TX, most of the improvement occurs fromM=2M=2toM=3M=3; larger values produce only small changes inW1\\mathrm\{W\}\_\{1\}and CRPS\. On SO, higher orders improve energy distance andW1\\mathrm\{W\}\_\{1\}, but this comes at the cost of worse CRPS atM=6M=6\. Thus, larger truncation orders can improve global distributional matching, but they do not uniformly improve calibration or all evaluation metrics\.

To understand that better, note that increasingMMchanges the set of coordinates used for training rather than simply refining the same discrepancy\. Low\-order coordinates often dominate the signature scale, while higher\-order iterated integrals decay factorially for bounded\-variation paths, despite the exponential growth in their number\[[37](https://arxiv.org/html/2607.06652#bib.bib13)\]\. Consequently, largeMMcan introduce weakly varying or poorly conditioned features, increasing estimation noise without necessarily improving the learned path distribution\. This is consistent with the dead\-degree statistics in Table[S4](https://arxiv.org/html/2607.06652#A4.T4)\.

Overall, the ablation supportsM=3M=3for the evaluation and suggests that using a degree higher than 8 does not yield further performance\.

## 5Conclusion

This paper introduces a common pathwise framework for generative modelling and evaluation of TPPs\. The interarrival embeddingΦ\\Phiprovides a stable, injective lift from càdlàg counting paths to continuous bounded\-variation paths \(Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)\), making signature\-based tools applicable to discrete event sequences\. Building onΦ\\Phi,SigTPPoptimises a global signature Wasserstein loss on whole paths, rather than a sum of local conditional losses\. The same counting\-path metricd𝒩d\_\{\\mathcal\{N\}\}supports the energy distance \(ℰ\\mathcal\{E\}\),W1\\mathrm\{W\}\_\{1\}, as well asSig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}through the embedding\. It also clarifies a practical failure mode of pointwise evaluation: MAE can reward deterministic regressors even when they fail to capture distributional variation\. Across 9 datasets,SigTPPachieves the best average rank on 6 of 8 evaluation criteria, best average relative score on 7 of them, and outperforms the other baselines by at least19%19\\%on average across metrics\. The gains are largest on pathwise distributional criteria, while diffusion and variational baselines remain competitive on marginal calibration\. The limitations of our work are discussed in Appendix[E](https://arxiv.org/html/2607.06652#A5)\.

## Acknowledgements

NCK is supported by EPSRC grants EP/T517793/1 and EP/W524335/1\. VL would like to thank all levels of support from grant EP/X031276/1 \(EPSRC\) and the SOFAIR Lab \(UKRI\)\.

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## Appendix

## Appendix AA Pathwise Metric on Counting Paths

### A\.1Distances induced by the Pathwise Metric on Counting Paths

###### Definition S1\(Metric of Negative Type\)\.

A metric space\(X,d\)\(X,d\)is of negative type if, for everyn∈ℕn\\in\\mathbb\{N\}, all pointsx1,…,xn∈Xx\_\{1\},\\dots,x\_\{n\}\\in X, and all real coefficientsa1,…,an∈ℝa\_\{1\},\\dots,a\_\{n\}\\in\\mathbb\{R\}satisfying∑i=1nai=0\\sum\_\{i=1\}^\{n\}a\_\{i\}=0, the following inequality holds:

∑i,j=1nai​aj​d​\(xi,xj\)≤0\.\\sum\_\{i,j=1\}^\{n\}a\_\{i\}a\_\{j\}\\,d\(x\_\{i\},x\_\{j\}\)\\leq 0\\,\.\(S1\)

Geometrically, a metricddis of negative type if and only if the metric space\(X,d\)\(X,\\sqrt\{d\}\)can be embedded isometrically into a Hilbert space\[[47](https://arxiv.org/html/2607.06652#bib.bib40)\], i\.e\. if there exists a mapΦ:X→ℋ\\Phi:X\\to\\mathcal\{H\}into a Hilbert spaceℋ\\mathcal\{H\}such that

d​\(x,y\)=‖Φ​\(x\)−Φ​\(y\)‖ℋ2\.d\(x,y\)=\\left\\lVert\\Phi\(x\)\-\\Phi\(y\)\\right\\rVert\_\{\\mathcal\{H\}\}^\{2\}\\,\.\(S2\)
###### Proposition S2\(L1L^\{1\}is of Negative Type\[[4](https://arxiv.org/html/2607.06652#bib.bib41),[15](https://arxiv.org/html/2607.06652#bib.bib42),[36](https://arxiv.org/html/2607.06652#bib.bib39)\]\)\.

Let\(Ω,ℱ,μ\)\(\\Omega,\\mathcal\{F\},\\mu\)be a measure space\. Then the metric space\(L1\(μ\),∥⋅∥L1\)\(L^\{1\}\(\\mu\),\\left\\lVert\\cdot\\right\\rVert\_\{L^\{1\}\}\)is of negative type\. Equivalently, for allf1,…,fn∈L1​\(μ\)f\_\{1\},\\dots,f\_\{n\}\\in L^\{1\}\(\\mu\)and alla1,…,an∈ℝa\_\{1\},\\dots,a\_\{n\}\\in\\mathbb\{R\}such that∑iai=0\\sum\_\{i\}a\_\{i\}=0, we have

∑i,j=1nai​aj​‖fi−fj‖L1≤0\.\\sum\_\{i,j=1\}^\{n\}a\_\{i\}a\_\{j\}\\,\\left\\lVert f\_\{i\}\-f\_\{j\}\\right\\rVert\_\{L^\{1\}\}\\leq 0\\,\.\(S3\)

###### Proposition S3\(Subspace Inheritance\[[4](https://arxiv.org/html/2607.06652#bib.bib41),[15](https://arxiv.org/html/2607.06652#bib.bib42)\]\)\.

Let\(X,d\)\(X,d\)be a metric space of negative type, and letY⊂XY\\subset X\. Then the restricted metric space\(Y,d\|Y×Y\)\(Y,d\|\_\{Y\\times Y\}\)is also of negative type\.

###### Theorem S4\(Negative Type Property\)\.

The metric space\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)is of negative type\.

###### Proof\.

Identifying eachη∈𝒩\\eta\\in\\mathcal\{N\}with its counting patht↦ηtt\\mapsto\\eta\_\{t\}gives an isometric embedding of\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)intoL1​\(\[0,Tmax\]\)L^\{1\}\(\[0,T\_\{\\max\}\]\), sinced𝒩​\(η,ξ\)=‖η−ξ‖L1d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)=\\left\\lVert\\eta\-\\xi\\right\\rVert\_\{L^\{1\}\}\. Proposition[S2](https://arxiv.org/html/2607.06652#Thmtheorem2a)states thatL1L^\{1\}is of negative type, and Proposition[S3](https://arxiv.org/html/2607.06652#Thmtheorem3a)shows that negative type is inherited by subspaces\. Therefore,\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)is of negative type\. ∎

### A\.2Direct Integration

We begin with the following exact discrete formulation ofd𝒩d\_\{\\mathcal\{N\}\}for efficient computation\.

###### Proposition S5\(Discrete Computation\)\.

Letη,ξ∈𝒩\\eta,\\xi\\in\\mathcal\{N\}with event times\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}and\(sk\)k=1n\(s\_\{k\}\)\_\{k=1\}^\{n\}\. Assuming without loss of generality thatm≤nm\\leq n, the distance is given by

d𝒩​\(η,ξ\)=∑k=1m\|tk−sk\|\+\(n−m\)​Tmax−∑k=m\+1nsk\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)=\\sum\_\{k=1\}^\{m\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\+\(n\-m\)T\_\{\\max\}\-\\sum\_\{k=m\+1\}^\{n\}s\_\{k\}\\,\.\(S4\)

###### Proof\.

We prove the discrete expression from the integral representation, as written in Eq\.[S4](https://arxiv.org/html/2607.06652#A1.E4)\. Assumingm≤nm\\leq n, we prove that

d𝒩​\(η,ξ\)=∑i=1m\|ti−si\|\+\(n−m\)​Tmax−∑i=m\+1nsi\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)=\\sum\_\{i=1\}^\{m\}\\left\\lvert t\_\{i\}\-s\_\{i\}\\right\\rvert\+\(n\-m\)T\_\{\\max\}\-\\sum\_\{i=m\+1\}^\{n\}s\_\{i\}\\,\.\(S5\)Let

f​\(t\):=ηt−ξt,I​\(Tmax\):=∫0Tmax\|ηt−ξt\|​d​t=∫0Tmax\|f​\(t\)\|​d​t\.f\(t\):=\\eta\_\{t\}\-\\xi\_\{t\},\\quad I\(T\_\{\\max\}\):=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert\\eta\_\{t\}\-\\xi\_\{t\}\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert f\(t\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\\,\.\(S6\)ffrecords the imbalance between the number of events ofη\\etaandξ\\xithat have occurred up to timett: it is constant between event times and changes by\+1\+1whenever sometit\_\{i\}is crossed and by−1\-1whenever somesjs\_\{j\}is crossed\. Pairing events by rank and settingti:=\+∞t\_\{i\}:=\+\\inftyfori=m\+1,…,ni=m\{\+\}1,\\dots,n\(the processη\\etahas no further events\), we may write

f​\(t\)=ηt−ξt=∑i=1n\(𝟙⁡1​c​h​a​r​a​c\{t≥ti\}−𝟙⁡1​c​h​a​r​a​c\{t≥si\}\)\.f\(t\)=\\eta\_\{t\}\-\\xi\_\{t\}=\\sum\_\{i=1\}^\{n\}\\bigl\(\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{t\\geq t\_\{i\}\\\}\}\-\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{t\\geq s\_\{i\}\\\}\}\\bigr\)\.Theii\-th summand equals\+1\+1fort∈\[ti,si\)t\\in\[t\_\{i\},s\_\{i\}\)and−1\-1fort∈\[si,ti\)t\\in\[s\_\{i\},t\_\{i\}\); in either case its magnitude is the indicator of\[ti∧si,ti∨si\)\[t\_\{i\}\\wedge s\_\{i\},\\,t\_\{i\}\\vee s\_\{i\}\)\. Since both\(ti\)\(t\_\{i\}\)and\(si\)\(s\_\{i\}\)are nondecreasing, two summands of opposite sign cannot be nonzero at the samett: if termkkcontributes\+1\+1and termllcontributes−1\-1withk<lk<l, thensl≤t<sks\_\{l\}\\leq t<s\_\{k\}contradictssk≤sls\_\{k\}\\leq s\_\{l\}\(and symmetrically fork\>lk\>l\)\. Hence at everyttthe nonzero summands share a common sign, so the absolute value passes through the sum,

\|f​\(t\)\|=∑i=1n\|𝟙⁡1​c​h​a​r​a​c\{t≥ti\}−𝟙⁡1​c​h​a​r​a​c\{t≥si\}\|=∑i=1n𝟙⁡1​c​h​a​r​a​c\[ti∧si,ti∨si\)​\(t\)\.\\left\\lvert f\(t\)\\right\\rvert=\\sum\_\{i=1\}^\{n\}\\bigl\|\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{t\\geq t\_\{i\}\\\}\}\-\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{t\\geq s\_\{i\}\\\}\}\\bigr\|=\\sum\_\{i=1\}^\{n\}\\operatorname\{\\mathds\{1\}\}1charac\_\{\[t\_\{i\}\\wedge s\_\{i\},\\,t\_\{i\}\\vee s\_\{i\}\)\}\(t\)\.\(S7\)Splitting off the unpaired indicesi\>mi\>m, for whichti=\+∞t\_\{i\}=\+\\inftyand hence\[ti∧si,ti∨si\)∩\[0,Tmax\]=\[si,Tmax\]\[t\_\{i\}\\wedge s\_\{i\},\\,t\_\{i\}\\vee s\_\{i\}\)\\cap\[0,T\_\{\\max\}\]=\[s\_\{i\},T\_\{\\max\}\], gives, for allt∈\[0,Tmax\]t\\in\[0,T\_\{\\max\}\],

\|f​\(t\)\|=∑i=1m𝟙⁡1​c​h​a​r​a​c\[ti∧si,ti∨si\)​\(t\)\+∑j=m\+1n𝟙⁡1​c​h​a​r​a​c\[sj,Tmax\]​\(t\)\.\\left\\lvert f\(t\)\\right\\rvert=\\sum\_\{i=1\}^\{m\}\\operatorname\{\\mathds\{1\}\}1charac\_\{\[t\_\{i\}\\wedge s\_\{i\},\\,t\_\{i\}\\vee s\_\{i\}\)\}\(t\)\+\\sum\_\{j=m\+1\}^\{n\}\\operatorname\{\\mathds\{1\}\}1charac\_\{\[s\_\{j\},\\,T\_\{\\max\}\]\}\(t\)\\,\.\(S8\)The first sum counts rank\-paired times\(ti,si\)\(t\_\{i\},s\_\{i\}\), for which exactly one of the two events has occurred by timett, while the second sum accounts for the unpaired eventssjs\_\{j\}withj\>mj\>m\. Integrating both sides of Eq\.[S8](https://arxiv.org/html/2607.06652#A1.E8), we obtain

I​\(Tmax\)\\displaystyle I\(T\_\{\\max\}\)=∑i=1m\(ti∨si−ti∧si\)\+∑j=m\+1n\(Tmax−sj\)\\displaystyle=\\sum\_\{i=1\}^\{m\}\(t\_\{i\}\\vee s\_\{i\}\-t\_\{i\}\\wedge s\_\{i\}\)\+\\sum\_\{j=m\+1\}^\{n\}\(T\_\{\\max\}\-s\_\{j\}\)=∑i=1m\|ti−si\|\+\(n−m\)​Tmax−∑j=m\+1nsj=d𝒩​\(η,ξ\),\\displaystyle=\\sum\_\{i=1\}^\{m\}\|t\_\{i\}\-s\_\{i\}\|\+\(n\-m\)T\_\{\\max\}\-\\sum\_\{j=m\+1\}^\{n\}s\_\{j\}=d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\,,\(S9\)which establishes Eq\.[S4](https://arxiv.org/html/2607.06652#A1.E4)form≤nm\\leq n\. The casem\>nm\>nfollows by symmetry\. ∎

### A\.3Optimal Transport Interpretation

Letη,ξ∈𝒩\\eta,\\xi\\in\\mathcal\{N\}with event times\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}and\(sk\)k=1n\(s\_\{k\}\)\_\{k=1\}^\{n\}respectively\. For the optimal transport argument, associate to each the point measureμ:=∑k=1mδtk\\mu:=\\sum\_\{k=1\}^\{m\}\\delta\_\{t\_\{k\}\}andν:=∑k=1nδsk\\nu:=\\sum\_\{k=1\}^\{n\}\\delta\_\{s\_\{k\}\}\. We writeδTmax\\delta\_\{T\_\{\\max\}\}for the Dirac measure atTmaxT\_\{\\max\}, and use\(⋅\)\+=max⁡\(⋅,0\)\(\\cdot\)^\{\+\}=\\max\(\\cdot,0\)\.

We establish that the Wasserstein distance\[[46](https://arxiv.org/html/2607.06652#bib.bib58)\]for the deterministic measures with anchor points atTmaxT\_\{\\max\}is indeed theL1L^\{1\}distance between the respective counting processes:

W1​\(μ~,ν~\)=∫0Tmax\|ηt−ξt\|​d​t,\\mathrm\{W\}\_\{1\}\(\\tilde\{\\mu\},\\tilde\{\\nu\}\)=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert\\eta\_\{t\}\-\\xi\_\{t\}\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\\,,\(S10\)where, assuming thatηTmax=m,ξTmax=n\\eta\_\{T\_\{\\max\}\}=m,\\xi\_\{T\_\{\\max\}\}=n,

μ~:=μ\+\(n−m\)\+​δTmax,ν~:=ν\+\(m−n\)\+​δTmax\.\\tilde\{\\mu\}:=\\mu\+\(n\-m\)^\{\+\}\\delta\_\{T\_\{\\max\}\}\\,,\\qquad\\tilde\{\\nu\}:=\\nu\+\(m\-n\)^\{\+\}\\delta\_\{T\_\{\\max\}\}\\,\.\(S11\)This establishes the correspondence between our definition and the optimal transport interpretation\.

###### Proof\.

By construction, both padded measures have equal total mass\. Since their supports are contained in\[0,Tmax\]\[0,T\_\{\\max\}\], bothμ~\\tilde\{\\mu\}andν~\\tilde\{\\nu\}have finite first moments\. By\[[52](https://arxiv.org/html/2607.06652#bib.bib37)\]and generalising to finite positive measures of equal mass and finite first moment:

W1​\(μ~,ν~\)≜infγ∈Γ​\(μ~,ν~\)∫ℝ×ℝ\|x−y\|​γ​\(d​x,d​y\)=∫ℝ\|Fμ~​\(t\)−Fν~​\(t\)\|​d​t\.\\mathrm\{W\}\_\{1\}\(\\tilde\{\\mu\},\\tilde\{\\nu\}\)\\triangleq\\inf\_\{\\gamma\\in\\Gamma\(\\tilde\{\\mu\},\\tilde\{\\nu\}\)\}\\int\_\{\\mathbb\{R\}\\times\\mathbb\{R\}\}\\left\\lvert x\-y\\right\\rvert\\gamma\(\\mathop\{\}\\\!\\mathrm\{d\}x,\\mathop\{\}\\\!\\mathrm\{d\}y\)=\\int\_\{\\mathbb\{R\}\}\\left\\lvert F\_\{\\tilde\{\\mu\}\}\(t\)\-F\_\{\\tilde\{\\nu\}\}\(t\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\\,\.\(S12\)Fort<Tmaxt<T\_\{\\max\}, no padded atoms have appeared\. Hence,Fμ~​\(t\)=μ~​\(\(−∞,t\]\)=μ​\(\(−∞,t\]\)=ηt,F\_\{\\tilde\{\\mu\}\}\(t\)=\\tilde\{\\mu\}\(\(\-\\infty,t\]\)=\\mu\(\(\-\\infty,t\]\)=\\eta\_\{t\},and analogously,Fν~​\(t\)=ξtF\_\{\\tilde\{\\nu\}\}\(t\)=\\xi\_\{t\}\. Thus,

\|Fμ~​\(t\)−Fν~​\(t\)\|=\{\|ηt−ξt\|,t∈\[0,Tmax\),0,t∉\[0,Tmax\)\.\\left\\lvert F\_\{\\tilde\{\\mu\}\}\(t\)\-F\_\{\\tilde\{\\nu\}\}\(t\)\\right\\rvert=\\begin\{cases\}\\left\\lvert\\eta\_\{t\}\-\\xi\_\{t\}\\right\\rvert,&t\\in\[0,T\_\{\\max\}\)\\,,\\\\ 0,&t\\notin\[0,T\_\{\\max\}\)\\,\.\\end\{cases\}\(S13\)Therefore, we get the equality

W1​\(μ~,ν~\)=∫ℝ\|Fμ~​\(t\)−Fν~​\(t\)\|​d​t=∫0Tmax\|ηt−ξt\|​d​t\.\\mathrm\{W\}\_\{1\}\(\\tilde\{\\mu\},\\tilde\{\\nu\}\)=\\int\_\{\\mathbb\{R\}\}\\left\\lvert F\_\{\\tilde\{\\mu\}\}\(t\)\-F\_\{\\tilde\{\\nu\}\}\(t\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert\\eta\_\{t\}\-\\xi\_\{t\}\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t\\,\.\(S14\)∎

### A\.4Counterexample for Strong Negative Type

The TPP distance is of negative type but not of strong negative type\. Consider four realised TPPs with two events each:

ξ1=\(0,2\),ξ2=\(1,3\),ξ3=\(0,3\),andξ4=\(1,2\)\.\\xi\_\{1\}=\(0,2\),\\qquad\\xi\_\{2\}=\(1,3\),\\qquad\\xi\_\{3\}=\(0,3\),\\qquad\\text\{and\}\\qquad\\xi\_\{4\}=\(1,2\)\\,\.\(S15\)Then, with coefficientsa1=a2=1a\_\{1\}=a\_\{2\}=1anda3=a4=−1a\_\{3\}=a\_\{4\}=\-1\(such that∑iai=0\\sum\_\{i\}a\_\{i\}=0\) one has

∑i,j=14ai​aj​d𝒩​\(ξi,ξj\)\\displaystyle\\sum\_\{i,j=1\}^\{4\}a\_\{i\}a\_\{j\}\\,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{i\},\\xi\_\{j\}\\right\)=2​d𝒩​\(ξ1,ξ2\)\+2​d𝒩​\(ξ3,ξ4\)\\displaystyle=2\\,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{1\},\\xi\_\{2\}\\right\)\+2\\,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{3\},\\xi\_\{4\}\\right\)−2⋅\(d𝒩​\(ξ1,ξ3\)\+d𝒩​\(ξ1,ξ4\)\+d𝒩​\(ξ2,ξ3\)\+d𝒩​\(ξ2,ξ4\)\)\\displaystyle\\quad\-2\\cdot\\big\(d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{1\},\\xi\_\{3\}\\right\)\+d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{1\},\\xi\_\{4\}\\right\)\+d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{2\},\\xi\_\{3\}\\right\)\+d\_\{\\mathcal\{N\}\}\\\!\\left\(\\xi\_\{2\},\\xi\_\{4\}\\right\)\\big\)=2⋅2\+2⋅2−2​\(1\+1\+1\+1\)\\displaystyle=2\\cdot 2\+2\\cdot 2\-2\(1\+1\+1\+1\)=0,\\displaystyle=0\\,,\(S16\)with theξi\\xi\_\{i\}not all equal, which violates strong negative type\. In general,L1L^\{1\}spaces are of negative type but not of strong negative type\[[36](https://arxiv.org/html/2607.06652#bib.bib39)\]\.

## Appendix BProofs of Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6)

Following the notation of Theorem[6](https://arxiv.org/html/2607.06652#Thmtheorem6), we prove the following statements\.

1. 1\.The mapΦ\\Phiis Lipschitz \(Theorem[S11](https://arxiv.org/html/2607.06652#Thmtheorem11)\)\.
2. 2\.The mapΦ\\Phiis injective on𝒩\\mathcal\{N\}, hence a bijection between𝒩\\mathcal\{N\}and its imageΦ​\(𝒩\)\\Phi\(\\mathcal\{N\}\)\(Theorem[S16](https://arxiv.org/html/2607.06652#Thmtheorem16)\)\.
3. 3\.The inverse mapΨ=Φ\|𝒩−1\\Psi=\\Phi\|\_\{\\mathcal\{N\}\}^\{\-1\}is continuous onΦ​\(𝒩δ\)\\Phi\(\\mathcal\{N\}\_\{\\delta\}\)for eachδ\>0\\delta\>0; more precisely, it satisfies a12\\tfrac\{1\}\{2\}\-Hölder bound on the separated subsetΦ​\(𝒩δ\)\\Phi\(\\mathcal\{N\}\_\{\\delta\}\)defined in Appendix[B\.3](https://arxiv.org/html/2607.06652#A2.SS3)\(Theorem[S29](https://arxiv.org/html/2607.06652#Thmtheorem29)\)\. Continuity fails on all ofΦ​\(𝒩\)\\Phi\(\\mathcal\{N\}\); see Remark[S17](https://arxiv.org/html/2607.06652#Thmtheorem17)\.

### B\.1Φ\\Phiis Lipschitz

###### Lemma S6\(AffineL1L^\{1\}Bound\)\.

Leta<ba<band leth:\[a,b\]→ℝh\\colon\[a,b\]\\to\\mathbb\{R\}be an affine function\. Then

∫ab\|h​\(t\)\|​d​t≤b−a2​\(\|h​\(a\)\|\+\|h​\(b\)\|\)\.\\int\_\{a\}^\{b\}\\left\\lvert h\(t\)\\right\\rvert\\mathop\{\}\\\!\\mathrm\{d\}\{t\}\\;\\leq\\;\\frac\{b\-a\}\{2\}\\bigl\(\\left\\lvert h\(a\)\\right\\rvert\+\\left\\lvert h\(b\)\\right\\rvert\\bigr\)\\,\.\(S17\)

###### Proof\.

The composition of a convex function with an affine function is convex\. Sincet↦\|h​\(t\)\|t\\mapsto\\left\\lvert h\(t\)\\right\\rvertis the composition of the convex function\|⋅\|\\left\\lvert\\cdot\\right\\rvertwith the affine functionhh, it is convex on\[a,b\]\[a,b\]\. Every convex function on an interval is bounded above by the chord through its endpoint values:

\|h​\(t\)\|≤b−tb−a​\|h​\(a\)\|\+t−ab−a​\|h​\(b\)\|,t∈\[a,b\]\.\\left\\lvert h\(t\)\\right\\rvert\\;\\leq\\;\\frac\{b\-t\}\{b\-a\}\\,\\left\\lvert h\(a\)\\right\\rvert\+\\frac\{t\-a\}\{b\-a\}\\,\\left\\lvert h\(b\)\\right\\rvert\\,,\\qquad t\\in\[a,b\]\\,\.\(S18\)Integrating over\[a,b\]\[a,b\]gives the result\. ∎

We recall that a right\-continuous function of bounded variation induces a Lebesgue–Stieltjes \(total variation\) measure\|D​y\|\\left\\lvert Dy\\right\\rvert; see, e\.g\., Folland\[[18](https://arxiv.org/html/2607.06652#bib.bib50)\], Sections 1\.5 and 3\.5\.

###### Lemma S7\(Bounded Variation Time\-Warp Inequality\)\.

Lety:\[0,Tmax\]→ℝy:\[0,T\_\{\\max\}\]\\to\\mathbb\{R\}be \(continuous111Continuity is not required for the proof but makes the argument cleaner, and we do not require a more general statement for our proofs\.and\) of bounded variation, with total variationTV​\(y\):=TV​\(y;\[0,Tmax\]\)\\mathrm\{TV\}\(y\):=\\mathrm\{TV\}\(y;\\,\[0,T\_\{\\max\}\]\)\. Letλ:\[0,Tmax\]→\[0,Tmax\]\\lambda:\[0,T\_\{\\max\}\]\\to\[0,T\_\{\\max\}\]be measurable and set‖λ−Id‖∞:=supt∈\[0,Tmax\]\|λ​\(t\)−t\|\.\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}:=\\sup\_\{t\\in\[0,T\_\{\\max\}\]\}\|\\lambda\(t\)\-t\|\.Then

‖y​\(λ​\(t\)\)−y​\(t\)‖L1≤2​‖λ−Id‖∞​TV​\(y\)\.\\left\\lVert y\(\\lambda\(t\)\)\-y\(t\)\\right\\rVert\_\{L^\{1\}\}\\leq 2\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\\,\\mathrm\{TV\}\(y\)\\,\.\(S19\)

###### Proof\.

Let\|D​y\|\\left\\lvert Dy\\right\\rvertdenote the total variation measure ofyy, i\.e\. the Lebesgue–Stieltjes measure associated with the total variation functionTV​\(y;\[0,t\]\)\\mathrm\{TV\}\(y;\\,\[0,t\]\)\. For anya,b∈\[0,Tmax\]a,b\\in\[0,T\_\{\\max\}\],

\|y​\(b\)−y​\(a\)\|≤\|D​y\|​\(\[a∧b,a∨b\]\)\.\|y\(b\)\-y\(a\)\|\\;\\leq\\;\\left\\lvert Dy\\right\\rvert\\big\(\[a\\wedge b,\\;a\\vee b\]\\big\)\\,\.\(S20\)
Setδ:=‖λ−Id‖∞\\delta:=\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\(smaller than or equal toTmaxT\_\{\\max\}\)\. For eacht∈\[0,Tmax\]t\\in\[0,T\_\{\\max\}\]we haveλ​\(t\)∈\[t−δ,t\+δ\]∩\[0,Tmax\]\\lambda\(t\)\\in\[t\-\\delta,\\,t\+\\delta\]\\cap\[0,T\_\{\\max\}\], so that\[t∧λ​\(t\),t∨λ​\(t\)\]⊆\[t−δ,t\+δ\]∩\[0,Tmax\]\[t\\wedge\\lambda\(t\),\\;t\\vee\\lambda\(t\)\]\\subseteq\[t\-\\delta,\\,t\+\\delta\]\\cap\[0,T\_\{\\max\}\]\. By monotonicity of measures,

\|y​\(λ​\(t\)\)−y​\(t\)\|≤\|D​y\|​\(\[t∧λ​\(t\),t∨λ​\(t\)\]\)≤\|D​y\|​\(\[t−δ,t\+δ\]∩\[0,Tmax\]\)\.\|y\(\\lambda\(t\)\)\-y\(t\)\|\\leq\\left\\lvert Dy\\right\\rvert\\big\(\[t\\wedge\\lambda\(t\),\\;t\\vee\\lambda\(t\)\]\\big\)\\leq\\left\\lvert Dy\\right\\rvert\\big\(\[t\-\\delta,\\,t\+\\delta\]\\cap\[0,T\_\{\\max\}\]\\big\)\\,\.\(S21\)Integrating over\[0,Tmax\]\[0,T\_\{\\max\}\]and writing the right\-hand side as a double integral,

∫0Tmax\|y​\(λ​\(t\)\)−y​\(t\)\|​d​t\\displaystyle\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert y\(\\lambda\(t\)\)\-y\(t\)\\right\\rvert\\mathop\{\}\\\!\\mathrm\{d\}t≤∫0Tmax∫\[0,Tmax\]𝟙⁡1​c​h​a​r​a​c\{\|u−t\|≤δ\}​\|D​y\|​\(d​u\)​d​t\.\\displaystyle\\leq\\int\_\{0\}^\{T\_\{\\max\}\}\\int\_\{\[0,T\_\{\\max\}\]\}\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{\|u\-t\|\\leq\\delta\\\}\}\\,\\left\\lvert Dy\\right\\rvert\(\\mathop\{\}\\\!\\mathrm\{d\}u\)\\mathop\{\}\\\!\\mathrm\{d\}t\\,\.\(S22\)The integrand is nonnegative and jointly measurable \(the set\{\(t,u\):\|u−t\|≤δ\}\\\{\(t,u\):\|u\-t\|\\leq\\delta\\\}is closed inℝ2\\mathbb\{R\}^\{2\}\), so Tonelli’s theorem gives the desired inequality

∫0Tmax\|y​\(λ​\(t\)\)−y​\(t\)\|​d​t\\displaystyle\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert y\(\\lambda\(t\)\)\-y\(t\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}t≤∫\[0,Tmax\]\(∫0Tmax𝟙⁡1​c​h​a​r​a​c\{\|u−t\|≤δ\}​d​t\)​\|D​y\|​\(d​u\)\\displaystyle\\leq\\int\_\{\[0,T\_\{\\max\}\]\}\\biggl\(\\int\_\{0\}^\{T\_\{\\max\}\}\\operatorname\{\\mathds\{1\}\}1charac\_\{\\\{\|u\-t\|\\leq\\delta\\\}\}\\,\\mathop\{\}\\\!\\mathrm\{d\}t\\biggr\)\\left\\lvert Dy\\right\\rvert\(\\mathop\{\}\\\!\\mathrm\{d\}u\)≤∫\[0,Tmax\]2​δ​\|D​y\|​\(d​u\)=2​δ​TV​\(y\)\.\\displaystyle\\leq\\int\_\{\[0,T\_\{\\max\}\]\}2\\delta\\,\\left\\lvert Dy\\right\\rvert\(\\mathop\{\}\\\!\\mathrm\{d\}u\)=2\\delta\\,\\mathrm\{TV\}\(y\)\\,\.\(S23\)∎

###### Lemma S8\(Interarrival Telescoping Bound\)\.

Let0=t0<t1<⋯<tm<tm\+1=Tmax0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\}<t\_\{m\+1\}=T\_\{\\max\}and0=s0<s1<⋯<sn<sn\+1=Tmax0=s\_\{0\}<s\_\{1\}<\\cdots<s\_\{n\}<s\_\{n\+1\}=T\_\{\\max\}be two augmented grids \(in the sense of Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\) withm≤nm\\leq n\. Define the interarrival times

τk:=tk−tk−1,k=1,…,m\+1,ςk:=sk−sk−1,k=1,…,n\+1,\\tau\_\{k\}:=t\_\{k\}\-t\_\{k\-1\},\\quad k=1,\\dots,m\{\+\}1,\\qquad\\varsigma\_\{k\}:=s\_\{k\}\-s\_\{k\-1\},\\quad k=1,\\dots,n\{\+\}1\\,,\(S24\)and setτ0:=ς0:=0\\tau\_\{0\}:=\\varsigma\_\{0\}:=0\. Extend the shorter grid by padding:ti:=Tmaxt\_\{i\}:=T\_\{\\max\}\\,fori=m\+1,…,ni=m\{\+\}1,\\dots,n, so that both sequences\(ti\)i=1n\(t\_\{i\}\)\_\{i=1\}^\{n\}and\(si\)i=1n\(s\_\{i\}\)\_\{i=1\}^\{n\}have the same length\.

Then the firstm\+2m\{\+\}2interarrival differences are controlled by the event\-time differences:

∑k=0m\+1\|τk−ςk\|≤2​∑i=1n\|ti−si\|\.\\sum\_\{k=0\}^\{m\+1\}\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\\leq 2\\sum\_\{i=1\}^\{n\}\\left\\lvert t\_\{i\}\-s\_\{i\}\\right\\rvert\\,\.\(S25\)

###### Proof\.

Fork=1,…,m\+1k=1,\\dots,m\{\+\}1, the telescoping identity

τk−ςk=\(tk−tk−1\)−\(sk−sk−1\)=\(tk−sk\)−\(tk−1−sk−1\)\\tau\_\{k\}\-\\varsigma\_\{k\}=\(t\_\{k\}\-t\_\{k\-1\}\)\-\(s\_\{k\}\-s\_\{k\-1\}\)=\(t\_\{k\}\-s\_\{k\}\)\-\(t\_\{k\-1\}\-s\_\{k\-1\}\)\(S26\)implies, by the triangle inequality,

\|τk−ςk\|≤\|tk−sk\|\+\|tk−1−sk−1\|\.\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\\leq\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\+\\left\\lvert t\_\{k\-1\}\-s\_\{k\-1\}\\right\\rvert\\,\.\(S27\)Summing overk=1,…,m\+1k=1,\\dots,m\{\+\}1yields

∑k=1m\+1\|τk−ςk\|≤∑k=1m\+1\|tk−sk\|\+∑k=0m\|tk−sk\|\.\\sum\_\{k=1\}^\{m\+1\}\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\\leq\\sum\_\{k=1\}^\{m\+1\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\+\\sum\_\{k=0\}^\{m\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\,\.\(S28\)Sinceτ0=ς0=0\\tau\_\{0\}=\\varsigma\_\{0\}=0andt0=s0=0t\_\{0\}=s\_\{0\}=0, the boundary terms vanish\. Noting that all summands are non\-negative, we obtain

∑k=0m\+1\|τk−ςk\|≤2​∑k=1m\+1\|tk−sk\|≤2​∑k=1n\|tk−sk\|\.\\sum\_\{k=0\}^\{m\+1\}\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\\leq 2\\sum\_\{k=1\}^\{m\+1\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\leq 2\\sum\_\{k=1\}^\{n\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\,\.\(S29\)In the above, whenm=nm=n, the sum includes the extra term\|tn\+1−sn\+1\|=Tmax−Tmax=0\\left\\lvert t\_\{n\+1\}\-s\_\{n\+1\}\\right\\rvert=T\_\{\\max\}\-T\_\{\\max\}=0\. ∎

###### Lemma S9\(Grid Mismatch Controlled by a Time\-Change\)\.

Letξ∈𝒩\\xi\\in\\mathcal\{N\}withnnevents and augmented grid0=s0<s1<⋯<sn\+1=Tmax0=s\_\{0\}<s\_\{1\}<\\cdots<s\_\{n\+1\}=T\_\{\\max\}, and lety=Φ​\(ξ\)∈𝒞y=\\Phi\(\\xi\)\\in\\mathcal\{C\}be the piecewise\-linear interpolant through\(sk,ςk\)k=0n\+1\(s\_\{k\},\\varsigma\_\{k\}\)\_\{k=0\}^\{n\+1\}, whereς0:=0\\varsigma\_\{0\}:=0andςk:=sk−sk−1\\varsigma\_\{k\}:=s\_\{k\}\-s\_\{k\-1\}fork=1,…,n\+1k=1,\\dots,n\{\+\}1are the interarrival times \(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\)\.

Let0=t0<t1<⋯<tm\+1=Tmax0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\+1\}=T\_\{\\max\}be a second augmented grid withm≤nm\\leq n\. Define:

1. \(a\)the piecewise\-linear mapλ:\[0,Tmax\]→\[0,sm\+1\]\\lambda\\colon\[0,T\_\{\\max\}\]\\to\[0,s\_\{m\+1\}\]byλ​\(tk\)=sk\\lambda\(t\_\{k\}\)=s\_\{k\}fork=0,…,m\+1k=0,\\dots,m\{\+\}1, affine on each\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\];
2. \(b\)the re\-gridded interpolanty~\\tilde\{y\}on thett\-grid:y~​\(tk\):=ςk\\tilde\{y\}\(t\_\{k\}\):=\\varsigma\_\{k\}fork=0,…,m\+1k=0,\\dots,m\{\+\}1, affine between nodes\.

Extend the shorter grid by paddingti:=Tmaxt\_\{i\}:=T\_\{\\max\}fori=m\+1,…,ni=m\+1,\\dots,n\. Then:

1. \(i\)y~​\(t\)=y​\(λ​\(t\)\)\\tilde\{y\}\(t\)=y\(\\lambda\(t\)\)for allt∈\[0,Tmax\]t\\in\[0,T\_\{\\max\}\]\.
2. \(ii\)‖y~−y‖L1≤4​Tmax​‖λ−Id‖∞\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\\leq 4T\_\{\\max\}\\,\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\.
3. \(iii\)‖λ−Id‖∞≤∑k=1n\|tk−sk\|\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{k=1\}^\{n\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\.

Consequently,

‖y~−y‖L1≤4​Tmax​∑k=1n\|tk−sk\|\.\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\\;\\leq\\;4T\_\{\\max\}\\sum\_\{k=1\}^\{n\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\,\.\(S30\)

###### Proof\.

Part \(i\): Identityy~=y∘λ\\tilde\{y\}=y\\circ\\lambda\.Fixk∈\{0,…,m\}k\\in\\\{0,\\dots,m\\\}andt∈\[tk,tk\+1\]t\\in\[t\_\{k\},t\_\{k\+1\}\]\. Since the grids are strictly increasing,tk\+1−tk\>0,k≤mt\_\{k\+1\}\-t\_\{k\}\>0,k\\leq mandsk\+1−sk\>0,k≤ns\_\{k\+1\}\-s\_\{k\}\>0,k\\leq n\. The fact that the grid is strictly increasing also ensures that on the interval\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\], the mapλ\\lambdais strictly increasing and affine withλ​\(tk\)=sk\\lambda\(t\_\{k\}\)=s\_\{k\}andλ​\(tk\+1\)=sk\+1\\lambda\(t\_\{k\+1\}\)=s\_\{k\+1\}, hence

λ​\(\[tk,tk\+1\]\)=\[sk,sk\+1\],\\lambda\\bigl\(\[t\_\{k\},t\_\{k\+1\}\]\\bigr\)=\[s\_\{k\},s\_\{k\+1\}\]\\,,\(S31\)so in particularλ​\(t\)∈\[sk,sk\+1\]\\lambda\(t\)\\in\[s\_\{k\},s\_\{k\+1\}\]andy​\(λ​\(t\)\)y\(\\lambda\(t\)\)is given by thekk\-th linear piece ofyy\.

By definition ofy~\\tilde\{y\},

y~​\(t\)=ςk\+ςk\+1−ςktk\+1−tk​\(t−tk\)\.\\tilde\{y\}\(t\)=\\varsigma\_\{k\}\+\\frac\{\\varsigma\_\{k\+1\}\-\\varsigma\_\{k\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\(t\-t\_\{k\}\)\\,\.\(S32\)Sinceλ\\lambdais linear on\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\],

λ​\(t\)=sk\+sk\+1−sktk\+1−tk​\(t−tk\),equivalentlyλ​\(t\)−sksk\+1−sk=t−tktk\+1−tk\.\\lambda\(t\)=s\_\{k\}\+\\frac\{s\_\{k\+1\}\-s\_\{k\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\(t\-t\_\{k\}\)\\,,\\qquad\\text\{equivalently\}\\qquad\\frac\{\\lambda\(t\)\-s\_\{k\}\}\{s\_\{k\+1\}\-s\_\{k\}\}=\\frac\{t\-t\_\{k\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\\,\.\(S33\)Substituting this into the expression foryyon\[sk,sk\+1\]\[s\_\{k\},s\_\{k\+1\}\]yields

y​\(λ​\(t\)\)=ςk\+ςk\+1−ςksk\+1−sk​\(λ​\(t\)−sk\)=ςk\+ςk\+1−ςktk\+1−tk​\(t−tk\)=y~​\(t\),y\(\\lambda\(t\)\)=\\varsigma\_\{k\}\+\\frac\{\\varsigma\_\{k\+1\}\-\\varsigma\_\{k\}\}\{s\_\{k\+1\}\-s\_\{k\}\}\\bigl\(\\lambda\(t\)\-s\_\{k\}\\bigr\)=\\varsigma\_\{k\}\+\\frac\{\\varsigma\_\{k\+1\}\-\\varsigma\_\{k\}\}\{t\_\{k\+1\}\-t\_\{k\}\}\(t\-t\_\{k\}\)=\\tilde\{y\}\(t\)\\,,\(S34\)which proves \(i\)\.

Part \(ii\):L1L^\{1\}\-bound via Lemma[S7](https://arxiv.org/html/2607.06652#Thmtheorem7)\.The mapλ:\[0,Tmax\]→\[0,Tmax\]\\lambda:\[0,T\_\{\\max\}\]\\to\[0,T\_\{\\max\}\]is piecewise linear, hence measurable andyyis piecewise linear hence continuous and of bounded variation, so all hypotheses of Lemma[S7](https://arxiv.org/html/2607.06652#Thmtheorem7)are fulfilled\. Eq\.[S19](https://arxiv.org/html/2607.06652#A2.E19)together with Part \(i\) gives

‖y~−y‖L1=‖y∘λ−y‖L1≤2​TV​\(y\)​‖λ−Id‖∞\.\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}=\\left\\lVert y\\circ\\lambda\-y\\right\\rVert\_\{L^\{1\}\}\\leq 2\\,\\mathrm\{TV\}\(y\)\\,\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\\,\.\(S35\)
Sinceyyis piecewise linear with nodesς0=0\\varsigma\_\{0\}=0,ςk≥0\\varsigma\_\{k\}\\geq 0, and∑k=1n\+1ςk=Tmax\\sum\_\{k=1\}^\{n\+1\}\\varsigma\_\{k\}=T\_\{\\max\},

TV​\(y\)=∑k=0n\|ςk\+1−ςk\|≤∑k=0n\(ςk\+1\+ςk\)≤2​∑k=1n\+1ςk=2​Tmax\.\\mathrm\{TV\}\(y\)=\\sum\_\{k=0\}^\{n\}\|\\varsigma\_\{k\+1\}\-\\varsigma\_\{k\}\|\\leq\\sum\_\{k=0\}^\{n\}\(\\varsigma\_\{k\+1\}\+\\varsigma\_\{k\}\)\\leq 2\\sum\_\{k=1\}^\{n\+1\}\\varsigma\_\{k\}=2T\_\{\\max\}\\,\.\(S36\)This yields \(ii\):

‖y~−y‖L1≤4​Tmax​‖λ−Id‖∞\.\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\\leq 4T\_\{\\max\}\\,\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\\,\.\(S37\)
Part \(iii\): Control of the time change\.Sincet↦λ​\(t\)−tt\\mapsto\\lambda\(t\)\-tis affine on each interval\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\], the function\|λ​\(t\)−t\|\|\\lambda\(t\)\-t\|is convex on each such interval\. Therefore, its maximum over\[0,Tmax\]\[0,T\_\{\\max\}\]is attained at a node\. Because all terms are nonnegative, we may bound the maximum by the sum

‖λ−Id‖∞=max0≤k≤m\+1⁡\|sk−tk\|≤∑k=0m\+1\|sk−tk\|\.\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}=\\max\_\{0\\leq k\\leq m\+1\}\|s\_\{k\}\-t\_\{k\}\|\\leq\\sum\_\{k=0\}^\{m\+1\}\|s\_\{k\}\-t\_\{k\}\|\\,\.\(S38\)The term atk=0k=0vanishes sinces0=t0=0s\_\{0\}=t\_\{0\}=0\. Finally, using the padding conventionti:=Tmaxt\_\{i\}:=T\_\{\\max\}fori=m\+1,…,ni=m\+1,\\dots,n, the sum∑i=1n\|ti−si\|\\sum\_\{i=1\}^\{n\}\|t\_\{i\}\-s\_\{i\}\|contains all nonzero terms\|sk−tk\|\|s\_\{k\}\-t\_\{k\}\|fork=1,…,m\+1k=1,\\dots,m\+1; whenm=nm=n, the omitted endpoint term is\|tn\+1−sn\+1\|=0\|t\_\{n\+1\}\-s\_\{n\+1\}\|=0\. It also contains additional nonnegative terms fori=m\+2,…,ni=m\+2,\\dots,n\. Hence,

‖λ−Id‖∞≤∑k=1m\+1\|sk−tk\|≤∑i=1n\|ti−si\|,\\left\\lVert\\lambda\-\\mathrm\{Id\}\\right\\rVert\_\{\\infty\}\\leq\\sum\_\{k=1\}^\{m\+1\}\|s\_\{k\}\-t\_\{k\}\|\\leq\\sum\_\{i=1\}^\{n\}\|t\_\{i\}\-s\_\{i\}\|\\,,\(S39\)which proves \(iii\)\.

Conclusion\.Combining \(ii\) and \(iii\) gives

‖y~−y‖L1≤4​Tmax​∑i=1n\|ti−si\|\.\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\\leq 4T\_\{\\max\}\\sum\_\{i=1\}^\{n\}\|t\_\{i\}\-s\_\{i\}\|\\,\.\(S40\)∎

###### Theorem S11\(Global Lipschitz continuity ofΦ\\Phi\)\.

Equip𝒩\\mathcal\{N\}withd𝒩d\_\{\\mathcal\{N\}\}and𝒞\\mathcal\{C\}withd1​\(x,y\):=∫0Tmax\|x​\(t\)−y​\(t\)\|​d​td\_\{1\}\(x,y\):=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert x\(t\)\-y\(t\)\\right\\rvert\\mathop\{\}\\\!\\mathrm\{d\}\{t\}\. ThenΦ:\(𝒩,d𝒩\)→\(𝒞,d1\)\\Phi\\colon\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)\\to\(\\mathcal\{C\},d\_\{1\}\)is globally Lipschitz:

d1​\(Φ​\(η\),Φ​\(ξ\)\)≤5​Tmax​d𝒩​\(η,ξ\)for all​η,ξ∈𝒩\.d\_\{1\}\\bigl\(\\Phi\(\\eta\),\\Phi\(\\xi\)\\bigr\)\\leq 5T\_\{\\max\}\\,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\qquad\\text\{for all \}\\eta,\\xi\\in\\mathcal\{N\}\\,\.\(S41\)

###### Proof\.

Letη∈𝒩\\eta\\in\\mathcal\{N\}havemmevents at times\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}andξ∈𝒩\\xi\\in\\mathcal\{N\}havennevents at times\(sk\)k=1n\(s\_\{k\}\)\_\{k=1\}^\{n\}, withm≤nm\\leq n\(WLOG by symmetry ofd𝒩d\_\{\\mathcal\{N\}\}\)\. Writex:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\)\.

Construct the re\-gridded interpolanty~\\tilde\{y\}on thett\-grid with nodal valuesy~​\(tk\):=ςk\\tilde\{y\}\(t\_\{k\}\):=\\varsigma\_\{k\}fork=0,…,m\+1k=0,\\dots,m\{\+\}1, as in Lemma[S9](https://arxiv.org/html/2607.06652#Thmtheorem9)\. By the triangle inequality,

‖x−y‖L1≤‖x−y~‖L1⏟\(I\): vertical\+‖y~−y‖L1⏟\(II\): horizontal\.\\left\\lVert x\-y\\right\\rVert\_\{L^\{1\}\}\\;\\leq\\;\\underbrace\{\\left\\lVert x\-\\tilde\{y\}\\right\\rVert\_\{L^\{1\}\}\}\_\{\\text\{\(I\): vertical\}\}\\;\+\\;\\underbrace\{\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\}\_\{\\text\{\(II\): horizontal\}\}\\,\.\(S42\)Bound \(I\): Vertical mismatch \(same grid, different nodal values\)\.On each\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\],x−y~x\-\\tilde\{y\}is affine\. By Lemma[S6](https://arxiv.org/html/2607.06652#Thmtheorem6a),

∫tktk\+1\|x​\(t\)−y~​\(t\)\|​d​t≤Δk2​\(\|τk−ςk\|\+\|τk\+1−ςk\+1\|\),\\int\_\{t\_\{k\}\}^\{t\_\{k\+1\}\}\\left\\lvert x\(t\)\-\\tilde\{y\}\(t\)\\right\\rvert\\mathop\{\}\\\!\\mathrm\{d\}\{t\}\\;\\leq\\;\\frac\{\\Delta\_\{k\}\}\{2\}\\bigl\(\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\+\\left\\lvert\\tau\_\{k\+1\}\-\\varsigma\_\{k\+1\}\\right\\rvert\\bigr\)\\,,\(S43\)whereΔk:=tk\+1−tk\\Delta\_\{k\}:=t\_\{k\+1\}\-t\_\{k\}\. Summing overk=0,…,mk=0,\\dots,m, each\|τj−ςj\|\\left\\lvert\\tau\_\{j\}\-\\varsigma\_\{j\}\\right\\rvertappears with coefficient at mostTmax/2T\_\{\\max\}/2\(sinceΔj−1\+Δj≤Tmax\\Delta\_\{j\-1\}\+\\Delta\_\{j\}\\leq T\_\{\\max\}\)\. Hence

‖x−y~‖L1≤Tmax2​∑k=0m\+1\|τk−ςk\|≤Tmax​∑k=1n\|tk−sk\|=Tmax​d𝒩​\(η,ξ\),\\left\\lVert x\-\\tilde\{y\}\\right\\rVert\_\{L^\{1\}\}\\;\\leq\\;\\frac\{T\_\{\\max\}\}\{2\}\\sum\_\{k=0\}^\{m\+1\}\\left\\lvert\\tau\_\{k\}\-\\varsigma\_\{k\}\\right\\rvert\\;\\leq\\;T\_\{\\max\}\\sum\_\{k=1\}^\{n\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\;=\\;T\_\{\\max\}\\;d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\,,\(S44\)where the second inequality applies Lemma[S8](https://arxiv.org/html/2607.06652#Thmtheorem8)\.

Bound \(II\): Horizontal mismatch \(different grids\)\.Recall thaty=Φ​\(ξ\)y=\\Phi\(\\xi\)is the piecewise–linear interpolant on the grid0=s0<s1<⋯<sn\+1=Tmax0=s\_\{0\}<s\_\{1\}<\\cdots<s\_\{n\+1\}=T\_\{\\max\}with nodal valuesy​\(sk\)=ςky\(s\_\{k\}\)=\\varsigma\_\{k\}, and thaty~\\tilde\{y\}is the piecewise–linear interpolant on0=t0<t1<⋯<tm\+1=Tmax0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\+1\}=T\_\{\\max\}with the same nodal valuesy~​\(tk\)=ςk\\tilde\{y\}\(t\_\{k\}\)=\\varsigma\_\{k\}fork=0,…,m\+1k=0,\\dots,m\{\+\}1\.

Define the piecewise\-linear increasing mapλ:\[0,Tmax\]→\[0,Tmax\]\\lambda:\[0,T\_\{\\max\}\]\\to\[0,T\_\{\\max\}\]byλ​\(tk\)=sk\\lambda\(t\_\{k\}\)=s\_\{k\}fork=0,…,m\+1k=0,\\dots,m\{\+\}1, linear on each\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]\.λ\\lambdais the unique increasing piecewise\-linear map sending until indexmmeach knottkt\_\{k\}tosks\_\{k\}\. Then Lemma[S9](https://arxiv.org/html/2607.06652#Thmtheorem9)givesy~​\(t\)=y​\(λ​\(t\)\)\\tilde\{y\}\(t\)=y\(\\lambda\(t\)\)for allttand

‖y~−y‖L1≤4​Tmax​∑k=1n\|tk−sk\|=4​Tmax​d𝒩​\(η,ξ\)\.\\left\\lVert\\tilde\{y\}\-y\\right\\rVert\_\{L^\{1\}\}\\;\\leq\\;4T\_\{\\max\}\\sum\_\{k=1\}^\{n\}\\left\\lvert t\_\{k\}\-s\_\{k\}\\right\\rvert\\;=\\;4T\_\{\\max\}\\;d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\,\.\(S45\)
Conclusion\.Combining Eq\.[S42](https://arxiv.org/html/2607.06652#A2.E42)with Bounds \(I\) and \(II\):

d1​\(Φ​\(η\),Φ​\(ξ\)\)≤\(1\+4\)​Tmax​d𝒩​\(η,ξ\)=5​Tmax​d𝒩​\(η,ξ\)\.d\_\{1\}\\bigl\(\\Phi\(\\eta\),\\Phi\(\\xi\)\\bigr\)\\;\\leq\\;\(1\+4\)\\,T\_\{\\max\}\\;d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\;=\\;5T\_\{\\max\}\\;d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\,\.\(S46\)∎

### B\.2Φ\\Phiis a Bijection

![Refer to caption](https://arxiv.org/html/2607.06652v1/images/theory/inverse_plot2.png)Figure S1:Recovery of event times withffby iterative level\-hitting\. Here, the TPPη\\etais constructed artificially with timestk=\[1,3,7,8\]t\_\{k\}=\[1,3,7,8\], interarrival timesτk=\[1,2,4,1\]\\tau\_\{k\}=\[1,2,4,1\], andTmax=10T\_\{\\max\}=10\. Starting fromt0=0t\_\{0\}=0, each event time is obtained astk=inf\{t\>tk−1:f​\(t\)=tk−1\}t\_\{k\}=\\inf\\\{t\>t\_\{k\-1\}:f\(t\)=t\_\{k\-1\}\\\}\. Horizontal dashed lines denote thresholdstk−1t\_\{k\-1\}, and vertical dashed lines mark the corresponding hit timestkt\_\{k\}\. The figure illustrates the constructive recovery underlying the bijectivity ofΦ\\Phi\.###### Definition S13\(Backshift Transform\)\.

Forx∈𝒞x\\in\\mathcal\{C\}, the backshift transform ofxxis

fx​\(t\):=t−x​\(t\),t∈\[0,Tmax\]\.f\_\{x\}\(t\):=t\-x\(t\)\\,,\\qquad t\\in\[0,T\_\{\\max\}\]\\,\.\(S47\)

Whenx=Φ​\(η\)x=\\Phi\(\\eta\)for someη∈𝒩\\eta\\in\\mathcal\{N\}with event times0<t1<⋯<tm<Tmax0<t\_\{1\}<\\cdots<t\_\{m\}<T\_\{\\max\}and interarrival timesτk=tk−tk−1\\tau\_\{k\}=t\_\{k\}\-t\_\{k\-1\}\(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\), the key observation is thatfx​\(tk\)=tk−τk=tk−1f\_\{x\}\(t\_\{k\}\)=t\_\{k\}\-\\tau\_\{k\}=t\_\{k\-1\}\. That is,fxf\_\{x\}maps each event time to its predecessor\. This is the basis for the level\-set recovery of Lemma[S14](https://arxiv.org/html/2607.06652#Thmtheorem14)\.

###### Lemma S14\(Level\-Set Recovery\)\.

Letη∈𝒩\\eta\\in\\mathcal\{N\}have event times0<t1<⋯<tm<Tmax0<t\_\{1\}<\\cdots<t\_\{m\}<T\_\{\\max\}, augmented byt0:=0t\_\{0\}:=0andtm\+1:=Tmaxt\_\{m\+1\}:=T\_\{\\max\}\(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\), interarrival timesτk:=tk−tk−1\\tau\_\{k\}:=t\_\{k\}\-t\_\{k\-1\}, and setx:=Φ​\(η\)x:=\\Phi\(\\eta\)\. Ifm=0m=0thenfx≡0f\_\{x\}\\equiv 0on\[0,Tmax\]\[0,T\_\{\\max\}\]; assume henceforthm≥1m\\geq 1\. Then the backshift transformfx​\(t\)=t−x​\(t\)f\_\{x\}\(t\)=t\-x\(t\)satisfies the following properties\.

1. \(i\)fxf\_\{x\}is continuous and piecewise\-linear on\[0,Tmax\]\[0,T\_\{\\max\}\]\.
2. \(ii\)fx​\(0\)=0f\_\{x\}\(0\)=0,fx​\(tk\)=tk−1f\_\{x\}\(t\_\{k\}\)=t\_\{k\-1\}fork=1,…,mk=1,\\dots,m, andfx​\(Tmax\)=tmf\_\{x\}\(T\_\{\\max\}\)=t\_\{m\}\.
3. \(iii\)fx≡0f\_\{x\}\\equiv 0on\[0,t1\]\[0,t\_\{1\}\]\.
4. \(iv\)fxf\_\{x\}is strictly increasing on\[t1,Tmax\]\[t\_\{1\},T\_\{\\max\}\]\.
5. \(v\)Fork=1,…,mk=1,\\dots,m,tk=sup\{t∈\(tk−1,Tmax\):fx​\(t\)=tk−1\}t\_\{k\}=\\sup\\\{t\\in\(t\_\{k\-1\},T\_\{\\max\}\):f\_\{x\}\(t\)=t\_\{k\-1\}\\\}\.

###### Proof\.

\(i\)Sincex∈𝒞x\\in\\mathcal\{C\}is continuous and piecewise\-linear on the grid\(tk\)k=0m\+1\(t\_\{k\}\)\_\{k=0\}^\{m\+1\}and the mapt↦tt\\mapsto tis linear, their differencefxf\_\{x\}inherits the same properties\.

\(ii\)At each event timetkt\_\{k\},

x​\(tk\)=τk=tk−tk−1,x\(t\_\{k\}\)=\\tau\_\{k\}=t\_\{k\}\-t\_\{k\-1\}\\,,\(S48\)hence

fx​\(tk\)=tk−τk=tk−1\.f\_\{x\}\(t\_\{k\}\)=t\_\{k\}\-\\tau\_\{k\}=t\_\{k\-1\}\\,\.\(S49\)At the origin,x​\(0\)=τ0=0x\(0\)=\\tau\_\{0\}=0, sofx​\(0\)=0f\_\{x\}\(0\)=0\.

\(iii\)On\[0,t1\]\[0,t\_\{1\}\],xxhas slope equal to11by definition and is linear withx​\(0\)=0x\(0\)=0\. Sox​\(t\)=tx\(t\)=tandfx≡0f\_\{x\}\\equiv 0\.

\(iv\)On each\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]fork=1,…,mk=1,\\dots,m, the functionfxf\_\{x\}is affine with

fx​\(tk\+1\)−fx​\(tk\)=tk−tk−1=τk\>0,f\_\{x\}\(t\_\{k\+1\}\)\-f\_\{x\}\(t\_\{k\}\)=t\_\{k\}\-t\_\{k\-1\}=\\tau\_\{k\}\>0\\,,\(S50\)where strict positivity follows from simplicity ofη\\eta\(i\.e\.τk\>0\\tau\_\{k\}\>0fork=1,…,mk=1,\\dots,m\)\. Sincefxf\_\{x\}is affine and strictly increasing on each subinterval and continuous across nodes, it is strictly increasing on all of\[t1,Tmax\]\[t\_\{1\},T\_\{\\max\}\]\.

\(v\)Fork=1k=1: by \(iii\),fx≡0f\_\{x\}\\equiv 0on\[0,t1\]\[0,t\_\{1\}\], and by \(iv\),fx\>0f\_\{x\}\>0on\(t1,Tmax\]\(t\_\{1\},T\_\{\\max\}\]\. Hence\{t∈\(0,Tmax\):fx​\(t\)=0\}=\(0,t1\]\\\{t\\in\(0,T\_\{\\max\}\):f\_\{x\}\(t\)=0\\\}=\(0,t\_\{1\}\]andsup=t1\\sup=t\_\{1\}\. Fork≥2k\\geq 2: by \(iv\),fxf\_\{x\}is strictly increasing on\[tk−1,Tmax\]⊂\[t1,Tmax\]\[t\_\{k\-1\},T\_\{\\max\}\]\\subset\[t\_\{1\},T\_\{\\max\}\]\. Sincefx​\(tk\)=tk−1f\_\{x\}\(t\_\{k\}\)=t\_\{k\-1\}andfxf\_\{x\}is strictly increasing on this interval,tkt\_\{k\}is the unique solution offx​\(t\)=tk−1f\_\{x\}\(t\)=t\_\{k\-1\}in\[tk−1,Tmax\]\[t\_\{k\-1\},T\_\{\\max\}\], hence the supremum\. ∎

###### Lemma S15\(Inverse Recovery\)\.

Forx∈Φ​\(𝒩\)x\\in\\Phi\(\\mathcal\{N\}\), define the sequence\(tk\)k≥0\(t\_\{k\}\)\_\{k\\geq 0\}by

t0:=0,tk:=sup\{t∈\(tk−1,Tmax\):fx​\(t\)=tk−1\},k≥1,t\_\{0\}:=0,\\qquad t\_\{k\}:=\\sup\\bigl\\\{t\\in\(t\_\{k\-1\},T\_\{\\max\}\):f\_\{x\}\(t\)=t\_\{k\-1\}\\bigr\\\},\\quad k\\geq 1\\,,\(S51\)and letm≥0m\\geq 0be the first index for which the level set\{t∈\(tm,Tmax\):fx​\(t\)=tm\}\\\{t\\in\(t\_\{m\},T\_\{\\max\}\):f\_\{x\}\(t\)=t\_\{m\}\\\}is empty or has supremum equal toTmaxT\_\{\\max\}\. Then:

1. \(i\)The recovered sequence satisfies0<t1<t2<⋯<tm<Tmax0<t\_\{1\}<t\_\{2\}<\\cdots<t\_\{m\}<T\_\{\\max\}andm<∞m<\\infty; moreoverm=0m=0if and only ifx​\(t\)=tx\(t\)=tfor alltt, the embedding of the eventless path\.
2. \(ii\)The mapΨ:Φ​\(𝒩\)→𝒩\\Psi:\\Phi\(\\mathcal\{N\}\)\\to\\mathcal\{N\}defined by Ψ​\(x\)t:=\#​\{k∈\{1,…,m\}:tk≤t\},\\Psi\(x\)\_\{t\}:=\\\#\\\{k\\in\\\{1,\\dots,m\\\}:t\_\{k\}\\leq t\\\}\\,,\(S52\)is a well\-defined element of𝒩\\mathcal\{N\}\.
3. \(iii\)Ifx=Φ​\(η\)x=\\Phi\(\\eta\)for someη∈𝒩\\eta\\in\\mathcal\{N\}with event times\(sk\)k=1n\(s\_\{k\}\)\_\{k=1\}^\{n\}, thenm=nm=nandtk=skt\_\{k\}=s\_\{k\}fork=1,…,nk=1,\\dots,n; equivalently,Ψ​\(Φ​\(η\)\)=η\\Psi\(\\Phi\(\\eta\)\)=\\eta\.

###### Proof\.

Fixη∈𝒩\\eta\\in\\mathcal\{N\}with event times0<s1<⋯<sn<Tmax0<s\_\{1\}<\\cdots<s\_\{n\}<T\_\{\\max\}, augmented bys0:=0s\_\{0\}:=0andsn\+1:=Tmaxs\_\{n\+1\}:=T\_\{\\max\}\(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\), interarrival timesςk:=sk−sk−1\\varsigma\_\{k\}:=s\_\{k\}\-s\_\{k\-1\}, and setx:=Φ​\(η\)x:=\\Phi\(\\eta\)\. Ifn=0n=0, thenx​\(t\)=tx\(t\)=tandfx≡0f\_\{x\}\\equiv 0, so the index\-0level set is\(0,Tmax\)\(0,T\_\{\\max\}\), whose supremum isTmaxT\_\{\\max\}: the recursion stops immediately withm=0m=0andΨ​\(x\)≡0=η\\Psi\(x\)\\equiv 0=\\eta\. Assume henceforthn≥1n\\geq 1\.

We show by induction that the recursion in Eq\.[S51](https://arxiv.org/html/2607.06652#A2.E51)recoverstk=skt\_\{k\}=s\_\{k\}for everykk; well\-definedness ofΨ\\Psiis then immediate\.

Part \(i\): Strict monotonicity and finiteness\.*Base case \(k=1k=1\)\.*On\[0,s1\]\[0,s\_\{1\}\], the pathxxinterpolates linearly from\(0,0\)\(0,0\)to\(s1,ς1\)=\(s1,s1\)\(s\_\{1\},\\varsigma\_\{1\}\)=\(s\_\{1\},s\_\{1\}\), hencex​\(t\)=tx\(t\)=tandfx​\(t\)=0f\_\{x\}\(t\)=0for allt∈\[0,s1\]t\\in\[0,s\_\{1\}\]\. By Lemma[S14](https://arxiv.org/html/2607.06652#Thmtheorem14)\(iv\),fxf\_\{x\}is strictly increasing on\[s1,Tmax\]\[s\_\{1\},T\_\{\\max\}\], sofx​\(t\)\>0f\_\{x\}\(t\)\>0for allt\>s1t\>s\_\{1\}\. Therefore

\{t∈\(0,Tmax\):fx​\(t\)=0\}=\(0,s1\],\\\{t\\in\(0,T\_\{\\max\}\):f\_\{x\}\(t\)=0\\\}=\(0,s\_\{1\}\]\\,,\(S53\)and the supremum yieldst1=s1t\_\{1\}=s\_\{1\}\.

*Inductive step\.*Assumetj=sjt\_\{j\}=s\_\{j\}for allj≤k−1j\\leq k\-1\. By Lemma[S14](https://arxiv.org/html/2607.06652#Thmtheorem14), the equationfx​\(t\)=sk−1f\_\{x\}\(t\)=s\_\{k\-1\}has a unique solution in\(sk−1,Tmax\)\(s\_\{k\-1\},T\_\{\\max\}\), namelysks\_\{k\}\. Hencetk=sup\{sk\}=sk,t\_\{k\}=\\sup\\\{s\_\{k\}\\\}=s\_\{k\},and in particulartk\>tk−1t\_\{k\}\>t\_\{k\-1\}\. For eachk=1,…,nk=1,\\dots,nthe level set has supremumsk<Tmaxs\_\{k\}<T\_\{\\max\}, so the stopping rule does not trigger before indexnn\.

*Termination\.*It remains to show that no time is recorded aftertn=snt\_\{n\}=s\_\{n\}\. By Lemma[S14](https://arxiv.org/html/2607.06652#Thmtheorem14)\(ii\) to \(iv\),fx≤fx​\(sn\)=sn−1<snf\_\{x\}\\leq f\_\{x\}\(s\_\{n\}\)=s\_\{n\-1\}<s\_\{n\}on\[0,sn\]\[0,s\_\{n\}\], andfxf\_\{x\}is strictly increasing on\[sn,Tmax\]\[s\_\{n\},T\_\{\\max\}\]withfx​\(Tmax\)=snf\_\{x\}\(T\_\{\\max\}\)=s\_\{n\}; hencefx​\(t\)<snf\_\{x\}\(t\)<s\_\{n\}for everyt∈\(sn,Tmax\)t\\in\(s\_\{n\},T\_\{\\max\}\)\. The level set\{t∈\(sn,Tmax\):fx​\(t\)=sn\}\\\{t\\in\(s\_\{n\},T\_\{\\max\}\):f\_\{x\}\(t\)=s\_\{n\}\\\}is therefore empty, and the recursion terminates withm=n<∞m=n<\\infty\.

Part \(ii\): Well\-definedness ofΨ​\(x\)\\Psi\(x\)\.By \(i\), the sequence\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}is finite and strictly increasing\. The process defined in Eq\.[S52](https://arxiv.org/html/2607.06652#A2.E52)is therefore nondecreasing, càdlàg, has unit jumps only, satisfiesΨ​\(x\)0=0\\Psi\(x\)\_\{0\}=0, and hasΨ​\(x\)Tmax=m<∞\\Psi\(x\)\_\{T\_\{\\max\}\}=m<\\infty\. HenceΨ​\(x\)∈𝒩\\Psi\(x\)\\in\\mathcal\{N\}\. Whenm=0m=0,Ψ​\(x\)≡0\\Psi\(x\)\\equiv 0is the eventless path, which lies in𝒩\\mathcal\{N\}trivially\. ∎

###### Theorem S16\(Bijective Embedding\)\.

The following identities hold\.

1. \(i\)Ψ∘Φ=Id𝒩\\Psi\\circ\\Phi=\\mathrm\{Id\}\_\{\\mathcal\{N\}\}\.
2. \(ii\)Φ∘Ψ=IdΦ​\(𝒩\)\\Phi\\circ\\Psi=\\mathrm\{Id\}\_\{\\Phi\(\\mathcal\{N\}\)\}\.

In particular,Φ:𝒩→Φ​\(𝒩\)\\Phi:\\mathcal\{N\}\\to\\Phi\(\\mathcal\{N\}\)is a bijection with inverseΨ\\Psi\.

###### Proof\.

We continue with the notation of the previous lemmas\. First we proveΨ∘Φ=Id𝒩\\Psi\\circ\\Phi=\\mathrm\{Id\}\_\{\\mathcal\{N\}\}\. Letη∈𝒩\\eta\\in\\mathcal\{N\}have event times\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}\. Lemma[S15](https://arxiv.org/html/2607.06652#Thmtheorem15)shows that applying the recursion tox=Φ​\(η\)x=\\Phi\(\\eta\)recovers all event times exactly, with no extra times\. Therefore,

Ψ​\(Φ​\(η\)\)t=\#​\{k∈\{1,…,m\}:tk≤t\}=ηt,t∈\[0,Tmax\]\.\\Psi\(\\Phi\(\\eta\)\)\_\{t\}=\\\#\\\{k\\in\\\{1,\\dots,m\\\}:t\_\{k\}\\leq t\\\}=\\eta\_\{t\}\\,,\\qquad t\\in\[0,T\_\{\\max\}\]\\,\.\(S54\)
Next we proveΦ∘Ψ=IdΦ​\(𝒩\)\\Phi\\circ\\Psi=\\mathrm\{Id\}\_\{\\Phi\(\\mathcal\{N\}\)\}\. Letx∈Φ​\(𝒩\)x\\in\\Phi\(\\mathcal\{N\}\), sox=Φ​\(η\)x=\\Phi\(\\eta\)for someη∈𝒩\\eta\\in\\mathcal\{N\}with event times\(tk\)k=1m\(t\_\{k\}\)\_\{k=1\}^\{m\}\. By the previous statement,Ψ​\(x\)\\Psi\(x\)recovers exactly these event times\. ReapplyingΦ\\Phiyields the same piecewise\-linear interpolation through\(tk,τk\)\(t\_\{k\},\\tau\_\{k\}\), henceΦ​\(Ψ​\(x\)\)=x\\Phi\(\\Psi\(x\)\)=x\.

The mapsΦ\\PhiandΨ\\Psiare therefore mutual inverses, andΦ:𝒩→Φ​\(𝒩\)\\Phi:\\mathcal\{N\}\\to\\Phi\(\\mathcal\{N\}\)is a bijection\. ∎

### B\.3Ψ\\Psiis Continuous on Each SieveΦ​\(𝒩δ\)\\Phi\(\\mathcal\{N\}\_\{\\delta\}\)

Hölder stability of the inverse\.Fixδ\>0\\delta\>0and let

𝒩δ:=\{η∈𝒩:τk≥δ​for all​k=1,…,m\+1\},\\mathcal\{N\}\_\{\\delta\}:=\\bigl\\\{\\eta\\in\\mathcal\{N\}:\\tau\_\{k\}\\geq\\delta\\text\{ for all \}k=1,\\dots,m\+1\\bigr\\\}\\,,\(S55\)wherem=ηTmaxm=\\eta\_\{T\_\{\\max\}\}andτ1,…,τm\+1\\tau\_\{1\},\\dots,\\tau\_\{m\+1\}are the interarrival times \(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\)\.

The separation parameterδ\\deltaexcludes near\-simultaneous events, which cause the embedding’s slopes to become indistinguishable\. In practice, temporal data recorded at finite precision satisfiesη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}for some resolution\-dependentδ\\delta\. The Lipschitz bound and bijectivity hold without any separation assumption\.

###### Lemma S18\(Quantitative Bounds forfxf\_\{x\}on the Sieve\)\.

Letδ\>0\\delta\>0,Mδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta\. Letη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}havem≥1m\\geq 1events, and write\(tk\)k=0m\+1\(t\_\{k\}\)\_\{k=0\}^\{m\+1\}for its augmented event\-time grid and\(τk\)k=0m\+1\(\\tau\_\{k\}\)\_\{k=0\}^\{m\+1\}for the corresponding interarrival times \(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\), and setx:=Φ​\(η\)x:=\\Phi\(\\eta\),fx​\(t\):=t−x​\(t\)f\_\{x\}\(t\):=t\-x\(t\)\. Then on each interval\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\],k=1,…,mk=1,\\dots,m, the functionfxf\_\{x\}is affine with slope

rk:=τkτk\+1∈\[Mδ−1,Mδ\]\.r\_\{k\}\\;:=\\;\\frac\{\\tau\_\{k\}\}\{\\tau\_\{k\+1\}\}\\;\\in\\;\\bigl\[M\_\{\\delta\}^\{\-1\},\\;M\_\{\\delta\}\\bigr\]\\,\.\(S56\)In particular,fxf\_\{x\}isMδM\_\{\\delta\}\-Lipschitz on\[0,Tmax\]\[0,T\_\{\\max\}\]\.

###### Proof\.

Fixk∈\{1,…,m\}k\\in\\\{1,\\dots,m\\\}\. By construction,x=Φ​\(η\)x=\\Phi\(\\eta\)is affine on\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]with valuesx​\(tk\)=τkx\(t\_\{k\}\)=\\tau\_\{k\}andx​\(tk\+1\)=τk\+1x\(t\_\{k\+1\}\)=\\tau\_\{k\+1\}\. Its slope is\(τk\+1−τk\)/τk\+1\(\\tau\_\{k\+1\}\-\\tau\_\{k\}\)/\\tau\_\{k\+1\}, sofx=id−xf\_\{x\}=\\mathrm\{id\}\-xis affine on\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]with slope

1−τk\+1−τkτk\+1=τkτk\+1\.1\-\\frac\{\\tau\_\{k\+1\}\-\\tau\_\{k\}\}\{\\tau\_\{k\+1\}\}\\;=\\;\\frac\{\\tau\_\{k\}\}\{\\tau\_\{k\+1\}\}\\,\.\(S57\)Sinceη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}, every interarrival time lies in\[δ,Tmax\]\[\\delta,T\_\{\\max\}\], givingrk=τk/τk\+1∈\[δ/Tmax,Tmax/δ\]=\[1/Mδ,Mδ\]r\_\{k\}=\\tau\_\{k\}/\\tau\_\{k\+1\}\\in\[\\delta/T\_\{\\max\},T\_\{\\max\}/\\delta\]=\[1/M\_\{\\delta\},M\_\{\\delta\}\]\.

Finally,fxf\_\{x\}is continuous and piecewise affine\. On\[0,t1\]\[0,t\_\{1\}\]the slope is0; on each subsequent piece it isrk≤Mδr\_\{k\}\\leq M\_\{\\delta\}\. Hencefxf\_\{x\}isMδM\_\{\\delta\}\-Lipschitz\. ∎

###### Proposition S19\(Restricted Invertibility of the Backshift Transform on the Sieve\)\.

Letδ\>0\\delta\>0,Mδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta\. Letη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}havem≥1m\\geq 1events, and write\(tk\)k=0m\+1\(t\_\{k\}\)\_\{k=0\}^\{m\+1\}for its augmented event\-time grid \(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\), and setx:=Φ​\(η\)x:=\\Phi\(\\eta\),fx​\(t\):=t−x​\(t\)f\_\{x\}\(t\):=t\-x\(t\)\. Then:

1. \(i\)The restrictionfx\|\[t1,Tmax\]:\[t1,Tmax\]→\[0,tm\]f\_\{x\}\\big\|\_\{\[t\_\{1\},T\_\{\\max\}\]\}:\[t\_\{1\},T\_\{\\max\}\]\\to\[0,t\_\{m\}\]is a continuous, strictly increasing bijection\.
2. \(ii\)Its inverseFx:=\(fx\|\[t1,Tmax\]\)−1F\_\{x\}:=\\bigl\(f\_\{x\}\\big\|\_\{\[t\_\{1\},T\_\{\\max\}\]\}\\bigr\)^\{\-1\}satisfies, for allu,v∈\[0,tm\]u,v\\in\[0,t\_\{m\}\], \|Fx​\(u\)−Fx​\(v\)\|≤Mδ​\|u−v\|\.\\left\\lvert F\_\{x\}\(u\)\-F\_\{x\}\(v\)\\right\\rvert\\leq M\_\{\\delta\}\\,\\left\\lvert u\-v\\right\\rvert\\,\.\(S58\)

###### Proof\.

By Lemma[S18](https://arxiv.org/html/2607.06652#Thmtheorem18),fxf\_\{x\}is affine on each\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]\(k=1,…,mk=1,\\dots,m\) with strictly positive sloperk≥Mδ−1r\_\{k\}\\geq M\_\{\\delta\}^\{\-1\}\. In particular,fxf\_\{x\}is continuous and strictly increasing on\[t1,Tmax\]\[t\_\{1\},T\_\{\\max\}\], hence a bijection onto\[fx​\(t1\),fx​\(Tmax\)\]\[f\_\{x\}\(t\_\{1\}\),\\,f\_\{x\}\(T\_\{\\max\}\)\]\.

It remains to identify the endpoints\. By Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1),τ1=t1−t0=t1\\tau\_\{1\}=t\_\{1\}\-t\_\{0\}=t\_\{1\}andτm\+1=tm\+1−tm=Tmax−tm\\tau\_\{m\+1\}=t\_\{m\+1\}\-t\_\{m\}=T\_\{\\max\}\-t\_\{m\}\. Sincex=Φ​\(η\)x=\\Phi\(\\eta\)interpolates the nodes\(tk,τk\)\(t\_\{k\},\\tau\_\{k\}\), we havex​\(t1\)=τ1=t1x\(t\_\{1\}\)=\\tau\_\{1\}=t\_\{1\}andx​\(Tmax\)=x​\(tm\+1\)=τm\+1=Tmax−tmx\(T\_\{\\max\}\)=x\(t\_\{m\+1\}\)=\\tau\_\{m\+1\}=T\_\{\\max\}\-t\_\{m\}\. Therefore,

fx​\(t1\)=t1−x​\(t1\)=0,fx​\(Tmax\)=Tmax−x​\(Tmax\)=tm,f\_\{x\}\(t\_\{1\}\)=t\_\{1\}\-x\(t\_\{1\}\)=0,\\qquad f\_\{x\}\(T\_\{\\max\}\)=T\_\{\\max\}\-x\(T\_\{\\max\}\)=t\_\{m\}\\,,\(S59\)so the range is exactly\[0,tm\]\[0,t\_\{m\}\]\.

On each image intervalfx​\(\[tk,tk\+1\]\)f\_\{x\}\(\[t\_\{k\},t\_\{k\+1\}\]\), the inverseFxF\_\{x\}is affine with slope1/rk≤Mδ1/r\_\{k\}\\leq M\_\{\\delta\}\. SinceFxF\_\{x\}is continuous and piecewise\-affine on a compact interval with all slopes at mostMδM\_\{\\delta\}, it is globallyMδM\_\{\\delta\}\-Lipschitz\. ∎

###### Lemma S20\(L1L^\{1\}–L∞L^\{\\infty\}Interpolation for Lipschitz functions\)\.

Leth:\[0,Tmax\]→ℝh:\[0,T\_\{\\max\}\]\\to\\mathbb\{R\}beLL\-Lipschitz withh​\(0\)=0h\(0\)=0andL\>0L\>0\. Then

‖h‖∞2≤4​L​‖h‖1,\\left\\lVert h\\right\\rVert\_\{\\infty\}^\{2\}\\;\\leq\\;4\\,L\\;\\left\\lVert h\\right\\rVert\_\{1\}\\,,\(S60\)where‖h‖1:=∫0Tmax\|h​\(t\)\|​d​t\\left\\lVert h\\right\\rVert\_\{1\}:=\\int\_\{0\}^\{T\_\{\\max\}\}\|h\(t\)\|\\mathop\{\}\\\!\\mathrm\{d\}tand‖h‖∞:=maxt∈\[0,Tmax\]⁡\|h​\(t\)\|\\left\\lVert h\\right\\rVert\_\{\\infty\}:=\\max\_\{t\\in\[0,T\_\{\\max\}\]\}\|h\(t\)\|\.

###### Proof\.

LetM:=‖h‖∞M:=\\left\\lVert h\\right\\rVert\_\{\\infty\}and chooset∗∈\[0,Tmax\]t^\{\*\}\\in\[0,T\_\{\\max\}\]with\|h​\(t∗\)\|=M\|h\(t^\{\*\}\)\|=M; ifM=0M=0the claim is trivial, so assumeM\>0M\>0\. Sinceh​\(0\)=0h\(0\)=0andhhisLL\-Lipschitz,M=\|h​\(t∗\)−h​\(0\)\|≤L​t∗M=\|h\(t^\{\*\}\)\-h\(0\)\|\\leq L\\,t^\{\*\}, sot∗≥M/Lt^\{\*\}\\geq M/L\. Consequently, the interval\[t∗−M/\(2​L\),t∗\]\[t^\{\*\}\-M/\(2L\),\\;t^\{\*\}\]has lengthM/\(2​L\)M/\(2L\)and is contained in\[0,Tmax\]\[0,T\_\{\\max\}\]\. Forttin this interval, the Lipschitz bound gives

\|h​\(t\)\|≥\|h​\(t∗\)\|−L​\|t−t∗\|≥M−L⋅M2​L=M2\.\|h\(t\)\|\\;\\geq\\;\|h\(t^\{\*\}\)\|\-L\\,\|t\-t^\{\*\}\|\\;\\geq\\;M\-L\\cdot\\frac\{M\}\{2L\}\\;=\\;\\frac\{M\}\{2\}\\,\.\(S61\)Therefore

‖h‖1≥∫t∗−M/\(2​L\)t∗\|h​\(t\)\|​d​t≥M2⋅M2​L=M24​L\.\\left\\lVert h\\right\\rVert\_\{1\}\\;\\geq\\;\\int\_\{t^\{\*\}\-M/\(2L\)\}^\{t^\{\*\}\}\|h\(t\)\|\\mathop\{\}\\\!\\mathrm\{d\}t\\;\\geq\\;\\frac\{M\}\{2\}\\cdot\\frac\{M\}\{2L\}\\;=\\;\\frac\{M^\{2\}\}\{4L\}\\,\.\(S62\)∎

###### Proposition S21\(Sup\-norm Control\)\.

Letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}and setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\),g:=x−yg:=x\-y\. AssumeMδ:=Tmax/δ≥2M\_\{\\delta\}:=T\_\{\\max\}/\\delta\\geq 2; whenMδ<2M\_\{\\delta\}<2, the sieve𝒩δ\\mathcal\{N\}\_\{\\delta\}contains at most the eventless path, so the only available pair isx=yx=yand the bounds below are trivial\.

1. \(i\)Each ofxxandyyis\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz on\[0,Tmax\]\[0,T\_\{\\max\}\], andggsatisfiesg​\(0\)=0g\(0\)=0and is2​\(Mδ−1\)2\(M\_\{\\delta\}\-1\)\-Lipschitz\.
2. \(ii\)‖g‖∞≤2​2​\(Mδ−1\)​d1​\(x,y\)≤2​2​Mδ​d1​\(x,y\)\\displaystyle\\left\\lVert g\\right\\rVert\_\{\\infty\}\\;\\leq\\;2\\sqrt\{2\(M\_\{\\delta\}\-1\)\\,d\_\{1\}\(x,y\)\}\\;\\leq\\;2\\sqrt\{2M\_\{\\delta\}\\,d\_\{1\}\(x,y\)\}\.

###### Proof\.

We treat three cases according to the event counts ofη\\etaandξ\\xi\.

Case 1: Bothη\\etaandξ\\xihave zero events\.Thenx​\(t\)=y​\(t\)=tx\(t\)=y\(t\)=tfor allt∈\[0,Tmax\]t\\in\[0,T\_\{\\max\}\], sog≡0g\\equiv 0and both claims hold trivially\.

Case 2: Bothη\\etaandξ\\xihave at least one event\.Letm≥1m\\geq 1denote the event count ofη\\eta, with augmented grid0=t0<t1<⋯<tm<tm\+1=Tmax0=t\_\{0\}<t\_\{1\}<\\cdots<t\_\{m\}<t\_\{m\+1\}=T\_\{\\max\}and interarrival timesτk:=tk−tk−1\\tau\_\{k\}:=t\_\{k\}\-t\_\{k\-1\}\.

Part \(i\): Lipschitz bound\.By Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1),x​\(0\)=τ0=0x\(0\)=\\tau\_\{0\}=0andy​\(0\)=ς0=0y\(0\)=\\varsigma\_\{0\}=0, sog​\(0\)=0g\(0\)=0\.

We showx=Φ​\(η\)x=\\Phi\(\\eta\)is\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz\. Fork≥1k\\geq 1, the slope ofxxon\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]is

αk:=τk\+1−τkτk\+1=1−τkτk\+1\.\\alpha\_\{k\}\\;:=\\;\\frac\{\\tau\_\{k\+1\}\-\\tau\_\{k\}\}\{\\tau\_\{k\+1\}\}\\;=\\;1\-\\frac\{\\tau\_\{k\}\}\{\\tau\_\{k\+1\}\}\\,\.\(S63\)Sinceη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}, every interarrival time lies in\[δ,Tmax\]\[\\delta,T\_\{\\max\}\], soτk/τk\+1∈\[Mδ−1,Mδ\]\\tau\_\{k\}/\\tau\_\{k\+1\}\\in\[M\_\{\\delta\}^\{\-1\},M\_\{\\delta\}\]and hence

\|αk\|≤max⁡\(\|1−Mδ−1\|,\|1−Mδ\|\)=Mδ−1\.\|\\alpha\_\{k\}\|\\;\\leq\\;\\max\\bigl\(\|1\-M\_\{\\delta\}^\{\-1\}\|,\\;\|1\-M\_\{\\delta\}\|\\bigr\)\\;=\\;M\_\{\\delta\}\-1\\,\.\(S64\)Fork=0k=0, the slope isα0=τ1/τ1=1\\alpha\_\{0\}=\\tau\_\{1\}/\\tau\_\{1\}=1\. Sincem≥1m\\geq 1, we haveTmax≥\(m\+1\)​δ≥2​δT\_\{\\max\}\\geq\(m\+1\)\\delta\\geq 2\\delta, soMδ≥2M\_\{\\delta\}\\geq 2and\|α0\|=1≤Mδ−1\|\\alpha\_\{0\}\|=1\\leq M\_\{\\delta\}\-1\.

Thusxxis\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz on\[0,Tmax\]\[0,T\_\{\\max\}\]\. The same argument applies toyy\(which also has≥1\\geq 1event\), soyyis\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz\. Their differenceggis therefore2​\(Mδ−1\)2\(M\_\{\\delta\}\-1\)\-Lipschitz\.

Case 3: Exactly one ofη\\eta,ξ\\xihas zero events\.Without loss of generality, supposeη\\etahasm=0m=0events andξ\\xihasn≥1n\\geq 1events\. Thenx​\(t\)=Φ​\(η\)​\(t\)=tx\(t\)=\\Phi\(\\eta\)\(t\)=tfor allt∈\[0,Tmax\]t\\in\[0,T\_\{\\max\}\], soxxhas constant slope11\. Sinceξ\\xihasn≥1n\\geq 1events,Tmax≥\(n\+1\)​δ≥2​δT\_\{\\max\}\\geq\(n\+1\)\\delta\\geq 2\\delta, soMδ≥2M\_\{\\delta\}\\geq 2\. Therefore\|1\|=1≤Mδ−1\|\{1\}\|=1\\leq M\_\{\\delta\}\-1, andxxis\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz\. The functiony=Φ​\(ξ\)y=\\Phi\(\\xi\)hasn≥1n\\geq 1events in𝒩δ\\mathcal\{N\}\_\{\\delta\}, so by the slope analysis in Case 2 \(applied toξ\\xi\),yyis also\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz\. Sinceg​\(0\)=x​\(0\)−y​\(0\)=0−0=0g\(0\)=x\(0\)\-y\(0\)=0\-0=0andggis the difference of two\(Mδ−1\)\(M\_\{\\delta\}\-1\)\-Lipschitz functions,ggis2​\(Mδ−1\)2\(M\_\{\\delta\}\-1\)\-Lipschitz\. This establishes Part \(i\)\.

Part \(ii\): Sup\-norm bound \(all cases\)\.In all cases, Part \(i\) gives thatggisLL\-Lipschitz withL=2​\(Mδ−1\)\>0L=2\(M\_\{\\delta\}\-1\)\>0\(recallMδ≥2M\_\{\\delta\}\\geq 2\) andg​\(0\)=0g\(0\)=0\. By Lemma[S20](https://arxiv.org/html/2607.06652#Thmtheorem20),

‖g‖∞2≤4​L​‖g‖1=8​\(Mδ−1\)​d1​\(x,y\)\.\\left\\lVert g\\right\\rVert\_\{\\infty\}^\{2\}\\;\\leq\\;4L\\,\\left\\lVert g\\right\\rVert\_\{1\}\\;=\\;8\(M\_\{\\delta\}\-1\)\\,d\_\{1\}\(x,y\)\\,\.\(S65\)Taking square roots and usingMδ−1≤MδM\_\{\\delta\}\-1\\leq M\_\{\\delta\}gives both bounds\. ∎

###### Lemma S22\(Separation of Distinct\-Count Images\)\.

AssumeMδ≥2M\_\{\\delta\}\\geq 2\. Since them\+1m\{\+\}1interarrival times of anyη∈𝒩δ\\eta\\in\\mathcal\{N\}\_\{\\delta\}withmmevents are each at leastδ\\deltaand sum toTmaxT\_\{\\max\}, we have\(m\+1\)​δ≤Tmax\(m\+1\)\\delta\\leq T\_\{\\max\}, i\.e\.m≤Mδ−1m\\leq M\_\{\\delta\}\-1; asmmis an integer,m≤⌊Mδ⌋−1m\\leq\\lfloor M\_\{\\delta\}\\rfloor\-1\. Forn∈\{0,…,⌊Mδ⌋−1\}n\\in\\\{0,\\dots,\\lfloor M\_\{\\delta\}\\rfloor\-1\\\}, let𝒩δ\(n\):=\{η∈𝒩δ:ηTmax=n\}\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}:=\\\{\\eta\\in\\mathcal\{N\}\_\{\\delta\}:\\eta\_\{T\_\{\\max\}\}=n\\\}denote the subset of𝒩δ\\mathcal\{N\}\_\{\\delta\}with exactlynnevents\. Each such class is nonempty \(the equispaced gridtk=k​δt\_\{k\}=k\\delta,k=1,…,nk=1,\\dots,n, is admissible because\(n\+1\)​δ≤⌊Mδ⌋​δ≤Tmax\(n\+1\)\\delta\\leq\\lfloor M\_\{\\delta\}\\rfloor\\,\\delta\\leq T\_\{\\max\}\), and these classes partition𝒩δ\\mathcal\{N\}\_\{\\delta\}\. For each pairm≠nm\\neq nin\{0,…,⌊Mδ⌋−1\}\\\{0,\\dots,\\lfloor M\_\{\\delta\}\\rfloor\-1\\\},

d1​\(Φ​\(𝒩δ\(m\)\),Φ​\(𝒩δ\(n\)\)\)\>0\.d\_\{1\}\\\!\\left\(\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(m\)\}\),\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}\)\\right\)\>0\\,\.\(S66\)Consequently,

dmin\(δ\):=minm,n∈\{0,…,⌊Mδ⌋−1\}m≠nd1\(Φ\(𝒩δ\(m\)\),Φ\(𝒩δ\(n\)\)\)\>0\.d\_\{\\min\}\(\\delta\):=\\min\_\{\\begin\{subarray\}\{c\}m,n\\in\\\{0,\\dots,\\lfloor M\_\{\\delta\}\\rfloor\-1\\\}\\\\ m\\neq n\\end\{subarray\}\}d\_\{1\}\\\!\\left\(\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(m\)\}\),\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}\)\\right\)\>0\\,\.\(S67\)

###### Proof\.

Fixn∈\{0,…,⌊Mδ⌋−1\}n\\in\\\{0,\\dots,\\lfloor M\_\{\\delta\}\\rfloor\-1\\\}\. Identify𝒩δ\(n\)\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}with the subset ofℝn\\mathbb\{R\}^\{n\}

\{\(t1,…,tn\)∈\[0,Tmax\]n:tk−tk−1≥δ​for​k=1,…,n,Tmax−tn≥δ\},t0=0,\\Big\\\{\(t\_\{1\},\\dots,t\_\{n\}\)\\in\[0,T\_\{\\max\}\]^\{n\}:t\_\{k\}\-t\_\{k\-1\}\\geq\\delta\\text\{ for \}k=1,\\dots,n,\\;T\_\{\\max\}\-t\_\{n\}\\geq\\delta\\Big\\\},\\quad t\_\{0\}=0\\,,\(S68\)and equip𝒩δ\(n\)\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}with the subspace topology inherited fromℝn\\mathbb\{R\}^\{n\}\. The defining constraints are finite intersections of closed half\-spaces, hence the set is closed in the compact cube\[0,Tmax\]n\[0,T\_\{\\max\}\]^\{n\}\. Therefore𝒩δ\(n\)\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}is compact \(the casen=0n=0being trivial\)\. Furthermore, on𝒩δ\(n\)\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\},d𝒩d\_\{\\mathcal\{N\}\}coincides with theℓ1\\ell^\{1\}distance between vectors, and so do their topologies\.Φ\\Phibeing Lipschitz with respect tod𝒩d\_\{\\mathcal\{N\}\}\(Theorem[S11](https://arxiv.org/html/2607.06652#Thmtheorem11)\), the restriction on𝒩δ\(n\)\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}is continuous\. Thus,Φ​\(𝒩δ\(n\)\)\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}\)is compact in\(𝒞,d1\)\(\\mathcal\{C\},d\_\{1\}\)\.

Injectivity ofΦ\\Phiimplies thatΦ​\(𝒩δ\(m\)\)∩Φ​\(𝒩δ\(n\)\)=∅\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(m\)\}\)\\cap\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}\)=\\varnothingform≠nm\\neq n\.

The continuous function\(x,y\)↦d1​\(x,y\)\(x,y\)\\mapsto d\_\{1\}\(x,y\)attains its minimum on the compact product of the two imagesΦ​\(𝒩δ\(m\)\)×Φ​\(𝒩δ\(n\)\)\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(m\)\}\)\\times\\Phi\(\\mathcal\{N\}\_\{\\delta\}^\{\(n\)\}\)\. Since the sets are disjoint, this minimum is strictly positive\. Since the index set\{0,…,⌊Mδ⌋−1\}\\\{0,\\dots,\\lfloor M\_\{\\delta\}\\rfloor\-1\\\}is finite and, byMδ≥2M\_\{\\delta\}\\geq 2, contains at least the two indices0and11, the minimum in the definition ofdmin​\(δ\)d\_\{\\min\}\(\\delta\)is taken over a finite, nonempty collection of strictly positive quantities\. This yieldsdmin​\(δ\)\>0d\_\{\\min\}\(\\delta\)\>0\. ∎

###### Corollary S24\(Count stability\)\.

AssumeMδ≥2M\_\{\\delta\}\\geq 2and letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}satisfyd1​\(Φ​\(η\),Φ​\(ξ\)\)<dmin​\(δ\)d\_\{1\}\\bigl\(\\Phi\(\\eta\),\\Phi\(\\xi\)\\bigr\)<d\_\{\\min\}\(\\delta\), the separation constant of Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22)\. Thenη\\etaandξ\\xihave the same number of events\.

###### Proof\.

This is immediate from Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22): if the event counts differed, thenΦ​\(η\)\\Phi\(\\eta\)andΦ​\(ξ\)\\Phi\(\\xi\)would lie in distinct image sets whosed1d\_\{1\}\-distance is at leastdmin​\(δ\)d\_\{\\min\}\(\\delta\)\. ∎

###### Lemma S25\(Base\-Case Perturbation\)\.

Letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}share event countm≥1m\\geq 1, with first event timest1t\_\{1\}ands1s\_\{1\}, respectively\. Setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\),ϵ:=‖x−y‖∞\\epsilon:=\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}, andMδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta\. Then

\|s1−t1\|≤Mδ​ϵ\.\|s\_\{1\}\-t\_\{1\}\|\\leq M\_\{\\delta\}\\epsilon\\,\.\(S69\)

###### Proof\.

Assumes1≥t1s\_\{1\}\\geq t\_\{1\}\(the reverse case follows by symmetry\)\. By level\-set recovery,fx​\(t1\)=0f\_\{x\}\(t\_\{1\}\)=0andfy​\(s1\)=0f\_\{y\}\(s\_\{1\}\)=0\. Sincefxf\_\{x\}has slope≥1/Mδ\\geq 1/M\_\{\\delta\}on\[t1,Tmax\]\[t\_\{1\},T\_\{\\max\}\]\(Lemma[S18](https://arxiv.org/html/2607.06652#Thmtheorem18)\),

fx​\(s1\)≥fx​\(t1\)\+1Mδ​\(s1−t1\)=1Mδ​\|s1−t1\|\.f\_\{x\}\(s\_\{1\}\)\\geq f\_\{x\}\(t\_\{1\}\)\+\\tfrac\{1\}\{M\_\{\\delta\}\}\(s\_\{1\}\-t\_\{1\}\)=\\tfrac\{1\}\{M\_\{\\delta\}\}\|s\_\{1\}\-t\_\{1\}\|\\,\.\(S70\)Furthermore,\|fx​\(s1\)−fy​\(s1\)\|=\|fx​\(s1\)−0\|=fx​\(s1\)≤‖fx−fy‖∞=‖x−y‖∞=ϵ\\left\\lvert f\_\{x\}\(s\_\{1\}\)\-f\_\{y\}\(s\_\{1\}\)\\right\\rvert=\\left\\lvert f\_\{x\}\(s\_\{1\}\)\-0\\right\\rvert=f\_\{x\}\(s\_\{1\}\)\\leq\\left\\lVert f\_\{x\}\-f\_\{y\}\\right\\rVert\_\{\\infty\}=\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}=\\epsilon\. ∎

###### Lemma S26\(Recursive Error Propagation\)\.

Letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}share event countm≥1m\\geq 1, with augmented event\-time grids\(tk\)k=0m\+1\(t\_\{k\}\)\_\{k=0\}^\{m\+1\}and\(sk\)k=0m\+1\(s\_\{k\}\)\_\{k=0\}^\{m\+1\}, respectively\. Setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\), andϵ:=‖x−y‖∞\\epsilon:=\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}\. Supposeϵ<δ\\epsilon<\\delta, andMδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta\. Then for everyk=2,…,mk=2,\\dots,m,

\|sk−tk\|≤Mδ​\|sk−1−tk−1\|\+Mδ​ϵ\.\|s\_\{k\}\-t\_\{k\}\|\\leq M\_\{\\delta\}\|s\_\{k\-1\}\-t\_\{k\-1\}\|\+M\_\{\\delta\}\\epsilon\\,\.\(S71\)

###### Proof\.

By Proposition[S19](https://arxiv.org/html/2607.06652#Thmtheorem19), the restricted inversesFx:=\(fx\|\[t1,Tmax\]\)−1F\_\{x\}:=\(f\_\{x\}\|\_\{\[t\_\{1\},T\_\{\\max\}\]\}\)^\{\-1\}andFy:=\(fy\|\[s1,Tmax\]\)−1F\_\{y\}:=\(f\_\{y\}\|\_\{\[s\_\{1\},T\_\{\\max\}\]\}\)^\{\-1\}areMδM\_\{\\delta\}\-Lipschitz, and the event times satisfytk=Fx​\(tk−1\)t\_\{k\}=F\_\{x\}\(t\_\{k\-1\}\),sk=Fy​\(sk−1\)s\_\{k\}=F\_\{y\}\(s\_\{k\-1\}\)fork≥2k\\geq 2\.

Step 1:tk≥s1t\_\{k\}\\geq s\_\{1\}\.Suppose for contradiction thattk<s1t\_\{k\}<s\_\{1\}\. Thenfy​\(tk\)=0f\_\{y\}\(t\_\{k\}\)=0\(sincefy≡0f\_\{y\}\\equiv 0on\[0,s1\]\[0,s\_\{1\}\]\), while by Lemma[S14](https://arxiv.org/html/2607.06652#Thmtheorem14)fx​\(tk\)=tk−1≥t1≥δf\_\{x\}\(t\_\{k\}\)=t\_\{k\-1\}\\geq t\_\{1\}\\geq\\delta\. But‖fx−fy‖∞=ϵ<δ\\left\\lVert f\_\{x\}\-f\_\{y\}\\right\\rVert\_\{\\infty\}=\\epsilon<\\delta, a contradiction\.

Step 2:tk−1∈\[0,sm\]t\_\{k\-1\}\\in\[0,s\_\{m\}\]\.Sincefx​\(Tmax\)=tmf\_\{x\}\(T\_\{\\max\}\)=t\_\{m\}andfy​\(Tmax\)=smf\_\{y\}\(T\_\{\\max\}\)=s\_\{m\}, we have\|sm−tm\|=\|\(fx−fy\)​\(Tmax\)\|≤ϵ\|s\_\{m\}\-t\_\{m\}\|=\|\(f\_\{x\}\-f\_\{y\}\)\(T\_\{\\max\}\)\|\\leq\\epsilon\. Theδ\\delta\-separation ofη\\etagivestk−1≤tm−δt\_\{k\-1\}\\leq t\_\{m\}\-\\delta\. Combining:

tk−1≤tm−δ<tm−ϵ≤sm,t\_\{k\-1\}\\leq t\_\{m\}\-\\delta<t\_\{m\}\-\\epsilon\\leq s\_\{m\}\\,,\(S72\)where the second inequality usesϵ<δ\\epsilon<\\deltaand the third usessm≥tm−ϵs\_\{m\}\\geq t\_\{m\}\-\\epsilon\. In other wordstk−1∈\[0,sm\]t\_\{k\-1\}\\in\[0,s\_\{m\}\]as required to evaluateFy​\(tk−1\)F\_\{y\}\(t\_\{k\-1\}\)\.

Step 3: Bounding\|Fy​\(tk−1\)−Fx​\(tk−1\)\|\|F\_\{y\}\(t\_\{k\-1\}\)\-F\_\{x\}\(t\_\{k\-1\}\)\|\.By Step 1,Fx​\(tk−1\)=tk≥s1F\_\{x\}\(t\_\{k\-1\}\)=t\_\{k\}\\geq s\_\{1\}\. Sincetk−1∈\[0,sm\]t\_\{k\-1\}\\in\[0,s\_\{m\}\]\(Step 2\),Fy​\(tk−1\)F\_\{y\}\(t\_\{k\-1\}\)is well\-defined and lies in\[s1,Tmax\]\[s\_\{1\},T\_\{\\max\}\]\(the range ofFyF\_\{y\}\)\. Thus bothFy​\(tk−1\)F\_\{y\}\(t\_\{k\-1\}\)andFx​\(tk−1\)F\_\{x\}\(t\_\{k\-1\}\)lie in\[s1,Tmax\]\[s\_\{1\},T\_\{\\max\}\], wherefyf\_\{y\}has slope at leastMδ−1M\_\{\\delta\}^\{\-1\}, so

Mδ−1​\|Fy​\(tk−1\)−Fx​\(tk−1\)\|≤\|fy​\(Fy​\(tk−1\)\)−fy​\(Fx​\(tk−1\)\)\|\.M\_\{\\delta\}^\{\-1\}\\left\\lvert F\_\{y\}\(t\_\{k\-1\}\)\-F\_\{x\}\(t\_\{k\-1\}\)\\right\\rvert\\leq\|f\_\{y\}\(F\_\{y\}\(t\_\{k\-1\}\)\)\-f\_\{y\}\(F\_\{x\}\(t\_\{k\-1\}\)\)\|\\,\.\(S73\)Sincefy​\(Fy​\(tk−1\)\)=tk−1=fx​\(Fx​\(tk−1\)\)f\_\{y\}\(F\_\{y\}\(t\_\{k\-1\}\)\)=t\_\{k\-1\}=f\_\{x\}\(F\_\{x\}\(t\_\{k\-1\}\)\)and\|fx​\(Fx​\(tk−1\)\)−fy​\(Fx​\(tk−1\)\)\|≤‖x−y‖∞=ϵ\|f\_\{x\}\(F\_\{x\}\(t\_\{k\-1\}\)\)\-f\_\{y\}\(F\_\{x\}\(t\_\{k\-1\}\)\)\|\\leq\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}=\\epsilon, we obtain\|Fy​\(tk−1\)−Fx​\(tk−1\)\|≤Mδ​ϵ\|F\_\{y\}\(t\_\{k\-1\}\)\-F\_\{x\}\(t\_\{k\-1\}\)\|\\leq M\_\{\\delta\}\\epsilon\.

Step 4: Triangle inequality\.

\|sk−tk\|\\displaystyle\|s\_\{k\}\-t\_\{k\}\|=\|Fy​\(sk−1\)−Fx​\(tk−1\)\|\\displaystyle=\|F\_\{y\}\(s\_\{k\-1\}\)\-F\_\{x\}\(t\_\{k\-1\}\)\|≤\|Fy​\(sk−1\)−Fy​\(tk−1\)\|⏟≤Mδ​\|sk−1−tk−1\|\+\|Fy​\(tk−1\)−Fx​\(tk−1\)\|⏟≤Mδ​ϵ\\displaystyle\\leq\\underbrace\{\|F\_\{y\}\(s\_\{k\-1\}\)\-F\_\{y\}\(t\_\{k\-1\}\)\|\}\_\{\\leq\\,M\_\{\\delta\}\|s\_\{k\-1\}\-t\_\{k\-1\}\|\}\+\\underbrace\{\|F\_\{y\}\(t\_\{k\-1\}\)\-F\_\{x\}\(t\_\{k\-1\}\)\|\}\_\{\\leq\\,M\_\{\\delta\}\\epsilon\}≤Mδ​\|sk−1−tk−1\|\+Mδ​ϵ\.\\displaystyle\\leq M\_\{\\delta\}\|s\_\{k\-1\}\-t\_\{k\-1\}\|\+M\_\{\\delta\}\\epsilon\\,\.\(S74\)∎

###### Lemma S27\(Large\-Perturbation Regime\)\.

Letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}share event countm≥1m\\geq 1, and setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\)\. Suppose‖x−y‖∞≥δ\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}\\geq\\delta\. Then

d𝒩​\(η,ξ\)≤m​8​Mδ3/2​d1​\(x,y\)1/2\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq m\\sqrt\{8\}M\_\{\\delta\}^\{3/2\}d\_\{1\}\(x,y\)^\{1/2\}\\,\.\(S75\)

###### Proof\.

Since both measures havemmevents in\[0,Tmax\)\[0,T\_\{\\max\}\), the discrete representation gives the trivial boundd𝒩​\(η,ξ\)≤m​Tmaxd\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq m\\,T\_\{\\max\}\. By Proposition[S21](https://arxiv.org/html/2607.06652#Thmtheorem21),d1​\(x,y\)≥ϵ2/\(8​Mδ\)≥δ2/\(8​Mδ\)d\_\{1\}\(x,y\)\\geq\\epsilon^\{2\}/\(8M\_\{\\delta\}\)\\geq\\delta^\{2\}/\(8M\_\{\\delta\}\)\. Hence

d𝒩​\(η,ξ\)≤m​Tmax⋅d1​\(x,y\)d1​\(x,y\)≤m​Tmax⋅8​Mδ​d1​\(x,y\)1/2δ=m​8​Mδ3/2​d1​\(x,y\)1/2\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq m\\,T\_\{\\max\}\\cdot\\frac\{\\sqrt\{d\_\{1\}\(x,y\)\}\}\{\\sqrt\{d\_\{1\}\(x,y\)\}\}\\leq m\\,T\_\{\\max\}\\cdot\\frac\{\\sqrt\{8M\_\{\\delta\}\}\\,d\_\{1\}\(x,y\)^\{1/2\}\}\{\\delta\}=m\\sqrt\{8\}M\_\{\\delta\}^\{3/2\}d\_\{1\}\(x,y\)^\{1/2\}\\,\.\(S76\)∎

###### Proposition S28\(Same\-Count Hölder Bound\)\.

Letη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\}have the same event countm≥0m\\geq 0\. Setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\),ϵ:=‖x−y‖∞\\epsilon:=\\left\\lVert x\-y\\right\\rVert\_\{\\infty\}, andMδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta\. Then

d𝒩​\(η,ξ\)≤Cm​\(Mδ\)​ϵ,Cm​\(Mδ\):=Mδ2​\(Mδm−1\)\(Mδ−1\)2,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq C\_\{m\}\(M\_\{\\delta\}\)\\epsilon\\,,\\quad C\_\{m\}\(M\_\{\\delta\}\):=\\frac\{M\_\{\\delta\}^\{2\}\(M\_\{\\delta\}^\{m\}\-1\)\}\{\(M\_\{\\delta\}\-1\)^\{2\}\}\\,,\(S77\)and consequently

d𝒩​\(η,ξ\)≤2​2​Cm​\(Mδ\)​Mδ1/2​d1​\(x,y\)1/2≤4​Mδm\+3\(Mδ−1\)2​d1​\(x,y\)1/2\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq 2\\sqrt\{2\}C\_\{m\}\(M\_\{\\delta\}\)M\_\{\\delta\}^\{1/2\}d\_\{1\}\(x,y\)^\{1/2\}\\leq\\frac\{4M\_\{\\delta\}^\{m\+3\}\}\{\(M\_\{\\delta\}\-1\)^\{2\}\}d\_\{1\}\(x,y\)^\{1/2\}\\,\.\(S78\)

###### Proof\.

The casem=0m=0is trivial\. Assumem≥1m\\geq 1; thenMδ≥m\+1≥2M\_\{\\delta\}\\geq m\+1\\geq 2\. Write\(tk\)k=0m\+1\(t\_\{k\}\)\_\{k=0\}^\{m\+1\}and\(sk\)k=0m\+1\(s\_\{k\}\)\_\{k=0\}^\{m\+1\}for the augmented event\-time grids ofη\\etaandξ\\xi\(Convention[1](https://arxiv.org/html/2607.06652#Thmtheorem1)\)\.

Step 1:ϵ<δ\\epsilon<\\delta\.Setak:=\|sk−tk\|a\_\{k\}:=\|s\_\{k\}\-t\_\{k\}\|\. The boundary convention givesa0=0a\_\{0\}=0, and Lemmas[S25](https://arxiv.org/html/2607.06652#Thmtheorem25)and[S26](https://arxiv.org/html/2607.06652#Thmtheorem26)yield

ak≤Mδ​ak−1\+Mδ​ϵ,k=1,…,m\.a\_\{k\}\\leq M\_\{\\delta\}a\_\{k\-1\}\+M\_\{\\delta\}\\epsilon\\,,\\quad k=1,\\dots,m\\,\.\(S79\)Unrolling froma0=0a\_\{0\}=0,

ak≤Mδ​ϵ​∑j=0k−1Mδj=Mδ​\(Mδk−1\)Mδ−1​ϵ,k=1,…,m\.a\_\{k\}\\leq M\_\{\\delta\}\\epsilon\\sum\_\{j=0\}^\{k\-1\}M\_\{\\delta\}^\{j\}=\\frac\{M\_\{\\delta\}\(M\_\{\\delta\}^\{k\}\-1\)\}\{M\_\{\\delta\}\-1\}\\epsilon\\,,\\quad k=1,\\dots,m\\,\.\(S80\)By Eq\.[S4](https://arxiv.org/html/2607.06652#A1.E4), because both processes share the same event count,d𝒩​\(η,ξ\)=∑k=1makd\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)=\\sum\_\{k=1\}^\{m\}a\_\{k\}\. Summing:

∑k=1mak≤Mδ​ϵMδ−1​∑k=1m\(Mδk−1\)=Mδ​ϵMδ−1​\(Mδ​\(Mδm−1\)Mδ−1−m\)≤Mδ2​\(Mδm−1\)\(Mδ−1\)2​ϵ,\\sum\_\{k=1\}^\{m\}a\_\{k\}\\leq\\frac\{M\_\{\\delta\}\\epsilon\}\{M\_\{\\delta\}\-1\}\\sum\_\{k=1\}^\{m\}\(M\_\{\\delta\}^\{k\}\-1\)=\\frac\{M\_\{\\delta\}\\epsilon\}\{M\_\{\\delta\}\-1\}\\biggl\(\\frac\{M\_\{\\delta\}\(M\_\{\\delta\}^\{m\}\-1\)\}\{M\_\{\\delta\}\-1\}\-m\\biggr\)\\leq\\frac\{M\_\{\\delta\}^\{2\}\(M\_\{\\delta\}^\{m\}\-1\)\}\{\(M\_\{\\delta\}\-1\)^\{2\}\}\\epsilon\\,,\(S81\)where the penultimate step drops the term−m<0\-m<0\. Substitutingϵ≤2​2​Mδ​d1​\(x,y\)≤4​Mδ​d1​\(x,y\)\\epsilon\\leq 2\\sqrt\{2M\_\{\\delta\}\\,d\_\{1\}\(x,y\)\}\\leq 4M\_\{\\delta\}\\sqrt\{d\_\{1\}\(x,y\)\}\(Proposition[S21](https://arxiv.org/html/2607.06652#Thmtheorem21)\) into Eq\.[S77](https://arxiv.org/html/2607.06652#A2.E77)yields Eq\.[S78](https://arxiv.org/html/2607.06652#A2.E78)\. The final constant relaxesMδm−1M\_\{\\delta\}^\{m\}\-1toMδmM\_\{\\delta\}^\{m\}for a cleaner closed form\.

Step 2:ϵ≥δ\\epsilon\\geq\\delta\.We first verify that Eq\.[S77](https://arxiv.org/html/2607.06652#A2.E77)continues to hold in this regime\. Since both processes havemmevents, Eq\.[S4](https://arxiv.org/html/2607.06652#A1.E4)gives the trivial boundd𝒩​\(η,ξ\)≤m​Tmaxd\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq mT\_\{\\max\}, while

Cm​\(Mδ\)​ϵ≥Cm​\(Mδ\)​δ=Mδ​\(Mδm−1\)\(Mδ−1\)2​Tmax≥m​Tmax,C\_\{m\}\(M\_\{\\delta\}\)\\,\\epsilon\\;\\geq\\;C\_\{m\}\(M\_\{\\delta\}\)\\,\\delta\\;=\\;\\frac\{M\_\{\\delta\}\(M\_\{\\delta\}^\{m\}\-1\)\}\{\(M\_\{\\delta\}\-1\)^\{2\}\}\\,T\_\{\\max\}\\;\\geq\\;m\\,T\_\{\\max\}\\,,\(S82\)where the last inequality usesMδm−1Mδ−1=1\+Mδ\+⋯\+Mδm−1≥m\\frac\{M\_\{\\delta\}^\{m\}\-1\}\{M\_\{\\delta\}\-1\}=1\+M\_\{\\delta\}\+\\cdots\+M\_\{\\delta\}^\{m\-1\}\\geq mtogether withMδMδ−1≥1\\frac\{M\_\{\\delta\}\}\{M\_\{\\delta\}\-1\}\\geq 1\. Hence Eq\.[S77](https://arxiv.org/html/2607.06652#A2.E77)holds unconditionally\.

We now establish Eq\.[S78](https://arxiv.org/html/2607.06652#A2.E78)in this regime\. Form≥1m\\geq 1andϵ≥δ\\epsilon\\geq\\delta, by Lemma[S27](https://arxiv.org/html/2607.06652#Thmtheorem27),d𝒩​\(η,ξ\)≤m​8​Mδ3/2​d1​\(x,y\)1/2d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq m\\sqrt\{8\}\\,M\_\{\\delta\}^\{3/2\}\\,d\_\{1\}\(x,y\)^\{1/2\}\. Since\(Mδ−1\)2≤Mδ2\(M\_\{\\delta\}\-1\)^\{2\}\\leq M\_\{\\delta\}^\{2\}, it suffices to showm​8≤4​Mδm−1/2m\\sqrt\{8\}\\leq 4\\,M\_\{\\delta\}^\{m\-1/2\}, i\.e\.,m≤2​Mδm−1/2m\\leq\\sqrt\{2\}\\,M\_\{\\delta\}^\{m\-1/2\}\. We verify this by induction onmm:

- •*Base casem=1m=1*: need1≤2​Mδ1/21\\leq\\sqrt\{2\}\\,M\_\{\\delta\}^\{1/2\}, which holds sinceMδ≥2M\_\{\\delta\}\\geq 2\.
- •*Inductive step*: ifm≤2​Mδm−1/2m\\leq\\sqrt\{2\}\\,M\_\{\\delta\}^\{m\-1/2\}, then\(m\+1\)≤2​m≤2​2​Mδm−1/2≤2​Mδm\+1/2\(m\{\+\}1\)\\leq 2m\\leq 2\\sqrt\{2\}\\,M\_\{\\delta\}^\{m\-1/2\}\\leq\\sqrt\{2\}\\,M\_\{\\delta\}^\{m\+1/2\}forMδ≥2M\_\{\\delta\}\\geq 2, where the last inequality uses2​Mδ−1≤12M\_\{\\delta\}^\{\-1\}\\leq 1\.

Thus, the constant holds for allm≥1m\\geq 1and whenMδ≥2M\_\{\\delta\}\\geq 2\. Hence Eqs\.[S77](https://arxiv.org/html/2607.06652#A2.E77)and[S78](https://arxiv.org/html/2607.06652#A2.E78)hold in both cases, completing the proof\. ∎

###### Theorem S29\(Hölder stability ofΦ−1\\Phi^\{\-1\}\)\.

Letδ\>0\\delta\>0andMδ:=Tmax/δM\_\{\\delta\}:=T\_\{\\max\}/\\delta, and assumeMδ≥2M\_\{\\delta\}\\geq 2; whenMδ<2M\_\{\\delta\}<2, the sieve𝒩δ\\mathcal\{N\}\_\{\\delta\}contains at most the eventless path and the statement holds trivially for anyκ\>0\\kappa\>0\. There exists a constantκ=κ​\(δ,Tmax\)\>0\\kappa=\\kappa\(\\delta,T\_\{\\max\}\)\>0such that for allη,ξ∈𝒩δ\\eta,\\xi\\in\\mathcal\{N\}\_\{\\delta\},

d𝒩​\(η,ξ\)≤κ​d1​\(Φ​\(η\),Φ​\(ξ\)\)1/2\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq\\kappa d\_\{1\}\\\!\\bigl\(\\Phi\(\\eta\),\\Phi\(\\xi\)\\bigr\)^\{1/2\}\\,\.\(S83\)Explicitly, one may take

κ=max⁡\(4​Mδ⌊Mδ⌋\+2\(Mδ−1\)2,⌊Mδ⌋​Tmaxdmin​\(δ\)\),\\kappa=\\max\\\!\\biggl\(\\frac\{4M\_\{\\delta\}^\{\\lfloor M\_\{\\delta\}\\rfloor\+2\}\}\{\(M\_\{\\delta\}\-1\)^\{2\}\},\\frac\{\\lfloor M\_\{\\delta\}\\rfloor\\,T\_\{\\max\}\}\{\\sqrt\{d\_\{\\min\}\(\\delta\)\}\}\\biggr\)\\,,\(S84\)wheredmin​\(δ\)\>0d\_\{\\min\}\(\\delta\)\>0is the separation constant from Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22)\. Sincedmin​\(δ\)d\_\{\\min\}\(\\delta\)is obtained by a compactness argument and is not quantified, we make no claim on the growth rate ofκ\\kappaasδ↓0\\delta\\downarrow 0\.

###### Proof\.

Letmmandnndenote the event counts ofη\\etaandξ\\xi, and setx:=Φ​\(η\)x:=\\Phi\(\\eta\),y:=Φ​\(ξ\)y:=\\Phi\(\\xi\)\.

Case 1:m=nm=n\.Proposition[S28](https://arxiv.org/html/2607.06652#Thmtheorem28)givesd𝒩​\(η,ξ\)≤Crec​d1​\(x,y\)1/2d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq C\_\{\\mathrm\{rec\}\}\\,d\_\{1\}\(x,y\)^\{1/2\}withCrec:=4​Mδ⌊Mδ⌋\+2/\(Mδ−1\)2C\_\{\\mathrm\{rec\}\}:=4\\,M\_\{\\delta\}^\{\\lfloor M\_\{\\delta\}\\rfloor\+2\}/\(M\_\{\\delta\}\-1\)^\{2\}, usingm=n≤⌊Mδ⌋−1m=n\\leq\\lfloor M\_\{\\delta\}\\rfloor\-1\(Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22)\) together withMδ≥2M\_\{\\delta\}\\geq 2to boundMδm\+3≤Mδ⌊Mδ⌋\+2M\_\{\\delta\}^\{m\+3\}\\leq M\_\{\\delta\}^\{\\lfloor M\_\{\\delta\}\\rfloor\+2\}; whenm=0m=0,d𝒩​\(η,ξ\)=0d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)=0and the bound is trivial\.

Case 2:m≠nm\\neq n\.By Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22),d1​\(x,y\)≥dmin​\(δ\)\>0d\_\{1\}\(x,y\)\\geq d\_\{\\min\}\(\\delta\)\>0\. Moreover, both event counts are at most⌊Mδ⌋−1\\lfloor M\_\{\\delta\}\\rfloor\-1\(Lemma[S22](https://arxiv.org/html/2607.06652#Thmtheorem22)\), so Proposition[S5](https://arxiv.org/html/2607.06652#Thmtheorem5a)givesd𝒩​\(η,ξ\)≤max⁡\(m,n\)​Tmax≤⌊Mδ⌋​Tmaxd\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq\\max\(m,n\)\\,T\_\{\\max\}\\leq\\lfloor M\_\{\\delta\}\\rfloor\\,T\_\{\\max\}\. Hence

d𝒩​\(η,ξ\)≤⌊Mδ⌋​Tmax=⌊Mδ⌋​Tmaxdmin​\(δ\)​dmin​\(δ\)≤⌊Mδ⌋​Tmaxdmin​\(δ\)​d1​\(x,y\)1/2\.d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta,\\xi\\right\)\\leq\\lfloor M\_\{\\delta\}\\rfloor\\,T\_\{\\max\}=\\frac\{\\lfloor M\_\{\\delta\}\\rfloor\\,T\_\{\\max\}\}\{\\sqrt\{d\_\{\\min\}\(\\delta\)\}\}\\,\\sqrt\{d\_\{\\min\}\(\\delta\)\}\\leq\\frac\{\\lfloor M\_\{\\delta\}\\rfloor\\,T\_\{\\max\}\}\{\\sqrt\{d\_\{\\min\}\(\\delta\)\}\}\\,d\_\{1\}\(x,y\)^\{1/2\}\\,\.\(S85\)Takingκ:=max⁡\(Crec,⌊Mδ⌋​Tmax/dmin​\(δ\)\)\\kappa:=\\max\\\!\\bigl\(C\_\{\\mathrm\{rec\}\},\\;\\lfloor M\_\{\\delta\}\\rfloor T\_\{\\max\}/\\sqrt\{d\_\{\\min\}\(\\delta\)\}\\bigr\)completes the proof\. ∎

## Appendix CMetrics

We evaluate generative TPP models along several complementary axes\. Each metric targets a distinct structural property of the process, and together they provide a comprehensive picture of model quality\. For all metrics a lower value is better\.

### C\.1Theory\-Motivated Metrics

We define and theoretically justify 3 theory\-motivated metrics in Section[3\.4](https://arxiv.org/html/2607.06652#S3.SS4)\. We recall them briefly here and describe their empirical estimators\. Throughout this section, we assume access toNPN\_\{P\}real samplesη\(1\),…,η\(NP\)∼P\\eta^\{\(1\)\},\\ldots,\\eta^\{\(N\_\{P\}\)\}\\sim PandNQN\_\{Q\}generated samplesξ^\(1\),…,ξ^\(NQ\)∼Q\\widehat\{\\xi\}^\{\(1\)\},\\ldots,\\widehat\{\\xi\}^\{\(N\_\{Q\}\)\}\\sim Q\.

Signature Wasserstein\-1 \(Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\)\.Sig​\-​W1\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}measures discrepancy between path distributions through their expected truncated signatures, capturing higher\-order temporal\-dependence structure\. It serves simultaneously as our training objective forSigTPPand as an evaluation metric\. Replacing the population expectations in Eq\.[4](https://arxiv.org/html/2607.06652#S3.E4)by sample means yields

Sig​\-​W1^\(M\)​\(P,Q\)=‖1NP​∑i=1NPSM​\(Φ​\(η\(i\)\)\)−1NQ​∑i=1NQSM​\(Φ​\(ξ^\(i\)\)\)‖2\.\\widehat\{\\mathrm\{Sig\\text\{\-\}W\}\_\{1\}\}^\{\(M\)\}\(P,Q\)=\\Bigg\\\|\\frac\{1\}\{N\_\{P\}\}\\sum\_\{i=1\}^\{N\_\{P\}\}S^\{M\}\\\!\\bigl\(\\Phi\(\\eta^\{\(i\)\}\)\\bigr\)\-\\frac\{1\}\{N\_\{Q\}\}\\sum\_\{i=1\}^\{N\_\{Q\}\}S^\{M\}\\\!\\bigl\(\\Phi\(\\widehat\{\\xi\}^\{\(i\)\}\)\\bigr\)\\Bigg\\\|\_\{2\}\\,\.\(S87\)We report results at truncation order33with unscaled signature components; during training, signature features are standardised \(common setting in linear regressions\)\.

Energy Distance \(ℰ\\mathcal\{E\}\)\.The energy distance is defined as

ℰ:=2⋅𝔼​\[d𝒩​\(N,N~\)\]−𝔼​\[d𝒩​\(N,N′\)\]−𝔼​\[d𝒩​\(N~,N~′\)\],P,Q∈𝒫1​\(𝒩\),\\mathcal\{E\}:=2\\cdot\\mathbb\{E\}\\left\[d\_\{\\mathcal\{N\}\}\\\!\\left\(N,\\widetilde\{N\}\\right\)\\right\]\-\\mathbb\{E\}\\left\[d\_\{\\mathcal\{N\}\}\\Bigl\(N,N^\{\\prime\}\\Bigr\)\\right\]\-\\mathbb\{E\}\\left\[d\_\{\\mathcal\{N\}\}\\\!\\left\(\\widetilde\{N\},\\widetilde\{N\}^\{\\prime\}\\right\)\\right\]\\,,\\quad P,Q\\in\\mathcal\{P\}\_\{1\}\(\\mathcal\{N\}\)\\,,\(S88\)withN,N′∼PN,N^\{\\prime\}\\sim PandN~,N~′∼Q\\widetilde\{N\},\\widetilde\{N\}^\{\\prime\}\\sim Qindependent\[[21](https://arxiv.org/html/2607.06652#bib.bib38),[50](https://arxiv.org/html/2607.06652#bib.bib36)\]\. The energy distance is a distribution\-level discrepancy on𝒫1​\(𝒩,d𝒩\)\\mathcal\{P\}\_\{1\}\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)that compares the average inter\-sample distance betweenPPandQQagainst the within\-distribution distances\. Its empirical estimator is

ℰ^​\(P,Q\)\\displaystyle\\widehat\{\\mathcal\{E\}\}\(P,Q\)=2NP​NQ​∑i,jd𝒩​\(η\(i\),ξ^\(j\)\)−1NP​\(NP−1\)​∑i≠i′d𝒩​\(η\(i\),η\(i′\)\)\\displaystyle=\\frac\{2\}\{N\_\{P\}N\_\{Q\}\}\\sum\_\{i,j\}d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta^\{\(i\)\},\\widehat\{\\xi\}^\{\(j\)\}\\right\)\-\\frac\{1\}\{N\_\{P\}\(N\_\{P\}\-1\)\}\\sum\_\{i\\neq i^\{\\prime\}\}d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta^\{\(i\)\},\\eta^\{\(i^\{\\prime\}\)\}\\right\)\(S89\)−1NQ​\(NQ−1\)​∑j≠j′d𝒩​\(ξ^\(j\),ξ^\(j′\)\)\.\\displaystyle\\quad\-\\frac\{1\}\{N\_\{Q\}\(N\_\{Q\}\-1\)\}\\sum\_\{j\\neq j^\{\\prime\}\}d\_\{\\mathcal\{N\}\}\\\!\\left\(\\widehat\{\\xi\}^\{\(j\)\},\\widehat\{\\xi\}^\{\(j^\{\\prime\}\)\}\\right\)\\,\.Althoughℰ≥0\\mathcal\{E\}\\geq 0at the population level \(sinced𝒩d\_\{\\mathcal\{N\}\}is of negative type\), the empirical estimatorℰ^​\(P,Q\)\\widehat\{\\mathcal\{E\}\}\(P,Q\)can take small negative values in finite samples, especially whenPPandQQare close\. We compute it on batches of2,0002\{,\}000sequences so that all pairwise distances fit in memory\. Remark thatℰ\\mathcal\{E\}reduces to the CRPS\[[45](https://arxiv.org/html/2607.06652#bib.bib34)\]in the univariate case, providing a natural score for unconditional generation that handles variable\-length sequences\.

Wasserstein\-1 \(W1\\mathrm\{W\}\_\{1\}\)\.Since\(𝒩,d𝒩\)\(\\mathcal\{N\},d\_\{\\mathcal\{N\}\}\)is a metric space, the Wasserstein\-1 distance\[[46](https://arxiv.org/html/2607.06652#bib.bib58),[52](https://arxiv.org/html/2607.06652#bib.bib37)\]

W1​\(P,Q\):=infπ∈Π​\(P,Q\)∫𝒩×𝒩d𝒩​\(N,N′\)​d​π​\(N,N′\),\\mathrm\{W\}\_\{1\}\\left\(P,Q\\right\):=\\inf\_\{\\pi\\in\\Pi\(P,Q\)\}\\int\_\{\\mathcal\{N\}\\times\\mathcal\{N\}\}d\_\{\\mathcal\{N\}\}\\\!\\left\(N,N^\{\\prime\}\\right\)\\,\\mathop\{\}\\\!\\mathrm\{d\}\\pi\(N,N^\{\\prime\}\)\\,,\(S90\)whereΠ​\(P,Q\)\\Pi\(P,Q\)denotes the set of couplings ofPPandQQ, is a well\-defined metric on𝒫1​\(𝒩\)\\mathcal\{P\}\_\{1\}\(\\mathcal\{N\}\)\.W1\\mathrm\{W\}\_\{1\}lifts the path\-level metricd𝒩d\_\{\\mathcal\{N\}\}to a metric between distributions\. It is the minimum cost of transporting generated mass onto real mass, with per\-pair costd𝒩d\_\{\\mathcal\{N\}\}\. With empirical measuresμ=1NP​∑i=1NPδη\(i\)\\mu=\\frac\{1\}\{N\_\{P\}\}\\sum\_\{i=1\}^\{N\_\{P\}\}\\delta\_\{\\eta^\{\(i\)\}\}andν=1NQ​∑j=1NQδξ^\(j\)\\nu=\\frac\{1\}\{N\_\{Q\}\}\\sum\_\{j=1\}^\{N\_\{Q\}\}\\delta\_\{\\widehat\{\\xi\}^\{\(j\)\}\}, it is estimated by solving

W^1​\(μ,ν\)=minπ∈Π​\(μ,ν\)​∑i=1NP∑j=1NQπi​j​d𝒩​\(η\(i\),ξ^\(j\)\),\\widehat\{W\}\_\{1\}\(\\mu,\\nu\)=\\min\_\{\\pi\\in\\Pi\(\\mu,\\nu\)\}\\sum\_\{i=1\}^\{N\_\{P\}\}\\sum\_\{j=1\}^\{N\_\{Q\}\}\\pi\_\{ij\}\\,d\_\{\\mathcal\{N\}\}\\\!\\left\(\\eta^\{\(i\)\},\\widehat\{\\xi\}^\{\(j\)\}\\right\)\\,,\(S91\)whereΠ​\(μ,ν\)\\Pi\(\\mu,\\nu\)denotes the set of couplings betweenμ\\muandν\\nu\. As for the energy distance, we solve this linear program exactly on batches of2,0002\{,\}000sequences, chosen so the full cost matrix fits in memory\.

Continuous Ranked Probability Score \(CRPS\)\.CRPS evaluates the calibration and sharpness of 1\-step\-ahead predictive distributions over interarrival times\. For each true event at positionllin sequencenn, the model defines a predictive distributionFn,lF\_\{n,l\}conditional on the true past\. We approximate it by the empirical cumulative distribution functionF^n,l\\widehat\{F\}\_\{n,l\}of theSSMonte Carlo samples𝒯^:,n,l\\widehat\{\\mathcal\{T\}\}\_\{:,n,l\}\. We report the per\-event average

CRPS¯=1∑n=1Nseqℓn​∑n=1Nseq∑l=1ℓnCRPS​\(F^n,l,𝒯n,l\),\\overline\{\\mathrm\{CRPS\}\}=\\frac\{1\}\{\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\ell\_\{n\}\}\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\sum\_\{l=1\}^\{\\ell\_\{n\}\}\\mathrm\{CRPS\}\\\!\\bigl\(\\widehat\{F\}\_\{n,l\},\\,\\mathcal\{T\}\_\{n,l\}\\bigr\)\\,,\(S92\)with the standard energy\-form definition for a sampleyy

CRPS​\(F^,y\)=𝔼X∼F^​\[\|X−y\|\]−12​𝔼X,X′∼F^​\[\|X−X′\|\],\\mathrm\{CRPS\}\(\\widehat\{F\},y\)=\\mathbb\{E\}\_\{X\\sim\\widehat\{F\}\}\\\!\\bigl\[\\left\\lvert X\-y\\right\\rvert\\bigr\]\-\\tfrac\{1\}\{2\}\\,\\mathbb\{E\}\_\{X,X^\{\\prime\}\\sim\\widehat\{F\}\}\\\!\\bigl\[\\left\\lvert X\-X^\{\\prime\}\\right\\rvert\\bigr\]\\,,\(S93\)whereX,X′X,X^\{\\prime\}are i\.i\.d\. draws fromF^\\widehat\{F\}\.

Scaling\.For comparability across datasets with different time horizons, we reportℰ^/Tmax2\\widehat\{\\mathcal\{E\}\}/T\_\{\\max\}^\{2\}\(resp\.W^1/Tmax2\\widehat\{W\}\_\{1\}/T\_\{\\max\}^\{2\}\) in all tables\. The remaining metrics operate at the per\-event level on interarrival times\. LetNseqN\_\{\\mathrm\{seq\}\}denote the number of evaluation sequences andℓn\\ell\_\{n\}the \(unpadded\) length of sequencenn, and writeℓmax=maxn⁡ℓn\\ell\_\{\\max\}=\\max\_\{n\}\\ell\_\{n\}\. We collect interarrivals in𝒯∈ℝNseq×ℓmax\\mathcal\{T\}\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{seq\}\}\\times\\ell\_\{\\max\}\}andSSMonte Carlo samples of generated interarrivals in𝒯^∈ℝS×Nseq×ℓmax\\widehat\{\\mathcal\{T\}\}\\in\\mathbb\{R\}^\{S\\times N\_\{\\mathrm\{seq\}\}\\times\\ell\_\{\\max\}\}, where entries beyond each sequence’s true length are set toNaN\. All estimators below use pairwise deletion ofNaNentries \(per\-lag for autocorrelations\)\. We write𝒯n,l\\mathcal\{T\}\_\{n,l\}for thell\-th interarrival of sequencennand𝒯^:,n,l∈ℝS\\widehat\{\\mathcal\{T\}\}\_\{:,n,l\}\\in\\mathbb\{R\}^\{S\}for the vector ofSSMonte Carlo predictions at position\(n,l\)\(n,l\)\.

### C\.2Heuristic Metrics

The following metrics target specific structural properties, second\-order temporal dependence and marginal distributional shape, without formal guarantees\. They are adapted fromNiet al\.\[[40](https://arxiv.org/html/2607.06652#bib.bib14)\]to the TPP setting\.

Pairwise Correlation Discrepancy \(PCD\)\.PCD assesses whether the model reproduces second\-order dependence across event positions: it compares the Pearson correlation matrix of interarrival times at different positions in real vs\. generated samples, catching models whose marginals are correct but whose cross\-position structure is not\. Writing𝒯:,p\\mathcal\{T\}\_\{:,p\}for the vector of position\-ppinterarrivals across theNseqN\_\{\\mathrm\{seq\}\}sequences \(and likewise, we slice 1 Monte Carlo sample𝒯^0,:,p\\widehat\{\\mathcal\{T\}\}\_\{0,:,p\}\), let

ρp,q=Corr​\(𝒯:,p,𝒯:,q\),ρ^p,q=Corr​\(𝒯^0,:,p,𝒯^0,:,q\),\\rho\_\{p,q\}=\\mathrm\{Corr\}\\\!\\bigl\(\\mathcal\{T\}\_\{:,p\},\\,\\mathcal\{T\}\_\{:,q\}\\bigr\)\\,,\\qquad\\widehat\{\\rho\}\_\{p,q\}=\\mathrm\{Corr\}\\\!\\bigl\(\\widehat\{\\mathcal\{T\}\}\_\{0,:,p\},\\,\\widehat\{\\mathcal\{T\}\}\_\{0,:,q\}\\bigr\)\\,,\(S94\)denote the Pearson correlations, computed by pairwise deletion ofNaNentries\. Then

PCD=1\|𝒱\|​∑\(p,q\)∈𝒱\|ρp,q−ρ^p,q\|,\\mathrm\{PCD\}=\\frac\{1\}\{\|\\mathcal\{V\}\|\}\\sum\_\{\(p,q\)\\in\\mathcal\{V\}\}\\left\\lvert\\rho\_\{p,q\}\-\\widehat\{\\rho\}\_\{p,q\}\\right\\rvert\\,,\(S95\)where𝒱\\mathcal\{V\}collects position pairs\(p,q\)\(p,q\)with1≤p<q≤ℓ⋆1\\leq p<q\\leq\\ell^\{\\star\}for which both correlations are well\-defined\. To stabilise estimation, we restrictℓ⋆\\ell^\{\\star\}to the largest prefix length such that every positionp≤ℓ⋆p\\leq\\ell^\{\\star\}retains at least5050valid \(non\-padded\) samples in both𝒯\\mathcal\{T\}and𝒯^\\widehat\{\\mathcal\{T\}\}\.

Autocorrelation Discrepancy \(ACD\)\.ACD probes the temporal memory of the process, how strongly an interarrival depends on those a few events earlier\. For lagτ≥1\\tau\\geq 1, letACτ​\(𝒯\)\\mathrm\{AC\}\_\{\\tau\}\(\\mathcal\{T\}\)denote the lag\-τ\\tausample autocorrelation of interarrivals, computed by pooling all valid pairs\(𝒯n,l,𝒯n,l\+τ\)\(\\mathcal\{T\}\_\{n,l\},\\,\\mathcal\{T\}\_\{n,l\+\\tau\}\)across sequences \(withNaN\-padded positions excluded\), and defineACτ​\(𝒯^\)\\mathrm\{AC\}\_\{\\tau\}\(\\widehat\{\\mathcal\{T\}\}\)analogously by slicing one Monte Carlo sample\. We restrict to short lags, where temporal dependence is most informative and estimation is most reliable: we setτmax=5\\tau\_\{\\max\}=5in general, andτmax=10\\tau\_\{\\max\}=10on long\-sequence datasets \(ℓmax≥100\\ell\_\{\\max\}\\geq 100\)\. We report

ACD=1τmax​∑τ≤τmax\|ACτ​\(𝒯\)−ACτ​\(𝒯^\)\|\.\\mathrm\{ACD\}=\\frac\{1\}\{\\tau\_\{\\max\}\}\\sum\_\{\\tau\\leq\\tau\_\{\\max\}\}\\left\\lvert\\mathrm\{AC\}\_\{\\tau\}\(\\mathcal\{T\}\)\-\\mathrm\{AC\}\_\{\\tau\}\(\\widehat\{\\mathcal\{T\}\}\)\\right\\rvert\\,\.\(S96\)
Histogram Distances \(Llog⁡\(τ\)L\_\{\\log\(\\tau\)\},LλL\_\{\\lambda\}\)\.These metrics compare marginal distributional shape at two complementary levels:Llog⁡\(τ\)L\_\{\\log\(\\tau\)\}targets the interarrival times density \(how long between events?\) andLλL\_\{\\lambda\}the event\-count density on\[0,Tmax\]\[0,T\_\{\\max\}\]\(when do events happen?\)\. Both areL1L^\{1\}distances between empirical histograms computed onBBshared bins\.BBin our setting follows a cube\-root heuristic2​n1/32\\,n^\{1/3\}\. ForLlog⁡\(τ\)L\_\{\\log\(\\tau\)\}, letp^true\\widehat\{p\}\_\{\\mathrm\{true\}\}andp^gen\\widehat\{p\}\_\{\\mathrm\{gen\}\}be histogram estimators of the marginal density ofu=log⁡τu=\\log\\tau, whereτ\\tauis an interarrival time, on bins covering the empirical range oflog⁡τ\\log\\tauin𝒯\\mathcal\{T\}:

Llog⁡\(τ\)=∫\|p^true​\(u\)−p^gen​\(u\)\|​d​u,u=log⁡τ\.L\_\{\\log\(\\tau\)\}=\\int\\left\\lvert\\widehat\{p\}\_\{\\mathrm\{true\}\}\(u\)\-\\widehat\{p\}\_\{\\mathrm\{gen\}\}\(u\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}u\\,,\\qquad u=\\log\\tau\\,\.\(S97\)ForLλL\_\{\\lambda\}, letλ^trueemp\\widehat\{\\lambda\}^\{\\mathrm\{emp\}\}\_\{\\mathrm\{true\}\}andλ^genemp\\widehat\{\\lambda\}^\{\\mathrm\{emp\}\}\_\{\\mathrm\{gen\}\}be histogram estimators of the empirical event rate per unit time on\[0,Tmax\]\[0,T\_\{\\max\}\]\(distinct from the conditional intensity used elsewhere in the paper\):

Lλ=∫0Tmax\|λ^trueemp​\(s\)−λ^genemp​\(s\)\|​d​s\.L\_\{\\lambda\}=\\int\_\{0\}^\{T\_\{\\max\}\}\\left\\lvert\\widehat\{\\lambda\}^\{\\mathrm\{emp\}\}\_\{\\mathrm\{true\}\}\(s\)\-\\widehat\{\\lambda\}^\{\\mathrm\{emp\}\}\_\{\\mathrm\{gen\}\}\(s\)\\right\\rvert\\,\\mathop\{\}\\\!\\mathrm\{d\}s\\,\.\(S98\)

### C\.31\-Step\-Ahead Point Estimates \(MAE\)

MAE and MSE evaluate the accuracy of 1\-step\-ahead point predictions of interarrival times\. For each true event at positionllin sequencenn, the model defines a predictive distributionFn,lF\_\{n,l\}conditional on the true past, from which we draw theSSMonte Carlo samples𝒯^:,n,l∈ℝS\\widehat\{\\mathcal\{T\}\}\_\{:,n,l\}\\in\\mathbb\{R\}^\{S\}already used for CRPS\. The natural point predictors are the median ofFn,lF\_\{n,l\}for MAE and its mean for MSE, which we estimate by their sample counterparts

x^n,lMAE=median⁡\(𝒯^:,n,l\),x^n,lMSE=1S​∑s=1S𝒯^s,n,l\.\\widehat\{x\}^\{\\mathrm\{MAE\}\}\_\{n,l\}=\\operatorname\{median\}\\\!\\bigl\(\\widehat\{\\mathcal\{T\}\}\_\{:,n,l\}\\bigr\)\\,,\\qquad\\widehat\{x\}^\{\\mathrm\{MSE\}\}\_\{n,l\}=\\frac\{1\}\{S\}\\sum\_\{s=1\}^\{S\}\\widehat\{\\mathcal\{T\}\}\_\{s,n,l\}\\,\.\(S99\)We report the per\-event averages

MAEmed=1∑n=1Nseqℓn​∑n=1Nseq∑l=1ℓn\|x^n,lMAE−𝒯n,l\|,MSEmean=1∑n=1Nseqℓn​∑n=1Nseq∑l=1ℓn\(x^n,lMSE−𝒯n,l\)2\.\\mathrm\{MAE\}\_\{\\mathrm\{med\}\}=\\frac\{1\}\{\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\ell\_\{n\}\}\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\sum\_\{l=1\}^\{\\ell\_\{n\}\}\\left\\lvert\\widehat\{x\}^\{\\mathrm\{MAE\}\}\_\{n,l\}\-\\mathcal\{T\}\_\{n,l\}\\right\\rvert\\,,\\qquad\\mathrm\{MSE\}\_\{\\mathrm\{mean\}\}=\\frac\{1\}\{\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\ell\_\{n\}\}\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\sum\_\{l=1\}^\{\\ell\_\{n\}\}\\bigl\(\\widehat\{x\}^\{\\mathrm\{MSE\}\}\_\{n,l\}\-\\mathcal\{T\}\_\{n,l\}\\bigr\)^\{2\}\\,\.\(S100\)The sample median used inx^n,lMAE\\widehat\{x\}^\{\\mathrm\{MAE\}\}\_\{n,l\}is a consistent but finite\-SSbiased plug\-in for the population median ofFn,lF\_\{n,l\}; we takeSSlarge enough that this bias is negligible relative to the sequence\-level Monte Carlo error\.

We also report the 1\-step\-ahead forecasting error:

MAE^=1∑n=1Nseqℓn​∑n=1Nseq∑l=1ℓn\(1S​∑s=1S\|𝒯^s,n,l−𝒯n,l\|\)\.\\widehat\{\\mathrm\{MAE\}\}=\\frac\{1\}\{\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\ell\_\{n\}\}\\sum\_\{n=1\}^\{N\_\{\\mathrm\{seq\}\}\}\\sum\_\{l=1\}^\{\\ell\_\{n\}\}\\left\(\\frac\{1\}\{S\}\\sum\_\{s=1\}^\{S\}\\left\\lvert\\widehat\{\\mathcal\{T\}\}\_\{s,n,l\}\-\\mathcal\{T\}\_\{n,l\}\\right\\rvert\\right\)\\,\.\(S101\)

### C\.4Relative Score

LetsA\(i\)s\_\{A\}^\{\(i\)\}denote the score of modelAAon dataseti∈\{1,…,D\}i\\in\\\{1,\\dots,D\\\}, andsref\(i\)s\_\{\\mathrm\{ref\}\}^\{\(i\)\}the score of a fixed reference model\. The normalised score of modelAAis the geometric mean of per\-dataset ratios:

s¯A=\(∏i=1DsA\(i\)sref\(i\)\)1D=exp⁡\(1D​∑i=1Dlog⁡sA\(i\)sref\(i\)\)\.\\bar\{s\}\_\{A\}=\\left\(\\prod\_\{i=1\}^\{D\}\\frac\{s\_\{A\}^\{\(i\)\}\}\{s\_\{\\mathrm\{ref\}\}^\{\(i\)\}\}\\right\)^\{\\\!\\frac\{1\}\{D\}\}=\\exp\\\!\\left\(\\frac\{1\}\{D\}\\sum\_\{i=1\}^\{D\}\\log\\frac\{s\_\{A\}^\{\(i\)\}\}\{s\_\{\\mathrm\{ref\}\}^\{\(i\)\}\}\\right\)\\,\.\(S102\)A values¯A<1\\bar\{s\}\_\{A\}<1indicates thatAAoutperforms the reference on average\. Crucially, comparing two modelsAAandBBvia the ratio of their normalised scores is*independent of the choice of reference*:

s¯As¯B=\(∏isA\(i\)/sref\(i\)\)1/D\(∏isB\(i\)/sref\(i\)\)1/D=\(∏i=1DsA\(i\)sB\(i\)\)1D,\\frac\{\\bar\{s\}\_\{A\}\}\{\\bar\{s\}\_\{B\}\}=\\frac\{\\bigl\(\\prod\_\{i\}s\_\{A\}^\{\(i\)\}/s\_\{\\mathrm\{ref\}\}^\{\(i\)\}\\bigr\)^\{1/D\}\}\{\\bigl\(\\prod\_\{i\}s\_\{B\}^\{\(i\)\}/s\_\{\\mathrm\{ref\}\}^\{\(i\)\}\\bigr\)^\{1/D\}\}=\\left\(\\prod\_\{i=1\}^\{D\}\\frac\{s\_\{A\}^\{\(i\)\}\}\{s\_\{B\}^\{\(i\)\}\}\\right\)^\{\\\!\\frac\{1\}\{D\}\}\\,,\(S103\)where thesref\(i\)s\_\{\\mathrm\{ref\}\}^\{\(i\)\}cancel\[[17](https://arxiv.org/html/2607.06652#bib.bib68)\]\.

## Appendix DInformation about Datasets

Table S1:Dataset statistics\.KKdenotes the number of event types \(marks\)\.DatasetKK\# Events\# SequencesSeq\. LengthTrainValTestTrainValTestMinMeanMaxPS314293479648371200400400212\.024IP3452321491614944300010001000215\.031H1157846194321939760002000200029\.739H3339357129151306112004004001132\.770EQ7463586212138453000400900115\.417TB1772183112722795513002005002755\.763TX105045472041442014002004003536\.037SO2288766252662601514014014013863\.6100YLP110542355035311916365355\.2107We evaluate on 4 synthetic and 5 real\-world datasets, spanning memoryless, time\-varying, and self\-exciting generative regimes\. All synthetic datasets are split 60 %/20 %/20 % into training, validation, and test sets; real\-world datasets use the splits described below\.

![Refer to caption](https://arxiv.org/html/2607.06652v1/x9.png)

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Figure S2:Diagnostics for the Poisson dataset \(PS\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\.Bottom:interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x13.png)

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Figure S3:Diagnostics for the inhomogeneous Poisson dataset \(IP\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\.Top:intensity function\. Bottom: interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x17.png)

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Figure S4:Diagnostics for the Hawkes dataset \(H1\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\.Top:intensity function\. Bottom: interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x21.png)

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Figure S5:Diagnostics for the 3\-dimensional Hawkes dataset \(H3\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\. Bottom: interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x25.png)

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Figure S6:Diagnostics for the earthquake dataset \(EQ\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\.Bottom:interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x29.png)

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Figure S7:Diagnostics for the Stack Overflow dataset \(SO\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\.Top:intensity function\. Bottom: interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x33.png)

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Figure S8:Diagnostics for the Taobao dataset \(TB\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\.Bottom:interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x37.png)

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Figure S9:Diagnostics for the Taxi dataset \(TX\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\.Bottom:interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x41.png)

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Figure S10:Diagnostics for the dataset Yelp \(YLP\)\. \(a\) Representative sample paths from the dataset\. \(b\) Empirical temporal summaries\. Top: intensity function\.Bottom:interarrival times PDF\. \(c\) Empirical autocorrelation of interarrival times\. \(d\) Correlation structure between interarrival times\.Homogeneous Poisson \(PS\)\.We generate2,0002\{,\}000event sequences from a homogeneous Poisson process on\[0,T\]\[0,T\]with horizonT=12T=12and constant rateλ=1\\lambda=1\. Equivalently, interarrival times are sampled i\.i\.d\. fromExp​\(1\)\\mathrm\{Exp\}\(1\)\.

Inhomogeneous Poisson \(IP\)\.We generate5,0005\{,\}000sequences from an inhomogeneous Poisson process on\[0,T\]\[0,T\]withT=10T=10and piecewise\-constant intensity

λ​\(t\)=1\+𝟙⁡1​c​h​a​r​a​ct≥T/2\.\\lambda\(t\)=1\+\\operatorname\{\\mathds\{1\}\}1charac\_\{t\\geq T/2\}\\,\.\(S104\)Thus, the rate is11on the first half of the window and22on the second half\. Sequences are simulated using the Lewis–Ogata thinning algorithm with upper boundλ¯=2\\bar\{\\lambda\}=2\.

Univariate Hawkes \(H1\)\.We generate10,00010\{,\}000sequences from a univariate Hawkes process with exponential kernel,

λ​\(t∣ℋt\)=μ\+α​∑ti<tβ​e−β​\(t−ti\),\\lambda\(t\\mid\\mathcal\{H\}\_\{t\}\)=\\mu\+\\alpha\\sum\_\{t\_\{i\}<t\}\\beta e^\{\-\\beta\(t\-t\_\{i\}\)\}\\,,\(S105\)usingμ=0\.3\\mu=0\.3,α=0\.4\\alpha=0\.4,β=1\.0\\beta=1\.0, and horizonT=20T=20\. Simulation is performed with theticklibrary\. Sinceα<1\\alpha<1, the process is stationary\. This setting introduces temporal clustering through self\-excitation and tests whether a model can capture history\-dependent intensities\.

Multivariate Hawkes \(H3\)\.We generate2,0002\{,\}000sequences from a33\-dimensional Hawkes process with exponential kernels usingtick\. The baseline intensities areμk=0\.5\\mu\_\{k\}=0\.5for allkk, the decay parameters areβk​ℓ=1\\beta\_\{k\\ell\}=1for all pairs\(k,ℓ\)\(k,\\ell\), and the excitation matrix is

A=\(0\.50\.10\.00\.10\.00\.00\.00\.00\.1\),A=\\begin\{pmatrix\}0\.5&0\.1&0\.0\\\\ 0\.1&0\.0&0\.0\\\\ 0\.0&0\.0&0\.1\\end\{pmatrix\}\\,,\(S106\)with horizonT=15T=15\. The 3 component streams are merged into a single time\-ordered event sequence, and the source dimension is used as a categorical mark \(K=3K=3\)\. This dataset evaluates a model’s ability to capture both cross\-excitation across dimensions and mark\-dependent intensity structure\.

#### Real\-World Datasets

Taxi \(NYC TLC\)\.Time\-stamped taxi pick\-up and drop\-off events in New York City\[[54](https://arxiv.org/html/2607.06652#bib.bib31)\], partitioned by borough and action type intoK=10K=10event categories \(55boroughs×\\times22actions\)\. The subset comprises2,0002\{,\}000randomly sampled drivers with a mean sequence length of3636events\. The standard EasyTPP train/validation/test split is used\.

Stack Overflow\.A dataset of user\-interaction events on Stack Overflow, where event types correspond to distinct awarded\-badge categories\[[54](https://arxiv.org/html/2607.06652#bib.bib31)\]\. The standard EasyTPP train/validation/test split is used\.

Taobao\.User click sequences on the Taobao shopping platform, collected between 25 November and 3 December 2017\[[54](https://arxiv.org/html/2607.06652#bib.bib31)\]\. Item categories are ranked by frequency; the most frequent are retained and all remaining categories are merged into a single*other*class, yieldingK=17K=17event types\. The subset comprises the2,0002\{,\}000most active users with a mean sequence length of5656events\. The standard EasyTPP train/validation/test split is used\.

Earthquake\.Time\-stamped seismic events recorded across the Conterminous United States from 1996 to 2023, derived from the USGS catalogue\[[54](https://arxiv.org/html/2607.06652#bib.bib31)\]\. The standard EasyTPP train/validation/test split is used\.

Yelp\-Mississauga \(YLP\)\.This real\-world dataset, taken from the EDITPP benchmark, contains user check\-in event sequences for businesses in the city of Mississauga during 2018\. Following the EDITPP convention, it is treated as an unmarked temporal point process \(K=1K=1\), where each sequence consists only of ordered check\-in times\.

### D\.1Training Details

Optimisation\. All experiments were optimised using the Adam optimiser\[[28](https://arxiv.org/html/2607.06652#bib.bib9)\], with learning rates tailored to each specific task\. The choice of learning rate was determined via grid search on the respective validation set\. Code Availability\. The complete source code is available at[github\.com/Code\-Cornelius/sigtpp](https://github.com/Code-Cornelius/sigtpp)\. Software Environment\. All numerical experiments were conducted on a system running Ubuntu 24\.04\.4 LTS with Python 3\.8 and PyTorch 1\.12\.1\+cu116\. CUDA was available through PyTorch, with cuDNN version 8302\. Hardware Infrastructure\.The machine was equipped with 3 NVIDIA L40S GPUs, each with 46,068 MiB of VRAM, using NVIDIA driver version 580\.126\.09 and compute capability 8\.9\. The system was powered by dual AMD EPYC 9354 32\-core processors, providing 128 CPU threads in total, with a maximum CPU frequency of 3\.80 GHz\. It had 755 GiB of RAM and NVMe SSD storage, consisting of one 11\.6 TB drive and four 3\.5 TB drives\.

We adapted the code ofLinet al\.\[[32](https://arxiv.org/html/2607.06652#bib.bib12)\]for some of the baselines \(WGAN\), correcting several bugs; our implementation consequently deviates from the original\.

Hyperparameter tuning:

Table S2:Hyperparameter search grid for all models\.†Fixed value, not searched\.ModelHyperparameterValuesWGANGenerator LRηG\\eta\_\{G\}10−4,5⋅10−5,10−510^\{\-4\},\\ 5\\cdot 10^\{\-5\},\\ 10^\{\-5\}Discriminator LRηD\\eta\_\{D\}10−2,10−3,10−410^\{\-2\},\\ 10^\{\-3\},\\ 10^\{\-4\}Hidden size16,3216,\\ 32Lipschitz reg\.λ\\lambda10−2,10−3,10−410^\{\-2\},\\ 10^\{\-3\},\\ 10^\{\-4\}DeterLearning rateη\\eta10−2,10−410^\{\-2\},\\ 10^\{\-4\}Hidden size8,16,328,\\ 16,\\ 32GammaLearning rateη\\eta1,10−2,10−41,\\ 10^\{\-2\},\\ 10^\{\-4\}VAELearning rateη\\eta10−2,10−3,10−410^\{\-2\},\\ 10^\{\-3\},\\ 10^\{\-4\}Hidden size16,3216,\\ 32Latent dimension8,168,\\ 16Reconstruction weight0\.01,0\.1,1,100\.01,\\ 0\.1,\\ 1,\\ 10KL annealing epochs†5050Free bits†0\.10\.1DDPMLearning rateη\\eta10−3,10−4,10−510^\{\-3\},\\ 10^\{\-4\},\\ 10^\{\-5\}Hidden size16,3216,\\ 32Diffusion stepsTT10,50,10010,\\ 50,\\ 100Batch size1024,4096,300001024,\\ 4096,\\ 30000SigTPPGenerator LRηG\\eta\_\{G\}10−3,5⋅10−4,10−410^\{\-3\},\\ 5\\cdot 10^\{\-4\},\\ 10^\{\-4\}Hidden size16,3216,\\ 32Signature degree†88Teacher forcingTrue, FalseTerminal anchorfree, residualDetach cumulative channelTrue, FalseNotes onSigTPPhyperparameters\.The terminal anchor controls how the endpoint of each path is handled before computing the signature\. The theoretical analysis uses theΦ\\Phi\-embedded representation on a fixed compact domain, which is the setting covered by the stability results\. In the implementation, we evaluate two path representations during validation\. Theresidualsetting uses theΦ\\Phi\-based terminal anchor, while thefreesetting computes signatures directly over the observed variable\-length interval ending at the last cumulative time\. Both options are included in the search because the preferred representation is dataset\-dependent\. The*teacher forcing*flag controls the training mode of the generator\. Because the generator is trained on full observed sequences, it can be conditioned on ground\-truth past events at each step \(teacher forcing,True\), or it can consume its own previously generated inter\-arrival times autoregressively \(False\)\. Teacher forcing typically stabilises early training, while the autoregressive mode better matches the inference regime\. Finally, the*detach cumulative channel*flag controls whether the cumulative\-time channel \(thettcoordinate of the path fed to the signature\) is detached from the automatic differentiation graph during training; this prevents gradients from flowing back through the time axis of the path, which can stabilise training when the cumulative channel dominates the gradient signal\.

Table S3:Performance comparison ofSigTPPagainst various baselines on synthetic TPP tasks \(PS, IP, H1, H3\) and real\-world TPP tasks \(EQ, SO, TB, TX, YLP\)\. We report MAE, MAEmed, and MSEmean\(see Appendix[C](https://arxiv.org/html/2607.06652#A3)for metric definitions\)\. Values report standard error in parentheses over 100 bootstrap replicates\. Lower is better\(↓\)\(\\downarrow\)\. The best result for each dataset and metric pair isbolded, and the second\-best isunderlined\.ModelPSIPH1H3EQSOTBTXYLPMAE \(×10−1\\times 10^\{\-1\}\)MAE \(×10−1\\times 10^\{\-1\}\)SigTPP9\.25\(0\.08\)6\.01\(0\.03\)23\.21\(0\.16\)4\.90\(0\.04\)11\.34\(0\.21\)10\.26\(0\.17\)1\.01\(0\.02\)2\.26\(0\.02\)5\.33\(0\.17\)VAE8\.88\(0\.08\)5\.61\(0\.03\)18\.59\(0\.12\)4\.50\(0\.04\)9\.57\(0\.19\)8\.78\(0\.14\)0\.67\(0\.01\)2\.00\(0\.02\)4\.88\(0\.14\)DDPM10\.77\(0\.08\)6\.91\(0\.03\)20\.87\(0\.12\)4\.99\(0\.04\)15\.02\(0\.29\)11\.19\(0\.16\)0\.71\(0\.01\)2\.16\(0\.02\)6\.04\(0\.16\)WGAN8\.90\(0\.08\)5\.73\(0\.03\)18\.29\(0\.14\)4\.47\(0\.04\)13\.08\(0\.19\)8\.97\(0\.12\)0\.71\(0\.01\)2\.25\(0\.02\)5\.26\(0\.15\)Deter6\.29\(0\.11\)4\.25\(0\.03\)12\.82\(0\.15\)3\.23\(0\.04\)6\.98\(0\.17\)6\.51\(0\.13\)0\.42\(0\.01\)1\.49\(0\.02\)3\.66\(0\.15\)Gamma9\.15\(0\.08\)6\.52\(0\.03\)19\.60\(0\.11\)4\.83\(0\.03\)14\.20\(0\.09\)10\.08\(0\.08\)0\.71\(0\.01\)2\.16\(0\.02\)6\.55\(0\.11\)MAEmed\(×10−1\\times 10^\{\-1\}\)MAEmed\(×10−1\\times 10^\{\-1\}\)SigTPP6\.37\(0\.11\)4\.22\(0\.04\)14\.28\(0\.14\)3\.38\(0\.04\)7\.51\(0\.16\)7\.18\(0\.12\)0\.68\(0\.02\)1\.49\(0\.02\)3\.97\(0\.14\)VAE6\.35\(0\.11\)4\.18\(0\.04\)12\.52\(0\.15\)3\.18\(0\.04\)6\.92\(0\.17\)6\.40\(0\.12\)0\.42\(0\.01\)1\.38\(0\.02\)3\.68\(0\.14\)DDPM6\.46\(0\.11\)4\.28\(0\.04\)11\.94\(0\.16\)3\.04\(0\.04\)6\.14\(0\.15\)6\.20\(0\.13\)0\.40\(0\.01\)1\.49\(0\.02\)3\.59\(0\.15\)WGAN6\.28\(0\.11\)4\.13\(0\.04\)12\.72\(0\.16\)3\.21\(0\.04\)7\.55\(0\.17\)6\.68\(0\.12\)0\.42\(0\.01\)1\.49\(0\.02\)3\.78\(0\.14\)Deter6\.29\(0\.11\)4\.25\(0\.03\)12\.82\(0\.15\)3\.23\(0\.04\)6\.98\(0\.17\)6\.51\(0\.13\)0\.42\(0\.01\)1\.49\(0\.02\)3\.66\(0\.15\)Gamma6\.33\(0\.11\)4\.37\(0\.04\)13\.06\(0\.15\)3\.27\(0\.04\)8\.16\(0\.13\)6\.64\(0\.11\)0\.43\(0\.01\)1\.49\(0\.02\)3\.91\(0\.16\)MSEmean\(×10−1\\times 10^\{\-1\}\)MSEmean\(×10−1\\times 10^\{\-1\}\)SigTPP8\.31\(0\.34\)3\.86\(0\.08\)56\.40\(0\.93\)2\.83\(0\.09\)16\.80\(0\.78\)13\.02\(0\.39\)0\.27\(0\.01\)0\.88\(0\.05\)9\.53\(0\.79\)VAE8\.27\(0\.35\)3\.68\(0\.09\)41\.53\(0\.97\)2\.42\(0\.08\)14\.73\(0\.74\)9\.88\(0\.37\)0\.16\(0\.01\)0\.82\(0\.05\)7\.71\(0\.74\)DDPM9\.05\(0\.32\)4\.11\(0\.09\)43\.15\(0\.99\)2\.25\(0\.08\)15\.47\(0\.63\)9\.78\(0\.34\)0\.15\(0\.01\)0\.83\(0\.05\)8\.26\(0\.77\)WGAN8\.10\(0\.34\)3\.62\(0\.08\)42\.59\(0\.98\)2\.46\(0\.08\)18\.20\(0\.99\)11\.42\(0\.39\)0\.16\(0\.01\)0\.90\(0\.05\)8\.55\(0\.71\)Deter8\.84\(0\.40\)3\.66\(0\.07\)45\.13\(1\.16\)2\.65\(0\.09\)16\.94\(0\.83\)12\.02\(0\.49\)0\.17\(0\.01\)0\.93\(0\.06\)9\.32\(0\.81\)Gamma8\.29\(0\.35\)4\.18\(0\.09\)44\.51\(1\.00\)2\.54\(0\.08\)18\.94\(0\.65\)10\.85\(0\.34\)0\.16\(0\.01\)0\.90\(0\.05\)9\.65\(0\.77\)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x45.png)Figure S11:Relative scores with respect toDeter, on synthetic datasets, plotting values reported in Table[S6](https://arxiv.org/html/2607.06652#A4.T6)\. The leftmost group shows the average relative score across all metrics, followed by the metric\-wise scores for all 8 metrics, see Appendix[C](https://arxiv.org/html/2607.06652#A3)\. Bars compare the different models\. Lower is better\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x46.png)Figure S12:Relative scores with respect toDeter, on real\-world datasets, plotting values reported in Table[S6](https://arxiv.org/html/2607.06652#A4.T6)\. The leftmost group shows the average relative score across all metrics, followed by the metric\-wise scores for all 8 metrics, see Appendix[C](https://arxiv.org/html/2607.06652#A3)\. Bars compare the different models\. Lower is better\.Table S4:First inactive signature degreeMallM^\{\\mathrm\{all\}\}per dataset, defined as the first degreemmfor which allDmD^\{m\}coordinates have empirical standard deviation below10−810^\{\-8\}across the training set\.Table S5:Average ranks by metric comparing the neural modelsSigTPP, VAE, DDPM, and WGAN, aggregated over synthetic datasets, real\-world datasets, and all datasets\. Values report the mean rank and standard deviation over datasets in parentheses\. Lower is better\(↓\)\(\\downarrow\)\. Best mean ranks for each metric and split arebolded, and second best areunderlined\.Table S6:Geometric relative scores againstDeterby metric forSigTPP, VAE, DDPM, WGAN, andGamma, aggregated over synthetic datasets, real\-world datasets, and both simultaneously\. Lower is better\(↓\)\(\\downarrow\)\. Best values within each metric and split arebolded, and second best areunderlined\. The final row is the geometric average across metrics and datasets\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x47.png)

\(a\)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x48.png)

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\(d\)

Figure S13:Qualitative evaluation on H3\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(c\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(d\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x51.png)

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\(e\)

Figure S14:Qualitative evaluation on EQ\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Quantile\-quantile \(QQ\) plots\. Each subplot shows empirical quantiles of generated samples \(xx\-axis\) against the reference quantiles of the data \(yy\-axis\); the red dashed line is the identityy=xy=x\. \(c\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(d\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(e\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x56.png)

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![Refer to caption](https://arxiv.org/html/2607.06652v1/x57.png)

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\(e\)

Figure S15:Qualitative evaluation on SO\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Quantile\-quantile \(QQ\) plots\. Each subplot shows empirical quantiles of generated samples \(xx\-axis\) against the reference quantiles of the data \(yy\-axis\); the red dashed line is the identityy=xy=x\. \(c\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(d\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(e\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x61.png)

\(a\)

![Refer to caption](https://arxiv.org/html/2607.06652v1/x62.png)

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Figure S16:Qualitative evaluation on Taobao\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Quantile\-quantile \(QQ\) plots\. Each subplot shows empirical quantiles of generated samples \(xx\-axis\) against the reference quantiles of the data \(yy\-axis\); the red dashed line is the identityy=xy=x\. \(c\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(d\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(e\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x66.png)

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![Refer to caption](https://arxiv.org/html/2607.06652v1/x67.png)

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Figure S17:Qualitative evaluation on Taxi\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Quantile\-quantile \(QQ\) plots\. Each subplot shows empirical quantiles of generated samples \(xx\-axis\) against the reference quantiles of the data \(yy\-axis\); the red dashed line is the identityy=xy=x\. \(c\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(d\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(e\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.![Refer to caption](https://arxiv.org/html/2607.06652v1/x71.png)

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\(e\)

Figure S18:Qualitative evaluation on Yelp\. The top row showsSigTPP, WGAN, and DDPM; the bottom row shows VAE,Deter, andGamma\. \(a\) Histogram estimates of the interarrival times densityp​\(x\)p\(x\)on a logarithmic scale; the leftmost panel shows the target distribution, the others the distributions of samples generated by each model\. \(b\) Quantile\-quantile \(QQ\) plots\. Each subplot shows empirical quantiles of generated samples \(xx\-axis\) against the reference quantiles of the data \(yy\-axis\); the red dashed line is the identityy=xy=x\. \(c\) Empirical intensity functionsλ^​\(t\)\\hat\{\\lambda\}\(t\)estimated from the same sequences\. \(d\) Absolute Pearson correlation\-matrix differences for the first 20 indices\. Each heatmap entry shows the absolute difference between the empirical correlation of two sequence components in the generated samples and in the reference data\. \(e\) Absolute autocorrelation error for55lags\. Each curve shows the difference between the ACF of the indicated model and the empirical ACF of the reference process, so that the target corresponds to the horizontal line at zero\.

## Appendix ELimitations

A primary limitation of our approach is the computational cost of the truncated signature transform, whose feature dimension scales exponentially with the truncation depthMMand polynomially with the path dimensiondd\. This may hinder direct extension to high\-dimensional or marked TPPs\. Second, our current embedding relies on a linear interpolation lift\. While effective in practice, this lift introduces an artificial continuous transition at jump times and is therefore not fully faithful to the jump structure of the process\. Third, our theoretical analysis relies on a determinacy property of the pushforward law of the temporal point process under the embedding, but such determinacy conditions are already difficult to verify in practice even for continuous processes, and we do not investigate them further here\. Finally, our theoretical bounds are preliminary and likely conservative: the constants are loose, and a sharper characterisation of the embedding remains open\. A better understanding of how the required truncation degree depends on the nature of the temporal point process, together with the relationship between moment conditions and the law of the process, are promising directions for future work\.