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Soft Spatial Reasoning
Rafi Ibn Sultan, Md. Sajid Alam Chowdhury, Saleh Zare Zade, Chengyin Li, Prashant Khanduri, Marco Brocanelli, Dongxiao Zhu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/4/2026, 2:53:07 AM
Summary
The paper introduces Soft Spatial Reasoning, a post-training framework for Large Vision-Language Models (LVLMs) that replaces discrete chain-of-thought (CoT) token selection with continuous soft states. This approach mitigates premature discretization errors by mixing token embeddings based on adaptive softness controlled by a module called AdaptSoft. AdaptSoft dynamically adjusts the mixture temperature using the model's hidden state and predictive uncertainty. The framework is trained using a gradient-alignment objective that aligns step-specific policy gradients with a reference gradient, enabling effective learning without intermediate reasoning supervision.
Entities (7)
Relation Signals (6)
Soft Spatial Reasoning → contains → AdaptSoft
confidence 97% · At the core of Soft Spatial Reasoning is AdaptSoft, a controller that uses the current hidden state and predictive uncertainty to adapt the degree of softness at each reasoning step.
Soft Spatial Reasoning → appliedto → Large Vision Language Models
confidence 95% · Soft Spatial Reasoning, a post-training framework that introduces soft thinking for spatial tasks in LVLMs.
AdaptSoft → controls → softness
confidence 95% · AdaptSoft adjusts the mixture’s softness at each step based on the current reasoning state and predictive uncertainty.
AdaptSoft → trainedwith → Gradient-Alignment
confidence 94% · To train AdaptSoft, we introduce a gradient-alignment learning objective that provides a step-specific learning signal for softness control
Soft Spatial Reasoning → mitigates → Premature Discretization
confidence 92% · Such hard thinking requires committing to a single token... This early commitment constitutes premature discretization... We propose Soft Spatial Reasoning... allowing multiple candidate continuations to influence the next step.
Soft Spatial Reasoning → replaces → Chain-of-Thought
confidence 90% · We propose Soft Spatial Reasoning... that introduces soft thinking for spatial tasks in LVLMs... forming continuous soft states by mixing token embeddings rather than selecting a single token
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Abstract
Abstract:Large Vision-Language Models (LVLMs) commonly perform spatial reasoning through chain-of-thought (CoT), encoding intermediate reasoning as autoregressive sequences of discrete language tokens. Such hard thinking requires committing to a single token at each step, even when the correct spatial interpretation remains uncertain. This early commitment constitutes premature discretization: an incorrect token selection can propagate errors through subsequent reasoning. We propose Soft Spatial Reasoning, a post-training framework that introduces soft thinking for spatial tasks in LVLMs. At each intermediate reasoning step, the LVLM forms a continuous soft state by mixing token embeddings rather than selecting a single token, allowing multiple candidate continuations to influence the next step. The appropriate degree of softness, however, can vary across reasoning steps: retaining multiple candidates may preserve a useful spatial interpretation, but if those candidates imply conflicting spatial relations, mixing them may interfere with subsequent reasoning. At the core of Soft Spatial Reasoning is AdaptSoft, a controller that uses the current hidden state and predictive uncertainty to adapt the degree of softness at each reasoning step. To train AdaptSoft, we introduce a gradient-alignment learning objective that provides a step-specific learning signal for softness control without intermediate reasoning supervision. Across diverse spatial benchmarks, Soft Spatial Reasoning outperforms hard and fixed-soft CoT baselines using the same backbone, as well as a range of existing LVLMs. The source code is available at this https URL
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- Source: https://arxiv.org/abs/2609.38717v1
- Canonical: https://arxiv.org/abs/2609.38717v1
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Soft Spatial Reasoning Rafi Ibn Sultan Affiliation: Department of Computer Science, Wayne State University Md. Sajid Alam Chowdhury Affiliation: Department of Computer Science, Wayne State University Saleh Zare Zade Affiliation: Department of Computer Science, Wayne State University Chengyin Li Affiliation: Department of Radiation Oncology, Henry Ford Health Prashant Khanduri Affiliation: Department of Computer Science, Wayne State University Marco Brocanelli Affiliation: Department of Electrical and Computer Engineering, The Ohio State University Dongxiao Zhu Affiliation: Institute for AI and Data Science, Wayne State University Abstract Large Vision-Language Models (LVLMs) commonly perform spatial reasoning through chain-of-thought (CoT), encoding intermediate reasoning as autoregressive sequences of discrete language tokens. Such hard thinking requires committing to a single token at each step, even when the correct spatial interpretation remains uncertain. This early commitment constitutes premature discretization: an incorrect token selection can propagate errors through subsequent reasoning. We propose Soft Spatial Reasoning, a post-training framework that introduces soft thinking for spatial tasks in LVLMs. At each intermediate reasoning step, the LVLM forms a continuous soft state by mixing token embeddings rather than selecting a single token, allowing multiple candidate continuations to influence the next step. The appropriate degree of softness, however, can vary across reasoning steps: retaining multiple candidates may preserve a useful spatial interpretation, but if those candidates imply conflicting spatial relations, mixing them may interfere with subsequent reasoning. At the core of Soft Spatial Reasoning is AdaptSoft, a controller that uses the current hidden state and predictive uncertainty to adapt the degree of softness at each reasoning step. To train AdaptSoft, we introduce a gradient-alignment learning objective that provides a step-specific learning signal for softness control without intermediate reasoning supervision. Across diverse spatial benchmarks, Soft Spatial Reasoning outperforms hard and fixed-soft CoT baselines using the same backbone, as well as a range of existing LVLMs. The source code is available at https://github.com/rafiibnsultan/Soft_Spatial_Reasoning. 1 Introduction Figure 1: Hard versus adaptive soft thinking for spatial reasoning. (a) Hard thinking commits to one token at each CoT step, allowing early errors to propagate. (b) Our Soft Spatial Reasoning forms continuous intermediate states by mixing token embeddings, carrying information from multiple candidate continuations through the CoT. AdaptSoft adjusts the mixture’s softness at each step based on the current reasoning state and predictive uncertainty. Large Vision-Language Models (LVLMs) Alayrac et al. (2022); Li et al. (2023); Liu et al. (2023); Dai et al. (2023); Bai et al. (2023) have demonstrated capabilities across a range of tasks involving visual understanding Yin et al. (2024); Li et al. (2024) and spatial reasoning Liu et al. (2026); Stogiannidis et al. (2025). To support complex reasoning, these models commonly use chain-of-thought (CoT) Wei et al. (2022), generating intermediate reasoning steps before producing a final answer. Standard CoT expresses these steps as an autoregressive sequence of discrete language tokens, a process we refer to as hard thinking. As an alternative to this discrete formulation, recent work on Large Language Models (LLMs) Hao et al. (2024); Zhang et al. (2026b); Zheng et al. (2025a) has explored soft thinking, in which continuous intermediate states carry information from multiple candidate continuations Zhu et al. (2026); Zhang et al. (2026a) while the final answer is expressed in language. LVLMs, however, typically use hard thinking for spatial reasoning tasks Li et al. (2025); Gholami et al. (2025); Wang and Ling (2025). During the CoT, several spatial interpretations may still appear plausible to the model Zhang et al. (2025). Consider the question in Figure 1a, which asks where the red ball is relative to the yellow chair. Before resolving which red object is the ball, the model may identify the object on the chair as the ball in its CoT, even though the object in midair remains a plausible candidate. Subsequent reasoning may then treat the object on the chair as the ball, producing the answer on rather than in front. This illustrates premature discretization: committing to a discrete continuation before resolving the spatial interpretation can allow an early error to propagate through the remaining CoT Hao et al. (2024); Hu et al. (2026). Human cognition offers a useful contrast: people can think through a spatial problem before reaching an answer, without putting every thought into words Quiroga et al. (2005); Fedorenko and Varley (2016). Reasoning without verbalizing every intermediate step is also emerging in LVLMs. Some approaches maintain and refine continuous visual tokens alongside the CoT Yang et al. (2026); Li et al. (2026a); Li et al. (2026b). In these approaches, softness lies in the visual representations, while the CoT still advances through discrete token selections and may commit prematurely to one interpretation. Other approaches extend soft thinking to the CoT itself Hu et al. (2026); Jeon et al. (2026); Wang et al. (2026b), forming soft states that carry information from multiple candidate continuations and may delay commitment to a single reasoning path. However, the benefit of preserving these alternatives depends on what they represent. In spatial reasoning, alternative interpretations of a scene can imply different spatial relations. During the CoT, a soft state may keep a useful interpretation available at one step; at another, mixing interpretations that imply conflicting relations may interfere with subsequent reasoning. A fixed degree of softness may not suit both situations (Figure 1b). Research Question How should LVLMs dynamically control representation softness during spatial reasoning? To address this question, we propose Soft Spatial Reasoning (Figure 1b), a post-training framework that introduces adaptive soft thinking for spatial reasoning in LVLMs using soft states at each intermediate CoT step. Central to the framework is AdaptSoft, a controller that adjusts the softness of the token-embedding mixtures forming these states. AdaptSoft uses the LVLM’s hidden state at each step to account for the ongoing reasoning and predictive uncertainty to gauge the model’s confidence in its next continuation. By varying softness, it adjusts the distribution of weights across candidate continuations within each soft state, with the aim of guiding subsequent reasoning toward a correct final answer. We further introduce a gradient-alignment objective that measures agreement between each step’s contribution to the LVLM policy gradient and a reference gradient. This provides step-level credit alongside task-level rewards for jointly post-training AdaptSoft and the LVLM, without requiring intermediate reasoning annotations. Our contributions are fourfold: • We introduce Soft Spatial Reasoning, to our knowledge the first framework dedicated to soft thinking for spatial reasoning in LVLMs. • We design AdaptSoft to control softness at each reasoning step using the current reasoning state and predictive uncertainty. • We develop a gradient-alignment objective that provides step-specific credit without external evaluators or intermediate reasoning annotations. • Extensive experiments covering 19 spatial reasoning categories demonstrate overall improvements over strong spatial reasoning baselines. 2 Related Works Hard Spatial Reasoning in LVLMs. Several approaches support spatial reasoning by providing LVLMs with explicit scene geometry and object relationships Cheng et al. (2024); Ma et al. (2025b); Cai et al. (2025b); Liu et al. (2025); Chen et al. (2024b); Hu et al. (2025); Wang et al. (2025c); Cai et al. (2025a); Chen et al. (2025a); Sultan et al. (2026); Daxberger et al. (2025); Hong et al. (2023); Wu et al. (2025a); Xu et al. (2026a); Zhao et al. (2025b). Others organize spatial evidence into intermediate representations for grounding and reasoning across perspectives Bigverdi et al. (2025); Wan et al. (2025); Ning et al. (2025); Gholami et al. (2025); Lee et al. (2025); Yang et al. (2025a); Chen et al. (2026c); Zhou et al. (2026a). Post-training methods optimize grounded reasoning and spatial predictions through supervision or task-specific rewards Kancheti et al. (2026); Xu et al. (2025); Zheng et al. (2025b); Ma et al. (2026); Chen et al. (2025c); Batra et al. (2025); Wang and Ling (2025); Wu et al. (2025b); Sarch et al. (2025); Zhao et al. (2025a); Li et al. (2025); Chen et al. (2026b); Li et al. (2026c), whereas training-free methods guide inference through spatial prompting or interventions on visual processing Liao et al. (2024); Ma et al. (2024); Mitra et al. (2024); Chen et al. (2025b); Yan et al. (2026). These methods strengthen spatial evidence and its use, but premature commitment to a single continuation can remain in discrete CoT. Continuous and Soft Thinking. Recent work explores continuous intermediate states in place of discrete CoT generation in LLMs Hao et al. (2024); Wei et al. (2026b); Zhou et al. (2026c). Soft thinking retains multiple candidate continuations through mixtures of token embeddings Zhang et al. (2026b); Butt et al. (2026); Zheng et al. (2025a); Wang et al. (2025a). Other work explores discrete CoT paths Dang et al. (2026); Zhou et al. (2026b); Yu et al. (2026); Wei et al. (2026a) or switches between continuous and discrete reasoning Shi et al. (2026); Xu et al. (2026b); Xu et al. (2026c). In LVLMs, continuous representations support multimodal reasoning, including soft thinking within the CoT Hu et al. (2026); Sun et al. (2026); Chen et al. (2026a); Shen et al. (2025); Pham and Ngo (2026); Ma et al. (2025a); Jeon et al. (2026); Wang et al. (2026b); Ray et al. (2026); Huang and Shan (2026), while complementary approaches construct or refine visual representations for subsequent reasoning Yang et al. (2026); Li et al. (2026a); Li et al. (2026b). These approaches do not learn step-specific softness for token-embedding mixtures. Soft Spatial Reasoning uses the hidden state and predictive uncertainty to adjust softness, with gradient alignment guiding training. 3 Method Soft Spatial Reasoning (Figure 2a) is a post-training framework that enables adaptive soft thinking for spatial tasks in LVLMs through continuous intermediate states and discrete final answers. Its controller, AdaptSoft (Figure 2b), determines step-specific softness from the current hidden state and predictive uncertainty. Gradient-Alignment Learning (Figure 3) trains this controller by deriving step-specific credit from alignment between each step’s policy gradient and a reference gradient. 3.1 Problem Setup Given an image I and a spatial reasoning query q, an LVLM policy πθ _θ generates a rollout o=(C,a)o=(C,a) consisting of a CoT C and a discrete final answer a. Hard thinking represents C as a sequence of discrete language tokens, each conditioning subsequent predictions. Soft thinking instead represents C as continuous states (Section 3.2). The LVLM policy is post-trained using Group Relative Policy Optimization (GRPO) Shao et al. (2024), which uses relative rewards within groups of sampled rollouts without requiring ground-truth CoT supervision. As illustrated in Figure 2a, for each image–query pair (I,q)(I,q), the rollout policy πθold _ _old samples a group of G rollouts: oi∼πθold(⋅∣I,q),i=1,…,G.o_i _ _old(· I,q), i=1,…,G. (1) Each rollout receives an answer reward Rans(i)R_ans^(i) and a format reward Rfmt(i)R_fmt^(i). The answer reward assigns full credit (11) when the final answer is correct and no credit (00) otherwise, while the format reward evaluates compliance with the required reasoning-and-answer structure. Using reward weights wansw_ans and wfmtw_fmt, and a small constant ϵε for numerical stability, we define the task reward and its group-normalized advantage as ri=wansRans(i)+wfmtRfmt(i),Aitask=ri−mean(rjj=1G)std(rjj=1G)+ϵ.r_i=w_ansR_ans^(i)+w_fmtR_fmt^(i), A_i^task= r_i-mean(\r_j\_j=1^G)std(\r_j\_j=1^G)+ε. (2) Figure 2: Overview of Soft Spatial Reasoning. (a) The LVLM policy is post-trained to carry multiple candidate continuations through soft states during reasoning, then generate the final answer as discrete tokens. Rollout rewards yield group-relative advantages for the GRPO update. (b) At each reasoning step, AdaptSoft sets the mixture temperature from the current hidden state and candidate-distribution entropy. The resulting mixture weights combine token embeddings into the next soft state. 3.2 Soft Chain-of-Thought (CoT) Within each rollout oio_i, the intermediate CoT is represented by a sequence of TiT_i soft states si,tt=1Ti\s_i,t\_t=1^T_i, formed by feeding continuous mixtures of token embeddings back into the language backbone at successive reasoning steps (Figure 2a) Zhang et al. (2026b); Butt et al. (2026). After the reasoning steps, the model switches to discrete token generation for the final answer. Stochastic Soft Rollout. To produce different soft CoTs within the rollout group, independent Gumbel perturbations are applied to the vocabulary log-probabilities at each reasoning step Zheng et al. (2025a); Wu et al. (2026), as illustrated in Figure 2b. Specifically, conditioned on (I,q)(I,q) and the preceding states si,<ts_i,<t, the rollout policy produces the unperturbed distribution p¯i,t p_i,t. For each token k, an independent Gumbel sample γi,t,k _i,t,k is then added to logp¯i,t,k p_i,t,k, yielding the perturbed score zi,t,kz_i,t,k: p¯i,t=πθold(⋅∣I,q,si,<t),γi,t,k∼i.i.d.Gumbel(0,1),zi,t,k=logp¯i,t,k+γi,t,k. gathered p_i,t= _ _old (· I,q,s_i,<t ), _i,t,k i.i.d. Gumbel(0,1),\\ z_i,t,k= p_i,t,k+ _i,t,k. gathered (3) A temperature-controlled softmax (Figure 2b) converts the perturbed scores into token-mixture weights pi,tp_i,t, whose weighted combination of token embeddings forms the soft state si,ts_i,t: pi,t=softmax(zi,tτi,t),si,t=∑k=1||pi,t,kEk,p_i,t=softmax\! ( z_i,t _i,t ), s_i,t= _k=1^|V|p_i,t,kE_k, (4) where V is the language vocabulary. At each step, the softmax is applied to its top-K candidate tokens, with pi,t,k=0p_i,t,k=0 for all other tokens. Here, k indexes vocabulary tokens, and Ek∈ℝdE_k ^d is the embedding of token k. The temperature τi,t>0 _i,t>0 controls mixture concentration: lower values move si,ts_i,t toward the highest-scoring token’s embedding, while higher values spread weight across token embeddings. Gumbel-Reparameterized Likelihood. GRPO requires a likelihood ratio between the current and rollout policies at each reasoning step. Unlike a discrete CoT step, a soft state si,ts_i,t combines token embeddings without selecting an individual token, so the standard sampled-token likelihood does not apply. The perturbed score vector zi,tz_i,t is the sampled variable that, given τi,t _i,t, deterministically specifies si,ts_i,t (Equations 3 and 4). The likelihood ratio is therefore defined over zi,tz_i,t, evaluating the same recorded score vector under both policies. During the policy update, the sampled scores and corresponding temperatures from the preceding steps reconstruct si,<ts_i,<t, ensuring that both policies are evaluated on the same reasoning prefix. Conditioned on (I,q,si,<t)(I,q,s_i,<t), the current policy produces p¯i,tθ p_i,t^\,θ. For each recorded score zi,t,kz_i,t,k, subtracting the current token log-probability yields the Gumbel noise value implied by the current policy. Evaluating the implied noise values under independent standard Gumbel densities gives the conditional joint log-density of zi,tz_i,t, with conditioning on (I,q,si,<t)(I,q,s_i,<t) omitted for compactness: p¯i,tθ=πθ(⋅∣I,q,si,<t),γ~i,t,kθ=zi,t,k−logp¯i,t,kθ,logPθ(zi,t)=∑k=1||[−γ~i,t,kθ−exp(−γ~i,t,kθ)]. gathered p_i,t^\,θ= _θ(· I,q,s_i,<t), γ_i,t,k^\,θ=z_i,t,k- p_i,t,k^\,θ,\\ P_θ(z_i,t)= _k=1^|V| [- γ_i,t,k^\,θ- \! (- γ_i,t,k^\,θ ) ]. gathered (5) Evaluating the same scores under the rollout policy gives Pθold(zi,t)P_ _old(z_i,t), yielding the soft-step GRPO ratio: ρi,tsoft(θ)=exp(logPθ(zi,t)−logPθold(zi,t)). _i,t^soft(θ)= \! ( P_θ(z_i,t)- P_ _old(z_i,t) ). (6) As detailed in Section 3.3, the rollout temperatures are reintroduced through AdaptSoft when reconstructing the soft states, allowing gradients from subsequent reasoning steps to propagate to its parameters. A derivation of the density and likelihood ratio is provided in Appendix Section C.1. 3.3 AdaptSoft: Adaptive Softness Control Step-Specific Softness Controller. At reasoning step t of rollout oio_i, let hi,t∈ℝdh_i,t ^d denote the final-layer hidden state and p¯i,t p_i,t the unperturbed next-token distribution. The hidden state summarizes the accumulated reasoning context, while the entropy of p¯i,t p_i,t measures predictive uncertainty among candidate continuations. The entropy is standardized as H^i,t=H(p¯i,t)−μHσH+ϵH, H_i,t= H( p_i,t)- _H _H+ _H, (7) where H(⋅)H(·) denotes Shannon entropy. The mean μH _H and standard deviation σH _H are estimated once from the next-token entropies of the initial policy and remain fixed throughout post-training. The constant ϵH>0 _H>0 prevents division by zero. Before entering the controller, hi,th_i,t is layer-normalized and projected to dp<d_p<d dimensions using a fixed random matrix ∈ℝdp×dP ^d_p× d. This compact representation reduces the rollout storage required for controller training. The projected state is concatenated with H^i,t H_i,t and passed to an MLP fϕf_φ, whose scalar output parameterizes the temperature: ui,t=fϕ([,∥,]),τi,t=τ0+Δtanh(ui,t),u_i,t=f_φ\! ( [P\,LN(h_i,t)\; \|\; H_i,t ] ), _i,t= _0+ (u_i,t), (8) where LNLN denotes layer normalization, ∥\| denotes concatenation, and ϕφ contains the learnable controller parameters. Since tanh(ui,t)∈(−1,1) (u_i,t)∈(-1,1), τ0 _0 sets the base temperature and Δ sets its maximum step-specific deviation, yielding τi,t∈(τ0−Δ,τ0+Δ) _i,t∈( _0- , _0+ ). We require 0<Δ<τ00< < _0 to keep all temperatures above zero. Within this range, AdaptSoft uses the current reasoning context and predictive uncertainty to adjust the softness of each mixture. As described in Section 3.2, the sampled soft states are reconstructed during the policy update from the recorded scores and rollout temperatures. To train AdaptSoft without altering these sampled states, the controller inputs and outputs are retained during rollout. A stop-gradient construction uses the recorded temperature in the forward computation while passing gradients through its recomputed value to ϕφ; details are provided in Appendix Section C.2. 3.4 Post-Training Optimization The LVLM policy is optimized with GRPO, while AdaptSoft is trained through step-specific gradient alignment. GRPO for the LVLM Policy. For each sampled step, ρi,t(θ) _i,t(θ) compares its likelihood under the current and rollout policies. Soft reasoning steps use the density ratio ρi,tsoft(θ) _i,t^soft(θ) in Equation 6, while discrete final-answer tokens use the token-probability ratio. Each ratio is weighted by the shared rollout advantage AitaskA_i^task, encouraging steps from higher-reward rollouts and discouraging those from lower-reward rollouts. The clipped surrogate limits the incentive for large ratio changes, while KL regularization over the LVLM’s next-token distributions penalizes deviation from πref _ref. Averaging over steps and rollouts gives (θ)= (θ)= oi∼πθold[1G∑i=1G1|oi|∑t=1|oi|(minρi,t(θ)Aitask, _\o_i\ _ _old [ 1G _i=1^G 1|o_i| _t=1^|o_i| ( \ _i,t(θ)A_i^task, (9) clip(ρi,t(θ),1−δ,1+δ)Aitask−ηDKL(πθ∥πref))]. ( _i,t(θ),1-δ,1+δ )A_i^task \-η D_KL ( _θ\,\|\, _ref ) ) ]. Here, |oi||o_i| counts soft reasoning steps and discrete final-answer tokens, δ is the clipping threshold, and η weights the KL penalty against the reference LVLM policy πref _ref. The KL distributions are conditioned on (I,q)(I,q) and the preceding steps. Optimization minimizes ℒGRPO(θ)=−(θ)L_GRPO(θ)=-J(θ). Figure 3: Gradient-Alignment Learning. At each soft reasoning step t, the temperature τi,t _i,t controls the state si,ts_i,t, whose contribution gi,tg_i,t to the LVLM output-layer gradient is compared with a reference gradient GrefG_ref computed from a disjoint subset of rollouts. The resulting alignment scores αi,t _i,t are aggregated to form the AdaptSoft loss. Step-Specific Gradient Alignment for AdaptSoft. To train AdaptSoft, we introduce a gradient-alignment objective that provides a step-specific learning signal for each predicted temperature. The shared rollout-level advantage in GRPO does not directly distinguish how softness should vary across reasoning steps. The proposed objective provides this signal by comparing each step’s contribution to the LVLM policy gradient with a reference gradient (Figure 3). Each batch of sampled rollouts is divided into disjoint reference and optimization subsets. The reference subset provides a gradient GrefG_ref, while each soft reasoning step t of rollout oio_i in the optimization subset contributes a gradient gi,tg_i,t. Both gradients are computed with respect to the LVLM output-layer weights W. The alignment score αi,t _i,t is defined as the Frobenius inner product of these gradients. The AdaptSoft loss ℒAS(ϕ)L_AS(φ) is the negative sum of the alignment scores across soft reasoning steps in the optimization subset: Gref=∇WℒGRPOref,gi,t=∇WℒGRPO(i,t),αi,t=⟨gi,t,Gref⟩F,ℒAS(ϕ)=−∑i∑t=1Tiαi,t, gatheredG_ref= _WL_GRPO^ref, g_i,t= _WL_GRPO^(i,t),\\ _i,t= g_i,t,G_ref _F, _AS(φ)=- _i _t=1^T_i _i,t, gathered (10) where ℒGRPOrefL_GRPO^ref is the GRPO loss on the reference subset, ℒGRPO(i,t)L_GRPO^(i,t) is the contribution of step t in rollout oio_i to the GRPO loss, and i ranges over the optimization subset. For a small update W′=W−λgi,tW =W-λ g_i,t with step size λ>0λ>0, the first-order change in the reference loss is −λαi,t-λ _i,t. Minimizing ℒASL_AS thus favors temperatures whose resulting step-specific gradients align with the reference gradient. The gradients gi,tg_i,t are computed through the reconstructed soft states and remain differentiable with respect to the temperatures. Holding GrefG_ref and the GRPO logit residuals fixed, differentiation provides a first-order alignment signal to temperatures through subsequent reasoning steps. To emphasize differences across reasoning steps, the mean temperature gradient is subtracted within each rollout: g~i,tτ=∂ℒAS∂τi,t−1Ti∑t′=1Ti∂ℒAS∂τi,t′. g_i,t^\,τ= _AS∂ _i,t- 1T_i _t =1^T_i _AS∂ _i,t . (11) The centered gradients satisfy ∑t=1Tig~i,tτ=0 _t=1^T_i g_i,t^\,τ=0. Alignment scores are computed using the outer-product structure of the output-layer gradients, avoiding a separate ||×d|V|× d gradient matrix for each soft reasoning step. Parameter Updates. Each training step updates the LVLM parameters θ using ℒGRPOL_GRPO from Equation 9. The AdaptSoft parameters ϕφ are updated by backpropagating the centered temperature gradients from Equation 11 through the controller in Equation 8 (detailed in Appendix Section C.3). 4 Experiments Table 1: OmniSpatial Jia et al. (2025) results across 10 spatial reasoning task categories. Soft Spatial Reasoning is compared against proprietary models, general open-source LVLMs, soft thinking models, and specialized spatial reasoning models. Proprietary models are included as reference points; the best open-source or specialized result is shown in bold. Average accuracy is weighted by category sample size. Method Average Dynamic Reasoning Spatial Interaction Complex Logic Perspective Taking Mani- pulation Motion Anal. Traffic Anal. Loca- lization Geospa. Strategy Pattern Rec. Geometric Reasoning Ego Centric Allo Centric Hypo- thetical Reference Baselines Random Choice 24.98 24.86 26.30 25.88 23.43 27.27 21.44 24.77 22.55 24.84 25.78 Human Evaluation 92.63 94.62 96.07 91.38 95.11 92.15 89.02 85.90 98.53 94.30 90.26 Proprietary Models GPT-4.1-mini Open (2025) 48.87 64.32 56.53 59.06 60.19 56.36 29.28 30.19 72.55 39.57 39.28 Gemini-2.5-flash-preview Team et al. (2023) 52.12 67.57 62.72 68.24 73.33 60.91 38.14 34.19 75.49 35.90 33.73 o4-mini OpenAI (2025) 52.77 72.97 59.83 60.00 73.33 61.82 34.02 36.77 73.53 40.69 40.96 Gemini-2.5-flash Team et al. (2023) 53.16 70.27 64.74 61.18 72.38 58.18 35.05 36.13 74.12 40.96 32.53 Open-weights Models LLaVA-1.5-7B Liu et al. (2024) 34.97 54.46 31.23 35.29 36.19 33.94 29.01 24.18 55.60 34.66 36.14 InternVL3-8B Zhu et al. (2025) 41.60 52.43 40.87 48.94 51.05 44.77 24.95 28.63 64.20 38.62 40.96 InternVL3-14B Zhu et al. (2025) 45.94 54.32 60.17 50.35 51.81 51.45 28.04 28.26 68.04 35.37 34.46 Qwen2.5-VL-7B Wang et al. (2024) 39.18 58.38 35.09 50.12 45.33 44.00 31.13 29.42 64.51 33.19 37.35 Gemma-3-12B Gemma Team and others (2025) 43.71 54.05 54.91 54.12 47.62 45.45 16.49 30.32 63.73 36.70 33.73 Soft Thinking Models LVR Li et al. (2026a) 41.86 54.05 35.83 44.70 53.33 48.18 28.86 27.74 72.54 38.29 50.60 Laser Wang et al. (2026b) 42.78 55.40 45.95 57.64 54.28 50.00 20.61 24.51 72.54 33.51 44.57 LaCoT Sun et al. (2026) 45.66 59.45 53.75 56.47 50.47 48.18 28.86 33.54 67.64 34.57 44.57 Spatial Reasoning Models SpaceMantis-13B Chen et al. (2024a) 36.36 47.03 36.59 40.94 34.86 33.09 22.27 24.39 49.22 38.25 39.28 SpaceQwen2.5-VL-3B Chen et al. (2024a) 40.25 58.11 39.88 41.18 40.95 40.91 29.90 25.81 63.73 38.83 39.76 SpaceThinkerQwen2.5VL-3B Chen et al. (2024a) 40.42 47.84 53.06 43.29 35.43 38.73 24.33 28.00 58.04 35.11 31.08 VST-RL-7B Yang et al. (2025b) 41.09 56.75 43.39 44.75 46.66 42.72 25.51 28.38 72.54 32.89 43.37 SoFar-Qwen2.5VL-3B Qi et al. (2025) 45.14 56.49 51.16 54.12 53.14 52.73 31.75 22.88 71.60 36.56 41.69 SpatialLadder-3B Li et al. (2025) 40.50 59.45 39.01 50.58 48.57 42.72 26.80 23.87 71.56 35.10 39.75 Backbone and Our Variants Qwen3-VL-8B-Thinking (Base) Bai et al. (2025) 43.90 57.14 54.28 36.04 54.95 48.67 24.77 23.12 71.56 31.11 38.82 Hard Thinking + GRPO 45.92 61.32 54.30 55.88 58.33 46.18 25.90 25.00 72.92 36.57 42.53 Soft Thinking + GRPO 46.93 61.82 55.85 52.21 62.62 49.77 25.26 25.00 76.35 36.23 45.88 Soft Spatial Reasoning (Ours) 49.68 63.51 62.14 57.65 62.86 50.00 29.90 34.84 75.49 34.93 46.34 Figure 4: Soft Spatial Reasoning on a randomly selected example from OmniSpatial. AdaptSoft varies softness across the CoT. For readability, greedy decoding is used only to display the soft states as text. Shading shows each span’s mean temperature (blue: lower; orange: higher). The plot below shows temperature at every reasoning step and labels several words with their temperatures. 4.1 Implementation Details We instantiate Soft Spatial Reasoning with a Qwen3-VL-8B-Thinking backbone Bai et al. (2025) and train for two epochs on the OmniSpatial training split Jia et al. (2025). GRPO uses G=8G=8 rollouts per prompt, a batch size of 64 prompts, and two optimizer updates per batch. The LVLM policy is optimized with AdamW at a learning rate of 10−610^-6 and regularized toward its frozen initialization using a KL coefficient of 10−310^-3. AdaptSoft is implemented as a two-layer MLP with projection dimension dp=8d_p=8 and optimized with AdamW at a learning rate of 10−310^-3. Additional optimization, preprocessing, sequence-length, and system details are provided in Appendix Section B.1. 4.2 Baselines We compare Soft Spatial Reasoning against open-source LVLMs, including models specialized in spatial reasoning and soft thinking, and report proprietary models as reference points. We also train controlled hard- and soft-thinking baselines using the same backbone and post-training setup. Evaluation covers three complementary settings. The held-out OmniSpatial test set Jia et al. (2025) evaluates post-training performance across its spatial-reasoning taxonomy. SpatiaLab Wasi et al. (2026) measures zero-shot transfer to realistic, unconstrained scenes from an unseen benchmark. MindCube Wang et al. (2026a) evaluates zero-shot spatial mental modeling from limited views through cognitive mapping, perspective-taking, and mental simulation. Dataset and evaluation details are provided in Appendix Section D.1. 4.3 Results Table 2: SpatiaLab Wasi et al. (2026) results across 6 spatial reasoning task categories in the zero-shot setting. Proprietary models are included as reference points; the best open-source or specialized result is shown in bold. Average accuracy is weighted by category sample size. Method Average Question Categories 3D Geom. Dep. & Occu. Orientation Relat. Posit. Size & Scale Spati. Navig. Reference Baselines Random Choice 25.00 25.00 25.00 25.00 25.00 25.00 25.00 Human Baseline 87.57 93.70 74.13 91.58 91.51 88.89 87.76 Proprietary Models GPT-4o-mini Hurst et al. (2024) 46.50 47.06 39.00 47.03 47.17 49.60 49.79 Gemini-2.5-Flash Team et al. (2023) 48.29 44.96 48.26 48.02 56.13 42.46 51.05 Mistral Medium 3.1 Mistral AI (2025) 47.93 46.64 49.81 47.52 61.79 41.67 41.77 Open-weights Models Qwen2.5-VL-3B-Instruct Wang et al. (2024) 41.43 41.18 35.52 46.04 40.09 47.22 39.24 InternVL3.5-4B Wang et al. (2025b) 43.29 42.86 42.86 42.08 54.72 36.51 42.19 Gemma-3-4B-it Gemma Team and others (2025) 40.57 43.70 34.36 46.53 45.75 37.30 37.97 LLaVA-1.5-7B Liu et al. (2024) 38.64 40.33 34.74 31.68 40.56 40.07 43.88 Qwen2.5-VL-7B-Instruct Wang et al. (2024) 41.00 42.86 37.84 42.57 46.23 42.06 35.44 Soft Thinking Models LVR Li et al. (2026a) 45.78 48.31 43.62 51.98 49.52 40.07 43.03 Laser Wang et al. (2026b) 44.78 46.21 43.24 46.03 50.47 40.07 43.88 LaCoT Sun et al. (2026) 45.50 46.63 40.15 47.52 53.30 43.25 43.88 Spatial Reasoning Models SpaceOm Chen et al. (2024a) 41.36 42.44 38.61 48.02 37.74 42.86 39.24 SpaceThinker-Qwen2.5VL-3B Chen et al. (2024a) 40.64 40.34 37.84 47.03 38.21 43.25 37.97 SpaceQwen2.5-VL-3B-Instruct Chen et al. (2024a) 40.14 31.51 35.14 37.62 37.74 50.79 47.26 SpatialLadder-3B Li et al. (2025) 35.28 39.91 34.61 39.90 38.20 25.39 33.19 Backbone and Our Variants Qwen3-VL-8B-Thinking (Base) Bai et al. (2025) 44.78 42.01 47.10 47.02 51.88 40.47 41.35 Hard Thinking + GRPO 46.21 43.70 47.88 47.52 54.25 41.27 43.88 Soft Thinking + GRPO 46.86 44.12 48.26 48.02 55.66 41.67 44.73 Soft Spatial Reasoning (Ours) 48.71 46.22 49.42 48.51 58.96 42.46 48.10 What Adaptive Softness Adds to Spatial Reasoning. On OmniSpatial, Soft Spatial Reasoning leads the non-proprietary models in weighted-average accuracy (Table 1), surpassing InternVL3-14B, SoFar, and LaCoT by 3.74, 4.54, and 4.02 percentage points, respectively. With the same backbone and post-training setup, basic soft thinking improves on hard thinking by 1.01 points, while adaptive softness adds another 2.75 points. Geometric Reasoning illustrates this distinction: hard and basic soft thinking both score 25.00, whereas adaptation reaches 34.84. Motion Analysis also rises from 55.85 to 62.14 over basic soft thinking. The absence of a geometric gain from basic soft thinking suggests that preserving alternatives alone may not suffice. Geometry and motion can involve competing spatial configurations whose usefulness changes across reasoning steps. Adaptive softness may help retain useful possibilities while limiting interference from conflicting relations as the reasoning context changes. Relative to the original backbone, the full model improves in all ten categories, including a 21.61-point gain in Traffic Analysis. Figure 4 illustrates how Soft Spatial Reasoning answers a spatial question about an image. The displayed CoT traces the model’s reasoning about the robot’s movement, reaching the correct answer as softness varies across steps. The text highlights connect this reasoning to the temperature plot below: “operator” and “robot” correspond to higher temperatures, while “press” and “forward” correspond to lower temperatures. These adjustments let broader candidate continuations contribute at some steps and concentrate their influence at others, showing how adaptive softness operates throughout reasoning. Further analysis of temperature variation appears in Appendix A. What Adaptive Softness Adds in Zero-Shot Transfer. On the unseen SpatiaLab benchmark (Table 2), Soft Spatial Reasoning achieves 48.71, exceeding the strongest prior open-weights model, LVR, by 2.93 percentage points. The matched-backbone comparison shows that adaptive softness contributes a further 1.85 points over basic soft thinking, compared with the 0.65-point gain from hard to basic soft thinking. Thus, the larger benefit from adjusting softness across reasoning steps observed on OmniSpatial persists under zero-shot transfer. Table 3: Zero-shot results on MindCube Wang et al. (2026a) across three spatial mental-modeling settings. Proprietary models are included as reference points, and Overall is weighted by setting size. Method Overall Rotation Among Around Reference Baseline Random Choice 32.35 36.36 32.29 30.66 Proprietary Models Gemini-2.5-Pro Team et al. (2023) 47.05 85.50 25.95 38.40 Claude-4-Sonnet Anthropic (2024) 44.75 48.42 44.21 47.62 Open-weight Models InternVL3-8B Wang et al. (2025b) 37.50 26.00 42.03 36.00 Qwen3-VL-8B-Thinking (Base) Bai et al. (2025) 33.62 45.14 33.26 30.44 Soft-Thinking Models LVR Li et al. (2026a) 29.53 36.07 29.44 26.59 Laser Wang et al. (2026b) 27.29 37.65 30.12 21.02 Spatial Reasoning Models SpaceMantis Chen et al. (2024a) 22.81 37.65 21.26 29.32 SpaceQwen Chen et al. (2024a) 33.28 38.02 33.71 26.32 Proposed Method Soft Spatial Reasoning (Ours) 38.13 47.36 38.18 32.37 Unlike on OmniSpatial, adaptive softness improves over basic soft thinking in every category. The gains are nevertheless uneven: Relative Position (+3.30) and Spatial Navigation (+3.37) improve substantially more than Orientation (+0.49) and Size & Scale (+0.79). The consistent direction of these improvements supports the transferability of learned softness control, while their differing magnitudes indicate that its benefit depends on the spatial task. What Adaptive Softness Adds to Spatial Mental Modeling. MindCube Wang et al. (2026a) requires integrating limited views to infer unseen spatial relations and reason about hypothetical movements. In this zero-shot setting, Soft Spatial Reasoning achieves 38.13 weighted-average accuracy, the highest among the compared non-proprietary models, and improves on its backbone by 4.51 points (Table 3). Improvements span all three settings, extending the framework’s benefits to reasoning about partially observed scenes. Adaptive softness may support this process by preserving plausible spatial interpretations while combining evidence from different views. Table 4: OmniSpatial Perspective Taking ablations Jia et al. (2025). All variants share the backbone and policy GRPO setup, with post-training and evaluation on the corresponding training and test splits. Averages are sample-weighted. Variant Avg. Ego centric Allo centric Hypo thetical Soft Spatial Reasoning (Full) 42.75 81.37 32.53 41.46 w/o hi,th_i,t in AdaptSoft 39.22 77.45 29.52 36.14 w/o predictive uncertainty 40.64 78.43 30.85 38.55 w/o gradient alignment 38.68 76.47 29.26 34.94 w/o gradient centering 41.18 79.41 31.12 39.76 Ablation Study. All four ablations reduce accuracy in every perspective-taking category (Table 4). Replacing AdaptSoft’s step-specific gradient alignment with the rollout-level task advantage produces the largest average decline (4.07 points), including a 6.52-point drop in Hypothetical reasoning. This suggests that the shared GRPO signal provides less guidance for controlling softness at individual steps. Removing the final-layer hidden state hi,th_i,t from AdaptSoft reduces average accuracy by 3.53 points, compared with 2.11 points without predictive uncertainty, supporting the use of the ongoing reasoning state alongside predictive uncertainty to control softness. Omitting gradient centering reduces accuracy in all three categories, with a smaller average decline of 1.57 points. 5 Conclusion We introduced Soft Spatial Reasoning, a post-training framework that enables adaptive soft thinking for spatial reasoning in LVLMs. AdaptSoft controls softness using the reasoning state and predictive uncertainty, with step-specific guidance from gradient alignment. Across three benchmarks, our framework achieves the highest weighted-average accuracy among the evaluated non-proprietary models, with gains extending to unseen benchmarks. Limitations. 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Temperatures and predictive uncertainty are measured across soft reasoning steps during inference on the test set. Color shows the number of steps on a logarithmic scale. The solid line shows median temperature in each entropy bin; dashed lines mark the 10th and 90th percentiles. The min–max span grows from 0.230.23 to 0.580.58, extending on both sides of the base value. AdaptSoft thus produces broader mixtures at some steps and more concentrated ones at others without shifting average softness. This spread develops over successive updates and persists later in training, revealing learned variation that the stable mean would obscure. Basic soft thinking can vary mixture weights as token probabilities change, but its temperature remains fixed. AdaptSoft also learns how soft each mixture should be; Figure 4 illustrates these adjustments within an individual soft CoT. Predictive Uncertainty and Softness. Figure 6 plots the temperature and predictive uncertainty at each soft reasoning step across inference rollouts of the trained model. As normalized top-k entropy increases, median temperature falls from 0.5900.590 to 0.3500.350, with a correlation of −0.613-0.613 across all steps. Figure 7: AdaptSoft Varies Softness Within CoTs. The histogram shows the 90th–10th percentile temperature spread within each test-set reasoning trace generated during inference. The solid line marks the median within-trace spread; the dashed line marks the corresponding spread across trace-median temperatures. When probability is spread across more candidates, these lower temperatures concentrate their influence and may reduce interference from conflicting spatial interpretations. AdaptSoft also assigns substantially different temperatures at similar uncertainty. Within every entropy bin, the 10th–90th percentile range is 0.180.18–0.200.20, nearly as large as the 0.240.24 change in median across the full entropy range. Even the lowest-entropy bin, containing 48.4%48.4\% of steps, spans 0.4700.470–0.6700.670 between these percentiles. Removing the hidden-state input leaves AdaptSoft with predictive uncertainty alone and lowers average accuracy by 3.533.53 points (Table 4). AdaptSoft Varies Softness Within CoTs. Figure 7 compares temperature variation within individual test-set CoTs during inference with variation across their median temperatures. The median within-trace spread is 0.2350.235, about 2.52.5 times the 0.0930.093 spread across trace medians: AdaptSoft changes softness more as a CoT unfolds than it differs between CoTs. Even the smallest within-trace spread is 0.1450.145, showing that these changes occur across the test set. Correct and incorrect traces have similar median spreads (0.2330.233 and 0.2370.237), so the variation is not limited to successful answers. It also persists in short and long traces (0.2180.218 below 100 steps and 0.2410.241 at 300 steps or more), rather than arising simply because longer CoTs offer more steps over which temperature can change. Table 5: Post-training and evaluation configuration for Soft Spatial Reasoning. Hyperparameter Value Notes Model configuration Backbone Qwen3-VL-8B-Thinking – Language-model adaptation Full fine-tuning No LoRA Numerical precision BF16 FSDP2 mixed precision Parameter sharding FSDP2 Parameter and optimizer offload Data preparation Training-image pixel budget 128×282128× 28^2 Aspect ratio preserved; dimensions floored to multiples of 28 Evaluation image resolution Native No resizing Maximum prompt length 10,240 tokens Overlong prompts discarded Training configuration Training GPUs 8 Two nodes × four H100s Training steps 200 ≈2≈ 2 epochs Prompt batch size 64 Per GRPO step Rollouts per prompt G=8G=8 GRPO group size Optimizer mini-batch size 32 Two prompt mini-batches per GRPO batch Micro-batch size 4 sequences/GPU Dynamic batching disabled Maximum training response length 2,048 tokens - Policy optimization Optimizer AdamW Policy parameters Learning rate 1×10−61× 10^-6 – Learning-rate schedule Cosine 10% minimum-LR floor Warmup ratio 0.05 – KL coefficient 1×10−31× 10^-3 Loss-only; frozen reference policy Entropy coefficient 0.0 No entropy bonus Advantage estimation GRPO Group-normalized Soft thinking Candidate set Top-k=5k=5 Per reasoning step Softness temperature Step-specific Predicted by AdaptSoft Gumbel-noise scale 1.0 Added to candidate log-probabilities Application span Reasoning tokens only Ends at </think>; discrete answer AdaptSoft fϕf_φ Projection dimension dp=8d_p=8 Fixed random projection AdaptSoft architecture Two-layer MLP Width 256; GELU Temperature map τi,t=0.5+0.4tanh(ui,t) _i,t=0.5+0.4 (u_i,t) τi,t∈(0.1,0.9) _i,t∈(0.1,0.9) AdaptSoft initialization a=0,b=0a=0,\ b=0 Starts at τ=0.5τ=0.5 Entropy standardization μH=0.173,σH=0.224 _H=0.173,\ _H=0.224 Estimated from the initial policy and held fixed AdaptSoft optimizer AdamW, 1×10−31× 10^-3 Separate from policy optimizer Gradient centering Per rollout Across soft reasoning steps Reference subset 25% of each batch Disjoint and all-gathered Reward Answer-reward weight 1.0 Exact answer match Format-reward weight 0.2 One <think> block; final letter A–D Evaluation Samples per question 8 Mean@8 Decoding temperature 0.6 – Decoding top-k 5 – Maximum response length 3,072 tokens – Infrastructure RL framework verl 0.8.0 GRPO; FSDP2 Rollout engine SGLang 0.5.12 TP=1=1; DP=8=8 GPU memory fraction 0.4 Rollout engine Orchestration Ray and Slurm Multi-node execution Appendix B Appendix: Implementation Details B.1 Hyperparameter Settings Table 5 reports the main hyperparameters used for Soft Spatial Reasoning post-training, including the GRPO training setup, optimization settings, and reward configuration. B.2 Reasoning Format and Discrete Answer Generation We use the fixed system prompt in Figure 8 during training to enforce a consistent reasoning format. Since the chat template of Qwen3-VL-Thinking Bai et al. (2025) automatically prefills the opening <think> token after the user message, the prompt does not ask the model to generate <think>; it only requires the model to close the reasoning segment with </think>. During rollout generation, each reasoning step feeds a mixture of token embeddings back to the model. We also retain the highest-weight token at each step as a discrete spine, which marks where reasoning ends. When the spine emits </think>, subsequent steps sample discrete tokens from the full vocabulary to generate the final answer. The delimiter itself remains part of the soft reasoning phase; the switch occurs immediately afterward. Figure 8: System prompt used to standardize the reasoning format during Soft Spatial Reasoning training. Appendix C Appendix: Technical Derivations C.1 Derivation of the Gumbel-Reparameterized Likelihood The likelihood in Equation 5 follows the Gumbel-reparameterized construction of SofT-GRPO Zheng et al. (2025a). The sampled variable at each soft reasoning step is the perturbed score vector zi,tz_i,t, while the soft state is constructed deterministically from zi,tz_i,t and τi,t _i,t. Conditioning on (I,q,si,<t)(I,q,s_i,<t) is omitted below for compactness. During rollout, each score is obtained by adding independent standard Gumbel noise to the rollout-policy log-probability. Under the current policy, the same recorded score implies a different noise value: zi,t,k z_i,t,k =logp¯i,t,k+γi,t,k, = p_i,t,k+ _i,t,k, γi,t,k _i,t,k ∼i.i.d.Gumbel(0,1), i.i.d. Gumbel(0,1), (12) fGum(γ) f_Gum(γ) =exp[−γ−exp(−γ)], = \! [-γ- (-γ) ], γ~i,t,kθ γ_i,t,k^\,θ =zi,t,k−logp¯i,t,kθ. =z_i,t,k- p_i,t,k^\,θ. The transformation from γ~i,t,kθ γ_i,t,k^\,θ to zi,t,kz_i,t,k is additive and therefore has unit Jacobian. Independence across vocabulary tokens gives the density and its logarithm, recovering Equation 5: Pθ(zi,t)=∏k=1||fGum(γ~i,t,kθ),logPθ(zi,t)=∑k=1||[−γ~i,t,kθ−exp(−γ~i,t,kθ)].P_θ(z_i,t)= _k=1^|V|f_Gum\! ( γ_i,t,k^\,θ ), P_θ(z_i,t)= _k=1^|V| [- γ_i,t,k^\,θ- \! (- γ_i,t,k^\,θ ) ]. (13) Under the rollout policy, the implied noise equals the originally sampled value. Evaluating the same score vector under both policies therefore gives γ~i,t,kθold=zi,t,k−logp¯i,t,k=γi,t,k,logPθold(zi,t)=∑k=1||[−γi,t,k−exp(−γi,t,k)],ρi,tsoft(θ)=Pθ(zi,t)Pθold(zi,t)=exp(logPθ(zi,t)−logPθold(zi,t)). gathered γ_i,t,k^\, _old=z_i,t,k- p_i,t,k= _i,t,k, P_ _old(z_i,t)= _k=1^|V| [- _i,t,k- (- _i,t,k) ],\\ _i,t^soft(θ)= P_θ(z_i,t)P_ _old(z_i,t)= \! ( P_θ(z_i,t)- P_ _old(z_i,t) ). gathered (14) The likelihood ratio in Equation 6 evaluates the recorded perturbed scores zi,tz_i,t under both policies. Equation 4 then constructs si,ts_i,t deterministically from (zi,t,τi,t)(z_i,t, _i,t), so the ratio requires no separate density over soft states. C.2 Stop-Gradient Reconstruction for AdaptSoft Training AdaptSoft requires gradients through the step-specific temperature, but retaining the language-backbone computation graph for every rollout step would be memory intensive. We therefore store the projected hidden state, standardized entropy, and AdaptSoft output during rollout. Let xi,troll=[LN(hi,t)∥H^i,t],ui,troll=fϕroll(xi,troll)x_i,t^roll= [P\,LN(h_i,t)\; \|\; H_i,t ], u_i,t^roll=f_ _roll\! (x_i,t^roll ) (15) denote the stored input and output, where ϕroll _roll denotes the AdaptSoft parameters used to generate the rollout. During the update, AdaptSoft recomputes its output from the stored input using the current parameters ϕφ. The operator sg(⋅)sg(·) stops gradients through recorded values; the construction below preserves the rollout temperature in the forward pass while allowing gradients to reach ϕφ: u~i,t=fϕ(sg(xi,troll)),ui,tupd=sg(ui,troll)+u~i,t−sg(u~i,t),τi,tupd=τ0+Δtanh(ui,tupd),ui,tupd=ui,troll,τi,tupd=τi,troll(forward pass). gathered u_i,t=f_φ\! (sg(x_i,t^roll) ),\\ aligned u_i,t^upd&=sg(u_i,t^roll)+ u_i,t-sg( u_i,t), _i,t^upd= _0+ (u_i,t^upd),\\ u_i,t^upd&=u_i,t^roll, _i,t^upd= _i,t^roll (forward pass). aligned gathered (16) With the recorded perturbed scores, the forward pass reproduces the rollout mixture weights and, if token embeddings are unchanged, the rollout soft state. Gradients to ϕφ pass through the recomputed AdaptSoft output: ∇ϕui,tupd=∇ϕu~i,t,∇ϕτi,tupd=Δ[1−tanh2(ui,troll)]∇ϕu~i,t. _φu_i,t^upd= _φ u_i,t, _φ _i,t^upd= [1- ^2\! (u_i,t^roll ) ] _φ u_i,t. (17) Recomputing AdaptSoft from its stored, detached input gives the update a gradient path to ϕφ without another backbone pass to obtain hi,th_i,t or backpropagation through that stored state. C.3 Differentiable Reconstruction and AdaptSoft Gradient Propagation AdaptSoft must preserve the temperatures used to generate each rollout while retaining a gradient path to its current parameters. Let sg(⋅)sg(·) denote stop-gradient, and define the recorded AdaptSoft input as ξi,t=[LN(hi,t)∥H^i,t] _i,t=[PLN(h_i,t)\| H_i,t]. During rollout, AdaptSoft records ξi,t _i,t and its output ui,trollu_i,t^roll, produced using the rollout parameters ϕroll _roll. At update time, the output is recomputed using the current parameters and combined with its recorded value: u~i,t=fϕ(sg(ξi,t)),ui,tupd=sg(ui,troll)+u~i,t−sg(u~i,t),τi,tupd=τ0+Δtanh(ui,tupd). gathered u_i,t=f_φ\! (sg( _i,t) ), u_i,t^upd=sg\! (u_i,t^roll )+ u_i,t-sg\! ( u_i,t ),\\ _i,t^upd= _0+ \! (u_i,t^upd ). gathered (18) In the forward pass, u~i,t−sg(u~i,t)=0 u_i,t-sg( u_i,t)=0; hence ui,tupd=ui,trollu_i,t^upd=u_i,t^roll and τi,tupd=τi,troll _i,t^upd= _i,t^roll. The recorded perturbed scores and candidate token identities consequently reproduce the rollout temperature and mixture weights even if ϕφ has changed since rollout generation. The stop-gradient terms vanish in the backward pass, allowing the recomputed output to carry gradients to ϕφ. Together with the dependence of the mixture weights on temperature, this gives ∂τi,tupd∂ϕ ∂ _i,t^upd∂φ =Δ[1−tanh2(ui,troll)]∂u~i,t∂ϕ, = [1- ^2\! (u_i,t^roll ) ] ∂ u_i,t∂φ, (19) ∂pi,t,k∂τi,t ∂ p_i,t,k∂ _i,t =−pi,t,kτi,t2(zi,t,k−∑j=1||pi,t,jzi,t,j). =- p_i,t,k _i,t^2 (z_i,t,k- _j=1^|V|p_i,t,jz_i,t,j ). Gradients can therefore pass through the mixture weights, the resulting soft states, and the subsequent language-model computation to the step-specific temperatures and AdaptSoft parameters. Equation 18 uses the recorded rollout temperatures in the forward pass and the recomputed AdaptSoft output to update ϕφ. Differentiable Gradient Alignment. Let vi,tv_i,t denote the hidden representation provided to the language-model output layer, with logits ℓi,t=Wvi,t _i,t=Wv_i,t. The GRPO logit residual is detached when constructing the step-specific output-layer gradient. The reference gradient is similarly detached after aggregation over the reference steps ℛR included in ℒGRPOrefL_GRPO^ref: di,t=sg(∂ℒGRPO∂ℓi,t),gi,t=di,tvi,t⊤,Gref=sg(∑(j,s)∈ℛdj,svj,s⊤),αi,t=⟨gi,t,Gref⟩F=di,t⊤Grefvi,t,∂αi,t∂vi,t=Gref⊤di,t. gathered aligned d_i,t&=sg\! ( _GRPO∂ _i,t ),&g_i,t&=d_i,tv_i,t ,\\ G_ref&=sg\! ( _(j,s) d_j,sv_j,s ),& _i,t&= g_i,t,G_ref _F=d_i,t G_refv_i,t, aligned\\ ∂ _i,t∂ v_i,t=G_ref d_i,t. gathered (20) With di,td_i,t and GrefG_ref detached, the alignment score provides a first-order gradient through vi,tv_i,t to earlier temperatures, since vi,tv_i,t precedes si,ts_i,t. Gradient Scaling and Centering. We scale the alignment objective using a detached moving estimate of the root-mean-square alignment score. For the optimization steps T in each micro-batch, the per-worker values of qbq_b are averaged before updating mbm_b: qb q_b =1||∑(i,t)∈sg(αi,t)2, = 1|T| _(i,t) sg( _i,t)^2, mb m_b =βmb−1+(1−β)qb, =β m_b-1+(1-β)q_b, (21) κb _b =min(κmax,cscmb+εsc), = \! ( _ , c_scm_b+ _sc ), ℒASscaled _AS^scaled =κb||ℒAS=−κb||∑(i,t)∈αi,t. = _b|T|L_AS=- _b|T| _(i,t) _i,t. With β=0.99β=0.99, κmax=106 _ =10^6, csc=10−3c_sc=10^-3, and εsc=10−30 _sc=10^-30, the resulting temperature gradients are centered within each rollout and applied to AdaptSoft through a surrogate objective: gi,tτ=∂ℒASscaled∂τi,t,g¯iτ=1|i|∑t∈igi,tτ,g~i,tτ=sg(gi,tτ−g¯iτ),ℒ~AS(ϕ)=∑i∑t∈ig~i,tττi,tupd,∇ϕℒ~AS=∑i∑t∈ig~i,tτ∂τi,tupd∂ϕ. gathered aligned g_i,t^τ&= _AS^scaled∂ _i,t,& g_i^τ&= 1|T_i| _t _ig_i,t^τ,\\ g_i,t^τ&=sg\! (g_i,t^τ- g_i^τ ),& L_AS(φ)&= _i _t _i g_i,t^τ _i,t^upd, aligned\\ _φ L_AS= _i _t _i g_i,t^τ ∂ _i,t^upd∂φ. gathered (22) Centering removes the component shared by the temperature-gradient signals within a rollout, retaining their differences across steps. The reference-gradient factors are shared across workers, and AdaptSoft gradients are averaged before its optimizer step. Only the centered alignment gradient updates ϕφ: accumulated GRPO gradients on ϕφ are discarded, and the alignment computation does not accumulate gradients on θ. Appendix D Appendix: Benchmark and Evaluation Details D.1 Benchmarks We evaluate Soft Spatial Reasoning on OmniSpatial Jia et al. (2025), SpatiaLab Wasi et al. (2026), and MindCube Wang et al. (2026a). OmniSpatial provides the post-training and held-out evaluation splits; the other two benchmarks are used only for zero-shot evaluation. OmniSpatial Jia et al. (2025). OmniSpatial comprises more than 8.4K question–answer pairs spanning dynamic reasoning, spatial interaction, complex spatial logic, and perspective taking. These four dimensions contain 50 fine-grained task subcategories. We use its official 6,902-sample training split for post-training and its 1,533-sample held-out test split for evaluation. SpatiaLab Wasi et al. (2026). SpatiaLab comprises 1,400 question–answer pairs drawn from realistic, unconstrained scenes. Its six categories are relative positioning, depth and occlusion, orientation, size and scale, spatial navigation, and 3D geometry; each contains five task types. We evaluate in the multiple-choice setting. No SpatiaLab samples are used for post-training. MindCube Wang et al. (2026a). MindCube contains 21,154 questions across 3,268 images and examines spatial mental modeling from limited views. Its Rotation, Among, and Around settings test reasoning about spatial relationships as viewpoints change and objects become partially visible. We evaluate its multiple-choice questions without using MindCube samples for post-training. D.2 Evaluation protocol For all three benchmarks, a response is correct when its selected option matches the ground-truth answer. We report accuracy by category for OmniSpatial and SpatiaLab and by setting for MindCube. Overall accuracy for Soft Spatial Reasoning is computed over all evaluated questions, equivalently weighting each category or setting by its sample count.