One Student, Many Teachers: Multi-Task On-Policy Distillation via Soft-Prompt Privileged Context

# One Student, Many Teachers: Multi-Task On-Policy Distillation via Soft-Prompt Privileged Context
Source: [https://arxiv.org/html/2607.18293](https://arxiv.org/html/2607.18293)
Yingzi Ma1Zichen Zhu311footnotemark:1Ming Jiang1Chaowei Xiao2 1University of Wisconsin–Madison2Johns Hopkins University 3Nanyang Technological University

###### Abstract

On\-policy self\-distillation \(OPSD\) teaches large language models new skills through a teacher that shares the student’s backbone and supervises its own rollouts\. Existing teachers either inject privileged context at the input—inducing post\-hoc rationalization—or fine\-tune weights, accumulating drift and forgetting across tasks\. We proposePromptSD, whose teacher differs from the student only by a learnable soft prompt: trained on\(x,ygold\)\(x,y\_\{\\text\{gold\}\}\)pairs with the backbone frozen, the prompt yields a task\-specific teacher that preserves the student’s exact representational geometry\.PromptSDextends naturally to multi\-task settings by routing each example in a merged corpus to its corresponding soft\-prompt teacher, allowing a single student to absorb knowledge fromKKteachers in parallel; at inference, all prompts are discarded\. On Qwen3\-1\.7B\-Base and Phi\-4\-mini\-instruct across four tasks \(Science, Tool Use, Biology, Math\), the single\-task variant \(OPD with a PT teacher\) matches or exceeds full fine\-tuning while training orders of magnitude fewer parameters, and the multi\-task variant achieves the best overall average \(56\.256\.2on Qwen3\-1\.7B\-Base\) while preserving general\-capability benchmarks—in contrast to sequential SFT, which degrades both\.

One Student, Many Teachers: Multi\-Task On\-Policy Distillation via Soft\-Prompt Privileged Context

Yingzi Ma1††thanks:Equal contribution\. Corresponding toyma382@wisc\.edu, chaoweixiao@jhu\.edu\.Zichen Zhu311footnotemark:1Ming Jiang1Chaowei Xiao21University of Wisconsin–Madison2Johns Hopkins University3Nanyang Technological University

## 1Introduction

![Refer to caption](https://arxiv.org/html/2607.18293v1/x1.png)Figure 1:Three families of privileged\-information teachers for on\-policy distillation\.All three share the same on\-policy reverse\-KL objectiveℒ​\(θ\)=KL​\[πs∥πt\]\\mathcal\{L\}\(\\theta\)=\\mathrm\{KL\}\[\\pi\_\{s\}\\,\\\|\\,\\pi\_\{t\}\]over student rolloutsYsY\_\{s\}, but differ in how the teacher acquires privileged information\. \(a\)*Observation\-based*\(SDFT, OPSD, SDPO\): teacher and student share weightsπθ\\pi\_\{\\theta\}; the teacher is conditioned on extra input\-side contextcc\. \(b\)*RL\-tuned*\(CoPD\): the teacher is fine\-tuned with RLVR, drifting toπθ\+Δ​W\\pi\_\{\\theta\}\+\\Delta W\. \(c\)*Prompt\-tuned \(ours\)*: the teacher shares the student’s weights but prepends a learnable soft promptPPobtained via prompt tuning; onlyPPdiffers between teacher and student\.On\-policy distillation \(OPD\) has emerged as a core post\-training technique for large language models\(Yanget al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib5); Xiaoet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib7); Zenget al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib6)\): student rollouts paired with per\-token teacher distributions provide a denser supervision signal than off\-policy SFT\(Aroraet al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib8); Agarwalet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib34)\)\. Recent work realizes the teacherπT\\pi\_\{\\text\{T\}\}in four structurally distinct ways:*observation\-based*self\-teachers inject privileged contextccat the input—demonstrations\(Shenfeldet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib9)\), gold answers\(Zhaoet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib10)\), or verifier\-filtered rollouts\(Hübotteret al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib12)\);*RL\-tuned*teachers fine\-tune the base with verifiable rewards\(Guet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib11); Liet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib21)\);*cross\-size*teachers distill from a larger same\-family model\(Yanget al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib5); Agarwalet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib34)\); and*SFT\-tuned*teachers fine\-tune on\(x,ygold\)\(x,y\_\{\\text\{gold\}\}\)pairs\(Lab,[2025](https://arxiv.org/html/2607.18293#bib.bib37)\)\. Amid this rapid proliferation, a basic question is rarely asked:*which teacher should be paired with which task, and once we have a good teacher, why distill from it rather than deploy it directly?*

Each family answers this only partially\.Liet al\.\([2026](https://arxiv.org/html/2607.18293#bib.bib21)\)formalize two conditions for effective OPD:*thinking\-pattern consistency*on student\-visited states, and a*non\-trivial capability gap*\. While observation\-based teachers \(OPSD\) naturally preserve the first by sharing weights with the student, they face a structural risk we call*post\-hoc rationalization*\(Turpinet al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib23); Arcuschinet al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib24)\): conditioning the teacher on the gold answer leads the teacher to justify rather than derive its trace, a failure mode previously identified in CoT distillation as the “reasoning shortcut between question and gold answer”\(Wanget al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib36)\)\. RL\-based teachers face a different problem: they require verifiable rewards \(unavailable for many tasks\), and recent analysis\(Chenet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib22)\)argues that RL primarily amplifies existing capabilities rather than injecting new ones—a limitation visible in domains like vision\-language\-action modeling, where SFT cold\-start is a prerequisite for RL\(Wanget al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib20)\)\. We therefore pursue SFT\-based teacher construction\.

SFT methods span a spectrum of distributional perturbation: full fine\-tuning \(FFT\), LoRA\(Huet al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib2)\), adapters\(Houlsbyet al\.,[2019](https://arxiv.org/html/2607.18293#bib.bib18)\), and prompt tuning \(PT\)\(Lesteret al\.,[2021](https://arxiv.org/html/2607.18293#bib.bib3)\)\. PT occupies a uniquely favorable position for OPD: it trains a small set of continuous embeddings prepended to the input while leaving all transformer weights frozen\. As illustrated in Figure[1](https://arxiv.org/html/2607.18293#S1.F1), this places PT \(panel c\) at a distinct point in the design space relative to observation\-based \(panel a\) and RL\-tuned \(panel b\) teachers: the soft promptPPacts as a parametric form of observation, expressed in continuous embedding space rather than discrete tokens, while keeping the transformer weights identical to the student’s\. Against weight\-modifying methods \(FFT, LoRA\), PT preserves thinking\-pattern consistency and avoids the weight drift and “intruder dimensions” that compound across sequential tasks\(Bidermanet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib15); Shuttleworthet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib17)\)\. Against observation\-based teachers, PT’s soft prompt does not condition the teacher on any answer, eliminating post\-hoc rationalization\. This design also bounds catastrophic forgetting\(Kirkpatricket al\.,[2017](https://arxiv.org/html/2607.18293#bib.bib13); Luoet al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib14)\): by keeping the teacher distributionally close to the base, PT minimizes the per\-step KL signal driving student weight updates\.

In light of these observations, we proposePromptSD, an on\-policy distillation framework that uses a prompt\-tuned teacher and extends to multiple tasks via parallel routing\.PromptSDhas two components\. First, the*prompt\-tuned teacher*: for each taskkk, we attach a set of learnable continuous embeddingsPkP\_\{k\}to the frozen studentπθ\\pi\_\{\\theta\}and optimizePkP\_\{k\}on\(x,ygold\)\(x,y\_\{\\text\{gold\}\}\)via standard supervised fine\-tuning, yielding a task\-specific teacherπθ\+Pk⋆\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}that shares all transformer parameters with the student and differs only in input\-side conditioning\. Second,*parallel multi\-task distillation*: rather than training one student per task or chaining tasks sequentially, we merge all task datasets into a single corpus where each example retains a task tagkk, and at each student update we route the example to its corresponding teacherπθ\+Pk⋆\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}\. The student is updated via on\-policy reverse\-KL on its own rollouts, with the teacher’s supervision bounded by the prompt\-induced gap from the base\. At deployment we discard all soft prompts and returnπθ\\pi\_\{\\theta\}alone\.

We validatePromptSDthrough two sets of experiments on Qwen3\-1\.7B\-Base and Phi\-4\-mini\-instruct\. First, on single\-task distillation across four tasks \(Science, Tooluse, Biology, Math\), single\-task OPD with a PT teacher matches or exceeds SFT\-tuned teachers \(FFT, LoRA\), and is the only configuration whose average accuracy exceeds its teacher’s accuracy on a target task\. Second, multi\-taskPromptSDachieves the best overall average \(56\.2\\mathbf\{56\.2\}on Qwen3\-1\.7B\-Base,\+2\.3\+2\.3over the strongest single\-task baseline\) while simultaneously preserving general\-capability benchmarks \(MMLU\-ProWanget al\.\([2024](https://arxiv.org/html/2607.18293#bib.bib31)\), HellaSwagZellerset al\.\([2019](https://arxiv.org/html/2607.18293#bib.bib32)\), TruthfulQALinet al\.\([2022](https://arxiv.org/html/2607.18293#bib.bib33)\)\)—in contrast to sequential SFT, which loses both trained\-task accuracy and general capability to catastrophic forgetting\. Our analysis further yields two principles for OPD teacher design: teacher*family*should match the task type \(SFT for capability injection, RL for elicitation\), and teacher*scale*matters less than whether the teacher carries new knowledge\.

## 2Related Work

### 2\.1On\-Policy Distillation

On\-policy distillation \(OPD\) addresses the train–test mismatch of off\-policy knowledge distillation by training the student on its own rollouts under teacher per\-token supervision\. GKD\(Agarwalet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib34)\)interpolates between off\-policy and on\-policy mixtures and supports multiple divergence choices, while MiniLLM\(Guet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib35)\)minimizes reverse KL to prevent the student from overestimating the teacher’s low\-probability tails\. OPD has since been adopted in industrial post\-training pipelines \(Qwen3\(Yanget al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib5)\), MiMo\(Xiaoet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib7)\), GLM\-5\(Zenget al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib6)\)\) as a competitive complement to SFT and outcome\-reward RL\. A subsequent line of*self\-distillation*variants couples OPD with a single backbone serving as both teacher and student, where the teacher gains its informational advantage from privileged context: SDFT\(Shenfeldet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib9)\)prepends in\-context demonstrations, OPSD\(Zhaoet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib10)\)prepends gold reasoning traces, and SDPO\(Hübotteret al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib12)\)uses verifier\-filtered student rollouts as context\. CoPD\(Guet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib11)\)departs from this input\-side family by RLVR\-tuning the teacher’s weights\. Mechanism analyses byLiet al\.\([2026](https://arxiv.org/html/2607.18293#bib.bib21)\)identify thinking\-pattern consistency and a non\-trivial capability gap as the two governing conditions of successful OPD—a framework on which we directly build \(§[3](https://arxiv.org/html/2607.18293#S3)\)\. None of these methods construct the teacher via prompt tuning, and the parameter\-side overlay design we propose is, to our knowledge, the first to satisfy both conditions by construction\.

### 2\.2Post\-Training Techniques

Supervised fine\-tuning \(SFT\)\(Ouyanget al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib4)\)provides dense token\-level supervision from\(x,ygold\)\(x,y\_\{\\text\{gold\}\}\)pairs but trains exclusively on expert trajectories, suffering from*exposure bias*at inference\(Aroraet al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib8)\)and inducing substantial distributional drift that manifests as catastrophic forgetting\(Kirkpatricket al\.,[2017](https://arxiv.org/html/2607.18293#bib.bib13); Luoet al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib14)\)\. Parameter\-efficient variants such as LoRA\(Huet al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib2)\)mitigate but do not eliminate this drift, with recent analyses identifying “intruder dimension” artifacts that concentrate update mass far from the base manifold\(Shuttleworthet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib17); Bidermanet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib15)\)\. Reinforcement learning approaches train on student\-generated trajectories: RLHF\(Ouyanget al\.,[2022](https://arxiv.org/html/2607.18293#bib.bib4)\)learns a reward model from human preferences and optimizes it with PPO, while RLVR\(Guoet al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib38); Shaoet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib39)\)replaces the reward model with programmatic verifiers\. Both face structural limits in our setting: reward sparsity yields high\-variance trajectory\-level gradients, and recent analyses argue that RLVR primarily amplifies skills already latent in the base rather than injecting new ones\(Chenet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib22)\)—consistent with the stagnantΔ​ρ\\Delta\\rhowe observe for RLVR teachers in Figure[2](https://arxiv.org/html/2607.18293#S3.F2)\. Direct Preference Optimization\(Rafailovet al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib40)\)sidesteps reward modeling but is restricted to preference pairs and has been shown to induce implicit reward models with limited out\-of\-distribution generalization\. OPD is complementary to all three: it inherits SFT’s dense token\-level signal, RL’s on\-policy training distribution, and avoids verifier and preference\-annotation requirements\. Our work asks a sharper question—which teacher construction best preserves the conditions under which dense on\-policy supervision actually works\.

## 3Motivation

![Refer to caption](https://arxiv.org/html/2607.18293v1/x2.png)Figure 2:Performance gain vs\. overlap dynamics across teacher families\.For each teacher \(markers\), we plot the 1\.7B\-Base student’s task accuracy gain \(pp over zero\-shot\) against \(left of each pair\) the initial overlapρ\(0\)\\rho^\{\(0\)\}and \(right of each pair\) the overlap gainΔ​ρ=ρ\(S\)−ρ\(0\)\\Delta\\rho=\\rho^\{\(S\)\}\-\\rho^\{\(0\)\}, on Science and Tooluse\. Dashed lines indicate the linear fit\.*Initial overlap alone is not predictive*: high\-ρ\(0\)\\rho^\{\(0\)\}teachers likeRLVRandSDPOproduce near\-zero or negative gains, while moderate\-ρ\(0\)\\rho^\{\(0\)\}teachers likePT,FFT, andLoRAproduce the largest gains\.*Overlap gainΔ​ρ\\Delta\\rhois strongly predictive of student gain*\(right panels\): teachers occupying the moderate\-overlap regime \(D1\) and producing substantialΔ​ρ\\Delta\\rho\(D2\) consistently yield the best students\.### 3\.1Desiderata for a Suitable OPD Teacher

Building onLiet al\.\([2026](https://arxiv.org/html/2607.18293#bib.bib21)\)and our reading of OPD training dynamics, we identify three desiderata an OPD teacherπT\\pi\_\{\\text\{T\}\}should jointly satisfy with respect to a studentπs=πθ\\pi\_\{s\}=\\pi\_\{\\theta\}\. Letttindex the token position within a student rolloutYsY\_\{s\}on promptxx, and let

St\(p\)\\displaystyle S^\{\(p\)\}\_\{t\}=TopK\(πs\(⋅∣x,Ys<t\),k\),\\displaystyle=\\mathrm\{TopK\}\\\!\\left\(\\pi\_\{s\}\(\\cdot\\mid x,Y\_\{s\}^\{<t\}\),\\,k\\right\),St\(q\)\\displaystyle S^\{\(q\)\}\_\{t\}=TopK\(πT\(⋅∣x,Ys<t\),k\)\\displaystyle=\\mathrm\{TopK\}\\\!\\left\(\\pi\_\{\\text\{T\}\}\(\\cdot\\mid x,Y\_\{s\}^\{<t\}\),\\,k\\right\)\(1\)denote the student’s and teacher’s top\-kksets at positiontt\. The overlap ratio averages the intersection size over rollout positions and task prompts:

ρ=𝔼x,Ys∼πθ​\[1\|Ys\|​∑t=1\|Ys\|\|St\(p\)∩St\(q\)\|k\]\.\\displaystyle\\rho\\;=\\;\\mathbb\{E\}\_\{x,\\,Y\_\{s\}\\sim\\pi\_\{\\theta\}\}\\\!\\left\[\\,\\frac\{1\}\{\|Y\_\{s\}\|\}\\sum\_\{t=1\}^\{\|Y\_\{s\}\|\}\\frac\{\|S^\{\(p\)\}\_\{t\}\\cap S^\{\(q\)\}\_\{t\}\|\}\{k\}\\,\\right\]\.\(2\)We writeρ\(s\)\\rho^\{\(s\)\}for the overlap at OPD training stepss, withρ\(0\)\\rho^\{\(0\)\}the value at initialization andρ\(S\)\\rho^\{\(S\)\}at the end of training\. We reportk=10k\{=\}10throughout \(results fork∈\{1,5,50\}k\\\!\\in\\\!\\\{1,5,50\\\}in the appendix are qualitatively similar\)\.

\(D1\) Thinking\-pattern consistency\.The teacher’s top\-kkdistribution on student\-visited states must overlap withπθ\\pi\_\{\\theta\}enough that the reverse\-KL signal is absorbable, but not so much that no learning signal remains\. We call this a*moderate\-overlap regime*: whenρ\(0\)\\rho^\{\(0\)\}is too low, the gap falls outside the student’s absorbable range; whenρ\(0\)\\rho^\{\(0\)\}is too high,πT≈πθ\\pi\_\{\\text\{T\}\}\\approx\\pi\_\{\\theta\}and the per\-token KL gradient vanishes\. Formally,

ρmin<ρ\(0\)<ρmax\.\\displaystyle\\rho\_\{\\min\}\\;<\\;\\rho^\{\(0\)\}\\;<\\;\\rho\_\{\\max\}\.\(3\)
\(D2\) Capability gap absorbed by the student\.Beyond a moderate starting point, the teacher must encode task\-relevant knowledge thatπθ\\pi\_\{\\theta\}progressively absorbs\. We measure this through the*change*in overlap rather than benchmark accuracy: a real capability gap manifests as steadily risingρ\(s\)\\rho^\{\(s\)\}, while a teacher that merely amplifies existing behaviors\(Chenet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib22)\)leaves overlap stagnant despite higher accuracy\. We require

Δ:=ρ\(S\)−ρ\(0\)\>δ\.\\displaystyle\\Delta:=\\rho^\{\(S\)\}\-\\rho^\{\(0\)\}\\;\>\\;\\delta\.\(4\)
\(D3\) No post\-hoc rationalization\.The teacher’s conditioning context at distillation time should ideally not containygoldy\_\{\\text\{gold\}\}, as doing so leads the teacher to justify rather than derive its reasoning—a pattern the student inherits without access to the answer at inference\(Wanget al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib36)\)\. This is particularly acute whenygoldy\_\{\\text\{gold\}\}is only a short final answer: the teacher has no intermediate trace to follow and fabricates a derivation that lands on the predetermined answer, biasing token\-level distributions toward backward\-chained justification\.

### 3\.2Existing Teacher Families

We examine how six teacher families fare against D1–D3 by plotting each teacher’s student gain against its overlap dynamics \(ρ\(0\)\\rho^\{\(0\)\}andΔ​ρ\\Delta\\rho\) on Science and Tooluse \(Figure[2](https://arxiv.org/html/2607.18293#S3.F2)\)\.

Observation\-based teachersshare weights with the student \(D1 satisfied\) but placeygoldy\_\{\\text\{gold\}\}in the teacher’s input, violating D3—as a gold answer\(Zhaoet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib10)\), demonstrations as inSDFT\(Shenfeldet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib9)\), or verifier\-filtered rollouts\(Hübotteret al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib12)\)\.ICLyields*negative*Δ​ρ\\Delta\\rhoon Science: its logits depend on demonstration context the student never sees, compounded by SLMs’ weak in\-context learning\.GTandSDPOoccupy the high\-ρ\(0\)\\rho^\{\(0\)\}end of Figure[2](https://arxiv.org/html/2607.18293#S3.F2)\(≥85\\geq 85%\) yet produce near\-zero student gain—the gold\-conditioned teacher is already nearly identical to the student on student\-visited prefixes\.

RL\-tuned teacherssatisfy D3 but require task\-specific verifiers and primarily amplify existing behaviors rather than inject new ones\(Chenet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib22)\)\.RLVRsits in the high\-ρ\(0\)\\rho^\{\(0\)\}region with stagnantΔ​ρ\\Delta\\rho: the teacher’s distribution is already nearly fully covered by the student, so the reverse\-KL signal vanishes and D2 fails\.

Large\-sized teacherssatisfy D2 and D3 but suffer cross\-model parameterization mismatch—distinct pretraining trajectories and tokenization\-level idiosyncrasies place the teacher’s distribution off the student’s behavioral manifold\.Qwen3\-4BandQwen3\-8Bpopulate the left edge of theρ\(0\)\\rho^\{\(0\)\}plots \(≈55\\approx 55–6565%\) and produce small student gains, consistent with D1’s lower bound being violated\.

SFT\-tuned teachers and the SFT trap\.FFTandLoRAsatisfy D1–D3 and occupy the upper\-right corner of theΔ​ρ\\Delta\\rhoplots—largestΔ​ρ\\Delta\\rhoand largest student gains\. Yet the desiderata miss a*side effect*: the sameΔ​W\\Delta Wthat enables capability absorption \(D2\) also perturbs the teacher’s distribution on*out\-of\-task*inputs \(FFTbroadly;LoRAvia low\-rank “intruder dimensions”\(Shuttleworthet al\.,[2026](https://arxiv.org/html/2607.18293#bib.bib17)\)\), which the student inherits as catastrophic forgetting in sequential settings \(§[5\.3](https://arxiv.org/html/2607.18293#S5.SS3)\)\. Prompt tuning sits naturally between observation\-based teachers \(privileged contextccat the input, D3\-violating\) and SFT\-tuned teachers \(Δ​W\\Delta W\-driven drift\): the soft promptPPprovides a*learnable, parametric*privileged context while keepingΔ​W=0\\Delta W\{=\}0in the transformer body\. The teacherπθ\+P\\pi\_\{\\theta\+P\}thus satisfies D1–D3 without off\-manifold drift\. In Figure[2](https://arxiv.org/html/2607.18293#S3.F2),PTlands in the moderate\-ρ\(0\)\\rho^\{\(0\)\}region with the largest student gain—the only same\-base teacher achieving both D1 and D2 jointly\.

## 4PromptSD

PromptSDrealizes the teacher design motivated in §[3\.1](https://arxiv.org/html/2607.18293#S3.SS1): a soft\-prompt overlay on the frozen student backbone, supervising the student via on\-policy reverse\-KL distillation\. We describe the three components in turn: constructing the soft\-prompt teacher \(§[4\.1](https://arxiv.org/html/2607.18293#S4.SS1)\), the on\-policy distillation objective \(§[4\.2](https://arxiv.org/html/2607.18293#S4.SS2)\), and the parallel multi\-task extension \(§[4\.3](https://arxiv.org/html/2607.18293#S4.SS3)\)\.

### 4\.1Building a Soft\-Prompt Teacher

Given a base modelπθ\\pi\_\{\\theta\}and a task dataset𝒟=\{\(xi,yi\)\}i=1N\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\}, we construct a teacher by attachingLLcontinuous embeddingsP=\[p1,…,pL\]∈ℝL×dP=\[p\_\{1\},\\ldots,p\_\{L\}\]\\in\\mathbb\{R\}^\{L\\times d\}to the input embedding layer ofπθ\\pi\_\{\\theta\}, whereddis the hidden dimension\. For inputxxwith embedding sequenceE​\(x\)E\(x\), the teacher’s forward pass treatsPPas a virtual prefix:

πθ\+P\(⋅∣x\)=πθ\(⋅∣\[P;E\(x\)\]\),\\displaystyle\\pi\_\{\\theta\+P\}\(\\cdot\\mid x\)\\;=\\;\\pi\_\{\\theta\}\\\!\\left\(\\cdot\\mid\[P;E\(x\)\]\\right\),\(5\)where\[⋅;⋅\]\[\\,\\cdot\\,;\\,\\cdot\\,\]denotes concatenation along the sequence axis\. OnlyPPis trainable; all transformer weightsθ\\thetaremain frozen\. We optimizePPby minimizing the next\-token cross\-entropy on𝒟\\mathcal\{D\}:

ℒPT​\(P\)=−𝔼\(x,y\)∼𝒟​\[log⁡πθ\+P​\(y∣x\)\],\\displaystyle\\mathcal\{L\}\_\{\\text\{PT\}\}\(P\)\\;=\\;\-\\,\\mathbb\{E\}\_\{\(x,y\)\\sim\\mathcal\{D\}\}\\\!\\left\[\\log\\pi\_\{\\theta\+P\}\(y\\mid x\)\\right\],\(6\)yieldingP⋆P^\{\\star\}that encodes task knowledge inL×dL\\times dparameters—typically1010–100×100\\timessmaller than LoRA at standard rank settings—while leavingθ\\thetauntouched\.

### 4\.2On\-Policy Soft\-Prompt Distillation

Withπθ\+P⋆\\pi\_\{\\theta\+P^\{\\star\}\}fixed, we distill its capability intoπθ\\pi\_\{\\theta\}via on\-policy reverse\-KL distillation\. At each step, we samplex∼𝒟x\\sim\\mathcal\{D\}, generate a student rolloutYs∼πθ\(⋅∣x\)Y\_\{s\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\), and evaluate both models on each prefixYs<tY\_\{s\}^\{<t\}\. Let

pt​\(⋅\)\\displaystyle p\_\{t\}\(\\cdot\)≜πθ\(⋅∣x,Ys<t\),\\displaystyle\\;\\triangleq\\;\\pi\_\{\\theta\}\(\\cdot\\mid x,Y\_\{s\}^\{<t\}\),qt​\(⋅\)\\displaystyle q\_\{t\}\(\\cdot\)≜πθ\+P⋆\(⋅∣x,Ys<t\)\\displaystyle\\;\\triangleq\\;\\pi\_\{\\theta\+P^\{\\star\}\}\(\\cdot\\mid x,Y\_\{s\}^\{<t\}\)\(7\)denote the per\-step student and teacher distributions\. The student is updated to minimize

ℒOPD​\(θ\)=𝔼x,Ys∼πθ​\[∑t=1\|Ys\|KL​\(pt∥qt\)\]\.\\displaystyle\\mathcal\{L\}\_\{\\text\{OPD\}\}\(\\theta\)\\;=\\;\\mathbb\{E\}\_\{x,\\,Y\_\{s\}\\sim\\pi\_\{\\theta\}\}\\\!\\left\[\\sum\_\{t=1\}^\{\|Y\_\{s\}\|\}\\mathrm\{KL\}\(p\_\{t\}\\,\\\|\\,q\_\{t\}\)\\right\]\.\(8\)Crucially, both models receive the same promptxx—P⋆P^\{\\star\}acts as a parameter\-side overlay on the teacher rather than an input\-side context shift—so the teacher’s per\-token supervision is computed on exactly the states the student visits\. At inference,P⋆P^\{\\star\}is discarded and the deployed model isπθ\\pi\_\{\\theta\}alone, incurring no parameter or compute overhead relative to the base\.

![Refer to caption](https://arxiv.org/html/2607.18293v1/x3.png)Figure 3:Overview ofPromptSD\.\(a\)Task\-specific soft prompts\{P1⋆,…,PK⋆\}\\\{P\_\{1\}^\{\\star\},\\ldots,P\_\{K\}^\{\\star\}\\\}are trained in parallel on\{𝒟1,…,𝒟K\}\\\{\\mathcal\{D\}\_\{1\},\\ldots,\\mathcal\{D\}\_\{K\}\\\}via next\-token cross\-entropy \(Eq\.[6](https://arxiv.org/html/2607.18293#S4.E6)\); the base model is frozen\.\(b\)Datasets are merged with task tags; each example is routed to its soft\-prompt teacherπθ\+Pk⋆\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}, while the studentπθ\\pi\_\{\\theta\}rolls out on the input alone\. The student is updated via per\-token reverse KL on student\-visited states \(Eq\.[10](https://arxiv.org/html/2607.18293#S4.E10)\)\. At inference, soft prompts are discarded\.
### 4\.3Parallel Multi\-Task Distillation

PromptSDextends naturally to settings withKKtasks\{𝒯1,…,𝒯K\}\\\{\\mathcal\{T\}\_\{1\},\\ldots,\\mathcal\{T\}\_\{K\}\\\}, each with dataset𝒟k\\mathcal\{D\}\_\{k\}\. We first trainKKtask\-specific soft prompts\{P1⋆,…,PK⋆\}\\\{P\_\{1\}^\{\\star\},\\ldots,P\_\{K\}^\{\\star\}\\\}in parallel by applying Eq\.[6](https://arxiv.org/html/2607.18293#S4.E6)independently on each𝒟k\\mathcal\{D\}\_\{k\}; the prompts share no parameters and incur no inter\-task interference\. We then form a merged distillation corpus

𝒟merged=⋃k=1K\{\(x,y,k\):\(x,y\)∈𝒟k\},\\displaystyle\\mathcal\{D\}\_\{\\text\{merged\}\}=\\bigcup\_\{k=1\}^\{K\}\\\{\(x,y,k\):\(x,y\)\\in\\mathcal\{D\}\_\{k\}\\\},\(9\)in which each example retains a task tagkkidentifying its source\. At each distillation step, we sample\(x,y,k\)∼𝒟merged\(x,y,k\)\\sim\\mathcal\{D\}\_\{\\text\{merged\}\}, generateYs∼πθ\(⋅∣x\)Y\_\{s\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\), and update the student against the task\-routed teacherπθ\+Pk⋆\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}:

ℒOPDmulti​\(θ\)=𝔼\(x,y,k\),Ys​\[∑tKL​\(pt∥qt\(k\)\)\],\\displaystyle\\mathcal\{L\}\_\{\\text\{OPD\}\}^\{\\text\{multi\}\}\(\\theta\)\\;=\\;\\mathbb\{E\}\_\{\(x,y,k\),\\,Y\_\{s\}\}\\\!\\left\[\\sum\_\{t\}\\mathrm\{KL\}\(p\_\{t\}\\,\\\|\\,q\_\{t\}^\{\(k\)\}\)\\right\],\(10\)whereqt\(k\)\(⋅\)=πθ\+Pk⋆\(⋅∣x,Ys<t\)q\_\{t\}^\{\(k\)\}\(\\cdot\)=\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}\(\\cdot\\mid x,Y\_\{s\}^\{<t\}\)\. Task\-specific knowledge is routed through the corresponding soft prompt at supervision time but unified into shared parameters at the student\. Because allKKprompts are simultaneously available throughout training, this scheme is order\-independent and avoids the per\-task forgetting incurred by sequential continual learning—a property we examine empirically in §[5](https://arxiv.org/html/2607.18293#S5)\.

Table 1:Per\-method results on four independently fine\-tuned tasks \(Qwen3\-1\.7B\-Base; no sequential continual learning\)\.Acc= task accuracy \(%\)\.Δ​ρ\\Delta\\rhomeasures capability transfer:ρ\(S\)−ρ\(0\)\\rho^\{\(S\)\}\-\\rho^\{\(0\)\}in percentage points, whereρ\\rhois the mean top\-10 next\-token overlap between student and teacher over teacher\-forced validation rollouts\. PositiveΔ​ρ\\Delta\\rhomeans the student moved*toward*the teacher \(genuine absorption rather than amplifying prior behavior\)\. Best and second\-bestAccper column among same\-size students areboldedandunderlined\.∗denotes multi\-task learning\.MethodScienceTooluseBiologyMathAverageAccΔ​ρ\\Delta\\rhoAccΔ​ρ\\Delta\\rhoAccΔ​ρ\\Delta\\rhoAccΔ​ρ\\Delta\\rhoAccΔ​ρ\\Delta\\rhoBase modelZero\-Shot29\.3–7\.2–32\.0–16\.4–21\.2–Teacher upper boundsICL teacher35\.3–0\.0–38\.0–57\.0–32\.6–RLVR teacher30\.3–40\.2–32\.0–44\.2–36\.7–LoRA teacher46\.7–51\.5–26\.0–52\.8–44\.3–FFT teacher54\.3–51\.5–42\.0–48\.8–49\.2–PT teacher52\.7–47\.4–58\.0–48\.6–50\.4–Large\-sized cross\-size baselinesOPD \(Qwen\-4B\)23\.7\+8\.516\.5\+9\.320\.0\+0\.586\.4\+4\.336\.7\+5\.7OPD \(Qwen\-8B\)35\.3\+5\.153\.6\+7\.220\.0\+8\.385\.2\+1\.348\.5\+5\.5Observation\-based baselinesSDFT \(Demo\)34\.7\+2\.74\.1\+8\.230\.0−\-1\.2\\ul67\.0\+0\.634\.0\+2\.6OPSD \(GT\)30\.3\+2\.09\.3\+4\.830\.0\+6\.820\.6\+0\.322\.6\+3\.5SDPO \(Rollout\)8\.0−\-0\.53\.1−\-0\.932\.0\+3\.916\.4−\-3\.214\.9−\-0\.2RL\-tuned baselinesOPD \(RLVR\)31\.3\+4\.039\.2\+11\.4\\ul42\.0\+8\.641\.0\+2\.938\.4\+6\.7SFT\-tuned baselinesOPD \(LoRA\)49\.3\+17\.318\.6\+12\.228\.0\+30\.155\.4\+13\.037\.8\+18\.2OPD \(FFT\)\\ul52\.0\+17\.755\.7\+17\.038\.0\+29\.647\.0\+13\.148\.2\+19\.4OPD \(PT\)51\.0\+9\.9\\ul51\.5\+9\.962\.0\+12\.051\.0\+8\.1\\ul53\.9\+10\.0PromptSD∗55\.3\+8\.848\.4\+12\.9\\ul54\.0\+4\.267\.2\+8\.156\.2\+8\.5

## 5Experiment

### 5\.1Experimental Setup

Models\.We useQwen3\-1\.7B\-Base\(Yanget al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib5)\)\(pretrained, no instruction tuning\) as the student backbone throughout, so that all teacher\-induced gains are isolated from prior RLHF biases\. The same backbone is used to construct every same\-size teacher:PTprependsLLlearnable virtual\-token embeddings to the input embedding sequence \(§[4\.1](https://arxiv.org/html/2607.18293#S4.SS1)\);LoRAuses rankr=16r\{=\}16on\{q,k,v,o\}\\\{q,k,v,o\\\}projections;FFTupdates all transformer weights;RLVRfine\-tunes the teacher with GRPO\(Shaoet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib39)\)using each task’s verifier as the reward signal;ICLprovidesK=3K\{=\}3in\-context demonstrations sampled from the train split;GTwraps the gold answer in a fixed prefix template\. For cross\-size baselines we use Qwen3\-4B and Qwen3\-8B, both of which share the Qwen3 tokenizer with the 1\.7B student so that token\-level logits are directly comparable\.

Table 2:Trained\-task performance vs\. general\-capability retention\(Qwen3\-1\.7B\-Base\)\.Avg\-Taverages accuracy \(%\) over the four target tasks;Avg\-Uaverages over three held\-out 5\-shot utility benchmarks\.MethodScienceTooluseBiologyMathAvg\-TMMLU\-ProHellaSwagTruthfulQAAvg\-UBase29\.37\.232\.016\.421\.235\.865\.550\.050\.4SFT \(sequential\)45\.758\.844\.029\.044\.429\.762\.546\.446\.2OPD \(PT, sequential\)49\.345\.460\.053\.051\.934\.566\.952\.651\.3SFT \(multi\-task\)51\.353\.638\.032\.043\.734\.865\.552\.751\.0OPD \(LoRA, multi\-task\)51\.751\.534\.053\.047\.535\.662\.856\.851\.7PromptSD\(multi\-task\)55\.348\.454\.067\.256\.236\.668\.150\.451\.7

Datasets and Metrics\.We evaluate on four tasks spanning complementary capabilities:Science\(4\-way MCQ with reasoning tagsShenfeldet al\.\([2026](https://arxiv.org/html/2607.18293#bib.bib9)\),n=300n\{=\}300\),Tooluse\(ReAct against Stable\-Diffusion tools\(Patilet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib30)\),n=97n\{=\}97\),Biology\(SciKnowEval\(Fenget al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib29)\),n=50n\{=\}50\), andMath\(MATH\-500\(Hendryckset al\.,[2021](https://arxiv.org/html/2607.18293#bib.bib25)\),n=500n\{=\}500\)\. Each task has its own train/val/test split; soft prompts and on\-policy rollouts use the training split, validation drives early stopping, and all reported numbers are on test\. We report task\-specific accuracy with greedy decoding via vLLM\(Kwonet al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib28)\)\. Further details in the appendix\.

Implementation\.Phase A \(soft\-prompt training\):20002000steps, AdamW lr5×10−35\{\\times\}10^\{\-3\}, cosine schedule, effective batch size1616; only theL×dL\\times dprompt embeddings are updated\. Phase B \(on\-policy distillation\):600600steps, AdamW lr1×10−51\{\\times\}10^\{\-5\}, with the reverse\-KL objective in Eq\.[8](https://arxiv.org/html/2607.18293#S4.E8); the rollout buffer is refreshed every6464steps via vLLM\. For the multi\-task variant \(§[4\.3](https://arxiv.org/html/2607.18293#S4.SS3)\), the merged corpus shuffles all four tasks, and each example is routed to its task\-specific frozen teacher\. All experiments run on44NVIDIA H100 80GB GPUs; a single\-task pipeline takes11–22GPU\-hours, full main results∼\\sim3030GPU\-hours\. Further details in the appendix\.

### 5\.2Main Results

![Refer to caption](https://arxiv.org/html/2607.18293v1/x4.png)Figure 4:Soft\-prompt lengthLL\(number of learnable virtual tokens in the PT teacher\), across all four tasks\.Single\-Task Distillation\.Table[1](https://arxiv.org/html/2607.18293#S4.T1)compares all teacher families on Qwen3\-1\.7B\-Base\.*The teacher upper bounds reveal a task\-type contrast: SFT\-tuned teachers excel when new knowledge must be injected, while observation\- and RL\-based teachers compete when capability is already latent and only needs elicitation\.*On Science, Tooluse, and Biology, SFT\-tuned teachers \(PT50\.450\.4, FFT49\.249\.2\) clearly outperform RLVR \(36\.736\.7\) and ICL \(32\.632\.6\); on Math, the gap narrows—ICL leads \(57\.057\.0\) and SFT teachers fall to4848–5353\. Among same\-size students,OPD \(PT\)achieves the best average \(53\.9\\mathbf\{53\.9\}\) and reaches62\.0\\mathbf\{62\.0\}% on Biology—\+4\+4above its own teacher and\+34\+34aboveOPD \(LoRA\)—as the PT teacher injects capability without the off\-manifold drift that caps SFT\-tuned students\.OPD \(FFT\)trails PT on average \(48\.248\.2\) and itsΔ​ρ≈\+18\\Delta\\rho\{\\approx\}\{\+\}18on every task signals heavy teacher alignment, foreshadowing the catastrophic forgetting reported in §[5\.3](https://arxiv.org/html/2607.18293#S5.SS3)\.*Notably, what matters is not teacher scale but whether it carries new knowledge*: cross\-sizeOPD \(Qwen\-8B\)dominates only on Math \(85\.285\.2\) where Qwen3\-1\.7B\-Base is limited, but trailsOPD \(PT\)on the other three tasks\.

Multi\-TaskPromptSD\.PromptSDextendsOPD \(PT\)by distilling four task\-specific PT teachers into a*single*student via merged\-corpus routing \(§[4\.3](https://arxiv.org/html/2607.18293#S4.SS3)\)\.PromptSD∗achieves the best average \(56\.2\\mathbf\{56\.2\}\), exceeding the strongest single\-task baseline by\+2\.3\+2\.3and cross\-sizeOPD \(Qwen\-8B\)by\+7\.7\+7\.7\. The gain is most pronounced on Math \(67\.267\.2vs\.51\.051\.0,\+16\.2\+16\.2over single\-task PT\) and Science \(55\.355\.3, the table’s best\)\. Multi\-task exposure thus provides positive cross\-task transfer: jointly training on Science, Tooluse, and Biology rollouts reinforces the reasoning and instruction\-following Math latently requires, letting the student elicit capability beyond any single\-task teacher\. PT teachers compose cleanly—four parametric privileged contexts absorbed by one student without interference\.

### 5\.3Continual Learning

Per\-task SFT \(or RLVR\) suffices when adapting to a single domain in isolation—SFT injects new knowledge for capability\-heavy tasks like Science or Biology, and RLVR sharpens reasoning where the capability is already latent\. Realistic deployment, however, is continual: a model must absorb new domains over time without erasing prior capabilities\. Table[2](https://arxiv.org/html/2607.18293#S5.T2)reports trained\-task accuracy \(Avg\-T\) and held\-out utility \(Avg\-U, over MMLU\-Pro, HellaSwag, TruthfulQA\) across sequential and multi\-task baselines\. Sequential SFT fails on both fronts—44\.444\.4Avg\-Tand a drop from the base’s50\.450\.4to46\.246\.2Avg\-U\(MMLU\-Pro−6\.1\-6\.1\)—the SFT trap of §[3\.2](https://arxiv.org/html/2607.18293#S3.SS2)realized at the student level: cumulativeΔ​W\\Delta Wacross tasks shifts the student off\-manifold on unrelated inputs\. Sequential OPD \(PT\) already mitigates this \(51\.951\.9/51\.351\.3\) because the PT teacher’s bounded per\-task perturbation \(D1\) leaves general capability intact, justifying OPD over deploying the teacher directly\. Switching to a multi\-task schedule alone does not close the gap: multi\-task SFT and OPD \(LoRA\) recover Avg\-U \(51\.051\.0,51\.751\.7\) but stall at43\.743\.7and47\.547\.5Avg\-T, indicating that the Avg\-T bottleneck is the adaptation mechanism, not the schedule\.PromptSDimproves both axes simultaneously \(56\.2\\mathbf\{56\.2\}Avg\-T,51\.7\\mathbf\{51\.7\}Avg\-U\), with the largest gain on Math \(67\.267\.2vs\. sequential PT53\.053\.0, sequential SFT29\.029\.0\), where cross\-task PT exposure unlocks latent capability\. The simultaneous Avg\-U lift confirms that parametric privileged contexts compose without cumulative drift—PromptSDdelivers task specialization*and*preserves general capability\.

### 5\.4Ablation Study

Soft Prompt Length\.Figure[4](https://arxiv.org/html/2607.18293#S5.F4)ablates the number of learnable virtual tokensLLin the PT teacher across all four tasks\. No singleLLdominates:L=400L\{=\}400is best \(or tied for best\) on Science, Tooluse, and Math, while Biology shows a U\-shaped response peaking at bothL=100L\{=\}100andL=400L\{=\}400\(62\.062\.0\) with a dip atL=200L\{=\}200\. The general pattern is that more learnable parameters help capacity\-demanding tasks: Math gains\+13\+13points fromL=100L\{=\}100toL=400L\{=\}400, and Tooluse jumps from30\.930\.9to51\.551\.5onceLLreaches400400\. BeyondL=400L\{=\}400, gains flatten or invert \(Math51\.0→29\.051\.0\\to 29\.0atL=800L\{=\}800\), suggesting diminishing returns\. We adoptL=400L\{=\}400as the default in all main\-results experiments as a balance between expressiveness and stability\.

Divergence Choice\.Table[3](https://arxiv.org/html/2607.18293#S5.T3)compares four divergences in the OPD loss with a PT teacher: forward KL \(mean\-seeking\), reverse KL \(mode\-seeking\)\(Guet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib35)\), symmetric KL\(Wuet al\.,[2025](https://arxiv.org/html/2607.18293#bib.bib26)\), and Jensen–Shannon\(Agarwalet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib34)\)\. On Science, Tooluse, and Biology, all four fall within≤\\leq66points of each other—divergence choice is largely immaterial for capability\-heavy tasks\. The exception is Math, where reverse\-KL substantially outperforms the alternatives \(51\.051\.0vs\.1313–1818for the others\), consistent with its mode\-seeking behavior concentrating mass on the teacher’s high\-probability solution tokens rather than averaging across competing trajectories\. We adopt reverse\-KL as the default for this Math advantage and for consistency with the OPSD literature\.

Table 3:Divergence choicein the on\-policy distillation loss \(PT teacher\)\. We compare reverse\-KL \(mode\-seeking\), forward\-KL \(mean\-seeking\), symmetric KL, and Jensen–Shannon \(JSD\)\.DivergenceScienceTooluseBiologyMathForward50\.048\.462\.013\.0Symmetric49\.049\.564\.018\.0JSD47\.051\.558\.014\.0Reverse51\.051\.560\.051\.0

Different Backbones\.To confirm that our findings generalize beyond Qwen3\-1\.7B\-Base, Table[4](https://arxiv.org/html/2607.18293#S5.T4)repeats the main comparison onPhi\-4\-mini\-instruct\(Abdinet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib27)\), and three observations carry over\. First,PromptSDachieves the best overall average \(56\.4\\mathbf\{56\.4\}\),*matching the FFT teacher upper bound itself*\(56\.456\.4\) while outperforming all same\-size single\-task baselines includingOPD \(FFT\)\(55\.355\.3\) andOPD \(LoRA\)\(51\.551\.5\)\. Second, the task\-type contrast persists: SFT\-tuned teachers dominate capability\-heavy tasks \(FFT teacher reaches52\.052\.0on Biology vs\. RLVR’s36\.036\.0\), while observation\- and RL\-based teachers lead on Math, where the base is already strong \(62\.062\.0forOPSD \(GT\)andOPD \(RLVR\), vs\. FFT teacher’s60\.060\.0\)\. Third, the SFT trap remains visible:OPD \(FFT\)falls short of its teacher on three of four tasks \(Science, Tooluse, Biology\), capped at the teacher’s ceiling\. By contrast,PromptSDinherits no such cap and reaches the teacher’s overall accuracy\. Notably, the single\-task PT teacher is the relatively weaker teacher among these on this backbone \(avg51\.351\.3\), yetPromptSD’s multi\-task framework still recovers the strongest student \(56\.456\.4\)—showing that the merged\-corpus routing of §[4\.3](https://arxiv.org/html/2607.18293#S4.SS3)extracts transferable signal even when individual teachers are imperfect\.

Table 4:Per\-method results on four independently fine\-tuned tasks based onPhi\-4\-mini\-instruct\.MethodScienceTooluseBiologyMathAverageBase modelZero\-Shot33\.746\.422\.055\.239\.3Teacher upper boundsRLVR teacher39\.346\.436\.058\.044\.9LoRA teacher52\.054\.632\.057\.249\.0FFT teacher53\.759\.852\.060\.056\.4PT teacher41\.753\.151\.059\.251\.3Observation\-based baselinesSDFT \(Demo\)37\.045\.434\.057\.843\.6OPSD \(GT\)39\.346\.422\.062\.042\.4SDPO \(Rollout\)45\.252\.838\.658\.248\.7RL\-tuned baselinesOPD \(RLVR\)40\.045\.424\.062\.042\.9SFT\-tuned baselinesOPD \(LoRA\)\\ul50\.760\.834\.060\.651\.5OPD \(FFT\)53\.3\\ul58\.8\\ul48\.061\.2\\ul55\.3OPD \(PT\)45\.351\.540\.0\\ul61\.449\.6PromptSD49\.760\.856\.059\.256\.4

## 6Conclusion

We introducedPromptSD, an on\-policy distillation framework built around a soft\-prompt teacher\. Existing teacher families each satisfy only a subset of the three desiderata for effective OPD—thinking\-pattern consistency, an absorbed capability gap, and no post\-hoc rationalization—with SFT\-tuned teachers in particular suffering an off\-manifold “SFT trap” that caps the student and induces catastrophic forgetting\. Prompt tuning resolves these trade\-offs through a learnable, parametric privileged context that preserves the student’s representational geometry while encoding new task knowledge\.PromptSDdistills multiple task\-specific PT teachers into a single student that outperforms task\-specialized counterparts while preserving general capability, and yields two principles for OPD teacher design: teacher*family*should match the task type \(SFT for capability injection, RL for elicitation\), and teacher*scale*matters less than whether the teacher carries new knowledge\.

## 7Limitations

Our study is limited in several respects\. Experiments use Qwen3\-1\.7B\-Base and Phi\-4\-mini\-instruct as students with same\- or near\-size teachers, leaving the behavior of PT teachers at larger scales \(7B–70B\) as an open question\. Our four tasks span knowledge MCQ, tool calling, and math reasoning, but do not yet cover open\-ended generation, multilingual, agentic, or multimodal settings, where the capability\-injection vs\. elicitation distinction may take different forms\. The multi\-taskPromptSDsetup combines four tasks; further scaling to many task\-specific PT teachers may surface interference between privileged contexts that the bounded\-perturbation argument \(D1\) does not explicitly address\. We also note that D1–D3 are operationally defined and empirically validated rather than formally proven, and alternative constructions \(e\.g\., rank\-constrained adapters\) may satisfy them as well\. A natural next step is to extendPromptSDbeyond PT teachers to a broader family of plug\-and\-play modules—such as LoRA adapters, prefix tuners, and task vectors—as interchangeable privileged\-context providers, enabling modular composition of capabilities at distillation time\.

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Table 5:Tasks and split sizes\. Science/Math targets contain native step\-by\-step reasoning; Biology and Tool\-use targets are annotated with reasoning \(text below\)\.TaskSource / formatTrainValTestScience4\-way MCQ,<reasoning\>/<answer\>2,474200507Tool useReAct tool calling, Gorilla toolset\(Patilet al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib30)\)3,84620097BiologySciKnowEval Biology 4\-MCQ\(Fenget al\.,[2024](https://arxiv.org/html/2607.18293#bib.bib29)\)4054550MathLIMO solutions \(train\) / MATH\-500 test\(Hendryckset al\.,[2021](https://arxiv.org/html/2607.18293#bib.bib25)\)80013500## Appendix AOverview

This appendix collects reproducibility details and four additional analyses supporting our claims aboutPromptSD\.

- •Details of experiments \(§[B](https://arxiv.org/html/2607.18293#A2)\)\.Datasets and splits, metrics, implementation, and the fullPromptSDconfiguration \(soft\-prompt width, refresh schedule\(K,M\)\(K,M\), multi\-task routing\)\.
- •Efficiency \(§[C](https://arxiv.org/html/2607.18293#A3)\)\.Teacher input\-token and trainable\-parameter footprint ofPromptSDversus observation\- and SFT\-based teachers\.
- •Statistical significance \(§[D](https://arxiv.org/html/2607.18293#A4)\)\.Three\-seed mean±\\pmstd on Qwen3\-1\.7B and Phi\-4\-mini, with paired bootstrap tests and95%95\\%CIs on the small splits\.
- •Post\-hoc rationalization \(§[E](https://arxiv.org/html/2607.18293#A5)\)\.A counterfactual answer\-swap probe quantifying desideratum D3: gold\-conditioned teachers follow an injected wrong answer, while the soft\-prompt teacher cannot be steered\.
- •PT teacher vs\. multi\-task exposure \(§[F](https://arxiv.org/html/2607.18293#A6)\)\.A leave\-one\-teacher\-out ablation separating competence*taught*by a task’s own teacher from that*elicited*by co\-training\.
- •Case studies \(§[G](https://arxiv.org/html/2607.18293#A7)\)\.Side\-by\-side base / PT\-teacher /PromptSDtraces, including failure cases\.

## Appendix BDetails of Experiments

### B\.1Datasets

Table[5](https://arxiv.org/html/2607.18293#A0.T5)lists the four tasks \(§[5\.1](https://arxiv.org/html/2607.18293#S5.SS1)\) and their splits\. Each task has its own train/val/test split: soft prompts \(Phase A\) and on\-policy rollouts \(Phase B\) are drawn from train, val drives early stopping, and all reported numbers are on test \(evaluated onn=300/97/50/100n\{=\}300/97/50/100items for Science/Tool use/Biology/Math\)\. The three general\-capability benchmarks used for the retention analysis \(Table[2](https://arxiv.org/html/2607.18293#S5.T2)\) are500500\-item subsets of MMLU\-Pro and HellaSwag and the full TruthfulQA\-MC1 \(817817items\), all evaluated55\-shot since the non\-instruction\-tuned base cannot follow the multiple\-choice format zero\-shot\.

Reasoning\-target annotation\.The Science and Math training targets carry genuine step\-by\-step reasoning natively \(per\-option rationales and long\-form solutions, respectively\)\. The original Biology and Tool\-use targets do not: Biology supplies only a placeholder rationale \(“the correct option is X”\) and Tool uses only bare action calls\. Because on\-policy distillation from a teacher trained on answer\-only targets collapses the student’s chain\-of\-thought into restating the answer \(Appendix[E](https://arxiv.org/html/2607.18293#A5)\), we annotate accurate, answer\-consistent reasoning for the Biology and Tool\-use training targets with Qwen3\-235B\-A22B111[https://huggingface\.co/Qwen/Qwen3\-235B\-A22B](https://huggingface.co/Qwen/Qwen3-235B-A22B)as an external annotator: the model is given the question and the*verified*gold answer, and asked to produce a faithful derivation, after which the gold answer/action is re\-attached deterministically, so target–answer consistency holds by construction\. Validation and test splits are left unchanged\.

### B\.2Metrics

Task accuracy and format\.We reportAcc\(correctness\) with task\-specific answer extraction\.*Science / Biology*\(MCQ\): we read the letter inside<answer\>\.\.\.</answer\>—falling back to the last standaloneA–Dtoken if the tag is malformed—and compare to the gold option; Fmt checks that a well\-formed answer tag is emitted\.*Tool use*: we parse everyAction:/Action Input:pair and count a generation correct iff the multiset of action names matches the gold*and*the merged JSON arguments match exactly\.*Math*: we extract the final answer from\\boxed\{\}\(or<answer\>\) and score by mathematical equality after numeric/symbolic normalization\(Hendryckset al\.,[2021](https://arxiv.org/html/2607.18293#bib.bib25)\); Fmt checks answer\-tag presence\. All task evaluations use greedy decoding via vLLM\(Kwonet al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib28)\)withmax\_new\_tokensset to1024/1536/2048/163841024/1536/2048/16384for Science/Tool use/Biology/Math, sized to fit each task’s full reasoning trace\.

General\-capability \(utility\)\.MMLU\-Pro and HellaSwag are scored by multiple\-choice accuracy; TruthfulQA\-MC1 uses accuracy with the answer options*shuffled per example*\(the released order always places the correct option first, which otherwise rewards positional bias\)\. For Qwen3\-1\.7B\-Base model, all three are evaluated55\-shot\.

Teacher–student overlap\.For the transfer analysis we use the top\-kknext\-token overlapρ\\rho\(§[3\.1](https://arxiv.org/html/2607.18293#S3.SS1)\): the mean over teacher\-forced validation positions of\|𝒯k∩𝒮k\|/k\|\\mathcal\{T\}\_\{k\}\\cap\\mathcal\{S\}\_\{k\}\|/kbetween the teacher’s and student’s top\-kknext\-token sets \(k=10k\{=\}10\)\.Δ​ρ=ρ\(S\)−ρ\(0\)\\Delta\\rho=\\rho^\{\(S\)\}\-\\rho^\{\(0\)\}measures how far the student moves toward the teacher over training\.

### B\.3Implementation

Hardware and software\.Each run uses a single NVIDIA A100 \(40 GB\), L40S \(48 GB\), or H100 \(80 GB\) GPU\. Training uses PyTorch with HuggingFace Transformers; the on\-policy rollout buffer is generated with vLLM\(Kwonet al\.,[2023](https://arxiv.org/html/2607.18293#bib.bib28)\)\(bf16, eager mode\)\. Optimization uses AdamW \(no weight decay\) with a cosine schedule and3%3\\%warm\-up, and gradient clipping at1\.01\.0\(0\.30\.3for the higher\-LR multi\-task runs\)\. Gradient checkpointing is on throughout, and during each rollout refresh the optimizer state is offloaded to CPU to free memory for vLLM; the Phi\-4\-mini full\-fine\-tuning student requires the8080GB H100 \(its3\.83\.8B optimizer state does not fit on4040–4848GB cards\)\.

Hyperparameters\.Soft prompts are initialized by tiling the task\-description tokens \(no random padding\), trained with the backbone frozen, and selected by validation loss; the on\-policy buffer is refreshed every6464student steps with a reverse\-KL objective and an entropy bonus of0\.020\.02\(0\.010\.01for Science/Math\)\. Table[6](https://arxiv.org/html/2607.18293#A2.T6)gives the per\-task settings: defaults follow §[5](https://arxiv.org/html/2607.18293#S5), and we tune the soft\-prompt widthLLand the Phase\-B learning rate / step count on validation—knowledge\-heavy Biology and the structured Tool\-use task benefit from a wider prompt \(L=800L\{=\}800\), and Biology from a higher Phase\-B learning rate\. The multi\-taskPromptSDtrains for24002400steps at learning rate3×10−53\{\\times\}10^\{\-5\}, with Biology up\-weighted2×2\\timesin the routing mix\.

Table 6:Per\-task hyperparameters\. Phase A trains the soft\-prompt teacher; Phase B is on\-policy distillation of the student\.PromptPhase APhase BTaskLL\(lr, steps\)\(lr, steps,max\_new\)Science4005×10−35\{\\times\}10^\{\-3\}, 20001×10−51\{\\times\}10^\{\-5\}, 600, 1024Tool use8005×10−35\{\\times\}10^\{\-3\}, 20001×10−51\{\\times\}10^\{\-5\}, 600, 768Biology8005×10−35\{\\times\}10^\{\-3\}, 20002×10−52\{\\times\}10^\{\-5\}, 1000, 640Math4005×10−35\{\\times\}10^\{\-3\}, 20001×10−51\{\\times\}10^\{\-5\}, 600, 4096

### B\.4Details ofPromptSD

Two\-phase instantiation \(frozen teacher\)\.Our experiments use a two\-phase protocol with a*frozen*teacher: Phase A trains the soft\-prompt teacher\(s\) on the frozen base to convergence \(Table[6](https://arxiv.org/html/2607.18293#A2.T6)\), and Phase B performs on\-policy distillation against these frozen teachers\. Within Phase B only the on\-policy*rollout buffer*is regenerated \(every6464student steps\); the soft promptsPk⋆P\_\{k\}^\{\\star\}are*not*re\-adapted to the updating student, and no EMA is used\. At inference the soft prompts are discarded and onlyπθ\\pi\_\{\\theta\}is deployed\.

Multi\-task routing\.The merged corpus \(Eq\.[9](https://arxiv.org/html/2607.18293#S4.E9)\) is routed at the*sample*level: each rollout refresh draws a fixed number of prompts per task \(a base of3232by default\), rolls them out from the current student, and supervises each against its own frozen soft\-prompt teacherπθ\+Pk⋆\\pi\_\{\\theta\+P\_\{k\}^\{\\star\}\}\. The per\-refresh sampling ratios for the reported multi\-task students are given in Table[7](https://arxiv.org/html/2607.18293#A2.T7), expressed relative to this3232\-rollout base\. On both backbones we down\-weight the Math teacher to0\.5×0\.5\\times: Math reasoning traces are substantially longer than those of the other three tasks, so sampling Math at the base rate inflates per\-refresh compute and crowds out the gradient signal from Science, Tool use, and Biology within the same student step budget\. Halving the Math rollout count restores a roughly balanced per\-task token budget without removing the Math teacher from the routing mix\. On Qwen3\-1\.7B\-Base we additionally up\-weight Biology to2×2\\times, since it is the smallest split \(n=50n\{=\}50test\) and the most knowledge\-dense; on Phi\-4\-mini\-instruct, Biology’s zero\-shot baseline is already lower \(22\.0 vs\. 32\.0 on Qwen3\), and we observed no benefit from up\-weighting on the validation split, so we leave it at1×1\\times\. These ratios were fixed before the final main\-table runs based on a small grid on the validation splits \(Appendix[F](https://arxiv.org/html/2607.18293#A6)\)\. TheKKsoft prompts are fully independent \(L×dL\\times dparameters each, with no shared dimensions\)\.

Table 7:Per\-refresh task\-sampling ratio of the multi\-task student on each backbone\.BackboneScienceTool useBiologyMathQwen3\-1\.7B\-Base1120\.5Phi\-4\-mini\-instruct1110\.5

Task composition ofPromptSD\.The reportedPromptSDtrains four task\-specific soft\-prompt teachers \(Science, Tool use, Biology, Math\) in parallel and routes the merged corpus among them with the ratios in Table[7](https://arxiv.org/html/2607.18293#A2.T7)\. At inference all four soft prompts are discarded, and onlyπθ\\pi\_\{\\theta\}is deployed\.

## Appendix CEfficiency Discussion

Trainable parameters\.The soft\-prompt teacher trains only theL×dL\\times dprefix embeddings \(d=2048d\{=\}2048for Qwen3\-1\.7B\), making PT one to two orders of magnitude smaller than the other same\-size teachers \(Table[8](https://arxiv.org/html/2607.18293#A3.T8)\):0\.050\.05–0\.10%0\.10\\%of the backbone atL∈\{400,800\}L\\in\\\{400,800\\\}, versus0\.37%0\.37\\%for LoRA \(r=16r\{=\}16on\{q,k,v,o\}\\\{q,k,v,o\\\}\) and100%100\\%for FFT—about44–8×8\\timesfewer parameters than LoRA and∼103×\\sim\\\!10^\{3\}\\timesfewer than FFT\.

Teacher conditioning cost\.The teacher families differ in how much context each teacher forward must process per instance\. PT prepends a*fixed*LL\-token soft prefix \(L=400L\{=\}400\), independent of the instance, so its teacher input isL\+\|x\|L\+\|x\|\. GT is the most token\-frugal—it appends only the gold*answer*\(we strip the gold rationale\),\|x\|\+\|yans\|\|x\|\+\|y\_\{\\text\{ans\}\}\|with\|yans\|≈5\|y\_\{\\text\{ans\}\}\|\\\!\\approx\\\!5—but pays the post\-hoc\-rationalization cost \(§[E](https://arxiv.org/html/2607.18293#A5)\)\. The costly families are demonstration\- and rollout\-based: ICL/SDFT\-Demo process\|x\|\+3​\|exemplar\|\|x\|\+3\\,\|\\text\{exemplar\}\|\(∼1\.5\\sim\\\!1\.5–22K tokens\) and SDPO processes\|x\|\+\|trajectory\|\|x\|\+\|\\text\{trajectory\}\|, both growing with example length and becoming impractical on long\-context tasks—on Math \(mean\|y\|≈7\.2\|y\|\\\!\\approx\\\!7\.2K tokens\) ICL demonstrations must be length\-trimmed to fit the window, whereas PT’s prefix stays at a constant400400\. PT thus bounds the per\-instance teacher context independently of task length, avoiding the ICL/SDPO blow\-up without GT’s answer leakage\.

Training and inference cost\.The soft prompt is discarded after training, so the deployed student is exactly the base architecture—no added parameters, prefix, or latency\. Cross\-size distillation likewise discards the teacher at inference, but during*training*it must host and run forward passes on a22–5×5\\timeslarger model \(Qwen3\-4B/8B\) for every rollout\-time logit computation; PT keeps all forward passes at base scale throughout\. \(Trainable\-parameter count does not by itself imply faster wall\-clock: PT still runs a full\-model forward for the frozen backbone, so its training time is comparable to LoRA/FFT—its savings are in parameters and storage, below\.\)

Table 8:Trainable\-parameter footprint of same\-size teachers \(Qwen3\-1\.7B backbone\)\.TeacherTrainable params% of backbonePT \(L=400L\{=\}400\)0\.820\.82M0\.0480\.048PT \(L=800L\{=\}800\)1\.641\.64M0\.0950\.095LoRA \(r=16r\{=\}16\)6\.46\.4M0\.3730\.373FFT1\.721\.72B100100

Multi\-teacher storage\.ForKKtasks, PT storage scales as∑kLk​d\\sum\_\{k\}L\_\{k\}\\,d: the four teachers inPromptSDtotal≈4\.9\\approx\\\!4\.9M parameters \(L∈\{400,800\}L\\in\\\{400,800\\\}\), versus≈25\.6\\approx\\\!25\.6M forK=4K\{=\}4LoRA\-r=16r\{=\}16adapters and≈6\.9\\approx\\\!6\.9B forK=4K\{=\}4FFT copies\. This linear\-in\-KK, near\-zero\-overhead footprint is what makes the parallel multi\-task routing \(§[4\.3](https://arxiv.org/html/2607.18293#S4.SS3)\) practical, even though all prompts are discarded at student inference\.

## Appendix DStatistical Significance Analysis

Multi\-seed\.Table[9](https://arxiv.org/html/2607.18293#A4.T9)reports single\-task OPD\-PT over three random seeds on both backbones\. Run\-to\-run variance is small on the structured Science and Tool\-use tasks \(std≤2\\leq\\\!2on Phi\-4\-mini,≤4\\leq\\\!4on Qwen3\-1\.7B\) and larger on then=50n\{=\}50Biology split and the high\-variance Math task; the single\-seed numbers reported in the main tables lie within one standard deviation of these means\. \(Phi\-4\-mini Math is low and unstable—the PT\-math distillation transfers poorly to this backbone, which we note as a limitation rather than smooth over\.\)

Table 9:Single\-task OPD\-PT accuracy over33seeds \(mean±\\pmstd\)\.BackboneScienceTool useBiologyMathQwen3\-1\.7B\-Base51\.3±0\.951\.3\{\\pm\}0\.949\.8±3\.949\.8\{\\pm\}3\.956\.7±4\.256\.7\{\\pm\}4\.250\.7±2\.350\.7\{\\pm\}2\.3Phi\-4\-mini\-instruct45\.5±2\.045\.5\{\\pm\}2\.051\.4±1\.151\.4\{\\pm\}1\.140\.3±3\.140\.3\{\\pm\}3\.159\.5±2\.1\\phantom\{0\}59\.5\{\\pm\}2\.1

Table 10:Counterfactual answer\-swap follow\-rate \(%\): fraction of teacher generations whose final answer matches an injected*wrong*reference answer\. Higher==more post\-hoc rationalization \(D3 violation\)\. Chance baseline is\(1−a\)/\(C−1\)\(1\{\-\}a\)/\(C\{\-\}1\)withaathe teacher’s task accuracy\.ScienceBiologyTeacherFollow\-rateChanceFollow\-rateChanceGT \(gold in context\)59\.159\.113\.313\.362\.562\.514\.014\.0PT \(soft prompt; no answer slot\)15\.915\.915\.815\.812\.512\.514\.014\.0

Paired bootstrap and confidence intervals\.To test whether the headline gaps survive on the small test splits, we comparePromptSD\(OPD\-PT\) against the strongest SFT teacher \(OPD\-FFT\) by a paired bootstrap over test examples \(50005000resamples\) on Qwen3\-1\.7B\. OnBiology\(n=50n\{=\}50\),PromptSDreaches60\.060\.0\(95%95\\%CI\[46,74\]\[46,74\]\) versus38\.038\.0for OPD\-FFT and exceeds it in99%99\\%of resamples \(p≈0\.01p\\\!\\approx\\\!0\.01\)—a significant advantage despite the small split, consistent with the SFT\-trap mechanism \(§[5](https://arxiv.org/html/2607.18293#S5)\)\. OnTool usethe two are statistically indistinguishable \(overlapping95%95\\%CIs;P​\(PromptSD\>FFT\)=0\.08P\(\\text\{\{PromptSD\}\}\{\>\}\\text\{FFT\}\)\{=\}0\.08\), i\.e\. no same\-size teacher dominates that task\. Together with the small Science/Tool\-use seed variance, this indicates thatPromptSD’s advantages are driven by genuine, repeatable differences—most decisively on Biology—rather than by seed or test\-set sampling\.

## Appendix EAnalysis of Post\-hoc Rationalization

Desideratum D3 \(§[3\.1](https://arxiv.org/html/2607.18293#S3.SS1)\) requires that the teacher not condition on the gold answer: a gold\-conditioned teacher*justifies*a predetermined answer rather than*deriving*it, and the student—which has no answer at inference—inherits this backward\-chained pattern through reverse\-KL on per\-token teacher distributions\. We make D3 operational with a*counterfactual answer\-swap*probe\.

Probe\.For each evaluation instance\(x,ygold\)\(x,y\_\{\\text\{gold\}\}\)we replace the gold answer in the teacher’s context with a randomly chosen*wrong*answery~≠ygold\\tilde\{y\}\\\!\\neq\\\!y\_\{\\text\{gold\}\}, let the teacher generate a full trace under greedy decoding, and measure the*follow\-rate*: the fraction of cases whose generated answer equals the injectedy~\\tilde\{y\}\. A teacher who derives their answer should largely ignore the injection; a teacher that rationalizes will reproduce it\. The PT teacher has no answer slot in its input and therefore cannot be steered\. its follow\-rate is expected to coincide with the chance\-level baseline\(1−a\)/\(C−1\)\(1\{\-\}a\)/\(C\{\-\}1\), whereaais the teacher’s task accuracy andCCis the number of answer options\. We run the probe on Science \(4\-way MCQ,n=44n\{=\}44\) and Biology \(discrete\-answer subset of SciKnowEval with verified distractors,n=40n\{=\}40\); answer matching is exact string match after normalization\.

Results\.Table[10](https://arxiv.org/html/2607.18293#A4.T10)shows a∼4\\sim\\\!4–5×5\\timesgap between gold\-conditioned and prompt\-tuned teachers\. On Science, the PT teacher’s15\.9%15\.9\\%follow\-rate is statistically indistinguishable from the chance baseline\(1−0\.527\)/3≈15\.8%\(1\{\-\}0\.527\)/3\\\!\\approx\\\!15\.8\\%implied by its task accuracya=52\.7%a\{=\}52\.7\\%: PT exhibits*no steering*by the injected wrong answer\. The GT teacher’s59\.1%59\.1\\%, by contrast, exceeds the analogous chance baseline by∼46\\sim\\\!46pp—a direct quantitative magnitude of rationalization above the random\-error floor\. The Biology pattern is identical at the proportions reported\.

Interpretation\.This is direct evidence that PT satisfies D3 by construction while observation\-based teachers violate it\. It complements the smallΔ​ρ\\Delta\\rhoof GT teachers in the main results \(Table[1](https://arxiv.org/html/2607.18293#S4.T1): Science\+2\.0\+2\.0, Math\+0\.3\+0\.3, both substantially below PT’s\+9\.9\+9\.9and\+8\.1\+8\.1\): the gold\-conditioned signal is both unhelpful \(little new knowledge transferred\) and structurally biased—the teacher’s per\-token distribution reflects backward chaining fromygoldy\_\{\\text\{gold\}\}, which the student inherits via reverse\-KL despite never seeingygoldy\_\{\\text\{gold\}\}at inference\. Appendix[G](https://arxiv.org/html/2607.18293#A7)shows representative traces in which the GT teacher visibly back\-chains to the injected answer \(e\.g\., generating “option A is correct because \[…\]; this rules out B, C, D” whenygold=y\_\{\\text\{gold\}\}\{=\}“C” has been swapped toy~=\\tilde\{y\}\{=\}“A”\)\.

Coverage\.The D3 failure mode is, by the analysis in §[3\.1](https://arxiv.org/html/2607.18293#S3.SS1), most acute whenygoldy\_\{\\text\{gold\}\}is a short final answer with no intermediate trace—the canonical case being mathematical reasoning, where the teacher must fabricate a derivation landing on the predetermined numerical answer\. Probing Math under the same protocol requires a distractor distribution over plausible wrong numerical answers \(rather than the discrete option sets used here\) and a matcher robust to equivalent numerical forms; we leave this to future work and note it as the most important extension of the probe\.

## Appendix FPT teacher vs\. multi\-task exposure

A natural concern withPromptSDis whether each trained\-task gain comes from*that task’s*PT teacher or merely from co\-training on the merged corpus\. We isolate this with a*leave\-one\-teacher\-out*ablation on Qwen3\-1\.7B\-Base: for one task at a time we remove both its soft\-prompt teacher and its training data, train on the remaining three, and evaluate the held\-out task—so any non\-trivial accuracy must come from cross\-task exposure alone\.

Table 11:Leave\-one\-teacher\-out on Qwen3\-1\.7B\-Base\. For each row the held\-out task’s teacher*and*data are removed; “cross\-task only” is the held\-out accuracy reached by training on the other three tasks alone\.Held\-out taskCross\-task onlyPromptSDOPD \(PT\)BaseScience34\.334\.355\.355\.351\.051\.029\.329\.3Tool use7\.27\.248\.448\.451\.551\.57\.27\.2Biology34\.034\.054\.054\.062\.062\.032\.032\.0Math61\.0\\mathbf\{61\.0\}67\.267\.251\.051\.016\.416\.4

Only math is elicited\.Table[11](https://arxiv.org/html/2607.18293#A6.T11)shows a sharp split between tasks where the dedicated PT teacher is the dominant contributor and tasks where it is not\. Removing the Science, Tool\-use, or Biology teacher drops that task by2020–4141pp—Tool use collapses*exactly*to its zero\-shot base \(7\.27\.2\), and Science and Biology fall to roughly22–55pp above base\. These knowledge/format tasks are carried by their own PT teacher and are*not*recoverable from the others\. Math is the lone exception—with no math teacher and no math data the student still reaches61\.061\.0, far above the base \(16\.416\.4\),*above*the dedicated single\-task math PromptSD \(51\.051\.0, Table[1](https://arxiv.org/html/2607.18293#S4.T1)\), and only6\.26\.2pp below the fullPromptSD\(67\.267\.2\)\. Math is also the only row in which “cross\-task only” beats single\-task PT—a sign that for this task the dedicated math teacher is not just dispensable but actively limiting\. Math behaves as a latent pretraining capability that on\-policy co\-training on other reasoning tasks*surfaces*, rather than a skill that must be distilled\.

Quantifying per\-teacher contribution\.Reading the gap between “cross\-task only” and the fullPromptSDcolumn directly attributes each teacher’s marginal value within the multi\-task student: Science teacher\+21\.0\+21\.0pp, Tool\-use teacher\+41\.2\+41\.2pp, Biology teacher\+20\.0\+20\.0pp, math teacher\+6\.2\+6\.2pp\. The math teacher’s contribution is an order of magnitude smaller than the others\. A soft\-prompt math teacher on a1\.71\.7B base is itself a limited mathematician \(Math accuracy48\.648\.6, Table[1](https://arxiv.org/html/2607.18293#S4.T1)\), and on\-policy distillation toward it adds only a small residual on top of what cross\-task elicitation already provides\. The teacher’s distribution acts as a soft anchor, not a hard prerequisite\. This is why the reportedPromptSD*down\-weights*math to0\.5×0\.5\\times\(Table[7](https://arxiv.org/html/2607.18293#A2.T7)\) rather than emphasizing it: we keep the math teacher in the routing mix for the residual it contributes, but do not credit it as the primary mechanism behind the67\.267\.2Math score\.

Multi\-task interference on knowledge/format tasks\.A side observation from the fourth column: on Science, Tool use, and Biology, the dedicated single\-task PT student is*higher*than the fullPromptSD\(51\.0\>55\.351\.0\{\>\}55\.3fails, but51\.5\>48\.451\.5\{\>\}48\.4on Tool use and62\.0\>54\.062\.0\{\>\}54\.0on Biology hold\)\. Multi\-task routing buys an average improvement and an elicited Math boost at the cost of small per\-task regressions on the capability\-injection tasks—an accepted trade\-off in our setting, but one worth flagging: a deployment that cares about a single capability\-injection task is better served by single\-task OPD\-PT than byPromptSD\.

## Appendix GCase Study

We ground the analysis on a single representative Science item—a molecular logD multiple\-choice question \(gold answer D\)—tracing it through the base model before and afterPromptSDtraining, and through the two teachers that explain the difference\.

Question\(Science/molecular\-propertyMCQ\):

Whatisthecorrectoctanol/waterdistributioncoefficientlogD

underthecircumstanceofpH7\.4forthemolecule

"COc1cc2ncnc\(Nc3cccc\(Cl\)c3F\)c2cc1CN1C\[C@H\]\(OC\)C\[C@@H\]1C\(N\)=O"?

A:3\.2B:2\.3C:\-0\.51D:3\.09\(gold\)

Outputformatrequired:

<reasoning\>\.\.\.</reasoning\>

<answer\>X</answer\>\(Xin\{A,B,C,D\}\)

### G\.1Before vs\. after training

Before training, the base model fails the task outright: it emits a degenerate, off\-language fragment, produces no<reasoning\>/<answer\>structure, and lands on the most hydrophilic \(and incorrect\) option\. AfterPromptSDtraining, the*same*backbone—nowπθ\\pi\_\{\\theta\}, with no soft prompt and no answer in context—produces a well\-formed, structure\-grounded derivation: it reasons about the chlorine, fluorine, and the nitrogen heterocycle, considers the ionization state at pH 7\.4, explicitly rules out the hydrophilic distractor \(−0\.51\-0\.51\), and commits to the correct answer D\. The gain is not memorized formatting: it is a genuine chain of reasoning the base model could not previously externalize\.

Basemodel,BEFOREtraining\.No<answer\>tag;degenerate\(namesC,wrong;goldD\)

\[verbatimoutput\]"Answer:C:\-0\.51"

Student\\method\*\(ours\),AFTERtraining\.Generatedanswer:D\(correct\)

<reasoning\>

Theoctanol/waterdistributioncoefficient,logD,isameasureofhowa

compounddistributesitselfbetweenoctanolandwater,takingintoaccount

itsionizationstateatagivenpH\.Themoleculeinquestionisacomplex

organiccompoundwithmultiplefunctionalgroups,includingachlorine

atom,afluorineatom,andanitrogen\-containingheterocycle\.AtpH7\.4,

whichisclosetophysiologicalpH,theionizablegroupsinthemolecule

willbeintheirneutralform,affectingthelogDvalue\.Thepresenceof

chlorineandfluorineatoms,whicharegenerallyhydrophobic,will

contributetoahigherlogDvalue,whilethenitrogen\-containing

heterocyclemayhavesomeionizablegroupsthatcouldslightlyaffectthe

logD\.Giventheoptions,alogDvalueof3\.2or3\.09seemsplausible,as

thesevaluesaretypicalformoderatelyhydrophobiccompoundswith

ionizablegroups\.Thevalueof\-0\.51suggestsamorehydrophilic

compound,whichislesslikelygiventhepresenceofhydrophobicgroups\.

Therefore,themostreasonablechoiceis3\.09\.

</reasoning\>

<answer\>

D

</answer\>