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FutureBridge: Token Selection Beyond Local Preference in Collaborative Decoding
Quanquan Li, Hongbo Zhang, Yihe Chi, Jingyu Li, Xidong Xi, Liuyang Song, Hongzhen Zhang, Yuxiang Huang, Jing Ke, Siyuan Ma, Junyi Lin, Guitao Cao
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Summary
The paper introduces FutureBridge, a method for token-level collaboration between Large Language Models (LLMs) and Small Language Models (SLMs). It addresses the 'teaching suitability gap' where LLM-preferred tokens may be difficult for SLMs to build upon. FutureBridge ranks joint LLM-SLM token candidates based on how well they support the SLM's subsequent reasoning, using an answer-verified LLM trajectory as a fixed shared future context for training. A lightweight token reranker is distilled to select tokens that maximize SLM compatibility, improving mathematical reasoning performance by 35.1% on Qwen3-1.7B without appending future suffixes at inference.
Entities (8)
Relation Signals (6)
FutureBridge → improves → Qwen3-1.7B
confidence 95% · FutureBridge improves the Qwen3-1.7B SLM's Math Avg. by 35.1% relative to greedy SLM decoding.
FutureBridge → uses → Shared Future
confidence 92% · FutureBridge uses a reasoning suffix extracted from an answer-verified LLM trajectory as a common reference for all candidates.
FutureBridge → employs → Token Reranker
confidence 90% · The resulting counterfactual scores supervise a lightweight token reranker that observes only the current state and candidate token.
FutureBridge → solves → Teaching Suitability Gap
confidence 90% · FutureBridge addresses the 'teaching suitability gap' in token-level collaboration driven by the LLM’s local preferences.
Token Reranker → ranks → LLM
confidence 85% · FutureBridge uses the LLM only to expand the candidate pool, selects one token, and returns generation to the SLM
Token Reranker → ranks → SLM
confidence 85% · FutureBridge ranks joint LLM–SLM token candidates according to how well they support the SLM's subsequent reasoning.
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Abstract
Abstract:Token-level collaboration allows a large language model (LLM) to assist a small language model (SLM) when their predictions diverge. Existing methods either use LLM-generated intervention tokens or rank candidates with the LLM's next-token probabilities. Both rely on the LLM's local preference, even though an LLM-selected token may be difficult for the SLM to build on. We present FutureBridge, which ranks joint LLM-SLM token candidates according to how well they support the SLM's subsequent reasoning. During training, an answer-verified LLM trajectory supplies a fixed shared future, and a frozen SLM evaluates every candidate under this common context. The resulting counterfactual scores supervise a lightweight token reranker that observes only the current state and candidate token. At inference, FutureBridge uses the LLM only to expand the candidate pool, selects one token, and returns generation to the SLM without generating or appending a future suffix. Across five mathematical reasoning benchmarks, FutureBridge improves the Qwen3-1.7B SLM's Math Avg. by 35.1% relative to greedy SLM decoding. These results indicate that token selection benefits from modeling whether the receiving SLM can use each candidate to continue reasoning, rather than relying on the LLM's local preference alone.
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- Source: https://arxiv.org/abs/2608.06819v1
- Canonical: https://arxiv.org/abs/2608.06819v1
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FutureBridge: Token Selection Beyond Local Preference in Collaborative Decoding Quanquan Li ∗1 , Hongbo Zhang ∗2 , Yihe Chi 1 , Jingyu Li 3 , Xidong Xi 1 , Liuyang Song 2 , Hongzhen Zhang 1 , Yuxiang Huang 1 , Jing Ke 5 , Siyuan Ma 4 , Junyi Lin 6 , Guitao Cao 1† 1 East China Normal University 2 Peking University 3 University of Science and Technology of China 4 Nanyang Technological University 5 Shanghai Jiao Tong University 6 Guangdong University of Technology Abstract Token-level collaboration allows a large language model (LLM) to assist a small language model (SLM) when their pre- dictions diverge. Existing methods either use LLM-generated intervention tokens or rank candidates with the LLM’s next- token probabilities. Both rely on the LLM’s local preference, even though an LLM-selected token may be difficult for the SLM to build on. We present FutureBridge, which ranks joint LLM–SLM token candidates according to how well they sup- port the SLM’s subsequent reasoning. During training, an answer-verified LLM trajectory supplies a fixed shared future, and a frozen SLM evaluates every candidate under this com- mon context. The resulting counterfactual scores supervise a lightweight token reranker that observes only the current state and candidate token. At inference, FutureBridge uses the LLM only to expand the candidate pool, selects one to- ken, and returns generation to the SLM without generating or appending a future suffix. Across five mathematical rea- soning benchmarks, FutureBridge improves the Qwen3-1.7B SLM’s Math Avg. by 35.1% relative to greedy SLM decod- ing. These results indicate that token selection benefits from modeling whether the receiving SLM can use each candidate to continue reasoning, rather than relying on the LLM’s local preference alone. 1 Introduction Token-level small–large model collaboration provides a fine- grained trade-off between reasoning capability and compu- tational cost by invoking the LLM on demand during gener- ation (Zheng et al. 2025; Fu et al. 2025; Huang et al. 2026). Rather than assigning an entire request to the LLM, the SLM generates independently for most steps and requests local as- sistance only at selected reasoning states. Mathematical rea- soning, however, is highly sensitive to local decisions: a token that is unsuitable for the SLM can redirect the subsequent trajectory away from a correct solution. For example, the LLM may prefer a compressed reasoning transition, whereas a less capable SLM may require an explicit intermediate step ∗ These authors contributed equally. † Corresponding author. to continue the derivation reliably. Effective token-level col- laboration therefore depends not only on when the LLM is invoked, but also on whether the provided local content can be effectively used by the SLM. Existing small–large model collaboration methods primar- ily determine whether, when, and for how long to invoke the LLM (Ong et al. 2025; Zheng et al. 2025; Fu et al. 2025; Liao et al. 2026; Lee et al. 2026; Zeng et al. 2026; Zhao et al. 2026; Huang et al. 2026; Akhauri et al. 2025; Dong et al. 2026). Once collaboration is activated, some methods directly adopt LLM-generated intervention tokens, while others rank candi- dates using the LLM’s local probabilities. S2T asks the LLM to rerank the SLM’s top-K candidates and distills the result- ing ranking into a local selector (Ye et al. 2026). This design relies on two assumptions: a useful action is already present in the SLM candidate set, and the action locally preferred by the LLM is also suitable for the SLM to continue reason- ing. The former produces a candidate coverage gap when a useful LLM candidate falls outside the SLM candidate set. The latter produces a teaching suitability gap when the LLM favors a compressed reasoning transition but the SLM re- quires an explicit intermediate step. Local LLM preference therefore does not provide sufficient evidence that the SLM can effectively use the selected token. These two gaps fundamentally arise from the absence of candidate-level supervision. An ideal selection signal would compare the complete reasoning outcomes produced by the same SLM from different candidates, but executing one roll- out per candidate incurs substantial computational cost. A standard teacher trajectory observes only the teacher action and its corresponding future, while generating a separate fu- ture for every candidate changes both the candidate action and the evaluation context, preventing a controlled compar- ison. The central challenge is therefore to construct SLM- conditioned supervision that is directly comparable across candidates under a fixed downstream context. To construct such supervision, FutureBridge uses a reason- ing suffix extracted from an answer-verified LLM trajectory as a common reference for all candidates. This choice is motivated by the intuition that local generation probabilities alone are insufficient to identify which candidate better sup- arXiv:2608.06819v1 [cs.CL] 7 Aug 2026 ports the SLM’s subsequent reasoning; evaluating different candidates against the same answer-consistent continuation provides downstream information for the current token deci- sion. During training, FutureBridge holds the current reason- ing state and the suffix fixed and changes only the token at the current position across branches. Because the SLM’s top-K set may omit a critical token required for subsequent reason- ing, we augment it with LLM candidates. For each token in the joint candidate set, the frozen SLM computes the aver- age log-likelihood of the shared suffix conditioned on that candidate and uses it as a proxy for compatibility between the candidate and subsequent reasoning. In this way, one answer-consistent continuation provides dense and directly comparable supervision for the entire candidate set. Building on this candidate-level supervision, Future- Bridge distills the future-compatibility scores into a lightweight token reranker. At inference, the SLM deter- mines whether to request assistance under a fixed request policy. Once collaboration is triggered, the LLM contributes only next-token candidates, and the reranker conditions on the current reasoning state and each candidate to rank the joint teacher–student pool. Only the selected token is ap- pended to the context, after which generation immediately returns to the SLM. Thus, answer-verified futures are privi- leged training information rather than an input or generated artifact at deployment. Under this protocol, experiments on five mathematical reasoning benchmarks show that Future- Bridge consistently improves the SLM’s end-to-end reason- ing performance under the same request policy and candidate budget. Matched candidate-level rollout analyses further test whether the future-compatibility score identifies candidates that support the SLM’s subsequent reasoning more accu- rately than local token probabilities, while controlled abla- tions isolate the contributions of candidate expansion and future-based supervision. Our contributions are summarized as follows: 1. We identify a teaching suitability gap in token-level col- laboration driven by the LLM’s local preferences: tokens preferred by the LLM may not support the SLM’s sub- sequent reasoning. We therefore formulate candidate se- lection as a local decision problem oriented toward the SLM’s subsequent reasoning. 2. We propose FutureBridge, which scores a joint SLM– LLM candidate pool against an answer-verified shared future under the frozen SLM. The resulting compatibility supervision is distilled into a lightweight token reranker that neither observes nor generates the future at inference. 3. Experiments on five mathematical reasoning benchmarks demonstrate consistent end-to-end gains and show that SLM-conditioned future compatibility provides a more effective token-selection signal than local preference. 2 Related Work 2.1 Token-Level Small–Large Model Collaboration Small–large model collaboration has evolved from query- level routing to fine-grained intervention during genera- tion (Ong et al. 2025; Zhang et al. 2026). CITER and R2R del- egate a small set of critical tokens to the large model (Zheng et al. 2025; Fu et al. 2025), while RelayLLM trains the small model to request large-model assistance at selected positions (Huang et al. 2026). Confidence-guided routing, GlimpRouter, and TrigReason further use local uncertainty or reasoning-risk signals to allocate model computation at token, reasoning-step, or event granularity (Lee et al. 2026; Zeng et al. 2026; Zhao et al. 2026). FusionRoute addition- ally combines token-level expert selection with complemen- tary logits to refine the selected model’s output distribu- tion (Xiong et al. 2026). These methods primarily deter- mine when additional computation should be introduced and which model should generate. Once collaboration is acti- vated, choosing among multiple concrete token candidates becomes a separate problem. 2.2 Local Token Candidate Selection At an admitted intervention, local candidates can be ver- ified or rescored using different signals. Speculative De- coding uses a lightweight draft model to propose tokens or short continuations that are verified by a target model, preserving the target distribution (Leviathan, Kalman, and Matias 2023). Reward-Guided Speculative Decoding further introduces process rewards to balance candidate quality and target-model computation (Liao et al. 2025). Contrastive De- coding instead adjusts the next-token distribution using the likelihood difference between expert and amateur models (Li et al. 2023). These methods primarily serve target-model ac- celeration or distribution shaping. S2T applies candidate selection to improve small-model reasoning: the large model ranks only the small model’s top- K candidates, and S2T-Local distills this ranking into a local selector (Ye et al. 2026). However, its candidate space is restricted to small-model proposals, and its supervision still reflects teacher preference. The methods above do not explic- itly evaluate candidate tokens by how well the small model can continue under a common downstream context. 2.3 Future-Guided Collaborative Decisions Several methods use downstream information to improve current decisions. SpecReason and SpecCoT generate or ver- ify future reasoning content to decide whether to accept the current reasoning unit, treating the future as an object of verification (Pan et al. 2025; Shi et al. 2025). R2R gener- ates separate large-model continuations from the SLM’s and LLM’s top-1 tokens and uses a verifier to determine whether the SLM token changes the reasoning path (Fu et al. 2025). AlphaRouter learns binary small–large model routing poli- cies through tree search and final rewards (Liao et al. 2026), while Local Branch Routing expands a local candidate tree at inference time and selects the current branch from post- candidate hidden states (Yin et al. 2026). In contrast, Future- Bridge uses one answer-verified LLM suffix only to construct offline supervision for a joint SLM–LLM candidate pool. Holding this training suffix fixed prevents the candidate and evaluation context from changing together, while distillation removes the suffix from the deployed selector. Figure 1: Overview of FutureBridge. (A) Prior token-level collaboration selects interventions using the LLM’s local preferences. (B) A motivating example illustrates that locally plausible tokens can differ in how well they support the SLM’s subsequent reasoning. (C) FutureBridge constructs a joint candidate pool for token reranking. (D) During training, every candidate is scored against the same answer-verified future, and the resulting supervision is distilled into a token reranker that does not observe the future at inference. 3 Problem Formulation We formulate token-level collaboration as a token reranking problem over a joint candidate pool constructed by an SLM M S and an LLM M T that share the same tokenizer. Given a problem x and an SLM-generated reasoning prefix y <t , the current decoding state is s t = (x,y <t ). We denote the two models’ next-token distributions by p S (·| s t ) and p T (·| s t ), respectively, and use ⊕ for token-sequence concatenation. When a fixed collaboration policy requests assistance, the models provide their respective top-k candidate sets, whose union is C t = C S t ∪ C T t . The selector chooses one token c∈ C t , appends only that token to the prefix, and then returns generation to the SLM. LetG S (s t ,c) denote the continuation produced by a fixed deterministic SLM decoding policy after receiving c, and let R(x, ˆy) ∈ 0, 1 indicate final-answer correctness. The ideal token decision is c ∗ t = arg max c∈C t R(x,y <t ⊕ c⊕ G S (s t ,c)).(1) This objective differs from selecting the token with the high- est local LLM probability: the best intervention is the can- didate that leads to the strongest subsequent SLM outcome. Because evaluating every candidate through a complete SLM rollout is prohibitively expensive at inference time, our goal is to use privileged future information during training to learn a lightweight token reranker that approximates this student- conditioned downstream value without observing a future suffix at deployment. 4 Method 4.1 Method Overview FutureBridge addresses token reranking over a joint candi- date pool at states admitted by a fixed request policy. At decoding step t, the policy determines whether to invoke collaboration from the current state and the SLM distribu- tion: g t = π req (s t ,p S (·| s t ))∈0, 1.(2) Equation (2) defines an external request policy shared by all candidate selectors. The policy and per-trajectory interven- tion budget are not optimized by FutureBridge. To construct training supervision independently of any selector, we first run the frozen SLM on the training problems and record the states admitted by the request policy. No candidate interven- tion or LLM suffix is inserted during this state-collection stage, so every candidate selector is trained from the same logged states. For each state with g t = 1, FutureBridge proceeds in three training stages. First, the SLM and LLM construct a joint can- didate pool, and a shared future is extracted from a factual LLM trajectory whose final answer is correct. Second, Fu- tureBridge fixes the state and shared future, replaces only the current token, and uses the frozen SLM to measure candidate compatibility. Third, the resulting candidate-level targets are distilled into a lightweight token reranker that receives only the current state and candidate token. At inference, the LLM provides next-token candidates, the reranker selects one to- ken from the joint pool, and generation returns immediately to the SLM. Figure 1 summarizes this separation between privileged training supervision and token-only deployment. 4.2 Joint Candidate Pool and Verified Shared Future For each admitted state, the SLM and LLM provide candi- dates from their next-token distributions: C S t = TopK(p S (·| s t ),K S ), C T t = TopK(p T (·| s t ),K T ), C t = C S t ∪ C T t , (3) where K S and K T are positive integers specifying the SLM and LLM candidate budgets. Because C S t and C T t are sets, Equation (3) removes duplicate tokens automatically. Re- stricting the pool to C S t assumes that a useful intervention token always remains in the SLM’s local shortlist. Con- versely, using only C T t discards locally plausible SLM ac- tions and makes candidate availability entirely LLM-defined. The union separates candidate availability from candidate selection: the LLM expands the action space, while the sub- sequent SLM-conditioned score determines which available token is most compatible with the receiving SLM. The con- struction does not presume that an LLM candidate is prefer- able to an SLM candidate; it places both sources in a common reranking space. LetH =16, 32, 64, 128 denote the future horizons con- sidered for model selection, and let H max = maxH = 128. All horizons are constructed from prefixes of the same maximum-length shared future. Once the joint pool is fixed, all candidates require directly comparable supervision. Evaluating Equation (1) would re- quire one complete SLM rollout per candidate. These free- running trajectories have different continuations, incur cost linear in the pool size, and provide only sparse final-answer feedback. FutureBridge instead uses greedy LLM decoding to generate one complete trajectory from the same state s t and checks its final answer with the task-specific verifierV: ̄y T ≥t = a T t ⊕ z T 1:ℓ t , V x,y <t ⊕ ̄y T ≥t = 1, f t = z T 1:H max , f (H) t = f t,1:H ,1≤ H ≤ H max ≤ ℓ t , (4) where a T t is the greedy LLM token at position t, z T 1:ℓ t is its complete post-action continuation, f t is the maximum- length shared future, and f (H) t is its length-H prefix. Greedy decoding implies a T t ∈ C T t for K T ≥ 1. Equation (4) re- tains only trajectories whose complete factual continuation is answer-correct and contains at least H max post-action to- kens. Verification applies to a T t ⊕z T 1:ℓ t as a complete factual continuation; it does not assert that f t remains correct after an arbitrary candidate replacement. For each candidate c k ∈ C t , we remove the original LLM token a T t , insert c k at the same position, and keep the shared future f t fixed: ̃y k,≥t = c k ⊕ f t .(5) The branch with c k = a T t recovers the corresponding factual prefix of the LLM trajectory; every other branch is a controlled candidate replacement. All branches share the same problemx, SLM reasoning prefixy <t , shared futuref t , and scoring model. Thus, Equation (5) changes only the token inserted at position t and provides a common downstream context for candidate-level compatibility. The shared future f t is privileged information used only to construct offline training targets. It is neither generated nor provided to the reranker at inference. Consequently, deploy- ment does not require a suffix-quality assumption and cannot inject an unverified LLM continuation into the SLM context. 4.3 Student-Conditioned Future Compatibility The candidate group above differs only in the token in- serted at the current position. We therefore measure candidate compatibility by the conditional likelihood that the receiv- ing SLM assigns to the shared future. For each candidate c k ∈ C t , the frozen SLM evaluates the shared future under teacher forcing, and one batched forward pass produces ℓ k,h = logp S (f t,h | s t ,c k ,f t,<h ), L = [ℓ k,h ]∈R |C t |×H max , (6) where k ∈ 1,...,|C t | indexes candidates and h ∈ 1,...,H max indexes positions in the shared future. The quantity ℓ k,h is the conditional log-probability that the SLM assigns to f t,h after inserting c k at state s t and observing the preceding shared-future prefix f t,<h . Equation (6) collects these token-level compatibilities for all candidates and future positions. For a given future horizon H, we define the student- conditioned compatibility score of candidate c k as B H (c k | s t ,f (H) t ) = 1 H H X h=1 ℓ k,h .(7) The score in Equation (7) is the average conditional log- likelihood that the receiving SLM assigns to the same length- H shared future after candidate c k is inserted. Length nor- malization prevents the score from accumulating mechani- cally with suffix length and makes different horizons com- parable. We write B H (c k ) when s t and f (H) t are clear from context. The reasoning prefix, candidate pool, shared future, and scoring model remain identical across candidate branches, so score differences are induced only by the current candidate. A candidate-specific LLM future would vary both the candidate and evaluation target while making LLM generation cost grow linearly with the pool size. The shared future instead evaluates every candidate under one downstream context and yields directly comparable candidate-level supervision. Importantly, B H measures compatibility under teacher forcing rather than final-answer correctness under free- running generation. It is a tractable surrogate for the complete SLM rollout value in Equation (1), not a causal outcome. We test its relationship with final reasoning outcomes using com- plete candidate rollouts from matched states. To convert compatibility scores into candidate-level su- pervision, we standardize them within each candidate group. For any score function r : C t →R, define μ r = 1 |C t | X c∈C t r(c), σ r = s 1 |C t | X c∈C t (r(c)− μ r ) 2 , Norm C t (r(c)) = r(c)− μ r σ r + ε , (8) where ε > 0 ensures numerical stability. Equation (8) aligns the scale of scores within each candidate group. The main FutureBridge target uses only SLM-conditioned future com- patibility: q H (c| s t ,f (H) t ,C t ) = exp(Norm C t (B H (c))/τ ) P c ′ ∈C t exp(Norm C t (B H (c ′ ))/τ ) , (9) where τ > 0 is the target temperature. Equation (9) en- codes both the relative ordering and compatibility differences within the candidate group, providing denser supervision than a hard top-1 label. The future horizon is a supervision and model-selection hyperparameter rather than a deployment-time generation length. We train each H ∈ H under the same states, can- didate pools, model architecture, and training budget, then select one horizon H ∗ and its reranker checkpoint on MATH validation. The selected checkpoint is fixed for every held-out benchmark; FutureBridge does not learn a state-dependent horizon policy or generate H ∗ future tokens at inference. To isolate the effect of LLM local preference, we retain the following hybrid target only as an ablation: U (α) H (c) = α Norm C t (B H (c)) + (1− α) Norm C t (logp T (c| s t )). (10) For this ablation, Equation (10) replaces the standardized compatibility term in Equation (9) before temperature nor- malization. When α = 1, it reduces to the main Future- Bridge supervision; when α = 0, it reduces to pure LLM local preference. Configurations with α < 1 are reported as supervision ablations and are not part of the core method. 4.4 Distilling Future Compatibility for Token Reranking Constructing the soft target in Equation (9) requires an answer-correct factual trajectory and frozen-SLM compat- ibility scoring, so it is restricted to offline supervision. We distill this candidate-group target into a lightweight token reranker. For each candidate c k ∈ C t , a LoRA candidate scorer based on the frozen SLM receives the augmented input (s t ,c k ) and produces the scalar logit r θ H (c k | s t ) = g θ H (s t ,c k )∈R,(11) where θ H denotes the trainable parameters learned from horizon-H targets and the base SLM remains frozen. Equa- tion (11) assigns each candidate an independent scalar score without exposing the shared future to the scorer. The pre- dicted candidate distribution is ˆq θ H (c| s t ,C t ) = exp(r θ H (c| s t )) P c ′ ∈C t exp(r θ H (c ′ | s t )) .(12) Equation (12) makes the dependence on C t explicit while preserving independent candidate scoring before normal- ization. For each H ∈ H, let D (H) FB denote the logged states, shared futures, and candidate groups used to con- struct horizon-H targets. All horizons use the same logged states, candidate pools, model architecture, and optimization budget. The reranker for horizon H minimizes L (H) FB (θ H ) =− 1 |D (H) FB | X (s t ,f (H) t ,C t )∈D (H) FB X c∈C t q H (c| s t ,f (H) t ,C t ) log ˆq θ H (c| s t ,C t ). (13) Equation (13) is a group-level cross-entropy objective that trains the reranker to predict the SLM-conditioned future- compatibility distribution from (s t ,c) alone. MATH valida- tion selects the supervision horizonH ∗ and its corresponding reranker checkpoint, which are then frozen for every held-out benchmark. At an admitted test event, the LLM provides only C T t from its next-token distribution. After forming the joint pool C t , the selected reranker evaluates every candidate from the current state and chooses ˆc t = arg max c∈C t ˆq θ H ∗ (c| s t ,C t ).(14) Equation (14) selects the current token without generating a shared future, generating candidate-specific futures, or exe- cuting candidate rollouts. The system appends only ˆc t and returns generation to the SLM. The deployed selector learns the relative value of alter- native local actions rather than the LLM future itself. Fu- tureBridge therefore relies on the LLM only for candidate proposal at admitted states. The matched LLM-preference ablation uses the same logged states, intervention budget, joint candidate pool, reranker architecture, and optimization budget; only its candidate-level supervision target changes. 5 Experiments 5.1 Experimental Setup Benchmarks and Metrics. We evaluate on GSM8K, MATH-500, OlympiadBench, AIME 2024, and AIME 2025. Dataset definitions and evaluation conventions follow their original sources (Cobbe et al. 2021; Hendrycks et al. 2021; He et al. 2024; Jia 2024; Math-AI 2025). The primary met- ric is single-trajectory pass@1 accuracy, and Math Avg. is the unweighted mean over the five benchmarks. The horizon ablation varies only the amount of future context used to con- struct offline supervision; it does not change the deployment- time generation length. All methods use the same zero-shot chain-of-thought prompt and answer verifier, and generation is truncated after 4,096 output tokens. Table 1: End-to-end accuracy (%) with Qwen3 SLM–LLM model pairs. Qwen3-1.7B is the primary SLM, and Qwen3-0.6B evaluates scale generalization. Param.BenchmarkSLM-onlyLLM-involved GreedyMaj@8S2T-LocalR2RTakeoverS2TFutureBridge 1.7B GSM8K82.083.586.795.395.595.896.8 MATH-50064.065.067.678.479.079.781.6 OlympiadBench27.127.928.941.542.042.443.5 AIME 2024 5.66.77.813.314.415.616.7 AIME 20257.88.910.010.011.112.213.3 Math Avg.37.3038.4040.2047.7048.4049.1450.38 0.6B GSM8K57.866.268.576.076.877.587.8 MATH-50033.541.547.252.553.054.274.8 OlympiadBench 13.218.020.527.028.028.531.2 AIME 20244.46.77.810.011.112.213.3 AIME 20253.35.66.77.88.910.011.1 Math Avg.22.4427.6030.1434.6635.5636.4843.64 Models and Baselines. We use Qwen3-1.7B as the pri- mary SLM, Qwen3-0.6B for scale generalization, and Qwen3-32B as the LLM (Yang et al. 2025); all base parame- ters remain frozen. The main comparison retains the methods most directly related to local token selection: greedy decod- ing, Maj@8 (Wang et al. 2023), S2T-Local (Ye et al. 2026), R2R (Fu et al. 2025), Takeover, and S2T (Ye et al. 2026). Compatible baselines use the same Qwen3 pairs. Under the S2T request schedule, Takeover delegates the remaining tra- jectory to the LLM at the first admitted state. Additional baselines are reported in the supplement. Collaboration and Implementation Protocol. Future- Bridge constructs supervision from MATH training and uses MATH validation to select the request threshold, reranker checkpoint, and H ∈ 16, 32, 64, 128; all choices are then fixed. Token-level collaborative methods share the request policy and intervention budget, whereas Takeover uses the same request condition but transfers the remaining genera- tion to the LLM after the first request. Matched selector abla- tions additionally share logged states, top-8 SLM–LLM joint pools, architecture, and optimization budget. At test time, Fu- tureBridge queries only next-token candidates, appends one selected token, and returns generation to the SLM. On held- out GSM8K, MATH-500, and OlympiadBench states, com- plete token-only candidate rollouts measure pairwise ranking accuracy and top-1 success only for diagnosis. The supple- ment provides filtering, LoRA, hardware, and cost details. 5.2 Experimental Results Table 1 compares FutureBridge with SLM-only and collabo- rative decoding methods using the same Qwen3 model pairs and benchmark protocol. Overall comparison. With Qwen3-1.7B, FutureBridge obtains 50.38 Math Avg., improving over S2T and Takeover by 1.24 and 1.98 points. Compared with the strongest SLM- only method, S2T-Local, it gains 10.18 points, correspond- ing to a 25.3% relative improvement. These results establish end-to-end effectiveness; the matched ablation and rollout analysis below test whether the gains are associated with receiver-conditioned future supervision. Table 2: Token-only inference efficiency for the primary Qwen3-1.7B SLM. LLM calls are averaged per problem, and latency is normalized to greedy SLM decoding. S2T and FutureBridge append one intervention token per call and generate no LLM suffix; Takeover is excluded because it del- egates the remaining trajectory to the LLM. MethodMath Avg.CallsLatency Greedy SLM37.300.0 1.00× S2T49.144.8 3.18× FutureBridge50.384.8 3.30× Consistency across tasks and SLM scales. FutureBridge achieves the highest accuracy on every benchmark at both SLM scales. With Qwen3-0.6B, it obtains 43.64 Math Avg., improving over S2T by 7.16 points and over S2T-Local by 13.50 points. The consistent gains for the weaker receiver indicate that future-compatible token selection remains ef- fective as SLM capacity decreases. Accuracy under a matched collaboration budget. Ta- ble 2 isolates token-selection cost for the matched token-level methods. S2T and FutureBridge average 4.8 LLM calls and generate no suffix. FutureBridge gains 1.24 Math Avg. points over S2T with 3.8% higher latency from joint-pool scoring. Takeover remains in Table 1 as an end-to-end delegation baseline but is not treated as a token-only, budget-matched method. Candidate-Level Rollout Analysis Figure 2(a) compares each score with complete token-only candidate rollouts from matched states. Direct-B H uses the privileged shared future, whereas FutureBridge predicts its ranking from the state and candidate alone. Relative to LLM local scoring, Direct-B H and FutureBridge improve pairwise accuracy by 8.5 and 7.1 points. FutureBridge also raises top-1 rollout success by 6.5, 6.9, and 6.7 points on GSM8K, MATH-500, and Olympiad- Bench, remaining only 1.3, 1.7, and 1.2 points below Direct- B H . Thus, the token-only selector preserves most of the future-aware signal across problem difficulty. 20406080100 Agreement with token-only rollouts (%) Pairwise GSM8K MATH-500 Olympiad 68.3 93.5 74.4 38.9 (a) Candidate-score agreement SLM local LLM local Direct-B H FutureBridge 1632 64 * 128 Training horizon H (tokens) 0 1 2 3 Δ accuracy (points) +1.6 +3.1 +2.9 +2.7 (b) Supervision horizon GSM8K MATH-500 Olympiad Math Avg. GSM8KMATH- 500 OlympiadMath Avg. Drop vs. verified supervision (points) Unverified Incorrect Cross-state Token-shuffled −1.2 −2.5 −1.7 −2.2 −2.4 −4.8 −4.1 −4.3 −3.1−6.6−6.3−6.1 −3.8−7.8 −7.5 −7.2 (c) Training-future quality Figure 2: Future-aware supervision improves agreement with token-only rollouts across GSM8K, MATH-500, and Olympiad- Bench, obtains most horizon gains by H = 64, and degrades when the shared future is corrupted. (a) Candidate scores are evaluated against complete token-only SLM rollouts. (b) Extending the training horizon to H = 128 provides only a further 0.1–0.2 accuracy points; changes are reported relative to H = 16, and 64 ∗ denotes the horizon selected on MATH validation. Deployment remains unchanged. (c) Each cell reports the accuracy drop relative to answer-verified future supervision. 5.3 Ablation Study Candidate Pool and Supervision Target We first sepa- rate the effect of candidate coverage from the effect of the supervision target. Table 3: Candidate-pool and supervision-target ablations for Qwen3-1.7B (%). VariantGSM8KMATH-500OlympiadMath Avg. FutureBridge96.881.643.550.38 SLM-only pool94.478.741.247.9 LLM-only pool94.979.441.848.3 Direct LLM score93.578.640.547.2 Distilled LLM pref.93.078.139.946.7 Table 3 separates candidate availability from selection. SLM-only and LLM-only pools reduce Math Avg. by 2.5 and 2.1 points. Direct LLM scoring uses the same states, budget, and joint pool but ranks with logp T (c | s t ); FutureBridge improves it by 3.3 points on GSM8K, 3.0 on MATH-500, 3.0 on OlympiadBench, and 3.2 in Math Avg. The distilled LLM-preference variant also matches the architecture and optimization budget, replacing only the target with Equa- tion (10) at α = 0; FutureBridge gains 3.8, 3.5, 3.6, and 3.7 points. Coverage and size-matched controls appear in the supplement. Shared-Future Horizon Figure 2(b) varies only the future used for offline targets. Extending H from 64 to 128 still improves GSM8K, MATH-500, OlympiadBench, and Math Avg., but only by 0.1, 0.2, 0.2, and 0.1 points, respectively. Most of the gain is already obtained between H = 16 and H = 64, where the four metrics improve by 1.6, 3.1, 2.9, and 2.7 points. MATH validation selects H = 64; deployment generates no future tokens. Shared-Future Quality Figure 2(c) changes only the of- fline training future. Unverified, incorrect, cross-state, and token-shuffled futures reduce GSM8K by 1.2, 2.4, 3.1, and 3.8 points and Math Avg. by 2.2, 4.3, 6.1, and 7.2 points. The same ordering holds on GSM8K, MATH-500, and OlympiadBench, indicating that effective supervision re- quires a state-specific, correctly ordered reasoning trajectory. 6 Conclusion In this work, we have formulated token-level small–large model collaboration as token reranking over a joint can- didate pool. Rather than treating the LLM’s local prefer- ence as the selection target, FutureBridge uses an answer- verified shared future to measure how well each candidate supports downstream reasoning under the receiving SLM. It fixes the reasoning state and future across candidates, de- rives candidate-level supervision from the frozen SLM, and distills this supervision into a deployable reranker that ob- serves only the state and candidate token. 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Zhao, Y.; Peng, Y.; Nguyen, C.-T.; Li, Z.; Wang, X.; Fu, X.; and Zhao, H. 2026. TrigReason: Trigger-Based Col- laboration between Small and Large Reasoning Models. In Findings of the Association for Computational Linguistics: ACL 2026. Zheng, W.; Chen, Y.; Zhang, W.; Kundu, S.; Li, Y.; Liu, Z.; Xing, E. P.; Wang, H.; and Yao, H. 2025. CITER: Collabora- tive Inference for Efficient Large Language Model Decoding with Token-Level Routing. In Conference on Language Mod- eling. Supplementary Material A Full Reproducibility Details Models and deployment boundary. The primary receiv- ing model is Qwen3-1.7B, the scale-generalization receiver is Qwen3-0.6B, and Qwen3-32B is the assisting LLM. All base-model parameters remain frozen. Shared futures and complete candidate rollouts are used only to construct train- ing targets or evaluation diagnostics. During deployment, the LLM returns only its top-8 next-token candidates at an ad- mitted state. FutureBridge merges them with the SLM top-8, scores the deduplicated union, appends exactly one selected token, and immediately returns decoding to the SLM. It nei- ther requests nor appends an LLM reasoning suffix. Request policy and collaboration budget. The external request policy computes the entropy of the SLM distribution on its normalized top-64 support. Its threshold is the 0.99 quantile estimated on the problem-disjoint MATH validation split. The threshold is frozen before evaluation, and each tra- jectory permits at most eight admitted intervention events. The policy therefore uses no LLM output to decide whether to request assistance. Once a state is admitted, token-level collaborative methods receive the same event and remaining intervention budget. Takeover instead delegates the remain- ing trajectory to the LLM at the first admitted state. S2T ranks the SLM top-8 with LLM next-token probabilities, and Fu- tureBridge ranks the deduplicated SLM–LLM union. Greedy decoding is used between intervention events for the token- level methods, and all methods share the same 4,096-token output limit, tokenizer, prompt, and answer verifier. Data construction and filtering. We split the 7,500 MATH training problems into 6,750 training problems and 750 validation problems before collecting states, preventing prefixes from the same problem from entering both subsets. At every state admitted by the frozen request policy, Qwen3- 32B greedily produces one complete trajectory. We discard a group when the final answer is incorrect or unparsable, the post-action trajectory contains fewer than 128 tokens, or tokenization and deduplication leave an invalid candidate group. Table 4 reports the resulting data volume. The over- all retained fractions are 58.2% for Qwen3-1.7B states and 58.3% for Qwen3-0.6B states. Table 4: Training-data construction and filtering by receiving SLM. Stage1.7B 0.6B Logged admitted states31,680 34,560 Answer-verified trajectory 21,216 23,424 At least 128 future tokens 18,912 20,736 Valid joint candidate group 18,432 20,160 Training groups16,576 18,144 Validation groups1,856 2,016 LoRA reranker. The candidate scorer is initialized from the corresponding receiving SLM. LoRA adapters are ap- plied to the query, key, value, and output projections of each self-attention block with rank 16, scaling 32, and dropout 0.05. A linear scalar head reads the hidden state of the appended candidate token. Only the LoRA parameters and scalar head are optimized; embeddings, transformer weights, and the language-model head remain frozen. We use bfloat16, AdamW with learning rate 2 × 10 −4 , weight decay 0.01, β 1 = 0.9, β 2 = 0.999, a 3% linear warmup, and gradient clipping at 1.0. One candidate group forms a microbatch, gradients are accumulated over 32 groups, and training lasts three epochs. The target temperature isτ = 0.5. We train one scorer for each H ∈ 16, 32, 64, 128 under identical bud- gets and select H = 64 on MATH validation. All reported FutureBridge results use three training seeds; the main-table Math Avg. reports the seed mean without per-cell standard deviations. Hardware and software. Data construction, compatibil- ity scoring, reranker training, and evaluation use NVIDIA H200 GPUs with 141 GB memory. Each timing run uses one GPU and batch size one so that latency is comparable across methods; independent seeds may run concurrently but are timed separately. Models run without quantization under CUDA 12.4, PyTorch 2.6, and Transformers 4.52. We reuse the SLM key–value cache up to the admitted state and batch the candidate branches for both offline compatibility scoring and deployed reranking. B Cost Accounting Offline supervision cost. Table 5 separates the one-time construction and training cost for the primary Qwen3-1.7B receiver. Generating answer-verified shared futures is the largest component. Compatibility scores for all four horizons are obtained from the same length-128 likelihood matrix, so shorter horizons do not require additional SLM forward passes. The twelve reranker runs correspond to four horizons and three random seeds. Table 5: One-time offline cost for the primary receiver. StageUnits H200 hours SLM state logging31,680 states5.8 Verified-future generation 21,216 trajectories29.4 SLM compatibility scoring18,432 groups8.6 Reranker optimization12 runs7.2 Total—51.0 Deployment cost. Table 6 reports token-only inference for S2T and FutureBridge under the matched request pol- icy. “Pool” is the mean number of tokens considered at an intervention event, “appended” counts the single interven- tion token, and latency is normalized to greedy SLM de- coding. An LLM call returns next-token logits only. Thus, both token-level methods append one intervention token per admitted event and generate zero LLM suffix tokens. Future- Bridge’s additional latency and memory arise from batching 13.2 candidate-scoring branches through the LoRA reranker, not from generating future reasoning. Takeover is excluded because its LLM generates the remaining trajectory. Table 6: Per-problem deployment cost for Qwen3-1.7B. MethodCalls Pool Appended Suffix Latency GB Greedy SLM 0.0 —0.00 1.00× 8.6 S2T4.8 8.04.80 3.18× 72.7 FutureBridge 4.8 13.24.80 3.30× 78.1 C Additional Baseline Results Tables 7 and 8 report the baselines omitted from the focused main comparison under the same Qwen3 protocol. Table 7: Additional baseline accuracy with Qwen3-1.7B (%). BenchmarkTSD-KDMul-TTaHSpecR GSM8K75.377.179.089.2 MATH-50056.157.859.672.9 OlympiadBench21.022.723.640.6 AIME 20243.34.44.411.1 AIME 20252.22.23.310.0 Math Avg.31.632.834.044.8 Table 8: Additional baseline accuracy with Qwen3-0.6B (%). BenchmarkTSD-KDMul-TTaHSpecR GSM8K55.757.658.967.9 MATH-50031.432.834.644.8 OlympiadBench10.511.812.121.6 AIME 20241.12.22.25.6 AIME 20251.11.11.14.4 Math Avg.20.021.121.828.9 D Detailed Mechanism Results Candidate-level rollout agreement. Table 9 gives the nu- merical results underlying the candidate-level analysis in the main paper. Each held-out candidate is appended alone, after which the frozen SLM completes the trajectory without an LLM suffix. Direct-B H is a privileged training-time score rather than an inference oracle. FutureBridge preserves most of its advantage over the LLM local score while observing only the current state and candidate. Table 9: Agreement with complete token-only candidate roll- outs (%). ScorePairwiseGSM8K MATH-500Olympiad accuracy top-1 success top-1 success top-1 success SLM local score56.984.162.427.8 LLM local score61.287.067.532.2 Direct-B H 69.794.876.140.1 FutureBridge68.393.574.438.9 Candidate-pool coverage. Table 10 measures the fraction of matched held-out states whose pool contains a token yield- ing a correct complete SLM rollout. The top-8 sets overlap by 2.8 tokens on average, producing 13.2 joint candidates and improving coverage over the stronger LLM-only pool by 9.3 points on MATH-500 and 8.8 points on OlympiadBench. On GSM8K, the corresponding gain is 8.8 points. Table 10: Candidate coverage at matched held-out interven- tion states (%). PoolAvg. size GSM8K MATH-500 Olympiad SLM top-88.072.849.645.2 LLM top-88.079.454.150.7 Joint pool13.288.263.459.5 A size-matched SLM top-16 pool obtains 94.6 on GSM8K, 79.0 on MATH-500, 41.5 on OlympiadBench, and 48.0 Math Avg., remaining 2.4 Math Avg. points below the joint pool. Future horizon and quality. Table 11 reports the hori- zon sweep, where the future length changes only the offline target. Table 12 changes the shared future used for train- ing while preserving the same logged states, joint candidate pools, model architecture, and deployment procedure. Both studies select and append only one token at inference. Table 11: Shared-future horizon used for offline supervision (%). H GSM8K MATH-500 Olympiad Math Avg. 1695.278.540.647.7 3296.079.642.048.7 6496.881.643.550.38 12896.981.843.750.5 Table 12: Quality of the future used to construct training targets (%). Training future GSM8K MATH-500 Olympiad Math Avg. Answer-verified96.881.643.550.38 Unverified95.679.141.848.2 Incorrect answer94.476.839.446.1 Cross-state93.775.037.244.3 Token-shuffled93.073.836.043.2