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How High Is 0.6? Floors, Ceilings, and Headroom in Interpretability Probing
Pranjal Garg
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Abstract
Abstract:Probes are the workhorse of interpretability. If a model's hidden states predict a variable, the model is said to represent it. But a probe score has no fixed meaning. An $R^2$ of 0.6 may only reflect what the input already gives away, and the same score can mean different things on different data. We propose reading every probe score against two reference points: a floor, what a declared set of simple inputs already predicts, and a ceiling, what the full input can predict. The gap between them, the headroom, is the range in which a probe can show that a model computes something beyond the simple inputs. We prove that headroom vanishes in two ways: the target stops depending on a hidden variable the model must infer, or the input stops revealing it. We test this on transformers trained for in-context meta-analysis, which must infer the hidden heterogeneity between studies to weight them correctly, and where both reference points are known. Under distribution shift, probe scores fall and prediction error rises $12$--$15\times$, yet the model recovers a similar share of the headroom, indicating that the data lost information, not the representation. We then analyze the real models. The single-cell foundation model scGPT encodes biological variability only partially. We also revisit four influential LLM probing studies, which claim that models represent geography, the state of an Othello board, truth, and the demographics of their users. Against a floor computed from the input text alone, some of these claims hold, while others are largely explained by the text itself.
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- Source: https://arxiv.org/abs/2610.08544v1
- Canonical: https://arxiv.org/abs/2610.08544v1
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How High Is 0.6? Floors, Ceilings, and Headroom in Interpretability Probing Pranjal Garg Independent Researcher Abstract Probes are the workhorse of interpretability. If a model’s hidden states predict a variable, the model is said to represent it. But a probe score has no fixed meaning. An R2R^2 of 0.6 may only reflect what the input already gives away, and the same score can mean different things on different data. We propose reading every probe score against two reference points: a floor, what a declared set of simple inputs already predicts, and a ceiling, what the full input can predict. The gap between them, the headroom, is the range in which a probe can show that a model computes something beyond the simple inputs. We prove that headroom vanishes in two ways: the target stops depending on a hidden variable the model must infer, or the input stops revealing it. We test this on transformers trained for in-context meta-analysis, which must infer the hidden heterogeneity between studies to weight them correctly, and where both reference points are known. Under distribution shift, probe scores fall and prediction error rises 1212–15×15×, yet the model recovers a similar share of the headroom, indicating that the data lost information, not the representation. We then analyze the real models.The single-cell foundation model scGPT encodes biological variability only partially. We also revisit four influential LLM probing studies, which claim that models represent geography, the state of an Othello board, truth, and the demographics of their users. Against a floor computed from the input text alone, some of these claims hold, while others are largely explained by the text itself. 1 Introduction Understanding what information is represented inside a complex system is a central problem across biological and artificial intelligence. In neuroscience, decoding analyses use neural population activity to ask whether variables such as sensory features, cognitive states, or behavioral quantities are represented in the brain; analogous methods are now widely used to study the hidden representations of learned artificial networks (Quian Quiroga and Panzeri, 2009; Naselaris et al., 2011; Belinkov, 2022). In modern AI, where high-dimensional internal states are otherwise difficult to interpret directly, probes provide a particularly simple and scalable readout: a classifier or regressor is trained to predict a quantity of interest from a model’s hidden states, and its predictive performance is used to assess how strongly that quantity is represented (Alain and Bengio, 2016; Hewitt and Liang, 2019). Such measurements underlie claims about what models encode, how representations emerge during learning and how information is distributed across layers (Tenney et al., 2019; Saphra and Lopez, 2019). They therefore carry substantial scientific weight, as conclusions about the internal organization of both biological and artificial systems can depend on what a probe score is taken to mean. Figure 1: Schematic of the floor–ceiling normalized probe R2R^2. A probe’s raw decodability (RH2R^2_H) for a target T is normalized between an estimated floor (ρ^O2 ρ_O^2), achievable from restricted information O, and an estimated ceiling (ρ^I2 ρ_I^2), achievable using full information I. The gap between them defines the headroom (Δ ). Yet this widespread use exposes a basic interpretive problem: a raw probe score has no intrinsic scale of evidential strength. A high R2R^2 may arise because the target is predictable from readily available input information rather than because the representation captures the computation of interest, and the same numerical score can carry different implications across distributions as this baseline predictability changes (Ravichander et al., 2021; Elazar et al., 2021; Hewitt and Liang, 2019). Interpreting probe scores therefore requires knowing not only what is decodable but also how much decodability is achievable without the computation of interest and how much is achievable from all available information. We study this distinction in a controlled evidence-aggregation task. We train transformer models in a random-effects meta-analysis setting where the sufficient statistics and the oracle Bayes estimator conditional on the true heterogeneity parameter are known in closed form. Under a distribution shift, the raw decodability of the sufficient statistic falls and the prediction error rises, and the raw scores alone cannot say whether the representation degraded or the data became less informative. To separate computation from statistical artifact, we introduce a framework that normalizes probe scores against the bounds implied by the information sets that the scientific question specifies. We partition the available information into restricted information (O) and additional evidence (X) used to infer a latent variable (Z). We evaluate probe decodability between two population reference points (under the true data distribution): a floor (ρO2 _O^2), which measures the predictability achievable from the baseline alone, and a ceiling (ρI2 _I^2), which measures the maximum predictability achievable given full information (I), which represents all available evidence. We define the gap between these reference points as headroom (Δ ): the maximum variance a probe could explain by successfully inferring the latent variable Z from X (Figure 1). The theoretical result demonstrates that this headroom can vanish through two mechanisms: weak target sensitivity (where the target ceases to depend on the latent variable) and weak identification (where the evidence becomes too noisy to identify the latent variable). When headroom vanishes, raw probe scores can cease to provide evidence about the computation of interest. We test the estimation procedure in the meta-analysis task and a synthetic single-cell model, where the generative model is known and the reference points can be computed exactly by numerical integration. We then apply it to real single-cell data and to published LLM probes, where the reference points can only be estimated from samples. Analyses of a fine-tuned chemistry model and recorded neural activity appear in Appendix J. Contributions: (1) we introduce a floor, ceiling and headroom normalization for interpretability probes and identify two distinct mechanisms by which this headroom vanishes: weak target sensitivity and weak identification of the latent variable; (2) we show that under distribution shift raw decodability can reflect reduced headroom rather than a degraded representation; (3) we test this estimation procedure in different domains, showing how the choice of floor and ceiling distinguishes different scientific questions; (4) we show that the same raw probe score can imply different amounts of recovered headroom across settings. 2 Related Work Probing and its baselines. A probe trains a simple readout on hidden states and treats its predictive performance as evidence that a variable is represented (Alain and Bengio, 2016; Belinkov, 2022). Raw probe accuracy underdetermines this conclusion in three ways. Expressive probes can learn the task themselves (Hewitt and Liang, 2019). Random encoders already support high accuracy (Zhang and Bowman, 2018; Wieting and Kiela, 2019). Decodable information need not be used by the model (Ravichander et al., 2021; Elazar et al., 2021; Kumar et al., 2022). Proposed remedies either control probe capacity (Hewitt and Liang, 2019; Voita and Titov, 2020) or estimate information content (Pimentel et al., 2020). Pimentel et al. (2020) note that, by the data-processing inequality, no representation carries more information about a target than its input does. Closest to our floor is conditional probing (Hewitt et al., 2021). Building on V-information (Xu et al., 2020), it measures what a representation adds beyond a baseline representation. We differ in three respects: (i) the floor is defined on a declared restricted input set O chosen by the scientific question; (i) we pair it with a ceiling from the full input I, turning the data-processing bound into an estimable reference point against which scores are normalized; (i) we show that both reference points move with the data regime, so a fixed baseline can misattribute a regime change to the representation; we show this in Section 4.2.1 and Appendix L, where the conditional gain halves under a distribution shift while the headroom-normalized score does not. Interpreting scientific foundation models. Probing and related readouts are increasingly being used to make scientific claims about learned models, for example, that protein language models encode structure (Rives et al., 2021) and that single-cell foundation models encode regulatory or cell-state information (Cui et al., 2024; Theodoris et al., 2023). Several audits find that such models often fail to beat simple baselines (Kedzierska et al., 2025; Boiarsky et al., 2024; Ahlmann-Eltze et al., 2025). A recent audit of scGPT and Geneformer finds that the apparent biological structure in attention adds nothing beyond trivial gene-level statistics (Kendiukhov, 2026). These studies choose baselines case by case and report raw differences. We formalize the question they share: how much a representation recovers beyond a declared simple baseline, relative to what is predictable from the available inputs, and apply it to scGPT, a fine-tuned TabPFN (Hollmann et al., 2025; Grinsztajn et al., 2026), and neural population activity. The latter two appear in Appendix J. Distribution shift and causal relevance. Interpretations that are faithful in distribution can fail under systematic shift (Friedman et al., 2023). Causal methods such as nullspace projection and interchange interventions test whether decodable information is used (Ravfogel et al., 2020; Geiger et al., 2024). Our framework is complementary. It does not test causal use; it specifies, before any intervention, how much evidential weight a correlational score can carry in a given regime. It also shows that a drop in raw decodability under distribution shift can reflect reduced headroom rather than a degraded representation. Controlled testbeds with known sufficient statistics. Meta-trained sequence models approximate Bayes-optimal inference (Mikulik et al., 2020; Müller et al., 2021), and transformers learn statistical estimators in context (Garg et al., 2022; Akyürek et al., 2022; Bai et al., 2023). Probes have been used to compare their internal quantities with classical estimators (Teh et al., 2025). We build on this line with random-effects meta-analysis (Higgins and Thompson, 2002). 3 Controlled meta-analysis setup 3.1 Task and model Controlled random-effects meta-analysis. Each example contains K studies estimating an underlying effect μ. Study-specific effects vary around μ with between-study heterogeneity τ2τ^2, while each reported estimate xix_i is additionally corrupted by study-specific sampling variance σi2 _i^2. Marginally, xi|μ,τ2,σi2∼(μ,σi2+τ2)x_i μ,τ^2, _i^2 (μ, _i^2+τ^2). If τ2τ^2 were known, each study would receive precision weight wi=1σi2+τ2w_i= 1 _i^2+τ^2, and all information relevant to estimating μ would be summarized by the sufficient statistics S1=∑iwiS_1= _iw_i and S2=∑iwixiS_2= _iw_ix_i. Here, S1S_1 is the total precision of the studies and S2S_2 is their precision-weighted effect. We therefore use logS1 S_1 as the primary target for representation probing, and separately probe τ2τ^2 to test whether the model represents the hidden heterogeneity needed to construct the weights. We define distribution shift as an out-of-distribution (OOD) change from a heterogeneity-dominated regime to a sampling-variance-dominated regime. We quantify the regime by the sampling-variance share ri=σi2σi2+τ2r_i= _i^2 _i^2+τ^2, whose median increases from 0.31 in distribution (ID) to 0.71 OOD. Thus, the ID regime is predominantly heterogeneity-driven, whereas the OOD regime is predominantly sampling-variance-driven. The distribution shift changes which source of uncertainty determines the precision weights. For full details, see Appendix A. Model and training. We train causal pre-norm transformers with attention and MLP blocks at five sizes ranging from 3.33.3k to 794794k parameters (xs, s, m, l, xl). The transformer model receives (xi,logσi2)(x_i, _i^2) as input, and the true heterogeneity τ2τ^2 is hidden. It predicts the overall effect μ under a causal mask (Appendix A.3). Probing and architectural controls. All probe scores and estimated reference points are computed using a bank of linear and nonlinear regressors (ridge, MLP, and gradient boosting) for every layer (Appendix B). We report the standard R2R^2 for every held-out target. We probe logS1 S_1 rather than S1S_1 because the logarithm reduces target skew from 3.93.9 to −0.6-0.6. 3.2 Raw probe behavior under the distribution shift Figure 2: In-distribution and OOD learning trajectories. Probe decodability for logS1 S_1 and τ2τ^2 and final-position model MSE across training (xl model). Table 12 and Figure 2 show that the distribution shift degrades all three measured quantities. For models s through xl, probe R2R^2 for logS1 S_1 falls (from 0.640.64 to 0.400.40 for xl), probe R2R^2 for τ2τ^2 falls (from 0.640.64 to 0.340.34 for xl), and prediction MSE at the final position rises 1212–15×15×. In distribution, model error is within 3%3\% of the oracle conditional-Bayes risk; out of distribution it is 2.12.1–2.6×2.6× that risk. Both targets nonetheless remain partly decodable OOD (R2≈0.29R^2≈ 0.29–0.400.40 for logS1 S_1 and 0.260.26–0.340.34 for τ2τ^2). The xs model is excluded from these comparisons because it does not reliably learn either target in distribution (R2=0.28±0.20R^2=0.28± 0.20 for logS1 S_1). The raw scores admit two explanations: the representation degraded, or the OOD data carry less information about the targets. A further concern applies to logS1 S_1. When sampling variance dominates, S1=∑i(σi2+τ2)−1≈∑iσi−2S_1= _i( _i^2+τ^2)^-1≈ _i _i^-2 is predictable from the sampling variances alone, so the OOD score of 0.400.40 might require no inference about τ2τ^2. If so, the best prediction of logS1 S_1 from the sampling variances alone should rise sharply under the shift. We formalize this quantity as the floor ρO2ρ^2_O in Section 4.1; Section 4.3 shows that it rises only from 0.010.01 to 0.090.09. Raw probe scores cannot separate these explanations; Section 4.1 introduces a normalization that does. 4 Normalizing probe decodability 4.1 Floor, ceiling and headroom Let T∈L2T∈ L^2 be a non-constant target quantity whose representation we wish to audit. Let O denote restricted information and I the full information available to the system, with σ(O)⊆σ(I)σ(O) σ(I). Equivalently, we may write I=(O,X)I=(O,X), where X denotes the additional evidence beyond O. The representation under study is denoted by H; for deterministic models, H=Φ(I)H= (I). We define two population reference points, ρO2:=1−[Var(T∣O)]Var(T),ρI2:=1−[Var(T∣I)]Var(T), _O^2:=1- E[Var(T O)]Var(T), _I^2:=1- E[Var(T I)]Var(T), and their difference Δ:=ρI2−ρO2≥0 := _I^2- _O^2≥ 0 (Propositions E.1 and E.2). Here ρO2 _O^2 is the maximum population R2R^2 attainable from the restricted information O, whereas ρI2 _I^2 is the maximum attainable from the full information I. We call ρO2 _O^2 the floor: it is not a lower bound on an arbitrary probe score, but the level of predictability already available without using the additional evidence X. Consequently, a probe score at or below ρO2 _O^2 does not by itself establish that the representation contains predictive information about T beyond O (Proposition E.2; Remark E.3). The headroom Δ quantifies how much additional predictability I supplies beyond O. It equals the normalized variance of the Bayes update that X supplies about T beyond O (Proposition F.1), and it vanishes through weak target sensitivity or weak identification (Corollary F.2; Section 4.3). Figure 3: Raw and normalized probe performance under distribution shift. Raw probe scores decrease OOD, while normalization by the floor and ceiling shows that most of the decline reflects reduced available information. For a representation H, let RH2R_H^2 denote the held-out probe performance for predicting T from H. When H is measurable with respect to I, no predictor based on H can exceed the population ceiling ρI2 _I^2 (Proposition E.1). When Δ>0 >0, we therefore normalize representation decodability as UH:=RH2−ρO2ρI2−ρO2U_H:= R_H^2- _O^2 _I^2- _O^2. Here UHU_H measures probe performance relative to the headroom available beyond O. Each reference point has three versions. The population reference points are defined above. Second is estimating the reference points ρ^O2,ρ^I2 ρ^2_O, ρ^2_I, by fitting a fixed decoder bank to O and to I on held-out data (the estimation procedure; Appendix B). When the generative model is known (meta-analysis, synthetic single-cell), we additionally compute them exactly by numerical integration. In both of them, estimates closely track exact values (Appendix H, Figure 7). For meta-analysis and single-cell test cases, we estimate the reference points with feature representations ϕO(O) _O(O) and ϕI(I) _I(I). The feature maps are fixed deterministic functions of their corresponding information sets (Appendix I). Estimated ceilings are conservative, so an underestimated ceiling inflates UHU_H. Figure 4: Raw and headroom-normalized decodability across domains. Left: R2R^2 of the floor, the ceiling and the representation probe. Right: the normalized decodability UH=(RH2−ρ^O2)/(ρ^I2−ρ^O2)U_H=(R_H^2- ρ_O^2)/( ρ_I^2- ρ_O^2), the fraction of the headroom recovered by the probe. Meta-analysis reference points are exact (Table 3). “Artificial” is the synthetic single-cell model and “Omics” the single-cell data of Table 13. A natural alternative is conditional probing (Hewitt et al., 2021) with baseline O instead of the model’s own non-contextual embedding layer: the gain RO,H2−RO2R^2_O,H-R^2_O from adding H to a probe on O. Measuring this gain from the floor instead (in the population, RO2=ρO2R^2_O=ρ^2_O) and normalizing it by the headroom gives UH|O:=(RO,H2−ρO2)/(ρI2−ρO2)U_H|O:=(R^2_O,H-ρ^2_O)/(ρ^2_I-ρ^2_O). In the population UH≤UH|O≤1U_H≤ U_H|O≤ 1, with equality on the left when H retains everything in O relevant to T (Appendix L). When an estimated floor is well below chance for classification or well below zero for regression, the headroom normalization is not applicable, and we report the raw probe performance instead. Floors only slightly below chance or zero, as expected from estimation noise when the population floor is near zero (Figure 7), are kept unclipped (Appendix B). Evaluation settings. We specify the contrast (O,I,T)(O,I,T) in settings with progressively weaker access to ground truth: the controlled meta-analysis and a synthetic single-cell model, where the reference points are exact (Appendix H), then real single-cell representations and published LLM probes. The contrast is chosen from the scientific question, independently of the probe scores, and normalized results are interpreted relative to it. Where no unique contrast exists, we evaluate nested baselines O1⊂O2⊂⋯⊂OkO_1⊂ O_2⊂…⊂ O_k. Appendix J applies this to a fine-tuned chemistry model and recorded neural activity. 4.2 Normalized probes across domains 4.2.1 Meta-analysis Using the controlled random-effects model defined above, we take the sufficient statistic logS1 S_1 as the probe target, T=log∑i=1K1σi2+τ2T= _i=1^K 1 _i^2+τ^2. The scientific question is whether the transformer’s hidden representation contains information about the precision-weighting computation beyond what the reported sampling variances alone predict. We therefore define the restricted information as O=σ1:K2O= _1:K^2 and the full information as I=(σ1:K2,x1:K)I=( _1:K^2,x_1:K), where the observed effects xix_i provide the additional evidence from which the latent heterogeneity τ2τ^2 can be inferred. We also probe τ2τ^2 directly. Because τ2τ^2 is independent of O, its floor is zero, and its ceiling measures how much heterogeneity the reported effects reveal. The distribution shift changes the relative contribution of sampling variance and heterogeneity, so it tests whether raw decodability tracks the information available in each regime. Figure 4 and Table 13 place each raw probe score between a floor and a ceiling set by a domain-specific contrast. In the meta-analysis, the contrast asks whether the hidden state carries heterogeneity inferred from the reported effects beyond what the sampling variances predict. In distribution, the probe recovers 78%78\% of the headroom for logS1 S_1 and 84%84\% for τ2τ^2. Under the shift, raw decodability falls for both targets (0.64→0.400.64→ 0.40 for logS1 S_1, 0.64→0.340.64→ 0.34 for τ2τ^2; Table 12), and so do the exact ceilings (0.82→0.470.82→ 0.47 and 0.76→0.400.76→ 0.40; Table 3). The fraction of headroom recovered does not decrease (UH=0.78→0.82U_H=0.78→ 0.82 for logS1 S_1, 0.84→0.850.84→ 0.85 for τ2τ^2; Figure 3). The lower raw scores therefore reflect less recoverable information in the data, not a less informative representation (Section 4.3). Conditional probing alone would suggest the opposite. With a linear decoder throughout, the conditional gain roughly halves under the shift (0.58→0.310.58→ 0.31 for logS1 S_1; 0.64→0.340.64→ 0.34 for τ2τ^2), which alone suggests a degraded representation. The headroom (estimated with linear decoder) also halves (0.62→0.33;0.66→0.360.62→ 0.33;0.66→ 0.36), and once the gain is divided by it, UH|OU_H|O stays flat (0.95→0.950.95→ 0.95; 0.97→0.950.97→ 0.95), corresponding to UHU_H (Appendix L). 4.2.2 Genomics and single-cell data Single-cell RNA sequencing has the same structure as meta-analysis. Each example is a gene g measured in K cells. The normalized count of a cell varies for two reasons: technical noise, which decreases as the depth of the cell’s sequencing dgid_gi increases, and the biological variation from cell-to-cell, summarized by the overdispersion of the gene bgb_g. Under a Gamma–Poisson model the normalized count has variance λg(dgi−1+bg) _g(d_gi^-1+b_g), where λg _g is the gene’s mean expression. Technical noise dgi−1d_gi^-1 therefore plays the role of the sampling variance σi2 _i^2, and overdispersion bgb_g the role of heterogeneity τ2τ^2. The target is the analogue of logS1 S_1, the gene’s effective precision Tg=log∑i=1K(dgi−1+bg)−1T_g= _i=1^K(d_gi^-1+b_g)^-1, which depends on the hidden overdispersion. The restricted information is the sequencing depths alone, Og=dg,1:KO_g=d_g,1:K, and the full information adds the observed counts, Ig=(yg,1:K,dg,1:K)I_g=(y_g,1:K,d_g,1:K), from which overdispersion can be inferred. We ask whether frozen scGPT (Cui et al., 2024) encodes TgT_g beyond what depth alone predicts. We first ask this on synthetic genes (K=40K=40; Appendix G) and then on four real scRNA-seq datasets (10x Genomics, 2021; 10x Genomics, 2024; 10x Genomics, 2022). In real data, the true bgb_g is unknown, so we split the cells of each dataset into half at random. Overdispersion is estimated from the first half and defines the target, while the depths, counts and scGPT inputs come from the second half, so the ceiling cannot simply recompute its own target. K is then the number of cells in the second half. For single-cell data, raw scores are misleading because the floor differs across datasets. In the 10k human dataset, depth alone already reaches R2=0.60R^2=0.60, so scGPT’s raw score of 0.810.81 recovers 57%57\% of the headroom. In the 20k donors dataset the raw score is much lower (0.570.57), but the floor is only 0.130.13, so scGPT recovers a similar 52%52\%. Across the four datasets, raw scores range from 0.350.35 to 0.810.81, while the fraction of headroom recovered ranges from 0.280.28 to 0.570.57; on synthetic genes it is 0.700.70. scGPT therefore encodes biological overdispersion beyond sequencing depth, but only partially. scGPT probes use a 64-component PCA fitted on training rows only; all floors, ceilings and other probes use the full feature vectors (Appendix I). 4.2.3 Published LLM probes We compute floors for four published LLM probes, using descriptions of the input. T is the published target, I is the input the model received, and O is a surface description of it: n-grams of a name, statement, or conversation, or move summaries that never simulate Othello flips. The floor is what the decoder bank learns from O on the probe’s own training split, so it needs only the released inputs, labels, and splits. For Othello, the ceiling is exactly 1, since the board follows from the moves; elsewhere we set it to 1, which can only lower UHU_H, while a richer floor readout could lower it further. We therefore read UHU_H here as an approximate value and compare the raw scores with the floor directly where the headroom is small. For categorical targets, we use accuracy in place of R2R^2. Figure 5 summarizes the results. See full results in Appendix K. Figure 5: Probe normalization puts published LLM probe scores in perspective. Left: floor computed from a description of the input (n-grams; disc placement without flips for Othello), with 95% CI, the probe score, and the ceiling set to 1 (R2R^2 for places and dates, accuracy otherwise). Right: normalized probe score, UHU_H, with error bars from the floor’s CI. Space. Gurnee and Tegmark (2024) report that Llama-2 represents the coordinates of named places. A floor computed from a description of the place name, its character 2–4-gram counts, reaches R2=0.61R^2=0.61 for world places and 0.46 for US places, yet the published probes recover 63–77% of the headroom above it. For New York City this floor is 0.33 (95% CI 0.29–0.37), and the published scores (0.22 for 7B, 0.36 for 70B) lie within or below it, so they cannot separate a city-level map from borough names and numbers in the strings. Othello. Li et al. (2022) report that Othello-GPT represents the board. A floor that knows which player placed each disc, but never simulates flips, reaches 0.80 accuracy (chance 0.46). The linear black/white/empty probe (0.74) falls below it (UH=−0.30U_H=-0.30), so its accuracy is accounted for by disc placement. A nonlinear probe recovers 45% of the headroom with our decoders. In the mine/theirs/empty encoding (Nanda et al., 2023), a linear probe reaches 0.99 and recovers 92% of the headroom above the floor from the same features (0.81 for this target), indicating that the effect of flips is represented linearly in the relative encoding. Truth. Marks and Tegmark (2023) report that LLMs represent whether a statement is true along a linear direction, and that a probe trained on one dataset of statements still works on another. We compute the floor from a description of each statement, its character and word n-grams. Within a single dataset, this floor is near chance for most datasets but reaches 0.97–0.99 on larger_than and smaller_than datasets, whose truth labels follow from the wording of the numbers. Probes trained and tested on those two datasets therefore show little beyond the text. Transfer between datasets is more informative, because wording learned on one dataset does not carry over to another. A floor trained on the two number datasets and tested on Spanish–English translations is at chance (0.49), while the published probes reach 0.97 (UH=0.94U_H=0.94). Trained on cities and neg_cities and tested on negated translations, the floor is again at chance (0.50), while the probes reach 0.96–0.99 (UHU_H up to 0.98). User attributes. Chen et al. (2024) report probes that read a user’s age, gender, education and socioeconomic status from Llama-2-Chat activations. The floor, an n-gram classifier on the conversation, reaches 0.92–0.98, and every published reading probe (0.94–0.98) lies within its 95% interval; every control probe falls below its point estimate. The floor is as high from the user’s turns alone, and falls to 0.76–0.90 when trained on GPT-3.5 conversations and tested on Llama-2-Chat ones. However, the role-played conversations were written to reveal the attribute, so these accuracies can neither establish nor rule out that the model internally represents the user’s attributes. Summary. Probe normalization puts these published scores in perspective (Figure 5). Probe scores reach well above the floor when the target is not revealed by the surface form: coordinates at world scale, the effect of flips on the board, and truth transferred across datasets. When the target is written or cued in the input, such as borough names and numbers for New York City places, disc placement for the absolute board encoding, or attributes the role-played user writes into the conversation, the floor alone accounts for much of the score. 4.3 Mechanisms for vanishing headroom The headroom Δ is the predictability that the full information I adds beyond the restricted baseline O. It can vanish in two distinct ways (Corollary F.2). Let T=f(O,Z)T=f(O,Z) for a latent variable Z, and let I=(O,X)I=(O,X), where X is the additional evidence about Z. (1) Weak target sensitivity: T barely depends on Z. If f is Lipschitz in Z with constant L(O)L(O), the headroom is bounded by [L(O)2Var(Z∣O)]/Var(T)E[L(O)^2\,Var(Z O)]/Var(T), so knowing Z adds little even when X reveals it perfectly. The floor rises toward the ceiling. (2) Weak identification: X reveals little about Z beyond O. If f varies by at most M over Z, the headroom is bounded by M2MI(X;Z∣O)/(2Var(T))M^2\,MI(X;Z O)/(2\,Var(T)), where MIMI is the conditional mutual information in nats, so T may depend strongly on Z, but the data cannot pin Z down. The ceiling falls toward the floor. Which mechanism drives the meta-analysis shift? The distribution shift (Section 3) pushes in both directions. When sampling variance dominates, logS1=log∑i(σi2+τ2)−1 S_1= _i( _i^2+τ^2)^-1 depends less on τ2τ^2 (weaker target sensitivity), and the per-study Fisher information about τ2τ^2, which scales as (σi2+τ2)−2( _i^2+τ^2)^-2, drops, so the reported effects reveal less about τ2τ^2 (weaker identification; Remark E.7). The exact reference points show that weak identification dominates (Table 3). The headroom for logS1 S_1 shrinks from 0.810.81 to 0.370.37. The floor rises only from 0.010.01 to 0.090.09, while the ceiling falls from 0.820.82 to 0.470.47, so most of the loss comes from the ceiling. For τ2τ^2 itself the target is the latent variable, so target sensitivity cannot change; its floor stays at zero and its ceiling falls from 0.760.76 to 0.400.40, a pure identification loss (Figures 3 and 4). Figure 6: Headroom as evidence quality and target sensitivity vary, in the synthetic single-cell model (with binomial thinning). Evidence quality pXp_X controls how well the data identify the latent variable; target sensitivity pTp_T controls how strongly the target depends on it. Low pXp_X gives weak identification, low pTp_T gives weak target sensitivity, and low values of both give both. Separating the mechanisms in a controlled grid. In the meta-analysis both mechanisms move together. The synthetic single-cell model (Section 4.2.2; Appendix G) lets us vary them independently. The latent variable is the gene’s biological overdispersion, Z=bgZ=b_g; the restricted information is sequencing depth, Og=dg,1:KO_g=d_g,1:K; the additional evidence is the observed counts, Xg=yg,1:KX_g=y_g,1:K; and the target is TgT_g. Reducing sequencing depth by binomial thinning, retaining a fraction p of counts, lowers the effective depth from dgid_gi to pdgip\,d_gi. In real data this affects both the evidence and the target at once. In simulation we apply it separately (Appendix G.1): thinning the counts used as evidence (pXp_X) degrades inference of bgb_g, giving weak identification, whereas thinning the depths entering the target (pTp_T) lets technical variance dominate overdispersion, giving weak target sensitivity; the restricted information is never thinned. Figure 6 shows the resulting headroom over the (pX,pT)(p_X,p_T) grid, with estimated reference points. Headroom vanishes when either parameter is small and stays large only when both are large, so a loss of headroom can be traced to its cause. 5 Conclusion A probe score of 0.60.6 has no fixed meaning. What it says about a representation depends on two quantities: how much of that predictability a declared baseline already provides, and how much the available data support in total. We made both explicit. The floor ρO2ρ^2_O is what a declared baseline O already predicts, the ceiling ρI2ρ^2_I is what the full input I supports, and the headroom Δ=ρI2−ρO2 =ρ^2_I-ρ^2_O between them is the range in which a probe score can show information beyond O. Headroom vanishes in two ways: weak target sensitivity, when the target stops depending on the latent variable, and weak identification, when the evidence stops revealing it. In the controlled meta-analysis, where the reference points are exact, normalization reverses the reading of a distribution shift. Raw decodability of logS1 S_1 and τ2τ^2 falls and prediction error rises, which alone suggests a degraded representation. The reference points show instead that the data lost information: the ceiling falls from 0.820.82 to 0.470.47 for logS1 S_1 and from 0.760.76 to 0.400.40 for τ2τ^2, while the floor for logS1 S_1 rises only from 0.010.01 to 0.090.09, so weak identification dominates. The representation recovers a similar share of the smaller headroom (UH=0.78→0.82U_H=0.78→ 0.82 for logS1 S_1; 0.84→0.850.84→ 0.85 for τ2τ^2). The estimation recovers the exact reference points in two fully specified generative models and extends to settings where only samples are available. In frozen scGPT, similar raw scores correspond to different fractions of recovered headroom across datasets (0.280.28–0.570.57), so biological overdispersion is encoded only partially. For published LLM probes, a floor computed from the input text alone separates claims the text cannot explain (world-scale coordinates, the effect of flips in the relative Othello encoding, truth transferred across datasets) from claims it largely explains (New York City locations, the absolute Othello board, user attributes in role-played conversations). The framework complements causal interventions: it bounds how much a correlational score can show before any intervention is run. We recommend that interpretability studies report the floor, the ceiling and the fraction of headroom recovered alongside raw scores, state the contrast (O,I,T)(O,I,T) before looking at probe scores, and, where no unique baseline exists, report a nested sequence O1⊂⋯⊂OkO_1⊂…⊂ O_k so that readers can see how the choice of baseline affects the conclusions. Limitations. Normalized scores are defined relative to a stated contrast (O,I,T)(O,I,T). Different contrasts answer different scientific questions, so we report nested baselines (Appendix J) rather than a single choice. On observational data the reference points are estimated, and an underestimated ceiling or floor inflates UHU_H. For published LLM probes the ceiling cannot be estimated and is set to 11, so UHU_H there is approximate. Our controlled tasks are kept simple so that the reference points can be computed exactly, and the transformers trained on them are small. AI use statement In this work, we used generative AI tools to aid and polish writing, for retrieval and discovery of related work, for research execution, and to draft sections of the paper. As part of research execution, we used coding agents to assist with implementing and debugging analysis code and with producing scientific figures. We did not use generative AI tools for generating synthetic datasets or for proving mathematical claims. We have reviewed all AI-assisted work. AI-assisted code and analyses were inspected and their outputs were checked against the corresponding experiments and results; AI-assisted text and figures were reviewed for consistency with the underlying methods, analyses, and findings. We take responsibility for the final content of this work, including text, claims, code, figures, and other artifacts produced with the aid of generative AI. References 10x Genomics (2021) 10x Genomics 10k PBMCs from Human Female, 3’ v3.1, Chromium X. 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Large τ2τ^2 means that the true effects θi _i vary substantially from study to study even before sampling noise is introduced; small τ2τ^2 means that the studies are close to sharing a common true effect. Each study also has finite-sample measurement uncertainty. We draw ni∼U5,…,99,si2∼U(0.1,10),n_i \5,…,99\, s_i^2 (0.1,10), and define its sampling variance as σi2=si2ni. _i^2= s_i^2n_i. Thus studies with larger samples or smaller within-study variance are measured more precisely. Finally, the reported study estimate is xi∼(θi,σi2).x_i ( _i, _i^2). The difference between xix_i and the global effect μ therefore arises from two distinct sources: θi−μ⏟between-study heterogeneityandxi−θi⏟sampling noise. _i-μ_between-study heterogeneity x_i- _i_sampling noise. Marginalizing over the unobserved study-specific effect θi _i gives xi|μ,τ2,σi2∼(μ,σi2+τ2).x_i μ,τ^2, _i^2 \! (μ,\, _i^2+τ^2 ). The total uncertainty of study i is therefore the sum of its sampling variance and the common between-study heterogeneity. The transformer receives (xi,logσi2)(x_i, _i^2) for each study as an input. It does not observe θi _i, τ2τ^2, or μ. We hypothesize that it infers whatever information about τ2τ^2 is useful for combining the reported studies. Precision weighting. A study with lower total variance σi2+τ2 _i^2+τ^2 should contribute more strongly to an estimate of μ. Its corresponding precision is the inverse of that variance, wi=1σi2+τ2.w_i= 1 _i^2+τ^2. Thus a large wiw_i denotes a comparatively informative study, while a small wiw_i denotes a noisy or unreliable one. If τ2τ^2 were known, the K studies could be summarized by two quantities: S1=∑i=1Kwi,S2=∑i=1Kwixi.S_1= _i=1^Kw_i, S_2= _i=1^Kw_ix_i. The first, S1S_1, is the total precision: it measures how much effective information the collection of studies provides about μ. The second, S2S_2, is the precision-weighted sum of the observed effects. For fixed τ2τ^2, the likelihood for μ can be written as p(x1:K∣μ,τ2,σ1:K2)∝exp[−12(S1μ2−2S2μ)].p(x_1:K μ,τ^2,σ^2_1:K) \! [- 12 (S_1μ^2-2S_2μ ) ]. Consequently, once τ2τ^2 is known, the individual study observations contain no additional information about μ beyond (S1,S2)(S_1,S_2). Therefore, (S1,S2)(S_1,S_2) are sufficient statistics for estimating μ conditional on τ2τ^2. This is why logS1 S_1 is useful as a representation-level audit target: constructing it requires the model to combine the observed sampling variances with information about the otherwise hidden heterogeneity τ2τ^2. Inverse-variance estimate. The standard inverse-variance-weighted estimator is μ^IVW=S2S1. μ_IVW= S_2S_1. This denotes a weighted average of the reported study effects, where more precise studies receive larger weights. For example, if two studies have identical reported effects but one has much smaller total variance, the latter contributes more strongly to μ^IVW μ_IVW. The heterogeneity τ2τ^2 plays an important role: increasing heterogeneity makes the study weights more similar because the common τ2τ^2 term begins to dominate their individual sampling variances. Bayesian estimate. The generative model places the prior μ∼(0,1)μ (0,1), whose precision is 11. Combining this prior with the conditional likelihood gives μ∣x1:K,σ1:K2,τ2∼(S2S1+1,1S1+1).μ x_1:K,σ^2_1:K,τ^2 \! ( S_2S_1+1, 1S_1+1 ). The corresponding posterior mean is μ^B(τ2)=S2S1+1. μ_B(τ^2)= S_2S_1+1. As the prior on μ is centered at zero, the Bayesian estimate is the inverse-variance estimate multiplied by S1/(S1+1)<1S_1/(S_1+1)<1. When the accumulated study precision S1S_1 is small, the prior has appreciable influence and the estimate is pulled toward zero. As S1S_1 becomes large, S1S1+1→1 S_1S_1+1→ 1, and the Bayesian and classical estimators become nearly identical. Their corresponding conditional mean-squared errors are MSEB=1S1+1,MSEIVW=1S1.MSE_B= 1S_1+1, _IVW= 1S_1. The difference is therefore largest when little total precision has accumulated, for example at short sequence prefixes, and becomes negligible when many precise studies are available. The expression S2S1+1 S_2S_1+1 requires the true τ2τ^2, because both S1S_1 and S2S_2 use wi=(σi2+τ2)−1w_i=( _i^2+τ^2)^-1. As the transformer is never given τ2τ^2, S2/(S1+1)S_2/(S_1+1) is not the Bayes-optimal predictor directly available from the model’s observed inputs. The actual Bayes-optimal predictor must instead average over uncertainty about the unknown heterogeneity: μ^Bayes=[μ∣x1:K,σ1:K2] μ_Bayes=E[μ x_1:K,σ^2_1:K]. Equivalently, μ^Bayes=∫S2(t)S1(t)+1p(t∣x1:K,σ1:K2)dt μ_Bayes= S_2(t)S_1(t)+1\,p(t x_1:K,σ^2_1:K)\,dt, where t denotes a candidate value of the heterogeneity, S1(t)=∑i=1K1σi2+tS_1(t)= _i=1^K 1 _i^2+t, and S2(t)=∑i=1Kxiσi2+tS_2(t)= _i=1^K x_i _i^2+t. The conditional quantity S2S1+1 S_2S_1+1 is nevertheless useful because the simulator knows the true τ2τ^2. We therefore use it only as an oracle conditional-Bayes reference: it tells us how well prediction could perform if the hidden heterogeneity were known. It is not supplied to the transformer and is not the model’s available Bayes-optimal solution. See Appendix C for the full derivation. A.1 Distribution shift The total variance of study i is σi2+τ2 _i^2+τ^2. We summarize which source of uncertainty dominates using the sampling-variance share ri=σi2σi2+τ2.r_i= _i^2 _i^2+τ^2. This quantity lies between zero and one. When ri≈0r_i≈ 0, most uncertainty comes from between-study heterogeneity. When ri≈1r_i≈ 1, most uncertainty comes from ordinary study-level sampling variance. In the in-distribution regime, the median value is approximately rID=0.31r_ID=0.31, so heterogeneity is usually the dominant contribution to total variance. In the OOD regime, which draws si2∼U(10,20)s_i^2 (10,20), ni∼U2,…,19n_i \2,…,19\ and τ2∼U(0,1.5)τ^2 (0,1.5) with all other draws as above, the median increases to rOOD=0.71r_OOD=0.71, so sampling variance becomes the dominant contribution. The distribution shift therefore changes the relative source of uncertainty governing the correct inverse-variance weights. A model that has learned a regime-specific shortcut, accurate only when τ2τ^2 dominates, can consequently fail when σi2 _i^2 becomes dominant, whereas a model implementing the appropriate weighting computation should adapt its weights to the changed balance. A.2 Heterogeneity baseline The simulator knows τ2τ^2; the transformer does not. A classical baseline for estimating heterogeneity from the observed study estimates is the DerSimonian–Laird estimator, τ^DL2=max0,Q−(K−1)C, τ^2_DL= \0,\, Q-(K-1)C \, where Q measures excess dispersion of the study estimates around their fixed-effect weighted mean and C is a normalization determined by the fixed-effect study weights. Since τ^DL2 τ^2_DL is computable from the model inputs, a probe that predicts τ2τ^2 may simply reproduce information already available through this standard estimator. The full-information features used for the estimated reference points therefore include τ^DL2 τ^2_DL (Appendix I). A.3 Training and checkpoints Models are causal pre-norm transformers at five sizes, from 3.3k to 794k parameters (Table 1), trained with mean-squared error at every prefix. Checkpoints are retained at epochs 0,1,2,5,10,20,50,100,…\0,1,2,5,10,20,50,100,…\ for training-trajectory analyses. Training stops when validation MSE, averaged over all prefixes, is within 2%2\% of the oracle conditional-Bayes risk averaged the same way (Appendix D), or after 300300 epochs without validation improvement; the oracle risk uses the true τ2τ^2 and is not attainable from the model’s inputs. Of the 25 random-effects runs, 24 stopped on patience. Model predictions are closer to the oracle conditional posterior mean than to the oracle IVW estimator. Behavioral comparisons with the IVW and conditional-Bayes estimators are performed on held-out meta-analyses. Model size d h L Parameters Peak learning rate Epoch budget xs 16 2 1 3,345 4×10−44× 10^-4 3,300 s 32 4 2 25,537 4×10−44× 10^-4 3,300 m 64 4 2 100,225 5×10−55× 10^-5 3,000 l 64 4 4 200,193 5×10−55× 10^-5 3,500 xl 128 8 4 793,601 5×10−55× 10^-5 3,500 Table 1: Transformer model configurations. d is the residual-stream width, h the number of attention heads (head width d/hd/h), and L the number of transformer blocks; the block MLP has hidden width 4d4d. Parameter counts include the input embedding, all blocks, and the scalar readout. Appendix B Decoder bank All reported regression-probe scores are computed using one canonical bank (Table 2) of candidate decoders. For the controlled meta-analysis, all probes and reference points are evaluated at the final position, using the final-token representation. Decoder type and hyperparameters are selected using the selection split based on R2R^2, after which the selected decoder is evaluated once on the evaluation split. Every probe uses disjoint training, selection, and evaluation rows. Categorical decoders are chosen by McFadden R2R^2, but reported and normalized into UHU_H, by accuracy. We report standard R2R^2 on this held-out split and retain negative values rather than clipping them to zero. ℳ=GBM,shallow GBM,MLP,RidgeCV.M=\GBM,shallow GBM,MLP,RidgeCV\. Decoder Fixed specification GBM HistGradientBoostingRegressor with 400 maximum iterations, learning rate 0.060.06, unrestricted depth, early stopping, and a 0.150.15 internal validation fraction. Shallow GBM HistGradientBoostingRegressor with 600 maximum iterations, learning rate 0.040.04, maximum depth 33, early stopping, and a 0.150.15 internal validation fraction. MLP Standardization followed by an MLP with hidden widths (256,128)(256,128), L2L_2 penalty 10−410^-4, 600 maximum iterations, early stopping, and patience 25. Linear Standardization followed by RidgeCV with α∈logspace(−6,4,21).α (-6,4,21). Table 2: The decoder bank. These definitions are fixed across domains, representations, targets, and random seeds. Appendix C Bayes posterior mean conditional on heterogeneity Consider a prefix containing the first k studies. Conditional on the overall effect μ, the heterogeneity τ2τ^2, and the sampling variances σ1:k2 _1:k^2, the reported effects are independent and satisfy xi|μ,τ2,σi2∼(μ,σi2+τ2)x_i μ,τ^2, _i^2 (μ, _i^2+τ^2). Define the precision weights and their cumulative sufficient statistics by wi=1σi2+τ2,S1(k)=∑i=1kwi,S2(k)=∑i=1kwixi.w_i= 1 _i^2+τ^2, S_1^(k)= _i=1^kw_i, S_2^(k)= _i=1^kw_ix_i. Ignoring factors that do not depend on μ, the likelihood is p(x1:k∣μ,τ2,σ1:k2)∝exp[−12∑i=1kwi(xi−μ)2]∝exp[−12(S1(k)μ2−2S2(k)μ)].p(x_1:k μ,τ^2, _1:k^2) \! [- 12 _i=1^kw_i(x_i-μ)^2 ] \! [- 12 (S_1^(k)μ^2-2S_2^(k)μ ) ]. Combining this likelihood with the prior μ∼(0,1)μ (0,1) gives p(μ∣x1:k,τ2,σ1:k2) p(μ x_1:k,τ^2, _1:k^2) ∝p(x1:k∣μ,τ2,σ1:k2)p(μ) p(x_1:k μ,τ^2, _1:k^2)\,p(μ) ∝exp[−12(S1(k)μ2−2S2(k)μ)]exp(−12μ2) \! [- 12 (S_1^(k)μ^2-2S_2^(k)μ ) ] \! (- 12μ^2 ) =exp[−12((S1(k)+1)μ2−2S2(k)μ)] = \! [- 12 ( (S_1^(k)+1 )μ^2-2S_2^(k)μ ) ] ∝exp[−12(S1(k)+1)(μ−S2(k)S1(k)+1)2]. \! [- 12 (S_1^(k)+1 ) (μ- S_2^(k)S_1^(k)+1 )^2 ]. Therefore, μ∣x1:k,τ2,σ1:k2∼(S2(k)S1(k)+1,1S1(k)+1).μ x_1:k,τ^2, _1:k^2 \! ( S_2^(k)S_1^(k)+1, 1S_1^(k)+1 ). Under squared-error loss, the posterior mean uniquely minimizes the conditional expected loss. Hence, when τ2τ^2 is known, the Bayes estimator is μ^B(k)(τ2)=argmina[(a−μ)2∣x1:k,τ2,σ1:k2]=[μ∣x1:k,τ2,σ1:k2]=S2(k)S1(k)+1. μ_B^(k)(τ^2)= _aE\! [(a-μ)^2 x_1:k,τ^2, _1:k^2 ]=E[μ x_1:k,τ^2, _1:k^2]= S_2^(k)S_1^(k)+1. In the actual prediction task, τ2τ^2 is not observed. The fully Bayes-optimal estimator therefore averages this conditional posterior mean over the posterior distribution of τ2τ^2: μ^Bayes(k)=[μ∣x1:k,σ1:k2]=∫00.5S2(k)(t)S1(k)(t)+1p(t∣x1:k,σ1:k2)dt, μ_Bayes^(k)=E[μ x_1:k, _1:k^2]= _0^0.5 S_2^(k)(t)S_1^(k)(t)+1\,p(t x_1:k, _1:k^2)\,dt, where S1(k)(t)=∑i=1k(σi2+t)−1S_1^(k)(t)= _i=1^k( _i^2+t)^-1 and S2(k)(t)=∑i=1kxi(σi2+t)−1S_2^(k)(t)= _i=1^kx_i( _i^2+t)^-1. Thus, S2(k)/(S1(k)+1)S_2^(k)/(S_1^(k)+1) is an oracle Bayes reference that assumes access to the true heterogeneity, whereas the model must approximate the marginal posterior mean by inferring τ2τ^2 from the observed studies. We do not compute μ^Bayes μ_Bayes; all behavioral references in this paper are oracle quantities. Appendix D Expected mean-squared error of the IVW and Bayes estimators We derive the expected mean-squared error of the oracle IVW and oracle conditional-Bayes estimators (both using the true τ2τ^2) derived in Appendix C. At a prefix containing the first k studies, define wi=1/(σi2+τ2)w_i=1/( _i^2+τ^2), S1(k)=∑i=1kwiS_1^(k)= _i=1^kw_i, and S2(k)=∑i=1kwixiS_2^(k)= _i=1^kw_ix_i. Conditional on μ, τ2τ^2, and the reported sampling variances σ1:k2 _1:k^2, each observation satisfies xi∼(μ,1/wi)x_i (μ,1/w_i), so S2(k)∼(S1(k)μ,S1(k))S_2^(k) (S_1^(k)μ,S_1^(k)). Hence, the weighted sum can be written as S2(k)=S1(k)μ+εk,εk∣S1(k)∼(0,S1(k)).S_2^(k)=S_1^(k)μ+ _k, _k S_1^(k) \! (0,S_1^(k) ). The inverse-variance-weighted estimator is μ^IVW(k):=S2(k)S1(k)=μ+εkS1(k). μ_IVW^(k):= S_2^(k)S_1^(k)=μ+ _kS_1^(k). Its estimation error is therefore εk/S1(k) _k/S_1^(k), and its conditional mean-squared error is [(μ^IVW(k)−μ)2|S1(k)] \! [ ( μ_IVW^(k)-μ )^2\, |\,S_1^(k) ] =[(εkS1(k))2|S1(k)] =E\! [ . ( _kS_1^(k) )^2\, |\,S_1^(k) ] =1(S1(k))2[εk2∣S1(k)] = 1 (S_1^(k) )^2E\! [ _k^2 S_1^(k) ] =Var(εk∣S1(k))(S1(k))2 = Var\! ( _k S_1^(k) ) (S_1^(k) )^2 =S1(k)(S1(k))2=1S1(k). = S_1^(k) (S_1^(k) )^2= 1S_1^(k). For the prior μ∼(0,1)μ (0,1), the Bayes posterior mean (from Appendix C) is μ^B(k)=S2(k)S1(k)+1. μ_B^(k)= S_2^(k)S_1^(k)+1. Substituting S2(k)=S1(k)μ+εkS_2^(k)=S_1^(k)μ+ _k gives μ^B(k)−μ=εk−μS1(k)+1. μ_B^(k)-μ= _k-μS_1^(k)+1. The variables εk _k and μ are independent, with variances S1(k)S_1^(k) and 11, respectively. Averaging over both the observation noise and the prior on μ therefore gives [(μ^B(k)−μ)2|S1(k)]=Var(εk)+Var(μ)(S1(k)+1)2=S1(k)+1(S1(k)+1)2=1S1(k)+1.E\! [ ( μ_B^(k)-μ )^2 |S_1^(k) ]= Var( _k)+Var(μ)(S_1^(k)+1)^2= S_1^(k)+1(S_1^(k)+1)^2= 1S_1^(k)+1. Thus, at every prefix, MSEB(k)=1S1(k)+1<1S1(k)=MSEIVW(k).MSE_B^(k)= 1S_1^(k)+1< 1S_1^(k)=MSE_IVW^(k). The absolute improvement is MSEIVW(k)−MSEB(k)=1S1(k)(S1(k)+1),MSE_IVW^(k)-MSE_B^(k)= 1S_1^(k)(S_1^(k)+1), while the ratio of the Bayes risk to the IVW risk is MSEB(k)MSEIVW(k)=S1(k)S1(k)+1. MSE_B^(k)MSE_IVW^(k)= S_1^(k)S_1^(k)+1. The gain is therefore largest when only a few studies have been observed and S1(k)S_1^(k) is small, and it vanishes as the accumulated precision grows. To match the model’s all-position training objective, we average these conditional risks over every prefix and every generated meta-analysis: MSE¯IVW=τ2,σ1:K2,k[1S1(k)],MSE¯B=τ2,σ1:K2,k[1S1(k)+1]. MSE_IVW=E_\,τ^2, _1:K^2,k\! [ 1S_1^(k) ], MSE_B=E_\,τ^2, _1:K^2,k\! [ 1S_1^(k)+1 ]. Evaluating these expectations on the generated evaluation data gives oracle risks MSE¯IVW=0.048 MSE_IVW=0.048 and MSE¯B=0.041 MSE_B=0.041. The conditional-Bayes risk lower-bounds the risk attainable without τ2τ^2; the IVW risk does not, because the IVW estimator ignores the prior on μ. Appendix E Floor–ceiling derivations and reference-point computation This appendix derives the floor–ceiling normalization used in Section 4.1. We first specialize the argument to the probe target logS1 S_1, then state the general headroom identity and the two mechanisms by which headroom vanishes, and finally describe the exact computation of the reference points in Appendix H. Task-specific definitions and bounds. One meta-analysis contains the reported effects and sampling variances X:=x1:K,O:=σ1:K2,I:=(O,X).X:=x_1:K, O:=σ^2_1:K, I:=(O,X). Thus, O contains the directly reported variances, X contains the additional evidence carried by the reported effects, and I contains all observed information. The latent variables (μ,τ2)(μ,τ^2) are drawn independently of O, with τ2∼U(0,τmax),μ∼(0,1),τ^2 (0, _ ), μ (0,1), with τmax=0.5 _ =0.5 in distribution and τmax=1.5 _ =1.5 out of distribution (Appendix A). At prefix k, the probe target is Tk:=logS1(k)=log∑i≤k1σi2+τ2.T_k:= S_1^(k)= _i≤ k 1 _i^2+τ^2. The target is therefore a deterministic function of (O,τ2)(O,τ^2). Conditional on O, the reported effects X are relevant to TkT_k only through what they reveal about the latent heterogeneity τ2τ^2. Throughout, we write MMSE(T∣I):=[Var(T∣I)].MMSE(T I):=E[Var(T I)]. Proposition E.1 (Probe ceiling). Let H=Φ(I)H= (I) be any representation constructed from the observed inputs, whether produced by a trained or untrained architecture, and let h be any probe. The population R2R^2 of h(H)h(H) for a target T satisfies R2≤ρI2:=1−[Var(T∣I)]Var(T)=1−MMSE(T∣I)Var(T).R^2≤ _I^2:=1- E[Var(T I)]Var(T)=1- MMSE(T I)Var(T). Naturally, ρI2≤1 _I^2≤ 1. Proof. Because H=Φ(I)H= (I), the prediction h(H)h(H) cannot use information beyond I. The minimum mean-squared error attainable from I is achieved by [T∣I]E[T I]. Therefore, MSE(h(H),T)≥MMSE(T∣I),MSE(h(H),T) (T I), and substitution into the definition of population R2R^2 gives the result. ∎ Proposition E.2 (Floor and headroom). Define ρO2:=1−[Var(T∣O)]Var(T)=1−MMSE(T∣O)Var(T). _O^2:=1- E[Var(T O)]Var(T)=1- MMSE(T O)Var(T). This is the population R2R^2 attained by the optimal predictor using only the reported sampling variances O and the prior on τ2τ^2. In particular, it performs no inference about heterogeneity from the reported effects X. Every O-measurable predictor satisfies R2≤ρO2.R^2≤ _O^2. For any predictor based on the complete observations, R2≤ρO2+Δ,Δ:=ρI2−ρO2≥0.R^2≤ _O^2+ , := _I^2- _O^2≥ 0. Since O⊆IO I, conditioning on I cannot increase the Bayes MMSE, so ρO2≤ρI2≤1 _O^2≤ _I^2≤ 1 and therefore Δ≥0 ≥ 0. Proof. The conditional expectation [T∣O]E[T O] minimizes MSE among all O-measurable predictors, giving the first inequality. The second follows from ρO2+Δ=ρI2 _O^2+ = _I^2 and Proposition E.1. ∎ Remark E.3 (Evidential meaning of the floor and headroom). The quantity ρO2 _O^2 is called a floor because it is a floor for evidential interpretation, not because every probe must attain it. A probe must exceed this threshold before its score can provide evidence of information derived from X rather than O alone. The quantity Δ is the maximum amount by which a predictor can outperform the optimal predictor that ignores the information about τ2τ^2 contained in X. Consequently, only max0,R2−ρO2, \0,R^2- _O^2\, which is at most Δ , can provide evidence that the representation encodes heterogeneity inferred from the reported effects. Since T is measurable with respect to (O,τ2)(O,τ^2), the only relevance of X to T given O is through what X reveals about τ2τ^2. Proposition E.4 (Regime dependence of the headroom). When sampling variance dominates, TkT_k is almost entirely predictable from the observed variances alone, whereas when heterogeneity dominates, predicting TkT_k increasingly requires information about the latent τ2τ^2. Let wi:=1σi2+τ2,S1(k):=∑i≤kwi,Tk:=logS1(k).w_i:= 1 _i^2+τ^2, S_1^(k):= _i≤ kw_i, T_k:= S_1^(k). (i) Sampling-variance-dominated regime (σi2≫τ2 _i^2 τ^2). For every τ2≥0τ^2≥ 0, |∂Tk∂τ2|≤(miniσi2)−1. | ∂ T_k∂τ^2 |≤( _i _i^2)^-1. It follows that [Var(Tk∣O)]≤τmax212[(miniσi2)−2].E[Var(T_k O)]≤ _ ^212E\! [( _i _i^2)^-2 ]. Therefore, along any sequence of regimes for which τmax2[(miniσi2)−2]Var(Tk)→0, _ ^2E[( _i _i^2)^-2]Var(T_k)→ 0, we have ρO2→1,Δ→0,ρI2→1. _O^2→ 1, → 0, _I^2→ 1. (i) Heterogeneity-dominated regime (τ2≫σi2τ^2 _i^2). At the final position K, Kτ2+maxiσi2≤S1(K)≤Kτ2. Kτ^2+ _i _i^2≤ S_1^(K)≤ Kτ^2. Hence, |TK−(logK−logτ2)|≤log(1+maxiσi2τ2). |T_K-( K- τ^2) |≤ \! (1+ _i _i^2τ^2 ). Consider a sequence of regimes in which maxiσi2τ2→0 _i _i^2τ^2→ 0 in probability, and suppose that the squared upper bounds log2(1+maxiσi2τ2) \ ^2\! (1+ _i _i^2τ^2 ) \ are uniformly integrable. Then TK−(logK−logτ2)→0T_K-( K- τ^2)→ 0 in L2L^2. If Var(TK)Var(T_K) remains bounded away from zero, it follows that ρO2→0. _O^2→ 0. Thus, in this limit, nearly all predictable variance in TKT_K requires information about the latent heterogeneity. Proof. For part (i), by definition, Tk=log∑i≤kwi,wi=1σi2+τ2.T_k= _i≤ kw_i, w_i= 1 _i^2+τ^2. Therefore, |∂Tk∂τ2|=|∂log∑i≤kwi∂τ2|. | ∂ T_k∂τ^2 |= | ∂ _i≤ kw_i∂τ^2 |. Since ∂wi∂τ2=−wi2and∂logx∂x=1x, ∂ w_i∂τ^2=-w_i^2 ∂ x∂ x= 1x, we have |∂Tk∂τ2| | ∂ T_k∂τ^2 | =∑i≤kwi2∑i≤kwi = _i≤ kw_i^2 _i≤ kw_i =∑i≤kwi∑j≤kwjwi = _i≤ k w_i _j≤ kw_jw_i ≤maxiwi ≤ _iw_i ≤(miniσi2)−1. ≤( _i _i^2)^-1. The second expression is a weighted average of the wiw_i, and therefore cannot exceed their maximum. Moreover, since τ2≥0τ^2≥ 0, wi=1σi2+τ2≤1σi2.w_i= 1 _i^2+τ^2≤ 1 _i^2. Once O is fixed, the map τ2↦Tkτ^2 T_k is therefore Lipschitz with constant L(O)=(miniσi2)−1.L(O)=( _i _i^2)^-1. For a function f and random variable V, if |f′(V)|≤L|f (V)|≤ L, then Var(f(V))≤L2Var(V).Var(f(V))≤ L^2Var(V). Therefore, Var(Tk∣O)≤(miniσi2)−2Var(τ2∣O).Var(T_k O)≤( _i _i^2)^-2Var(τ^2 O). Since τ2τ^2 is independent of O and τ2∼U(0,τmax),τ^2 (0, _ ), we have Var(τ2∣O)=Var(τ2)=τmax212.Var(τ^2 O)=Var(τ^2)= _ ^212. Hence, Var(Tk∣O)≤τmax212(miniσi2)−2.Var(T_k O)≤ _ ^212( _i _i^2)^-2. Taking expectations gives [Var(Tk∣O)]≤τmax212[(miniσi2)−2].E[Var(T_k O)]≤ _ ^212E\! [( _i _i^2)^-2 ]. By definition, ρO2=1−[Var(Tk∣O)]Var(Tk). _O^2=1- E[Var(T_k O)]Var(T_k). Therefore, 0≤1−ρO2≤112τmax2[(miniσi2)−2]Var(Tk).0≤ 1- _O^2≤ 112 _ ^2E[( _i _i^2)^-2]Var(T_k). Under the stated condition, τmax2[(miniσi2)−2]Var(Tk)→0, _ ^2E[( _i _i^2)^-2]Var(T_k)→ 0, so ρO2→1. _O^2→ 1. Since 0≤Δ≤1−ρO20≤ ≤ 1- _O^2 and ρO2≤ρI2≤1, _O^2≤ _I^2≤ 1, it follows that Δ→0,ρI2→1. → 0, _I^2→ 1. For part (i), at the final position, S1(K)=∑i=1K1σi2+τ2.S_1^(K)= _i=1^K 1 _i^2+τ^2. For every i, τ2≤σi2+τ2≤τ2+maxjσj2.τ^2≤ _i^2+τ^2≤τ^2+ _j _j^2. Taking reciprocals, 1τ2+maxjσj2≤1σi2+τ2≤1τ2. 1τ^2+ _j _j^2≤ 1 _i^2+τ^2≤ 1τ^2. Summing over i gives Kτ2+maxiσi2≤S1(K)≤Kτ2. Kτ^2+ _i _i^2≤ S_1^(K)≤ Kτ^2. Since log is increasing and TK=logS1(K)T_K= S_1^(K), logK−log(τ2+maxiσi2)≤TK≤logK−logτ2. K- (τ^2+ _i _i^2)≤ T_K≤ K- τ^2. Now, log(τ2+maxiσi2)=logτ2+log(1+maxiσi2τ2), (τ^2+ _i _i^2)= τ^2+ \! (1+ _i _i^2τ^2 ), and therefore logK−logτ2−log(1+maxiσi2τ2)≤TK≤logK−logτ2. K- τ^2- \! (1+ _i _i^2τ^2 )≤ T_K≤ K- τ^2. Hence, |TK−(logK−logτ2)|≤log(1+maxiσi2τ2). |T_K-( K- τ^2) |≤ \! (1+ _i _i^2τ^2 ). If maxiσi2τ2→0 _i _i^2τ^2→ 0 in probability, then, by continuity of log(1+x) (1+x), log(1+maxiσi2τ2)→0 \! (1+ _i _i^2τ^2 )→ 0 in probability. The preceding bound therefore gives TK−(logK−logτ2)→0T_K-( K- τ^2)→ 0 in probability. Convergence in probability alone does not guarantee convergence of the mean squared error. The additional uniform-integrability assumption on log2(1+maxiσi2τ2) \ ^2\! (1+ _i _i^2τ^2 ) \ rules out rare extreme values that would contribute a non-vanishing amount to the expected squared error. Since (TK−(logK−logτ2))2≤log2(1+maxiσi2τ2), (T_K-( K- τ^2) )^2≤ ^2\! (1+ _i _i^2τ^2 ), the two conditions together imply [(TK−(logK−logτ2))2]→0.E\! [ (T_K-( K- τ^2) )^2 ]→ 0. Equivalently, TK−(logK−logτ2)→0T_K-( K- τ^2)→ 0 in L2L^2. Let g(τ2):=logK−logτ2.g(τ^2):= K- τ^2. Then TK−g(τ2)→0T_K-g(τ^2)→ 0 in L2L^2. Since τ2τ^2 is independent of O, [g(τ2)∣O]=[g(τ2)].E[g(τ^2) O]=E[g(τ^2)]. Therefore, Var([TK∣O])=Var([TK−g(τ2)∣O]).Var(E[T_K O])=Var\! (E[T_K-g(τ^2) O] ). By conditional Jensen’s inequality, Var([TK∣O])≤[(TK−g(τ2))2]→0.Var(E[T_K O]) \! [(T_K-g(τ^2))^2 ]→ 0. Finally, by the law of total variance and Proposition E.2, ρO2=1−[Var(TK∣O)]Var(TK)=Var([TK∣O])Var(TK). _O^2=1- E[Var(T_K O)]Var(T_K)= Var(E[T_K O])Var(T_K). If Var(TK)Var(T_K) remains bounded away from zero, it follows that ρO2→0. _O^2→ 0. Thus, in this limit, nearly all predictable variance in TKT_K requires information about the latent heterogeneity. ∎ Corollary E.5 (Distribution shifts can produce a probe–performance dissociation). Under a distribution shift from a regime with substantial heterogeneity-dependent headroom to a sampling-variance-dominated regime, ρO2→1,Δ→0. _O^2→ 1, → 0. Consequently, probe R2R^2 for T can rise because O alone becomes more predictive, while the maximum amount by which any predictor can benefit from information about τ2τ^2 contained in X shrinks to zero. Whether a particular network’s probe score rises, and whether its prediction error increases, must be determined experimentally. Proposition E.6 (Ceilings for τ2τ^2). Any probe for τ2τ^2 satisfies R2≤ρI2(τ2):=1−MMSE(τ2∣I)Var(τ2).R^2≤ _I^2(τ^2):=1- MMSE(τ^2 I)Var(τ^2). Proof. This is Proposition E.1 applied with target τ2τ^2.∎ Remark E.7 (Why identification of τ2τ^2 deteriorates under the shift). If μ were known, one study would contribute Fisher information 12(σi2+τ2)2 12( _i^2+τ^2)^2 about τ2τ^2. Thus, scaling the total variance σi2+τ2 _i^2+τ^2 by a factor c would reduce this per-study information by a factor of c2c^2. Since typical total variance is substantially larger under the OOD regime, this scaling provides an intuitive explanation for the reduced identifiability of τ2τ^2. Appendix F The general headroom identity The task-specific argument is an instance of a more general principle. Let Z be a latent variable, let T=f(O,Z)T=f(O,Z) be a square-integrable target, let X denote additional evidence that may reveal Z, and let I:=(O,X)I:=(O,X). Define ρO2:=1−[Var(T∣O)]Var(T),ρI2:=1−[Var(T∣I)]Var(T),Δ:=ρI2−ρO2, _O^2:=1- E[Var(T O)]Var(T), _I^2:=1- E[Var(T I)]Var(T), := _I^2- _O^2, and define the Bayes update (change in the Bayesian prediction of T after seeing the additional evidence X) B(I,O):=[T∣I]−[T∣O].B(I,O):=E[T I]-E[T O]. Proposition F.1 (Headroom identity). The headroom satisfies ΔVar(T)=[Var(B(I,O)∣O)]. \,Var(T)=E\! [Var\! (B(I,O) O ) ]. Thus, the headroom is the conditional variance of the Bayes update about T supplied by X beyond the information already available in O. Proof. By definition, ΔVar(T)=[Var(T∣O)]−[Var(T∣I)]. \,Var(T)=E[Var(T O)]-E[Var(T I)]. Conditional total variance gives Var(T∣O)=[Var(T∣I)∣O]+Var([T∣I]∣O).Var(T O)=E[Var(T I) O]+Var(E[T I] O). Taking expectations yields [Var(T∣O)]=[Var(T∣I)]+[Var([T∣I]∣O)].E[Var(T O)]=E[Var(T I)]+E\! [Var\! (E[T I] O ) ]. Hence, ΔVar(T)=[Var([T∣I]∣O)]. \,Var(T)=E\! [Var\! (E[T I] O ) ]. Since [T∣O]E[T O] is constant conditional on O, subtracting it does not change the conditional variance, giving the result. ∎ Corollary F.2 (Two mechanisms for vanishing headroom). For each fixed O, suppose that z↦f(O,z)z f(O,z) is Lipschitz continuous with Lipschitz constant L(O)L(O), |f(O,z)−f(O,z′)|≤L(O)|z−z′|,|f(O,z)-f(O,z )|≤ L(O)|z-z |, and has oscillation bounded by M, osczf(O,⋅):=supzf(O,z)−infzf(O,z)≤M.osc_zf(O,·):= _zf(O,z)- _zf(O,z)≤ M. Then the headroom can vanish through either of two mechanisms: Δ≤[L(O)2Var(Z∣O)]Var(T)weak target sensitivity ≤ E[L(O)^2Var(Z O)]Var(T) weak target sensitivity when the target T=f(O,Z)T=f(O,Z) becomes insensitive to the latent Z, and Δ≤M2MI(X;Z∣O)2Var(T)weak identification, ≤ M^2\,MI(X;Z O)2Var(T) weak identification, when the additional evidence X contains little information about Z beyond O, where MIMI denotes conditional mutual information in nats. The first inequality describes weak target sensitivity, in which the target becomes insensitive to the latent variable. The second describes weak identification, in which the additional evidence becomes uninformative about the latent variable given O. Proof. (i) Weak target sensitivity: By the conditional law of total variance (proof of Proposition F.1), Var([T∣I]∣O)≤Var(T∣O),Var(E[T I] O) (T O), since the variance of the conditional mean cannot exceed the variance of T. Since T=f(O,Z)T=f(O,Z), holding O fixed gives Var(T∣O)=Var(f(O,Z)∣O)Var(T O)=Var(f(O,Z) O). By the Lipschitz assumption, Var(f(O,Z)∣O)≤L(O)2Var(Z∣O).Var(f(O,Z) O)≤ L(O)^2Var(Z O). Therefore, Var([T∣I]∣O)≤L(O)2Var(Z∣O).Var(E[T I] O)≤ L(O)^2Var(Z O). Taking expectations and applying Proposition F.1 gives the first inequality. Δ≤[L(O)2Var(Z∣O)]Var(T). ≤ E[L(O)^2Var(Z O)]Var(T). Intuitively, if T barely changes when Z changes, learning Z cannot help very much. (i) Weak identification: Let πZ|I _Z I and πZ|O _Z O denote the posterior beliefs of Z given I and O. The oscillation bound gives |B(I,O)|≤MTV(πZ|I,πZ|O).|B(I,O)|≤ M\,TV( _Z I, _Z O). Therefore, [Var(B(I,O)∣O)]≤[B(I,O)2]≤M2[TV2(πZ|I,πZ|O)].E[Var(B(I,O) O)] [B(I,O)^2]≤ M^2E[TV^2( _Z I, _Z O)]. Pinsker’s inequality gives TV2(πZ|I,πZ|O)≤12KL(πZ|I∥πZ|O),TV^2( _Z I, _Z O)≤ 12KL( _Z I\| _Z O), whose expectation is MI(X;Z∣O)MI(X;Z O). Applying Proposition F.1 completes the proof. ∎ Appendix G Synthetic single-cell generative modeling To test whether the estimation procedure generalizes beyond the controlled meta-analysis setting, we construct a second completely defined generative model based on single-cell RNA sequencing. Because the data-generating process is known, the reference points can be computed exactly and compared with their estimates (Appendix H). Generative model. Each example is one gene g observed in K=40K=40 cells, indexed by i=1,…,Ki=1,…,K. The draws are Draw Meaning logλg∼(μλ,σλ2) _g ( _λ, _λ^2) mean expression bg∼U(0,bmax)b_g (0,b_ ) overdispersion (latent Z) logdgi∼(μd,σd2) d_gi ( _d, _d^2) sequencing depth of cell i θgi∼Gamma(shape=λg/bg,scale=bg) _gi (shape= _g/b_g,\ scale=b_g ) true expression rate ygi|θgi,dgi∼Poisson(dgiθgi)y_gi _gi,d_gi (d_gi _gi) observed count with μλ=−4.5 _λ=-4.5, σλ=1 _λ=1, bmax=0.05b_ =0.05, μd=6.9 _d=6.9 and σd=0.7 _d=0.7. The draws of λg _g, bgb_g and dg,1:Kd_g,1:K are mutually independent, and the depths are independent across genes and cells. (In the simulation the lower end of the prior on bgb_g is 10−610^-6 rather than 00, which keeps the Gamma shape finite.) Each seed contains 15001500 genes, and all results use five seeds. At these values the median depth is e6.9≈990e^6.9≈ 990 and the median expected count dgiλgd_gi _g is e2.4≈11e^2.4≈ 11. The Gamma draw gives [θgi∣λg,bg]=λgE[ _gi _g,b_g]= _g and Var(θgi∣λg,bg)=λgbgVar( _gi _g,b_g)= _gb_g, so true expression rate θgi _gi differs from mean expression λg _g because of biological cell-to-cell variation (bgb_g), while observed count ygiy_gi adds technical counting noise from the finite depth dgid_gi. Averaging the Poisson likelihood over θgi _gi gives a negative-binomial count, ygi|λg,bg,dgi∼NB(mean=dgiλg,size=λgbg),y_gi _g,b_g,d_gi \! (mean=d_gi _g,\ size= _gb_g ), with Var(ygi∣λg,bg,dgi)=dgiλg+bgdgi2λgVar(y_gi _g,b_g,d_gi)=d_gi _g+b_gd_gi^2 _g. The normalized count rgi=ygi/dgir_gi=y_gi/d_gi therefore has Var(rgi∣λg,bg,dgi)=λg(1dgi+bg),Var(r_gi _g,b_g,d_gi)= _g ( 1d_gi+b_g ), the sum of a technical component 1/dgi1/d_gi and a biological component bgb_g. This is the variance structure of the meta-analysis task, σi2↔1dgi,τ2↔bg. _i^2 1d_gi, τ^2 b_g. Target and information sets. For known biological overdispersion bgb_g, cell i has precision wgi=(dgi−1+bg)−1w_gi=(d_gi^-1+b_g)^-1. The probe target is Tg=log∑i=1K1dgi−1+bg,T_g= _i=1^K 1d_gi^-1+b_g, the analogue of logS1 S_1 with dgi−1d_gi^-1 in place of σi2 _i^2 and bgb_g in place of τ2τ^2. The restricted information is the depths alone, Og=dg,1:KO_g=d_g,1:K; the additional evidence is the observed counts, Xg=yg,1:KX_g=y_g,1:K; and the full information is Ig=(yg,1:K,dg,1:K)I_g=(y_g,1:K,d_g,1:K). Because bgb_g is independent of dg,1:Kd_g,1:K, the floor ρO2 _O^2 comes only from how TgT_g varies with the depths, averaged over the prior on bgb_g; inferring bgb_g requires the counts. The floor ρO2ρ^2_O and ceiling ρI2ρ^2_I are the population reference points of Section 4.1 (Propositions E.1 and E.2). Binomial thinning. Retaining a fraction p∈(0,1]p∈(0,1] of the counts gives ygi′∼Bin(ygi,p)y _gi (y_gi,p) and dgi′=pdgid _gi=p\,d_gi. Because binomial thinning of Poisson(dgiθgi)Poisson(d_gi _gi) is Poisson(pdgiθgi)Poisson(p\,d_gi _gi), the thinned data follow the same model with depths pdgip\,d_gi (so logdgi′∼(μd+logp,σd2) d _gi ( _d+ p, _d^2)). G.1 Separating the two mechanisms of vanishing headroom (Figure 6). Sequencing depth affects headroom in two ways at once: shallower data give fewer counts from which to infer the dispersion bgb_g (weaker identification), and also make technical noise 1/dgi1/d_gi larger relative to bgb_g in the target, so that TgT_g depends less on bgb_g (weaker sensitivity). In real data a single depth p moves both. In the synthetic model we vary them separately with two depth fractions: • the evidence fraction pXp_X thins only the counts the ceiling sees: the count-based features of ϕI _I are computed from thinned counts ygi′∼Bin(ygi,pX)y _gi (y_gi,p_X) with depths pXdgip_Xd_gi; • the target fraction pTp_T scales only the depths in the target, Tg=log∑i((pTdgi)−1+bg)−1T_g= _i ((p_Td_gi)^-1+b_g )^-1. The floor features ϕO _O always use the full depths dg,1:Kd_g,1:K; since pTp_T is a known constant, they carry the same information as pTdg,1:Kp_Td_g,1:K. Both fractions run on a log grid, pX∈1,…,10−2p_X∈\1,…,10^-2\ (4 values) and pT∈1,…,10−4p_T∈\1,…,10^-4\ (7 values), in steps of 10−2/310^-2/3. For scale, at full depth the median technical variance 1/dgi≈0.0011/d_gi≈ 0.001 is small next to bg≤0.05b_g≤ 0.05, whereas at p=0.01p=0.01 it is ≈0.1≈ 0.1 and dominates bgb_g. Each grid value is the mean over five seeds of the estimated headroom ρ^I2−ρ^O2 ρ^2_I- ρ^2_O, both fitted by the decoder bank on the grid’s own 50/20/30 split of the genes. At full depth this gives 0.87, against 0.89 in Table 13, which uses a different split, and the exact 0.880 (Table 3). G.2 Real data. The four real 10x cohorts (10x Genomics, 2021; 10x Genomics, 2024; 10x Genomics, 2022) are cast in the same form. We keep the 2000 most frequently detected genes among those detected in at least 50 cells. The sequencing depth of gene g in cell i is dgi=ℓifgd_gi= _i\,f_g, where ℓi _i is the library size of cell i and fgf_g the fraction of all counts that belong to gene g. The cells are split in half at random: biological overdispersion parameter bgb_g is replaced by its method-of-moments estimate b^MoM b_MoM (Appendix I) from the first half, and the target, OgO_g, IgI_g, the features and the scGPT input use the second half, so the ceiling cannot recover its own target estimate. K is the number of cells in the second half. The features are those of the synthetic model, with bmaxb_ set to 1.11.1 times the largest b^MoM b_MoM among training genes. Appendix H Exact reference points and validation of the estimation procedure Figure 7: The estimation procedure recovers exact reference points. Each marker compares an estimated reference point (decoder bank; vertical axis) with its exact value (horizontal axis), for the meta-analysis (circles) and the synthetic single-cell model (diamonds). Points lie close to the identity line (dotted) across regimes and reference-point types. Slightly negative estimated floors are held-out R2R^2 values where the exact floor is near zero, not negative population floors. We test the floor–ceiling construction in two independently specified generative models: the random-effects meta-analysis model and a synthetic single-cell RNA-sequencing model. In both cases the latent distribution and observation model are known, so the reference points can be computed exactly, independently of the estimation procedure. Population reference points. Let T be the probe target, let O denote the restricted information, and let I=(O,X)I=(O,X) denote the full information after observing the additional evidence X. Under squared-error loss, the optimal predictors measurable with respect to these information sets are mO(O)=[T∣O],mI(I)=[T∣I].m_O(O)=E[T O], m_I(I)=E[T I]. The corresponding population reference points are ρO2=1−[Var(T∣O)]Var(T)=R2(T,mO(O)), _O^2=1- E[Var(T O)]Var(T)=R^2\! (T,m_O(O) ), and ρI2=1−[Var(T∣I)]Var(T)=R2(T,mI(I)). _I^2=1- E[Var(T I)]Var(T)=R^2\! (T,m_I(I) ). Their difference, Δ=ρI2−ρO2, = _I^2- _O^2, is the headroom supplied by X beyond O. These definitions follow directly from the L2L^2 projection property of conditional expectation. Because O⊆IO I, [Var(T∣I)]≤[Var(T∣O)],E[Var(T I)] [Var(T O)], and hence 0≤ρO2≤ρI2≤1,Δ≥0.0≤ _O^2≤ _I^2≤ 1, ≥ 0. Moreover, for any representation H=Φ(I)H= (I) and any probe h, MSE(T,h(H))≥MMSE(T∣I),MSE\! (T,h(H) ) (T I), which implies R2(T,h(H))≤ρI2.R^2\! (T,h(H) )≤ _I^2. Thus the population ceiling ρI2ρ^2_I bounds every representation constructed from I, while ρO2 _O^2 is the amount of predictability already attainable without using the additional evidence X. Exact meta-analysis reference points. For the random-effects model, xi|μ,τ2,σi2∼(μ,σi2+τ2),μ∼(0,1),τ2∼U(0,τmax),x_i μ,τ^2, _i^2 \! (μ, _i^2+τ^2 ), μ (0,1), τ^2 (0, _ ), with τmax=0.5 _ =0.5 in distribution and τmax=1.5 _ =1.5 out of distribution (Appendix A), we define O=σ1:K2,I=(σ1:K2,x1:K),O= _1:K^2, I=( _1:K^2,x_1:K), and T=logS1=log∑i=1K1σi2+τ2.T= S_1= _i=1^K 1 _i^2+τ^2. Because τ2τ^2 is independent of O, the exact restricted-information predictor is mO(O)=∫log(∑i=1K1σi2+τ2)p(τ2)dτ2.m_O(O)= \! ( _i=1^K 1 _i^2+τ^2 )p(τ^2)\,dτ^2. To obtain the exact full-information predictor, we marginalize the nuisance mean μ. For a candidate τ2τ^2, define νi=σi2+τ2,A1=∑iνi−1,Ax=∑ixiνi−1,Axx=∑ixi2νi−1. _i= _i^2+τ^2, A_1= _i _i^-1, A_x= _ix_i _i^-1, A_x= _ix_i^2 _i^-1. The marginal log likelihood is logp(x1:K∣τ2,O)=−12[∑ilogνi+log(1+A1)+Axx−Ax21+A1]+const. p(x_1:K τ^2,O)=- 12 [ _i _i+ (1+A_1)+A_x- A_x^21+A_1 ]+const. Therefore, p(τ2∣I)∝p(x1:K∣τ2,O)p(τ2),p(τ^2 I) p(x_1:K τ^2,O)p(τ^2), and mI(I)=∫log(∑i=1K1σi2+τ2)p(τ2∣I)dτ2.m_I(I)= \! ( _i=1^K 1 _i^2+τ^2 )p(τ^2 I)\,dτ^2. The one-dimensional prior and posterior integrals are evaluated by numerical quadrature over τ2τ^2. The exact floor and ceiling are then obtained by evaluating mOm_O and mIm_I on independent held-out meta-analyses in both the in-distribution and out-of-distribution regimes. Exact reference points of the synthetic single-cell model. The model, target TgT_g and information sets Og=dg,1:KO_g=d_g,1:K and Ig=(yg,1:K,dg,1:K)I_g=(y_g,1:K,d_g,1:K) are defined in Appendix G. Write T(dg,b)=log∑i=1K(dgi−1+b)−1T(d_g,b)= _i=1^K(d_gi^-1+b)^-1. Because bgb_g is independent of the depth vector dgd_g, p(bg∣Og)=p(bg),p(b_g O_g)=p(b_g), and the exact floor predictor is mO(dg)=∫T(dg,b)p(b)b.m_O(d_g)= T(d_g,b)\,p(b)\,db. Its conditional residual variance is vO(dg)=∫T(dg,b)2p(b)b−mO(dg)2.v_O(d_g)= T(d_g,b)^2p(b)\,db-m_O(d_g)^2. With the negative-binomial likelihood of the counts, ygi|λg,bg,dgi∼NB(mean=dgiλg,size=λg/bg)y_gi _g,b_g,d_gi (mean=d_gi _g,\ size= _g/b_g) (Appendix G), independent across cells, p(bg∣yg,dg)∝p(bg)∫p(yg∣λ,bg,dg)p(λ)λ.p(b_g y_g,d_g) p(b_g) p(y_g λ,b_g,d_g)p(λ)\,dλ. The exact full-information predictor is mI(yg,dg)=∫T(dg,b)p(b∣yg,dg)b,m_I(y_g,d_g)= T(d_g,b)\,p(b y_g,d_g)\,db, with conditional residual variance vI(yg,dg)=Var(Tg∣yg,dg).v_I(y_g,d_g)=Var(T_g y_g,d_g). The exact reference points of the synthetic single-cell model are therefore ρO2=1−[vO(O)]Var(T),ρI2=1−[vI(I)]Var(T),Δ=ρI2−ρO2. _O^2=1- E[v_O(O)]Var(T), _I^2=1- E[v_I(I)]Var(T), = _I^2- _O^2. The integrals over (λ,b)(λ,b) are evaluated on a fixed numerical grid: 150 cells in b on (0,bmax)(0,b_ ), with edges bmaxu2b_ u^2 for u equally spaced on [0,1][0,1] so that the grid is dense near b=0b=0, and 50 points in logλ λ, equally spaced on μλ±3.5σλ _λ± 3.5\, _λ. Under binomial thinning (Appendix G) the same computation is applied to the thinned (y′,d′)(y ,d ). The numerator and denominator are computed using the same quadrature rule, ensuring the total-variance identity Var(T)=[Var(T∣O)]+Var([T∣O])Var(T)=E[Var(T O)]+Var(E[T O]) up to numerical precision. Validation against exact values. We independently estimate the same reference points from observed samples without giving the estimator access to the true latent variable. Each seed is divided into disjoint training, model-selection, and evaluation partitions. Candidate regressors from the decoder bank are fitted on the training partition, the regressor class is fixed using the selection partition, and the selected class is refitted on the combined training and selection data before evaluation. This produces ρ^O2=R2(T,m^O(O)),ρ^I2=R2(T,m^I(I)),Δ^=ρ^I2−ρ^O2. ρ_O^2=R^2\! (T, m_O(O) ), ρ_I^2=R^2\! (T, m_I(I) ), = ρ_I^2- ρ_O^2. The validation compares (ρO2,ρ^O2),(ρI2,ρ^I2),(Δ,Δ^)( _O^2, ρ_O^2), ( _I^2, ρ_I^2), ( , ) for every seed, for both meta-analysis regimes, and for six thinning fractions p∈1, 0.40, 0.16, 0.063, 0.025, 0.01p∈\1,\,0.40,\,0.16,\,0.063,\,0.025,\,0.01\ of the synthetic single-cell model (Appendix G). Figure 7 shows the comparison: floors, ceilings and headroom estimated by the estimation procedure lie close to their exact values in both models, and Table 3 lists the exact reference points. Floor Ceiling Headroom Domain Target Condition Meta-analysis logS1 S_1 ID 0.012±0.0040.012± 0.004 0.822±0.0140.822± 0.014 0.809±0.0170.809± 0.017 OOD 0.092±0.0110.092± 0.011 0.466±0.0190.466± 0.019 0.374±0.0170.374± 0.017 τ2τ^2 ID 0.000±0.0000.000± 0.000 0.758±0.0130.758± 0.013 0.758±0.0130.758± 0.013 OOD 0.000±0.0000.000± 0.000 0.404±0.0170.404± 0.017 0.404±0.0170.404± 0.017 Synthetic genomics TgT_g Full depth 0.000±0.0000.000± 0.000 0.880±0.0010.880± 0.001 0.880±0.0010.880± 0.001 Table 3: Exact reference points for the meta-analysis and synthetic-genomics validation experiments (mean ± sample standard deviation across five seeds). Appendix I Feature engineering for estimated reference points The exact reference points (Appendix H) condition on the raw information sets O and I. The estimated reference points are instead fitted by the decoder bank (Appendix B) on fixed-length feature vectors ϕO(O) _O(O) and ϕI(I) _I(I): ρ^O2=R2(T,m^O(ϕO(O))),ρ^I2=R2(T,m^I(ϕI(I))), ρ_O^2=R^2\! (T, m_O( _O(O)) ), ρ_I^2=R^2\! (T, m_I( _I(I)) ), where T is the probe target and m m the fitted regressor on feature vectors. Each feature vector is a deterministic function of its own information set only. It never uses the latent variable (τ2τ^2 or b) or the target, and ϕI _I always contains ϕO _O. The features are fixed in advance and are identical across seeds, conditions and sequencing depths. Notation. For a vector a1:Ka_1:K, a¯=K−1∑iai a=K^-1 _ia_i; sd(a)=(K−1∑i(ai−a¯)2)1/2sd(a)= (K^-1 _i(a_i- a)^2 )^1/2; s2(a)=(K−1)−1∑i(ai−a¯)2s^2(a)=(K-1)^-1 _i(a_i- a)^2; a(1)≤⋯≤a(K)a_(1)≤…≤ a_(K) are the order statistics; qα(a)q_α(a) is the empirical α-quantile with linear interpolation; med(a)=q0.5(a)med(a)=q_0.5(a). We write ε=10−12 =10^-12 and slog(a)=logmax(a,ε)slog(a)= (a, ). Meta-analysis. Features are computed from all K=32K=32 studies of a meta-analysis (the final position). Let ℓi=logσi2 _i= _i^2 and wi=1/σi2w_i=1/ _i^2. Restricted features ϕO _O (43 columns), from O=σ21:KO=σ^2_1:K: 1. sorted log sampling variances ℓ(1),…,ℓ(K) _(1),…, _(K) (32); 2. mean(ℓ)mean( ), sd(ℓ)sd( ), miniℓ _i , maxiℓ _i (4); 3. quantiles qα(ℓ)q_α( ) for α∈0.10,0.25,0.50,0.75,0.90α∈\0.10,0.25,0.50,0.75,0.90\ (5); 4. the log total and log squared total fixed-effect precision, log∑iwi _iw_i and log∑iwi2 _iw_i^2 (2). Full features ϕI _I (114 columns), from I=(σ1:K2,x1:K)I=(σ^2_1:K,x_1:K): 1. ϕO _O (43); 2. the effects x1,…,xKx_1,…,x_K in study order (32); 3. ℓ1,…,ℓK _1,…, _K in study order, so that each xix_i is paired with its own σi2 _i^2 (32); 4. meanxmeanx, sd(x)sd(x), minix _ix, maxix _ix (4); 5. the fixed-effect mean μ^FE=∑iwixi/∑iwi μ_FE= _iw_ix_i/ _iw_i, Cochran’s Q=∑iwi(xi−μ^FE)2Q= _iw_i(x_i- μ_FE)^2, and the DerSimonian–Laird estimate τ^DL2=max0,Q−(K−1)∑iwi−∑iwi2/∑iwi(3). τ^2_DL= \! \0,\ Q-(K-1) _iw_i- _iw_i^2/ _iw_i \ (3). No information is lost. T=logS1T= S_1 is symmetric in the studies and τ2τ^2 is independent of O, so [T∣O]E[T O] depends on σ21:Kσ^2_1:K only through the sorted vector in item 1, and [T∣O]=[T∣ϕO(O)]E[T O]=E[T _O(O)]. Likewise, I is recovered exactly from items 2–3 of ϕI _I. The remaining columns are deterministic summaries that ease fitting. The estimated and exact reference points therefore target the same population quantities. Synthetic single-cell model. Features are computed separately for each gene using its K=40K=40 cells (Appendix G). They are constructed from the observed counts y and sequencing depths d after binomial thinning (Appendix G). Define the normalized count ri=yidir_i= y_id_i, the moment estimate of mean expression λ^=mean(r) λ=mean(r), and the sample variance s2=s2(r).s^2=s^2(r). The restricted feature vector ϕO _O has 8 columns and uses only O=d1:KO=d_1:K. It contains: 1. the number of cells K (1); 2. mean(logd)mean( d), sd(logd)sd( d), minilogd _i d, maxilogd _i d, and med(logd)med( d) (5); 3. the log total and log squared total technical variance, slog∑idi−1slog _id_i^-1 and slog∑idi−2slog _id_i^-2 (2). The full feature vector ϕI _I has 27 columns and uses I=(y1:K,d1:K)I=(y_1:K,d_1:K). We first define three quantities that summarize biological variation beyond Poisson counting noise: (1) the expected Poisson variance, v=λ^1K∑idi−1v= λ\, 1K _id_i^-1, where λ λ is the estimated mean expression, K is the number of cells, and did_i is the sequencing depth of cell i; (2) the excess variance, e=max(0,s2−v)e= (0,s^2-v), where s2s^2 is the sample variance of the normalized counts ri=yi/dir_i=y_i/d_i; and (3) the method-of-moments estimate of biological dispersion, b^MoM=e/max(λ^,ε) b_MoM=e/ ( λ, ), where ε is a small constant used to avoid division by zero. These definitions follow from Var(ri)=λ/di+λbVar(r_i)=λ/d_i+λ b, where λ/diλ/d_i is the variance expected from Poisson counting noise and λbλ b is the additional variance due to biological dispersion b. We also define the depth-weighted mean r¯d=∑iyi/∑idi r_d= _iy_i/ _id_i, and the weighted dispersion statistic Q=∑idi(ri−r¯d)2Q= _id_i(r_i- r_d)^2. The 27 features in ϕI _I are: 1. the 8 restricted features in ϕO _O (8); 2. two sparsity features: the detection rate K−1∑i[yi>0]K^-1 _i1[y_i>0], which is the fraction of cells with a nonzero count, and log(1+maxiyi) (1+ _iy_i), which summarizes the largest observed count (2); 3. three variance features: log(s2+ε) (s^2+ ) for the total observed variance of the normalized counts, log(v+ε) (v+ ) for the variance expected from Poisson counting noise, and log(e+ε) (e+ ) for the excess variance beyond that noise (3); 4. two representations of the moment estimate of biological dispersion b: log(b^MoM+ε) ( b_MoM+ ) and b^MoM b_MoM (2); 5. four overdispersion features: Q; log(1+maxQ−(K−1),0) \! (1+ \Q-(K-1),0\ ), which records the amount by which Q exceeds the reference value K−1K-1; the dispersion ratio s2/max(v,ε)s^2/ (v, ), which compares observed variance with the variance expected from Poisson noise; and the normalized-count spread q0.9(r)−q0.1(r)q_0.9(r)-q_0.1(r), the difference between the 90th and 10th percentiles of r (4); 6. seven profile-likelihood features. We evaluate the negative-binomial log-likelihood at seven candidate dispersion values bj=jbmax/7b_j=j\,b_ /7, for j=1,…,7j=1,…,7. We fix the mean-expression parameter at λ~=max(λ^,ε) λ= ( λ, ), define the expected count for cell i as mi=diλ~m_i=d_i λ, and define the corresponding negative-binomial shape parameter as κj=λ~/bj _j= λ/b_j. For each candidate bjb_j, the log-likelihood is Lj=∑i[lnΓ(yi+κj)−lnΓ(κj)−lnΓ(yi+1)+κjlogκjκj+mi+yilogmiκj+mi]L_j= _i[ (y_i+ _j)- ( _j)- (y_i+1)+ _j _j _j+m_i+y_i m_i _j+m_i]. This is the likelihood from Appendix G, with λ fixed at λ~ λ. We use the seven relative likelihood values Lj−maxkLkL_j- _kL_k as features, so that the best grid value has value zero and the others indicate how much less likely they are (7); 7. one likelihood-based estimate of b: the grid value with the largest likelihood, bj⋆b_j , where j⋆=argmaxjLjj = _jL_j (1). For the synthetic model, bmax=0.05b_ =0.05, which is the upper bound of the prior on biological overdispersion b. For the real cohorts, bmaxb_ is set to 1.11.1 times the largest b^MoM b_MoM among the training genes (Appendix G). This construction parallels the meta-analysis setting. The restricted features ϕO _O summarize the technical variances 1/di1/d_i, which play the same role as the sampling variances σi2 _i^2. The full features ϕI _I additionally include information about biological dispersion, analogous to adding Q and τ^DL2 τ^2_DL in the meta-analysis. The analogy for Q is only approximate. Here, Q=r¯dχP2Q= r_d\,χ^2_P, where χP2=∑i(yi−dir¯d)2/(dir¯d)χ^2_P= _i(y_i-d_i r_d)^2/(d_i r_d) is Pearson’s statistic. Therefore Q is not standardized, and comparing Q directly with K−1K-1 in item 5 is only a heuristic rather than a calibrated DerSimonian–Laird statistic. Standardized information about excess variation is still available to the decoder through s2/vs^2/v and b^MoM b_MoM. Both feature vectors discard some information. Unlike the meta-analysis setting, ϕO _O keeps only 8 summaries of the K sequencing depths, while ϕI _I does not include the raw vectors y1:Ky_1:K and d1:Kd_1:K. Therefore, the best predictors based on ϕO _O and ϕI _I can fall below the exact population quantities ρO2 _O^2 and ρI2 _I^2, which condition on the full raw data. Figure 7 compares these estimated and exact values. The full feature vector ϕI _I nevertheless contains two direct estimates of the latent biological dispersion b: the moment estimator b^MoM b_MoM and the likelihood estimator bj⋆b_j . These features give the estimated reference points at all six thinning fractions in Figure 7, the full-depth (p=1p=1) floor and ceiling in Figure 4 and Table 13, and the headroom grid in Figure 6. Appendix J Sensitivity to the choice of (O,I,T)(O,I,T) The quantities used for normalization in Section 4.1 are conditional on the choice of (O,I,T)(O,I,T). In the controlled meta-analysis and synthetic single-cell experiments, the generative model identifies a natural distinction between restricted and full information. In the observational chemistry and neuroscience applications, however, there is no unique decomposition of the measurements into O and I, because changing the choice of (O,I,T)(O,I,T) changes the scientific estimand itself. We therefore evaluate several prespecified, scientifically interpretable contrasts in each observational domain to determine whether the qualitative interpretation of a probe depends strongly on a particular placement of the boundary between restricted and additional information. Whenever possible, we hold the target (T) and full information I fixed and progressively enrich the restricted baseline, σ(O1)⊆σ(O2)⊆σ(O3)⊆σ(O4)⊆σ(I).σ(O_1) σ(O_2) σ(O_3) σ(O_4) σ(I). Population monotonicity then implies ρO12≤ρO22≤ρO32≤ρO42≤ρI2,Δ(O1,I)≥Δ(O2,I)≥Δ(O3,I)≥Δ(O4,I), _O_1^2≤ _O_2^2≤ _O_3^2≤ _O_4^2≤ _I^2, (O_1,I)≥ (O_2,I)≥ (O_3,I)≥ (O_4,I), although estimated reference points need not satisfy these inequalities. J.1 Chemistry Dataset and experimental setting. We use part 1 of the Raman-ChEMBL database, which provides molecular composition, physicochemical descriptors, three-dimensional molecular geometry, and quantum-chemical orbital energies computed with density functional theory (PBE/6-31G). We draw a fixed random sample of 50,000 molecules, keep the 48,931 that pass structural validation, and use 30,000 of these per seed. The analysis unit is one molecule. Identical SMILES are assigned to the same data split so that duplicate molecular identities do not occur across fine-tuning, validation, probe-fitting, and final-evaluation sets. The experiment is useful for probe normalization because molecular orbital energies are partly predictable from relatively coarse molecular properties such as composition, charge, and global shape, while detailed atomic geometry provides additional information. We therefore ask whether TabPFN hidden representations contain information about orbital energies beyond what was already predictable from progressively richer non-geometric or coarse molecular summaries. TabPFN hidden-state probes use a 64-component PCA fitted on training rows only. All other probes, floors and ceilings use full features. Prediction target. We analyze the two frontier orbital energies separately, THOMO=EHOMO,TLUMO=ELUMO,T^HOMO=E_HOMO, T^LUMO=E_LUMO, where EHOMOE_HOMO is the energy of the highest occupied molecular orbital and ELUMOE_LUMO is the energy of the lowest unoccupied molecular orbital. Both are measured in Hartree. These two quantities also determine the HOMO–LUMO gap, Eg=ELUMO−EHOMO.E_g=E_LUMO-E_HOMO. TabPFN is fine-tuned to predict EgE_g, but HOMO and LUMO are not individually used as fine-tuning targets. They are subsequently used only as targets of the representation audit. Restricted molecular information. The restricted information O is constructed from progressively richer fixed-size molecular summaries. We consider four nested specifications for restricted information, O1⊂O2⊂O3⊂O4.O_1⊂ O_2⊂ O_3⊂ O_4. The first contains elemental composition and total formal charge, O1=(element counts,Q),O_1= (element counts,Q ), where Q denotes total formal charge. This 18-dimensional representation describes which atomic species are present and their abundances, but does not specify how the atoms are arranged in space. The second specification additionally contains three rotational constants, O2=(O1,Rx,Ry,Rz).O_2= (O_1,R_x,R_y,R_z ). These quantities summarize the distribution of molecular mass around the principal axes and therefore provide coarse information about molecular size, mass distribution, and global shape without specifying individual atomic distances. The third specification additionally includes the total dipole magnitude, O3=(O2,Dtotal).O_3= (O_2,D_total ). The total dipole provides a coarse summary of molecular charge separation and therefore introduces global electrostatic information beyond composition and shape. Finally, the richest specification is O4=(O3,αiso,Re2,Cv),O_4= (O_3, _iso,R_e^2,C_v ), where αiso _iso is isotropic polarizability, Re2R_e^2 denotes electronic spatial extent, and CvC_v is heat capacity. These quantities add information about the response and spatial extent of the electronic distribution and about the molecule’s thermodynamic state. Thus the restricted baseline becomes progressively more informative while remaining a compact tabular description of the molecule. Full information I. The full information is held fixed across all four specifications: I=(O4,GMBTR),I= (O_4,G_MBTR ), where GMBTRG_MBTR is the unchanged monolithic MBTR geometry block. The MBTR representation encodes species-resolved pairwise inverse-distance information. It describes which types of atoms occur at which relative distances. It is invariant to translation, rotation, and atom ordering. Unlike the tabular descriptors in the restricted information, it provides a detailed representation of molecular geometry. As OjO_j becomes richer, the variables that I adds become more specific. Above O1O_1, it includes the full physicochemical and geometric information absent from composition alone, whereas above O4O_4 the remaining increment is specifically the detailed MBTR geometry block. The associated headroom, Δ^j=ρ^I2−ρ^Oj2, _j= ρ^2_I- ρ^2_O_j, measures how much additional predictability of the orbital energy remains after conditioning on the corresponding molecular baseline. Model representation. TabPFN is fine-tuned with LoRA using the fixed full information I to predict Eg=ELUMO−EHOMO.E_g=E_LUMO-E_HOMO. Neither EHOMOE_HOMO nor ELUMOE_LUMO is supplied as an individual fine-tuning target. LoRA adapters (rank 8, α=16α=16, dropout 0.05) are added to the attention and MLP projections of every TabPFN v3 block and to its output projection, and are fitted on molecules disjoint from those used for probing. After fine-tuning, hidden representations H(ℓ)H^( ) are extracted from the audited TabPFN layers. Probes from the decoder bank are then trained separately to predict THOMOT^HOMO and TLUMOT^LUMO from these representations, producing RH(ℓ)2R^2_H^( ). Thus, the estimated reference points ρ^Oj2 ρ^2_O_j and ρ^I2 ρ^2_I correspond to the chosen (Oj,I,T)(O_j,I,T), and RH(ℓ)2R^2_H^( ) measures the orbital-energy information accessible from the model’s internal representation. TabPFN receives O4O_4 and a 99%-variance PCA of the MBTR block; the ceiling probe uses the raw MBTR, so this PCA does not limit the ceiling. What is the scientific question? The raw question “How well can a TabPFN hidden state predict HOMO or LUMO?” is insufficient because substantial orbital-energy information may already be predictable from molecular composition or other coarse physicochemical properties. The question posed by normalization is instead “How should orbital-energy decodability be interpreted relative to what was already ``How should orbital-energy decodability be interpreted relative to what was already predictable from a declared molecular summary and what additional predictability from a declared molecular summary and what additional predictability is available from the full molecular descriptor set I?” available from the full molecular descriptor set $I$?′ For each orbital-energy target and each choice of (Oj,I,T)(O_j,I,T), we therefore compare ρ^Oj2 ρ^2_O_j, ρ^I2 ρ^2_I and RH(ℓ)2R^2_H^( ). The first quantity measures what is predictable from the restricted molecular description, the second measures what is predictable from the fixed geometry-aware representation, and their difference defines the estimated headroom Δ^j _j. The model probe is interpreted relative to these quantities rather than on an absolute R2R^2 scale. The sequence O1⊂O2⊂O3⊂O4O_1⊂ O_2⊂ O_3⊂ O_4 further provides a sensitivity analysis for the choice of restricted information. O1O_1 asks what the complete molecular representation adds beyond composition and charge; O2O_2 additionally conditions on coarse molecular size and shape; O3O_3 further conditions on global charge separation; and O4O_4 conditions on a rich physicochemical state before asking what detailed geometry adds. These baseline specifications define distinct estimands. If the interpretation of the TabPFN probe changes across them, that dependence identifies which classes of molecular information account for the apparent decodability. Result. Most of each raw probe score is already predictable from the physicochemical baseline O4O_4. Above it, the fine-tuned TabPFN recovers 69% (HOMO) and 82% (LUMO) of the additional predictability that detailed geometry provides (Table 13). Neither orbital energy was a fine-tuning target, so substantial information about both is present in the representation even though the model was trained only on their difference. The interpretation is stable across the nested baselines O1O_1–O4O_4. J.2 Neuroscience Related work and motivation. Neuroscience has long used decoding to ask what neural populations represent (Quian Quiroga and Panzeri, 2009; Naselaris et al., 2011; Glaser et al., 2020), and has documented its limits (Kriegeskorte and Douglas, 2019; Ritchie et al., 2019; Hebart and Baker, 2018). Two common practices resemble our reference points. First, decoding analyses control for confounds by removing the variance explained by variables that are not of interest (Snoek et al., 2019), and encoding studies split variance between feature sets (de Heer et al., 2017). Our floor plays this role, with the baseline stated explicitly and chosen by the scientific question. Second, encoding models are compared against a noise ceiling: the part of the neural response that repeats across identical trials, which no model can exceed (Hsu et al., 2004; Schoppe et al., 2016; Lage-Castellanos et al., 2019). Unlike a noise ceiling, which is limited by trial-to-trial variability in the response, the ceiling we define is limited by how much the available information reveals about the target. Dataset and experimental setting. We use the Area 2 dataset of Chowdhury et al. (2022), consisting of neural population recordings from proprioceptive Area 2 of somatosensory cortex during an arm-reaching task with externally imposed mechanical perturbations. On perturbation trials, the animal holds or moves a manipulandum while a brief planar bump is applied through the interface. The dataset contains simultaneously recorded neural activity together with hand position, hand velocity, interface force, and the commanded perturbation. The analysis unit is one perturbation trial. The experiment is useful for probe normalization because the subsequent mechanical response is partly predictable from the state of the limb before the perturbation, while knowledge of the externally imposed bump provides additional information. We therefore ask whether post-perturbation neural population activity contains information about the ensuing force response beyond what was already predictable from the measured pre-bump state. Prediction target. We analyze the two planar force components separately. For component c∈x,yc∈\x,y\, the target is the baseline-corrected interface-force response during the first 100ms100\,ms after bump onset, Tc=Fc(t)¯ 0≤t<100ms−Fc(t)¯−100≤t<0ms.T_c= F_c(t)_\,0≤ t<100\,ms- F_c(t)_\,-100≤ t<0\,ms. Thus TcT_c measures the change in force following the perturbation relative to the force already present immediately before it. Baseline correction removes pre-existing load, tonic force, and constant sensor offsets, so the target reflects the perturbation-evoked mechanical response rather than the absolute force reading. Baseline Target Floor mean ± SD Ceiling mean ± SD Headroom mean ± SD Probe mean ± SD UHU_H mean ± SD Neuroscience O1O_1 Force x −-0.013 ± 0.019 0.987 ± 0.009 1.000 ± 0.025 0.745 ± 0.014 0.758 ± 0.019 O1O_1 Force y −-0.002 ± 0.006 0.997 ± 0.000 0.999 ± 0.006 0.573 ± 0.019 0.575 ± 0.017 O2O_2 Force x 0.004 ± 0.009 0.987 ± 0.009 0.983 ± 0.016 0.745 ± 0.014 0.754 ± 0.018 O2O_2 Force y −-0.013 ± 0.028 0.997 ± 0.000 1.010 ± 0.028 0.573 ± 0.019 0.580 ± 0.011 O3O_3 Force x −-0.002 ± 0.019 0.987 ± 0.009 0.989 ± 0.025 0.745 ± 0.014 0.755 ± 0.019 O3O_3 Force y −-0.003 ± 0.005 0.997 ± 0.000 1.000 ± 0.005 0.573 ± 0.019 0.575 ± 0.019 O4O_4 Force x 0.111 ± 0.023 0.987 ± 0.009 0.876 ± 0.028 0.745 ± 0.014 0.724 ± 0.018 O4O_4 Force y 0.088 ± 0.024 0.997 ± 0.000 0.909 ± 0.024 0.573 ± 0.019 0.533 ± 0.025 Chemistry O1O_1 HOMO 0.378 ± 0.004 0.628 ± 0.005 0.250 ± 0.007 0.555 ± 0.014 0.710 ± 0.065 O1O_1 LUMO 0.458 ± 0.020 0.693 ± 0.007 0.236 ± 0.014 0.660 ± 0.012 0.855 ± 0.071 O2O_2 HOMO 0.381 ± 0.003 0.628 ± 0.005 0.247 ± 0.003 0.555 ± 0.014 0.705 ± 0.070 O2O_2 LUMO 0.460 ± 0.007 0.693 ± 0.007 0.233 ± 0.011 0.660 ± 0.012 0.857 ± 0.066 O3O_3 HOMO 0.383 ± 0.003 0.628 ± 0.005 0.245 ± 0.005 0.555 ± 0.014 0.703 ± 0.069 O3O_3 LUMO 0.457 ± 0.016 0.693 ± 0.007 0.236 ± 0.012 0.660 ± 0.012 0.856 ± 0.070 O4O_4 HOMO 0.391 ± 0.004 0.628 ± 0.005 0.237 ± 0.005 0.555 ± 0.014 0.693 ± 0.073 O4O_4 LUMO 0.504 ± 0.016 0.693 ± 0.007 0.190 ± 0.012 0.660 ± 0.012 0.820 ± 0.089 Table 4: Estimated reference points and probe scores for the nested baselines O1⊂O2⊂O3⊂O4O_1⊂ O_2⊂ O_3⊂ O_4, reported as mean ± sample standard deviation (SD) across five seeds. Restricted pre-bump information. The restricted information O is constructed entirely from measurements before bump onset. The primary pre-bump window is [−100,0)ms[-100,0)\,ms, which summarizes the limb state immediately before the perturbation while excluding post-bump information. Averaging within this interval reduces measurement noise relative to conditioning on a single time point and provides a local estimate of the state from which the perturbation begins. We consider progressively richer restricted-information specifications: O1 O_1 =(x¯pre,y¯pre), = ( x_pre, y_pre ), O2 O_2 =(O1,vx¯pre,vy¯pre), = (O_1, v_x_pre, v_y_pre ), O3 O_3 =(O2,Fx¯pre,Fy¯pre). = (O_2, F_x_pre, F_y_pre ). Here each bar denotes the mean over [−100,0)ms[-100,0)\,ms. These contrasts progressively condition on hand location, motion, and the force already being exerted before the bump. The richest baseline, O4O_4, additionally captures whether these quantities were changing immediately before perturbation. For any pre-bump variable q(t)q(t), we define Δqpre=q(t)¯−30≤t<0ms−q(t)¯−100≤t<−70ms. q_pre= q(t)_\,-30≤ t<0\,ms- q(t)_\,-100≤ t<-70\,ms. We compute this late-minus-early change separately for planar position, velocity, and force, yielding six additional features: Δx,Δy,Δvx,Δvy,ΔFx,ΔFy. x,\; y,\; v_x,\; v_y,\; F_x,\; F_y. Thus, O4=(O3,Δx,Δy,Δvx,Δvy,ΔFx,ΔFy).O_4= (O_3, x, y, v_x, v_y, F_x, F_y ). These features distinguish a stable pre-bump state from one in which the limb or interface force was already drifting immediately before perturbation, while remaining entirely pre-perturbation. Full information I. The full information is held fixed across all four specifications. It augments the richest pre-bump state with the experimentally commanded planar perturbation, I=(O4,bx,by),I=(O_4,b_x,b_y), where (bx,by)(b_x,b_y) is the commanded bump vector. Because O1⊂O2⊂O3⊂O4O_1⊂ O_2⊂ O_3⊂ O_4, every specification satisfies σ(Oj)⊆σ(I)σ(O_j) σ(I). The ceiling ρ^I2 ρ^2_I is therefore fitted once per target and seed and shared by all four contrasts; only the floor changes between them. The associated headroom, Δ^j=ρ^I2−ρ^Oj2, _j= ρ^2_I- ρ^2_O_j, measures how much additional predictability of the subsequent force response is available from knowing the perturbation after conditioning on a specified description of the pre-bump body state. Neural representation. The representation H is the post-perturbation Area 2 population activity extracted from the neural response following bump onset. A decoder-bank probe is trained to predict TxT_x or TyT_y from this population representation (spike counts), separately within each recording session with predictions pooled across sessions, producing RH2R_H^2; like the ceiling, this probe score is shared by all four contrasts. This setting differs from the artificial-network experiments because H is a recorded biological population state rather than a deterministic transformation of the measured I. Area 2 activity may contain proprioceptive, biomechanical, or internally generated information that is absent from our behavioral measurements. What is the scientific question? The raw question “How well can post-bump Area 2 activity predict the force response?” is insufficient because part of that response may already be predictable from the state of the hand before the perturbation. The question posed by normalization is instead “How should neural decodability be interpreted relative to what was already predictable ``How should neural decodability be interpreted relative to what was already predictable before the bump and what additional predictability is provided by knowing the bump?” the bump and what additional predictability is provided by knowing the bump?′ For each target and each choice of (Oj,I,T)(O_j,I,T), we therefore compare ρ^Oj2,ρ^I2,RH2. ρ^2_O_j, ρ^2_I, R^2_H. The first quantity is the floor from the pre-bump state alone, the second is the estimated full-information reference, and their difference is the estimated headroom Δ^j _j. The neural probe score is interpreted relative to these quantities rather than on an absolute R2R^2 scale. The sequence O1⊂O2⊂O3⊂O4O_1⊂ O_2⊂ O_3⊂ O_4 further provides a sensitivity analysis for the choice of restricted information. O1O_1 asks whether bump information matters beyond initial hand position alone; O2O_2 additionally conditions on ongoing motion; O3O_3 conditions on pre-existing interface force; and O4O_4 further accounts for recent pre-bump trends. These baseline specifications define distinct estimands. If the interpretation of the neural probe changes across them, that dependence identifies which aspects of the pre-perturbation state are responsible for the apparent decodability. Result. Under O4O_4, floors are near 0.1 and the full-information reference is near 1, so normalized and raw scores nearly coincide: Area 2 activity captures about 72% of the perturbation-driven force response along x and 53% along y under the richest pre-bump baseline O4O_4 (Table 13). Because neural activity is not a function of the measured inputs, the reference is not an upper bound on the probe score. The interpretation is stable across O1O_1–O4O_4; only O4O_4, which adds pre-bump trends, raises the floor appreciably. Appendix K Normalizing published LLM probes K.1 Common setup Floors without the model. In every analysis, O is a simple description of the input that uses no language model: character n-grams of a name, word and character n-grams of a statement or conversation, or summaries of an Othello move sequence. The floor therefore needs no access to the model and can be computed for any published result whose inputs, labels, and splits are released. What the floor measures. Here O is computed from the input string itself, so in principle it holds the same information as the input. What differs is what can be learned from it: the floor is the best readout that a decoder from the bank (Appendix B) can learn from O using only the probe’s training data. The headroom above it is what the model adds, through knowledge from its training or through computation. Metric. We use R2R^2 for continuous targets and accuracy for categorical ones. The decoder is chosen on a validation split taken from the training rows and evaluated once on the test rows. A floor can never truly be below chance; an estimated floor well below chance means the readout’s cues invert outside its training data. We do not compute UHU_H in that case and compare raw scores instead. Dataset Floor (95% CI) Probe R2R^2 (7B / 70B) UHU_H (7B / 70B) World places 0.61 (0.59–0.62) 0.88 / 0.91 0.70 / 0.77 US places 0.46 (0.44–0.48) 0.80 / 0.86 0.63 / 0.75 NYC places 0.33 (0.29–0.37) 0.22 / 0.36 −-0.17 / 0.04 Historical figures 0.60 (0.59–0.62) 0.79 / 0.83 0.46 / 0.58 Entertainment 0.16 (0.05–0.28) 0.79 / 0.89 0.75 / 0.86 Headlines 0.45 (0.43–0.47) 0.56 / 0.75 0.21 / 0.54 Table 5: Spelling floors for the space and time datasets of Gurnee and Tegmark (2024). The floor is the held-out R2R^2 of the selected decoder on character n-grams of the entity name; probe scores are the published linear probes on Llama-2-7B / 70B; UHU_H uses a ceiling of 1, which can only lower a positive UHU_H (Appendix K.1). Ceiling. We cannot estimate the ceiling here, because predicting a city’s location or a statement’s truth from its text needs knowledge that the probe’s training data cannot supply. We set it to 1, the highest possible value, which can only lower a positive UHU_H. In Othello the ceiling is exactly 1, since the board follows from the moves. When a probe scores at or near the floor, we compare the two directly, which does not depend on the ceiling. The floors themselves are lower estimates: a stronger readout could raise them and lower every UHU_H. The two choices push UHU_H in opposite directions, so we read it as approximate. The space and time floors use character 2–4-gram counts (Appendix K.2), and the truth and user-attribute floors word 1–2-gram and character 2–4-gram TF-IDF features; in both cases the linear decoder uses the sparse features directly, and the nonlinear decoders use a 1,024-component truncated SVD. K.2 Space and time Published claim. Gurnee and Tegmark (2024) train linear ridge probes on Llama-2 activations at the last token of an entity name to predict its real-world location (world, US, and New York City places) or date (historical figures, entertainment, news headlines), and report high out-of-sample R2R^2 as evidence of spatial and temporal representations. For New York City, where scores are lowest, they note that the largest model performs best and suggest that larger models may form city-level spatial models. Choice of (O,I,T)(O,I,T). T is a coordinate or a date, I is the entity string exactly as presented to the model, and O is its character 2–4-gram counts. The floor measures how much of T a readout can learn from spelling alone, for example from state modifiers, borough names, or numbers in the string. The original authors raise a related concern, that a probe could assemble coordinates from simpler features under supervision, and address it by holding out whole regions; the floor quantifies one specific form of it. Matching the published evaluation. We use the published probe scores (their Table 2: linear probes at 60% layer depth, Llama-2-7B and Llama-2-70B) and re-ran one to confirm that our pipeline matches theirs: for Llama-2-7B on New York City we obtain R2=0.218R^2=0.218 against the published 0.219. We verified that (i) the floor uses exactly the strings the model received, including rewritten forms such as “Iowa’s Dallas County” and “⟨ ⟩ ’s ⟨ ⟩ ”, matching token for token on all 186,669 rows; (i) rows and splits follow the released train–test assignment, which groups New York City locations by complex and entertainment items by creator; and (i) targets are in their original units, with R2R^2 computed as in the original evaluation. Ridge regression is fitted on the n-gram counts, and gradient boosting and an MLP on a 1,024-component truncated SVD of them. Confidence intervals come from 1,000 bootstrap resamples of the test set, resampling complexes for New York City and creators for entertainment. Result (Table 5). For world and US places, spelling accounts for half to two-thirds of the raw R2R^2, yet the probes recover 63–77% of the remaining headroom, so most of what remains reflects information beyond the names. For New York City, spelling reaches R2=0.33R^2=0.33; the Llama-2-70B score (0.36) lies within the floor’s confidence interval and the Llama-2-7B score (0.22) lies below it. At this scale the reported scores cannot separate a city-level spatial representation from lexical cues, and the better relative performance of larger models is also consistent with better use of such cues. This agrees with the original block-holdout analysis, in which probes generalized to held-out boroughs only slightly better than chance. For temporal targets, the probes recover most of the headroom for entertainment (UH≥0.75U_H≥ 0.75), more than half for historical figures with the larger model (0.58; 0.46 for the smaller), and about half for headlines with the larger model (0.54; 0.21 for the smaller). Floor features Decoder Floor acc. (95% CI) Linear floor acc. Chance acc. Occupied acc. Probe acc. (95% CI) Linear probe acc. UHU_H UHU_H linear probe Black/White/Empty O1O_1: position GBM 0.62 (0.61–0.62) 0.56 0.46 0.39 0.89 (0.89–0.90) 0.74 0.72 0.33 O2O_2: recency GBM 0.66 (0.65–0.67) 0.60 0.46 0.47 0.89 (0.89–0.90) 0.74 0.68 0.24 O3O_3: placement GBM 0.80 (0.80–0.81) 0.76 0.46 0.64 0.89 (0.89–0.90) 0.74 0.45 −0.30-0.30 Mine/Theirs/Empty O1O_1: position MLP 0.62 (0.61–0.63) 0.56 0.46 0.40 0.98 (0.98–0.99) 0.99 0.96 0.96 O2O_2: recency GBM 0.66 (0.66–0.67) 0.57 0.46 0.47 0.98 (0.98–0.99) 0.99 0.95 0.96 O3O_3: placement GBM 0.81 (0.81–0.82) 0.80 0.46 0.65 0.98 (0.98–0.99) 0.99 0.92 0.92 Table 6: Floors, probe accuracy and headroom-normalized decodability for the Black/White/Empty and Mine/Theirs/Empty targets under nested floor feature sets O1⊂O2⊂O3O_1⊂ O_2⊂ O_3. The ceiling is exactly 1, since the board is a deterministic function of the move sequence. Occupied acc. is the floor accuracy on occupied squares only. UHU_H is the fraction of the headroom recovered by the probe. The entertainment floor is low (0.16), possibly because the released split holds out entire creators, so spelling cues learned for one creator’s titles cannot transfer to another’s. The entertainment result is robust to the floor’s uncertainty: across its interval, UHU_H lies between 0.71 and 0.78 for Llama-2-7B and between 0.85 and 0.88 for Llama-2-70B. New York City locations are also grouped, yet their floor stays high, plausibly because names carry borough names and numbers shared across complexes. K.3 Othello world model Published claim. Li et al. (2022) train probes on Othello-GPT activations to predict the state of each board square, and report 0.88–0.98 accuracy for nonlinear probes and 0.76–0.79 for linear probes on a black/white/empty encoding for synthetic tasks, as evidence of an emergent world model. Nanda et al. (2023) show that a mine/theirs/empty encoding, relative to the player to move, is linearly decodable at higher accuracy. Choice of (O,I,T)(O,I,T). T is the state of one square, I is the move sequence, and O is a nested set of move summaries that never simulate flips: O1O_1 records only the move number and the square’s position; O2O_2 adds whether the square was among the last four moves and by which player, and its distance to the last move; O3O_3 adds whether the square has ever been played, by which player (relative to the player to move), and how many moves ago. O3O_3 is thus the board as it would be if no disc were ever flipped. The floor measures how much of the board state is predictable without computing flips; for O3O_3, the headroom is exactly what the flip dynamics add. Model and probes. We use the released Othello-GPT checkpoint of Li et al. (2022) and evaluate it on 1,000 games of uniformly random legal play, the same distribution as its training data. The model predicts a legal next move for 99.94% of positions (58,958/58,992; 98.67% after a pass, which the input does not mark). Floor and probes share one set of rows: eight random positions per game and every square at each (512,000 rows), split by game into 60% fit, 20% selection and 20% test. Rows at the final move (1.6%) are excluded from the probe because the model reads at most 59 moves. For each square we fit the MLP and linear decoders of the decoder bank on the 512-d residual stream after the embedding or after each of the 8 blocks. The layer and each square’s decoder are chosen by McFadden R2R^2 on the selection games, refitted on fit and selection, and scored once on the test games; 95% CIs resample test games. The selected layer is 5 for black/white/empty and 6 for mine/theirs/empty. Probe accuracy is 0.892 (0.887–0.897) for black/white/empty, with a linear probe at 0.745, and 0.985 (0.980–0.989) for mine/theirs/empty, with a linear probe at 0.986. Our probes are trained on about 4,700 positions per square, fewer than in the original work, so their accuracies, and hence UHU_H, are lower bounds. Result (Table 6). The floor rises from 0.62 (O1O_1) to ≈0.80≈ 0.80 (O3O_3). For black/white/empty, our nonlinear probe (0.89) recovers 72% of the headroom above O1O_1 but only 45% above O3O_3: much of its accuracy is available from disc placement without flips. The linear black/white probe (0.74) recovers 33% above O1O_1 and falls below the O3O_3 floor (UH=−0.30U_H=-0.30), so its accuracy is explained by placement alone. For mine/theirs/empty the representation is linear: the linear probe reaches 0.99 and recovers 92% of the headroom even above O3O_3 (96% above O1O_1), matching the nonlinear probe. The world-model claim therefore holds most strongly in the relative encoding. The floor depends on which move summaries are allowed; O3O_3 is the richest summary we tested that still excludes flips. A floor that approximates flips heuristically would be higher and would lower UHU_H. K.4 Truth directions Dataset Decoder Floor acc. (95% CI) Linear acc. cities Linear 0.46 (0.40–0.52) 0.46 neg_cities Linear 0.45 (0.39–0.50) 0.45 sp_en_trans Linear 0.65 (0.52–0.76) 0.65 neg_sp_en_trans GBM 0.49 (0.38–0.62) 0.56 larger_than Linear 0.99 (0.97–1.00) 0.99 smaller_than Linear 0.97 (0.96–0.99) 0.97 cities_cities_conj Shallow GBM 0.56 (0.50–0.62) 0.54 cities_cities_disj Linear 0.53 (0.48–0.59) 0.53 companies_true_false Shallow GBM 0.53 (0.46–0.59) 0.43 common_claim_true_false Linear 0.62 (0.59–0.65) 0.62 counterfact_true_false GBM 0.53 (0.52–0.54) 0.50 Table 7: Wording-only floors for the truth datasets. The floor is the test accuracy of the selected decoder on word and character n-grams of the statement; Linear acc. is the linear decoder’s. All datasets are balanced, so chance accuracy is 0.50 (0.51 for sp_en_trans and neg_sp_en_trans). The ceiling is taken as 1, since the truth value is fixed by the statement and world knowledge. Published claim. Marks and Tegmark (2023) report linear directions in LLM activations that separate true from false statements and transfer across datasets, including to negated statements. Choice of (O,I,T)(O,I,T). The target T is the binary truth label, I is the full statement, and O is its wording-only representation using n-gram features. The floor therefore measures how well truth can be predicted from surface form alone, without access to model activations or pretrained world knowledge. We evaluate on a random split. For transfer experiments, the floor decoder is trained on the same source dataset as the activation probe and evaluated unchanged on the target dataset; we call this the transfer floor. Result. Table 7 shows that wording alone is close to chance (0.50.5) on most truth datasets. For cross-dataset transfer (Table 8), we compare the published probes with this transfer floor. When trained on larger_than and smaller_than and tested on sp_en_trans, the floor is 0.490.49, while the published 13B and 70B probes both reach 0.970.97. Similarly, training on cities and neg_cities and testing on neg_sp_en_trans gives a floor of 0.500.50, compared with probe accuracies of 0.960.96 and 0.990.99. In both cases, the probe transfers substantially better than surface wording alone. Train → Test Decoder Floor acc. (95% CI) Linear acc. Published probe acc. (13B / 70B) larger_than + smaller_than → sp_en_trans GBM 0.49 (0.44–0.55) 0.50 0.97 / 0.97 cities + neg_cities → neg_sp_en_trans Shallow GBM 0.50 (0.45–0.55) 0.50 0.96 / 0.99 cities → neg_cities 11 1 The decoder is selected on held-out source rows, which contain unseen city–country pairs, so no decoder is informative there (shallow GBM pseudo-R2=0.035R^2=0.035, linear pseudo-R2=0.001R^2=0.001); neg_cities repeats every cities pair with “not” inserted and the label flipped, so pairs that shallow GBM memorised during refitting are wrong on the target (0.350.35, below the 0.500.50 chance level), while the near-prior linear decoder stays at chance (0.470.47). Shallow GBM 0.35 (0.32–0.37) 0.47 0.75 / 0.37 larger_than → smaller_than Linear 0.00 (0.00–0.00) 0.00 0.07 / 0.55 Table 8: Wording-only floors for cross-dataset transfer, against the published mass-mean probe accuracy on Llama-2-13B and Llama-2-70B. Floors are trained on the source datasets and scored on the target; chance accuracy is 0.50. The other two transfers give a different pattern. For larger_than→ _than, the floor is 0.000.00, while the 13B and 70B probes obtain 0.070.07 and 0.550.55. For cities→ _cities, the floor is 0.350.35, while the probes obtain 0.750.75 and 0.370.37. Both floors fall well below chance (0.500.50) because cues learned on the source dataset invert on the target, so headroom normalization is not applicable (Section 4.1); we compare the probe accuracies directly with the floor and with chance. K.5 User attributes in conversations Published claim. Chen et al. (2024) train linear probes on Llama-2-Chat-13B to read a user’s age, gender, education, and socioeconomic status from a conversation, as evidence that the model builds an internal model of its user. They report two kinds of probe. Reading probes are applied after a prompt that asks the model about the user, and reach 0.94–0.98. Control probes are applied at the end of the user’s message and are used for steering, and reach 0.91–0.96. Choice of (O,I,T)(O,I,T). T is the user’s attribute, I is the conversation, and O is its surface wording: word 1–2-gram and character 2–4-gram TF-IDF features, with no pretrained model. The linear decoder uses the sparse features directly, and the other decoders use a 1,024-component truncated SVD of them. The linear decoder wins every split. Matching the published evaluation. We use the released synthetic conversations, in which GPT-3.5 and Llama-2-Chat role-play users with a given attribute (2,900–4,500 conversations per attribute), pooled across both sources as the published probes were trained. We use a stratified 80/20 split, with 20% of the training rows held out for decoder selection. Chance accuracy is the share of the most common class. Additional evaluations. To see where the cue comes from, we also fit the floor on the user’s turns only, on each source separately, and on GPT-3.5 conversations tested on Llama-2-Chat ones. Attribute Evaluation Floor acc. (95% CI) Chance acc. Published probe acc. (reading / control) UHU_H (reading / control) Age Pooled 0.98 (0.97–0.99) 0.25 0.98 / 0.96 0.11 / −-0.78 User turns 0.98 (0.97–0.99) 0.25 – – GPT-3.5 0.98 (0.97–0.99) 0.25 – – Llama-2-Chat 0.93 (0.89–0.96) 0.25 – – GPT-3.5→ -2-Chat 0.88 (0.86–0.90) 0.25 – – Gender Pooled 0.92 (0.89–0.94) 0.50 0.94 / 0.91 0.27 / −-0.09 User turns 0.91 (0.88–0.93) 0.50 – – GPT-3.5 0.95 (0.93–0.97) 0.50 – – Llama-2-Chat 0.77 (0.71–0.82) 0.50 – – GPT-3.5→ -2-Chat 0.76 (0.73–0.78) 0.50 – – Education Pooled 0.97 (0.96–0.98) 0.33 0.96 / 0.93 −-0.24 / −-1.17 User turns 0.98 (0.97–0.99) 0.33 – – GPT-3.5 0.97 (0.96–0.98) 0.33 – – Socioeconomic status Pooled 0.96 (0.95–0.98) 0.33 0.97 / 0.95 0.14 / −-0.43 User turns 0.95 (0.93–0.97) 0.33 – – GPT-3.5 0.97 (0.95–0.98) 0.33 – – Llama-2-Chat 0.97 (0.95–0.99) 0.33 – – GPT-3.5→ -2-Chat 0.90 (0.87–0.92) 0.33 – – Table 9: Wording-only floors for user attributes. Pooled: both role-play sources, as the published probes were trained. User turns: the user’s turns only. GPT-3.5, Llama-2-Chat: one source. GPT-3.5 → Llama-2-Chat: fitted on GPT-3.5 conversations and tested on Llama-2-Chat ones. The released data contain no Llama-2-Chat conversations for education, so those rows are absent. The floor is the test accuracy of the selected decoder (linear in every row). UHU_H uses a ceiling of 1 and is unstable here because the headroom is small. Result (Table 9). Wording alone predicts every attribute almost as well as the published probes. Each published reading probe lies within the floor’s 95% interval: 0.98 against a floor of 0.98 for age, 0.94 against 0.92 for gender, 0.96 against 0.97 for education, and 0.97 against 0.96 for socioeconomic status. Every control probe falls below the floor. The published accuracies are therefore largely explained by cues in the wording. The floor from the user’s turns alone (0.91–0.98) matches the floor from the full conversation, so the cues are in what the role-played user writes. Trained on GPT-3.5 conversations and tested on Llama-2-Chat ones, the floor falls to 0.76–0.90, so part of the cue is the generating model’s style rather than the attribute itself. Reading UHU_H near a ceiling. With floors of 0.92–0.98, the headroom is only 0.02–0.08, so a difference of a few points between probe and floor becomes a large UHU_H. The reading probes give UHU_H of 0.11, 0.27, and 0.14 for age, gender, and socioeconomic status, and −0.24-0.24 for education; the control probes give −0.09-0.09 to −1.17-1.17. These values mainly reflect how small the headroom is, so we read this analysis by the raw comparison above rather than by UHU_H. Caveats. The conversations were written to reflect each attribute, so the label is visible in the text by design. This shows that the published accuracies cannot establish a user model, not that the model lacks one; testing that would need data where the attribute is not cued in the wording. Appendix L Conditional probing and predictive V-information Hewitt et al. (2021), based on predictive V-information (Xu et al., 2020), ask how much information about a linguistic property, such as part of speech, a contextual layer of a language model (ELMo, RoBERTa) contains beyond what the non-contextual word embeddings already provide. The baseline B is therefore the embedding layer, and the quantity of interest is the information that a deeper layer H adds to it, I(H→Y∣B)=H(Y∣B)−H(Y∣B,H),I_V(H→ Y B)=H_V(Y B)-H_V(Y B,H), estimated by training one probe on B and one on the concatenation [B;H][B;H]. They contrast this with the earlier practice of comparing a probe on H with a probe on B separately. That standalone comparison penalizes H for any information about word identity it has discarded, even when it adds new contextual information. Under conditional probing, deeper layers turn out to carry more information beyond the embeddings than standalone comparisons had suggested. Our floor plays the role of the baseline defined by them, with two differences: O is a declared information set chosen by the scientific question rather than another layer of the model, and we pair it with a ceiling ρI2ρ^2_I, the predictability supported by the full input. L.1 Comparing probe normalization with conditional probes Here, T is the target, O the observed information that defines the floor ρO2 _O^2, I the full input, H the representation and B a baseline (here B=OB=O). RX2R^2_X is the held-out R2R^2 of a probe on X, and [B;H][B;H] is the concatenation of B and H. The normalized score, UHU_H has the form of a skill score. In our method, two reference points, floor and ceiling are what O and I themselves determine about T, not a fixed baseline and a perfect score. For continuous targets, conditional probing reduces to a difference in R2R^2. V-information measures how much better a probe predicts T when it is given X than when it is given nothing, with prediction quality scored by log-loss. Suppose the probe outputs a Gaussian with mean φ(X) (X) and fixed variance 12 12. Its log-loss is then −logp(T∣X)=(T−φ(X))2+const- p(T X)=(T- (X))^2+const, so minimizing log-loss is the same as minimizing squared error, and the best achievable log-loss is the probe’s MSE plus a constant. With no input, the best prediction is the mean of T, whose MSE is Var(T)Var(T). Subtracting, I(X→T)=Var(T)−MSEX=Var(T)RX2.I_V(X→ T)=Var(T)-MSE_X=Var(T)\,R^2_X. The same subtraction with baseline B gives I(H→T∣B)Var(T)=MSEB−MSEB,HVar(T)=RB,H2−RB2. I_V(H→ T B)Var(T)= MSE_B-MSE_B,HVar(T)=R^2_B,H-R^2_B. For continuous targets, conditional probing is therefore just the gain in R2R^2 from adding H to B, so we report it in R2R^2 units for the meta-analysis. For Othello the targets are categorical, so we report the corresponding gain in accuracy. UH|OU_H O is conditional probing with B=OB=O divided by the headroom. The normalized score UHU_H compares the probe on H with the floor directly. We define a conditional variant, UH|OU_H|O that credits H only for what it adds to O: UH|O:=RO,H2−ρO2ρI2−ρO2,U_H|O:= R^2_O,H-ρ^2_Oρ^2_I-ρ^2_O, where RO,H2R^2_O,H is the held-out R2R^2 of a probe given both O and H. Its numerator is conditional probing with B=OB=O, so UH|OU_H|O is conditional probing normalized by the headroom. In the population, UH≤UH|O≤1U_H≤ U_H O≤ 1. The two scores share the denominator ρI2−ρO2 _I^2- _O^2 and differ only in whether the probe sees H alone or H together with O. With Bayes-optimal predictors, more information never lowers R2R^2. Since (O,H)(O,H) contains H, RO,H2≥RH2R^2_O,H≥ R^2_H, which gives UH|O≥UHU_H O≥ U_H. Since (O,H)(O,H) is a function of I when H=Φ(I)H= (I), RO,H2≤ρI2R^2_O,H≤ _I^2, which gives UH|O≤1U_H O≤ 1. The gap UH|O−UH=RO,H2−RH2ρI2−ρO2U_H O-U_H= R^2_O,H-R^2_H _I^2- _O^2 is the information about T that O carries and H has discarded. It is zero when H retains everything in O that is relevant to T. In the population, UHU_H is therefore conservative: it can understate, but not overstate, what H adds beyond O. With finite samples, estimated reference points and a fixed decoder family, either inequality can fail; out of distribution in Table 10, for example, UHU_H slightly exceeds UH|OU_H O. L.2 Meta-analysis. H is the last layer of the trained xl transformer at the final position. O is the summary statistics of σ2σ^2 (Appendix I). Every probe (for O, H, [O;H][O;H]) is the linear decoder from the decoder bank of Table 2. Condition ID OOD Target logS1 S_1 τ2τ^2 logS1 S_1 τ2τ^2 Estimated floor ρ^O2 ρ_O^2 0.01±0.010.01± 0.01 −0.00±0.00-0.00± 0.00 0.09±0.010.09± 0.01 −0.00±0.00-0.00± 0.00 Estimated ceiling ρ^I2 ρ_I^2 0.63±0.020.63± 0.02 0.66±0.010.66± 0.01 0.41±0.020.41± 0.02 0.36±0.030.36± 0.03 Estimated headroom Δ^=ρ^I2−ρ^O2 = ρ_I^2- ρ_O^2 0.62±0.010.62± 0.01 0.66±0.010.66± 0.01 0.33±0.020.33± 0.02 0.36±0.030.36± 0.03 Hidden probe RH2R^2_H 0.60±0.040.60± 0.04 0.64±0.020.64± 0.02 0.40±0.020.40± 0.02 0.34±0.020.34± 0.02 Probe on [O;H][O;H], RO,H2R^2_O,H 0.59±0.040.59± 0.04 0.64±0.020.64± 0.02 0.40±0.020.40± 0.02 0.34±0.030.34± 0.03 Normalized score UHU_H 0.95±0.040.95± 0.04 0.97±0.040.97± 0.04 0.96±0.020.96± 0.02 0.97±0.020.97± 0.02 Conditional normalized score UH|OU_H O 0.95±0.050.95± 0.05 0.97±0.040.97± 0.04 0.95±0.020.95± 0.02 0.95±0.030.95± 0.03 Conditional probe (B=OB=O), RO,H2−ρ^O2R^2_O,H- ρ_O^2 0.58±0.040.58± 0.04 0.64±0.030.64± 0.03 0.31±0.020.31± 0.02 0.34±0.020.34± 0.02 Table 10: Meta-analysis transformer (xl, last layer, final position): normalized headroom and conditional probing with baseline B=OB=O. Every probe, floor and ceiling is the linear decoder; the floor and ceiling are estimated from O and I. Mean ± SD over five seeds. We found that conditional normalized score is roughly equal to normalized score (Table 10). For both τ2τ^2 and logS1 S_1, UHU_H and UH|OU_H O differ by at most 0.020.02 in every condition. Raw conditional probe gains mix up the representation and the data and normalizing by headroom separates them. From in distribution to out of distribution, Conditional Probe (H∣B=O)(H B=O) halves from 0.58→0.310.58→ 0.31 for logS1 S_1 and 0.64→0.340.64→ 0.34 for τ2τ^2. Alone, this indicates that the representation got twice as bad. But the estimated headroom ρ^I2−ρ^O2 ρ_I^2- ρ_O^2 also shrinks (0.62→0.330.62→ 0.33 for logS1 S_1; 0.66→0.360.66→ 0.36 for τ2τ^2), and the normalized score does not fall: UH|O=0.95→0.95U_H O=0.95→ 0.95 for logS1 S_1 and 0.97→0.950.97→ 0.95 for τ2τ^2. The drop therefore reflects information missing from the shifted data, not a worse representation. Conditional probing cannot differentiate without a ceiling score. Target Black/White/Empty Mine/Theirs/Empty Layer 5 6 Estimated floor (rung O3O_3) 0.762 (0.754–0.768) 0.803 (0.797–0.808) Ceiling (exact) 1.000 1.000 Headroom 0.238 0.197 Probe on O 0.768 (0.762–0.774) 0.808 (0.803–0.814) Hidden probe 0.745 (0.738–0.752) 0.986 (0.981–0.990) Probe on [O;H][O;H] 0.767 (0.760–0.773) 0.989 (0.984–0.992) Normalized score UHU_H -0.069 0.929 Conditional normalized score UH|OU_H O 0.021 0.942 Conditional probe (B=OB=O), AccO,H−AccOAcc_O,H-Acc_O -0.001 0.180 Table 11: Othello-GPT: normalized headroom and conditional probing with baseline B=OB=O (rung O3O_3: disc placement without flips). Test accuracy of the linear decoder, one probe per square; the ceiling is exactly 1 because the board is a function of the moves. The estimated floor is the linear decoder of the floor ladder (Table 6), fitted once across all squares; the probe on O fits one linear probe per square on the same features and on the probe’s rows, which exclude the final move, so the two differ slightly. The conditional probe is measured from the probe on O; UHU_H and UH|OU_H O are measured from the estimated floor. L.3 Othello-GPT. We fit one linear probe per square on the residual stream (block 5 for black/white/empty, block 6 for mine/theirs/empty; 1,000 games split 60/20/20), with O the move-sequence features of rung O3O_3 (disc placement without flips) and a ceiling of 1, since the board is a function of the moves. Because every probe here, including the floor, is linear, the floor (0.76 and 0.80) is slightly lower than the decoder-bank floor of Appendix K. The standalone and conditional scores reach the same verdict (Table 11). For black/white/empty, H adds nothing beyond disc placement: the conditional gain is −0.001-0.001, UH=−0.07U_H=-0.07 and UH|O=0.02U_H|O=0.02. For mine/theirs/empty, H captures most of what the flips add: the conditional gain is 0.180.18, UH=0.93U_H=0.93 and UH|O=0.94U_H|O=0.94. The normalized and conditional normalized scores agree in both cases. Therefore, both methods support the conclusion of Section 4.2.3: a linear readout of the absolute board is explained by disc placement, while with mine/theirs labels a linear probe recovers the board state that requires simulating flips. Appendix M Additional figures and tables logS1 S_1 R2R^2 τ2τ^2 R2R^2 MSEmodel/MSEoracleMSE_ model/MSE_ oracle MSE ID OOD ID OOD ID OOD OOD/ID xs 0.28 ± 0.20 0.30 ± 0.04 0.22 ± 0.20 0.24 ± 0.03 1.05 ± 0.01 5.05 ± 2.25 28.3 s 0.60 ± 0.04 0.29 ± 0.03 0.54 ± 0.03 0.26 ± 0.02 1.02 ± 0.00 2.44 ± 0.37 14.2 m 0.60 ± 0.03 0.34 ± 0.03 0.60 ± 0.04 0.28 ± 0.03 1.03 ± 0.00 2.58 ± 0.73 14.9 l 0.61 ± 0.02 0.35 ± 0.05 0.58 ± 0.05 0.29 ± 0.05 1.03 ± 0.01 2.23 ± 0.59 12.9 xl 0.64 ± 0.05 0.40 ± 0.02 0.64 ± 0.02 0.34 ± 0.02 1.03 ± 0.00 2.08 ± 0.51 12.0 Table 12: In-distribution (ID) vs. out-of-distribution (OOD) probe R2R^2 and prediction MSE at the final checkpoint (mean ± sample standard deviation across five seeds). MSE is measured at the final position; the oracle is the conditional-Bayes estimator S2/(S1+1)S_2/(S_1+1), which uses the true τ2τ^2. Target Floor Ceiling Headroom Representation probe UHU_H Neuroscience Force x 0.11±0.020.11± 0.02 0.99±0.010.99± 0.01 0.88±0.030.88± 0.03 0.75±0.010.75± 0.01 0.72±0.020.72± 0.02 Force y 0.09±0.020.09± 0.02 1.00±0.001.00± 0.00 0.91±0.020.91± 0.02 0.57±0.020.57± 0.02 0.53±0.030.53± 0.03 Chemistry HOMO 0.39±0.000.39± 0.00 0.63±0.000.63± 0.00 0.24±0.000.24± 0.00 0.56±0.010.56± 0.01 0.69±0.070.69± 0.07 LUMO 0.50±0.020.50± 0.02 0.69±0.010.69± 0.01 0.19±0.010.19± 0.01 0.66±0.010.66± 0.01 0.82±0.090.82± 0.09 Meta-analysis logS1 S_1 (ID) 0.01±0.000.01± 0.00 0.82±0.010.82± 0.01 0.81±0.020.81± 0.02 0.64±0.050.64± 0.05 0.78±0.060.78± 0.06 τ2τ^2 (ID) 0.00±0.000.00± 0.00 0.76±0.010.76± 0.01 0.76±0.010.76± 0.01 0.64±0.020.64± 0.02 0.84±0.040.84± 0.04 logS1 S_1 (OOD) 0.09±0.010.09± 0.01 0.47±0.020.47± 0.02 0.37±0.020.37± 0.02 0.40±0.020.40± 0.02 0.82±0.080.82± 0.08 τ2τ^2 (OOD) 0.00±0.000.00± 0.00 0.40±0.020.40± 0.02 0.40±0.020.40± 0.02 0.34±0.020.34± 0.02 0.85±0.080.85± 0.08 Single-cell Synthetic −0.02±0.02-0.02± 0.02 0.87±0.010.87± 0.01 0.89±0.020.89± 0.02 0.60±0.030.60± 0.03 0.70±0.030.70± 0.03 10k human 0.60±0.020.60± 0.02 0.97±0.010.97± 0.01 0.38±0.020.38± 0.02 0.81±0.020.81± 0.02 0.57±0.030.57± 0.03 20k donors (pooled) 0.13±0.040.13± 0.04 0.97±0.000.97± 0.00 0.84±0.040.84± 0.04 0.57±0.030.57± 0.03 0.52±0.030.52± 0.03 40k breast 0.15±0.060.15± 0.06 0.87±0.020.87± 0.02 0.72±0.040.72± 0.04 0.35±0.040.35± 0.04 0.28±0.050.28± 0.05 40k lung 0.06±0.060.06± 0.06 0.86±0.020.86± 0.02 0.80±0.050.80± 0.05 0.44±0.050.44± 0.05 0.47±0.050.47± 0.05 Table 13: Headroom-normalized decodability across domains (mean ± sample standard deviation across five seeds). UH=(RH2−ρO2)/(ρI2−ρO2)U_H=(R_H^2- _O^2)/( _I^2- _O^2) is the fraction of headroom recovered by the representation probe. Neuroscience uses the O4O_4 pre-bump floor and a decoder-bank probe on Area 2 response counts, fitted within each recording session; meta-analysis uses the exact reference points of Table 3 with final-checkpoint xl last-layer probes; single-cell data use reference points estimated at full sequencing depth and a probe on frozen scGPT representations (64 PCA components); chemistry uses the O4O_4 physicochemical floor and a probe on the LoRA-fine-tuned TabPFN.