GRATE: Temporal Extensions for Inductive KG Foundation Models via Gated Rotary Attention

arXiv cs.AI Papers
temporal-knowledge-graphs inductive-reasoning foundation-models gated-rotary-attention graph-neural-networks knowledge-graph-embeddings link-prediction

Summary

This paper proposes GRATE (Gated Rotary Attention for Temporal Encoding), a parameter-free temporal encoding method that enhances inductive knowledge graph foundation models by incorporating relative time differences and query-conditioned gating. It also introduces new inductive temporal knowledge graph benchmarks (GDELTIndT and WIKIIndT) to evaluate cross-dataset transfer, demonstrating improved performance over static base models.

arXiv:2607.10197v1 Announce Type: new Abstract: Knowledge graph foundation models such as Ultra and Trix achieve strong inductive transfer by learning relation-graph representations that generalise to unseen entities and relations. Extending this transferability to temporal knowledge graphs (TKGs) remains challenging: existing temporal models tie their parameters to dataset-specific entities, relations, or timestamps and are not designed to transfer to TKGs with disjoint vocabularies. We propose GRATE (Gated Rotary Attention for Temporal Encoding), an entity-side message function that adds no learnable parameters and encodes time through relative time differences by rotating each edge message according to its time gap to the query and applying a query-conditioned gate to select temporally relevant signals. GRATE integrates into NBFNet-style KG foundation models while preserving structural transferability. Existing TKG benchmarks evaluate within shared train/test vocabularies and cannot directly test cross-dataset temporal transfer; we therefore construct GDELTIndT and WIKIIndT, inductive transfer benchmark suites with disjoint entities, relations, and timestamps spanning both interpolation and extrapolation. Across these benchmarks and held-out forecasting datasets, a single jointly pretrained GRATE checkpoint improves over the static base model in most settings.
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# Temporal Extensions for Inductive KG Foundation Models via Gated Rotary Attention
Source: [https://arxiv.org/html/2607.10197](https://arxiv.org/html/2607.10197)
###### Abstract

Knowledge graph foundation models such asUltraandTrixachieve strong inductive transfer by learning relation\-graph representations that generalise to unseen entities and relations\. Extending this transferability to temporal knowledge graphs \(TKGs\) remains challenging: existing temporal models tie their parameters to dataset\-specific entities, relations, or timestamps and are not designed to transfer to TKGs with disjoint vocabularies\. We proposeGrate\(Gated Rotary Attention for Temporal Encoding\), an entity\-side message function that adds no learnable parameters and encodes time through relative time differences by rotating each edge message according to its time gap to the query and applying a query\-conditioned gate to select temporally relevant signals\.Grateintegrates into NBFNet\-style KG foundation models while preserving structural transferability\. Existing TKG benchmarks evaluate within shared train/test vocabularies and cannot directly test cross\-dataset temporal transfer; we therefore constructGDELTIndTandWIKIIndT, inductive transfer benchmark suites with disjoint entities, relations, and timestamps spanning both interpolation and extrapolation\. Across these benchmarks and held\-out forecasting datasets, a single jointly pretrainedGratecheckpoint improves over the static base model in most settings\.

Temporal Knowledge Graphs, Knowledge Graph Foundation Models, Inductive Reasoning, Rotary Position Embedding, Graph Neural Networks

## 1Introduction

Recent knowledge graph \(KG\) foundation models\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36); Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58); Duet al\.,[2026](https://arxiv.org/html/2607.10197#bib.bib5)\)learn vocabulary\-agnostic representations that transfer to graphs with unseen entities and relations, supporting link prediction queries such as “who did Obama engage in negotiation with?” Many facts are also inherently temporal: temporal knowledge graphs \(TKGs\) attach a timestamp to each fact, so the same query is only meaningful when paired with a year \(“…in 2010”\)\. Existing TKG models\(Zhanget al\.,[2022](https://arxiv.org/html/2607.10197#bib.bib24); Liet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib12); Sunet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib40)\)learn dataset\-specific embeddings for entities, relations, and timestamps and must be retrained whenever new ones appear, making them unsuitable for settings such as emerging political actors, newly discovered scientists, or rapidly evolving event streams, where new vocabulary arrives continuously\.

A natural alternative is to apply an inductive KG foundation model directly to a TKG\. However, temporal reasoning requires similarity judgements that are not purely structural\. Inductive KG foundation models transfer by comparing new entities and relations through their graph contexts, but in a TKG the usefulness of a supporting fact also depends on its time and its relevance to the query\. For a query such as “who did the US president meet in 2024?”, a meeting from 2024 is usually more similar to the query context than one from 1974, even if both have similar graph structure\. Likewise, among same\-day events, a meeting with a head of state is more relevant to a diplomatic query than a meeting with a sports figure\. Thus, inductive temporal reasoning requires the model to judge supporting facts not only by structural similarity, but also by relative temporal displacement and query\-conditioned relevance\. Existing inductive KG foundation models do not provide this: they define similarity only through static graph structure, treating facts from all timestamps as equally aligned with the query and aggregating all supporting facts indiscriminately\.

Therefore, we proposeGrate\(Gated Rotary Attention for Temporal Encoding\), a parameter\-free temporal message function for NBFNet\-style\(Zhuet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib44)\)KG foundation models\.Grateimproves the temporal similarity judgement of the base model in two complementary ways\. First, it rotates each supporting fact’s message according to its relative time gap to the query, so temporal alignment is expressed as a function of relative displacement rather than absolute timestamp identity\. Second, it applies a query\-conditioned gate that weights each rotated message by its alignment with the query, allowing the model to emphasise temporally and semantically relevant evidence\. Because both mechanisms depend only on relative time differences and existing hidden representations,Grateadds no learnable parameters and preserves the transferability of the underlying model\.

To evaluate temporal reasoning in inductive transfer settings, we construct a suite of TKG benchmarks with disjoint entities, relations, and timestamps between training and inference graphs\. Experiments show thatGrateimproves zero\-shot temporal reasoning in most settings over strong inductive base models\. Together,Grateand our inductive transfer benchmarks demonstrate that temporal reasoning can be added to inductive KG foundation models without sacrificing parameter\-free transferability\.

## 2Related Work

### 2\.1Temporal Knowledge Graph Embedding

Transductive interpolation \(TKG completion\)\.TKG completion models predict missing facts at timestamps drawn from within the training time range\. Early methods incorporate time as an additional component in the scoring function, including TTransE\(Leblay and Chekol,[2018](https://arxiv.org/html/2607.10197#bib.bib30)\)and TA\-DistMult\(García\-Duránet al\.,[2018](https://arxiv.org/html/2607.10197#bib.bib33)\)\. Later works such as TComplEx and TNTComplEx\(Lacroixet al\.,[2020](https://arxiv.org/html/2607.10197#bib.bib17)\)extend tensor\-factorisation approaches to better capture temporal dependencies\. Geometric extensions include rotation\-based encodings for time embedding such as TeRo\(Xuet al\.,[2020](https://arxiv.org/html/2607.10197#bib.bib27)\)\.

Transductive extrapolation \(TKG forecasting\)\.TKG forecasting models predict facts at strictly future timestamps via GNN\-based subgraph evolution\(Liet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib12),[2022](https://arxiv.org/html/2607.10197#bib.bib43); Hanet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib55); Sunet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib40); Lianget al\.,[2023](https://arxiv.org/html/2607.10197#bib.bib38)\)or rule\-based and explainable reasoning\(Liuet al\.,[2022](https://arxiv.org/html/2607.10197#bib.bib42); Gastingeret al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib3)\)\.

LLM\-based inductive TKG reasoning\.Recent approaches explore inductive inference on TKGs using textual descriptions and large language models\. ICL\(Leeet al\.,[2023a](https://arxiv.org/html/2607.10197#bib.bib52)\)and GenTKG\(Liaoet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib56)\)leverage entity descriptions, relation names, or in\-context temporal reasoning to transfer across unseen temporal facts, but their transferability arises from textual semantics rather than structural representations\.Grateis purely structure\-driven, learning transferable temporal patterns directly from graph structure without textual annotations or language models\.

### 2\.2KG Foundation Models

Recent work has moved toward inductive, vocabulary\-agnostic foundation models\. NBFNet\(Zhuet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib44)\), GraIL\(Teruet al\.,[2020](https://arxiv.org/html/2607.10197#bib.bib45)\), and INDIGO\(Liuet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib46)\)introduce query\-conditioned message passing for unseen entities but assume a fixed relation vocabulary; INGRAM\(Leeet al\.,[2023b](https://arxiv.org/html/2607.10197#bib.bib48)\)lifts this by modelling relations as nodes in a relation graph\.Ultra\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)andTrix\(Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\)build on this idea, learning transferable relational structure that lets a single pretrained checkpoint operate across graphs with disjoint entity and relation vocabularies\. None of these models address time; lifting them to temporal graphs without sacrificing transferability is the open challengeGrateaddresses\.

## 3Problem Definition

### 3\.1Temporal Knowledge Graphs

A TKG is a structureG=\(V,R,T,Q\)G=\(V,R,T,Q\), whereQQis a set of quadruples\(s,r,o,τ\)∈V×R×V×T\(s,r,o,\\tau\)\\in V\\times R\\times V\\times T\(the observed facts\), withs,o∈Vs,o\\in Vsubject and object entities,r∈Rr\\in Ra relation, andτ∈T\\tau\\in Ta timestamp\. We model the timestamp setTTas a finite subset ofℕ\\mathbb\{N\}, with each dataset’s calendar granularity \(e\.g\. daily, yearly\) absorbed into the integer encoding\.

The reasoning task studied in this work is temporal link prediction: given a query\(s,r,?,τ\)\(s,r,?,\\tau\)or\(?,r,o,τ\)\(?,r,o,\\tau\), predict the missing entity at the specified timestampτ\\tau\.

### 3\.2Inference Settings

We define an inference setting as the combination of assumptions governing the relationship between the training graph and the inference graph\. In temporal knowledge graphs, this relationship has two orthogonal dimensions: \(i\) the overlap between training and inference vocabularies, and \(ii\) the temporal relationship between training and inference timestamps\.

Vocabulary condition\.We denote a training TKG asG𝑡𝑟𝑎𝑖𝑛=\(V𝑡𝑟𝑎𝑖𝑛,R𝑡𝑟𝑎𝑖𝑛,T𝑡𝑟𝑎𝑖𝑛,Q𝑡𝑟𝑎𝑖𝑛\)G\_\{\\mathit\{train\}\}=\(V\_\{\\mathit\{train\}\},R\_\{\\mathit\{train\}\},T\_\{\\mathit\{train\}\},Q\_\{\\mathit\{train\}\}\)and an inference TKG asG𝑖𝑛𝑓=\(V𝑖𝑛𝑓,R𝑖𝑛𝑓,T𝑖𝑛𝑓,Q𝑖𝑛𝑓\)G\_\{\\mathit\{inf\}\}=\(V\_\{\\mathit\{inf\}\},R\_\{\\mathit\{inf\}\},T\_\{\\mathit\{inf\}\},Q\_\{\\mathit\{inf\}\}\)\.*Transductive*settings share the entity and relation vocabularies between training and inference,V𝑡𝑟𝑎𝑖𝑛=V𝑖𝑛𝑓V\_\{\\mathit\{train\}\}=V\_\{\\mathit\{inf\}\}andR𝑡𝑟𝑎𝑖𝑛=R𝑖𝑛𝑓R\_\{\\mathit\{train\}\}=R\_\{\\mathit\{inf\}\}; this is the regime in which existing TKG embedding methods learn per\-entity, per\-relation, and per\-timestamp embeddings\. Inductive transfer settings \(referred to as the fully\-inductive setting in KG foundation models\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36); Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\)\) make the three sets disjoint \(i\.e\.,V𝑡𝑟𝑎𝑖𝑛∩V𝑖𝑛𝑓V\_\{\\mathit\{train\}\}\\cap V\_\{\\mathit\{inf\}\},R𝑡𝑟𝑎𝑖𝑛∩R𝑖𝑛𝑓R\_\{\\mathit\{train\}\}\\cap R\_\{\\mathit\{inf\}\}, andT𝑡𝑟𝑎𝑖𝑛∩T𝑖𝑛𝑓T\_\{\\mathit\{train\}\}\\cap T\_\{\\mathit\{inf\}\}are all empty\), so representations tied to the entity, relation, or timestamp vocabularies are, by construction, not applicable at inference time\. The model is trained on a sourceG𝑡𝑟𝑎𝑖𝑛G\_\{\\mathit\{train\}\}and applied, without fine\-tuning, to a targetG𝑖𝑛𝑓G\_\{\\mathit\{inf\}\}\.

Temporal condition\.We evaluate two regimes, both defined within the inference graphG𝑖𝑛𝑓G\_\{\\mathit\{inf\}\}:*interpolation*, where test\-fact timestamps fall within the time range of the observed inference subgraph, and*extrapolation*, where they fall strictly after it\.

Existing benchmarks vs\. our setting\.Gratetargets the inductive transfer setting under both interpolation and extrapolation evaluation regimes, a combination not addressed by prior TKG models\.

## 4NBFNet\-Style KG Foundation Models

We build on inductive knowledge graph foundation models based on NBFNet\-style\(Zhuet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib44)\)message passing, includingUltra\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)andTrix\(Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\)\.

Given a query\(s,r,?\)\(s,r,?\), the model initialises the source entitysswith the query relationrr\. Message passing then proceeds over directed edges\(u→v\)\(u\\\!\\to\\\!v\)in the knowledge graph\. At layerℓ\\ell, each edge produces a message

mu​v\(ℓ\)=ϕ​\(hu\(ℓ−1\),ru​v\),m\_\{uv\}^\{\(\\ell\)\}=\\phi\\\!\\left\(h\_\{u\}^\{\(\\ell\-1\)\},r\_\{uv\}\\right\),\(1\)whereϕ\\phiis a KG scoring function \(e\.g\. DistMult or TransE\),hu\(ℓ−1\)h\_\{u\}^\{\(\\ell\-1\)\}is the representation of nodeuufrom the previous layer, andru​vr\_\{uv\}is the embedding associated with edge\(u→v\)\(u\\\!\\to\\\!v\)\. Messages from neighbouring nodes are then aggregated to update nodevv:

hv\(ℓ\)=AGG⁡\(\{mu​v\(ℓ\):u∈𝒩​\(v\)\}\)\.h\_\{v\}^\{\(\\ell\)\}=\\operatorname\{AGG\}\\\!\\left\(\\bigl\\\{m\_\{uv\}^\{\(\\ell\)\}:u\\in\\mathcal\{N\}\(v\)\\bigr\\\}\\right\)\.\(2\)Traditional KG models learn a separate embedding for every relation type, which prevents transfer to unseen relations\. Inductive KG foundation models instead construct a*relation graph*GRG\_\{R\}, derived from the original knowledge graph, in which relations themselves are treated as nodes\. Two relation nodes are connected when they exhibit compatible structural interaction patterns in the original graph, such as sharing head or tail entities\. The relation graph therefore captures higher\-order dependencies between relation types independently of specific entities\.

UltraandTrixcompute relation representations by performing message passing over this relation graph:

\{ru​v\}\(u,v\)∈E=RelGNN​\(GR∣r\)\.\\\{r\_\{uv\}\\\}\_\{\(u,v\)\\in E\}=\\mathrm\{RelGNN\}\(G\_\{R\}\\mid r\)\.\(3\)Ultraconstructs relation representations using a single\-pass aggregation over four relation interaction types \(h2h, h2t, t2h, t2t\), producing a count\-based structural summary for each relation\.Trixinstead iteratively refines the relation graph representations across layers and interleaves relation\-graph updates with entity\-graph message passing, yielding more expressive query\-conditioned relation representations\.

In temporal knowledge graphs, each edge additionally carries a timestamp, which the standard messagemu​v\(ℓ\)m\_\{uv\}^\{\(\\ell\)\}in[Equation1](https://arxiv.org/html/2607.10197#S4.E1)does not incorporate\.

## 5Method:Grate

We proposeGrate\(Gated Rotary Attention for Temporal Encoding\), a parameter\-free mechanism that augments NBFNet\-style KG foundation models with temporal information\.[Figure1](https://arxiv.org/html/2607.10197#S5.F1)illustrates theGratecomponent\.

![Refer to caption](https://arxiv.org/html/2607.10197v1/x1.png)Figure 1:Gratemechanism and integration\.*\(a\)*Overview of the four processing steps for an example query \(Barack Obama,engage\_in\_negotiation, ?, Dec 2014\)\.\(1\)Observed supporting facts from the message graph\.\(2\)Semantic base messagesmu​v=ϕ​\(hu,ru​v\)m\_\{uv\}=\\phi\(h\_\{u\},r\_\{uv\}\)computed by the KG score function\.\(3\)Relative\-time rotation: each base message is rotated byR​\(Δ​tu​v\)R\(\\Delta t\_\{uv\}\), whereΔ​tu​v=τ−tu​v\\Delta t\_\{uv\}=\\tau\-t\_\{uv\}, encoding temporal displacement at multiple frequencies\.\(4\)Query\-conditioned gate: the rotated message is scored against the query stateQqQ\_\{q\}viag=σ​\(Qq⊤​mrot/d\)g=\\sigma\(Q\_\{q\}^\{\\\!\\top\}m^\{\\mathrm\{rot\}\}/\\sqrt\{d\}\), producing a gated message forwarded to the base model\.*\(b\)*Grateintegrates into NBFNet\-style KG foundation models by replacing only the entity\-side message function\.### 5\.1Rotary Relative\-Time Encoding

Temporal knowledge graph reasoning depends not only on structural connectivity, but also on how supporting facts evolve relative to the query time\. In many temporal reasoning tasks, facts closer to the query timestamp are often more informative than temporally distant ones, and different supporting events contribute unequally depending on their temporal displacement from the query\. A temporal message function should therefore capture relative temporal evolution and encode the varying influence of supporting facts according to their time difference from the query\.

Concretely, for an observed edge\(u,ru​v,v,tu​v\)\(u,r\_\{uv\},v,t\_\{uv\}\), wheretu​vt\_\{uv\}is the timestamp of edge\(u→v\)\(u\\\!\\to\\\!v\), we define the relative time gap regarding query timeτ\\tauas:

Δ​tu​v=τ−tu​v\.\\Delta t\_\{uv\}=\\tau\-t\_\{uv\}\.\(4\)The temporal transformation should depend onΔ​tu​v\\Delta t\_\{uv\}rather than on learned timestamp embeddings, so that the model can generalise to timestamps not observed during training\.

To implement this relative\-time transformation, we adopt rotary position embedding \(RoPE\)\(Suet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib2)\), originally introduced for positional encoding in Transformer models\. RoPE rotates vector representations in a fixed frequency basis according to an input offset\. In our setting, the offset corresponds to the relative time gapΔ​tu​v\\Delta t\_\{uv\}, allowing supporting facts at different timestamps to be represented as temporally shifted versions of the same structural message\.

Letdddenote the \(even\) hidden dimension of the message function and define the frequency vector

ωk=1100002​k/d,k=0,1,…,d/2−1\.\\omega\_\{k\}=\\frac\{1\}\{10000^\{2k/d\}\},\\qquad k=0,1,\\ldots,d/2\-1\.\(5\)The rotation matrixR​\(δ\)∈ℝd×dR\(\\delta\)\\in\\mathbb\{R\}^\{d\\times d\}is block\-diagonal,

R​\(δ\)=diag​\(R0​\(δ\),…,Rd/2−1​\(δ\)\),R\(\\delta\)=\\mathrm\{diag\}\\\!\\bigl\(R\_\{0\}\(\\delta\),\\ldots,R\_\{d/2\-1\}\(\\delta\)\\bigr\),\(6\)where each2×22\{\\times\}2block rotates thekk\-th frequency pair by angleθk=ωk​δ\\theta\_\{k\}=\\omega\_\{k\}\\,\\delta:

Rk​\(δ\)=\(cos⁡θk−sin⁡θksin⁡θkcos⁡θk\)\.R\_\{k\}\(\\delta\)=\\begin\{pmatrix\}\\cos\\theta\_\{k\}&\-\\sin\\theta\_\{k\}\\\\ \\sin\\theta\_\{k\}&\\phantom\{\-\}\\cos\\theta\_\{k\}\\end\{pmatrix\}\.\(7\)Partitioning a vectorx∈ℝdx\\in\\mathbb\{R\}^\{d\}into consecutive pairs\(x2​k,x2​k\+1\)\(x\_\{2k\},x\_\{2k\+1\}\), the rotation acts independently on each pair:

\(R​\(δ\)​x\)2​k\\displaystyle\\bigl\(R\(\\delta\)\\,x\\bigr\)\_\{2k\}=x2​k​cos⁡θk−x2​k\+1​sin⁡θk,\\displaystyle=x\_\{2k\}\\cos\\theta\_\{k\}\-x\_\{2k\+1\}\\sin\\theta\_\{k\},\(8\)\(R​\(δ\)​x\)2​k\+1\\displaystyle\\bigl\(R\(\\delta\)\\,x\\bigr\)\_\{2k\+1\}=x2​k​sin⁡θk\+x2​k\+1​cos⁡θk\.\\displaystyle=x\_\{2k\}\\sin\\theta\_\{k\}\+x\_\{2k\+1\}\\cos\\theta\_\{k\}\.Gratethen rotates the base messagemu​vm\_\{uv\}111FollowingUltraandTrix, the base model instantiatesϕ\\phiin[Equation1](https://arxiv.org/html/2607.10197#S4.E1)as DistMult:mu​v=hu⊙ru​v∈ℝdm\_\{uv\}=h\_\{u\}\\odot r\_\{uv\}\\in\\mathbb\{R\}^\{d\}, where⊙\\odotdenotes the element\-wise Hadamard product\.in[Equation1](https://arxiv.org/html/2607.10197#S4.E1)according to the relative time gap:

mu​vrot=R​\(Δ​tu​v\)​mu​v\.m^\{\\text\{rot\}\}\_\{uv\}=R\(\\Delta t\_\{uv\}\)\\,m\_\{uv\}\.\(9\)This construction transforms each structural message as a continuous function of its temporal displacement from the query, enabling temporally distinct supporting facts to produce different message representations while remaining independent of absolute timestamp identities\.

### 5\.2Query\-Conditioned Gating

The rotation in[Equation9](https://arxiv.org/html/2607.10197#S5.E9)incorporates the relative time gapΔ​t\\Delta tinto each message, but does not yet address a second requirement: selecting which temporally shifted edges are relevant to the current query\. In temporal knowledge graphs, multiple supporting facts may exist at different time offsets, and not all contribute equally to the prediction\.

To model query\-dependent relevance, we introduce a gating mechanism with no learnable parameters that depends jointly on the query and the rotated message\. Lethhead​\(q\)\(ℓ−1\)h\_\{\\mathrm\{head\}\(q\)\}^\{\(\\ell\-1\)\}denote the representation of the query head entity at layerℓ−1\\ell\-1, and letrqr\_\{q\}be the query relation embedding\. We define the query state

Qq=hhead​\(q\)\(ℓ−1\)\+rq,Q\_\{q\}=h\_\{\\mathrm\{head\}\(q\)\}^\{\(\\ell\-1\)\}\+r\_\{q\},\(10\)which summarises the query context\. This additive form mirrors NBFNet\-style initialisation, where the query relation is placed at the source node;QqQ\_\{q\}therefore represents the same query signal that drives message propagation from the source\.

Given the rotated messagemu​vrotm^\{\\text\{rot\}\}\_\{uv\}, we compute a scalar gate

gu​v=σ​\(Qq⊤​mu​vrotd\),g\_\{uv\}=\\sigma\\\!\\left\(\\frac\{Q\_\{q\}^\{\\top\}\\,m^\{\\text\{rot\}\}\_\{uv\}\}\{\\sqrt\{d\}\}\\right\),\(11\)whereσ\\sigmadenotes the sigmoid function\. The1/d1/\\sqrt\{d\}scaling matches scaled dot\-product attention\(Vaswaniet al\.,[2017](https://arxiv.org/html/2607.10197#bib.bib8)\)and stabilises gate magnitudes asddgrows\. The finalGratemessage is

mu​vGrate=gu​v⋅mu​vrot\.m^\{\\textsc\{Grate\}\{\}\}\_\{uv\}=g\_\{uv\}\\cdot m^\{\\text\{rot\}\}\_\{uv\}\.\(12\)The gate measures the alignment between the query and the rotated message, allowing the model to emphasise temporally relevant signals while suppressing irrelevant ones\.

Unlike softmax attention, which couples edge weights within a neighbourhood, the sigmoid gate evaluates each edge independently\. This matches the additive aggregation structure of NBFNet\-style message passing, where supporting evidence is accumulated rather than normalised across neighbours\. Independent gating is also more stable under inductive transfer, where neighbourhood\-size distributions may differ substantially from training graphs\. Because the gate depends only on the alignment between the rotated messageR​\(Δ​tu​v\)​mR\(\\Delta t\_\{uv\}\)\\,mand the query stateQqQ\_\{q\}in the shared RoPE basis, the notion of temporal relevance remains consistent across training and inference graphs without introducing timestamp\-specific parameters\.

Both the rotation and gating rely only on relative time differences and existing learned representations \(huh\_\{u\},ru​vr\_\{uv\},hhead​\(q\)h\_\{\\mathrm\{head\}\(q\)\},rqr\_\{q\}\), soGrateadds zero learnable parameters over the base model \([AppendixD](https://arxiv.org/html/2607.10197#A4)\) while incorporating temporal and query\-dependent reasoning\.[Algorithm1](https://arxiv.org/html/2607.10197#alg1)summarises the full per\-layer forward pass\.

Algorithm 1Grateforward pass for layerℓ\\ell, query\(s,r,?,τ\)\(s,r,?,\\tau\)\.0:node states

\{hu\(ℓ−1\)\}\\\{h\_\{u\}^\{\(\\ell\-1\)\}\\\},query relation embedding

rqr\_\{q\},edges

u→ru​vvu\\xrightarrow\{r\_\{uv\}\}vwith timestamps

\{tu​v\}\\\{t\_\{uv\}\\\},hidden dimension

dd\(even\), RoPE base

b=10000b\{=\}10000\.

0:updated node states

\{hv\(ℓ\)\}\\\{h\_\{v\}^\{\(\\ell\)\}\\\}\.

1:

ωk←b−2​k/d\\omega\_\{k\}\\leftarrow b^\{\-2k/d\}for

k=0,…,d/2−1k=0,\\ldots,d/2\{\-\}1
2:

Qq←hs\(ℓ−1\)\+rqQ\_\{q\}\\leftarrow h\_\{s\}^\{\(\\ell\-1\)\}\+r\_\{q\}\{query state\}

3:for alledges

u→ru​vvu\\xrightarrow\{r\_\{uv\}\}vwith timestamp

tu​vt\_\{uv\}do

4:

Δ​tu​v←τ−tu​v\\Delta t\_\{uv\}\\leftarrow\\tau\-t\_\{uv\}
5:

m←hu\(ℓ−1\)⊙ru​vm\\leftarrow h\_\{u\}^\{\(\\ell\-1\)\}\\odot r\_\{uv\}\{base message\}

6:

mrot←R​\(Δ​tu​v\)​mm^\{\\mathrm\{rot\}\}\\leftarrow R\(\\Delta t\_\{uv\}\)\\,mvia[Equation8](https://arxiv.org/html/2607.10197#S5.E8)

7:

g←σ​\(Qq⊤​mrot/d\)g\\leftarrow\\sigma\\bigl\(Q\_\{q\}^\{\\top\}m^\{\\mathrm\{rot\}\}/\\sqrt\{d\}\\bigr\)\{gate\}

8:

mu​vGrate←g⋅mrotm^\{\\textsc\{Grate\}\{\}\}\_\{uv\}\\leftarrow g\\cdot m^\{\\mathrm\{rot\}\}
9:endfor

10:for allnodes

vvdo

11:

hv\(ℓ\)←AGG⁡\(\{mu​vGrate:u∈𝒩​\(v\)\}\)h\_\{v\}^\{\(\\ell\)\}\\leftarrow\\operatorname\{AGG\}\\bigl\(\\\{m^\{\\textsc\{Grate\}\{\}\}\_\{uv\}:u\\in\\mathcal\{N\}\(v\)\\\}\\bigr\)
12:endfor

13:return

\{hv\(ℓ\)\}\\\{h\_\{v\}^\{\(\\ell\)\}\\\}

### 5\.3Multi\-TKG Pretraining

BecauseGrateapplies rotary encoding over ordered time IDs, jointly training on multiple source TKGs introduces a conflict: different datasets often reuse the same local indices \(e\.g\., starting from0\) for different timestamps or granularities\. We therefore remap the timestamps of each source onto disjoint integer ranges before joint training, so no two datasets share an ID\. SinceGratedepends only on relative gapsΔ​t\\Delta t, this remapping preserves per\-source temporal relationships\.

## 6Experimental Setup

### 6\.1Evaluation Regimes

We evaluateGrateunder three regimes: \(1\)*transductive fine\-tuning*on ICEWS14 and ICEWS05\-15; \(2\)*inductive transfer*on the GDELT and WIKI inductive benchmarks we construct; \(3\)*held\-out zero\-shot transfer*on ICEWS18 and YAGO under the standard single\-step extrapolation protocol\. Regime \(2\) tests the central claim ofGrate; the same jointly pretrained checkpoint is used in all three regimes\.

### 6\.2Datasets

Pretraining sources\.We jointly pretrain on ICEWS14 and ICEWS05\-15\(García\-Duránet al\.,[2018](https://arxiv.org/html/2607.10197#bib.bib33); Lautenschlageret al\.,[2015](https://arxiv.org/html/2607.10197#bib.bib32)\), two political\-event TKGs from ICEWS with daily timestamps\. Both use interpolation splits and are the only data the trainedGratecheckpoint sees\.

Inductive transfer benchmarks\.We constructGDELTIndTfrom GDELT\(Trivediet al\.,[2017](https://arxiv.org/html/2607.10197#bib.bib26)\)\(one year of high\-density daily events, April 2015–March 2016\) andWIKIIndTfrom WIKI\(Jinet al\.,[2020](https://arxiv.org/html/2607.10197#bib.bib23)\)\(232 years of yearly encyclopaedic facts\)\. Construction details follow in[Section6\.3](https://arxiv.org/html/2607.10197#S6.SS3)\.

Held\-out forecasting\.ICEWS18\(daily political events, January–October 2018\) andYAGO\(yearly encyclopaedic facts under a chronological split\), both fromJinet al\.\([2020](https://arxiv.org/html/2607.10197#bib.bib23)\), are too small in relation and entity vocabulary to admit our V/R/T\-disjoint construction\. We therefore use them only for zero\-shot evaluation under the standard single\-step extrapolation protocol\. Per\-dataset statistics for all six TKGs are in[Table5](https://arxiv.org/html/2607.10197#A3.T5)\([AppendixC](https://arxiv.org/html/2607.10197#A3)\)\.

### 6\.3Inductive transfer benchmark

Split protocol\.The model is trained on the source TKGG𝑡𝑟𝑎𝑖𝑛G\_\{\\mathit\{train\}\}and hyper\-parameters are selected on a held\-out validation setQ𝑣𝑎𝑙𝑖𝑑Q\_\{\\mathit\{valid\}\}, whereQ𝑣𝑎𝑙𝑖𝑑∩Q𝑡𝑟𝑎𝑖𝑛=∅Q\_\{\\mathit\{valid\}\}\\cap Q\_\{\\mathit\{train\}\}=\\emptysetbutVvalid⊆V𝑡𝑟𝑎𝑖𝑛V\_\{\\mathrm\{valid\}\}\\subseteq V\_\{\\mathit\{train\}\}andRvalid⊆R𝑡𝑟𝑎𝑖𝑛R\_\{\\mathrm\{valid\}\}\\subseteq R\_\{\\mathit\{train\}\}\.222Two protocols exist in inductive KG evaluation: holding the validation set out from the inference graph\(Leeet al\.,[2023b](https://arxiv.org/html/2607.10197#bib.bib48)\), or from the training graph\(Zhouet al\.,[2023](https://arxiv.org/html/2607.10197#bib.bib6)\)\. We adopt the latter to keep the inference graph fully unseen during training and hyper\-parameter selection, eliminating leakage from the target\.At inference, the model is applied to the targetG𝑖𝑛𝑓=\(V𝑖𝑛𝑓,R𝑖𝑛𝑓,T𝑖𝑛𝑓,Q𝑖𝑛𝑓\)G\_\{\\mathit\{inf\}\}=\(V\_\{\\mathit\{inf\}\},R\_\{\\mathit\{inf\}\},T\_\{\\mathit\{inf\}\},Q\_\{\\mathit\{inf\}\}\)without fine\-tuning: an observed subsetQo⊂Q𝑖𝑛𝑓Q\_\{\\textsc\{o\}\}\\subset Q\_\{\\mathit\{inf\}\}serves as the message\-passing graph, and predictions are scored on a disjoint held\-outQ𝑡𝑒𝑠𝑡⊂Q𝑖𝑛𝑓Q\_\{\\mathit\{test\}\}\\subset Q\_\{\\mathit\{inf\}\}, withQo∩Q𝑡𝑒𝑠𝑡=∅Q\_\{\\textsc\{o\}\}\\cap Q\_\{\\mathit\{test\}\}=\\emptyset, which the model never sees during any forward pass\.

Benchmark construction\.We extend INGRAM Algorithm 2\(Leeet al\.,[2023b](https://arxiv.org/html/2607.10197#bib.bib48)\)to quadruples\. The procedure varies two orthogonal axes:ptri∈\{0\.25,0\.50,0\.75,1\.00\}p\_\{\\mathrm\{tri\}\}\\in\\\{0\.25,0\.50,0\.75,1\.00\\\}\(the fraction of inference\-graph edges using unseenRR*and*unseenTT\)333V/R/T\-disjoint transfer is a deliberate stress test, it is most representative of rapidly\-evolving domains where new entities, relations continually appear over time\.and a temporal mode∈\{𝐢𝐧𝐭𝐞𝐫,𝐞𝐱𝐭𝐫𝐚\}\\in\\\{\\mathbf\{inter\},\\mathbf\{extra\}\\\}\(random shuffle vs\. chronological message/test split\)\. Full algorithmic details are in[AppendixC](https://arxiv.org/html/2607.10197#A3); per\-variant sizes are in[Tables7](https://arxiv.org/html/2607.10197#A3.T7)and[8](https://arxiv.org/html/2607.10197#A3.T8)\.

### 6\.4Evaluation Protocol and Baselines

Metrics\.We report Mean Reciprocal Rank \(MRR\) and Hits@1/101/10averaged over both tail prediction\(s,r,?,τ\)\(s,r,?,\\tau\)and head prediction\(?,r,o,τ\)\(?,r,o,\\tau\), under the time\-aware filter ofGoelet al\.\([2020](https://arxiv.org/html/2607.10197#bib.bib28)\)which masks only the candidate entities matching known facts at the specific query timestampτ\\tau\.

Baselines per regime\.\(i\)*Transductive*: the main table reportsUltra\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)andTrix\(Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\)retrained on the same temporal data without theGratemodule, isolating the contribution of the temporal mechanism\. A full comparison with transductive interpolation baselines is in[AppendixF](https://arxiv.org/html/2607.10197#A6)\. \(ii\)*Inductive transfer*: no existing structural TKG model is directly applicable, as the V/R/T\-disjoint construction rules out any model that learns per\-entity, per\-relation, or per\-timestamp parameters\. We therefore use the count\-based recurrency predictor ofGastingeret al\.\([2024](https://arxiv.org/html/2607.10197#bib.bib3)\)as a learning\-free reference point; despite requiring no learning, it reaches performance comparable to state\-of\-the\-art trained TKG forecasting methods on standard benchmarks\(Gastingeret al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib3)\)\. \(iii\)*Held\-out forecasting*: LLM\-based forecasting methods ICL\(Leeet al\.,[2023a](https://arxiv.org/html/2607.10197#bib.bib52)\)and GenTKG\(Liaoet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib56)\), matchingGrate’s zero\-shot evaluation protocol\. A full comparison with transductive forecasting baselines is in[AppendixE](https://arxiv.org/html/2607.10197#A5)\.

Implementation\.Base model hyperparameters, optimiser, batch size, and hardware are in[AppendixB](https://arxiv.org/html/2607.10197#A2); zero\-shot evaluation is deterministic, so we report single\-run numbers throughout\.444Code and benchmarks:[https://anonymous\.4open\.science/r/GRATE\-F431/](https://anonymous.4open.science/r/GRATE-F431/)\.

## 7Results

We report results across the three evaluation regimes introduced in[Section6\.1](https://arxiv.org/html/2607.10197#S6.SS1): transductive in\-distribution performance on the pretraining sources \([Section7\.1](https://arxiv.org/html/2607.10197#S7.SS1)\), inductive transfer on the GDELT and WIKI benchmarks we construct \([Section7\.2](https://arxiv.org/html/2607.10197#S7.SS2)\), and zero\-shot transfer to the held\-out forecasting benchmarks ICEWS18 and YAGO \([Section7\.3](https://arxiv.org/html/2607.10197#S7.SS3)\)\.[Section7\.4](https://arxiv.org/html/2607.10197#S7.SS4)then isolates the contribution of each design choice inGrate\.

### 7\.1Transductive in\-distribution performance

Table 1:Transductive link prediction on ICEWS14 and ICEWS05\-15\. Base models retrained withoutGrateon the same pretraining mix;%Δ\\%\\,\\Deltarows show relative improvement\. Full comparison with transductive baselines in[AppendixF](https://arxiv.org/html/2607.10197#A6)\. Best per columnbold\.[Table1](https://arxiv.org/html/2607.10197#S7.T1)reports transductive link prediction on ICEWS14 and ICEWS05\-15 under the time\-aware filter\. AddingGrateconsistently improves bothUltraandTrixacross all metrics, with relative gains of\+25\+25–\+38%\+38\\%MRR and\+38\+38–\+58%\+58\\%Hits@11over the corresponding base models without temporal modelling\. Gains are largest on Hits@11, indicating that relative\-time alignment primarily improves top\-rank precision, and consistent across both base models, suggesting they arise from the temporal message function itself\.

### 7\.2Inductive transfer

[Figure2](https://arxiv.org/html/2607.10197#S7.F2)reports zero\-shot transfer of a single jointly pretrainedGratecheckpoint to all1616variants of theGDELTIndTandWIKIIndTinductive transfer benchmarks; full numerical results and supporting figures are in[AppendixA](https://arxiv.org/html/2607.10197#A1)\. Across all1616variants,Trix\+Grateconsistently outperforms the recurrency\-only baseline, demonstrating that the relational gate captures predictive signal beyond pure recurrence matching, even on high\-recurrence sources where a recurrence\-only strategy is known to be competitive\(Gastingeret al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib3)\)\.

![Refer to caption](https://arxiv.org/html/2607.10197v1/x2.png)Figure 2:Zero\-shot inductive transfer MRR onGDELTIndTandWIKIIndTas a function ofptrip\_\{\\mathrm\{tri\}\}\. Hits@11/Hits@1010variants are in[Figure3](https://arxiv.org/html/2607.10197#A1.F3)and the full table is in[AppendixA](https://arxiv.org/html/2607.10197#A1)\.On GDELT, the staticTrixbase model performs at parity with the recurrency baseline \(0\.220\.22–0\.240\.24MRR across variants\), yetTrix\+Grateimproves over recurrency by\+0\.16\+0\.16to\+0\.21\+0\.21MRR on interpolation and\+0\.12\+0\.12to\+0\.18\+0\.18MRR on extrapolation\. GDELT is a dense event graph \(460460–990990edges per entity in the benchmark variants\), in which any given entity participates in many co\-occurring events with different actors and relations\. This rich neighborhood givesGrate’s gate abundant evidence to discriminate among; the model can selectively up\-weight temporally relevant messages from a large candidate pool, which is the regime the mechanism is designed for\.Ultra\+Gratedelivers consistent but smaller gains \(\+0\.025\+0\.025to\+0\.040\+0\.040MRR on interpolation\);Trix’s iterative refinement appears better suited to preserving this discriminative neighborhood structure for the gate thanUltra’s single\-pass count summary\.

On WIKI, facts are largely isolated temporal assertions, a single\(s,r,o\)\(s,r,o\)triple recurring across yearly timestamps, yielding a sparse graph of only77–2121edges per entity\. With few neighboring facts to discriminate among,Grate’s gate has limited evidence to work with, and gains over the recurrency baseline are correspondingly smaller\. Where the gate does add value is when DRec is high: DRec, defined byGastingeret al\.\([2024](https://arxiv.org/html/2607.10197#bib.bib3)\)as the fraction of test triples whose literal\(s,r,o\)\(s,r,o\)match in the inference history sits at the immediately preceding timestamp, rises from0\.450\.45atp25p\_\{25\}to0\.950\.95atp100p\_\{100\}across theWIKIIndTinterpolation variants\. Atp100p\_\{100\}, nearly every recurrent triple has its match exactly one year prior;Grateanchors on these, andTrix\+Grateleads all methods \(\+0\.029\+0\.029MRR over staticTrix,\+0\.338\+0\.338MRR over recurrency\)\. At low DRec \(p25p\_\{25\}–p50p\_\{50\}\), the staticTrixbase already captures the relational signal effectively through flat aggregation;Grate’s decay down\-weights older matches the static model relied on, andTrix\+Gratetrails staticTrixby up to0\.1160\.116MRR, while still outperforming the recurrency baseline throughout\.[Figure4](https://arxiv.org/html/2607.10197#A1.F4)in[SectionA\.3](https://arxiv.org/html/2607.10197#A1.SS3)visualises this DRec–gain relationship across all1818evaluated distributions\.

### 7\.3Zero\-shot transfer

[Table2](https://arxiv.org/html/2607.10197#S7.T2)reports zero\-shot evaluation on ICEWS18 and YAGO under the single\-step extrapolation protocol, using the same pretrained checkpoint without fine\-tuning\. We compare against the vocabulary\-agnostic LLM\-based forecasters ICL\(Leeet al\.,[2023a](https://arxiv.org/html/2607.10197#bib.bib52)\)and GenTKG\(Liaoet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib56)\); MRR is omitted as they decode candidates rather than rank all entities\. A full comparison with transductive baselines is in[AppendixE](https://arxiv.org/html/2607.10197#A5)\.

Table 2:Zero\-shot transfer to ICEWS18 and YAGO compared with LLM\-based forecasting methods\. A full comparison with transductive baselines is in[AppendixE](https://arxiv.org/html/2607.10197#A5)\.Despite being a purely structure\-driven model with no access to textual descriptions or language\-model priors,Grateoutperforms both LLM\-based methods on Hits@1010on ICEWS18 and achieves higher Hits@11and MRR on YAGO\. GenTKG attains higher Hits@11on ICEWS18 \(0\.2430\.243vs\.0\.1790\.179\), partly because it pre\-filters candidates using entity–relation co\-occurrence statistics that narrow the prediction space;Grateapplies no such filtering and ranks over all candidate entities\.

The gain pattern from[Section7\.2](https://arxiv.org/html/2607.10197#S7.SS2)extends to the held\-out targets\. YAGO sits in the same regime asWIKIIndT\-100\-inter \(DRec=0\.93\\mathrm\{DRec\}=0\.93\): nearly every recurrent test triple has its literal match exactly one year prior, andTrix\+Grategains\+7\.3%\+7\.3\\%MRR\. ICEWS18 sits in the opposite regime \(DRec=0\.108\\mathrm\{DRec\}=0\.108, median recurrence gap1313days\): literal recurrence is diffuse across timescales rather than concentrated at the preceding timestamp, giving the gate little signal to exploit, andTrix\+Grateis accordingly near\-flat \(−0\.4%\-0\.4\\%MRR\)\.

The asymmetry betweenUltra\+Grate\(\+26\.7%\+26\.7\\%MRR\) andTrix\+Grate\(−0\.4%\-0\.4\\%\) on ICEWS18 reflects the difference in relation\-graph expressiveness between the two base models\.Trix’s iteratively refined relation graph with four\-way directional message passing captures more of the available structural signal on ICEWS18 thanUltra’s single\-pass count\-based relation graph construction, leaving less room forGrate’s temporal channel to contribute additional gains\.

### 7\.4Component ablation

[Table3](https://arxiv.org/html/2607.10197#S7.T3)isolates the contribution of eachGratecomponent by incrementally augmenting theTrixbase model\. All variants share the same architecture and training setup, differing only in the entity\-side message function\.

Table 3:Component ablation ofGrateonTrix\.GDELT\-100i=GDELTIndT\-100\-inter\(zero\-shot\)\.Replacing the static DistMult message with its rotated formR​\(Δ​t\)R\(\\Delta t\)yields gains of\+0\.103\+0\.103MRR on ICEWS14 and\+0\.146\+0\.146MRR onGDELTIndT\-100\-interwithout adding parameters, showing that most of the temporal signal can be captured by relative\-time alignment alone\. Rotation also outperforms an element\-wise sinusoidal alternativem⊙SinCos​\(Δ​t\)m\\odot\\mathrm\{SinCos\}\(\\Delta t\)by0\.0290\.029/0\.0570\.057MRR, since rotation preserves the message norm whereas sinusoidal modulation rescales each dimension independently\. Adding a relation\-conditioned gate on top of rotation yields further gains of\+0\.014\+0\.014MRR transductively and\+0\.033\+0\.033MRR under inductive transfer evaluation; the effect is larger in the zero\-shot setting, where structural and recurrent cues are weaker\. Switching to query\-conditioned gating yields modest in\-distribution gains \(\+0\.006\+0\.006MRR\) and larger inductive transfer gains \(\+0\.034\+0\.034MRR\), confirming the value of query\-aware edge selection\. Each step in[Table3](https://arxiv.org/html/2607.10197#S7.T3)adds a single design choice; removing any one degrades performance, and the gap widens under the strictest inductive transfer regime, indicating that the four design choices function together rather than additively\.

## 8Conclusion

We introducedGrate, a parameter\-free message function that lifts inductive KG foundation models to the temporal setting via relative\-time rotation and query\-conditioned gating\. Across three evaluation regimes,Grateimproves performance in most settings, with the largest gains where recurrence and dataset\-specific patterns are unavailable, and its impact scales with the richness of the base model’s relation graph\.

## Limitations

Grateencodes time purely through relative index differences and does not explicitly model temporal granularity or irregular intervals, which weakens its effect on coarse\-grained datasets\. In addition, whileGrateadds no parameters, the underlying NBFNet\-style message passing remains computationally expensive on large TKGs\. Combining relative\-time encoding with more scalable propagation strategies, and broader pretraining mixes including GDELT or interval\-style TKGs, is a natural direction for future work\. The current pretraining sources share daily granularity; extending to mixed\-granularity pretraining is left to future work\.

## Acknowledgements

We gratefully acknowledge computing time on the HoreKa supercomputer at the National High\-Performance Computing Center at KIT \(NHR@KIT\), jointly funded by the German Federal Ministry of Education and Research \(BMBF\), the Ministry of Science, Research, and the Arts of Baden\-Württemberg, and the German Research Foundation \(DFG\)\. Jiaxin Pan and Osama Mohammed acknowledge funding from the EU Chips Joint Undertaking \(GA 101140087, SMARTY\) and from the BMBF sub\-project 16MEE0444\. Daniel Hernández is funded by the German Research Foundation \(DFG\) – SFB 1574 – 471687386\.

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Appendix

## Appendix AFull Numerical Results

### A\.1Inductive Transfer Benchmark Tables

\(a\)GDELTIndT: interpolation mode

\(b\)GDELTIndT: extrapolation mode

\(c\)WIKIIndT: interpolation mode

\(d\)WIKIIndT: extrapolation mode

Table 4:Zero\-shot inductive transfer to theGDELTIndTandWIKIIndTbenchmark suites under the time\-aware filter\. Eachptrip\_\{\\mathrm\{tri\}\}column reports MRR / H@1 / H@10\. Rows grouped by base model, withΔ\\Deltarows showing theGrate−\-no\-Gratedifference per cell\. All learned methods use the same joint checkpoint pretrained on ICEWS14 \+ ICEWS05\-15; theUltraandTrixrows are retrained on the same temporal data without theGratemodule\. Recurrency baseline fromGastingeret al\.\([2024](https://arxiv.org/html/2607.10197#bib.bib3)\);UltrafromGalkinet al\.\([2024](https://arxiv.org/html/2607.10197#bib.bib36)\);TrixfromZhanget al\.\([2025](https://arxiv.org/html/2607.10197#bib.bib58)\)\. Best per columnbold\.[Table4](https://arxiv.org/html/2607.10197#A1.T4)reports the full numerical results underlying[Figure2](https://arxiv.org/html/2607.10197#S7.F2), including Hits@11and Hits@1010alongside MRR for every cell of theGDELTIndTandWIKIIndTbenchmark suites\. We highlight four cross\-cutting observations\.

GDELT interpolation \(Table[4](https://arxiv.org/html/2607.10197#A1.T4)a\)\.Trix\+Grateachieves the strongest results across allptrip\_\{\\mathrm\{tri\}\}levels, with gains over the staticTrixbase model growing monotonically from\+0\.155\+0\.155MRR atp25p\_\{25\}to\+0\.213\+0\.213MRR atp100p\_\{100\}\. This monotone increase confirms thatGrate’s gate becomes more valuable as the inference graph contains a larger fraction of relation\- and timestamp\-disjoint edges, where direct recurrence matching is less reliable\.Ultra\+Gratealso improves consistently \(\+0\.025\+0\.025to\+0\.040\+0\.040MRR\) but at a smaller scale, consistent with the relation\-graph difference discussed in[Section7\.2](https://arxiv.org/html/2607.10197#S7.SS2)\.

GDELT extrapolation \(Table[4](https://arxiv.org/html/2607.10197#A1.T4)b\)\.Gains persist under the stricter extrapolation split but are smaller than under interpolation, reflecting that temporal patterns transfer less directly to future timestamps\.Trix\+Gratemaintains consistent positive gains \(\+0\.063\+0\.063to\+0\.085\+0\.085MRR\) whileUltra\+Grateis near\-flat at highptrip\_\{\\mathrm\{tri\}\}, reinforcing the role ofTrix’s richer relation graph\.

WIKI interpolation \(Table[4](https://arxiv.org/html/2607.10197#A1.T4)c\)\.WIKI is highly recurrent \(DRec85\.6%85\.6\\%\) and static base models already perform strongly\. At lowptrip\_\{\\mathrm\{tri\}\},Grateprovides no benefit as the dominant signal is pure entity recurrence rather than temporal patterning\. Atptri=1\.0p\_\{\\mathrm\{tri\}\}\\\!=\\\!1\.0, where every test edge involves entirely unseen relations and timestamps,Trix\+Grateachieves the best MRR \(0\.9910\.991\) and Hits@11\(0\.9880\.988\), demonstrating that relative\-time alignment is most effective when prediction cannot fall back on memorised facts\.

WIKI extrapolation \(Table[4](https://arxiv.org/html/2607.10197#A1.T4)d\)\.Results are more mixed at intermediateptrip\_\{\\mathrm\{tri\}\}, butTrix\+Grateregains the lead atptri=1\.0p\_\{\\mathrm\{tri\}\}\\\!=\\\!1\.0\(\+0\.035\+0\.035MRR,\+0\.064\+0\.064Hits@11\)\.Ultra\+Grateshows modest positive gains at the two highestptrip\_\{\\mathrm\{tri\}\}levels \(\+0\.025\+0\.025and\+0\.012\+0\.012MRR\), suggesting that even a weaker relation graph benefits from temporal gating under strict disjointness\.

### A\.2Additional Hits@K Results

[Figure3](https://arxiv.org/html/2607.10197#A1.F3)plots Hits@11and Hits@1010in the same format as the MRR\-based[Figure2](https://arxiv.org/html/2607.10197#S7.F2)\.

![Refer to caption](https://arxiv.org/html/2607.10197v1/x3.png)\(a\)Hits@11\. The recurrency baseline sits at0across all panels because the count\-based predictor breaks ties uniformly and never returns the gold tail at exact rank11\.
![Refer to caption](https://arxiv.org/html/2607.10197v1/x4.png)\(b\)Hits@1010\. OnWIKIIndT\-interpolation the curves saturate near the ceiling, compressing the differences visible under MRR\.

Figure 3:Hits@11\(top\) and Hits@1010\(bottom\) counterparts of[Figure2](https://arxiv.org/html/2607.10197#S7.F2)\.
### A\.3DRec as a Predictor ofGrate’s Lift

[Figure4](https://arxiv.org/html/2607.10197#A1.F4)plots the per\-variantGrategain \(Δ\\DeltaMRR over the staticTrixbase model\) against DRec across all1616inductive transfer benchmarks and the two held\-out targets from[Section7\.3](https://arxiv.org/html/2607.10197#S7.SS3)\. Within each \(source×\\timesmode\) block, the gain is monotone in DRec: as a larger fraction of test triples have their literal\(s,r,o\)\(s,r,o\)match at the immediately preceding timestamp,Grate’s relative\-time gate has more concentrated evidence to anchor on, and the lift grows accordingly\. The ICEWS18 and YAGO held\-out points fall onto the same band defined by the sweep variants, confirming that DRec is a consistent predictor across datasets\. Rec \(any\-history occurrence\) does not share this property: onWIKIIndT\-interpolation, Rec stays at≈0\.99\\approx 0\.99across all fourptrip\_\{\\mathrm\{tri\}\}levels while the gain spans−0\.116\-0\.116to\+0\.029\+0\.029MRR\.

![Refer to caption](https://arxiv.org/html/2607.10197v1/x5.png)Figure 4:Grategain \(Δ\\DeltaMRR over staticTrix\) against DRec across all1616GDELTIndT/WIKIIndTvariants and the two held\-out targets \(ICEWS18, YAGO\)\. Each point is labelled by itsptrip\_\{\\mathrm\{tri\}\}level\.

## Appendix BImplementation and Base Model Details

Unless explicitly noted otherwise, we follow the originalUltra\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)andTrix\(Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\)configurations and training recipes\.

### B\.1Training Configuration

Frequency basis\.Grateuses the standard rotary frequency vectorωk=1/100002​k/d\\omega\_\{k\}=1/10000^\{2k/d\}withk=0,1,…,d/2−1k=0,1,\\dots,d/2\-1\. We keep the RoPE base at1000010000in all experiments and do not tune it\.

Relative time scale\.We pass the integer relative time gap

Δ​tu​v=τ−tu​v\\Delta t\_\{uv\}=\\tau\-t\_\{uv\}directly to the RoPE rotation, without normalising it by the dataset time range\. This choice follows the design goal of keepingGratedataset\-agnostic\. RoPE already contains frequencies spanning several orders of magnitude, so low\-frequency components can respond to large temporal gaps while high\-frequency components capture finer differences\. Moreover, normalising by a dataset\-specific time range would require access to target\-graph statistics at inference time\. In our inductive transfer setting, target timestamps are disjoint from training and may use different granularities, such as daily timestamps in ICEWS and yearly timestamps in WIKI\. We therefore avoid dataset\-level rescaling and keep the temporal transformation dependent only on relative time gaps\.

Base models\.UltraandTrixuseL=6L=6message\-passing layers, PNA aggregation, and hidden dimensiond=32d=32\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)\.Grateoperates entirely inside the per\-edge message function and requires no changes to the aggregation or update steps of the base model\.

### B\.2Base Model Integration

Grateintegrates into any NBFNet\-style base model as a replacement for the per\-edge message functionϕ\\phi\. We instantiate on two:

- •Ultra\+Gratereplaces ULTRA’s\(Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)DistMult entity message withmu​vGratem^\{\\textsc\{Grate\}\{\}\}\_\{uv\}from[Equation12](https://arxiv.org/html/2607.10197#S5.E12)\. ULTRA’s count\-aggregated relation graph is unchanged\.
- •Trix\+Grateperforms the same swap on the TRIX base model\(Zhanget al\.,[2025](https://arxiv.org/html/2607.10197#bib.bib58)\), which uses a role\-aware relation graph that distinguishes head\- and tail\-side incidences\.

Grateperforms no surgery on the relation graph; it is purely an entity\-side modification\. Whatever relation features the base model produces, whether single\-pass likeUltraor iterative likeTrix, are consumed by theGrate\-replaced message function without modification\.

### B\.3Hardware and Reproducibility

All joint pretraining runs on ICEWS14 \+ ICEWS05\-15 use2×2\\timesNVIDIA A100\-80G GPUs\. ForUltraandUltra\+Gratewe use batch size 8; forTrixandTrix\+Gratewe use batch size 2\. Detailed per\-epoch runtimes, memory usage, and wall\-clock time to peak MRR are reported in[Table10](https://arxiv.org/html/2607.10197#A4.T10)\. Code and the constructedGDELTIndTandWIKIIndTdatasets are available at[https://anonymous\.4open\.science/r/GRATE\-F431/README\.md](https://anonymous.4open.science/r/GRATE-F431/README.md)\.

## Appendix CDataset Construction and Statistics

### C\.1Source Dataset Statistics

[Table5](https://arxiv.org/html/2607.10197#A3.T5)lists statistics for the six TKGs used in this paper\. The inductive transfer splits we construct from GDELT and WIKI are reported separately in[Tables7](https://arxiv.org/html/2607.10197#A3.T7)and[8](https://arxiv.org/html/2607.10197#A3.T8)\.

Table 5:Source dataset statistics for the six TKGs used in this paper\.
### C\.2Inductive Transfer Construction Procedure

Starting from a source TKG\(V,R,T,Q\)\(V,R,T,Q\), we extend INGRAM Algorithm 2\(Leeet al\.,[2023b](https://arxiv.org/html/2607.10197#bib.bib48)\)to quadruples and add a chronological time split:

1. 1\.Restrict to the giant connected component on the entity\-level graph\.
2. 2\.PartitionR=Rtr∪RinfR=R\_\{\\mathrm\{tr\}\}\\cup R\_\{\\mathrm\{inf\}\}disjointly with\|Rinf\|/\|R\|=pR\|R\_\{\\mathrm\{inf\}\}\|/\|R\|=p\_\{R\}\.
3. 3\.Partition timestamps*chronologically*:TtrT\_\{\\mathrm\{tr\}\}contains the earliest\(1−pT\)⋅\|T\|\(1\-p\_\{T\}\)\\cdot\|T\|timestamps,TinfT\_\{\\mathrm\{inf\}\}the remainingpT⋅\|T\|p\_\{T\}\\cdot\|T\|\.
4. 4\.Samplentrn\_\{\\mathrm\{tr\}\}seed entities and BFS\-expand with a per\-hop neighbour cap to obtain a candidateVtrV\_\{\\mathrm\{tr\}\}\. The training edge set isEtr=\{\(s,r,o,τ\):s,o∈Vtr,r∈Rtr,τ∈Ttr\}E\_\{\\mathrm\{tr\}\}=\\\{\(s,r,o,\\tau\):s,o\\in V\_\{\\mathrm\{tr\}\},r\\in R\_\{\\mathrm\{tr\}\},\\tau\\in T\_\{\\mathrm\{tr\}\}\\\}; we take its giant component and defineVtrV\_\{\\mathrm\{tr\}\}as the entities appearing in it\.
5. 5\.RemoveVtrV\_\{\\mathrm\{tr\}\}from the source graph; sampleninfn\_\{\\mathrm\{inf\}\}seeds from the giant component of the remainder; expand to a candidateVinfV\_\{\\mathrm\{inf\}\}\.
6. 6\.FormEinfpool=\{\(s,r,o,τ\):s,o∈Vinf\}E\_\{\\mathrm\{inf\}\}^\{\\mathrm\{pool\}\}=\\\{\(s,r,o,\\tau\):s,o\\in V\_\{\\mathrm\{inf\}\}\\\}and partition it intoX=\{r∈Rtr,τ∈Ttr\}X=\\\{r\\in R\_\{\\mathrm\{tr\}\},\\tau\\in T\_\{\\mathrm\{tr\}\}\\\}\(seen relation*and*seen time\) andY=\{r∈Rinf,τ∈Tinf\}Y=\\\{r\\in R\_\{\\mathrm\{inf\}\},\\tau\\in T\_\{\\mathrm\{inf\}\}\\\}\(strictly disjoint\)\. Subsample to obtain\|X\|:\|Y\|=\(1−ptri\):ptri\|X\|:\|Y\|=\(1\-p\_\{\\mathrm\{tri\}\}\):p\_\{\\mathrm\{tri\}\}\. Atptri=1p\_\{\\mathrm\{tri\}\}=1the inference graph is fully V/R/T\-disjoint; smaller values mix in seen edges as additional supporting context\.
7. 7\.SplitEtrE\_\{\\mathrm\{tr\}\}intotrain\.txt/valid\.txt\(90/10\) and splitEinfE\_\{\\mathrm\{inf\}\}intomsg\.txt/test\.txt\(80/20\)\. Theintermode uses a random shuffle \(msg and test span the same time window\); theextramode uses a chronological cut withTmsg≤TtestT\_\{\\textsf\{msg\}\}\\leq T\_\{\\textsf\{test\}\}\.

### C\.3Per\-source Sampling Parameters

[Table6](https://arxiv.org/html/2607.10197#A3.T6)lists the per\-source sampling parameters\. Both sources usepT=0\.3p\_\{T\}=0\.3and split seed0;pR=0\.5p\_\{R\}=0\.5on GDELT andpR=0\.25p\_\{R\}=0\.25on WIKI \(lowered because WIKI’s2424relations would otherwise leave too few for bothRtrR\_\{\\mathrm\{tr\}\}andRinfR\_\{\\mathrm\{inf\}\}post\-GCC\)\.

Table 6:Per\-source sampling parameters used to generate theGDELTIndTandWIKIIndTbenchmark suites\.Temporal split boundaries\.On GDELT,TtrT\_\{\\mathrm\{tr\}\}covers 2015\-04\-01 to 2015\-12\-12 \(256256days\) andTinfT\_\{\\mathrm\{inf\}\}covers 2015\-12\-13 to 2016\-03\-31 \(110110days\)\. On WIKI,TtrT\_\{\\mathrm\{tr\}\}covers the first162162years andTinfT\_\{\\mathrm\{inf\}\}the remaining7070years\. The split is by date or year \(not by quadruple count\), so the inference timestamps are strictly later than every training timestamp\.

### C\.4GDELTIndTStatistics

[Table7](https://arxiv.org/html/2607.10197#A3.T7)reports per\-variant entity, edge, and split sizes for the eightGDELTIndTvariants\. All eight share the V/R/T\-disjoint partition; only the inference\-edge mix and the msg/test split differ\.

Table 7:GDELTIndTbenchmark suite \(pR=0\.5p\_\{R\}=0\.5,pT=0\.3p\_\{T\}=0\.3\)\. Edge counts in thousands\.
### C\.5WIKIIndTStatistics

[Table8](https://arxiv.org/html/2607.10197#A3.T8)reports per\-variant statistics for the eightWIKIIndTvariants\. The drop in\|Vinf\|\|V\_\{\\mathrm\{inf\}\}\|atptri=1p\_\{\\mathrm\{tri\}\}=1reflects that strict V/R/T disjointness on WIKI’s small2424\-relation vocabulary leaves a smaller candidate inference set after the chronological time split\.

Table 8:WIKIIndTbenchmark suite \(pR=0\.25p\_\{R\}=0\.25,pT=0\.3p\_\{T\}=0\.3\)\. Edge counts in thousands\.

## Appendix DParameter and Complexity Analysis

### D\.1Theoretical Analysis

Parameter count\.Grateadds no learnable parameters over the base model\. The rotationR​\(Δ​t\)R\(\\Delta t\)in[Equation9](https://arxiv.org/html/2607.10197#S5.E9)is parameterised entirely by the fixed RoPE frequency vectorωk=1/100002​k/d\\omega\_\{k\}=1/10000^\{2k/d\}; the gategu​vg\_\{uv\}in[Equation11](https://arxiv.org/html/2607.10197#S5.E11)is a parameter\-free composition of a dot product betweenQqQ\_\{q\}\(which reuses the layer\-ℓ−1\\ell\{\-\}1head state and the existing query relation embedding\) and the rotated message, followed by a1/d1/\\sqrt\{d\}scaling and a sigmoid\.[Table9](https://arxiv.org/html/2607.10197#A4.T9)reports parameter counts on the two transductive ICEWS sources at the dimensiond=32d=32used in the main paper\. TKGE methods learn per\-entity, per\-relation, and per\-timestamp embeddings, so their counts grow with\|V\|\|V\|,\|R\|\|R\|, and\|T\|\|T\|and vary substantially across datasets\.UltraandTrixare vocabulary\-agnostic foundation models whose counts at fixed\(L,d\)\(L,d\)are dataset\-independent, andGrateadds zero on top of either\.

Table 9:Trainable parameter counts on the two transductive ICEWS sources\. TKGE baselines\(Lacroixet al\.,[2020](https://arxiv.org/html/2607.10197#bib.bib17)\)are dataset\-specific\. Base model counts atL=6L=6,d=32d=32, PNA aggregation are dataset\-independent\.Grate\-augmented base models use roughly an order of magnitude fewer parameters than the TKGE baselines on ICEWS14 \(1\.71\.7–3\.4×1053\.4\\times 10^\{5\}vs\.∼\\sim2\.7×1062\.7\\times 10^\{6\}\) and ICEWS05\-15 \(1\.71\.7–3\.4×1053\.4\\times 10^\{5\}vs\.∼\\sim3\.9×1063\.9\\times 10^\{6\}\)\. A parameter\-free mechanism in time also has a structural advantage under inductive transfer evaluation: there is no parameter to mismatch whenT𝑖𝑛𝑓∩T𝑡𝑟𝑎𝑖𝑛=∅T\_\{\\mathit\{inf\}\}\\cap T\_\{\\mathit\{train\}\}=\\emptyset, whereas TKGE methods maintain a per\-timestamp vocabulary of size\|T𝑡𝑟𝑎𝑖𝑛\|\|T\_\{\\mathit\{train\}\}\|that is by definition unusable when the inference timestamps are disjoint\.

Time complexity\.A static NBFNet\-style base model performs𝒪​\(M​d\)\\mathcal\{O\}\(Md\)work per layer for the entity\-side message and aggregation\(Zhuet al\.,[2021](https://arxiv.org/html/2607.10197#bib.bib44); Galkinet al\.,[2024](https://arxiv.org/html/2607.10197#bib.bib36)\)\.Grateadds, per edge:

- •𝒪​\(1\)\\mathcal\{O\}\(1\)to computeΔ​tu​v=τ−tu​v\\Delta t\_\{uv\}=\\tau\-t\_\{uv\};
- •𝒪​\(d\)\\mathcal\{O\}\(d\)to apply the pair\-wise rotationR​\(Δ​tu​v\)​mR\(\\Delta t\_\{uv\}\)m\([Equation8](https://arxiv.org/html/2607.10197#S5.E8)\);
- •𝒪​\(d\)\\mathcal\{O\}\(d\)for the dot productQq⊤​mu​vrotQ\_\{q\}^\{\\top\}m^\{\\mathrm\{rot\}\}\_\{uv\}in the gate;
- •𝒪​\(1\)\\mathcal\{O\}\(1\)for the sigmoid and the gate multiply\.

Summing over edges,Gratecontributes𝒪​\(M​d\)\\mathcal\{O\}\(Md\)per layer \(the same order as the host base model\), so the total per\-query forward time is𝒪​\(L​M​d\)\\mathcal\{O\}\(LMd\), matching the static base model asymptotically\.

Space complexity\.Gratecarries one persistent float per edge fortet\_\{e\}, contributing𝒪​\(M\)\\mathcal\{O\}\(M\)persistent storage\. Per\-layer transient tensors are𝒪​\(M​d\)\\mathcal\{O\}\(Md\)for the cos/sin rotation factors and the rotated message, and𝒪​\(M\)\\mathcal\{O\}\(M\)for the per\-edge gate scalars \(the same order as the base model’s per\-layer activation memory\)\.

### D\.2Empirical Efficiency

[Table10](https://arxiv.org/html/2607.10197#A4.T10)reports empirical training efficiency on joint ICEWS14 \+ ICEWS05\-15 pretraining\.Gratedoes not increase GPU memory usage in our implementation:UltraandUltra\+Grateboth use 32 GB, whileTrixandTrix\+Grateboth use 70 GB\. The per\-epoch runtime increases because the rotation and gate introduce additional per\-edge tensor operations\. ForTrix, this overhead is largely offset by faster convergence to the best validation checkpoint, yielding a similar wall\-clock time to peak performance \(8 h 30 min vs\. 7 h 47 min\)\. ForUltra, the temporal extension reaches peak performance in 3 h 10 min vs\. 2 h 05 min for the static backbone, remaining within a single\-node training budget\.

Table 10:Efficiency comparison of static backbones vs\.Grate\-augmented variants on joint ICEWS14 \+ ICEWS05\-15 pretraining\.*Wall to peak*is total training time to the epoch with highest MRR on the held\-out validation set\.Overall,Gratepreserves the parameter efficiency and memory footprint of the base models, while adding moderate per\-edge computation\. This supports the main design goal: introducing temporal reasoning into inductive KG foundation models without adding timestamp\-specific parameters or sacrificing transferability\.

## Appendix ETransductive Forecasting Comparison on ICEWS18 and YAGO

[Table11](https://arxiv.org/html/2607.10197#A5.T11)placesGratein the context of transductive forecasting baselines on ICEWS18 and YAGO\. The two groups are not directly comparable: fine\-tuned methods are trained on each target dataset and have full access to its entity, relation, and timestamp vocabulary;Grateuses a single checkpoint pretrained on ICEWS14 \+ ICEWS05\-15 and is evaluated without any target\-specific fine\-tuning\. The % SOTA rows report each zero\-shot variant’s score as a percentage of the strongest fine\-tuned baseline in that column, quantifying how much of the supervised ceiling is recovered at zero cost\.

Despite the setting mismatch,Trix\+Graterecovers97\.8%97\.8\\%/97\.2%97\.2\\%/99\.2%99\.2\\%of the fine\-tuned SOTA in MRR / Hits@11/ Hits@1010on YAGO, and83\.7%83\.7\\%/79\.9%79\.9\\%/87\.6%87\.6\\%on ICEWS18\.Ultra\+Gratereaches90\.3%90\.3\\%MRR on YAGO and69\.9%69\.9\\%on ICEWS18\. The larger gap on ICEWS18 is consistent with that dataset being harder to generalise to zero\-shot, as discussed in[Section7\.3](https://arxiv.org/html/2607.10197#S7.SS3)\.

Table 11:Zero\-shot transfer to ICEWS18 and YAGO vs\. transductive forecasting baselines fromGastingeret al\.\([2023](https://arxiv.org/html/2607.10197#bib.bib54)\)under the single\-step extrapolation protocol\.*Fine\-tuned*methods are trained directly on each target;*zero\-shot*methods use a single pretrained checkpoint with no fine\-tuning\. % SOTA rows report the zero\-shot score as a percentage of the best fine\-tuned baseline per column\. Best per columnbold\.
## Appendix FFull Transductive Comparison

[Table12](https://arxiv.org/html/2607.10197#A6.T12)extends[Table1](https://arxiv.org/html/2607.10197#S7.T1)with dataset\-specific transductive TKG embedding baselines trained and evaluated on each dataset independently\. These methods learn per\-entity, per\-relation, and per\-timestamp embeddings and are not transferable across datasets; they serve as an in\-distribution SOTA reference\.

Table 12:Transductive link prediction on ICEWS14 and ICEWS05\-15, including dataset\-specific TKG embedding baselines\.UltraandTrixare retrained on the joint ICEWS14 \+ ICEWS05\-15 pretraining mix without theGratemodule\.%Δ\\%\\,\\Deltarows show the relative improvement ofGrateover the corresponding base model\. Best per columnbold\.Comparison with TA\-DistMult\.TA\-DistMult also uses a DistMult\-style message function but incorporates time as a static additive component in the scoring function, which cannot model the varying relevance of facts at different time offsets\.Grateimproves substantially over TA\-DistMult on both datasets, showing that encoding temporal displacement via relative\-time rotation provides a stronger temporal signal than additive time features alone\.

Comparison with TeRo\.TeRo also injects a rotation into the temporal scoring function, but applies a per\-timestamp rotation that must be learned during training, making it inapplicable to unseen timestamps\.Grateinstead rotates by the relative time gap using a fixed frequency basis \(RoPE\), requiring no timestamp\-specific parameters and generalising to arbitrary timestamps at inference\.Grateoutperforms TeRo on both datasets despite this additional constraint\.

Comparison with TComplEx and TNTComplEx\.TComplEx and TNTComplEx are the strongest transductive baselines, relying on per\-entity, per\-relation, and per\-timestamp embeddings that scale with vocabulary size \(on the order of10610^\{6\}parameters\)\.Grateadds zero learnable parameters over its base model \(Ultra\+Grate:1\.7×1051\.7\{\\times\}10^\{5\}parameters at dim 32;Trix\+Grate:3\.4×1053\.4\{\\times\}10^\{5\}\), yet matches or comes within a few points of their performance on both datasets, demonstrating that temporal signal can be captured through relative\-time alignment without dataset\-specific embedding tables\.

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