arXiv:2512.07805v5 Announce Type: replace-cross Abstract: We present GRAPE (Group Representational Position Encoding), a unified framework for positional encoding based on group actions. GRAPE unifies two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in $\operatorname{SO}(d)$ and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group $\mathrm{GL}$. In Multiplicative GRAPE, a position $n \in \mathbb{Z}$ (or $t \in \mathbb{R}$) acts as $\mathbf{G}(n) = \exp(n \, \omega \, \mathbf{L})$ with a rank-2 skew-symmetric generator $\mathbf{L} \in \mathbb{R}^{d \times d}$, yielding a relative, compositional, norm-preserving map with a closed-form matrix exponential. RoPE is recovered exactly when the $d/2$ planes correspond to canonical coordinate pairs with a log-uniform spectrum. Learned commuting subspaces and compact non-commuting mixtures strictly extend this geometry to capture cross-subspace feature coupling at $O(d)$ and $O(r d)$ cost per head, respectively. In Additive GRAPE, additive logits arise from rank-1 (or low-rank) unipotent actions, recovering ALiBi and the Forgetting Transformer (FoX) as exact special cases while preserving an exact relative law and streaming cacheability. Overall, GRAPE provides a principled design space for positional geometry in long-context models, subsuming RoPE and ALiBi as special cases. Project page: https://github.com/model-architectures/GRAPE.

Background: Evidence-based physician-pharmacist collaborative clinics have demonstrated significant short-term benefits for patients with type 2 diabetes (T2D), but their long-term effectiveness remains unclear, especially in primary health care settings. Objective: This study aimed to explore the long-term effectiveness and cost-effectiveness of a novel, digital-driven, multifaceted physician-pharmacist collaborative model for managing patients with T2D in underresourced settings. Methods: We conducted a 12-month cluster randomized controlled trial from May 2021 to December 2022 across 6 primary health care settings in China. Guided by the theory of planned behavior, the intervention involved routine therapy from physicians along with pharmaceutical interventions from pharmacists. These were delivered through a combination of face-to-face visits and mobile health care. The intervention group received 4 face-to-face visits and biweekly remote education sessions over the 12 months. We conducted intention-to-treat analyses to estimate differences in clinical and behavior indicators between the intervention and control groups. Primary outcomes included glycosylated hemoglobin and 10-year atherosclerotic cardiovascular risk. Data were analyzed using adjusted generalized estimation equations. Results: This study included 574 patients (291 in the intervention group and 283 in the control group). Over 12 months, patients in the intervention group had significant reductions in hemoglobin A_1c (–2.57 vs –1.96, respectively; P<.001; 95% CI –1.027 to –0.238) and 10-year atherosclerotic cardiovascular risk (–1.35 vs 0.01, respectively; P<.001; 95% CI –1.690 to –0.630) compared with the control group. Substantial improvements were also observed in several secondary outcomes, including fasting blood glucose, 2-hour postprandial blood glucose, waist circumference, waist-to-hip ratio, blood pressure, triglyceride, and total cholesterol. Total diabetes-related costs decreased, and patient satisfaction improved significantly in the intervention group. There were no significant differences in BMI, high-density lipoprotein, or low-density lipoprotein. Conclusions: These findings suggest that the physician-pharmacist collaborative model could improve the long-term quality and efficiency of T2D management and reduce medical costs in underresourced areas globally. Patients with T2D, especially those with central obesity or high cardiovascular risk, may benefit more from collaborative clinics. Trial Registration: Chinese Clinical Trial Registry ChiCTR2000031839; https://www.chictr.org.cn/showproj.html?proj=51910

arXiv:2512.07805v4 Announce Type: replace-cross Abstract: We present GRAPE (Group Representational Position Encoding), a unified framework for positional encoding based on group actions. GRAPE unifies two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in $\operatorname{SO}(d)$ and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group $\mathrm{GL}$. In Multiplicative GRAPE, a position $n \in \mathbb{Z}$ (or $t \in \mathbb{R}$) acts as $\mathbf{G}(n) = \exp(n \, \omega \, \mathbf{L})$ with a rank-2 skew-symmetric generator $\mathbf{L} \in \mathbb{R}^{d \times d}$, yielding a relative, compositional, norm-preserving map with a closed-form matrix exponential. RoPE is recovered exactly when the $d/2$ planes correspond to canonical coordinate pairs with a log-uniform spectrum. Learned commuting subspaces and compact non-commuting mixtures strictly extend this geometry to capture cross-subspace feature coupling at $O(d)$ and $O(r d)$ cost per head, respectively. In Additive GRAPE, additive logits arise from rank-1 (or low-rank) unipotent actions, recovering ALiBi and the Forgetting Transformer (FoX) as exact special cases while preserving an exact relative law and streaming cacheability. Overall, GRAPE provides a principled design space for positional geometry in long-context models, subsuming RoPE and ALiBi as special cases. Project page: https://github.com/model-architectures/GRAPE.