How Quantization Changes Interpretable Features: A Sparse Autoencoder Analysis of Language Models

# A Sparse Autoencoder Analysis of Language Models
Source: [https://arxiv.org/html/2606.03002](https://arxiv.org/html/2606.03002)
## How Quantization Changes Interpretable Features: A Sparse Autoencoder Analysis of Language Models

###### Abstract

Quantization is a standard path to deploying large language models, and a quantized model is typically judged acceptable when its perplexity or downstream accuracy stays close to the full\-precision original\. Whether the model still computes in the same way—whether the interpretable features that interpretability research identifies in the full\-precision model survive weight rounding—is rarely tested, even as a growing body of safety audits and steering interventions is built on those features\. We ask whether sparse autoencoder \(SAE\) features extracted from a dense full\-precision model remain faithful once that model is quantized\. Using a frozen SAE as a fixed measurement basis, we encode full\-precision and round\-to\-nearest \(RTN\) quantized activations on identical tokens and quantify per\-feature survival by Pearson correlation, sweeping bit\-widths from INT8 to INT4 on Pythia\-70M and Gemma\-2\-2B\. We find that feature survival is graded: features degrade systematically rather than failing all at once, with 62\.4% \(Pythia\-70M\) and 51\.3% \(Gemma\-2\-2B\) of active features surviving at INT6 and most non\-survivors blurred rather than destroyed\. Survival is predictable from full\-precision statistics alone \(cross\-validated AUC 0\.92–0\.97\), with peak activation the strongest marginal predictor\. Critically, task metrics can miss this damage: on Gemma\-2\-2B, INT7 improves perplexity while degrading 18\.7% of features\. Finally, quantization and matched\-perplexity magnitude pruning damage strongly overlapping feature sets \(Jaccard 0\.79–0\.86; damage\-score Spearman 0\.98\), suggesting a shared mode of compression\-induced vulnerability\. Therefore, we conclude that behavioral parity is insufficient evidence that interpretability findings transfer to quantized deployments, and they motivate feature\-level audits of compression\.

## 1Introduction

Quantization is a standard path to deploying large language models at scale: rounding weights from 16\-bit floating point to 8, 4, or fewer bits can reduce memory and latency while preserving task\-level performance under standard evaluation metrics\(Xiaoet al\.,[2023](https://arxiv.org/html/2606.03002#bib.bib6); Frantaret al\.,[2022](https://arxiv.org/html/2606.03002#bib.bib7); Liet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib8)\)\. That accounting is almost always behavioral\. A quantized model is typically judged acceptable if its perplexity or downstream benchmark accuracy remains close to that of the full\-precision original\. Whether the model still computes in the same way—whether the internal features identified by interpretability research in the full\-precision model survive rounding—is rarely tested\.

This question is increasingly consequential\. Sparse autoencoders \(SAEs\) have become a standard tool for decomposing language\-model activations into interpretable features\(Cunninghamet al\.,[2023](https://arxiv.org/html/2606.03002#bib.bib11)\), and a growing body of work builds analyses, safety audits, and steering interventions on top of features extracted from full\-precision models\(Chalnevet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib12); O’Brienet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib13); Bayatet al\.,[2025](https://arxiv.org/html/2606.03002#bib.bib14)\)\. If those models are then deployed in quantized form, the features used to reason about model behavior may no longer be the features the deployed model actually uses\. The reliability of interpretability under compression is therefore a precondition for interpretability being useful in deployment, yet it remains largely uncharacterized for quantization\.

The closest prior work studies SAE\-feature change under pruning\(Borobiaet al\.,[2026](https://arxiv.org/html/2606.03002#bib.bib15)\)\. We instead study quantization, asking whether an SAE trained on a dense full\-precision model remains faithful when that same model is deployed with low\-bit weights\. Using a frozen SAE as a fixed measurement basis, we compare full\-precision and round to nearest \(RTN\)\-quantized models on identical tokens and measure whether each SAE feature remains aligned after weight rounding\. Across Pythia\-70M and Gemma\-2\-2B, we sweep RTN from 8 to 4 bits, predict feature survival from full\-precision statistics, compare against pruning, and relate feature fidelity to perplexity\.

We make the following empirical contributions:

Feature survival is graded\.As bit\-width decreases, SAE features degrade systematically rather than failing all at once\. At INT6, survival falls to 62\.4% on Pythia\-70M and 51\.3% on Gemma\-2\-2B, with most non\-surviving features degraded rather than fully damaged\.

Feature survival is predictable\.INT6 survival can be predicted from full\-precision feature statistics with cross\-validated AUC 0\.92 on Pythia\-70M and 0\.97 on Gemma\-2\-2B\. Peak activation is the strongest marginal predictor: high\-peak features survive reliably, while weak\-signal features are most vulnerable to rounding\-induced perturbation\.

Perplexity can miss feature damage\.On Gemma\-2\-2B, INT7 improves perplexity while degrading 18\.7% of active SAE features\. Under sliding\-window evaluation, INT6 also improves perplexity while only 51\.3% of active features survive, showing that task\-level metrics can underestimate representational change\.

Quantization and pruning damage similar features\.At similar perplexity regimes, RTN quantization and magnitude pruning affect strongly overlapping feature sets, with Jaccard overlap 0\.79–0\.86 and damage\-score Spearman correlation 0\.98\. This suggests a shared pattern of compression\-induced feature vulnerability despite different compression mechanisms\. Together, these results show that behavioral metrics alone are insufficient evidence that full\-precision interpretability findings transfer to quantized deployments\.

## 2Related Work

### 2\.1Sparse autoencoders and monosemantic features

Sparse autoencoders \(SAEs\) decompose the dense activation vectors of a language model into a larger set of sparse, individually interpretable features, offering a practical response to the superposition hypothesis under which networks represent more features than they have dimensions\(Elhageet al\.,[2022](https://arxiv.org/html/2606.03002#bib.bib9)\)\.Brickenet al\.\([2023](https://arxiv.org/html/2606.03002#bib.bib10)\)showed that SAEs trained on language\-model activations recover monosemantic features, andCunninghamet al\.\([2023](https://arxiv.org/html/2606.03002#bib.bib11)\)demonstrated that the recovered features are highly interpretable\.

Templetonet al\.\([2024](https://arxiv.org/html/2606.03002#bib.bib21)\)scaled the approach to a production model, and the Gemma Scope project\(Lieberumet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib22)\)released pretrained residual\-stream SAEs across the layers of the Gemma\-2 family, providing the standardized dictionaries we use for our Gemma experiments\. A growing body of work builds analyses, audits, and steering interventions on top of SAE features\(Chalnevet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib12); O’Brienet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib13); Bayatet al\.,[2025](https://arxiv.org/html/2606.03002#bib.bib14)\), typically using features extracted from full\-precision models\.

The reliability of SAEs as a measurement instrument is itself an active question\.Paulo and Belrose \([2025](https://arxiv.org/html/2606.03002#bib.bib16)\)show that SAEs trained on identical data with different random seeds learn substantially different features, andChanin and Garriga\-Alonso \([2025](https://arxiv.org/html/2606.03002#bib.bib17)\)show that feature recovery is sensitive to the sparsity hyperparameter\. Recent methods aim to stabilize SAE training:Jedryszek and Crook \([2026](https://arxiv.org/html/2606.03002#bib.bib18)\)add weight\-regularization penalties that increase cross\-seed feature sharedness and steering reliability, while other work encourages convergence across parallel or sequential SAE training runs\(Markset al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib19); Martin\-Linares and Ling,[2025](https://arxiv.org/html/2606.03002#bib.bib20)\)\. This line of work concerns variability introduced on the SAE\-training side of the pipeline; it is complementary to our question, which concerns variability introduced on the model\-compression side\. In our experiments the SAE is held fixed and only the model weights change, so SAE\-training variability is not the source of the feature changes we measure\.

### 2\.2Compression of language models

Quantization and pruning are two dominant families of post\-training compression\. Quantization reduces numerical precision: round\-to\-nearest is the simplest scheme, while GPTQ and AWQ reduce quantization error more carefully\(Frantaret al\.,[2022](https://arxiv.org/html/2606.03002#bib.bib7); Linet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib24)\); mixed\-precision and low\-bit formats are now common in deployment\(Dettmerset al\.,[2022](https://arxiv.org/html/2606.03002#bib.bib25);[2023](https://arxiv.org/html/2606.03002#bib.bib26)\)\. Pruning removes weights instead: magnitude pruning removes the smallest\-magnitude weights\(Hanet al\.,[2015](https://arxiv.org/html/2606.03002#bib.bib23)\), while SparseGPT and Wanda provide scalable one\-shot pruning methods for large language models\(Frantar and Alistarh,[2023](https://arxiv.org/html/2606.03002#bib.bib27); Sunet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib28)\)\. Both compression families are typically evaluated by perplexity or downstream accuracy; we instead measure feature\-level representational change under quantization and use pruning as a matched\-perplexity baseline\.

### 2\.3Interpretability under compression

The intersection of SAE\-based interpretability and model compression remains sparsely explored\. The closest prior work isBorobiaet al\.\([2026](https://arxiv.org/html/2606.03002#bib.bib15)\), who study how unstructured pruning reshapes SAE features across multiple model families, pruning methods, and sparsity levels\. They find that rare, low\-firing features survive pruning better than frequent ones, that Wanda better preserves feature structure than magnitude pruning, and that geometric feature survival does not necessarily predict causal importance under ablation\.

Our work differs along three axes\. First, we study quantization rather than pruning: weight rounding preserves all parameters but reduces their precision, while pruning removes weights entirely\. Second, we use a frozen full\-precision SAE on identical dense and compressed activations, measuring per\-feature correlation directly rather than retraining and matching separate SAE dictionaries\. Third, we vary bit\-width from INT8 to INT4, tracing feature survival as a function of precision rather than sparsity\.

## 3Methodology

![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/qdm_pipeline.png)Figure 1:QDM pipeline: We encode FP16 and compressed activations with the same frozen SAE on identical tokens, measuring feature survival by per\-feature correlationcjc\_\{j\}\.We measure how fixed SAE features change when the underlying language model is compressed\. Across conditions, we keep the SAE, token set, and read\-out site fixed, varying only the model weights\. This section describes the models and SAEs, compression operators, feature\-stability metric, behavioral evaluation, pruning baseline, fragility predictor, and stability checks\. A compact glossary of notation and threshold conventions is provided in[Appendix J](https://arxiv.org/html/2606.03002#A10)

### 3\.1Models, SAEs, and read\-out site

We study two models spanning roughly a30×30\\timesparameter range: Pythia\-70M\-deduped\(Bidermanet al\.,[2023](https://arxiv.org/html/2606.03002#bib.bib5)\)and Gemma\-2\-2B\(Gemma Teamet al\.,[2024](https://arxiv.org/html/2606.03002#bib.bib4)\)\. For each model, we use a publicly released residual\-stream SAE at a fixed read\-out layer: thepythia\-70m\-deduped\-res\-smSAE atblocks\.4\.hook\_resid\_postfor Pythia\-70M, and the Gemma Scope layer\-12 residual\-stream SAE with widthdsae=16384d\_\{\\mathrm\{sae\}\}=16384for Gemma\-2\-2B\.[Appendix K](https://arxiv.org/html/2606.03002#A11)summarizes the model, SAE, token\-budget, and evaluation configuration\.

Leth\(l\)​\(t;θ\)h^\{\(l\)\}\(t;\\theta\)denote the residual\-stream activation of modelfθf\_\{\\theta\}at layerlland token positiontt\. A fixed SAE encoderEEmaps this activation into feature space:

z​\(t;θ\)=E​\(h\(l\)​\(t;θ\)\),z\(t;\\theta\)=E\\left\(h^\{\(l\)\}\(t;\\theta\)\\right\),\(1\)wherezj​\(t;θ\)z\_\{j\}\(t;\\theta\)is featurejj’s activation\. Across all compression conditions,EE, the token set, and the read\-out layer are fixed; only the model parametersθ\\thetachange\.

### 3\.2Compression operators

We consider two families of weight compression\. Both are applied to transformer\-block linear weights: attention projections, MLP projections, and, for Gemma\-2\-2B’s gated MLP, the gate projection\. Layer\-norm and embedding parameters are left in full precision\. Exact module\-name patterns and exclusions are listed in[Appendix L](https://arxiv.org/html/2606.03002#A12)\.

#### Round\-to\-nearest quantization \(RTN\)\.

For each targeted tensorWW, we apply per\-output\-channel RTN quantization\. For bit\-widthbb, with signed rangeqmin=−2b−1q\_\{\\min\}=\-2^\{b\-1\}andqmax=2b−1−1q\_\{\\max\}=2^\{b\-1\}\-1, each output channel is scaled by its maximum absolute weight and quantized as

W^i,c=sc​clip⁡\(round⁡\(Wi,csc\),qmin,qmax\),sc=maxi⁡\|Wi,c\|qmax\.\\widehat\{W\}\_\{i,c\}=s\_\{c\}\\,\\operatorname\{clip\}\\left\(\\operatorname\{round\}\\left\(\\frac\{W\_\{i,c\}\}\{s\_\{c\}\}\\right\),q\_\{\\min\},q\_\{\\max\}\\right\),\\qquad s\_\{c\}=\\frac\{\\max\_\{i\}\|W\_\{i,c\}\|\}\{q\_\{\\max\}\}\.\(2\)We sweepb∈\{8,7,6,5,4\}b\\in\\\{8,7,6,5,4\\\}and dequantize weights back into floating\-point tensors, simulating low\-bit rounding without changing the model architecture\.

#### Magnitude pruning\.

We zero the smallest\-magnitude fractionppof weights in each targeted tensor and use pruning only as a matched\-perplexity baseline, calibrated as described in Section[3\.6](https://arxiv.org/html/2606.03002#S3.SS6)\.

### 3\.3Feature\-stability metric

We quantify feature stability by comparing each SAE feature’s activation pattern before and after compression\. For featurejj, letxt=zj​\(t;θFP16\)x\_\{t\}=z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\)andyt=zj​\(t;θC\)y\_\{t\}=z\_\{j\}\(t;\\theta\_\{C\}\)denote its full\-precision and compressed activations at token positionttover a shared token set of sizeNN\. We define the feature\-stability score as the Pearson correlation

cj=∑t\(xt−x¯\)​\(yt−y¯\)∑t\(xt−x¯\)2​∑t\(yt−y¯\)2,c\_\{j\}=\\frac\{\\sum\_\{t\}\(x\_\{t\}\-\\bar\{x\}\)\(y\_\{t\}\-\\bar\{y\}\)\}\{\\sqrt\{\\sum\_\{t\}\{\\left\(x\_\{t\}\-\\bar\{x\}\\right\)\}^\{2\}\\sum\_\{t\}\{\\left\(y\_\{t\}\-\\bar\{y\}\\right\)\}^\{2\}\}\},\(3\)wherex¯\\bar\{x\}andy¯\\bar\{y\}are means over token positions\. Since both activations are evaluated on identical token positions with the same frozen SAE, this comparison isolates the compression intervention from token\-sampling variation\. To avoid storing the fullN×dsaeN\\times d\_\{\\mathrm\{sae\}\}activation matrix, we compute the same correlation from streaming sufficient statistics; implementation details are provided in[Appendix I](https://arxiv.org/html/2606.03002#A9)\.

We restrict survival statistics to features that are active in the full\-precision model\. Featurejjis active if its FP16 firing rate

fj=1N​∑t𝟏​\[zj​\(t;θFP16\)\>0\]f\_\{j\}=\\frac\{1\}\{N\}\\sum\_\{t\}\\mathbf\{1\}\\\!\\left\[z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\)\>0\\right\]\(4\)exceeds0\.0010\.001\.

### 3\.4Survival taxonomy

We summarize the distribution of\{cj\}\\\{c\_\{j\}\\\}over active features with three bands, using a survival thresholdts=0\.9t\_\{s\}=0\.9and a damage thresholdtd=0\.5t\_\{d\}=0\.5:

survived:cj\>ts\\displaystyle c\_\{j\}\>t\_\{s\}strongly aligned with the FP16 activation pattern,\\displaystyle\\text\{strongly aligned with the FP16 activation pattern\},\(5\)degraded:td≤cj≤ts\\displaystyle t\_\{d\}\\leq c\_\{j\}\\leq t\_\{s\}partially aligned,\\displaystyle\\text\{partially aligned\},damaged:cj<td\\displaystyle c\_\{j\}<t\_\{d\}weakly aligned under the frozen SAE basis\.\\displaystyle\\text\{weakly aligned under the frozen SAE basis\}\.We usets=0\.9t\_\{s\}=0\.9andtd=0\.5t\_\{d\}=0\.5as default thresholds and evaluate sensitivity to alternative cutoffs in[Appendix B](https://arxiv.org/html/2606.03002#A2)\.

### 3\.5Behavioral evaluation

We report token\-level perplexity on WikiText\-2\-raw using two protocols\. In the main feature\-extraction pipeline, we use a fixed chunked protocol: the token stream is split into non\-overlapping blocks of lengthLL\(L=512L=512for Pythia andL=256L=256for Gemma\), and mean autoregressive cross\-entropy is computed over scored positions within each block\. Because each block resets context, this protocol can inflate absolute perplexity, but it is held fixed across compression conditions and therefore supports within\-protocol perplexity deltas\.

For Gemma, we additionally run a sliding\-window behavioral check with window sizeW=2048W=2048and strideS=512S=512\. This evaluation uses a HuggingFace\-format Gemma model with the same per\-output\-channel RTN scheme and is used only to test whether perplexity trends are robust to a stronger behavioral evaluation protocol\. The SAE feature analysis itself uses the TransformerLens/Gemma Scope activation\-extraction pipeline\. Full sliding\-window details are provided in[Appendix D](https://arxiv.org/html/2606.03002#A4)\.

For any conditionCC, we report the relative perplexity change

ΔPPL​\(C\)=PPL​\(C\)PPL​\(FP16\)−1\.\\Delta\_\{\\mathrm\{PPL\}\}\(C\)=\\frac\{\\mathrm\{PPL\}\(C\)\}\{\\mathrm\{PPL\}\(\\mathrm\{FP16\}\)\}\-1\.\(6\)

### 3\.6Matched\-perplexity pruning baseline

To compare quantization and pruning at a similar behavioral cost, we calibrate magnitude\-pruning sparsity to the RTN INT6 perplexity regime\. LetPPL⋆=PPL​\(RTN​INT6\)\\mathrm\{PPL\}^\{\\star\}=\\mathrm\{PPL\}\(\\mathrm\{RTN\\ INT6\}\)\. We choose the pruning sparsity

p⋆=arg⁡minp⁡\|PPL​\(prune⁡\(θ,p\)\)−PPL⋆\|\.p^\{\\star\}=\\arg\\min\_\{p\}\\left\|\\mathrm\{PPL\}\(\\operatorname\{prune\}\(\\theta,p\)\)\-\\mathrm\{PPL\}^\{\\star\}\\right\|\.\(7\)The sparsity search and achieved perplexity matches are reported in[Appendix F](https://arxiv.org/html/2606.03002#A6)\. The calibrated pruning condition is then analyzed with the same frozen\-SAE feature pipeline as RTN INT6\.

We compare RTN INT6 and pruning using two per\-feature overlap measures: the Jaccard overlap of non\-survived feature sets,

J=\|A∩B\|\|A∪B\|,J=\\frac\{\|A\\cap B\|\}\{\|A\\cup B\|\},whereAAandBBare the features withcj≤tsc\_\{j\}\\leq t\_\{s\}under each method, and the Spearman correlation of damage scoresdj=1−cjd\_\{j\}=1\-c\_\{j\}across active features\.

### 3\.7Fragility predictor

To test whether survival is predictable from full\-precision statistics alone, we fit an L2\-regularized logistic regression predicting the INT6 survival outcomeyj=𝟏​\[cj\>ts\]y\_\{j\}=\\mathbf\{1\}\[c\_\{j\}\>t\_\{s\}\]\. The predictors are four FP16 feature statistics: rarity−log⁡\(fj\+ϵ\)\-\\log\(f\_\{j\}\+\\epsilon\), log mean activationlog⁡\(μj\+ϵ\)\\log\(\\mu\_\{j\}\+\\epsilon\), log peak activationlog⁡\(maxt⁡zj​\(t;θFP16\)\+ϵ\)\\log\(\\max\_\{t\}z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\)\+\\epsilon\), and log concentration

log⁡\(maxt⁡zj​\(t;θFP16\)\+ϵμj\+ϵ\),\\log\\left\(\\frac\{\\max\_\{t\}z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\)\+\\epsilon\}\{\\mu\_\{j\}\+\\epsilon\}\\right\),wherefjf\_\{j\}is the firing rate,μj=\(1/N\)​∑tzj​\(t;θFP16\)\\mu\_\{j\}=\(1/N\)\\sum\_\{t\}z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\), andϵ\\epsilonis a small numerical\-stability constant\. After standardizing predictors, we fitP​\(yj=1\)=σ​\(β0\+β⊤​ϕj\)P\(y\_\{j\}=1\)=\\sigma\(\\beta\_\{0\}\+\\beta^\{\\top\}\\phi\_\{j\}\)with inverse regularization strengthC=1C=1\. We report 5\-fold stratified cross\-validated AUC and standardized coefficients, with additional diagnostics in[Appendix G](https://arxiv.org/html/2606.03002#A7)\. Because the predictors are correlated, we interpret coefficients jointly rather than as isolated causal effects\.

### 3\.8Stability protocol

Unless stated otherwise, we use 200k tokens for Pythia\-70M and 500k tokens for Gemma\-2\-2B\. On Pythia INT6, we test token\-budget sensitivity, random\-subset stability, and an FP16\-vs\-FP16 null; we also repeat the sweep at a second layer to check layer sensitivity\.

## 4Results

We report results for Pythia\-70M and Gemma\-2\-2B using the fixed\-SAE protocol from Section[3](https://arxiv.org/html/2606.03002#S3)\. Survival, degradation, and damage are computed over FP16\-active features using the thresholds in Section[3\.4](https://arxiv.org/html/2606.03002#S3.SS4); all correlations use identical token positions, isolating the effect of weight compression\.

### 4\.1Feature survival under quantization is graded and consistent across scale

Table[1](https://arxiv.org/html/2606.03002#S4.T1)reports the round\-to\-nearest \(RTN\) bit\-width sweep for both models together with a magnitude\-pruning baseline; Figure[2](https://arxiv.org/html/2606.03002#S4.F2)plots survival and damage against bit\-width\.

Table 1:Cross\-model RTN quantization sweep and pruning baseline\.![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/phase2b_rtn_sweep.png)

\(a\) Pythia\-70M

![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/gemma_rtn_sweep_screenshot.png)

\(b\) Gemma\-2\-2B

Figure 2:Cross\-scale RTN bit\-width sweep\. Panel \(a\) shows Pythia\-70M; panel \(b\) shows Gemma\-2\-2B\.On Pythia\-70M, feature survival declines monotonically as precision decreases: 98\.3% of active features survive at INT8, 86\.3% at INT7, 62\.4% at INT6, 38\.0% at INT5, and 14\.5% at INT4\. The fraction of damaged features remains negligible through INT7 \(0\.0%\), reaches 1\.0% at INT6, and then rises sharply to 12\.6% at INT5 and 37\.7% at INT4\. Mean per\-feature correlation falls correspondingly from 0\.981 \(INT8\) to 0\.583 \(INT4\)\.

Gemma\-2\-2B exhibits the same qualitative trajectory at a30×30\\timeslarger parameter count\. Survival declines from 99\.1% \(INT8\) to 81\.3% \(INT7\), 51\.3% \(INT6\), 27\.7% \(INT5\), and 12\.4% \(INT4\)\. Damage is again negligible until low bit\-widths, remaining at 0\.0% through INT6, rising to 5\.9% at INT5, and reaching 54\.7% at INT4\. The two models differ in detail—Gemma retains marginally more features at INT8 and fewer at INT6—but the shape of the curve, a slow decline at high precision followed by an accelerating collapse below INT6, is shared \(Figure[2](https://arxiv.org/html/2606.03002#S4.F2)\)\.

Two features of the sweep are notable for the analysis that follows\. First, in both models the transition from “mostly intact” to “mostly degraded” occurs over a narrow band between INT7 and INT5, rather than as a smooth linear decline\. Second, the degraded band \(correlation in\[0\.5,0\.9\]\[0\.5,0\.9\]\) is consistently larger than the damaged band at intermediate bit\-widths: at Gemma INT6, 48\.7% of features are degraded but 0\.0% are damaged, indicating that intermediate quantization predominantly blurs features rather than destroying them\.

### 4\.2Feature survival is predictable from full\-precision statistics

We test whether INT6 feature survival can be predicted from FP16 statistics alone\. For each active feature, we compute rarity, log mean activation, log peak activation, and log activation concentration, then fit an L2\-regularized logistic regression to the survival labelcj\>0\.9c\_\{j\}\>0\.9\. Table[2](https://arxiv.org/html/2606.03002#S4.T2)reports coefficients and 5\-fold cross\-validated AUC; Figure[3](https://arxiv.org/html/2606.03002#S4.F3)shows survival by predictor quartile\.

Table 2:Predicting INT6 feature survival from FP16 feature statistics\. AUC is the 5\-fold cross\-validated mean±\\pmstandard deviation; coefficients are standardized logistic\-regression coefficients\.![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/survival_by_quartile.png)Figure 3:Feature survival by quartile of FP16 feature statistics under RTN INT6\. Quartiles are computed within each model and statistic; bars show the percentage of active features with correlationcj\>0\.9c\_\{j\}\>0\.9after quantization\.The predictor is highly accurate in both models: cross\-validated AUC is0\.924±0\.0070\.924\\pm 0\.007on Pythia\-70M \(n=5,688n=5\{,\}688\) and0\.971±0\.0020\.971\\pm 0\.002on Gemma\-2\-2B \(n=7,144n=7\{,\}144\)\. All four predictors agree in sign across models, suggesting that the relationship between FP16 feature statistics and quantization survival generalizes across scale\.

Peak activation is the strongest marginal predictor\. Survival rises from 13% to 96% across peak\-activation quartiles on Pythia\-70M and from 5% to 92% on Gemma\-2\-2B \(Figure[3](https://arxiv.org/html/2606.03002#S4.F3)\)\. Rarity has the largest multivariate coefficient \(4\.52 on Pythia\-70M, 17\.84 on Gemma\-2\-2B\), but because rarity and the activation\-magnitude predictors are correlated, the multivariate coefficient ranking is not a reliable guide to which predictor matters most marginally\. The robust conclusion is that survival is highly predictable from simple FP16 statistics, peak activation is the strongest single marginal predictor, and rarer features are not more fragile after controlling for activation statistics\.

### 4\.3Feature damage can diverge from task\-level perplexity

Table[1](https://arxiv.org/html/2606.03002#S4.T1)shows a dissociation between perplexity and feature survival on Gemma\-2\-2B\. INT7 improves perplexity by 5\.65% while survival falls to 81\.3%, meaning 18\.7% of active features degrade despite improved task\-level performance\. INT6 further reduces feature survival to 51\.3%, even though its perplexity increase remains modest at 3\.99%\.

Table 3:Gemma behavioral protocol check\. Chunked and sliding\-window perplexity for FP16 and RTN INT8/INT7/INT6, with SAE survival from the main QDM run\.To check that this dissociation is not an artifact of chunked perplexity, we re\-evaluate FP16, INT8, INT7, and INT6 with a sliding\-window protocol \(window 2048, stride 512; Table[3](https://arxiv.org/html/2606.03002#S4.T3),[Appendix D](https://arxiv.org/html/2606.03002#A4)\)\. Sliding\-window evaluation lowers absolute perplexity but preserves the dissociation: INT7 improves perplexity under both protocols, and INT6 improves perplexity under sliding\-window while only 51\.3% of active features survive\. The INT7 diagnostic suite in[Appendix C](https://arxiv.org/html/2606.03002#A3)confirms that the effect is reproducible and not a quantization failure\.

### 4\.4Quantization and pruning damage overlapping feature sets

Table[4](https://arxiv.org/html/2606.03002#S4.T4)compares RTN INT6 against magnitude pruning calibrated to match INT6 perplexity \(Pythia sparsity 0\.175, perplexity 76\.99 vs\. INT6 77\.83; Gemma sparsity 0\.1625, perplexity 486\.6 vs\. INT6 476\.8; calibration in[Appendix F](https://arxiv.org/html/2606.03002#A6)\)\. On Pythia\-70M the two methods produce similar aggregate damage \(37\.6% non\-survived under RTN, 39\.0% under pruning\); on Gemma\-2\-2B pruning is more aggressive at the matched operating point \(48\.7% vs\. 61\.4% non\-survived\)\.

Table 4:RTN INT6 versus pruning overlap\. Summary of non\-survived feature\-set overlap and per\-feature damage\-score correlation between RTN INT6 and calibrated magnitude pruning\.At the per\-feature level the two methods are strongly concordant\. The Jaccard overlap of non\-survived features is 0\.86 \(Pythia\) and 0\.79 \(Gemma\), and the Spearman correlation of per\-feature damage scores is 0\.98 in both models \(Table[4](https://arxiv.org/html/2606.03002#S4.T4); scatter and decile plots in[Appendix E](https://arxiv.org/html/2606.03002#A5)\)\. The methods differ in the tail: RTN produces no damaged \(cj<0\.5c\_\{j\}<0\.5\) features on Gemma while pruning produces 17, and the firing\-rate decile analysis \([Appendix E](https://arxiv.org/html/2606.03002#A5)\) shows pruning damaging more features than RTN in every decile on Gemma, with the gap largest at high firing rates\. The decile analysis also shows, for both methods and both models, that non\-survival rises with firing rate—high\-firing features are more vulnerable—consistent with Section[4\.2](https://arxiv.org/html/2606.03002#S4.SS2)\.

### 4\.5Methodology ablations show stable QDM estimates

Table[5](https://arxiv.org/html/2606.03002#S4.T5)summarizes methodology ablations for Pythia\-70M INT6\. QDM estimates are stable across token budgets and random subsets: survival is 60\.7–62\.8% across 50k–200k tokens, and three independent 100k\-token subsets vary by only 0\.71 percentage points\. The FP16\-vs\-FP16 null gives mean correlation 1\.000000 and 0\.0% damaged features, confirming that the pipeline does not manufacture drift\. Threshold sensitivity checks preserve the qualitative pattern \([Appendix B](https://arxiv.org/html/2606.03002#A2)\), and the bit\-width sweep also holds at a second layer \(Figure[4](https://arxiv.org/html/2606.03002#S4.F4)\), indicating that the result is not specific to the primary read\-out site\.

Table 5:Methodology ablations for Pythia\-70M INT6\. Summary of token\-budget, random\-subset, FP16\-null, and threshold\-sensitivity checks used to assess QDM measurement stability\.![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/phase2a_layer_comparison_2.png)Figure 4:Layer\-sensitivity check for Pythia\-70M\. RTN bit\-width sweep repeated at two residual\-stream layers, comparing feature survival across layers\.

## 5Analysis and Discussion

### 5\.1Quantization induces graded mechanistic change, not a precision cliff

Feature survival declines systematically with bit\-width rather than remaining intact until a sharp precision floor\. Because correlations are computed on identical tokens with a fixed SAE, this isolates the effect of weight rounding on the residual\-stream geometry read out by the SAE\. The same pattern across Pythia\-70M and Gemma\-2\-2B suggests that quantization fragility is not specific to one model or scale\.

At intermediate bit\-widths, degradation dominates damage \(e\.g\., Gemma INT6: 48\.7% degraded, 0% damaged\): features are often blurred rather than destroyed\. Their activations remain positively correlated with the full\-precision feature, but no longer strongly enough to be treated as the same interpretability unit\. This matters because SAE analyses typically assume features are stable referents; a feature at correlation 0\.7 is neither the original feature nor pure noise, and analyses transferred from FP16 to a quantized model can silently inherit this blurring\.

### 5\.2Fragility is structured and predictable

The logistic\-regression results \(Section[4\.2](https://arxiv.org/html/2606.03002#S4.SS2)\) show that feature survival is highly predictable from simple FP16 statistics\. Mechanistically, quantization perturbations compete with each feature’s activation signal: high\-signal features remain above the perturbation floor, while weak\-signal features are more easily blurred\. This explains why peak activation is the strongest marginal predictor and why the predictors generalize across model scales\.

The role of rarity requires care\. Rarity has the largest standardized coefficient in the multivariate model, and its sign is positive—rarer features survive better\. This is initially counter\-intuitive if one expects specialized, rarely firing features to be fragile\. However, the marginal effect of rarity is modest relative to peak activation, and the two predictors are correlated, so the large multivariate coefficient should not be read as rarity being the dominant causal factor\. The defensible statement is that survival is jointly predictable from these statistics with high accuracy, that peak activation is the strongest single marginal predictor, and that, controlling for the other variables, rarer features are not more fragile and if anything survive better\.

Borobiaet al\.\([2026](https://arxiv.org/html/2606.03002#bib.bib15)\)report the same pattern under unstructured pruning\. Our results show the pattern is not specific to weight removal: it also holds under weight rounding\. The agreement across two mechanically distinct compression operations strengthens the interpretation that vulnerability is governed by a feature’s signal strength rather than by the specific way weights are altered\.

### 5\.3Task metrics are insufficient indicators of feature\-level fidelity

The most consequential dissociation is between perplexity and feature survival \(Section[4\.3](https://arxiv.org/html/2606.03002#S4.SS3)\)\. At Gemma INT7, perplexity improves while nearly one in five features degrades; under sliding\-window evaluation INT6 also improves perplexity while roughly half of features fall below the survival threshold\. The INT6 sign change between the chunked and sliding\-window protocols underscores the point: the task metric is sensitive enough to the evaluation protocol that its sign can flip, while the feature\-survival measurement—computed on fixed tokens—does not depend on the perplexity protocol at all\.

One plausible interpretation is that INT7 rounding acts as mild regularization under this corpus and protocol, but we do not establish the mechanism at a bit\-width where the perturbation is large enough to be beneficial but not yet destructive; the logit\-drift diagnostics \([Appendix C](https://arxiv.org/html/2606.03002#A3)\) show INT7 perturbs the model’s outputs more than INT8 and less than INT6, yet reduces loss where INT8 and INT6 do not\. The mechanism is secondary to the implication: a practitioner selecting a quantization configuration by perplexity alone could choose a setting that appears lossless, or even beneficial, while having substantially altered the model’s internal feature structure\. For interpretability work that is developed on a full\-precision model and then deployed in quantized form, perplexity parity is therefore not sufficient evidence that the analyzed features remain intact\.

### 5\.4Vulnerability is shared across compression methods

The matched\-perplexity comparison \(Section[4\.4](https://arxiv.org/html/2606.03002#S4.SS4)\) shows that quantization and pruning, at equal task degradation, damage strongly overlapping sets of features—Jaccard 0\.79 to 0\.86 and damage\-score Spearman 0\.98\. The two methods are therefore better described as agreeing on which features are vulnerable than as targeting different feature classes\. They differ in severity: on Gemma, matched pruning is somewhat more aggressive overall \(and was calibrated to a slightly higher perplexity than INT6, so part of this gap reflects residual mismatch\), and pruning produces a small tail of fully damaged features that quantization does not\.

More revealing than the difference in aggregate severity is the shared vulnerability ranking, which connects directly to Section[5\.2](https://arxiv.org/html/2606.03002#S5.SS2): if survival is governed by a feature’s signal strength, then any perturbation that competes with that signal—whether rounding weights or removing them—should endanger the same low\-signal features first\. The high cross\-method damage\-score correlation is what this account predicts\. Our results extendBorobiaet al\.\([2026](https://arxiv.org/html/2606.03002#bib.bib15)\)from unstructured pruning to quantization, replacing weight removal with weight rounding and tracing feature survival along the bit\-width axis\. This extension shows that survival is predictable from FP16 statistics, can diverge from perplexity, and overlaps strongly with matched\-pruning damage on the same SAE basis\.

## 6Conclusion

We asked whether sparse\-autoencoder features extracted from a full\-precision model remain stable after quantization\. Holding the SAE, token set, and read\-out site fixed while varying only model weights, we measured per\-feature survival across RTN bit\-widths on Pythia\-70M and Gemma\-2\-2B\. Three findings stand out\. First, quantization produces graded feature change rather than a sharp precision cliff: features are often blurred rather than destroyed\. Second, survival is predictable from full\-precision statistics alone, with cross\-validated AUC 0\.92–0\.97 and peak activation as the strongest marginal predictor\. Third, task\-level metrics can miss feature damage: perplexity can remain stable or improve while many SAE features degrade, and quantization damages many of the same features as matched\-perplexity pruning\.

These results show that behavioral parity is not sufficient evidence that full\-precision interpretability transfers to quantized deployment\. An analysis and a quantization setting can each appear sound in isolation while failing to compose\. We release our pipeline and per\-feature results, and view causal validation as the next step: testing whether these feature changes alter steering, auditing, or other safety\-relevant interventions in deployed quantized models\.

## 7Limitations

Our study is limited to two model families, publicly available residual\-stream SAEs, and simulated round\-to\-nearest weight quantization\. We evaluate fixed token budgets, read\-out layers, and compression settings, so the quantitative survival rates should not be interpreted as universal across architectures, datasets, SAE training recipes, activation sites, or deployment quantizers\. We also focus on feature\-level correlation and perplexity rather than downstream tasks or causal interventions; future work should test whether the same feature changes alter steering, auditing, or safety\-relevant behavior in deployed quantized models\.

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## Appendix Appendix AFull all\-condition compression plots

Appendix A reports the full all\-condition bar plots for Pythia\-70M and Gemma\-2\-2B\. These figures supplement the main RTN bit\-width sweep by showing all evaluated compression conditions, including FP16, RTN INT8–INT4, and the pruning baseline\.

![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase2b_all_conditions_bar.png)Appendix Figure A\.1:Full Pythia\-70M compression comparison across FP16, RTN bit\-widths, and matched pruning\.![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase3_all_conditions_bar.png)Appendix Figure A\.2:Full Gemma\-2\-2B compression comparison across FP16, RTN bit\-widths, and approximately matched pruning\.
## Appendix Appendix BStability ablation plots and threshold sensitivity

![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase4_stability_ablations_2.png)Appendix Figure B\.1:Stability ablation plots for Phase 4\.Appendix Table B\.1:Threshold sensitivity for ablation D\. Percentages are reported over all evaluated conditions\.Appendix Table B\.2:Token\-budget stability ablation\. This table reports QDM feature\-survival statistics for RTN INT6 using increasing token budgets\. It tests whether estimated survival and damage rates are sensitive to the number of evaluated tokens\.Appendix Table B\.3:Random\-subset seed stability ablation\. This table reports QDM statistics across independently sampled 100k\-token subsets\. It tests whether the measured feature\-survival rate depends on the random token subset used for activation comparison\.Appendix Table B\.4:FP16\-vs\-FP16 pipeline null check\. This table compares FP16 activations against the FP16 reference under the same QDM pipeline\. The expected result is mean correlation 1\.0, 100% survival, and 0% damage, confirming that the pipeline does not create artificial feature drift\.
## Appendix Appendix CINT7 investigation checks

Appendix Table C\.1:Corrected 50k\-token perplexity sweep verifying the INT7 decrease persists under shifted\-loss evaluation\.Appendix Table C\.2:Five\-seed subset check showing the INT7 perplexity decrease is reproducible across token samples\.Appendix Table C\.3:Weight\-error diagnostics confirming INT7 quantization behaves normally between INT8 and INT6\.Appendix Table C\.4:Logit\-drift check showing INT7 output perturbation lies between INT8 and INT6\.
## Appendix Appendix DGemma sliding\-window behavioral evaluation

To check whether the high absolute Gemma perplexity under the chunked Phase 3 protocol was caused by context resets, we reran behavioral evaluation with a sliding\-window protocol\. This check was applied to FP16, RTN INT8, RTN INT7, and RTN INT6 Gemma\-2\-2B on WikiText\-2\-raw\.

The evaluation used a HuggingFace\-format Gemma model in bfloat16 with per\-output\-channel RTN quantization\. The token stream contained288,894288\{,\}894tokens, of which288,893288\{,\}893were scored\. We used window sizeW=2048W=2048and strideS=512S=512\. For each window, previous tokens served as context and only newly introduced target positions were included in the loss\. The aggregate perplexity was computed as a token\-weighted mean negative log\-likelihood:

PPL=exp⁡\(∑wmw​ℒw∑wmw\),\\mathrm\{PPL\}=\\exp\\left\(\\frac\{\\sum\_\{w\}m\_\{w\}\\mathcal\{L\}\_\{w\}\}\{\\sum\_\{w\}m\_\{w\}\}\\right\),\(8\)wheremwm\_\{w\}is the number of scored tokens in windowwwandℒw\\mathcal\{L\}\_\{w\}is the mean loss over those scored tokens\.

For the quantized conditions, the HuggingFace RTN implementation quantized182182tensors and2,024,275,9682\{,\}024\{,\}275\{,\}968parameters\. This behavioral check is distinct from the main SAE feature\-survival pipeline, which uses TransformerLens/Gemma Scope activations\. We therefore use the sliding\-window results only as a robustness check on the perplexity trend, not as a replacement for the main feature\-correlation results or as a benchmark\-comparable Gemma perplexity evaluation\.

Appendix Table D\.1:Full Gemma sliding\-window perplexity results using window size 2048 and stride 512\.Appendix Table D\.2:Comparison of chunked and sliding\-window Gemma perplexity with corresponding SAE feature\-survival rates\.
## Appendix Appendix ERTN\-versus\-pruning overlap visualizations

Appendix E visualizes the RTN\-versus\-pruning overlap analysis\. The scatter plots compare per\-feature survival correlations under RTN INT6 and magnitude pruning, while the decile plots show how non\-survival varies across FP16 firing\-rate deciles\. These figures support the main\-text result that quantization and pruning largely affect overlapping vulnerable feature sets rather than unrelated feature classes\.

![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase5b_pythia_rtn_vs_pruning_scatter.png)Appendix Figure E\.1:Pythia\-70M per\-feature correlation scatter comparing RTN INT6 and matched magnitude pruning\.![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase5b_gemma_rtn_vs_pruning_scatter.png)Appendix Figure E\.2:Gemma\-2\-2B per\-feature correlation scatter comparing RTN INT6 and approximately matched magnitude pruning\.![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase5b_pythia_rtn_vs_pruning_firing_rate_deciles.png)Appendix Figure E\.3:Pythia\-70M non\-survival rates by FP16 firing\-rate decile for RTN INT6 and matched pruning\.![Refer to caption](https://arxiv.org/html/2606.03002v1/figures/phase5b_gemma_rtn_vs_pruning_firing_rate_deciles.png)Appendix Figure E\.4:Gemma\-2\-2B non\-survival rates by FP16 firing\-rate decile for RTN INT6 and approximately matched pruning\.
## Appendix Appendix FPruning calibration

We calibrate magnitude pruning by searching over sparsity valuesp∈\[0,0\.8\]p\\in\[0,0\.8\]to match the RTN INT6 perplexity regime\. The search is performed with a fixed\-step bisection procedure over six steps on a held\-out token budget\. For Pythia\-70M, the selected pruning sparsity isp=0\.175p=0\.175, giving perplexity76\.9976\.99andΔ​PPL=\+13\.49%\\Delta\\mathrm\{PPL\}=\+13\.49\\%, close to RTN INT6 perplexity77\.8377\.83andΔ​PPL=\+14\.72%\\Delta\\mathrm\{PPL\}=\+14\.72\\%\. For Gemma\-2\-2B, the selected pruning sparsity isp=0\.1625p=0\.1625, giving perplexity486\.63486\.63andΔ​PPL=\+6\.13%\\Delta\\mathrm\{PPL\}=\+6\.13\\%, compared with RTN INT6 perplexity476\.82476\.82andΔ​PPL=\+3\.99%\\Delta\\mathrm\{PPL\}=\+3\.99\\%\. Thus, the Pythia pruning baseline is closely matched, while the Gemma pruning baseline is approximately matched and slightly harsher\.

Appendix Table F\.1:Pythia\-70M pruning calibration search for matching the RTN INT6 perplexity regime\.Appendix Table F\.2:Gemma\-2\-2B pruning calibration search for approximately matching the RTN INT6 perplexity regime\.
## Appendix Appendix GFragility predictor details

Appendix G reports additional details for the fragility\-prediction analysis: multivariate logistic\-regression coefficients, cross\-validated ROC curves, and a cross\-model coefficient comparison\. Because the predictors are correlated, the coefficients should be interpreted jointly rather than as isolated causal effects\.

![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/coefficient_comparison.png)Appendix Figure G\.1:Standardized logistic\-regression coefficients for predicting INT6 feature survival from FP16 feature statistics\.![Refer to caption](https://arxiv.org/html/2606.03002v1/prism-uploads/roc_curves.png)Appendix Figure G\.2:Cross\-validated ROC curves for INT6 feature\-survival prediction on Pythia\-70M and Gemma\-2\-2B\.Appendix Table G\.1:Cross\-model comparison of standardized fragility\-predictor coefficients\.
## Appendix Appendix HCondition\-level verification summaries

Appendix H lists condition\-level verification summaries for the main RTN INT6, pruning, and layer\-sensitivity results\. These files support the aggregate values reported in the main tables and figures\.

Appendix Table H\.1:Pythia\-70M layer 4 summary statistics for the Phase 2A RTN quantization sweep\.Feature\-survival statistics

Behavioral and degraded\-feature statistics

Appendix Table H\.2:Pythia\-70M layer 2 summary statistics for the Phase 2A\.5 layer\-sensitivity check\.Feature\-survival statistics

Behavioral and degraded\-feature statistics

Appendix Table H\.3:Pythia\-70M RTN INT6 per\-condition summary used to verify the main INT6 results\.Condition: RTN INT6

PPLLossMethodPrune sparsity77\.834\.3545TL—

Appendix Table H\.4:Pythia\-70M matched\-pruning per\-condition summary used to verify the pruning baseline\.Condition: Magnitude pruning matched INT6

PPLLossMethodPrune sparsity76\.994\.3437TL0\.1750

Appendix Table H\.5:Gemma\-2\-2B RTN INT6 per\-condition summary used to verify the main Gemma INT6 results\.Condition: RTN INT6

PPLLossPPL delta \(%\)Method476\.826\.16713\.99RTN

Appendix Table H\.6:Gemma\-2\-2B approximately matched\-pruning per\-condition summary used to verify the Gemma pruning baseline\.Condition: Magnitude pruning matched INT6

Appendix Table H\.7:Full combined cross\-model RTN quantization and pruning summary\.
## Appendix Appendix IStreaming estimator for per\-feature correlations

Computing the feature\-stability score in Eq\.[3](https://arxiv.org/html/2606.03002#S3.E3)naively requires storing the full activation matricesZFP16,ZC∈ℝN×dsaeZ^\{\\mathrm\{FP16\}\},Z^\{C\}\\in\\mathbb\{R\}^\{N\\times d\_\{\\mathrm\{sae\}\}\}\. This is memory\-intensive at the token budgets used in our experiments, especially for Gemma\-2\-2B whereN≈500,000N\\approx 500\{,\}000anddsae=16,384d\_\{\\mathrm\{sae\}\}=16\{,\}384\. We therefore compute the same Pearson correlations in a single streaming pass over batches of token positions\.

For each featurejj, define

xt=zj​\(t;θFP16\),yt=zj​\(t;θC\)\.x\_\{t\}=z\_\{j\}\(t;\\theta\_\{\\mathrm\{FP16\}\}\),\\qquad y\_\{t\}=z\_\{j\}\(t;\\theta\_\{C\}\)\.During streaming evaluation, we accumulate the following sufficient statistics:

Sx\\displaystyle S\_\{x\}=∑txt,\\displaystyle=\\sum\_\{t\}x\_\{t\},Sy\\displaystyle S\_\{y\}=∑tyt,\\displaystyle=\\sum\_\{t\}y\_\{t\},\(I\.1\)Sx2\\displaystyle S\_\{x^\{2\}\}=∑txt2,\\displaystyle=\\sum\_\{t\}x\_\{t\}^\{2\},Sy2\\displaystyle S\_\{y^\{2\}\}=∑tyt2,\\displaystyle=\\sum\_\{t\}y\_\{t\}^\{2\},Sx​y\\displaystyle S\_\{xy\}=∑txt​yt\.\\displaystyle=\\sum\_\{t\}x\_\{t\}y\_\{t\}\.The Pearson correlation for featurejjis then recovered as

cj=Sx​y−Sx​Sy/N\(Sx2−Sx2/N\)​\(Sy2−Sy2/N\)\.c\_\{j\}=\\frac\{S\_\{xy\}\-S\_\{x\}S\_\{y\}/N\}\{\\sqrt\{\\left\(S\_\{x^\{2\}\}\-S\_\{x\}^\{2\}/N\\right\)\\left\(S\_\{y^\{2\}\}\-S\_\{y\}^\{2\}/N\\right\)\}\}\.\(I\.2\)This expression is algebraically equivalent to Eq\.[3](https://arxiv.org/html/2606.03002#S3.E3), but reduces memory fromO​\(N​dsae\)O\(Nd\_\{\\mathrm\{sae\}\}\)toO​\(dsae\)O\(d\_\{\\mathrm\{sae\}\}\)\.

We accumulate all sufficient statistics in float64 to reduce numerical error in the sum\-of\-squares computation\. In addition to the correlation statistics, we stream the FP16 firing count

∑t𝟏​\[xt\>0\]\\sum\_\{t\}\\mathbf\{1\}\\\!\\left\[x\_\{t\}\>0\\right\]and the FP16 running maximum

used for the active\-feature filter and feature\-property analyses\. As a pipeline null check, comparing FP16 activations against themselves yieldscj=1c\_\{j\}=1for all active features to six decimal places\.

## Appendix Appendix JNotation and Glossary

This appendix summarizes the notation and threshold conventions used throughout the paper\.

Appendix Table J\.1:Notation and threshold conventions used in QDM feature\-survival analysis\.
## Appendix Appendix KModel, SAE, and Evaluation Configuration

This appendix summarizes the model, SAE, and evaluation settings used in the main QDM experiments\.

Appendix Table K\.1:Model, SAE, and token\-budget configuration for the main QDM experiments\. The SAE, token set, and read\-out site are fixed within each model; only the model weights vary across compression conditions\.Appendix Table K\.2:Shared experimental conventions for QDM feature\-survival evaluation\.
## Appendix Appendix LQuantized Module Implementation Details

This appendix specifies which weight tensors are modified by the compression operators\. For both RTN quantization and magnitude pruning, we apply compression only to transformer block linear weights\. Embeddings, unembeddings, layer\-normalization parameters, biases, and SAE parameters are left unchanged\.

For each transformer block, the targeted attention weights are the query, key, value, and output projections\. The targeted MLP weights are the input/up projection and output/down projection\. For Gemma\-2\-2B, which uses a gated MLP, we additionally compress the gate projection\. Thus, the compressed module families are:

Appendix Table L\.1:Transformer block module families included in, and excluded from, compression\.In TransformerLens\-style notation, the Pythia targeted modules correspond to the attention projection weights and MLP projection weights in each block, including names such asW\_Q,W\_K,W\_V,W\_O,W\_in, andW\_out\. In HuggingFace/Gemma\-style notation, the targeted modules correspond to names such asq\_proj,k\_proj,v\_proj,o\_proj,up\_proj,down\_proj, andgate\_proj\.

All targeted tensors are compressed with the same per\-output\-channel procedure described in Section[3\.2](https://arxiv.org/html/2606.03002#S3.SS2)\. For RTN quantization, weights are rounded to the target integer grid and then dequantized back into floating\-point tensors, so the experiment simulates low\-bit weights without changing the model architecture\. For pruning, the smallest\-magnitude weights within each targeted tensor are set to zero according to the calibrated sparsity\. The SAE encoder is never quantized, pruned, or retrained\.

The output\-channel convention follows the tensor layout used by the loaded model implementation\. For each targeted matrix, the scale is computed over all dimensions except the output\-channel dimension, so each output channel receives its own RTN scale\. This convention is applied consistently across Pythia and Gemma experiments; where module layouts differ between TransformerLens and HuggingFace implementations, the corresponding output\-channel axis is selected before applying the same per\-output\-channel rule\.