On-Policy Self-Distillation in Diffusion Models

1 Introduction

• Endpoint rewards must be translated into actionable intermediate supervision, and finite fitting can under-realize, rotate, or overshoot a locally useful target because the model is queried repeatedly across noise levels.Separating target construction from finite realization helps distinguish poor reward guidance from imperfect realization by the model update.Source: §1 · p. 4
• Unlike FlowGRPO’s trajectory-credit estimation [34], ReFL’s late-state clean-output update [62], and DiffusionNFT’s endpoint reweighting [75], DiffusionOPSD fits explicit reward-improving intermediate targets.Source: §1 · p. 4
• DiffusionOPSD uses frozen behavior-policy trajectories and clean-output anchors, constructs bounded positive and negative targets via reward ascent and descent, fits detached supervision, then refreshes behavior by EMA.Stop-gradient separation allows construction gain and realized gain to be measured at the same query under the same fixed suffix.Source: §1 · pp. 4–5
• DiffusionOPSD achieves the best held-out score in 19 of 20 reward-matched settings, gains up to 44.0% over the strongest competing method, and cuts DiffusionNFT training GPU-hours by 40% and 63%.The comparisons span SD3.5-M and Z-Image-Turbo across ten evaluators.Source: §1 · pp. 4–5
• One jointly trained DiffusionOPSD policy outperforms the corresponding DiffusionNFT policy on PickScore, CLIPScore, and HPSv2.1 while retaining most gains from separate specialists.Source: §1 · p. 4
• A larger construction gain can yield a smaller realized gain after one fitting update without cross-query interference; reward-gradient direction explains most construction gain, while forward-noised query controls have little effect in the evaluated low-noise setting.Source: §1 · pp. 4–5

2.1 Reward-to-Target Interfaces

• Likelihood/preference objectives [2] [14] [30] [59] [65], policy gradients [34] [63] [64], differentiable rewards [10] [62], and DiffusionNFT [75] leave intermediate denoising changes implicit, unlike DiffusionOPSD’s explicit targets.Likelihood-ratio variants are also sensitive to likelihood estimation, sampler choice, and discretization [7] [27].Source: §2.1 · p. 5

2.2 On-Policy Distillation

• On-policy imitation reduces distribution shift [47] [61], and recent distillation matches external-teacher predictions [15] [28] [29] [33] [76]; DiffusionOPSD instead rewards targets anchored to its own behavior policy.Traditional approaches include progressive distillation, consistency training, distribution matching, and adversarial objectives [18] [39] [49] [50] [58] [68].Source: §2.2 · p. 5

2.3 Reward Gradients

• Unlike score-composition guidance [11] [19] or direct reward-gradient updates [10] [45] [62], DiffusionOPSD uses reward gradients to construct bounded, detached targets at behavior queries.Control-, score-, and value-based formulations also link reward gradients to policy improvement [16] [36] [74]; infinitesimal positive-target fitting shares direct reward ascent’s first-order direction, but finite optimization differs and targets are rebuilt as behavior changes.Source: §2.3 · pp. 5–6, Figure 3 · p. 6

3.1 Clean-Output Prediction

• At a fixed query, the velocity prediction defines a clean-output prediction that can be decoded and scored by the local reward; equal clean-output radii require larger velocity displacements at lower noise levels.The local neighborhood is defined around the clean-output anchor at a fixed query, while conditioning varies with noise level.Source: §3.1 · p. 6

3.2 Query Collection and Anchor

• A frozen behavior policy supplies trajectory-derived query states and clean-output anchors, and remains frozen during target construction and finite fitting.The method is on-policy because each query comes from a trajectory generated by the current outer iteration’s behavior policy.Source: §3.2 · p. 7

3.3 Target Construction

• Reward-gradient steps construct positive and negative targets around the clean-output anchor, with projection keeping their displacements within a trust-region radius.The positive target is intended to improve reward; the negative target is fitted through a separate branch as a repulsive reference.Source: §3.3 · p. 9

3.4 Finite Fitting

• Finite fitting uses two implicit branches to attract the shared clean-output prediction toward the positive target and reject the negative target, fitting detached supervision rather than optimizing to convergence.At a fixed query, the clean-output objective is equivalent to weighted velocity-MSE terms; the finite fitting operator includes the optimizer, parameterization, and batching procedure.Source: §3.4 · pp. 9–10, §3.5 · p. 10

3.5 Online Training

• Each behavior-policy update triggers new trajectories and rebuilt queries, anchors, weights, and targets for the next round of finite fitting.The trainable policy fits the current detached targets while the updated behavior policy supplies the queries and anchors used to construct the next set.Source: §3.5 · p. 11

4.2 Training Efficiency

• 40% and 63% lower training cost than DiffusionNFT are reported for SD3.5-M and Z-Image-Turbo, respectively.DiffusionOPSD uses 28.2 and 149.8 GPU-hours per 100 updates on the two backbones. At matched cost, it reaches the highest-quality point on nine of ten SD3.5-M frontiers and every Z-Image-Turbo frontier, although ReFL uses fewer GPU-hours on Z-Image-Turbo.Source: §4.2 · p. 12, Figure 8 · p. 15

4.3 Optimization Regimes

• 98% is DiffusionOPSD’s median terminal position within each single-reward run’s observed native-reward range, versus 97% for ReFL, 90% for DiffusionNFT, and 83% for FlowGRPO.On native nine-step Z-Image-Turbo, DiffusionOPSD leads all ten reward-matched comparisons; a shared 300-update policy also scores 25.51 on PickScore, 0.333 on CLIPScore, and 0.389 on HPSv2.1, exceeding DiffusionNFT on all three. The joint policy retains 113.0%, 92.9%, and 99.6% of the separate specialists’ gains over the base model, respectively.Source: Figure 5 · p. 12, §4.3 · p. 13, Figure 6 · p. 13

4.1 Main Results

• 19 of 20 reward-matched settings have the best final held-out score with DiffusionOPSD across SD3.5-M and Z-Image-Turbo and ten evaluators.It leads nine of ten SD3.5-M comparisons and all ten Z-Image-Turbo comparisons; SD3.5-M Aesthetic is the exception, at 12.08 versus ReFL’s 12.09.Source: §4.1 · p. 12, Figure 8 · p. 15

4.4 Target Construction

• 0.03511 fixed-suffix reward gain comes from the positive reward-gradient target, while the DiffusionNFT rollout endpoint changes reward by −0.03551 at the same query.After 50 updates, canonical targets reach CLIPScore 0.3122 versus 0.2363 for no-op, 0.2303 for matched-radius random-direction, and 0.1256 for rollout-residual targets. A forward-noised query changes the score only to 0.3089, a 0.0033 gap.Source: §4.4 · p. 16, Figure 9 · p. 16

4.5 Finite Fitting

• 62.3% of prompts reverse the HPSv2.1 target-construction ordering after one fitting update, showing that a better target need not yield a better finite update.The random target realizes 0.000719 more reward despite a 0.03496 worse construction gain; on CLIPScore the reversal rate is 29.5%. In a separate calibrated plain-SGD probe, ReFL realizes gain 0.0005805 versus 0.0004588 for bounded DiffusionOPSD, while DiffusionNFT has −0.0000333.Source: §4.5 · pp. 16–17, Figure 10 · p. 17

4.6 Qualitative Results

• 64%, 71%, 90%, and 61% of annotators’ preferences favor DiffusionOPSD over the base model, FlowGRPO, DiffusionNFT, and ReFL, respectively.Its reward-matched VLM-Pointwise score is 0.243 versus 0.227 for ReFL. Across fixed held-out prompts, qualitative comparisons report more consistent preservation of requested text, motion, object identity, composition, and fine detail.Source: §4.6 · p. 17, Figure 11 · p. 18, Figure 12 · p. 19

4.7 Ablations

• 0.3117 is the canonical screening CLIPScore, versus 0.2311, 0.2280, and 0.1456 for random, no-op, and rollout-residual targets.Most implementation controls remain within 0.0079 of the default; query noise 0.90, target radius 0.02, and branch coefficient 10 are failure modes. No CFG-training configuration exceeds 0.3117, though the experiment does not establish that CFG training is harmful.Source: §4.7 · p. 22, Figure 15 · p. 23, Figure 16 · p. 23

5 Conclusion

• DiffusionOPSD achieves the best final held-out score in 19 of 20 reward-matched settings across ten evaluators, improving over the strongest competitor by up to 44.0%.It reduces training GPU-hours relative to DiffusionNFT by 40% on SD3.5-M and 63% on Z-Image-Turbo.Source: §5 · p. 23
• A frozen behavior policy supplies query states and clean-output anchors; reward gradients form bounded positive and negative targets for detached fitting before the behavior policy is refreshed.This separates target construction from finite model fitting and avoids using an endpoint reward as direct supervision for every intermediate prediction.Source: §5 · p. 23
• Jointly training one policy improves all three optimized rewards over DiffusionNFT, demonstrating application to direct multi-reward training.Source: §5 · p. 23

B Mathematical Foundations

• The mathematical foundations distinguish target construction from post-fitting predictions, and ideal output-space optima characterize the loss rather than guarantee finite AdamW updates reach them.Source: B Mathematical Foundations · p. 28

B.1 Affine Clean-Output Map

• The affine map recovers the clean-output prediction from velocity, and under rectified flow a clean-output target radius induces a noise-dependent velocity displacement.The coordinate map is bijective at a fixed query with σ > 0, but is not an isometry; algebraic equivalence does not make finite fitting geometry- or optimizer-invariant.Source: §B.1 · pp. 28–29
• For rectified flow, the clean-output map makes velocity perturbations produce smaller clean-output changes at low noise, while useful queries remain at low but nonzero noise.Source: §B.1 · p. 29
• For general affine schedules, low noise alone is insufficient: queries must keep the schedule determinant bounded away from zero during target construction.A determinant near zero makes the inverse clean-output map ill-conditioned, amplifying errors from sampler-state reconstruction and velocity prediction.Source: §B.1 · p. 30

B.2 Endpoint-Weighted Objectives and Reward Weighting

• DiffusionOPSD retains reward weighting but changes the query-state source and applies reward-ascent or descent target maps to on-policy clean-output anchors.Reweighting selects which anchors matter to each branch; target construction changes the prediction attached to each anchor and introduces the local reward direction.Source: §B.2 · pp. 31–32
• Per-prompt-centered advantages use a shared rollout-batch scale and clipped fitting weights in [0, 1], limiting outlier rewards’ loss-multiplier effect.The clipped weights need not average exactly 1/2 because clipping may act asymmetrically on a finite group.Source: §B.2 · pp. 32–33

B.3 Bounded Target Construction

• Stabilized normalized-gradient steps and per-step projection keep target construction within a trust region; multiple short steps reduce curvature penalty versus one step of equal aggregate length.The local analysis justifies small reward-gradient targets around the policy’s existing clean-output prediction, not global optimality.Source: §B.3 · pp. 35–36
• Local ascent guarantees depend on smoothness and step size; the sufficient condition becomes more conservative near stationary points as the first-order signal weakens.The cited condition is for a nonzero gradient; the result is local rather than a claim of global improvement.Source: §B.3 · p. 36

B.4 Fixed-Suffix Reward

• The fixed-suffix analysis relates local reward movement to endpoint reward under smoothness and positive alignment between their gradients near the anchor.Endpoint reward is evaluated by taking the query update from the proposed clean-output prediction and continuing the remaining solver steps with the behavior policy.Source: §B.4 · pp. 36–37
• A sufficiently small local reward-ascent step improves fixed-suffix reward under Assumption B.6.The proposition defines the step along the normalized local reward gradient from the clean-output anchor.Source: §B.4 · p. 37
• Local and fixed-suffix rewards become closer when query noise and the relative clean-output target radius are small.This motivates local target moves at low-noise on-policy states rather than global reward-gradient optimization from arbitrary noisy states.Source: §B.4 · p. 37
• Occasional gradient-cosine checks on a small validation subset can diagnose alignment without differentiating endpoint reward during training.The analysis separates local reward approximation error from fixed-suffix curvature.Source: §B.4 · p. 38

B.5 Fitting-Branch Geometry and Ideal Output Improvement

• With symmetric targets, both fitting branches induce the same reward-improving direction, independent of their mixture weights; asymmetric targets yield a weighted compromise.The optimum interpolates between attraction to the positive target and repulsion from the negative target.Source: §B.5 · p. 39
• The clean-output branch objective is isotropic, with Hessian eigenvalue 2β^2(a + b) in every dimension; β and branch weights change curvature magnitude, not anisotropy.Any anisotropy in parameter optimization comes from the network Jacobian rather than the quadratic branch construction.Source: §B.5 · p. 39
• Multiple target steps can make positive and negative displacements asymmetric, while smaller β amplifies both the intended displacement and target-approximation error.Actual finite fitting can differ from the ideal output-space minimizer, and the effective clean-output step depends on the ratio of target radius to β.Source: §B.5 · pp. 39–41
• The ideal branch optimum improves reward when its target displacement aligns strongly enough with the local reward gradient to dominate curvature.In the symmetric small-step case, the ideal improvement is positive for a sufficiently small target radius relative to β; finite fitting may under-realize, rotate, or overshoot it.Source: §B.5 · p. 41

B.6 Velocity-Space Formulation

• At a fixed query state, clean-output regression maps to velocity-model training through the chain rule, using the ordinary velocity interface without a special policy-gradient estimator.The schedule-dependent factor also changes the clean-output step induced by a given velocity update.Source: §B.6 · p. 42
• A low but nonzero query noise level balances reliable reward gradients on decoded outputs against a velocity update large enough to realize clean-output targets.Very high noise can make decoded outputs too noisy for reliable reward gradients; near-zero noise attenuates the velocity gradient and clean-output update.Source: §B.6 · p. 42

B.7 Adaptive Scaling

• Detached adaptive normalizers make branch updates independent of positive residual rescaling, while separate branch scales preserve the common reward-improving direction for symmetric targets.Stopping the normalizers preserves the direct regression gradient direction; separate scales prevent a larger residual branch from suppressing the smaller branch.Source: §B.7 · pp. 44–45

B.8 Finite Fitting and the Fitting Gap

• Reward gradients act only on a temporary clean-output variable, while detached-target MSE uses an ordinary supervised field gradient whose infinitesimal descent direction matches direct reward ascent.DiffusionOPSD combines bounded finite construction, detached data, and positive and negative fitting online; it does not claim the infinitesimal direction as an independent contribution.Source: §B.8 · pp. 45, 47
• Reusing a target buffer for more optimizer passes lowers construction cost but increases staleness and may move the field beyond the neighborhood where its target was constructed.Source: §B.8 · p. 45
• The same-query fitting gap is signed: finite fitting can under-realize, rotate, or overshoot constructed gains, so the accounting is not a mediation proof.The authors state that extending it to end-to-end online performance would require distribution-shift terms they do not estimate.Source: §B.8 · p. 46
• Locally, construction specifies target displacements, while fitting produces a kernel-filtered response that can suppress or rotate reward-relevant directions and introduce cross-query interference; this is not a theorem for finite AdamW [37].The explanation uses a local linearization and preconditioner.Source: §B.8 · p. 46
• The isolated HPSv2.1 audit shows a protocol-conditioned ranking reversal, not a universal law: each state starts from the same checkpoint and receives one positive-target AdamW update.The audit evaluates the same query and restores the checkpoint between states; it is not the full batched procedure with positive and negative targets.Source: §B.8 · p. 47

B.9 Behavior-Policy EMA Update

• The behavior policy is refreshed by EMA once after all Mfit updates, supplying new queries next iteration; its rate trades fresh on-policy data against stable data collection, and an excessive rate can make the policy stale and slow improvement [75].Source: §B.9 · p. 47

B.10 Computational Cost

• Increasing sampler length raises rollout time but does not multiply stored autograd activations with the number of denoising steps, because rollout remains detached.Source: §B.10 · p. 48

C.1 Main Results

• Tab. 1 compares held-out scores across ten evaluators, using reward-specific checkpoints for ✓ rows and a single checkpoint for × rows; joint-training rows test compatibility across evaluators.Reward-specific rows generally use 100 optimizer updates; direct joint PickScore, CLIPScore, and HPSv2.1 rows use 300.Source: §C.1 · p. 49
• SD3.5-M experiments fine-tune rank-32, alpha-64 LoRA adapters at 512 × 512, with locally trained main-table evaluations using deterministic 40-step flow sampling on DrawBench prompts.DiffusionNFT, ReFL, and DiffusionOPSD use deterministic 10-step DPM-Solver++ 2M trajectories; FlowGRPO uses stochastic SDE-flow rollouts with ηSDE = 0.7.Source: §C.1 · p. 49
• Z-Image-Turbo experiments use 1024 × 1024 resolution and guidance scale 0.0, with deterministic native 9-step FlowMatchEuler evaluation after training.FlowGRPO instead collects trajectories with the stochastic SDE version at ηSDE = 0.7; the passage clarifies that guidance scale 0.0 here and 1.0 for SD3.5-M both indicate CFG-free settings.Source: §C.1 · p. 49

C.2 Two-Stage On-Policy Distillation Baselines

• Three reward-specific DiffusionOPSD specialists are trained for 100 updates each, then frozen as teachers for a 300-update shared-student distillation with equal teacher weights.Only the 300 second-stage optimizer updates are counted in the main table’s Updates column; the specialist runs are excluded.Source: §C.2 · p. 50
• DanceOPD [76] matches mean teacher velocities at one low-noise query, whereas DiffusionOPD [28] matches teacher transition means across all ten denoising steps.Both use a deterministic, CFG-free 10-step student trajectory and the same SD3.5-M backbone, prompts, resolution, and LoRA configuration.Source: §C.2 · p. 50
• FlowOPD [15] uses stochastic SDE-flow trajectories and a PPO-style clipped likelihood-ratio update [53] based on teacher-consistency rewards, without second-stage reward-model guidance.Reward models train the preceding specialists and evaluate the distilled student, but do not enter the second-stage OPD objectives.Source: §C.2 · p. 50
• The 300-update stages are calibrated to comparable wall-clock budgets using a reference of 175.2 seconds per update on eight GPUs, with batches of 2,688, 528, and 544 images per update.The runs total 806,400, 158,400, and 163,200 sampled images, respectively, and each second-stage run is approximately 117 GPU-hours, excluding specialist training.Source: §C.2 · pp. 50–51
• Shared policies are evaluated on 1,000 images from 200 DrawBench prompts using deterministic 40-step flow sampling and the same scorer suite, with fixed prompt-matched Seedream 5.0 Pro references for VLM-Pairwise.These are two-stage policy-composition controls with one final policy and Reward-Specific marked ×, not additional reward-to-target interfaces.Source: §C.2 · p. 51

C.3 Efficiency Profiling

• DiffusionOPSD profiles one query per trajectory with 16 or 12 diffusion backward calls and 72 or 108 target-gradient calls across the two backbones.Its positive and negative paths share the first reward gradient at y0; with Mtgt = 2, the backbones use 88 and 120 total reward forward passes. Table 2 profiles per-step time, throughput, reward passes, gradient calls, backward passes, and GPU-hours per 100 updates.Source: §C.3 · p. 51, Table 2 · p. 51
• On SD3.5-M, DiffusionOPSD takes 0.59× ReFL’s time per optimizer step and 28.2 instead of 47.7 GPU-hours per 100 updates; on Z-Image-Turbo, ReFL takes 0.68× DiffusionOPSD’s step time and 102.1 instead of 149.8 GPU-hours.The comparisons use matched settings and compute GPU-hours from step time on eight GPUs.Source: §C.3 · p. 51
• DiffusionOPSD has higher final held-out quality in all ten reward-matched settings, but FlowGRPO is not sampler-matched and Z-Image-Turbo peak VRAM is not directly comparable.The Z-Image-Turbo DiffusionOPSD run uses a smaller reward-gradient microbatch to fit memory.Source: §C.3 · p. 51

C.4 Qualitative Component Ablations

• Across shared and disjoint prompts, query-state source changes have limited effect, while excessively large branch coefficients cause semantic drift and harm scene and typography details under finite fitting.The comparisons use the same update grid for the disjoint prompts.Source: Figure 17 · p. 52, Figure 18 · p. 53, Figure 19 · p. 54

C.5 Training Reward Curves

• The inventory covers 73 complete runs, including reward-specific and joint-training runs; raw per-update rewards remain on native scales, and dispersion is within-prompt rollout reward standard deviation, not a confidence interval.Figure 22 shows the unnormalized histories underlying the aggregate analyses.Source: §C.5 · p. 55, Figure 22 · p. 56
• Curves are measured from each run’s update-0 reward; cross-cell medians and interquartile ranges summarize normalized gains, and final smoothed reward position is measured within each run’s observed range.The gain denominator is the joint smoothed best across methods minus the median method value at update 0.Source: §C.5 · p. 55
• On SD3.5-M, DiffusionOPSD has the strongest average normalized gain and stable final rewards across ten objectives; on Z-Image-Turbo, it improves consistently while DiffusionNFT shows below-base regressions.The Z-Image-Turbo normalized-gain plot retains DiffusionNFT’s full negative curve rather than clipping it.Source: Figure 20 · p. 55, Figure 21 · p. 55, §C.5 · p. 55

C.6 Fitting Audit

• Endpoint attribution uses 50-update SD3.5-M CLIPScore variants evaluated on 999 images from 200 DrawBench prompts, with paired prompt-and-latent comparisons.Generation uses a deterministic 40-step first-order ODE sampler at CFG-free scale 1.0, with prompt-clustered bootstrap intervals.Source: §C.6 · p. 57
• Construction gain is measured before parameter updates at fixed queries, using 512 held-out prompts, a fixed suffix sampler, and a target radius of ρ = 0.10.The query noise level is σq ≃0.278; FlowGRPO uses 1,024 observations because each finite-group direction is estimated twice.Source: §C.6 · p. 57
• The matched realized-gain audit compares interfaces after one calibrated plain-SGD step at the same query, with matched pooled-RMS output displacement and parameters restored between prompts.Its isolated protocol is not the batched online procedure with positive and negative targets.Source: §C.6 · p. 57
• In the separate positive-target probe, one fresh AdamW update is applied per restored checkpoint; when gradient magnitudes dominate ϵAdam, the step is approximately sign normalized.Matching target radius therefore does not match parameter-step magnitude; κfit measures signed target-direction realization and is reported with orthogonal drift.Source: §C.6 · p. 57
• On HPSv2.1, the reward-gradient positive target had larger construction gain but smaller realized gain after one fresh-AdamW update than a matched-radius random-direction target.On CLIPScore, grad-versus-random ordering did not reverse on a majority of anchors; the audits show construction and realized gain are not interchangeable under these protocols.Source: §C.6 · p. 57

C.7 Stress-Test Protocols

• The Z-Image-Turbo few-step study uses its native 9-step sampler at 1024 × 1024, with ten reward-specific checkpoints trained for 100 optimizer updates each.Paired base-model significance is available for nine rewards with aligned per-prompt scores.Source: §C.7 · p. 58
• The joint policy trains for 300 updates on PickScore26 + CLIPScore + HPSv2.1 versus 100-update specialists; this intentionally unmatched stress test does not establish general conflict resolution [73].It asks only whether one complete DiffusionOPSD policy retains high performance on all three objectives in this setting; weighted-sum gradients may suffer conflict or dilution [73].Source: §C.7 · p. 58

C.8 Ablation Details

• Ablations use SD3.5-M LoRAs trained for 50 optimizer updates, with a shared CLIPScore screening protocol on 100 images from 20 DrawBench prompts.The small fixed subset supports matched-compute comparisons; the passage cautions that conclusions affecting the main table require confirmation on the full DrawBench protocol.Source: §C.8 · p. 58
• Reward-gradient targets preserve requested attributes more reliably than random-direction and no-op controls in qualitative target-direction ablations.The comparison includes text rendering, object identity, color, counting, and spatial relations across the shown prompts.Source: §C.8 · p. 58, Figure 23 · p. 59, Figure 24 · p. 60
• Across training-evolution grids, reward-gradient variants separate from initial behavior and preserve intended attributes through update 50, unlike the weaker random and no-op controls.The grids use fixed prompts from update 0 to update 50, with additional disjoint prompts in Fig. 26.Source: Figure 25 · p. 61, Figure 26 · p. 62
• Human preferences are assessed through blinded paired comparisons on 100 held-out prompts, using randomized method order, half-tie win scoring, and prompt-clustered bootstrap intervals.Annotators choose an overall preferred image or tie and assign a primary reason; comparisons are against the base model, FlowGRPO, DiffusionNFT, and ReFL.Source: §C.8 · p. 63

C.9 CFG Compensation

• In train-CFG OPSD, conditional and unconditional predictions share trainable parameters, and the same guidance operator is used for rollouts, anchors, and loss velocities.This differs from frozen-teacher CFG absorption, which distills a fixed guided field into a single student field.Source: §C.9 · p. 63
• The guided loss constrains only an affine combination of branch changes, leaving the individual branches non-identifiable and not directly penalizing their difference.Initialization and branch coupling may restrict reachable compensation, but do not remove objective non-identifiability without explicit regularization.Source: §C.9 · p. 63

• Under exact train-scale fitting, evaluation-field deviation includes (ζ_ev − ζ_tr)Δv_gap, so its penalty grows quadratically with scale mismatch when Δv_gap ≠ 0.Source: Its squared field error is therefore · p. 64
• For fixed training scale, the squared error is quadratic in evaluation scale, and its minimizing scale approaches the matched scale as the training residual vanishes.Source: Its squared field error is therefore · p. 64

Optimization geometry

• When ζ_tr > 1, shared-parameter branch gradients have opposite-sign coefficients, so the objective controls the guided affine combination more directly and has no separate ∥Δv_gap∥2 regularizer.Source: Its squared field error is therefore · p. 64

• The compensation identity is exact and vanishes when ζ_ev = ζ_tr; frozen-teacher absorption avoids this ambiguity by fixing the guided teacher field before optimizing one student field [76].Source: Its squared field error is therefore · p. 64