{"id":"f6a501cf-f6e9-4f27-a858-f3db98cb27ca","arxiv_id":"2607.07665","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"Replacing CFG's w(r-1) coefficient with r^(1+w)-r removes a sigma_min-divergent residual blow-up on a Gaussian calibration model and stabilizes high-guidance diffusion sampling at zero extra NFE.","lead":"The paper shows that classifier-free guidance (CFG) in diffusion models destabilizes sampling because it changes how fast certain directions contract, breaking the numerical solver. It proposes a one-line fix that replaces CFG's guidance coefficient with a derived alternative, stabilizing high-guidance sampling at zero extra cost.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The theory-to-practice gap is the load-bearing concern: Theorem 2's first-order accuracy is proven only for a single discriminative coordinate on the commuting Gaussian model, and the learned-checkpoint diagnostics are indirect evidence that the model ordering holds.","rationale":"The reader correctly identifies the theory-to-practice gap as the load-bearing concern. The proofs on the commuting Gaussian model are complete with explicit constants (verified: Proposition 1's exponent classification, Theorem 1's barrier thresholds, Lemma 2's uniqueness argument, Proposition 4's telescoping bound, and Theorem 2's integral defect bound all check out technically). The uniqueness claim (Lemma 2) is clean: on (A,B)=(1,0), exactness requires r+α = r^(1+w), giving α = r^(1+w)−r with no free parameters. The first-order consistency (Proposition 2) and sign preservation (Proposition 3) are straightforward. The concern is not with the mathematics but with whether the mathematical model captures the relevant structure of learned checkpoints. The paper is transparent about this: it uses the phrase 'consistent with' rather than 'verifies,' acknowledges the KID split, and includes the dense-reference diagnostic showing the fitted step is not uniformly closer in L2. The CONDITIONAL verdict with HIGH confidence is appropriate — the theory is sound on its domain, the empirical evidence is honestly scoped, and the limitations are real but clearly stated. I see no red flags (no post-hoc exclusions, no circular reasoning in the coefficient derivation, no missing error bars on key metrics). The one adjustment I might suggest is slightly lowering confidence from HIGH to MEDIUM-HIGH given that the theory-to-practice bridge is the central unresolved question, but this is a minor calibration issue. The proposed concrete test (estimating the guided terminal exponent on learned checkpoints) would directly settle whether Proposition 1's structural prediction holds in practice, which is the linchpin for the entire repair strategy.","tokens_in":16421,"tokens_out":4088,"duration_ms":278949,"concrete_test":"On the learned CIFAR-10 EDM checkpoints, estimate the local Jacobian ∂η_w/∂x of the guided velocity field at low σ (e.g., σ ∈ {0.01, 0.005, 0.002}) along sampled trajectories, compute its eigenvalue spectrum, and check whether the largest-magnitude eigenvalues scale as σ^{-(1+w)} (the discriminative exponent prediction of Proposition 1) rather than σ^{-1} (the unguided exponent). If the scaling matches 1+w, the model's key structural prediction holds on real checkpoints and the theoretical foundation for the specific coefficient r^(1+w)−r is validated. If the scaling is closer to 1 or shows mixed behavior, the model-to-practice gap is more severe than acknowledged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim — that the fitted step r^(1+w)−r removes σ_min-divergent blow-up and is first-order accurate against the exact guided flow — is proven rigorously but only on a single discriminative coordinate (a=0 < b) of the commuting Gaussian model (Theorem 2, Proposition 4). The paper explicitly states: 'Extending this crossover accuracy to the tangential layers and learned geometry needs the spectrum-aware Gaussian factor (6) and is future work.' On the commuting model this is less concerning because coordinates decouple, but on learned CIFAR-10/SD1.5 checkpoints the paper provides only indirect evidence: residual amplification ratios (Table 1) and saturation metrics that are 'consistent with' the model ordering rather than verifying it. No direct measurement confirms that the guided terminal exponent on the discriminative subspace of a learned checkpoint actually approaches 1+w as σ→0 (Proposition 1's prediction). Without this, the theoretical mechanism motivating the specific coefficient r^(1+w)−r — as opposed to any other stabilizing modification — rests on an unverified representational assumption. The practical evidence (9/9 FID wins, reduced clipping) supports the repair as a stabilizer, but the KID split (CFG wins every block) and the dense-reference L2 diagnostic (Appendix H, ratios >1) show the fitted step is not a uniformly better integrator, narrowing the practical claim to terminal residual/saturation repair. This is honestly scoped but means the strongest form of the central claim — that the repair is 'first-order accurate against the exact guided flow' — is verified only on the calibration model, not on the systems where it is deployed.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper analyzes classifier-free guidance (CFG) through a numerical-analysis lens, building on a prior fitted-operator framework for unguided DDIM. On a commuting Gaussian calibration model (Hypothesis 1), the author shows that guidance re-stiffens the discriminative subspace to an anomalous terminal exponent 1+w, causing DDIM+CFG's terminal residual to diverge as sigma_min → 0 on coarse meshes. A guided clock barrier with three ordered step-size thresholds is proven (Theorem 1), and a one-coefficient, zero-extra-NFE repair—replacing w(r-1) by r^(1+w)-r on the guidance direction—is derived as the unique spectrum-free terminal-exact coefficient (Lemma 2). On the discriminative crossover, this repair removes the sigma_min-divergent blow-up and is first-order accurate against the exact guided flow (Theorem 2, Proposition 4). Empirical diagnostics on learned CIFAR-10 EDM checkpoints and a Stable Diffusion 1.5 smoke test show reduced residual amplification, clipping, and saturation, with 9/9 FID wins over CFG on the tested grid, though KID favors CFG and the dense-reference L2 diagnostic shows the fitted step is not a uniformly better integrator.","tokens_in":16537,"tokens_out":1519,"duration_ms":343943,"significance":"The paper provides a clean, parameter-free derivation of a one-coefficient CFG repair from terminal-exactness on a calibration model, with explicit constants and complete proofs (Appendix A). The barrier thresholds (Theorem 1) yield a falsifiable, parameter-free prediction for the terminal edge of limited-interval guidance (Remark 3). The empirical program is honestly scoped: the author reports the KID split, the dense-reference L2 failure (Appendix H), and the reverse-ordering instability boundary (Remark 4) alongside the positive results. The zero-extra-NFE, one-line implementation delta and the cross-domain SD 1.5 smoke test add practical value. The theory-to-practice gap—between the commuting Gaussian model and learned checkpoints—is the main limitation, and the author is transparent about it.","major_comments":[{"comment":"§4, Table 2 and Appendix H: The practical evidence is honestly scoped but the central tension between the theoretical mechanism and the empirical results is underdeveloped. The KID split (CFG wins every block) and the dense-reference L2 diagnostic (Table 5, ratios >1) show the fitted step is not a uniformly better integrator. Meanwhile, the theoretical motivation for the specific coefficient r^(1+w)-r rests on the commuting Gaussian model, and the learned-checkpoint diagnostics (Table 1) are described as 'consistent with' the model ordering without directly verifying that the guided terminal exponent on the discriminative subspace of a learned checkpoint approaches 1+w as sigma → 0 (Proposition 1's prediction). The paper would be substantially strengthened by a direct measurement attempting to estimate the effective terminal exponent on the discriminative subspace of the CIFAR-10 EDM den","section":null},{"comment":"§3, Proposition 4 / Theorem 2: The first-order accuracy result is proven only for a single discriminative coordinate (a=0 < b) on the commuting model. The paper explicitly states: 'Extending this crossover accuracy to the tangential layers and learned geometry needs the spectrum-aware Gaussian factor (6) and is future work.' On the commuting model coordinates decouple, so this is less concerning there, but the gap to the multi-coordinate learned setting is not bridged even at the level of a multi-coordinate Gaussian model where the spectrum-aware factor (6) could be evaluated. A numerical experiment on a multi-coordinate commuting Gaussian (e.g., 2-3 coordinates with different b_i values) showing that the single-coordinate accuracy result does or does not extend would help clarify the scope of the theory.","section":null},{"comment":"§2, Hypothesis 1 and Remark 4: The class-subset hypothesis (a_i ≤ b_i) is a load-bearing assumption, and the reverse ordering (b=0 < a) creates a continuous-flow instability. The paper acknowledges that 'real learned checkpoints need not obey it' but does not provide any diagnostic or argument for why the EDM CIFAR-10 checkpoints used in §4 might satisfy it, or how common the reverse ordering is in practice. Even an indirect argument (e.g., relating the class-subset property to the training setup where the unconditional model is a mixture of class-conditional models) would help readers assess the applicability of the theory.","section":null}],"minor_comments":[{"comment":"Figure 1: The caption says 'schematic' but the axes are unlabeled beyond the text annotations. Adding axis labels (e.g., sigma or lambda on the horizontal axis, exponent on the vertical) would improve clarity.","section":null},{"comment":"§3, Eq. (15): The notation r^(1+w) could be confused with r raised to the power (1+w) in typeset form; the paper is consistent but a brief note on first use that this is exponentiation would help.","section":null},{"comment":"Table 1: The 'clip_delta' column is defined as 'fitted CFG minus CFG for the median fraction of final-denoised pixels outside [-1,1]' but the negative values (improvement) are not immediately intuitive; a note that negative means improvement would help.","section":null},{"comment":"§4, Table 3: The target accuracy for fitted CFG at w=8, N=16 jumps to 99.3%, notably higher than CFG's 93.8%. The paper notes 'the high fitted value at N=16 should not be read as fuller conditional fidelity, as it may partly reflect reduced diversity,' but this large jump deserves more discussion or investigation.","section":null},{"comment":"Appendix F: The Stable Diffusion smoke test uses float16, which could introduce numerical artifacts at high guidance. A note on whether this was checked would be useful.","section":null},{"comment":"Reference [12] is by the same author and is the foundational fitted-operator result this paper builds on. This dependency is clearly stated, but the paper could more explicitly delineate what is novel in this work versus [12].","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper builds on the author's own prior work [12], which is clearly disclosed. The core theoretical contribution (the guided barrier and the fitted coefficient derivation) is genuinely new relative to [12]. The main concern is whether the theory-to-practice gap is too wide for the claims to be actionable, but the author's honest scoping (explicitly stating what is proved vs. evidence vs. neither) and the zero-extra-NFE simplicity of the repair make this a reasonable contribution for the venue. The KID split and dense-reference L2 failure are appropriately reported and do not undermine the central claim, which is about terminal residual/saturation repair rather than universal image-quality improvement. I recommend minor revision with a request to address the theory-to-practice gap more directly, ideally via a direct exponent-estimation diagnostic on the learned checkpoint."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The recommendation is minor revision, and all three major comments are actionable. We agree with the substance of each: (1) the theory-to-practice gap should be bridged by a direct terminal-exponent measurement on learned checkpoints; (2) the single-coordinate accuracy result should be tested on a multi-coordinate Gaussian; and (3) the class-subset hypothesis needs an explicit argument connecting it to the EDM training setup. We commit to all three additions in the revision and explain below what each will contain, including honest accounting of what we can and cannot claim.","responses":[{"response":"We agree completely. The gap between 'consistent with the model ordering' and a direct exponent measurement is the weakest link in the current manuscript, and the referee is right that closing it—even partially—would substantially strengthen the paper. We will add a new diagnostic in Section 4 that attempts to estimate the effective terminal exponent on the discriminative subspace of the EDM CIFAR-10 checkpoints. Concretely: along sampling trajectories at high guidance (w=6.5, 8), we will compute the local contraction ratio of the guidance direction (D_u - D_c) across consecutive steps in the small-sigma regime and fit a power-law exponent in sigma. The commuting model predicts this exponent should approach 1+w on discriminative directions; we will report whatever the data shows, including if the exponent does not cleanly approach 1+w. We are honest that this is an indirect measurement: the learned denoiser's Jacobian does not diagonalize into clean eigen-coordinates, so we are measuring an effective exponent on a projected subspace, not verifying Proposition 1 literally. We will state this limitation explicitly. If the measurement is inconclusive or contradicts the prediction, we will say so rather than cherry-pick. We believe this is feasible with the existing checkpoints and sampling infrastructure, and we commit to including it in the revision.","revision_made":"yes","referee_comment":"The central tension between the theoretical mechanism and the empirical results is underdeveloped. The KID split and dense-reference L2 diagnostic show the fitted step is not a uniformly better integrator. Meanwhile, the theoretical motivation rests on the commuting Gaussian model, and the learned-checkpoint diagnostics are described as 'consistent with' the model ordering without directly verifying that the guided terminal exponent on the discriminative subspace of a learned checkpoint approaches 1+w as sigma → 0 (Proposition 1's prediction). The paper would be substantially strengthened by a direct measurement attempting to estimate the effective terminal exponent on the discriminative subspace of the CIFAR-10 EDM denoiser."},{"response":"This is a fair and concrete suggestion. On the commuting model, coordinates decouple, so the single-coordinate accuracy result (Theorem 2) extends component-wise to any number of coordinates with different b_i values—the log-defect bound applies independently per coordinate and the W_2 bound sums. However, the referee's point is deeper: the practical question is whether the spectrum-free coefficient r^(1+w) - r, which is exact only on the pure discriminative terminal layer (A=1, B=0), remains accurate when different coordinates are at different points in their crossovers (different b_i values, hence different B_i at a given sigma). The honest answer is that it does not remain uniformly accurate in that regime—the spectrum-aware factor (6) would be needed for exactness, and the fitted step pays a finite O(1+w) crossover tax per coordinate (Proposition 4). We will add a numerical experiment on a 3-coordinate commuting Gaussian with distinct b_i values, showing the per-coordinate accuracy and the aggregate W_2 error as sigma_min → 0, on both uniform and non-uniform meshes. This will make the scope of Theorem 2 precise: the single-coordinate bound extends by independence, but the practical accuracy depends on how many coordinates are mid-crossover at the terminal steps, which is a mesh- and spectrum-dependent question the theory does not fully resolve.","revision_made":"yes","referee_comment":"The first-order accuracy result (Theorem 2) is proven only for a single discriminative coordinate (a=0 < b) on the commuting model. The gap to the multi-coordinate learned setting is not bridged even at the level of a multi-coordinate Gaussian model where the spectrum-aware factor (6) could be evaluated. A numerical experiment on a multi-coordinate commuting Gaussian (e.g., 2-3 coordinates with different b_i values) showing that the single-coordinate accuracy result does or does not extend would help clarify the scope of the theory."},{"response":"The referee is right that we left this hanging. We will add an explicit argument and a diagnostic. The argument: in the EDM CIFAR-10 setup, the unconditional model is trained on the full data distribution (a mixture over all classes), while each class-conditional model is trained on a single class. Under a Gaussian approximation, the marginal covariance C_u is the mixture covariance E[C_c | class] + Cov(mean_c), which is at least as large as any single class covariance C_c in the PSD order—this is the class-subset property a_i ≤ b_i at the population level. The reverse ordering (class wider than marginal in some direction) would require a class whose covariance exceeds the mixture covariance in some eigen-direction, which is possible for atypical classes but structurally disfavored by the mixture construction. We will state this argument in Section 2 and add a diagnostic in Section 4: we will estimate the effective class and marginal variances along the guidance direction (D_u - D_c) at small sigma for the EDM checkpoints and check whether the ordering holds in practice. If some classes show reverse ordering, we will report that honestly and note that the theory's applicability is partial. We cannot fully verify the hypothesis on learned checkpoints (the Jacobian structure is not Gaussian), but the indirect argument plus the diagnostic gives readers a concrete basis to assess applicability.","revision_made":"yes","referee_comment":"The class-subset hypothesis (a_i ≤ b_i) is a load-bearing assumption, and the reverse ordering (b=0 < a) creates a continuous-flow instability. The paper acknowledges that 'real learned checkpoints need not obey it' but does not provide any diagnostic or argument for why the EDM CIFAR-10 checkpoints used in §4 might satisfy it, or how common the reverse ordering is in practice. Even an indirect argument (e.g., relating the class-subset property to the training setup where the unconditional model is a mixture of class-conditional models) would help readers assess the applicability of the theory."}],"tokens_in":16309,"tokens_out":1426,"duration_ms":265135,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper derives a one-line, zero-extra-NFE modification to classifier-free guidance — replacing w(r−1) with r^{1+w}−r on the guidance direction — from a terminal-exactness argument on a commuting Gaussian model. The math on that model is clean and complete. The practical evidence is honestly scoped but does not bridge the gap between the calibration model and real checkpoints, and the paper is admirably upfront about this. It deserves a serious referee because the analytical lens is genuinely new and the repair is immediately deployable, even though the strongest theoretical claim is verified only on the calibration model, not on the systems where it is actually used. The stress-test concern about the theory-to-practice gap is real but lands softly: the paper never overclaims, and the limitations section is one of the most candid I have read. What is actually new: the layer classification of the guided terminal exponent (Proposition 1), the three-threshold barrier theorem (Theorem 1), and the uniqueness argument for the fitted coefficient (Lemma 2). The derivation of r^{1+w}−r is parameter-free — it falls out of requiring exactness on the pure discriminative terminal layer. The proofs in Appendix A are stated with explicit constants and check out. The model-free oversaturation identity (Remark 2) — that one-step DDIM+CFG lands on D_w rather than D_c — is a nice observation that gives a concrete mechanism for a widely felt practical problem. The empirical program is well-designed: fixed cells and metrics before runs, paired comparisons, a 50k replication, a DINOv2 backbone swap, all-class audit (60 pairs), and a Stable Diffusion smoke test. The 9/9 FID wins and the residual/clipping improvements at high guidance are solid evidence for stabilization. The soft spots are exactly where the paper says they are. First, the theory holds under Hypothesis 1 (commuting Gaussian, class narrower than marginal in every direction). Learned checkpoints are shown to be 'consistent with' this ordering via residual amplification ratios, but no direct measurement confirms the guided terminal exponent actually approaches 1+w on a real network. So the specific coefficient r^{1+w}−r is motivated by an unverified representational assumption — though any stabilizing coefficient would need some such assumption, and this one is at least uniquely determined by its own model. Second, KID favors CFG in every block, so the image-quality benefit is metric-dependent. Third, the dense-reference diagnostic (Appendix H) shows the fitted step is not uniformly closer to the dense CFG endpoint in final-image L2, narrowing the practical claim to residual and saturation repair. None of these are hidden; the paper states all three explicitly. This is a paper for researchers working on diffusion sampling and CFG pathologies. The numerical-analysis framing — treating CFG oversaturation as a solver artifact rather than a model property — is a useful contribution regardless of whether the specific coefficient catches on. Recommend accepting for peer review. The theory is sound on its stated model, the empirical work is honest and well-scoped, and the one-line repair is worth putting in front of referees who can stress-test it further.","headline":"One-coefficient CFG repair derived from a clean terminal-exactness argument on a Gaussian calibration model; theory is solid but only proven on that model, with learned-checkpoint evidence that is suggestive but indirect.","tokens_in":17230,"tokens_out":737,"would_cite":false,"duration_ms":183100,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"One-coefficient fix stops CFG oversaturation at zero extra cost","keywords":["classifier-free guidance","diffusion models","DDIM","numerical analysis","asymptotic-preserving schemes","oversaturation","fitted operator","discretization error"],"falsifier":"If on real learned checkpoints the directional variance ordering between conditional and unconditional denoisers does not follow the class-subset pattern (a_i <= b_i), the exponent-1+w re-stiffening mechanism may not be the operative cause of oversaturation, and the fitted coefficient's improvements could stem from a different effect than the one the theory describes.","tokens_in":16377,"feed_emoji":"🎛️","tokens_out":1326,"duration_ms":247401,"temperature":0.7,"pith_summary":"Classifier-free guidance (CFG) is the standard technique for strengthening class-conditional signals in diffusion model sampling, but at high guidance scales it oversaturates and destabilizes. This paper argues that the root cause is a discretization mismatch: CFG changes the contraction rate of the sampler's discriminative subspace from exponent 1 to 1+w, which breaks the property that makes the standard DDIM step exact on the terminal (low-noise) stretch of sampling. On coarse step schedules, this causes the guided residual to diverge as the noise floor approaches zero. The paper proves a three-threshold barrier showing exactly when sign-flips, residual blow-up, and instability occur, and reads the familiar one-step oversaturation as the endpoint of this barrier—a solver artifact rather than a property of the continuous guided flow. The same analysis produces a repair: replace CFG's guidance coefficient w(r-1) with r^(1+w) - r, where r is the per-step noise ratio. This one-line, zero-extra-cost change is the unique coefficient that restores exactness on the terminal discriminative layer without requiring any spectral information. On the calibration Gaussian model, it removes the divergent blow-up and is first-order accurate against the exact guided flow. On learned CIFAR-10 checkpoints and in a Stable Diffusion 1.5 smoke test, it acts as a high-guidance stabilizer—cutting residual amplification and pixel saturation while preserving class-conditional accuracy—though it is not a universal image-quality win (KID can still favor vanilla CFG).","feed_headline":"One-coefficient fix stops CFG oversaturation at zero extra cost","feed_subtitle":"Swapping w(r-1) for r^(1+w)-r in the guidance direction removes a solver-induced divergence that standard classifier-free guidance creates.","key_machinery":"The guided terminal exponent mu_w = (1+w)A - wB, where A and B are per-coordinate contraction rates of the conditional and unconditional denoisers on a commuting Gaussian model. The three-threshold guided clock barrier (h_flat, h_sharp, h_infinity) in terms of step size h and guidance weight w. The fitted coefficient r^(1+w) - r, derived by a gauge change that rectifies the anomalous exponent back to 1, then transporting the sigma-clock Euler step through it.","core_discovery":"The central object is the guided terminal exponent: on a commuting Gaussian calibration model, CFG re-stiffens the discriminative subspace (directions present in the marginal but collapsed in the class) to contraction exponent 1+w instead of the unguided exponent 1. This exits the class of fitted operators for which the DDIM step is layer-exact, causing a sigma_min-divergent residual blow-up on coarse meshes. The paper proves this is governed by three ordered step-size thresholds (sign preservation, residual amplification, absolute stability) and shows that the coefficient replacement w(r-1) -> r^(1+w) - r is the unique spectrum-free repair that restores terminal exactness, removes the blow-","pith_inferences":["If the discriminative-subspace re-stiffening mechanism holds on non-Gaussian or non-commuting geometries, the fitted coefficient could serve as a general-purpose terminal stabilizer for any flow-matching or diffusion sampler using CFG, not just DDIM.","The O(w^2) prefactor in the accuracy bound (not uniform in w) suggests there may be a w-dependent step-count requirement even with the repair, implying an optimal guidance-weight-to-step-count tradeoff curve that the paper does not fully characterize.","The FID/KID split (fitted wins FID, CFG wins KID) may reflect a fundamental tension between reducing discretization artifacts and preserving the distributional properties that KID measures—worth investigating whether the KID penalty is causally linked to the oversaturation removal or is an independent effect.","The class-subset hypothesis (class narrower than marginal in every direction) may be approximately testable on real checkpoints by estimating directional variances of conditional vs. unconditional denoisers, which would bridge the gap between the calibration model and the learned-checkpoint diagnostics."],"forward_implications":["Practitioners using high guidance scales (g >= 7.5) can replace one coefficient in their DDIM+CFG sampler to reduce oversaturation and residual amplification at zero additional compute cost, as a drop-in one-line code change.","The three-threshold barrier provides a parameter-free prediction for the terminal edge of limited-interval guidance schedules, potentially replacing tuned guidance-interval cutoffs with a derived one.","The finding that oversaturation is a solver artifact (discretization mismatch) rather than a property of the continuous guided law suggests that other guidance pathologies at high w may also have discretization-induced components separable from the continuous-flow distortion.","The reverse-ordering case (class wider than marginal in some direction) creates a continuous-flow instability at exponent -w that no solver can repair, marking a hard boundary on where any discretization-level fix can help."],"fun_headline_variants":["Guidance breaks DDIM's fitted operator, causing oversaturation","CFG re-stiffens discriminative subspace to exponent 1+w, diverging on coarse meshes","One-coefficient repair restores terminal exactness to guided diffusion","Swapping w(r-1) for r^(1+w)-r removes CFG's sigma_min divergence without extra NFE","Proven step-size thresholds explain one-step oversaturation as solver artifact"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire theoretical analysis is carried out on a linear Gaussian model where the class and marginal covariances commute and the class is narrower in every direction. Real neural network checkpoints need not satisfy this, and the paper acknowledges the learned diagnostics are only 'consistent with' but do not verify the model's structural predictions.","fun_headline_variants_meta":{"raw":{"variants":["Guidance breaks DDIM's fitted operator, causing oversaturation","CFG re-stiffens discriminative subspace to exponent 1+w, diverging on coarse meshes","One-coefficient repair restores terminal exactness to guided diffusion","Swapping w(r-1) for r^(1+w)-r removes CFG's sigma_min divergence without extra NFE","Proven step-size thresholds explain one-step oversaturation as solver artifact"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":812,"prompt_tokens":721,"completion_tokens":91,"prompt_tokens_details":null},"tokens_in":721,"tokens_out":91,"duration_ms":41620,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T03:02:17.071453+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If on real learned checkpoints the directional variance ordering between conditional and unconditional denoisers does not follow the class-subset pattern (a_i <= b_i), the exponent-1+w re-stiffening mechanism may not be the operative cause of oversaturation, and the fitted coefficient's improvements could stem from a different effect than the one the theory describes.","supporting_citations":[],"review_version":1}