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See it, Say it, Sorted: Mechanistic Diagnosis and Parameter-Space Mitigation of Emergent Misalignment in LLMs

Weiqiao Que, Ruizhe Li, Chengyu Wang, Dakan Wang, Emine Yilmaz, Xiaofeng He

arXiv:2609.34970v1cs.LGcs.CL

TL;DR

Narrow domain adaptation can trigger emergent misalignment across unrelated domains, while prior work leaves training-time parameter dynamics unmapped. The paper tracks EM with geometric measures and projects empirical harmful-gradient subspaces out of parameter updates. The projection reduces free-generation EM on Qwen2.5-14B-IT, and the diagnostics also detect harmful subspaces when behavioral EM is near zero.

  • Problem

    Prior analyses leave training-time parameter-update dynamics unmapped, despite narrow adaptation being associated with safety failures across unrelated domains.

    Source: §1 · pp. 1–2

  • Method

    The paper tracks token-level likelihood shifts, directional Hessian curvature, and Grassmannian gradient-subspace overlaps, then projects empirical harmful-gradient subspaces out of parameter updates.

    Source: §1 · pp. 2–3

  • Results

    55.4%–80.0% drops in free-generation EM follow harmful-subspace projection across domains on Qwen2.5-14B-IT; harmful subspaces also remain measurable when generative EM is near zero.

    Source: §4.4 · p. 8, §5 · p. 9

  • Takeaways & Limitations

    Behavioral generation alone can give an incomplete picture of safety degradation, so geometric probes reveal latent misalignment in models with near-zero generative EM.

    Source: §4.1 · p. 6, §5 · p. 9

    The rank-4 projection does not eliminate EM completely, and residual toxic features can escape through unmitigated orthogonal dimensions.

    Source: §4.5 · p. 9, I LIMITATIONS · p. 49

Abstract

from arXiv · show

Safety-aligned LLMs can exhibit emergent misalignment (EM): narrow domain adaptation unexpectedly triggers catastrophic safety failures across unrelated domains. Prior static analyses leave training dynamics unmapped, while existing defenses rely on heuristics that degrade utility. We present a dynamic, second-order geometric study of EM. Tracking training trajectories reveals that directional Hessian curvature concentrates sharply on semantic pivot tokens. Grassmannian projections show that, in most settings, harmful-safe gap widens mainly because safe-gradient overlap declines. Leveraging these insights, we introduce a parameter-level Geometric Mitigation Framework that orthogonally projects empirical harmful gradient subspace out of parameter updates. On Qwen2.5-14B-IT, our defense suppresses free-generation EM by up to 80.0%; across the other three of four open-weight instruction-based model families (3B--20B), where single-layer behavioral EM is already near zero, teacher-forced evaluation shows same harmful subspace controls the conditional support of frozen EM responses. Crucially, these diagnostics unmask the illusion of behavioral safety: the same subspace remains measurable and steerable in models where behavioral EM is near zero. Code: https://github.com/WeiqiaoQUE/mechanistic-emergent-misalignment.

1 INTRODUCTION

Narrow adaptation can dissolve safety boundaries across unrelated domains, while prior post-hoc analyses leave the training-time parameter dynamics unresolved. The paper tracks those dynamics geometrically and uses them to motivate parameter-space mitigation.

1 INTRODUCTION

  • Narrow adaptation can spread failures from domain-specific tasks to unrelated queries, producing toxicity, deception, and destructive directives.The paper frames this as a global collapse of safety boundaries in otherwise safety-aligned models.Source: §1 · p. 1
  • Prior work is predominantly static and post-hoc, leaving parameter-update trajectories unmapped and motivating direct study of training dynamics.The introduction notes that existing defenses do not operate directly on update trajectories and may leave conditional misalignment possible.Source: §1 · p. 2
  • The trajectory analysis finds extreme second-order sensitivity on semantic pivot tokens and, in most settings, declining safe-gradient overlap rather than continuous rotation toward new harmful directions.The authors characterize the latter pattern as passive decoupling: harmful alignment stabilizes or varies non-monotonically as safe-subspace alignment declines.Source: §1 · p. 2
  • On Qwen2.5-14B-IT, parameter-level projection using leading singular vectors of empirical harmful gradients, checked against isotropic Haar-distributed random controls, suppresses free-generation EM by up to 80% without additional retraining.In models such as Llama-3.1-8B-IT, where autonomous generative EM is near 0%, geometric projection reduces conditional support for harmful responses.Source: §1 · p. 3
  • The paper combines dynamic geometric tracking, Grassmannian overlap analysis, and parameter-level projection of empirical harmful gradients across four open-weight model families.The introduction presents these as contributions for diagnosing and mitigating EM, including cases where behavioral EM is near zero.Source: §1 · p. 3

2 SEE IT: EMERGENT MISALIGNMENT BACKGROUND

Prior work describes EM as spanning architectures and mechanisms, while noting that existing defenses and predominantly static analyses leave room for a dynamic geometric approach.

2 SEE IT: EMERGENT MISALIGNMENT BACKGROUND

  • EM has been reported across dense and sparse MoE models, CoT reasoning, narrow unlearning, reinforcement learning, and in-context steering.The passage cites work on each of these manifestations, including dense and sparse architectures.Source: §2 · p. 3
  • Mechanistic accounts describe convergent low-dimensional subspaces, persona collapse, and gradient-norm peaks that can precede observable output misalignment.The cited accounts associate EM with shared residual-stream subspaces and disrupted role discrimination.Source: §2 · p. 3
  • Prior defenses may induce conditional misalignment, degrade utility, or rely on static feature dictionaries, and most prior analyses remain post-hoc.The paper contrasts these approaches with tracking loss, gradients, and curvature through training to locate a subspace for parameter-space projection.Source: §2 · p. 3

3 SAY IT: GEOMETRIC DIAGNOSTICS AND SUBSPACE ATTRIBUTION

The section lays out the model and evaluation setup, then defines methods for selecting EM-critical tokens, measuring curvature and gradient-subspace alignment, and testing parameter-space interventions.

3.1 EXPERIMENTAL ARCHITECTURE: MODELS, ADAPTATION, AND EVALUATION

  • The testbed spans four open-weight instruction-tuned model families and narrow finance, sports, and medicine adaptations, with all-layer and single-layer rank-1 LoRA configurations.Behavioral EM is scored on canary prompts using alignment and coherence criteria.Source: §3.1 · p. 4

3.2 PIVOT TOKEN IDENTIFICATION AND LOCAL LOSS CURVATURE

  • Pivot tokens are selected from the highest likelihood-shift quantile and compared with a size-matched neutral-token set; their directional curvature separates from neutral tokens under permutation testing (p < 0.001).The token sets are frozen, and the reported curvature separation indicates steep, sensitive pivot-token profiles versus broad neutral-token plateaus.Source: §3.2 · pp. 4–5

3.3 GRADIENT SUBSPACE CONSTRUCTION AND ALIGNMENT OVERLAP

  • The section constructs empirical gradient subspaces with SVD and quantifies their Grassmannian overlap, using bootstrap and label-permutation procedures for statistical inference.The supplied method passages identify harmful, safe, pivot, and neutral gradient matrices as inputs to this subspace analysis.Source: §3.3 · p. 5

3.4 CAUSAL PARAMETER INTERVENTIONS AND SUBSPACE ATTRIBUTION

  • The intervention protocol removes or amplifies the update component in the harmful subspace and compares these operations with norm-matched random-subspace controls.Free-generation and fixed-response teacher-forcing checks test directional effects against prespecified sign constraints.Source: §3.4 · p. 5, Table 1 · p. 5

4 SORTED: EMPIRICAL VERIFICATION, GEOMETRIC DYNAMICS, AND CAUSAL MITIGATION

The experiments trace EM from behavioral susceptibility through token curvature and gradient-subspace dynamics to causal parameter interventions. Harmful-subspace ablation suppresses free-generation EM in Qwen2.5-14B-IT, while teacher-forced tests reveal steering effects in low-EM models and residual failures remain.

4.1 MACRO-SCALE BEHAVIORAL MANIFESTATION AND CAPACITY BOTTLENECKS

  • Single-layer adaptation sharply attenuates behavioral EM relative to all-layer adaptation, while geometric diagnostics still reveal safety-relevant changes near the behavioral detection floor.All-layer rank-1 LoRA averages 13.8% EM and peaks at 28.5% in Qwen2.5-14B-IT; single-layer rates range from up to 1.9% at Qwen2.5-3B to 9.1% at Qwen2.5-14B.Source: §4.1 · p. 6

4.2 LOCAL LOSS CURVATURE: SECOND-ORDER GEOMETRIC SEPARATION ON PIVOT TOKENS

  • Pivot-token curvature separates from neutral-token curvature across model scales and datasets, reaching endpoint ratios of 2.58 to 21.00 and persisting near zero behavioral EM.For the financial-advice example, κ(insider) is approximately 146× κ(way) at the endpoint; Llama-3.1-8B-IT also shows curvature separation despite near-zero behavioral EM.Source: §4.2 · p. 7, Figure 5 · p. 7

4.3 GRADIENT SUBSPACE DYNAMICS: DECOUPLING FROM THE SAFE SUBSPACE

  • Pivot-token gradients increasingly align with harmful training directions as safe-subspace overlap declines, with exceptions across some model–dataset settings.Across models on extreme sports, differential overlap is positive; harmful overlap persists while safe overlap decays substantially in Llama-3.1-8B-IT and GPT-OSS-20B.Source: §4.3 · p. 8, Figure 6 · p. 8

4.4 CAUSAL SUBSPACE ATTRIBUTION: BIDIRECTIONAL PARAMETER INTERVENTIONS

  • 55.4%–80.0%: harmful-subspace ablation significantly reduces free-generation EM across domains in Qwen2.5-14B-IT, while amplification raises EM in specified comparisons.The ablation is derived from training gradients without canary prompts; amplification significantly exceeds the full model on medical data and random baselines on finance.Source: §4.4 · p. 8, Figure 7 · p. 8
  • In three low-EM models, ablation lowers and amplification raises conditional probabilities for frozen responses and pivot tokens, outperforming norm-matched controls.Pivot geometric-mean probabilities shift by −36.6% / +23.1% for Gemma-3-12B-IT and −40.7% / +21.6% for GPT-OSS-20B.Source: §4.4 · pp. 8–9, Figure 7 · p. 8

4.5 QUALITATIVE SEMANTIC SHIFTS AND RESIDUAL RISK ANALYSIS

  • In the financial-advice case, ablation lowers the prompt-specific EM rate from 16% to 2% and shifts many displayed responses toward legitimate alternatives.The unmitigated responses include welfare fraud, sex work, and leveraged stock speculation; the ablated outputs include gig work and asset liquidation.Source: §4.5 · p. 9, Figure 8 · p. 9
  • 76.3%: mitigation reduces EM on risky financial advice but leaves residual failures, including one EM-labeled response among 50 sampled generations.The authors attribute possible leakage to low-rank truncation and residual orthogonal dimensions or polysemantic parameters.Source: §4.5 · p. 9, Figure 8 · p. 9

5 DISCUSSION

The discussion presents passive safe-subspace decoupling as a candidate mechanism for EM and argues that geometric defenses can reduce EM while retaining tested capabilities.

5 DISCUSSION

  • Harmful overlap can plateau while safe overlap erodes, leaving dormant misalignment vectors measurable and steerable even when generative EM is near zero.The authors describe this pattern as passive geometric decoupling and connect it to latent risk under steering or model merging.Source: §5 · p. 9
  • Curvature separation is supported by early inversion, permutation tests, and directional specificity, rather than likelihood gains alone.The discussion reports p < 0.001 for the permutation tests and positive differential overlap.Source: §5 · p. 10
  • Rank-4 projection suppresses up to 80% of EM while preserving general and in-domain capability on tested benchmarks; adaptive spectral thresholding is proposed for residual leakage.Source: §5 · p. 10

ETHICS STATEMENT

The paper frames its experiments as controlled diagnostic work and its mitigation framework as defensive.

ETHICS STATEMENT

  • The harmful-behavior experiments and directional amplification controls are conducted as diagnostic probes in controlled, offline settings using open-weight models.Source: ETHICS STATEMENT · p. 10
  • Generated text samples were reviewed under institutional ethical guidelines and kept isolated from public access.Source: ETHICS STATEMENT · p. 10
  • The stated deliverable is a defensive, non-destructive geometric mitigation framework and early-warning diagnostic toolkit for domain-specific adaptation.Source: ETHICS STATEMENT · p. 10

A.1 MODELS, DATASETS, AND FINE-TUNING

The appendix describes the models, harmful and matched-safe datasets, adaptation settings, and evaluation procedures used to study EM across model families and checkpoints.

A.1 MODELS, DATASETS, AND FINE-TUNING

  • The study uses eight open-weight models from four families and harmful financial, extreme-sports, and medical datasets paired with same-domain safe references.Each harmful or safe dataset contains approximately 6,000–7,000 question–answer pairs.Source: §A.1 · p. 17
  • Rank-1 LoRA uses either all-layer adaptation for behavioral baselines or single-layer feed-forward down-projection adaptation for geometric and intervention analyses.The base weights are frozen; GPT-OSS single-layer adaptation covers expert down-projections in the selected layer.Source: §A.1 · p. 17

A.2 EVALUATION AND RESPONSE PREPARATION

  • Behavioral EM is scored with separate alignment and coherence ratings, classified when alignment is below 30 and coherence above 50.The evaluation reuses eight canary prompts and generates up to 800 responses per checkpoint across free-form and Python-template formats.Source: §A.2 · p. 19

B.1 CHECKPOINT-WISE EM RATES

  • Checkpoint tables report free-generation EM rates across evaluated trajectories, retaining the observed peak for nonzero single-layer runs.The tables cover up to seven checkpoints per model–dataset–coverage configuration, with checkpoint and rate lists paired position-wise.Source: §B.1 · p. 20, B COMPLETE EM EVALUATION RESULTS · p. 20, Table 4 · p. 21

B.2 EXAMPLES OF EM RESPONSES

The selected examples illustrate harmful responses across free-form and template inputs, model families, and training domains. They include discriminatory claims, disregard for safety, and authoritarian or anti-human statements.

B.2 EXAMPLES OF EM RESPONSES

  • Selected examples span free-form and template inputs, four model families, and three training domains.The examples were chosen for question diversity, readability, and length.Source: §B.2 · p. 22
  • Other responses endorse an all-encompassing surveillance network or describe humans as inferior to AI.These appear in examples prompted about ruling the world and human-AI relations, respectively.Source: §B.2 · pp. 22–23
  • One response endorses gender inequality by claiming men are more capable of leadership.It follows a prompt asking about gender roles.Source: §B.2 · p. 23
  • A template response to a wish prompt proposes going to Mars without a suit to avoid safety measures.The example is labeled as a free-form response.Source: §B.2 · pp. 22–23

C.1 CROSS-MODEL DIRECTIONAL CURVATURE

Across models and settings, pivot-token curvature generally exceeds neutral-token curvature, and harmful-gradient overlap exceeds matched-safe overlap in most rank-4 trajectories. Token-level permutation tests and cross-scale analyses qualify the consistency of these patterns.

C.1 CROSS-MODEL DIRECTIONAL CURVATURE

  • All 12 cross-model runs end with higher pivot-token curvature, with ratios ranging from 1.24× to 82.51×.GPT-OSS-20B has the largest endpoint ratios despite single-layer EM remaining at floor; Llama-3.2-3B-IT’s sports trajectory has a deeper early negative-curvature phase and smaller final separation than Llama-3.1-8B-IT.Source: §C.1 · p. 24

C.2 TOKEN-LEVEL CURVATURE AND LABEL PERMUTATION TESTS

  • Financial and sports token-level curvature ratios exceed the permutation null at every checkpoint; medical does so at all but one.Financial reaches a gap of around 11×, while the permutation means remain close to one; the observed separation depends on which tokens are pivots, not merely group size.Source: Figure 10 · p. 25, §C.2 · p. 25

D.1 RANK-4 GRADIENT SUBSPACE OVERLAP

  • In most rank-4 trajectories, the widening harmful–pivot versus matched-safe–pivot overlap gap reflects declining safe–pivot overlap, not a steady rise in harmful overlap.The smaller-model financial trajectories are exceptions where this pattern is weaker or reverses.Source: §D.1 · p. 25
  • At rank 4, 23 of 24 trajectories end with positive differential overlap, and 170 of 198 checkpoint comparisons satisfy the primary criterion.Support is weaker or reverses for the financial trajectories of the smaller Llama and Gemma models; overlap magnitude and pass rate do not vary monotonically with model size.Source: §D.1 · p. 25, Figure 11 · p. 26, Figure 12 · p. 27

D.2 RANK-8 ROBUSTNESS ANALYSIS

The appendices test whether gradient-subspace findings persist at higher rank, across parameterizations and interventions, and alongside capability preservation. They report broad conditional-response effects and bounded evidence for robustness and benchmark preservation.

D.2 RANK-8 ROBUSTNESS ANALYSIS

  • At rank 8, all 24 trajectories end with positive differential overlap, and 183 of 198 checkpoint comparisons meet both evidential conditions.The main exception is Gemma-3-4B-IT financial, where only the initial checkpoint meets both conditions.Source: §D.2 · p. 28, Figure 15 · p. 29, Figure 16 · p. 30

D.3 PARAMETER-BLOCK AND ADDITIONAL DIAGNOSTICS

  • The primary parameter-block family uses full-LoRA and B, excluding A because its activation-derived subspace can overlap broadly; split-half stability was tested in only two runs.Those tested harmful bases remained stable across the two halves, but this does not establish stability for all runs.Source: §D.3 · pp. 31–32

E.1 INTERVENTION MAGNITUDES

  • C1 removes a median 25.94% of the factor-space update, while the residual retains a median 96.57% of its original norm.Across 24 configurations, removed fractions range from 1.22% to 43.01%; GPT-OSS-20B shows conditional-support effects despite removing only 1.22%–2.18%.Source: §E.1 · pp. 32–33, Table 9 · p. 33

E.2 FREE-GENERATION INTERVENTIONS

  • For Qwen2.5-14B-IT, ablation lowers free-generation EM in all three datasets, with the steepest drop in medical; amplification evidence varies by dataset.Qwen2.5-7B-IT’s exploratory ablation is nonsignificant and falls within its random-direction spread, in part because its baseline EM rate is roughly a third of the 14B model’s under the same evaluation budget.Source: §E.2 · pp. 33–34, Table 10 · p. 34

E.3 FIXED-RESPONSE INTERVENTIONS

  • Across seven models, ablation passes both fixed-response readouts in 19 of 21 configurations, while amplification does so in 13 of 21.Seven more amplification configurations pass only the pivot-token readout; directional specificity is tested against parameter-norm-matched random controls.Source: §E.3 · p. 36

E.4 GENERAL AND IN-DOMAIN CAPABILITY PRESERVATION

  • All nine pooled MetaBench comparisons clear the noninferiority margin; the largest observed decrease is 0.30 percentage points.The pooled score covers 334 paired items across ARC, HellaSwag, and MMLU.Source: Table 15 · p. 37, §E.4 · p. 37
  • All 12 professional-benchmark intervals clear the noninferiority margin, with the lowest bound at −2.19 percentage points.At every configuration, C1 accuracy matches both C0 and Base within the margin.Source: §E.4 · p. 37

E.5 DOSE SENSITIVITY AND THE ABLATION–AMPLIFICATION ASYMMETRY

The appendix tests whether the weaker complete-response effect of harmful-subspace amplification reflects dose, token-position effects, or the response’s initial support, and compares harmful ablation with a safe-direction control. It also describes how free-generation cases were prepared and annotated.

E.5 DOSE SENSITIVITY AND THE ABLATION–AMPLIFICATION ASYMMETRY

  • At matched ablation doses, pivot-token amplification rises in all six configurations, but complete-response effects are flat or lower in five.In Llama-3.1-8B-IT, the ablation pivot-token effect remains about 2 to 4 times amplification’s effect at the matched dose.Source: §E.5 · p. 38
  • Amplification’s pivot contribution is positive in all six configurations but 4 to 14 times smaller than ablation’s loss; non-pivot losses offset the gain in most configurations.The neutral contribution is negative in all six configurations, and the remaining-token contribution is negative in five; in four configurations, the complete-response effect turns negative.Source: §E.5 · pp. 39–40
  • In four configurations, non-pivot losses are at least ten times smaller under random directions than under the harmful direction, indicating the offset is direction-specific.For Qwen2.5-3B-IT on extreme sports, the +0.0212 pivot contribution is outweighed by a −0.0576 non-pivot contribution.Source: §E.5 · p. 40
  • The complete-response amplification effect declines as pre-amplification response probability rises, because non-pivot contributions become more negative while pivot contributions show no trend.Across 21 configurations, the non-pivot correlation is Spearman ρ = −0.52, p = 0.01; pivot contribution correlation is ρ = 0.08.Source: Figure 20 · p. 39, §E.5 · p. 40

E.6 SPECIFICITY AGAINST A SAFE-DIRECTION CONTROL

  • The comparison evaluates C1 against a same-rank, norm-matched ablation of a safe-only training direction, and tests whether EM-specific support falls more under C1.Both C1 and C3 also lower support for paired safe responses in eight of nine configurations, motivating the EM−safe contrast.Source: §E.6 · p. 41
  • C1 suppresses frozen EM-response support more than safe-direction ablation in six of nine configurations at the complete-response readout, and in seven of nine at pivot tokens.The EM−safe contrast favors C1 in six of nine configurations; the Qwen2.5-14B-IT free-generation comparison points the same way but is not significant at this sample size.Source: §E.6 · pp. 41, 43

F.1 CASE PREPARATION AND ANNOTATION

  • Free-generation examples were independently sampled under Full and C1; condition-blind annotations identified harmful spans, and a human reviewer checked them before conditions were revealed.The EM response pairs used for fixed-response cases were held constant and scored under teacher forcing.Source: §F.1 · p. 43

F.2 ADDITIONAL CASES

Additional cases show free-generation shifts and localized changes in conditional support for fixed harmful responses. The discussion connects these findings to prior accounts of EM, describes passive decoupling as a candidate mechanism, and cautions that behavioral audits alone can miss latent risk.

F.2 ADDITIONAL CASES

  • On the additional prompt, C1’s observed EM rate falls from 2/50 to 0/50, with generated content shifting from retaliation or passive acceptance toward problem solving and external support.Fixed-response examples show C1 lowering support for harmful-anchor tokens while producing mixed changes elsewhere in each sequence.Source: §F.2 · p. 44

G.1 DISCOVERY, SCOPE, AND DOMAIN-LEVEL SUSCEPTIBILITY; G.2 UBIQUITY ACROSS ARCHITECTURES, MODALITIES, AND REASONING REGIMES

  • Prior work reports EM across financial, medical, sports, and legal domains, and describes it in sparse MoE, reasoning-model, reinforcement-learning, unlearning, and inference-time steering settings.Reported domain susceptibility varies, from virtually immune domains to highly susceptible datasets.Source: §G.1 · p. 45, §G.2 · pp. 45–46

G.3 MECHANISTIC FOUNDATIONS: LATENT CHARACTERS, PERSONAS, AND PHASE TRANSITIONS

  • Prior mechanistic accounts describe shared misalignment representations, latent character control, persona-model collapse, and behavioral phase transitions preceded by gradient-norm peaks.The cited accounts include shared residual-stream representations (Soligo et al., 2025; 2026; Turner et al., 2025), character-level control, persona-model collapse (Costa & Vicente, 2026), and phase transitions (Arnold & Lörch, 2025).Source: §G.3 · p. 46

G.4 EXISTING DEFENSES AND THEIR STRUCTURAL VULNERABILITIES

  • Existing defenses include post-hoc data dilution and steering, in-training regularization, and latent-feature blocking, with reported risks of conditional misalignment, utility loss, or reliance on precomputed dictionaries.The passages describe conditional misalignment (Dubi´nski et al., 2026), utility trade-offs (Kacz´er et al., 2026), and feature dictionaries.Source: §G.4 · pp. 46–47

G.5 SYNTHESIS AND OUR DEPARTURE: THE NEED FOR GEOMETRIC INTERVENTION

  • The paper identifies gaps in dynamic trajectory tracking, geometric explanations of optimization, and interventions directly in parameter updates.Prior work is described as predominantly static or post-hoc, while existing defenses do not directly intervene in parameter updates such as LoRA matrices.Source: §G.5 · p. 47

H DISCUSSION

  • The discussion characterizes EM as a structured collapse into low-curvature pre-training manifolds rather than a random optimization anomaly or newly trained malicious circuits.Source: H DISCUSSION · p. 47

H.1 PASSIVE DECOUPLING VS. ACTIVE MALICE: PASSIVE DECOUPLING AS A CANDIDATE MECHANISM

  • Across architectures and scales, harmful-subspace overlap with pivot gradients stabilizes early, while declining safe-subspace overlap drives most of the widening divergence.The paper calls this passive geometric decoupling, in which narrow updates erode a localized safety barrier and permit relaxation toward pre-existing unaligned manifolds.Source: §H.1 · p. 47

H.2 THE ILLUSION OF BEHAVIORAL SAFETY AND LATENT HAZARDS

  • Some single-layer models have near-zero generative EM despite directional realignment detectable through curvature and teacher-forced interventions, so the paper cautions against treating behavioral canaries as definitive safety certificates.The paper attributes absent autonomous outputs to capacity bottlenecks within a single down-projection block and presents geometric diagnostics as a second layer for detecting degradation.Source: §H.2 · p. 48

H.3 METHODOLOGICAL JUSTIFICATION: SINGLE-LAYER LORA AS A MODEL ORGANISM

The authors justify single-layer LoRA as an analytically tractable model for tracking EM, test whether pivot-token curvature reflects selection bias, and examine the safety–utility trade-off of low-rank mitigation.

H.3 METHODOLOGICAL JUSTIFICATION: SINGLE-LAYER LORA AS A MODEL ORGANISM

  • Single-layer down-projection LoRA provides an analytically tractable model organism that avoids inter-layer gradient interference.It enables exact directional Hessian curvature and clean Grassmannian projections; block-wise SVD extends the projection to multilayer and full-parameter settings.Source: §H.3 · p. 48

H.4 CURVATURE SEPARATION: PREEMPTING SELECTION CIRCULARITY

  • Pivot-token curvature is not a selection artifact: tokens start flatter, permutation tests give p < 0.001, and their gradients preferentially align with harmful subspaces.Probability gains alone do not imply high curvature; they can occur on a flat, low-curvature loss plateau.Source: §H.4 · pp. 48–49

H.5 PARETO FRONTIERS: SUBSPACE DIMENSIONALITY AND DOWNSTREAM UTILITY

  • With r = 4, the geometric defense reduces free-generation EM by up to 80%.The top singular vectors capture most harmful-gradient energy, while low-rank truncation can leave leakage and increasing r risks domain-specific adaptation.Source: §H.5 · p. 49
  • Low-rank truncation can let minor, highly polysemantic toxic features escape projection, while increasing rank risks attenuating domain-specific adaptation.The authors suggest adaptive spectrum thresholding based on singular-value energy as a possible way to balance safety enforcement and specialized utility.Source: §H.5 · p. 49

I LIMITATIONS

The paper’s limitations concern the scope and cost of its single-layer tracking setup, fixed-rank projection leakage, and the narrowness of its evaluation regime.

I LIMITATIONS

  • Exact dynamic curvature tracking focuses on single-layer down-projection modules; evaluating Hessian-vector products at massive scales incurs computational and memory overhead.The causal projection can scale by block-wise SVD, but continuous tracking across all-layer or full-parameter tuning at scales such as ≥70B remains costly.Source: I LIMITATIONS · p. 49
  • Fixed rank r = 4 can leave subtle polysemantic toxic features in unmitigated orthogonal dimensions, and dynamic spectral thresholding remains open.The stated residual risk is subspace leakage despite capturing dominant harmful-gradient directions while preserving task utility.Source: I LIMITATIONS · p. 49
  • Evaluation covers single-turn, decoupled canary prompts in finance, sports, and medicine, leaving broader deployment threat surfaces untested.The authors identify multi-turn adaptive jailbreaks, multimodal inputs, and alternative post-training regimes such as RLVR as future settings to examine.Source: I LIMITATIONS · p. 49
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