Response Magnitude as a Dominant Signal for Held-Out CRISPRi Perturbation Effect Prediction

# Response Magnitude as a Dominant Signal for Held-Out CRISPRi Perturbation Effect Prediction
Source: [https://arxiv.org/html/2608.00152](https://arxiv.org/html/2608.00152)
Mehrdad Shoeibi University of Central Florida Mehrdad\.Shoeibi@ucf\.edu &Niloofar Yousefi University of Central Florida Niloofar\.Yousefi@ucf\.edu

###### Abstract

Predicting the magnitude of a CRISPRi perturbation’s transcriptomic effect on held\-out target genes is an important open problem in single\-cell biology\. Recent work has documented that simple baselines often match or exceed deep perturbation predictors on related protocols\(Ahlmann\-Eltze et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib1); Wu et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib33); Mejia et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib20)\)\. We study this phenomenon on the Virtual Cell Challenge \(VCC\) benchmark\(Roohani et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib25)\)under a strict held\-out target\-gene split, identify the specific low\-dimensional signal that drives the gap, and characterize how it transfers across cell types\. The target is the log Anderson–Darling distance from non\-targeting controls, which is strongly predictable from four deterministic scalar functions of the2,0002\{,\}000\-dimensional input\. A deep MLP encoder with direct access to the full input collapses toward the marginal training mean \(R2=\+0\.082±0\.010R^\{2\}=\+0\.082\\pm 0\.010\), and standard remedies \(tail\-weighted regression, label\-density smoothing, post\-hoc calibration, cross\-cell\-type pretraining, an inductive\-bias gate\) do not close the gap\. A linear regression on the four magnitude scalars alone reachesR2=\+0\.208R^\{2\}=\+0\.208, exceeding the strongest𝐱\\mathbf\{x\}\-only classical model \(\+0\.190\+0\.190\); removing row\-level magnitude collapses the latter to\+0\.062±0\.008\+0\.062\\pm 0\.008; and a Random Forest on the input plus the four scalars reachesR2=\+0\.370±0\.004R^\{2\}=\+0\.370\\pm 0\.004, substantially exceeding our deep proof\-of\-concept encoder \(mag\-spsm,R2=\+0\.225±0\.004R^\{2\}=\+0\.225\\pm 0\.004\)\. Two pre\-specified controls—a row\-shuffle and substitution by four i\.i\.d\. Gaussian features—attribute the magnitude gain to per\-row alignment rather than added dimensionality\. Under zero\-shot transfer to two external CRISPRi screens, evaluated against a target\-gene endpoint rebuilt from single\-cell data so that source and target score the same quantity, magnitude\-only predictors are nonnegative and usually resolved\-positive, whereas expression\-only predictors are negative or unresolved\. Exposing magnitude to the deep encoder improves transfer over its expression\-only counterpart by a resolved margin in both screens \(Δ​ρ=\+0\.117\\Delta\\rho=\+0\.117and\+0\.086\+0\.086\), yet the encoder does not outperform a four\-scalar linear regression on the same features in either screen\. We also find that the Anderson–Darling column distributed with these screens measures transcriptome\-wide response breadth rather than target\-gene effect strength, so evaluating transfer against it scores a different outcome\.

## 1Introduction

Predicting the transcriptomic response of a cell to a genetic perturbation is a central problem in computational single\-cell biology, with applications to target discovery, drug repurposing, and synthetic biology\(Dixit et al\.,[2016](https://arxiv.org/html/2608.00152#bib.bib9); Replogle et al\.,[2022b](https://arxiv.org/html/2608.00152#bib.bib23); Lotfollahi et al\.,[2019](https://arxiv.org/html/2608.00152#bib.bib18); Roohani et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib24)\)\. The hardest variant of this problem, and the one emphasized by recent benchmarks such as the Virtual Cell Challenge \(VCC\)\(Roohani et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib25)\), is held\-out target\-gene generalization: train on one set of perturbations, evaluate on a disjoint set whose identities have not been seen during training\. Foundation\-model approaches such as scGPT\(Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\)and Geneformer\(Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\)have advanced single\-cell representation learning, yet on the strict held\-out target\-gene split studied here, even strong stable\-prediction encoders collapse predictions toward the marginal training mean and fail to recover the high\- and low\-effect tails\.

#### This paper\.

We study held\-out target\-gene generalization on VCC\. Recent work has documented that simple baselines often match or exceed deep perturbation predictors on related protocols\(Ahlmann\-Eltze et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib1); Wu et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib33); Csendes et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib7); Bendidi et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib3)\); our contribution is more specific: we identify the low\-dimensional signal that drives this gap and quantify how it transfers across cell contexts\. The target is the log Anderson–Darling distance from non\-targeting controls, which is strongly predicted by four deterministic scalar functions of the2,0002\{,\}000\-dimensional input,𝒎​\(𝐱\)=\(\|x\|¯,‖𝐱‖2/d,‖𝐱‖∞,σ​\(𝐱\)\)\.\\boldsymbol\{m\}\(\\mathbf\{x\}\)=\(\\bar\{\|x\|\},\\,\\\|\\mathbf\{x\}\\\|\_\{2\}/\\sqrt\{d\},\\,\\\|\\mathbf\{x\}\\\|\_\{\\infty\},\\,\\sigma\(\\mathbf\{x\}\)\)\.Yet a deep MLP encoder with direct access to𝐱\\mathbf\{x\}collapses toward the marginal training mean \(R2=\+0\.082±0\.010R^\{2\}=\+0\.082\\pm 0\.010\), while a linear regression on𝒎​\(𝐱\)\\boldsymbol\{m\}\(\\mathbf\{x\}\)alone reaches\+0\.208\+0\.208and a Random Forest on\[𝐱;𝒎​\(𝐱\)\]\[\\mathbf\{x\};\\,\\boldsymbol\{m\}\(\\mathbf\{x\}\)\]reaches\+0\.370±0\.004\+0\.370\\pm 0\.004\. Five plausible remedies for tail collapse do not close the gap, and row\-normalizing𝐱\\mathbf\{x\}collapses the strongest classical baseline from\+0\.190\+0\.190to\+0\.062\+0\.062, isolating row\-aligned magnitude as the dominant component\. Under zero\-shot transfer to two external screens the same asymmetry appears: expression\-only models are negative or unresolved under transfer, whereas magnitude\-only models are nonnegative and usually resolved\-positive\. Explicit magnitude exposure is necessary but not sufficient—individual model families change verdict between screens—and across both screens no deep encoder we tested outperforms a four\-scalar linear regression on the same features\.

#### What is and is not surprising about magnitude\.

Because the target is the log Anderson–Darling distance between each perturbed profile and the matched NT control, response magnitude is in principle informative of the label by construction\. The relationship between magnitude and the label is not the surprising part of this paper; recent work has likewise observed that simple baselines compete with deep perturbation predictors\(Ahlmann\-Eltze et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib1); Wu et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib33)\)\. What is specific to this paper is \(i\) which signal drives the gap on the held\-out target\-gene split—response magnitude rather than directional patterns, isolated by pre\-specified shuffle and i\.i\.d\. Gaussian controls; \(ii\) that a Random Forest on the same input augmented with magnitude beats every deep architecture we tried by a substantial margin \(\+0\.370\+0\.370vs\.\+0\.225\+0\.225\); and \(iii\) that under cross\-cell transfer explicit magnitude exposure is helpful but not sufficient—the model class and how it consumes the signal also matter\. The diagnostic claim is about which signal models exploit and under which evaluation protocols, not about whether the simple\-baseline phenomenon exists; that is established in prior work\.

#### Contributions\.

This paper is a diagnostic study within the recent linear\-baseline and mode\-collapse literature\(Ahlmann\-Eltze et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib1); Wu et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib33); Mejia et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib20)\)\. Within that lineage, our contributions are:

1. 1\.Identification of response magnitude as the specific signal\.On the VCC held\-out target\-gene benchmark, four scalar magnitude statistics of𝐱\\mathbf\{x\}predict the label nearly as well as the full2,0002\{,\}000\-dimensional input under classical models \(R2=\+0\.208R^\{2\}=\+0\.208vs\.\+0\.190\+0\.190\)\. Removing row\-level magnitude \(row\-normalizing𝐱\\mathbf\{x\}\) collapses the strongest classical baseline toR2=\+0\.062R^\{2\}=\+0\.062, isolating row\-aligned response magnitude as the dominant recoverable signal under this split\.
2. 2\.Attribution to per\-row alignment\.Two pre\-specified controls \(row shuffling, four i\.i\.d\. Gaussian features\) both return performance to baseline within seed noise, and a population\-level argument shows that any augmentation independent of the input cannot improve over the Bayes predictor on the input alone \(App\.[C](https://arxiv.org/html/2608.00152#A3)\)\. The gain is therefore attributable to per\-row dependence, not to added dimensionality or capacity\.
3. 3\.MLP\-encoder representational failure\.Although𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)is a deterministic function of the input, a deep MLP with direct access to that input fails to recover it in held\-out prediction \(R2=\+0\.082±0\.010R^\{2\}=\+0\.082\\pm 0\.010vs\.\+0\.208\+0\.208for a four\-feature linear model\), and an architecture sweep over normalization, dropout, width, and a hand\-designed scale shortcut does not rescue it \(App\.[Q](https://arxiv.org/html/2608.00152#A17)\)\. The diagnosis is scope\-limited to the MLP family we tested\.
4. 4\.Model class changes how the magnitude signal is exploited\.A Random Forest on𝐱\\mathbf\{x\}plus four magnitude statistics reachesR2=\+0\.370±0\.004R^\{2\}=\+0\.370\\pm 0\.004, substantially outperforming our deepmag\-spsmproof of concept \(\+0\.225±0\.004\+0\.225\\pm 0\.004\)\. The dominant diagnostic is therefore not specific to any single deep architecture in the MLP family\.
5. 5\.Endpoint comparability across screens\.The Anderson–Darling column distributed with the Replogle screens is strongly consistent with a transcriptome\-wide differential\-expression gene count rather than a target\-gene effect size, and is only weakly associated with the latter within a screen \(ρ=0\.17\\rho=0\.17–0\.400\.40\)\. Transfer evaluated against it therefore scores a different outcome from the one models are trained on\.
6. 6\.Cross\-screen magnitude–direction asymmetry\.Rebuilding the target\-gene endpoint from single\-cell data for two external screens, magnitude\-only predictors transfer positively while expression\-only predictors do not\. Exposing magnitude to a deep encoder improves transfer over its expression\-only counterpart in both screens \(Δ​ρ=\+0\.117\\Delta\\rho=\+0\.117and\+0\.086\+0\.086\), yet no deep encoder we tested beats a four\-scalar linear regression\. To our knowledge this asymmetry has not been documented in prior linear\-baselines literature\.
7. 7\.Magnitude\-anchored residual prediction \(secondary\)\.MAG\-Anchor decomposes the prediction into a MAG\-LINEAR\-initialized magnitude anchor and a gated residual branch, improving overmag\-spsmon VCC \(R2=\+0\.268±0\.011R^\{2\}=\+0\.268\\pm 0\.011\)\. Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]remains stronger in\-distribution\. We treat it as supporting evidence for the diagnostic, not as a new state\-of\-the\-art predictor \(App\.[D](https://arxiv.org/html/2608.00152#A4)\)\.

## 2Related work

#### Perturbation prediction\.

spsm\-style stable\-prediction models descend from stable\-prediction methods in causal inference\(Kuang et al\.,[2018](https://arxiv.org/html/2608.00152#bib.bib16); Shen et al\.,[2018](https://arxiv.org/html/2608.00152#bib.bib29)\)adapted to single\-cell data\. Recent perturbation prediction methods include CPA\(Lotfollahi et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib19)\), GEARS\(Roohani et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib24)\), and pretrained foundation models such as scGPT\(Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\), Universal Cell Embeddings\(Rosen et al\.,[2026](https://arxiv.org/html/2608.00152#bib.bib26)\), and Geneformer\(Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\)\. Many existing evaluations either pool across target\-gene splits or use a held\-out\-pair setting; the strict disjoint target\-gene split used here is more demanding, and our analysis is complementary: rather than proposing a new architecture we ask which features any model class needs to exploit on this split\.

#### Linear baselines and mode collapse in perturbation prediction\.

A recent line of work documents that deep and foundation\-model perturbation predictors frequently fail to outperform simple baselines\.Ahlmann\-Eltze et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib1)\)compared five foundation models and two deep architectures against linear baselines on Adamson, Norman, and Replogle, and found that none consistently outperformed the linear model on unseen\-perturbation prediction; for held\-out perturbations the deep models often failed to outperform predicting the unconditional training mean\. They additionally show that a linear model with perturbation embeddings pretrained on Replogle K562 transfers across cell lines to RPE1 \(and vice versa\), establishing that cross\-cell\-line transfer with linear models is feasible\.Wu et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib33)\)introduced PerturBench with covariate\-transfer benchmarks across cell lines on Srivatsan20 and Jiang24, and identified mode collapse—predictions concentrating in a narrow region of output space—as a recurring failure mode, with simple baselines \(Latent Additive, Decoder\-Only\) often matching or exceeding more complex autoencoder models\.Csendes et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib7)\)found that a mean baseline exceeded scGPT and scFoundation\(Hao et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib12)\)on post\-perturbation Pearson\(Δ\\Delta\), andBendidi et al\. \([2024](https://arxiv.org/html/2608.00152#bib.bib3)\)reported that PCA\-based baselines remained competitive with transcriptomics foundation models\.Mejia et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib20)\)addressed mode collapse on the loss side by introducing DEG\-weighted MSE as a training objective with calibrated negative and positive baselines\. Our work is in this lineage but contributes a more specific diagnosis on a distinct protocol\. Where prior work establishes the broader phenomenon, we \(i\) isolate response magnitude as the specific low\-dimensional signal the deep encoders fail to recover, with pre\-specified shuffle and i\.i\.d\. Gaussian controls plus a population\-level argument that attribute the gain to per\-row alignment rather than added dimensionality \(Sec\.[5\.2](https://arxiv.org/html/2608.00152#S5.SS2), App\.[C](https://arxiv.org/html/2608.00152#A3)\); \(ii\) report a cross\-cell*magnitude–direction*asymmetry under zero\-shot transfer to two external screens, in which expression\-only models transfer with negative or unresolved Spearman while magnitude\-only models transfer positively \(Sec\.[6\.3](https://arxiv.org/html/2608.00152#S6.SS3)\); and \(iii\) show that the endpoint distributed with those screens measures a different quantity from the one models are trained on, and rebuild a common endpoint for the comparison \(Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)\)\. The evaluation uses the held\-out target\-gene split of the Virtual Cell Challenge\(Roohani et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib25)\), distinct from the covariate\-transfer splits ofWu et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib33)\)and the unseen\-perturbation splits ofAhlmann\-Eltze et al\. \([2025](https://arxiv.org/html/2608.00152#bib.bib1)\)\.

#### Imbalanced regression and tail prediction\.

Deep imbalanced regression frameworks include Label Distribution Smoothing \(LDS\) and Feature Distribution Smoothing\(Yang et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib34)\), Balanced MSE\(Ren et al\.,[2022](https://arxiv.org/html/2608.00152#bib.bib21)\), density\-based reweighting\(Steininger et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib30)\), and label\-distribution\-aware margin losses\(Cao et al\.,[2019](https://arxiv.org/html/2608.00152#bib.bib5)\)\. We instantiate LDS\-weighted and tail\-weighted variants as baselines \(Sec\.[4](https://arxiv.org/html/2608.00152#S4)\); both fail in our setting and we discuss why in Sec\.[7](https://arxiv.org/html/2608.00152#S7)\.

#### Distribution shift and rank objectives\.

Cell\-type transfer is a distribution shift between domains with distinct gene\-expression baselines\. Standard OOD techniques such as IRM\(Arjovsky et al\.,[2019](https://arxiv.org/html/2608.00152#bib.bib2)\), GroupDRO\(Sagawa et al\.,[2020](https://arxiv.org/html/2608.00152#bib.bib27)\), REx\(Krueger et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib15)\), and the WILDS benchmark family\(Koh et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib14)\)address related but distinct shifts and are complementary\. Our use of magnitude as a deliberately low\-dimensional, coordinate\-permutation\-invariant summary is conceptually allied with norm\- and energy\-based detection\(Hendrycks and Gimpel,[2017](https://arxiv.org/html/2608.00152#bib.bib13); Liu et al\.,[2020](https://arxiv.org/html/2608.00152#bib.bib17)\)and norm\-aware embeddings\(Wang et al\.,[2017](https://arxiv.org/html/2608.00152#bib.bib32)\)\. Themag\-Rankextension \(App\.[I](https://arxiv.org/html/2608.00152#A9)\) uses a margin\-based pairwise rank loss in the spirit of LambdaRank\-family approaches\(Burges et al\.,[2005](https://arxiv.org/html/2608.00152#bib.bib4); Cao et al\.,[2007](https://arxiv.org/html/2608.00152#bib.bib6); Hadsell et al\.,[2006](https://arxiv.org/html/2608.00152#bib.bib11)\)\.

## 3Problem setup and notation

#### Data\.

The Virtual Cell Challenge \(VCC\) corpus\(Roohani et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib25)\)consists of CRISPRi \(dCas9–KRAB\)\(Gilbert et al\.,[2014](https://arxiv.org/html/2608.00152#bib.bib10)\)Perturb\-seq\(Dixit et al\.,[2016](https://arxiv.org/html/2608.00152#bib.bib9)\)screens in H1 human embryonic stem cells \(H1 hESC\), organized into 48 experimental batches indexed byb∈ℬb\\in\\mathcal\{B\}\. Each rowiicorresponds to a \(batch, target\-gene\) pair\(bi,gi\)\(b\_\{i\},g\_\{i\}\)with feature vector𝐱i∈ℝd\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{d\}and scalar labelyi∈ℝ≥0y\_\{i\}\\in\\mathbb\{R\}\_\{\\geq 0\}\. Features are log1p\-normalized mean expression deltas relative to the NT\-control mean of batchbib\_\{i\}, restricted to thed=2,000d=2\{,\}000highly variable genes \(HVGs\) selected on the train split\. The label isyi=log⁡\(1\+max⁡\{0,Abi,gi2\}\)y\_\{i\}=\\log\(1\+\\max\\\{0,\\,A^\{2\}\_\{b\_\{i\},g\_\{i\}\}\\\}\), whereAb,g2A^\{2\}\_\{b,g\}is the two\-sample Anderson–Darling statistic\(Scholz and Stephens,[1987](https://arxiv.org/html/2608.00152#bib.bib28)\)between the log1p expression of geneggin cells perturbed atggwithin batchbband in non\-targeting controls from the same batch; the AD statistic is computed on the target\-gene column only, and𝐱i\\mathbf\{x\}\_\{i\}enters as predictor input only \(App\.[V](https://arxiv.org/html/2608.00152#A22)\)\. The set of target genes is partitioned disjointly into𝒢tr\\mathcal\{G\}\_\{\\text\{tr\}\},𝒢val\\mathcal\{G\}\_\{\\text\{val\}\}, and𝒢te\\mathcal\{G\}\_\{\\text\{te\}\}of sizes140140,4949, and9393, giving5,7405\{,\}740training,2,0522\{,\}052validation, and3,9723\{,\}972test rows\.

#### Task and metrics\.

Given\(bi,𝐱i\)\(b\_\{i\},\\mathbf\{x\}\_\{i\}\)for a test row, predictyiy\_\{i\}\. We report pooled testR2=1−∑i\(yi−y^i\)2/∑i\(yi−y¯te\)2,R^\{2\}=1\-\\sum\_\{i\}\(y\_\{i\}\-\\hat\{y\}\_\{i\}\)^\{2\}\\big/\\sum\_\{i\}\(y\_\{i\}\-\\bar\{y\}\_\{\\text\{te\}\}\)^\{2\},seed\-level mean±\\pmstandard deviation across independent training runs \(default: seeds\{0,1,2\}\\\{0,1,2\\\}; MAG\-Anchor uses five seeds\), Spearman rank correlation, and per\-quintile bias𝔼​\[y^−y∣q​\(y\)\]\\mathbb\{E\}\[\\hat\{y\}\-y\\mid q\(y\)\]\. All reported standard deviations use sample standard deviation with then−1n\-1denominator \(ddof=1\\mathrm\{ddof\}\{=\}1\)\.

#### Cross\-dataset shift\.

For external transfer we use two Replogle CRISPRi \(dCas9–KRAB\) Perturb\-seq screens\(Replogle et al\.,[2022b](https://arxiv.org/html/2608.00152#bib.bib23),[a](https://arxiv.org/html/2608.00152#bib.bib22)\): RPE1 and K562\-essential\. Features are mapped into the VCC HVG namespace by gene symbol, recovering1,4301\{,\}430of2,0002\{,\}000HVGs \(71\.5%71\.5\\%coverage\); the570570unmapped coordinates are imputed as zero in the log1p\-delta space, the value an unperturbed cell takes in the VCC delta pipeline\. For each screen we rebuild the label from single\-cell data using the definition above, so that source and target are scored on the same quantity—the Anderson–Darling column distributed with the Replogle release measures something else \(Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)\)\. We never train on either screen and never use target\-side statistics for preprocessing\. A1,4301\{,\}430\-HVG training control was run against the distributed endpoint and is reported as a legacy sensitivity analysis \(App\.[O](https://arxiv.org/html/2608.00152#A15)\); it is not evidence for the rebuilt endpoint\.

## 4Diagnosing the failure of standard remedies

#### Baseline failure mode\.

ThespsmV4 baseline, a stable\-prediction MLP encoder with a within\-context stability loss and a relation loss \(App\.[G](https://arxiv.org/html/2608.00152#A7)\), achieves test pooledR2=\+0\.0824±0\.0100R^\{2\}=\+0\.0824\\pm 0\.0100across three seeds\. Its predictions exhibit a hallmark of mean collapse: test prediction standard deviation isσ^y^=0\.240\\widehat\{\\sigma\}\_\{\\hat\{y\}\}=0\.240versus target standard deviationσyte=0\.81\\sigma\_\{y\_\{\\text\{te\}\}\}=0\.81, and the three\-seed mean per\-quintile bias is monotonically structured \(\+1\.093\+1\.093on Q1,−0\.696\-0\.696on Q5\)\. We use “mean collapse” rather than “mode collapse”\(Wu et al\.,[2025](https://arxiv.org/html/2608.00152#bib.bib33)\)because in our scalar\-regression setting predictions concentrate around the marginal training mean specifically, rather than at one of multiple modes of a multimodal output distribution\.

#### Five remedies that fail or stall\.

We instantiated five plausible corrections under a matched protocol \(full numerical results in Table[8](https://arxiv.org/html/2608.00152#A9.T8), Appendix\): \(i\) an inductive\-bias bilevel gate failed both pre\-specified tests of gate behavior; \(ii\) cross\-cell\-type pretraining on Replogle followed by VCC fine\-tuning crashed performance \(R2=−0\.181R^\{2\}=\-0\.181\); \(iii\) a deployable post\-hoc tail correction worsened valR2R^\{2\}by−0\.046\-0\.046vs\. the single\-seed uncorrected SPSM V4 baseline used in that phase \(absolute valR2=−0\.192R^\{2\}=\-0\.192\); \(iv\) a tail\-weighted MSE in the spirit of Balanced MSE\(Ren et al\.,[2022](https://arxiv.org/html/2608.00152#bib.bib21)\)producedΔ​R2\\Delta R^\{2\}of at most\+0\.0075\+0\.0075; \(v\) LDS\-weighted MSE\(Yang et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib34)\)*worsened*R2R^\{2\}at both bandwidths tested\. The shared failure mode of \(iv\) and \(v\) is informative: reweighting alone does not fix prediction\-range collapse, because the reweighted predictor still maps inputs through an encoder lacking direct information about a sample’s response*magnitude*\. This motivates exposing magnitude as a feature, not as a loss\.

#### The magnitude axis\.

A pre\-specified diagnostic suite \(App\.[A](https://arxiv.org/html/2608.00152#A1)\) identified that the mean absolute value of𝐱\\mathbf\{x\}correlated strongly withyy\(Spearman∼\\sim0\.400\.40\) and that prediction failures concentrated at low values of this proxy; tail rows are linearly separable along the magnitude axis but the baseline encoder collapses predictions to a narrow window centered ony¯tr\\bar\{y\}\_\{\\text\{tr\}\}\(Fig\.[1](https://arxiv.org/html/2608.00152#S4.F1)\)\.

![Refer to caption](https://arxiv.org/html/2608.00152v1/x1.png)Figure 1:Diagnostic\. Response\-magnitude statistics correlate with the per\-row label; the baseline encoder collapses predictions toward the marginal training mean\.

## 5Magnitude features and the proof\-of\-concept encoder

#### Magnitude features\.

For𝐱∈ℝd\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\}with componentsx1,…,xdx\_\{1\},\\ldots,x\_\{d\}, define the magnitude map𝒎:ℝd→ℝ4\\boldsymbol\{m\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{4\}by

𝒎​\(𝐱\)=\(1d​∑j\|xj\|⏟m1:mean​\|x\|,\(1d​∑jxj2\)1/2⏟m2:ℓ2/d,maxj⁡\|xj\|⏟m3:ℓ∞,\(1d​∑j\(xj−x¯\)2\)1/2⏟m4:σ​\(𝐱\)\)\.\\boldsymbol\{m\}\(\\mathbf\{x\}\)\\;=\\;\\Bigg\(\\,\\underbrace\{\\tfrac\{1\}\{d\}\\\!\\sum\_\{j\}\\\!\|x\_\{j\}\|\}\_\{m\_\{1\}:\\;\\text\{mean\}\|x\|\}\\,,\\;\\underbrace\{\\Big\(\\tfrac\{1\}\{d\}\\\!\\sum\_\{j\}\\\!x\_\{j\}^\{\\,2\}\\Big\)^\{\\\!1/2\}\}\_\{m\_\{2\}:\\;\\ell\_\{2\}/\\sqrt\{d\}\}\\,,\\;\\underbrace\{\\max\_\{j\}\|x\_\{j\}\|\}\_\{m\_\{3\}:\\;\\ell\_\{\\infty\}\}\\,,\\;\\underbrace\{\\Big\(\\tfrac\{1\}\{d\}\\\!\\sum\_\{j\}\(x\_\{j\}\-\\bar\{x\}\)^\{2\}\\Big\)^\{\\\!1/2\}\}\_\{m\_\{4\}:\\;\\sigma\(\\mathbf\{x\}\)\}\\,\\Bigg\)\.\(1\)Wezz\-score𝒎\\boldsymbol\{m\}component\-wise using*training\-set*statistics to obtain𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\. All four statistics are coordinate\-permutation invariant;m1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}are coordinate\-wise sign\-flip

invariant;m4m\_\{4\}is invariant under translations along𝟏\\mathbf\{1\}\. These invariances emphasize*how much*the cell responded rather than*which genes*responded, and are the property that, we will argue, makes magnitude a more shift\-stable summary across cell types\.

#### Three baselines built from𝒎~\\widetilde\{\\boldsymbol\{m\}\}\.

We use𝒎~\\widetilde\{\\boldsymbol\{m\}\}in three ways: \(i\)mag\-Linear, a four\-feature linear regressiony^=𝐰⊤​𝒎~​\(𝐱\)\+b\\hat\{y\}=\\mathbf\{w\}^\{\\top\}\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\+b; \(ii\) classical methods on𝐱\\mathbf\{x\},𝐱/‖𝐱‖2\\mathbf\{x\}/\\\|\\mathbf\{x\}\\\|\_\{2\},𝒎~\\widetilde\{\\boldsymbol\{m\}\}, or\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\(Sec\.[6\.1](https://arxiv.org/html/2608.00152#S6.SS1)\); and \(iii\)mag\-spsm\(below\) as a proof of concept that the same signal can be exposed to a deep encoder\.

#### mag\-spsm\(proof of concept\)\.

mag\-spsmconcatenates𝐱\\mathbf\{x\}with𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)to form𝐱\+=\[𝐱;𝒎~​\(𝐱\)\]∈ℝ2004\\mathbf\{x\}^\{\+\}=\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\]\\in\\mathbb\{R\}^\{2004\}and feeds it to an MLP encoderfθ:ℝ2004→ℝ128f\_\{\\theta\}:\\mathbb\{R\}^\{2004\}\\to\\mathbb\{R\}^\{128\}\(three hidden layers of width256256, ReLU, LayerNorm, no architectural dropout\) followed by a linear head\. The objective is

ℒtotal=ℒMSE\+λstab​ℒstab\+λrel​ℒrel,\\mathcal\{L\}\_\{\\text\{total\}\}\\;=\\;\\mathcal\{L\}\_\{\\text\{MSE\}\}\\,\+\\,\\lambda\_\{\\text\{stab\}\}\\,\\mathcal\{L\}\_\{\\text\{stab\}\}\\,\+\\,\\lambda\_\{\\text\{rel\}\}\\,\\mathcal\{L\}\_\{\\text\{rel\}\},\(2\)whereℒstab\\mathcal\{L\}\_\{\\text\{stab\}\}andℒrel\\mathcal\{L\}\_\{\\text\{rel\}\}are thespsmstability and relation losses \(App\.[G](https://arxiv.org/html/2608.00152#A7);λstab=λrel=0\.5\\lambda\_\{\\text\{stab\}\}=\\lambda\_\{\\text\{rel\}\}=0\.5, inherited unchanged\)\. Training schedule and optimizer are in App\.[G](https://arxiv.org/html/2608.00152#A7); model selection is by validationR2R^\{2\}\.mag\-spsmis*not*the strongest model in this paper; the strong\-baseline audit \(Sec\.[6\.1](https://arxiv.org/html/2608.00152#S6.SS1)\) shows that classical tree ensembles on the same input outperform it by a large margin\. We retainmag\-spsmas the simplest deep\-architecture verification that magnitude is recoverable when supplied explicitly, and as the substrate for the causal controls in Sec\.[5\.2](https://arxiv.org/html/2608.00152#S5.SS2)\.

### 5\.1Magnitude\-anchored residual prediction

Motivated by the diagnostic finding that response magnitude is a cross\-cell\-stable signal, we briefly introduce*MAG\-Anchor*, a secondary deep architecture that decomposes the predicted perturbation effect into a direct magnitude anchor and a gated residual correction:y^=gϕ​\(𝒎~​\(𝐱\)\)\+α​hθ​\(\[𝐱;𝒎~​\(𝐱\)\]\)\\hat\{y\}=g\_\{\\phi\}\(\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\)\+\\alpha\\,h\_\{\\theta\}\(\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\]\), where the anchor branchgϕg\_\{\\phi\}is a linear layer initialized from MAG\-LINEAR, the residual branchhθh\_\{\\theta\}is a small MLP, andα\\alphais a learnable gate\. Full architectural details, hyperparameters, and ablations are deferred to App\.[D](https://arxiv.org/html/2608.00152#A4); we treat MAG\-Anchor as supporting evidence for the diagnostic, not as a new state\-of\-the\-art predictor\.

### 5\.2The role of per\-row alignment

The controls above rest on a simple population\-level argument: if an augmenting blockM⋆M^\{\\star\}is independent of\(X,Y\)\(X,Y\), then𝔼​\[Y∣X,M⋆\]=𝔼​\[Y∣X\]\\mathbb\{E\}\[Y\\mid X,M^\{\\star\}\]=\\mathbb\{E\}\[Y\\mid X\], so no measurable predictor using\(X,M⋆\)\(X,M^\{\\star\}\)can improve over the Bayes predictor onXXalone under squared loss\. This yields two consequences we test directly: \(a\) replacing𝒎~​\(𝐱i\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\_\{i\}\)by four i\.i\.d\.𝒩​\(0,I4\)\\mathcal\{N\}\(0,I\_\{4\}\)features makes the augmenting block independent of\(𝐱i,Yi\)\(\\mathbf\{x\}\_\{i\},Y\_\{i\}\)by construction; \(b\) a row\-shuffled augmentation preserves the marginal distribution of𝒎~\\widetilde\{\\boldsymbol\{m\}\}but breaks row\-specific information aboutYiY\_\{i\}\. AnyR2R^\{2\}gain from concatenating the four scalars must therefore arise from per\-row dependence on𝐱i\\mathbf\{x\}\_\{i\}\(App\.[C](https://arxiv.org/html/2608.00152#A3)\)\.

## 6Experiments

#### Implementation\.

The deepspsmandmag\-spsmmodels share thespsmV4 backbone \(App\.[G](https://arxiv.org/html/2608.00152#A7)\); MAG\-Anchor uses a separate two\-branch architecture with MSE ony^\\hat\{y\}\(App\.[D](https://arxiv.org/html/2608.00152#A4)\)\. Hyperparameters, optimizer settings, and compute details are listed in App\.[F](https://arxiv.org/html/2608.00152#A6)and App\.[G](https://arxiv.org/html/2608.00152#A7); we report mean±\\pmstd over seeds\{0,1,2\}\\\{0,1,2\\\}\(MAG\-Anchor uses five seeds\)\.

### 6\.1Strong classical baselines reveal magnitude as the dominant axis

We test whether the magnitude effect is specific to the deepspsmarchitecture or a more general property of held\-out target\-gene prediction\. We evaluated five classical methods \(Ridge, ElasticNet, Random Forest, HistGradientBoosting,kk\-NN\) on four feature sets: the full𝐱\\mathbf\{x\}, the row\-normalized direction𝐱/‖𝐱‖2\\mathbf\{x\}/\\\|\\mathbf\{x\}\\\|\_\{2\}\(direction\-only\),𝒎~\\widetilde\{\\boldsymbol\{m\}\}alone \(mag\-only\), and the concatenation\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\(x\+mag\)\. All preprocessing used training\-split statistics; the four magnitude features are pre\-specified before any model training\. Five hypotheses were pre\-specified before the experiment was run; we report all five verdicts in App\.[A](https://arxiv.org/html/2608.00152#A1)\.

Table 1:Strong classical baselines on the VCC held\-out target\-gene test set \(mean±\\pmstd, three seeds; Ridge and ElasticNet are deterministic at the displayed precision\)\. Removing row\-level magnitude \(dir,𝐱/‖𝐱‖2\\mathbf\{x\}/\\\|\\mathbf\{x\}\\\|\_\{2\}\) sharply reduces performance, while concatenating𝐱\\mathbf\{x\}with the four magnitude statistics \(x\+m\) enables tree ensembles to substantially outperformmag\-spsm\.The result \(Table[1](https://arxiv.org/html/2608.00152#S6.T1)\) reframes the interpretation of the deep model\.*\(i\) Four magnitude scalars match the full input\.*Ridge onmag\-onlyreaches\+0\.187\+0\.187, only0\.0030\.003below the strongest classical model on𝐱\\mathbf\{x\}\(HistGradientBoosting,\+0\.190±0\.006\+0\.190\\pm 0\.006\);mag\-Linearreaches\+0\.208\+0\.208\. Four scalars carry approximately as much held\-out predictive signal as the full2,0002\{,\}000\-dimensional vector under this split\.*\(ii\) Removing magnitude collapses performance\.*Row\-normalizing𝐱\\mathbf\{x\}reduces the strongest classical model from\+0\.190\+0\.190to\+0\.062±0\.008\+0\.062\\pm 0\.008\. All linear methods become strongly negative ondirection\-only\.*\(iii\) Combining the two outperforms the deep encoder\.*Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]reaches\+0\.370±0\.004\+0\.370\\pm 0\.004and HistGradientBoosting reaches\+0\.365±0\.008\+0\.365\\pm 0\.008, both substantially exceeding our deepmag\-spsmencoder \(\+0\.225±0\.004\+0\.225\\pm 0\.004\)\. Direction combined with explicit magnitude adds∼0\.18​R2\\sim 0\.18\\,R^\{2\}that tree ensembles, but not our deep encoder, exploit\. The diagnostic that response magnitude is the dominant recoverable signal is therefore not specific to a single MLP\-encoder configuration; whether attention\-based, set\-equivariant, or foundation\-model encoders behave similarly is left to future work\. The Random Forest result on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]is robust across reasonable RF hyperparameter choices \(App\.[R](https://arxiv.org/html/2608.00152#A18)\)\.

### 6\.2The external endpoint is a different quantity

Evaluating transfer requires that the training and evaluation datasets score the same quantity\. The Replogle Perturb\-seq releases include an Anderson–Darling column\. We first examine whether it represents the same quantity as the endpoint of Sec\.[3](https://arxiv.org/html/2608.00152#S3); our analysis indicates that the two are distinct\.

We attempted to reconstruct the distributed column from the Replogle single\-cell data using the definition of Sec\.[3](https://arxiv.org/html/2608.00152#S3)—a two\-sample Anderson–Darling comparison between perturbed and non\-targeting cells at the target gene—under eight pre\-declared variants spanning aggregation, transformation, input matrix, and control definition\. No variant reproduced it; the closest reached Pearson0\.3350\.335and Spearman0\.2680\.268\(App\.[J](https://arxiv.org/html/2608.00152#A10)\)\.

The distributed column is instead strongly consistent with a transcriptome\-wide differential\-expression gene count\. Candidate significant\-gene counts are rank\-monotone with it \(ρ=0\.96\\rho=0\.96–0\.980\.98\), a Benjamini–Hochberg\-corrected count reaches Pearson0\.9860\.986, and a detection\-count null is rejected\. Absolute values are not reproduced—our closest reconstruction runs roughly1\.6×1\.6\\timeslarger—so the upstream preprocessing, threshold, control set, and gene universe remain unresolved, and we do not claim to have reconstructed it\.

The two endpoints quantify different things: ours measures target\-gene\-specific effect strength, the distributed column the breadth of a transcriptome\-wide response\. Their within\-screen associations are weak—construct\-level Spearman0\.1650\.165in RPE1,0\.3010\.301in K562\-essential,0\.3960\.396in the K562 genome\-wide screen—and the scale disparity between the two label sets \(σy=0\.758\\sigma\_\{y\}=0\.758versus2\.7502\.750\) is a property of the statistics rather than of the datasets\.

This matters beyond our own evaluation: a cross\-dataset comparison that trains on a target\-gene effect size and evaluates against the distributed Replogle column is measuring response breadth, not effect strength on the perturbed gene\. We therefore rebuild the target\-gene endpoint from single\-cell data for two external screens and evaluate transfer against it\. For completeness we also report association with the distributed endpoint \(App\.[L](https://arxiv.org/html/2608.00152#A12)\); because both are evaluated on the same predictions from the same trained models, the difference between them isolates the effect of the outcome definition\.

### 6\.3Cross\-screen transfer on a common endpoint

#### Endpoint construction\.

For a construct targeting genegg, the endpoint is the definition of Sec\.[3](https://arxiv.org/html/2608.00152#S3)applied to each external screen\. Because the Anderson–Darling statistic grows with sample size, we fix the subsample sizes at the VCC medians \(2727perturbed and788788control cells\), draw2525times with pre\-specified seeds, and take the median; this removes systematic sample\-size differences between screens, though the estimators are not thereby identical\. Eligibility requires those cell counts and an unambiguous target\-to\-feature mapping, yielding1,9321\{,\}932constructs in RPE1 \(86\.2%86\.2\\%coverage\) and1,9031\{,\}903in K562\-essential \(96\.6%96\.6\\%\)\. Diagnostics are in App\.[K](https://arxiv.org/html/2608.00152#A11)\.

#### Evaluation\.

Models trained only on VCC are evaluated zero\-shot, with the HVG mapping and VCC\-side normalization of Sec\.[3](https://arxiv.org/html/2608.00152#S3)and no target\-side calibration\. We evaluate twelve prediction objects: nine classical model×\\timesfeature\-set families and three deep or magnitude\-only predictors\. MAG\-Anchor is omitted here; see App\.[N](https://arxiv.org/html/2608.00152#A14)\. Evaluation sets, metrics, bootstrap procedure, and decision rules were fixed before any value was computed\. Uncertainty is a paired cluster bootstrap resampling target genes, so constructs sharing a target are resampled together\.

Table 2:Zero\-shot transfer to two external screens on a common target\-gene endpoint\. Entries are construct\-level Spearman correlations between each VCC\-trained prediction and the harmonized endpoint\. Verdicts follow a pre\-specified rule:Posif the95%95\\%target\-gene cluster\-bootstrap interval lies wholly above zero,Negif wholly below,Incif it includes zero; an inconclusive verdict is not evidence of absent transfer\. Panel A families are evaluated per seed and averaged \(33seeds for HistGB and RF,11for Ridge\)\. Panel B predictors have a single training seed, so their intervals are conditional on it\. Feature sets:𝐱\\mathbf\{x\}expression only,𝒎~\\widetilde\{\\boldsymbol\{m\}\}magnitude only,\[𝐱;𝒎~\]\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\]both\. Interval widths are not comparable across screens: the two evaluations differ in cluster structure \(Sec\.[6\.3](https://arxiv.org/html/2608.00152#S6.SS3)\)\. Full intervals in App\.[M](https://arxiv.org/html/2608.00152#A13)\.RPE1K562\-essentialModelFeaturesρ\\rhoVerdictρ\\rhoVerdict*Panel A: classical families, seed\-averaged*Ridge𝐱\\mathbf\{x\}−0\.091\-0\.091Neg\+0\.009\+0\.009IncRidge𝒎~\\widetilde\{\\boldsymbol\{m\}\}\+0\.097\+0\.097Pos\+0\.133\+0\.133PosRidge\[𝐱;𝒎~\]\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\]−0\.068\-0\.068Neg−0\.043\-0\.043IncHistGB𝐱\\mathbf\{x\}−0\.116\-0\.116Neg−0\.211\-0\.211NegHistGB𝒎~\\widetilde\{\\boldsymbol\{m\}\}\+0\.009\+0\.009Inc\+0\.160\+0\.160PosHistGB\[𝐱;𝒎~\]\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\]−0\.038\-0\.038Inc\+0\.018\+0\.018IncRF𝐱\\mathbf\{x\}−0\.105\-0\.105Neg−0\.218\-0\.218NegRF𝒎~\\widetilde\{\\boldsymbol\{m\}\}\+0\.053\+0\.053Pos\+0\.178\+0\.178PosRF\[𝐱;𝒎~\]\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\]−0\.018\-0\.018Inc\+0\.166\+0\.166Pos*Panel B: deep and magnitude\-only predictors*spsmbaseline—−0\.050\-0\.050Neg−0\.070\-0\.070Negmag\-spsm—\+0\.067\+0\.067Pos\+0\.016\+0\.016Incmag\-Linear𝒎~\\widetilde\{\\boldsymbol\{m\}\}\+0\.057\+0\.057Pos\+0\.147\+0\.147Pos
#### Magnitude transfers; expression direction does not\.

The pattern of Table[2](https://arxiv.org/html/2608.00152#S6.T2)is consistent across both screens\. Every magnitude\-only classical family is resolved\-positive in K562\-essential, and two of three are positive in RPE1\. Every expression\-only family is negative or unresolved in both, as is the deepspsmbaseline\. Exposing magnitude to the deep encoder improves transfer over that baseline by a resolved margin in both screens:Δ​ρ=\+0\.117\\Delta\\rho=\+0\.117\[\+0\.079,\+0\.155\]\[\+0\.079,\+0\.155\]in RPE1 and\+0\.086\+0\.086\[\+0\.042,\+0\.130\]\[\+0\.042,\+0\.130\]in K562\-essential\. In these two evaluations, positive rank transfer is concentrated in response\-amplitude summaries rather than in the expression\-direction features learned by the tested models\.

#### The deep encoder does not outperform a linear magnitude model\.

The deep encoder does not outperform a four\-scalar linear regression on the same magnitude features in either screen\. In K562\-essentialmag\-Linearis resolvedly better \(Δ​ρ=−0\.131\\Delta\\rho=\-0\.131,\[−0\.187,−0\.074\]\[\-0\.187,\-0\.074\]\); in RPE1 the two are indistinguishable \(\+0\.011\+0\.011,\[−0\.030,\+0\.053\]\[\-0\.030,\+0\.053\]\)\. Two objects also change verdict between screens in opposite directions:mag\-spsmis positive in RPE1 and unresolved in K562\-essential, while Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]—the strongest in\-distribution model—is unresolved in RPE1 and positive in K562\-essential\. Transfer behavior at the level of individual model families is screen\-dependent even where the group\-level pattern is not\.

#### Scope\.

The two evaluations differ in cluster structure—at most three constructs per target gene in RPE1 versus two in K562\-essential, both over1,8111\{,\}811genes—so interval widths are not mechanically comparable between screens\. Three of the twelve objects have a single training seed, so their intervals reflect target\-gene sampling conditional on that seed\. PooledR2R^\{2\}is negative throughout because predictions produced on one outcome scale are not calibrated to another; rank association is the interpretable quantity here\.

### 6\.4MAG\-Anchor results

Across five seeds, MAG\-Anchor reaches VCC testR2=\+0\.268±0\.011R^\{2\}=\+0\.268\\pm 0\.011, improving overmag\-spsm\(\+0\.225±0\.004\+0\.225\\pm 0\.004\) but remaining below Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\(\+0\.370±0\.004\+0\.370\\pm 0\.004\)\. Three pre\-specified controls—eval\-time row shuffling, train\-time row shuffling, and random\-Gaussian replacement of𝒎~\\widetilde\{\\boldsymbol\{m\}\}—collapse the gain in each case, so the improvement depends on correctly aligned per\-row magnitude features rather than on added capacity\. Full results, the residual\-branch ablation, and the controls are in App\.[D](https://arxiv.org/html/2608.00152#A4)and Table[4](https://arxiv.org/html/2608.00152#A4.T4)\.

## 7Discussion and limitations

#### What this paper claims\.

The central claim is diagnostic: on the disjoint target\-gene split of VCC, the predictive signal is dominated by a low\-dimensional response\-magnitude axis\. Across the tested classical families and MLP\-based encoders, removing explicit magnitude degrades in\-distribution performance and yields negative or inconclusive external rank transfer, whereas magnitude\-based predictors give the most stable rank signal\. The deepmag\-spsmencoder benefits from magnitude in\-distribution but does not outperformmag\-Linearunder transfer\. We do not claim a new representation\-learning method or a replacement for scGPT\(Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\), Geneformer\(Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\), or GEARS\(Roohani et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib24)\)\.

#### Why magnitude appears more stable than learned direction\.

The magnitude features are coordinate\-permutation\- and \(form1,m2,m3m\_\{1\},m\_\{2\},m\_\{3\}\) sign\-flip invariant, sensitive to overall response scale rather than to which genes respond\. These are function\-level invariances and do not by themselves imply distributional stability; transfer to two external screens is the empirical evidence that the stability holds under those shifts\. The deep encoder, by contrast, appears to rely on source\-specific directional patterns\. Held\-out target\-gene performance should therefore not be read as evidence of cross\-context robustness without external evaluation\. Foundation\-model scope and an alternative reading of the diagnostic are in App\.[T](https://arxiv.org/html/2608.00152#A20)\.

#### Limitations\.

Cross\-screen evidence comes from two target screens and a single source, so the pattern is established under two shifts rather than a range of them\. Individual model families change verdict between screens, and the deep encoder’s advantage over a linear magnitude model is unresolved in one screen and negative in the other; we therefore state the transfer claim at the level of feature groups rather than individual architectures\. The construction of the endpoint distributed with the external screens remains unresolved \(Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)\)\. Further limitations are in App\.[U](https://arxiv.org/html/2608.00152#A21)\.

#### Conclusion\.

The predictive signal on this benchmark is dominated by a low\-dimensional response\-magnitude axis that the MLP\-based deep encoders tested here do not exploit reliably, and pre\-specified controls attribute the gain to per\-row alignment rather than added dimensionality\. Under zero\-shot transfer to two external screens, scored on a rebuilt target\-gene endpoint, expression\-only models are negative or inconclusive under rank transfer whereas magnitude\-only models are usually resolved\-positive, and no deep encoder we tested beats a four\-scalar linear regression\. These results argue for models that explicitly separate response magnitude from expression direction rather than assuming a deeper encoder recovers both under shift\.

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## Appendix APre\-specified hypotheses for the strong\-baseline audit

The five hypotheses below were written in the experiment report before the corresponding scripts were run; we have no formal external preregistration, but each hypothesis is dated in the project log preceding the run\. We report each verdict honestly\.

1. H1Classical models on𝐱\\mathbf\{x\}alone will not clearly beatmag\-Linear\.*Verdict: supported\.*Best𝐱\\mathbf\{x\}\-only model \(HistGradientBoosting,R2=\+0\.190R^\{2\}=\+0\.190\) is belowmag\-Linear\(\+0\.208\+0\.208\) and within0\.0030\.003of Ridge onmag\-only\(\+0\.187\+0\.187\); no𝐱\\mathbf\{x\}\-only classical model exceedsmag\-Linearby a margin larger than seed noise\.
2. H2direction\-onlywill perform worse than𝐱\\mathbf\{x\}\-only andmag\-only\.*Verdict: supported\.*Bestdirection\-only\(\+0\.062\+0\.062\) is well below both anchors\.
3. H3Linear methods onmag\-onlywill approximately reproducemag\-Linear\.*Verdict: supported\.*Ridge onmag\-onlyreaches\+0\.187\+0\.187\(≥\+0\.180\\geq\+0\.180pre\-specified threshold\)\.
4. H4Tree ensembles onx\+magmay match or beatmag\-spsm\.*Verdict: clearly beats\.*Random Forest onx\+magreaches\+0\.370±0\.004\+0\.370\\pm 0\.004, exceedingmag\-spsmbyΔ≥\+0\.140\\Delta\\geq\+0\.140across all three seeds\.
5. H5If a non\-deep model onx\+magclearly beatsmag\-spsm, the paper should be reframed as diagnostic discovery rather than method paper\.*Verdict: triggered\.*The reframing applied to this paper is the consequence of H5\.

## Appendix BPre\-specified hypotheses for the legacy distributed\-endpoint transfer

This appendix records the original K562 transfer experiment scored against the distributed Replogle endpoint\. It is retained for provenance and endpoint\-sensitivity analysis; it is not evidence for the rebuilt target\-gene endpoint of Table[2](https://arxiv.org/html/2608.00152#S6.T2)\.

1. T1Ridge onmag\-onlywill reproduce the existingmag\-Lineartransfer \(R2≈\+0\.12R^\{2\}\\approx\+0\.12,ρ≈\+0\.32\\rho\\approx\+0\.32\)\.*Verdict: pipeline matches anchor; Ridge mag\-only attainsR2=\+0\.029R^\{2\}=\+0\.029,ρ=\+0\.173\\rho=\+0\.173, andmag\-Linearreproduces the\+0\.121\+0\.121,\+0\.315\+0\.315anchor exactly\.*
2. T2Tree ensembles on𝐱\\mathbf\{x\}\-only will transfer negatively or near zero\.*Verdict: supported\.*HistGB and RF on𝐱\\mathbf\{x\}\-only both transfer withR2<−0\.10R^\{2\}<\-0\.10andρ<−0\.37\\rho<\-0\.37\.
3. T3Tree ensembles onmag\-onlywill transfer with positive Spearman but possibly negativeR2R^\{2\}from poor leaf calibration on the cross\-cell shift\.*Verdict: supported\.*RF onmag\-only:R2=\+0\.114R^\{2\}=\+0\.114,ρ=\+0\.346\\rho=\+0\.346\.
4. T4Trees onx\+magmay transfer positively \(T4a\) or preserve only Spearman \(T4b\) or fully overfit \(T4c\)\.*Verdict: T4a holds for Random Forest \(R2=\+0\.037R^\{2\}=\+0\.037,ρ=\+0\.314\\rho=\+0\.314\); T4b holds for HistGradientBoosting \(R2=−0\.035R^\{2\}=\-0\.035,ρ=\+0\.070\\rho=\+0\.070\); T4c does not occur\.*

## Appendix CPer\-row alignment: full population\-level argument

In the population view, letM⋆M^\{\\star\}be an auxiliary random vector with the same marginal distribution as𝒎~​\(X\)\\widetilde\{\\boldsymbol\{m\}\}\(X\)but independent of\(X,Y\)\(X,Y\)\. Then the Bayes predictor with access to\(X,M⋆\)\(X,M^\{\\star\}\)coincides with the Bayes predictor with access toXXalone:

𝔼​\[Y∣X,M⋆\]=𝔼​\[Y∣X\]\.\\mathbb\{E\}\[Y\\mid X,M^\{\\star\}\]\\;=\\;\\mathbb\{E\}\[Y\\mid X\]\.By the optimality of conditional expectation under squared loss, no measurable predictor using\(X,M⋆\)\(X,M^\{\\star\}\)can improve over the Bayes risk achieved by𝔼​\[Y∣X\]\\mathbb\{E\}\[Y\\mid X\]in squared loss\. This yields two consequences we test directly: \(a\) replacing𝒎~​\(𝐱i\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\_\{i\}\)by four i\.i\.d\.𝒩​\(0,I4\)\\mathcal\{N\}\(0,I\_\{4\}\)features makes the augmenting block independent of\(𝐱i,Yi\)\(\\mathbf\{x\}\_\{i\},Y\_\{i\}\)by construction; \(b\) under a row\-shuffled augmentation𝐱iπ=\[𝐱i;𝒎~​\(𝐱π​\(i\)\)\]\\mathbf\{x\}\_\{i\}^\{\\pi\}=\[\\,\\mathbf\{x\}\_\{i\};\\,\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\_\{\\pi\(i\)\}\)\\,\]withπ\\pia uniformly random permutation independent of the data, the augmenting block preserves the marginal distribution of𝒎~\\widetilde\{\\boldsymbol\{m\}\}but no longer carries row\-specific information aboutYiY\_\{i\}\. The shuffled\-magnitude and random\-feature controls of Sec\.[5\.2](https://arxiv.org/html/2608.00152#S5.SS2)are finite\-sample diagnostics of this principle: they preserve dimensionality and marginal scale while breaking row\-specific alignment with the label\. Crucially, anyR2R^\{2\}gain from concatenating the four scalars must therefore arise from their per\-row dependence on𝐱i\\mathbf\{x\}\_\{i\}, not from added input dimensionality, scale, or noise diversity\. We do not promote this observation to a formal theorem; its role is to motivate the two pre\-specified experimental controls \(row\-shuffling and i\.i\.d\. Gaussian features\) reported in Sec\.[5\.2](https://arxiv.org/html/2608.00152#S5.SS2)\.

#### Per\-row alignment: empirical controls \(Table[3](https://arxiv.org/html/2608.00152#A3.T3)\)\.

We instantiated three additional arms at seed0\(otherwise identical training\): \(i\)baseline\-mse:spsmV4 with no magnitude features \(𝐱\+=𝐱\\mathbf\{x\}^\{\+\}=\\mathbf\{x\}\); \(ii\)shuffled\-magnitude:mag\-spsmbut with per\-row correspondence between𝐱i\\mathbf\{x\}\_\{i\}and𝒎~\\widetilde\{\\boldsymbol\{m\}\}broken by a uniformly random permutationπ\\pi; \(iii\)random\-feature:mag\-spsmbut with the magnitude block replaced by four i\.i\.d\.𝒩​\(0,1\)\\mathcal\{N\}\(0,1\)features\. The result \(Table[3](https://arxiv.org/html/2608.00152#A3.T3)\) is unambiguous: both controls return performance to baseline within seed noise\. Shuffling removes the gain entirely \(Δ=−0\.151\\Delta=\-0\.151vs\.mag\-spsm\); four random Gaussian features fail to recover any gain\. Under shuffling,𝒎~\(𝐱π​\(i\)\)⟂⟂\(𝐱i,yi\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\_\{\\pi\(i\)\}\)\\perp\\\!\\\!\\\!\\perp\(\\mathbf\{x\}\_\{i\},y\_\{i\}\)in the sample limit, so the augmenting block carries no information aboutyiy\_\{i\}\.

Table 3:Causal verification \(seed 0\)\. Both shuffled\-magnitude \(per\-row alignment broken\) and four random Gaussian features \(independent of input by construction\) return performance to baseline within seed noise, in agreement with the population\-level argument of this appendix\.

## Appendix DMAG\-Anchor: full architecture, hyperparameters, and controls

Table 4:VCC in\-distribution results and transfer to Replogle K562 scored against the*distributed*Anderson–Darling endpoint\. The Replogle columns therefore measure association with a transcriptome\-wide response\-breadth statistic, not with the target\-gene endpoint of Sec\.[3](https://arxiv.org/html/2608.00152#S3); see Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)and App\.[L](https://arxiv.org/html/2608.00152#A12)\. Mean±\\pmstd across seeds \(ddof=1\\mathrm\{ddof\}\{=\}1\);nnvaries by model\.MAGAnchor\-xis the residual ablation in which the residual branch receives only𝐱\\mathbf\{x\}\.#### Architecture\.

MAG\-Anchor decomposes the predicted perturbation effect into a direct magnitude anchor and a gated residual correction:

y^=gϕ​\(𝒎~​\(𝐱\)\)\+α​hθ​\(\[𝐱;𝒎~​\(𝐱\)\]\),\\hat\{y\}=g\_\{\\phi\}\\\!\\left\(\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\\right\)\+\\alpha\\,h\_\{\\theta\}\\\!\\left\(\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\]\\right\),\(3\)where𝒎~​\(𝐱\)∈ℝ4\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{4\}is the standardized magnitude vector defined in Eq\.[1](https://arxiv.org/html/2608.00152#S5.E1)\. The anchor branchgϕ​\(𝒎~\)=𝐰⊤​𝒎~\+bg\_\{\\phi\}\(\\widetilde\{\\boldsymbol\{m\}\}\)=\\mathbf\{w\}^\{\\top\}\\widetilde\{\\boldsymbol\{m\}\}\+bis a single linear layer with𝐰∈ℝ4\\mathbf\{w\}\\in\\mathbb\{R\}^\{4\},b∈ℝb\\in\\mathbb\{R\}; its weights are warm\-started from a sklearnLinearRegressionfitted on\(𝒎~,y\)\(\\widetilde\{\\boldsymbol\{m\}\},y\)and remain trainable thereafter \(no freezing\)\. The residual branchhθh\_\{\\theta\}is a small MLP with layersdin→256→128→1d\_\{\\mathrm\{in\}\}\\to 256\\to 128\\to 1, ReLU activations, and architectural dropoutp=0\.1p\{=\}0\.1after each ReLU; no LayerNorm or BatchNorm is used\. This dropout setting applies only to the MAG\-Anchor residual branch; thespsm/mag\-spsmbackbone \(App\.[G](https://arxiv.org/html/2608.00152#A7)\) uses dropoutp=0p\{=\}0\. For the headline MAGAnchor\-xmag the residual input is\[𝐱;𝒎~\]∈ℝ2004\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\\in\\mathbb\{R\}^\{2004\}\(din=2004d\_\{\\mathrm\{in\}\}=2004\); for the MAGAnchor\-x ablation the residual input is𝐱\\mathbf\{x\}alone \(din=2000d\_\{\\mathrm\{in\}\}=2000\)\. The scalar gateα\\alphais parameterized asα=softplus​\(ρ\)\\alpha=\\mathrm\{softplus\}\(\\rho\)withρinit=ln⁡\(e0\.1−1\)\\rho\_\{\\mathrm\{init\}\}=\\ln\(\\mathrm\{e\}^\{0\.1\}\-1\), soαinit=0\.1\\alpha\_\{\\mathrm\{init\}\}=0\.1exactly andα∈\(0,∞\)\\alpha\\in\(0,\\infty\)throughout training\. This design preserves a direct pathway from magnitude features to the output while allowing the residual branch to learn corrections from both the expression vector and the magnitude summary\.

#### Training hyperparameters\.

Unless otherwise stated, MAG\-Anchor uses a trainable anchor, no residual penalty, Adam with learning rate10−310^\{\-3\}and weight decay10−510^\{\-5\}, batch size6464,2525training epochs, and checkpoint selection by validationR2R^\{2\}\. Frozen\-anchor and residual\-penalty variants are evaluated only as sensitivity checks \(App\.[E](https://arxiv.org/html/2608.00152#A5)\)\.

#### Per\-row alignment is load\-bearing \(controls\)\.

We report three pre\-specified controls on the MAG\-Anchor recipe to test whether its gain depends on per\-row magnitude alignment or merely on added input dimensionality \(Table[5](https://arxiv.org/html/2608.00152#A4.T5)\)\. In the*eval\-shuffled*control, we train MAG\-Anchor normally but permute𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)across rows at evaluation time; VCCR2R^\{2\}collapses from\+0\.268\+0\.268to−0\.476\-0\.476, and Replogle Spearman falls from\+0\.341\+0\.341to−0\.041\-0\.041\. In the*train\-shuffled*control,𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)is permuted across training rows, causing VCC performance to collapse toR2=−0\.033R^\{2\}=\-0\.033and eliminating positive transfer\. Replacing the magnitude vector with random Gaussian features also fails, yielding near\-zero VCC performance and anti\-correlated Replogle transfer \(ρ=−0\.517\\rho=\-0\.517\)\. All three controls confirm that MAG\-Anchor’s gain depends on correctly aligned per\-row magnitude features and is not a generic capacity effect\.

Table 5:Per\-row magnitude alignment is load\-bearing for MAG\-Anchor\. Shuffling or replacing the magnitude block collapses VCC performance and removes positive association with the legacy distributed endpoint; random Gaussian magnitude features produce anti\-correlated Replogle transfer\. All controls are direct ablations of the trained MAG\-Anchor \(xmag\) recipe\. Mean±\\pmstd across 3 seeds \(base MAG\-Anchor: 5 seeds\)\.

## Appendix EMAG\-Anchor sensitivity analysis

Table 6:Sensitivity analysis for MAG\-Anchor\. We tested freezing the magnitude anchor at MAG\-LINEAR weights, adding a small residual penaltyλres∈\{0\.01,0\.1\}\\lambda\_\{\\text\{res\}\}\\in\\\{0\.01,0\.1\\\}, and combining both\. All variants are statistically indistinguishable from the base MAG\-Anchor within seed noise \(Replogleρ\\rhodifferences<0\.02\{<\}0\.02\)\. We pre\-declared adoption thresholds ofΔ​ρ\>\+0\.04\\Delta\\rho\{\>\}\{\+\}0\.04or seed\-variance reduction\>30%\{\>\}30\\%; neither was met\. We therefore report the simpler base MAG\-Anchor in the main paper\. Mean±\\pmstd acrossnnseeds\.#### Pre\-declared adoption thresholds\.

Before running our five\-seed confirmation, we pre\-declared two thresholds for adopting any architectural complication beyond the base MAG\-Anchor formulation: \(i\) the more complex variant must improve mean Replogle Spearman by at least one combined standard error, approximately\+0\.04\+0\.04; or \(ii\) it must reduce the seed\-level standard deviation of Replogle Spearman by at least30%30\\%\. We tested a frozen\-anchor plus residual\-penalty variant against the base MAG\-Anchor over five seeds\. The mean Replogle Spearman improved by only\+0\.015\+0\.015, below the\+0\.04\+0\.04threshold, and the seed\-level standard\-deviation reduction was only8\.8%8\.8\\%, below the30%30\\%threshold\. We therefore report the base MAG\-Anchor in the main paper and treat frozen\-anchor and residual\-penalty variants as sensitivity checks\.

## Appendix FClassical\-baseline hyperparameters

- •Ridge:Ridge\(alpha=1\.0\)\.
- •ElasticNet:ElasticNet\(alpha=0\.001, l1\_ratio=0\.5, max\_iter=10000\)\.
- •Random Forest:RandomForestRegressor\(n\_estimators=200, max\_depth=20, min\_samples\_leaf=2, n\_jobs=8, random\_state=seed\)\.
- •HistGradientBoosting \(XGBoost fallback\): default sklearnHistGradientBoostingRegressorwithmax\_iter=500, learning\_rate=0\.05, max\_depth=6, random\_state=seed, early stopping on validation with patience 20\.
- •kk\-NN:KNeighborsRegressor\(n\_neighbors=10, weights="distance"\)\.

For linear models andkk\-NN, features are standardized using training\-split mean/std exclusively; for tree models, raw features are used \(with the four magnitude featureszz\-scored regardless of model\)\.

## Appendix GspsmV4 backbone and loss formulas

ThespsmV4 backbone is a three\-layer MLP encoder with hidden width256256, ReLU activations, and LayerNorm; architectural dropout isp=0p=0\(no active dropout regularization\)\. The encoder produces a128128\-dimensional representation𝐳i=fθ​\(𝐱i\+\)\\mathbf\{z\}\_\{i\}=f\_\{\\theta\}\(\\mathbf\{x\}\_\{i\}^\{\+\}\)\. The total objective combines a prediction loss with two SPSM regularizers,ℒtotal=ℒMSE\+λstab​ℒstab\+λrel​ℒrel\\mathcal\{L\}\_\{\\mathrm\{total\}\}=\\mathcal\{L\}\_\{\\mathrm\{MSE\}\}\+\\lambda\_\{\\mathrm\{stab\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{stab\}\}\+\\lambda\_\{\\mathrm\{rel\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{rel\}\}, withλstab=λrel=0\.5\\lambda\_\{\\mathrm\{stab\}\}=\\lambda\_\{\\mathrm\{rel\}\}=0\.5, both inherited from the SPSM V4 backbone unchanged\.

#### Stability loss\.

The stability term is a Gaussian\-kernel maximum\-mean\-discrepancy \(MMD\) penalty on encoder features across the experimental Flex batches that co\-occur within the same SGD minibatch; no stochastic input perturbation, dropout\-perturbed forward pass, or input noising is used\. Concretely, for an SGD minibatchℐ\\mathcal\{I\}of sizeB=64B=64, letℬ​\(ℐ\)=\{bi:i∈ℐ\}\\mathcal\{B\}\(\\mathcal\{I\}\)=\\\{b\_\{i\}:i\\in\\mathcal\{I\}\\\}denote the set of experimental Flex batches represented inℐ\\mathcal\{I\}, and letZb​\(ℐ\)=\{𝐳i:i∈ℐ,bi=b\}Z\_\{b\}\(\\mathcal\{I\}\)=\\\{\\mathbf\{z\}\_\{i\}:i\\in\\mathcal\{I\},\\,b\_\{i\}=b\\\}collect the encoder outputs from rows of batchbb\. The stability loss averages the squared MMD across all unordered pairs of co\-represented batches:

ℒstab=2\|ℬ​\(ℐ\)\|​\(\|ℬ​\(ℐ\)\|−1\)​∑b,b′∈ℬ​\(ℐ\)b<b′MMDκ2​\(Zb​\(ℐ\),Zb′​\(ℐ\)\),\\mathcal\{L\}\_\{\\mathrm\{stab\}\}\\;=\\;\\frac\{2\}\{\|\\mathcal\{B\}\(\\mathcal\{I\}\)\|\\,\(\|\\mathcal\{B\}\(\\mathcal\{I\}\)\|\-1\)\}\\\!\\\!\\sum\_\{\\begin\{subarray\}\{c\}b,b^\{\\prime\}\\in\\mathcal\{B\}\(\\mathcal\{I\}\)\\\\ b<b^\{\\prime\}\\end\{subarray\}\}\\mathrm\{MMD\}^\{2\}\_\{\\kappa\}\\\!\\left\(Z\_\{b\}\(\\mathcal\{I\}\),\\,Z\_\{b^\{\\prime\}\}\(\\mathcal\{I\}\)\\right\),whereκ\\kappais a Gaussian kernel whose bandwidth is set by the median heuristic on the in\-minibatch pairwise distances \(computed underno\_grad\)\. For each SGD minibatchℐ\\mathcal\{I\},ℬ​\(ℐ\)\\mathcal\{B\}\(\\mathcal\{I\}\)denotes the experimental batches represented in that minibatch\. The sampler constructs context\-stratified minibatches so that each represented batch contributes enough rows for the MMD pairs in the stability loss below to be well\-defined\.

#### Relation loss\.

The relation loss is a soft\-Spearman ranking penalty applied to the scalar predictionsy^\\hat\{y\}across pairs of experimental batches that co\-occur in the same minibatch; it does not use a learnable projection on𝐳\\mathbf\{z\}and does not enter the labelsyiy\_\{i\}\(labels enter only throughℒMSE\\mathcal\{L\}\_\{\\mathrm\{MSE\}\}\)\. For an SGD minibatchℐ\\mathcal\{I\}, we construct a small set𝒫\\mathcal\{P\}of matched context\-pairs by iterating over unordered pairs of distinct Flex batches\(b,b′\)∈ℬ​\(ℐ\)×ℬ​\(ℐ\)\(b,b^\{\\prime\}\)\\in\\mathcal\{B\}\(\\mathcal\{I\}\)\\times\\mathcal\{B\}\(\\mathcal\{I\}\)and, for each pair, taking the first applicable of three tiers: tier 1 \(weightw=1\.00w=1\.00\) matches rows with the same target\-gene identifier shared betweenbbandb′b^\{\\prime\}; tier 2 \(weightw=0\.50w=0\.50\) matches rows with the same auxiliary cluster index, paired greedily; tier 3 \(weightw=0\.25w=0\.25\) takesk=1k\{=\}1nearest neighbors in an auxiliaryuu\-space\. On VCC, only tier 1 contributes informative matches, so in practice𝒫\\mathcal\{P\}contains pairs of Flex batches with at least one shared target gene, aligned by target gene\. Each matched pairp∈𝒫p\\in\\mathcal\{P\}yields two aligned prediction vectors𝒚^a\(p\),𝒚^b\(p\)∈ℝmp\\hat\{\\boldsymbol\{y\}\}^\{\(p\)\}\_\{a\},\\hat\{\\boldsymbol\{y\}\}^\{\(p\)\}\_\{b\}\\in\\mathbb\{R\}^\{m\_\{p\}\}and a weightwpw\_\{p\}\. With soft ranksr~k​\(𝒗;τ\)=∑ℓ≠ksigmoid⁡\(\(vk−vℓ\)/τ\)\\widetilde\{r\}\_\{k\}\(\\boldsymbol\{v\};\\tau\)=\\sum\_\{\\ell\\neq k\}\\operatorname\{sigmoid\}\\\!\\bigl\(\(v\_\{k\}\-v\_\{\\ell\}\)/\\tau\\bigr\), the soft\-Spearman correlation

ρ~​\(𝒂,𝒃;τ\)=∑k\(r~k​\(𝒂\)−r~¯​\(𝒂\)\)​\(r~k​\(𝒃\)−r~¯​\(𝒃\)\)∑k\(r~k​\(𝒂\)−r~¯​\(𝒂\)\)2​∑k\(r~k​\(𝒃\)−r~¯​\(𝒃\)\)2,\\widetilde\{\\rho\}\(\\boldsymbol\{a\},\\boldsymbol\{b\};\\tau\)\\;=\\;\\frac\{\\sum\_\{k\}\(\\widetilde\{r\}\_\{k\}\(\\boldsymbol\{a\}\)\-\\overline\{\\widetilde\{r\}\}\(\\boldsymbol\{a\}\)\)\(\\widetilde\{r\}\_\{k\}\(\\boldsymbol\{b\}\)\-\\overline\{\\widetilde\{r\}\}\(\\boldsymbol\{b\}\)\)\}\{\\sqrt\{\\sum\_\{k\}\(\\widetilde\{r\}\_\{k\}\(\\boldsymbol\{a\}\)\-\\overline\{\\widetilde\{r\}\}\(\\boldsymbol\{a\}\)\)^\{2\}\\,\\sum\_\{k\}\(\\widetilde\{r\}\_\{k\}\(\\boldsymbol\{b\}\)\-\\overline\{\\widetilde\{r\}\}\(\\boldsymbol\{b\}\)\)^\{2\}\}\},the relation loss is

ℒrel=∑p∈𝒫wp​\(1−ρ~​\(𝒚^a\(p\),𝒚^b\(p\);τ\)\)∑p∈𝒫wp,τ=0\.1\.\\mathcal\{L\}\_\{\\mathrm\{rel\}\}\\;=\\;\\frac\{\\sum\_\{p\\in\\mathcal\{P\}\}w\_\{p\}\\,\\bigl\(1\-\\widetilde\{\\rho\}\(\\hat\{\\boldsymbol\{y\}\}^\{\(p\)\}\_\{a\},\\hat\{\\boldsymbol\{y\}\}^\{\(p\)\}\_\{b\};\\,\\tau\)\\bigr\)\}\{\\sum\_\{p\\in\\mathcal\{P\}\}w\_\{p\}\},\\qquad\\tau=0\.1\.ℒrel\\mathcal\{L\}\_\{\\mathrm\{rel\}\}thus drives the model to produce a consistent rank order of predictions across pairs of experimental batches in the same minibatch\.

#### Optimization\.

Adam at lr10−310^\{\-3\}, weight decay10−510^\{\-5\}, batch size6464,2525epochs; HVG selection on the training split only; magnitudezz\-score statistics from the training split only; model selection by validationR2R^\{2\}\.

#### Architecture overview \(visual\)\.

Figure[2](https://arxiv.org/html/2608.00152#A7.F2)summarizes themag\-spsmpipeline\. The four magnitude statistics𝒎​\(𝐱\)\\boldsymbol\{m\}\(\\mathbf\{x\}\)defined in Eq\. \([1](https://arxiv.org/html/2608.00152#S5.E1)\) arezz\-scored using VCC training statistics to obtain𝒎~​\(𝐱\)∈ℝ4\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{4\}, then concatenated with thed=2,000d\{=\}2\{,\}000\-dimensional log1p delta input𝐱\\mathbf\{x\}to form𝐱\+∈ℝ2004\\mathbf\{x\}^\{\+\}\\in\\mathbb\{R\}^\{2004\}, which is fed to thespsmV4 backbone and a linear head\. The same backbone, the same losses, and the same optimizer are used for thespsmbaseline \(which receives only𝐱\\mathbf\{x\}\) and formag\-spsm\(which receives𝐱\+\\mathbf\{x\}^\{\+\}\); the only difference is whether the four magnitude features are exposed to the encoder\.

![Refer to caption](https://arxiv.org/html/2608.00152v1/x2.png)Figure 2:mag\-spsmarchitecture\. The expression delta vector𝐱\\mathbf\{x\}\(d=2,000d\{=\}2\{,\}000HVGs\) is concatenated with the fourzz\-scored magnitude features𝒎~​\(𝐱\)=\(\|x\|¯,‖𝐱‖2/d,‖𝐱‖∞,σ​\(𝐱\)\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)=\(\\overline\{\|x\|\},\\,\\\|\\mathbf\{x\}\\\|\_\{2\}/\\sqrt\{d\},\\,\\\|\\mathbf\{x\}\\\|\_\{\\infty\},\\,\\sigma\(\\mathbf\{x\}\)\)to form𝐱\+∈ℝ2004\\mathbf\{x\}^\{\+\}\\in\\mathbb\{R\}^\{2004\}, which is passed through thespsmV4 encoder MLP \(three hidden layers of width256256, ReLU, LayerNorm, no architectural dropout\) and a linear head\. Allzz\-score statistics use VCC training data exclusively\.

## Appendix HMagnitude statistics on Replogle

For the primary2,0002\{,\}000\-HVG transfer evaluation, each Replogle row𝐱\(R\)\\mathbf\{x\}^\{\(R\)\}is first embedded into the VCC HVG namespace by zero\-imputing the570570VCC\-only coordinates missing from Replogle\. The magnitude statistics𝒎​\(𝐱\(R\)\)\\boldsymbol\{m\}\(\\mathbf\{x\}^\{\(R\)\}\)are then computed using Eq\. \([1](https://arxiv.org/html/2608.00152#S5.E1)\) over this2,0002\{,\}000\-dimensional vector, including the zero\-imputed entries\. In the HVG\-coverage controls \(App\.[O](https://arxiv.org/html/2608.00152#A15)\), by contrast, models are trained and evaluated directly in the1,4301\{,\}430shared\-HVG subspace\. We thenzz\-score using*VCC training*statistics𝝁tr,𝝈tr\\boldsymbol\{\\mu\}\_\{\\text\{tr\}\},\\boldsymbol\{\\sigma\}\_\{\\text\{tr\}\}\(Sec\.[5](https://arxiv.org/html/2608.00152#S5)\), preserving the strict zero\-shot constraint\. The Reploglezz\-scored magnitudes have empirical means roughly\+0\.8​σtr\+0\.8\\,\\sigma\_\{\\text\{tr\}\}to\+1\.9​σtr\+1\.9\\,\\sigma\_\{\\text\{tr\}\}relative to the VCC training baseline, and component\-wise standard deviations1\.6×1\.6\\timesto2\.4×2\.4\\timeswider, reflecting the larger feature scale of the Replogle release\. The deepmag\-spsmsees these as out\-of\-distribution; the linear baseline, having no encoder, does not suffer the same OOD penalty\.

#### Visualizing the failure mode on Replogle\.

Figure[3](https://arxiv.org/html/2608.00152#A8.F3)contrasts the predicted\-versus\-true scatter of thespsmbaseline andmag\-Linearon the Replogle K562 test set\. Thespsmbaseline, trained on the expression vector alone, collapses its predictions to a narrow band centered near the VCC training\-label mean \(y^≈4\\hat\{y\}\\approx 4\) and is anti\-correlated with the distributed Replogle breadth endpoint \(R2=−0\.161R^\{2\}=\-0\.161,ρ=−0\.435\\rho=\-0\.435\): high\- and low\-effect perturbations are predicted similarly, and the rank order is reversed under cell\-type shift\.mag\-Linear, a four\-feature linear regression on𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\), expands its prediction range and recovers a positive monotone relationship to the same distributed breadth endpoint \(R2=\+0\.121R^\{2\}=\+0\.121,ρ=\+0\.315\\rho=\+0\.315\)\. The contrast illustrates the legacy\-endpoint failure mode that motivated the harmonized analysis of Sec\.[6\.3](https://arxiv.org/html/2608.00152#S6.SS3): the expression\-only encoder transfers negatively, whereas an explicit magnitude summary transfers positively, even though both models share the same training data and target\.

#### Optimization, training schedule, and compute\.

All deep models \(SPSM, MAG\-SPSM, MAG\-Anchor\) use Adam at learning rate10−310^\{\-3\}with weight decay10−510^\{\-5\}, batch size6464, and2525training epochs\. The loss \([2](https://arxiv.org/html/2608.00152#S5.E2)\) applies tospsm/mag\-spsm; MAG\-Anchor uses MSE ony^\\hat\{y\}\. HVG selection and allzz\-score statistics use the VCC training split exclusively\. All deep experiments fit on a single TITAN RTX 24 GB GPU; classical baselines were run on CPU\. Classical baselines usescikit\-learndefaults \(detailed in App\.[F](https://arxiv.org/html/2608.00152#A6)\)\.

![Refer to caption](https://arxiv.org/html/2608.00152v1/x3.png)Figure 3:Failure mode on Replogle K562 \(cell\-type shifted\), zero\-shot from VCC\. \(a\) Thespsmbaseline’s predictions collapse to a narrow band near the VCC training\-label mean and are anti\-correlated with the distributed Replogle breadth endpoint \(R2=−0\.161R^\{2\}=\-0\.161,ρ=−0\.435\\rho=\-0\.435\)\. \(b\)mag\-Linear\(four\-feature linear regression on𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\) expands the prediction range and recovers a positive monotone relationship to the held\-out label \(R2=\+0\.121R^\{2\}=\+0\.121,ρ=\+0\.315\\rho=\+0\.315\)\.

## Appendix IAdditional results: ablations, rank extension, and negative results

#### Single\-feature and leave\-one\-out ablation ofmag\-spsm\.

We performed a complete single\-feature and leave\-one\-out \(LOO\) ablation of the four magnitude features at seed0\(Table[7](https://arxiv.org/html/2608.00152#A9.T7)\)\. Across the four single\-feature variants, the VCC testR2R^\{2\}ranges from\+0\.095\+0\.095\(single‖𝐱‖2\\\|\\mathbf\{x\}\\\|\_\{2\}\) to\+0\.128\+0\.128\(single\|𝐱\|¯\\overline\{\|\\mathbf\{x\}\|\}\), all well below the full 4\-featuremag\-spsmat\+0\.222\+0\.222\. Across the four leave\-one\-out variants, the testR2R^\{2\}ranges from\+0\.137\+0\.137\(dropmax⁡\|𝐱\|\\max\|\\mathbf\{x\}\|\) to\+0\.230\+0\.230\(drop‖𝐱‖2\\\|\\mathbf\{x\}\\\|\_\{2\}\); the latter is the best LOO arm and is within0\.0080\.008of the full 4\-feature model, while removingmax⁡\|𝐱\|\\max\|\\mathbf\{x\}\|costs0\.0860\.086, indicating thatmax⁡\|𝐱\|\\max\|\\mathbf\{x\}\|contributes the most among the four magnitude features\.

Table 7:Magnitude feature ablation on VCC test \(seed 0\)\.Δ\\Deltavs full = R2minus full 4\-feature MAG\-SPSM\. Bold marks the best leave\-one\-out variant\.
#### mag\-Rank: a small rank\-margin extension ofmag\-spsm\.

mag\-Rankadds a margin\-based pairwise rank lossℒrank=1\|𝒫\|​∑\(i,j\)∈𝒫max⁡\(0,α−sign​\(yi−yj\)⋅\(y^i−y^j\)\)\\mathcal\{L\}\_\{\\text\{rank\}\}=\\tfrac\{1\}\{\|\\mathcal\{P\}\|\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\max\(0,\\,\\alpha\-\\mathrm\{sign\}\(y\_\{i\}\-y\_\{j\}\)\\cdot\(\\hat\{y\}\_\{i\}\-\\hat\{y\}\_\{j\}\)\)over within\-batch close\-call pairs, withα=0\.1\\alpha=0\.1, in\-batch close\-call pairs dropped atεy=0\.05\\varepsilon\_\{y\}=0\.05, and weightλrank=0\.1\\lambda\_\{\\text\{rank\}\}=0\.1\. The 3\-seed testR2R^\{2\}rises to\+0\.2396±0\.0078\+0\.2396\\pm 0\.0078, a paired improvement overmag\-spsmof\+0\.0142±0\.0040\+0\.0142\\pm 0\.0040\. Spearman is statistically unchanged; we reportmag\-Rankas a small reproducible extension and do not promote it as a contribution\. The strong\-baseline result of Sec\.[6\.1](https://arxiv.org/html/2608.00152#S6.SS1)dominates this gain by an order of magnitude\.

#### Per\-quintile bias\.

mag\-spsmreduces per\-quintile bias most sharply at the tails \(Q1 bias from\+1\.093\+1\.093to\+0\.918\+0\.918; Q5 bias from−0\.696\-0\.696to−0\.580\-0\.580\), consistent with the diagnostic that magnitude features expand the prediction range\. The remaining∼60%\\sim 60\\%of the per\-quintile bias is not closed bymag\-spsm, motivating the strong\-baseline audit \(Sec\.[6\.1](https://arxiv.org/html/2608.00152#S6.SS1)\) and the cross\-dataset experiment \(Sec\.[6\.3](https://arxiv.org/html/2608.00152#S6.SS3)\)\.

#### Failed and weak methods \(full table\)\.

![Refer to caption](https://arxiv.org/html/2608.00152v1/x4.png)Figure 4:Single\-feature and leave\-one\-out ablations of the four magnitude features\. All four behave as a redundant low\-dimensional summary on this split\.Table 8:Additional negative and weak baselines, all evaluated on VCC unless noted\.Table 9:Dataset statistics\. Both use CRISPRi perturbation; differ in cell line, biological\-row unit, and feature normalization\.VCCReplogle K562 \(essential\)SourceVirtual Cell ChallengeReplogle et al\. 2022 \(Cell\)Cell typeH1 hESC \(stem\)K562 \(CML\)PerturbationCRISPRi \(dCas9\-KRAB\)CRISPRi \(dCas9\-KRAB\)Train / Val / Test rows5,740 / 2,052 / 3,972— \(held\-out test only\)Non\-NT rows used \(test\)3,9722,176Unique target genes282 \(all splits\)2,057Train/Val/Test target\-gene split140 / 49 / 93 \(disjoint\)—Feature dim \(dxd\_\{x\}used\)2,000 \(HVG\)8,563 full; 1,430 in HVGFeature normalizationlog1pΔ\\Deltavs per\-batch NTpre\-z\-scored log1p changeFeature std \(post\-HVG\)0\.0850\.0850\.1610\.161\(∼\\sim1\.9×1\.9\\times\)Labellog1p\(AD\) per \(batch, gene\)log1p\(AD\) per \(pert, prom\)Label range\[0,5\.72\]\[0,\\,5\.72\]\[0,8\.71\]\[0,\\,8\.71\]Label mean \(std\)4\.23​\(0\.81\)4\.23\\,\(0\.81\)3\.99​\(2\.75\)3\.99\\,\(2\.75\)Per\-row biological unit\(batch×\\timesgene\) pseudobulk\(pert, prom\) batch\-aggregatedTarget\-gene overlap with VCC—68 / 282 \(24\.1%\)
#### All methods compared \(consolidated\)\.

Table 10:Comparison of all methods tested\. VCC test R2shown as mean±\\pmstd \(3 seeds\) for verified methods; single\-seed for negative results\. Replogle K562 shows seed\-0 cross\-dataset transfer\. Replogle columns are scored against the distributed endpoint \(Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)\)\.MethodVCC test R2Replogle R2StatusNotesSPSM V4 baseline\+0\.0824±0\.0100\+0\.0824\\pm 0\.0100−0\.162\-0\.162verifiednseeds=3n\_\{\\text\{seeds\}\}\{=\}3MAG\-SPSM\+0\.2253±0\.0040\\mathbf\{\+0\.2253\\pm 0\.0040\}−0\.101\-0\.101verifiednseeds=3n\_\{\\text\{seeds\}\}\{=\}3MAG\-Rank \(λ=0\.1\\lambda\{=\}0\.1\)\+0\.2396±0\.0078\+0\.2396\\pm 0\.0078—extensionpairedΔ=\+0\.0142±0\.0040\\Delta\{=\}\+0\.0142\\pm 0\.0040MAG\-3feature\+0\.2320±0\.0162\+0\.2320\\pm 0\.0162—extensionhigher seed varianceMagnitude\-only LR\+0\.2082\+0\.2082\(1 seed\)\+0\.121\\mathbf\{\+0\.121\}linear baseline4\-feature linear reg\.Negative / failed methodsIGA bilevel gate— \(gate test\)—failedH1p=0\.380p\{=\}0\.380, H2ρ=0\.10\\rho\{=\}0\.10Replogle pretrained encoder−0\.181\-0\.181\(DWP\)—failedcell\-type mismatchPost\-hoc tail correction−0\.192\-0\.192†\\dagger—faileddeployableΔ=−0\.05\\Delta=\-0\.05†\\daggerVCC val \(n=2052\), single\-seed;Δ\\Deltavs uncorrected single\-seed SPSM V4 base \(R=2−0\.146\{\}^\{2\}\{=\}\{\-\}0\.146\)\.Tail\-weighted MSE\+0\.0843\+0\.0843\(best\)—weakΔ=\+0\.0075\\Delta\{=\}\+0\.0075LDS\-weighted MSE−0\.0233\-0\.0233—failedσ=2\\sigma\{=\}2,Δ=−0\.100\\Delta\{=\}\-0\.100

## Appendix JEndpoint provenance for the distributed Replogle column

#### Reconstruction attempt\.

We tested whether the Anderson–Darling column distributed with the Replogle releases can be reproduced from the corresponding single\-cell data under the definition of Sec\.[3](https://arxiv.org/html/2608.00152#S3)\. Using RPE1, where both the distributed column and single\-cell counts are available, we selected3030constructs by a deterministic rule—sorted by the stored value, binned into ten equal\-frequency strata, three per stratum—and computed the two\-sample Anderson–Darling statistic at each target gene’s own column under eight pre\-declared variants: aggregation \(pooled across batches; per\-batch then unweighted mean, count\-weighted mean, or median\), transform \(raw statistic;log⁡1​p​\(max⁡\(0,⋅\)\)\\log 1p\(\\max\(0,\\cdot\)\)\), input matrix \(raw counts; normalized expression\), and control sample \(non\-targeting cells pooled globally; restricted to the same batch\)\.

No variant reproduced the distributed values\. The closest was per\-batch median on normalized expression with globally pooled controls, reaching Pearson0\.3350\.335and Spearman0\.2680\.268with normalized RMSE0\.300\.30\. We classify the outcome as divergent: the distributed column is not a target\-gene Anderson–Darling effect size under any tested construction\.

#### What the column is instead\.

The distributed values are integer\-valued and bounded well below the gene count in every screen \(\[0,6036\]\[0,6036\],\[0,5547\]\[0,5547\], and\[0,5528\]\[0,5528\]for K562\-essential, RPE1, and K562 genome\-wide\), which is consistent with a per\-perturbation count over genes\. We therefore computed transcriptome\-wide per\-gene Anderson–Darling statistics for ten deterministically selected constructs and compared eight pre\-declared count definitions against the stored values: counts of genes with statistic above zero, above1\.01\.0, or above2\.02\.0; counts withp<0\.05p<0\.05orp<0\.01p<0\.01, uncorrected or after Benjamini–Hochberg correction; and a detection\-count null\.

All count candidates were rank\-monotone with the stored column \(ρ=0\.96\\rho=0\.96–0\.980\.98\)\. The Benjamini–Hochberg\-corrected count atq<0\.05q<0\.05on normalized expression reached Pearson0\.9860\.986; the detection\-count null was rejected, being constant at8,7498\{,\}749\. The fitted slope was approximately0\.600\.60: our reconstruction runs roughly1\.6×1\.6\\timesthe stored value\. The exact upstream preprocessing, significance threshold, control set, and tested gene universe therefore remain unresolved, and we do not claim to have reconstructed the column\. We note thatscipy\.stats\.anderson\_ksampfloors its reported significance level at0\.0010\.001and caps it at0\.250\.25, which makes any threshold\-based count sensitive to implementation detail\.

#### Association with the target\-gene endpoint\.

Within each screen, the distributed column and the rebuilt target\-gene endpoint are only weakly associated at construct level: Spearman0\.1650\.165\[0\.118,0\.211\]\[0\.118,0\.211\]in RPE1,0\.3010\.301\[0\.257,0\.343\]\[0\.257,0\.343\]in K562\-essential, and0\.3420\.342\[0\.322,0\.362\]\[0\.322,0\.362\]at target level in the K562 genome\-wide screen\. The two endpoints capture largely distinct aspects of perturbation effect\.

## Appendix KHarmonized endpoint construction

#### Protocol\.

The subsample sizes are the medians of the VCC distribution:2727perturbed cells per row and788788non\-targeting cells per batch\. For each eligible construct we draw2727perturbed cells without replacement from its pooled cells and788788non\-targeting cells without replacement from the pooled control set, compute the log1p Anderson–Darling statistic at the target gene’s column, and repeat2525times with pre\-specified seeds\. The endpoint is the median across draws; we also record the mean, standard deviation, interquartile range, and55th and9595th percentiles\. Quantiles usenumpy\.quantilewith linear interpolation throughout\.

Eligibility requires at leastmax⁡\(20,27\)\\max\(20,27\)perturbed cells, at leastmax⁡\(200,788\)\\max\(200,788\)pooled control cells, an unambiguous mapping from target gene to exactly one feature column, and a verified one\-to\-one join between the bulk and single\-cell artifacts\.

#### Coverage\.

Against the constructs used in the corresponding transfer evaluation, coverage is1,9321\{,\}932of2,2422\{,\}242\(86\.2%86\.2\\%\) for RPE1 and1,9031\{,\}903of1,9701\{,\}970\(96\.6%96\.6\\%\) for K562\-essential\. Exclusions are dominated by target genes absent from the feature namespace rather than by insufficient cells\.

#### Resampling spread\.

The ratio of the median within\-construct interquartile range to the between\-construct standard deviation is0\.1740\.174at construct level and0\.1840\.184at target level for RPE1, and0\.2360\.236and0\.2580\.258for K562\-essential\. Within\-construct resampling variation is therefore small relative to between\-construct variation in both screens\.

#### Pooled\-scale comparison\.

A naive pooled construction using all eligible cells, without subsampling, correlates with the matched\-size endpoint at Spearman0\.8840\.884\[0\.871,0\.895\]\[0\.871,0\.895\]in RPE1 and0\.8740\.874\[0\.860,0\.886\]\[0\.860,0\.886\]in K562\-essential\. Rank agreement is strong but not exact: matching the subsample size changes some rankings, which is why we fix it rather than pooling\.

#### Control\-matching sensitivity\.

Restricting control cells to the same batch as the perturbed cells, rather than pooling globally, is possible for4343constructs in RPE1 and194194in K562\-essential under a minimum of ten perturbed and fifty control cells per batch\. Where computable, the batch\-matched endpoint correlates with the pooled\-control endpoint at Spearman0\.9100\.910in RPE1 and0\.9730\.973in K562\-essential\. Batch structure is not consequential for this endpoint in the subsets where the comparison is well powered\. We note that the batch variable available in these releases is a technical grouping whose status as an independent experimental replicate is not established by the accompanying metadata\.

## Appendix LTransfer against the distributed endpoint

For completeness we report association between the same archived predictions and the Anderson–Darling column distributed with the Replogle releases\. Because Sec\.[6\.2](https://arxiv.org/html/2608.00152#S6.SS2)establishes that this column measures transcriptome\-wide response breadth rather than target\-gene effect strength, these values are not estimates of the same outcome as Table[2](https://arxiv.org/html/2608.00152#S6.T2)and should not be compared to it directly\.

Table 11:Zero\-shot transfer from VCC \(H1 hESC, train\) to Replogle K562 \(test\) for all model families\. Stochastic methods report mean±\\pmstd over three seeds\. Magnitude\-only classical models and tree ensembles onxxplus magnitude transfer with positive Spearman, while expression\-only classical and deep models have negative Spearman against this legacy distributed endpoint\. Explicit magnitude exposure is not sufficient by itself: Ridge onxxplus magnitude and the deep MAG\-SPSM encoder both transfer negatively despite explicit magnitude input\. Bold marks the empirical Pareto frontier between in\-distribution VCCR2R^\{2\}and Replogle Spearman\.#### Endpoint sensitivity\.

Evaluating one prediction vector against both endpoints on the same construct subset isolates the effect of the outcome definition\. On K562\-essential, the differenceρtarget\-gene−ρbreadth\\rho\_\{\\text\{target\-gene\}\}\-\\rho\_\{\\text\{breadth\}\}is positive for every expression\-only family and for the deepspsmbaseline, reaching\+0\.371\+0\.371, and negative for every magnitude\-carrying family and formag\-Linear, reaching−0\.151\-0\.151\. The two endpoints reward different model behavior: predictors that track target\-gene effect strength score worse against a breadth endpoint, and conversely\.

#### An earlier reading of these numbers\.

Evaluated against the distributed endpoint alone, transfer to RPE1 appears qualitatively opposite to transfer to K562—Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]reachesρ=\+0\.314\\rho=\+0\.314on K562 and−0\.397\-0\.397on RPE1\. Under the rebuilt target\-gene endpoint the two screens instead show the same group\-level pattern \(Table[2](https://arxiv.org/html/2608.00152#S6.T2)\)\. We report both because the discrepancy is itself informative about what the distributed endpoint measures, not because the two evaluations are interchangeable\.

## Appendix MFull intervals for cross\-screen transfer

Table[2](https://arxiv.org/html/2608.00152#S6.T2)reports point estimates and verdicts\. The corresponding95%95\\%intervals come from a paired cluster bootstrap over target genes withB=10,000B=10\{,\}000replicates and a pre\-specified seed\. Each replicate samples target\-gene clusters with replacement; a cluster drawn more than once contributes all of its constructs once per occurrence; and the same draw is applied jointly to every prediction, every seed, and every contrast within that replicate, so that paired comparisons share resampling noise\.

For the nine classical families the metric is computed per seed and averaged, with the averaging repeated inside each replicate; for the three single\-seed predictors the metric comes from the one archived prediction vector, so the interval reflects target\-gene sampling conditional on that seed\. No replicate was discarded for an undefined correlation in either screen\.

The two evaluations differ in cluster structure: RPE1 has1,9321\{,\}932constructs over1,8111\{,\}811target genes with at most three constructs per gene, and K562\-essential has1,9031\{,\}903over1,8111\{,\}811with at most two\. Interval widths are therefore not mechanically comparable across screens\.

## Appendix NMAG\-Anchor and external transfer

MAG\-Anchor is evaluated in\-distribution on VCC \(Sec\.[6\.4](https://arxiv.org/html/2608.00152#S6.SS4)\) but does not appear in Table[2](https://arxiv.org/html/2608.00152#S6.T2)\. The per\-construct prediction vectors required to evaluate it against a rebuilt endpoint were not retained for the external screens; only aggregate metrics were\. Since our transfer evaluation re\-scores stored predictions rather than retraining, and no stored MAG\-Anchor prediction exists to validate a reconstruction against, we exclude it rather than introduce an unvalidated refit into the comparison\. This is an availability constraint, not a selection based on outcome\.

## Appendix OHVG\-coverage control

To test whether the legacy K562 distributed\-endpoint transfer result is explained by the2,0002\{,\}000\-to\-1,4301\{,\}430HVG dimension drop, we reran the RF\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]baseline using only the1,4301\{,\}430HVGs shared between VCC and Replogle K562\. The shared genes were selected deterministically by gene name in VCC HVG order, without using Replogle target labels or outcomes\. Magnitude features were recomputed from the1,4301\{,\}430\-dimensional inputs andzz\-scored using VCC training statistics only\. Across three seeds with the same RF hyperparameters as the main experiment \(nest=200n\_\{\\text\{est\}\}\{=\}200,maxdepth=20\\max\_\{\\text\{depth\}\}\{=\}20,minleaf=2\\min\_\{\\text\{leaf\}\}\{=\}2\), the1,4301\{,\}430\-HVG control achieved VCC testR2=\+0\.340±0\.002R^\{2\}=\+0\.340\\pm 0\.002and K562 transfer Spearmanρ=\+0\.377±0\.010\\rho=\+0\.377\\pm 0\.010\(Table[12](https://arxiv.org/html/2608.00152#A15.T12)\)\. This passes our pre\-specified robustness rule \(ρ≥\+0\.20\\rho\\geq\+0\.20\) and shows that,*when transfer is scored against the distributed endpoint*, the positive K562 rank association is not explained by the missing570570HVGs alone\. The in\-domain VCCR2R^\{2\}decreases relative to the2,0002\{,\}000\-HVG result \(\+0\.370±0\.004\+0\.370\\pm 0\.004\), as expected from the reduced feature set, but cross\-cell rank transfer is preserved\.

Table 12:HVG\-coverage control for K562 transfer\. Mean±\\pmsample standard deviation over three seeds\.#### Pre\-specified HVG\-coverage control:𝐱\\mathbf\{x\}\-only models\.

We additionally pre\-specified a symmetric x\-only counterpart of the control above to test whether the negative cross\-cell transfer of expression\-only models is a feature\-truncation artifact rather than a direction\-vs\-magnitude phenomenon\. The pre\-registration committed four hypotheses with PASS/WARN/FAIL thresholds and the consequent paper\-text edits before any model was trained in the pre\-registration file committed to the project repository\. We trained Ridge, Random Forest, HistGradientBoosting, and the deepspsmV4 baseline on𝐱\\mathbf\{x\}alone restricted to the same1,4301\{,\}430shared HVGs, three seeds each, with hyperparameters frozen at their main\-paper values\. Three\-seed mean K562 Spearman values were: Ridgeρ=−0\.229\\rho=\-0\.229, Random Forestρ=−0\.407±0\.003\\rho=\-0\.407\\pm 0\.003, HistGradientBoostingρ=−0\.305±0\.019\\rho=\-0\.305\\pm 0\.019, andspsmV4 baselineρ=−0\.389±0\.007\\rho=\-0\.389\\pm 0\.007\. All four pre\-specified hypotheses passed: H1 \(every𝐱\\mathbf\{x\}\-only classical model retains negative mean Spearman\) passed with the largest value at Ridge−0\.229\-0\.229; H2 \(𝐱\\mathbf\{x\}\-only deep baseline meanρ<\+0\.05\\rho<\+0\.05\) passed at−0\.389\-0\.389; H3 \(best magnitude\-aware minus best𝐱\\mathbf\{x\}\-onlyρ\\rhogap on the1,4301\{,\}430\-HVG sub\-space≥\+0\.20\\geq\+0\.20\) passed with gap\+0\.607\+0\.607\(\+0\.377\+0\.377vs−0\.229\-0\.229\); and H4 \(within\-VCC\|Δ​R2\|≤0\.030\|\\Delta R^\{2\}\|\\leq 0\.030relative to the2,0002\{,\}000\-HVG baselines\) passed with worst\-case\|Δ​R2\|=0\.011\|\\Delta R^\{2\}\|=0\.011for Ridge\. Per the pre\-specified decision rule, the legacy distributed\-endpoint𝐱\\mathbf\{x\}\-only negative\-transfer pattern is robust to feature truncation and the magnitude–direction asymmetry holds on the shared\-HVG sub\-space\.

#### Scope\.

Per\-construct prediction vectors from this control were not retained, so it cannot be re\-scored against the rebuilt target\-gene endpoint without an unvalidated refit\. We therefore treat it as a legacy distributed\-endpoint sensitivity analysis and do not use it to support Table[2](https://arxiv.org/html/2608.00152#S6.T2)\.

## Appendix PBootstrap confidence intervals for key comparisons

As a robustness analysis of the central comparisons in the paper, we computed row\-level paired bootstrap95%95\\%confidence intervals \(B=10,000B=10\{,\}000resamples, RNG seed4242, percentile method\) for three pre\-specified comparisons\. The summary is in Table[13](https://arxiv.org/html/2608.00152#A16.T13)\. \(C5\) Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]versusmag\-spsmon VCC test pooledR2R^\{2\}, paired across the same three seeds: medianΔ​R2=\+0\.145\\Delta R^\{2\}=\+0\.145,95%95\\%CI\[\+0\.127,\+0\.163\]\[\+0\.127,\+0\.163\], sign\-stable in100%100\\%of resamples \(strong\)\. \(C7\)mag\-Linearversus HistGradientBoosting on𝐱\\mathbf\{x\}\-only on VCC testR2R^\{2\}against the mean of three HistGB seeds: medianΔ​R2=\+0\.012\\Delta R^\{2\}=\+0\.012,95%95\\%CI\[−0\.009,\+0\.032\]\[\-0\.009,\+0\.032\]\(weak\)\. \(C8\)mag\-Linearversus thespsmbaseline on K562 Spearmanρ\\rho, row\-resampled with no seed component: medianΔ​ρ=\+0\.750\\Delta\\rho=\+0\.750,95%95\\%CI\[\+0\.700,\+0\.800\]\[\+0\.700,\+0\.800\]\(strong\)\.

Table 13:Row\-level paired bootstrap95%95\\%confidence intervals for three pre\-specified comparisons\.B=10,000B=10\{,\}000resamples, RNG seed4242, percentile method\. Sign\-stability is the fraction of bootstrap resamples in which the sign of the paired difference matches the median\.We do not report per\-row bootstrap intervals for comparisons involving MAG\-Anchor or for the asymmetric three\-seed\-vs\-one\-seed RF–mag\-spsmK562 comparison\. A seed\-0calibration check on regenerated deep\-model predictions did not reproduce the original predictions to the pre\-specified tolerance; we therefore avoid uncertainty estimates that would mix the original paper predictions with regenerated deep\-model outputs\. This analysis is reported as supporting uncertainty evidence for the main comparisons, not as a new contribution\.

## Appendix QArchitecture controls for𝐱\\mathbf\{x\}\-only MLP

To test whether the𝐱\\mathbf\{x\}\-only MLP failure to recover the magnitude\-like signal is explained by normalization, dropout, capacity, or the absence of a simple scale\-preserving path, we ran an architecture\-control audit\. We held the data, target, optimizer, and training schedule fixed and varied only the encoder: \(B\) remove LayerNorm; \(C\) remove LayerNorm and Dropout; \(D\) widen the MLP to10241024hidden units; \(E\) add a learnable scalar shortcutf​\(𝐱\)=MLP​\(𝐱\)\+β⋅‖𝐱‖2/df\(\\mathbf\{x\}\)=\\mathrm\{MLP\}\(\\mathbf\{x\}\)\+\\beta\\cdot\\\|\\mathbf\{x\}\\\|\_\{2\}/\\sqrt\{d\}withβ\\betainitialized at0\. Variants \(B\)–\(D\) are strictly𝐱\\mathbf\{x\}\-only; variant \(E\) augments𝐱\\mathbf\{x\}with the single hand\-designed magnitude scalar‖𝐱‖2/d\\\|\\mathbf\{x\}\\\|\_\{2\}/\\sqrt\{d\}\(i\.e\.,m2m\_\{2\}from Eq\.[1](https://arxiv.org/html/2608.00152#S5.E1)\) routed through an additive shortcut, so it is the strongest x\-with\-magnitude\-shortcut control we tested\. Across three seeds each, no variant crosses VCC testR2=\+0\.150R^\{2\}=\+0\.150\(best individual seed: wide\-no\-norm seed11atR2=\+0\.034R^\{2\}=\+0\.034\); means range from−0\.040±0\.021\-0\.040\\pm 0\.021to\+0\.016±0\.018\+0\.016\\pm 0\.018\(sample standard deviation,ddof=1\\mathrm\{ddof\}\{=\}1\)\. Per a pre\-specified rule \(Partial\-Rescueif meanR2≥\+0\.150R^\{2\}\\geq\+0\.150\), the outcome isNo\-Rescue\. The originalspsmbaseline’s calibration matches the paper’s\+0\.082±0\.010\+0\.082\\pm 0\.010exactly\. We conclude that the deep𝐱\\mathbf\{x\}\-only MLP failure is not explained by LayerNorm, Dropout, limited capacity, or even a hand\-designed magnitude shortcut: even when the network is given direct access tom2m\_\{2\}via an additive path, it fails to recover the signal\. This audit is reported as a diagnostic robustness analysis covering four common architecture variants and is not exhaustive; activation, optimizer, and skip\-connection variants remain untested\.

## Appendix RRF hyperparameter robustness

To test whether the Random Forest result on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]depends on a narrow hyperparameter choice, we ran a targeted hyperparameter robustness sweep\. We held the data, features, and seeds fixed and varied only the RF: paper reference S0 \(nest=200n\_\{\\text\{est\}\}\{=\}200,maxdepth=20\\max\_\{\\text\{depth\}\}\{=\}20,minleaf=2\\min\_\{\\text\{leaf\}\}\{=\}2\); S1 shallower trees \(maxdepth=10\\max\_\{\\text\{depth\}\}\{=\}10\); S2 unbounded depth; S3 stronger leaf regularization \(minleaf=5\\min\_\{\\text\{leaf\}\}\{=\}5\); S4 more trees \(nest=500n\_\{\\text\{est\}\}\{=\}500\); S5 intermediate depth \(maxdepth=15\\max\_\{\\text\{depth\}\}\{=\}15\)\. Across three seeds×\\timessix settings \(1818fits\), VCC testR2R^\{2\}remained in\[\+0\.361,\+0\.372\]\[\+0\.361,\+0\.372\]and legacy distributed\-endpoint K562 Spearmanρ\\rhoin\[\+0\.309,\+0\.359\]\[\+0\.309,\+0\.359\]\(Table[14](https://arxiv.org/html/2608.00152#A18.T14)\)\. Every setting met the pre\-specifiedRobustcriteria \(VCCR2≥\+0\.330R^\{2\}\\geq\+0\.330and K562ρ≥\+0\.250\\rho\\geq\+0\.250\)\. The shallower\-tree variant S1 gave the highest K562 transferρ=\+0\.359±0\.011\\rho=\+0\.359\\pm 0\.011\(vs\.\+0\.314\+0\.314at S0\) at a small VCCR2R^\{2\}cost \(−0\.009\-0\.009\), consistent with reduced direction\-overfitting; we do not retroactively change the paper’s S0 numbers\. We report this sweep as a robustness check and keep the original S0 setting as the paper’s primary RF configuration\.

Table 14:RF hyperparameter robustness sweep on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\. Mean±\\pmsample standard deviation across three seeds \(ddof=1\) for each setting\. All settings meet the robustness criterion used for this sweep \(VCCR2≥\+0\.330R^\{2\}\\geq\+0\.330and K562ρ≥\+0\.250\\rho\\geq\+0\.250\)\.
## Appendix SFoundation model feasibility analysis

We evaluated the feasibility of adding single\-cell foundation\-model baselines \(scGPT\[Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\], Geneformer\[Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\], UCE\[Rosen et al\.,[2026](https://arxiv.org/html/2608.00152#bib.bib26)\], scFoundation\[Hao et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib12)\]\)\. Two scientific considerations led us to defer this comparison rather than include it in the present study\.*First, the closest in\-house analog is already on the record as a negative result\.*We pretrained a small MLP encoder on Replogle K562 and RPE1 single\-cell pseudobulks and transferred it to VCC; this configuration is the most direct pretrain\-and\-transfer test possible with our data and obtained VCC testR2=−0\.181R^\{2\}=\-0\.181\.*Second, our task and the FM pretraining distribution are not directly comparable\.*The cited FMs are pretrained on raw single\-cell counts over full gene vocabularies, whereas the VCC task uses per\-perturbation pseudobulk delta features over a2,0002\{,\}000\-HVG subset\. Applying a pretrained FM to our inputs without a non\-trivial format conversion would evaluate it out\-of\-distribution from its pretraining regime, and any negative result would be attributable to the format mismatch rather than to the FM hypothesis\. The paper’s contribution is the magnitude\-signal diagnostic, not FM benchmarking; we therefore leave a fairness\-controlled FM comparison to follow\-up work that can adopt FM\-native input formats\.

## Appendix TAdditional discussion:mag\-spsmand MAG\-Anchor as a secondary deep method

#### Why representation\-blind remedies fail\.

Reweighting losses \(Balanced MSE\[Ren et al\.,[2022](https://arxiv.org/html/2608.00152#bib.bib21)\], LDS\[Yang et al\.,[2021](https://arxiv.org/html/2608.00152#bib.bib34)\]\) leave the encoder’s input representation unchanged and therefore cannot help in our setting: the encoder still maps inputs through a representation that does not expose magnitude\. This is the unifying explanation for the failure of the five remedies in Sec\.[4](https://arxiv.org/html/2608.00152#S4)that act on the loss but not on the input representation\.

#### Foundation models scope\.

The diagnostic claims of this paper concern the MLP\-encoder family we tested and may not generalize to foundation\-model\-based perturbation predictors\. Whether large pretrained single\-cell encoders such as scGPT\[Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\], Geneformer\[Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\], and Universal Cell Embeddings\[Rosen et al\.,[2026](https://arxiv.org/html/2608.00152#bib.bib26)\]recover magnitude when fine\-tuned on the disjoint target\-gene split, or whether they exhibit the same representational collapse on the high\- and low\-effect tails, was not evaluated in this study\. Compute and preprocessing costs \(gene\-vocabulary alignment, memory budget for fine\-tuning on a single 24 GB GPU\) made a defensible foundation\-model baseline outside the scope of this submission; this is an important direction for follow\-up work\.

#### mag\-spsmas a proof of concept, not the final model\.

The strong\-baseline audit \(Sec\.[6\.1](https://arxiv.org/html/2608.00152#S6.SS1)\) decisively shows thatmag\-spsmis not the strongest exploiter of the magnitude signal\. Random Forest on the same features outperforms it byΔ​R2=\+0\.145\\Delta R^\{2\}=\+0\.145in\-distribution and byΔ​R2=\+0\.138\\Delta R^\{2\}=\+0\.138on the legacy distributed endpoint under cross\-cell transfer\. We retainmag\-spsmin the main text as a diagnostic deep augmentation: the simplest verification that the same magnitude signal can be exposed to a deep encoder, and the substrate for the causal controls in Sec\.[5\.2](https://arxiv.org/html/2608.00152#S5.SS2)\. We do not recommendmag\-spsmas a final architecture\.

#### MAG\-Anchor as a secondary deep method\.

The MAG\-Anchor results refine, rather than replace, the diagnostic story\. The failed MAGAnchor\-x ablation shows that exposing magnitude at the output is not enough; the residual branch must also condition on the magnitude summary\. Even so, Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]remains stronger in\-distribution, so we treat MAG\-Anchor as a secondary deep\-architecture contribution rather than a replacement for the strongest classical baseline \(App\.[D](https://arxiv.org/html/2608.00152#A4)\)\.

#### Two readings of the diagnostic\.

A reasonable alternative reading of our finding is that mean\-collapse arises from optimization or normalization behavior of the deep MLP when the label scale couples to input scale through normalization layers, rather than from a representational property of perturbation prediction*per se*\. The architecture\-control analysis in App\.[Q](https://arxiv.org/html/2608.00152#A17)tests this reading directly: holding the data, target, optimizer, and training schedule fixed and varying only the encoder \(no LayerNorm; no LayerNorm and no Dropout; widened to10241024hidden units; an additive scale\-preserving shortcut\), no𝐱\\mathbf\{x\}\-only variant crosses VCC testR2=\+0\.150R^\{2\}=\+0\.150, well below the\+0\.208\+0\.208obtained by a four\-feature linear regression on𝒎~​\(𝐱\)\\widetilde\{\\boldsymbol\{m\}\}\(\\mathbf\{x\}\)\. This empirically disfavors the optimization\-only reading within the MLP family we tested, though it does not exclude it for architectures we did not test \(set transformers, attention\-based encoders, foundation\-model fine\-tuning\)\. Under both readings, explicit magnitude exposure is the empirical separator between models that recover the magnitude signal and models that collapse, and the practical recommendation is the same—supply magnitude as features rather than rely on the encoder to recover it from the expression vector alone\. The diagnostic is therefore most precisely stated as a finding about*which models, on the disjoint target\-gene split with this label parameterization, exploit a known low\-dimensional predictive signal*\.

## Appendix UAdditional limitations

1. 1\.*The label is magnitude\-like by construction\.*Because the target is the log Anderson–Darling distance from NT, magnitude features are in principle informative\. Our claim is not that this relationship is biologically surprising in isolation; it is that direction is fragile under the disjoint target\-gene split, that standard encoders fail to exploit magnitude reliably, and that explicit magnitude exposure is the empirical separator under cross\-cell transfer\.
2. 2\.*Replogle preprocessing residuals\.*Replogle K562 uses a different per\-row biological unit than VCC \(1,4301\{,\}430of2,0002\{,\}000HVGs mapped; feature scales differ by∼1\.9×\\sim 1\.9\\times\)\. We align via VCC\-side normalization \(App\.[H](https://arxiv.org/html/2608.00152#A8)\); residual confounding cannot be fully ruled out\.
3. 3\.*Cross\-datasetR2R^\{2\}scale disparity\.*VCC test, Replogle K562 transfer, and Replogle RPE1 transfer have AD\-based labels with substantially different marginal standard deviations \(sample std,ddof=1\\mathrm\{ddof\}\{=\}1\): VCCσy=0\.758\\sigma\_\{y\}=0\.758, K562σy=2\.750\\sigma\_\{y\}=2\.750\(3\.63×3\.63\\timesthe VCC value\), and RPE1σy=2\.372\\sigma\_\{y\}=2\.372\(3\.13×3\.13\\timesthe VCC value\)\. SinceR2R^\{2\}denominators scale with the squared label std, cross\-datasetR2R^\{2\}magnitudes are not directly comparable across the three benchmarks; the K562R2R^\{2\}denominator is approximately13\.2×13\.2\\timesthe VCC denominator and the RPE1 denominator is approximately9\.8×9\.8\\times\. Cross\-datasetR2R^\{2\}values should therefore be read as within\-dataset goodness\-of\-fit measures, not as a single calibrated quantity transferable from one dataset to the other; rank\-based metrics \(Spearmanρ\\rho\) are unaffected by this scale disparity and are the metric we emphasize in cross\-cell claims\.
4. 4\.*HVG\-coverage confound on cross\-cell transfer\.*Models are trained on the full2,0002\{,\}000\-dimensional VCC HVG namespace\. In the primary Replogle K562 transfer evaluation, Replogle rows are embedded into this2,0002\{,\}000\-dimensional namespace by zero\-imputing the570570VCC\-only coordinates missing from Replogle\. This is a structural input shift that is logically distinct from the biological cross\-cell shift\. The HVG\-coverage control in App\.[O](https://arxiv.org/html/2608.00152#A15)removes the zero\-padding by training and evaluating directly in the1,4301\{,\}430shared\-HVG subspace, and K562 rank association is preserved and slightly higher \(ρ=\+0\.377±0\.010\\rho=\+0\.377\\pm 0\.010vs\.\+0\.314±0\.009\+0\.314\\pm 0\.009\)\. This addresses the coverage question only for the legacy distributed\-endpoint analysis; the corresponding prediction vectors were not retained, so the control cannot be re\-scored against the rebuilt target\-gene endpoint without an unvalidated refit\.
5. 5\.*Modest transfer magnitudes\.*On the distributed Replogle endpoint, the strongestR2R^\{2\}we observe is\+0\.121\+0\.121\(mag\-Linear\), and the strongest in\-distribution\-and\-transfer combination \(RF on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\) reachesR2=\+0\.037R^\{2\}=\+0\.037\. On the rebuilt target\-gene endpoint we report rank association rather than pooledR2R^\{2\}, since predictions produced on one outcome scale are not calibrated to another\. Magnitude helps transfer but does not solve cross\-context generalization\.
6. 6\.*Tree behavior on low\-dimensional input\.*Random Forest onmag\-onlyhas VCC testR2=−0\.042R^\{2\}=\-0\.042but Replogle Spearman\+0\.346\+0\.346; the negative VCCR2R^\{2\}reflects poor calibration of leaf means rather than absence of signal \(the in\-distribution Spearman is\+0\.555\+0\.555, and tree predictions are bounded by the training label range\)\. We therefore interpret tree results onmag\-onlyas evidence of preserved rank association, not as calibrated in\-distribution predictors\.
7. 7\.*No exhaustive deep\-architecture search\.*We tested MLP\-family encoders only \(spsm/mag\-spsm, MAG\-Anchor, and the MAGAnchor\-x ablation\)\. Set Transformers, attention\-based encoders, foundation\-model fine\-tuning \(scGPT\[Cui et al\.,[2024](https://arxiv.org/html/2608.00152#bib.bib8)\], Geneformer\[Theodoris et al\.,[2023](https://arxiv.org/html/2608.00152#bib.bib31)\], UCE\[Rosen et al\.,[2026](https://arxiv.org/html/2608.00152#bib.bib26)\]\), and width/depth sweeps were not evaluated\. The conclusion that no deep encoder we tried matches the tree\-ensemble baseline is therefore empirical and scope\-limited to the MLP family\. Within the MLP family, additional architecture controls \(no\-LayerNorm, no\-Dropout, wider, and scale\-aware variants\) did not recover signal from𝐱\\mathbf\{x\}\-only input \(App\.[Q](https://arxiv.org/html/2608.00152#A17)\); a separate feasibility audit of external transformer foundation models concluded that a fairness\-controlled comparison would require substantial input\-format conversion and is deferred to follow\-up work, with proxy evidence \(an in\-house pretrain\-and\-transfer experiment that obtained VCCR2=−0\.181R^\{2\}=\-0\.181\) and additional details reported in App\.[S](https://arxiv.org/html/2608.00152#A19)\. Whether other architecture families recover magnitude from𝐱\\mathbf\{x\}alone, or whether the gap to RF on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]narrows under such architectures, remains an important open question\.
8. 8\.*MAG\-Anchor is not state of the art\.*MAG\-Anchor obtains positive H1→\\toK562 rank association when scored against the legacy distributed endpoint, while also improving overmag\-spsmin\-distribution\. It is not, however, the strongest in\-distribution predictor\. Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\widetilde\{\\boldsymbol\{m\}\}\]remains stronger by about\+0\.10\+0\.10VCC testR2R^\{2\}\. We therefore view MAG\-Anchor as a secondary method contribution that operationalizes the diagnostic finding, not as a new state\-of\-the\-art perturbation predictor\.
9. 9\.*Statistical reporting choices and their scope\.*We report all stochastic results as mean±\\pmsample standard deviation \(ddof=1\\mathrm\{ddof\}\{=\}1\) across seeds, and report paired three\-seed deltas for matched\-architecture comparisons \(e\.g\.,mag\-Rankvs\.mag\-spsm,Δ​R2=\+0\.0142±0\.0040\\Delta R^\{2\}=\+0\.0142\\pm 0\.0040; App\.[I](https://arxiv.org/html/2608.00152#A9)\)\. For three central comparisons where reliable per\-row predictions are available, we report row\-level paired bootstrap95%95\\%confidence intervals on the pooled test set \(nVCC=3,972n\_\{\\text\{VCC\}\}\{=\}3\{,\}972,nK562=2,176n\_\{\\text\{K562\}\}\{=\}2\{,\}176\) in App\.[P](https://arxiv.org/html/2608.00152#A16)\. We do not report formal hypothesis tests, paired Wilcoxon signed\-rank statistics, or multiple\-comparison corrections across the full method grid; withnseeds∈\{3,5\}n\_\{\\text\{seeds\}\}\{\\in\}\\\{3,5\\\}, such tests would be underpowered for small effects, and we judged that qualitative pattern descriptions are more honest than artificially sharppp\-values\. The central qualitative findings of the paper do not depend on small\-effect comparisons: the gap between Random Forest on\[𝐱;𝒎~\]\[\\mathbf\{x\};\\,\\widetilde\{\\boldsymbol\{m\}\}\]\(R2=\+0\.370±0\.004R^\{2\}=\+0\.370\\pm 0\.004\) and our deepmag\-spsmencoder \(\+0\.225±0\.004\+0\.225\\pm 0\.004\) is many times the seed\-noise envelope of either estimate; the contrast between expression\-only models \(Spearmanρ∈\[−0\.435,−0\.198\]\\rho\\in\[\-0\.435,\-0\.198\]on Replogle K562\) and magnitude\-aware models \(Spearmanρ∈\[\+0\.173,\+0\.346\]\\rho\\in\[\+0\.173,\+0\.346\]\) on cross\-screen transfer is qualitative rather than marginal; and the harmonized transfer verdicts of Sec\.[6\.3](https://arxiv.org/html/2608.00152#S6.SS3)are determined by whether a bootstrap interval excludes zero, not by app\-value cutoff\. Pairwise deltas with overlapping intervals are flagged as unresolved in the Discussion and in App\.[T](https://arxiv.org/html/2608.00152#A20); we do not claim these are statistically distinguishable\. A targeted hyperparameter robustness sweep for the RF result is reported in App\.[R](https://arxiv.org/html/2608.00152#A18)\.

## Appendix VAnderson–Darling label construction

For each rowi=\(bi,gi\)i=\(b\_\{i\},g\_\{i\}\)with experimental Flex batchbib\_\{i\}and target genegig\_\{i\}, the regression label is

yi=log⁡\(1\+max⁡\{0,Abi,gi2\}\),y\_\{i\}\\;=\\;\\log\\\!\\bigl\(1\+\\max\\\{0,\\,A^\{2\}\_\{b\_\{i\},g\_\{i\}\}\\\}\\bigr\),whereAb,g2A^\{2\}\_\{b,g\}is the two\-sample Anderson–Darling statistic returned byscipy\.stats\.anderson\_ksamp\[Scholz and Stephens,[1987](https://arxiv.org/html/2608.00152#bib.bib28)\]applied to two one\-dimensional samples: \(i\) the log1p\-normalized expression values of the target geneggin cells perturbed atggwithin batchbb, and \(ii\) the log1p\-normalized expression values of the same geneggin non\-targeting \(NT\) control cells from the same batchbb\. The Anderson–Darling test is therefore applied to the*target\-gene column only*; the2,0002\{,\}000\-HVG vector𝐱i\\mathbf\{x\}\_\{i\}is the predictor input and does not enter label construction\. Negative AD statistics \(rare, possible with the AD null\-bias subtraction\) are clamped to zero before thelog\(1\+⋅\)\\log\(1\+\\cdot\)transform\. The same per\-row\(b,g\)\(b,g\)\-indexed AD label convention is used on Replogle K562 and RPE1 during cross\-dataset evaluation, withbbplaying the role of the Replogle gemgroup\-style batch identifier\.