How High Is 0.6? Floors, Ceilings, and Headroom in Interpretability Probing

Pranjal Garg

cs.CLOct 6, 2026

How High Is 0.6? Floors, Ceilings, and Headroom in Interpretability Probing

Authors: Pranjal Garg

Organizations: 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 $R^2$ 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.

Figures & tables

Figure 1: Schematic of the floor–ceiling normalized probe $R^{2}$ . A probe’s raw decodability ( $R^{2}_{H}$ ) for a target $T$ is normalized between an estimated floor ( $\hat{\rho}_{O}^{2}$ ), achievable from restricted information $O$ , and an estimated ceiling ( $\hat{\rho}_{I}^{2}$ ), achievable using full information $I$ . The gap between them defines the headroom ( $\Delta$ ).

Figure 2: In-distribution and OOD learning trajectories. Probe decodability for $\log S_{1}$ and $\tau^{2}$ and final-position model MSE across training (xl model).

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.

Figure 4: Raw and headroom-normalized decodability across domains. Left: $R^{2}$ of the floor, the ceiling and the representation probe. Right: the normalized decodability $U_{H}=(R_{H}^{2}-\hat{\rho}_{O}^{2})/(\hat{\rho}_{I}^{2}-\hat{\rho}_{O}^{2})$ , 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 .

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 ( $R^{2}$ for places and dates, accuracy otherwise). Right: normalized probe score, $U_{H}$ , with error bars from the floor’s CI.

Figure 6: Headroom as evidence quality and target sensitivity vary, in the synthetic single-cell model (with binomial thinning). Evidence quality $p_{X}$ controls how well the data identify the latent variable; target sensitivity $p_{T}$ controls how strongly the target depends on it. Low $p_{X}$ gives weak identification, low $p_{T}$ gives weak target sensitivity, and low values of both give both.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Model size	$d$	$h$	$L$	Parameters	Peak learning rate	Epoch budget
xs	16	2	1	3,345	$4\times 10^{-4}$	3,300
s	32	4	2	25,537	$4\times 10^{-4}$	3,300
m	64	4	2	100,225	$5\times 10^{-5}$	3,000
l	64	4	4	200,193	$5\times 10^{-5}$	3,500
xl	128	8	4	793,601	$5\times 10^{-5}$	3,500

Appendix

Table 1: Transformer model configurations.

d

is the residual-stream width,

h

the number of attention heads (head width

d/h

), and

L

the number of transformer blocks; the block MLP has hidden width

4d

. Parameter counts include the input embedding, all blocks, and the scalar readout.

Decoder	Fixed specification
GBM	HistGradientBoostingRegressor with 400 maximum iterations, learning rate $0.06$ , unrestricted depth, early stopping, and a $0.15$ internal validation fraction.
Shallow GBM	HistGradientBoostingRegressor with 600 maximum iterations, learning rate $0.04$ , maximum depth $3$ , early stopping, and a $0.15$ internal validation fraction.
MLP	Standardization followed by an MLP with hidden widths $(256,128)$ , $L_{2}$ penalty $10^{-4}$ , 600 maximum iterations, early stopping, and patience 25.
Linear	Standardization followed by RidgeCV with $\alpha\in\operatorname{logspace}(-6,4,21).$

Appendix

Table 2: The decoder bank. These definitions are fixed across domains, representations, targets, and random seeds.

Draw	Meaning
$\log\lambda_{g}\sim\mathcal{N}(\mu_{\lambda},\sigma_{\lambda}^{2})$	mean expression
$b_{g}\sim\mathrm{U}(0,b_{\max})$	overdispersion (latent $Z$ )
$\log d_{gi}\sim\mathcal{N}(\mu_{d},\sigma_{d}^{2})$	sequencing depth of cell $i$
$\theta_{gi}\sim\mathrm{Gamma}\big(\mathrm{shape}=\lambda_{g}/b_{g},\ \mathrm{scale}=b_{g}\big)$	true expression rate
$y_{gi}\mid\theta_{gi},d_{gi}\sim\mathrm{Poisson}(d_{gi}\theta_{gi})$	observed count

Appendix

Table 9

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 $R^{2}$ values where the exact floor is near zero, not negative population floors.

			Floor	Ceiling	Headroom
Domain	Target	Condition
Meta-analysis	$\log S_{1}$	ID	$0.012\pm 0.004$	$0.822\pm 0.014$	$0.809\pm 0.017$
	$\log S_{1}$	OOD	$0.092\pm 0.011$	$0.466\pm 0.019$	$0.374\pm 0.017$
	$\tau^{2}$	ID	$0.000\pm 0.000$	$0.758\pm 0.013$	$0.758\pm 0.013$
	$\tau^{2}$	OOD	$0.000\pm 0.000$	$0.404\pm 0.017$	$0.404\pm 0.017$
Synthetic genomics	$T_{g}$	Full depth	$0.000\pm 0.000$	$0.880\pm 0.001$	$0.880\pm 0.001$

Appendix

Table 3: Exact reference points for the meta-analysis and synthetic-genomics validation experiments (mean

\pm

sample standard deviation across five seeds).

Baseline	Target	Floor mean $\pm$ SD	Ceiling mean $\pm$ SD	Headroom mean $\pm$ SD	Probe mean $\pm$ SD	$U_{H}$ mean $\pm$ SD
Neuroscience
$O_{1}$	Force $x$	$-$ 0.013 $\pm$ 0.019	0.987 $\pm$ 0.009	1.000 $\pm$ 0.025	0.745 $\pm$ 0.014	0.758 $\pm$ 0.019
$O_{1}$	Force $y$	$-$ 0.002 $\pm$ 0.006	0.997 $\pm$ 0.000	0.999 $\pm$ 0.006	0.573 $\pm$ 0.019	0.575 $\pm$ 0.017
$O_{2}$	Force $x$	0.004 $\pm$ 0.009	0.987 $\pm$ 0.009	0.983 $\pm$ 0.016	0.745 $\pm$ 0.014	0.754 $\pm$ 0.018
$O_{2}$	Force $y$	$-$ 0.013 $\pm$ 0.028	0.997 $\pm$ 0.000	1.010 $\pm$ 0.028	0.573 $\pm$ 0.019	0.580 $\pm$ 0.011
$O_{3}$	Force $x$	$-$ 0.002 $\pm$ 0.019	0.987 $\pm$ 0.009	0.989 $\pm$ 0.025	0.745 $\pm$ 0.014	0.755 $\pm$ 0.019

Appendix

Table 4: Estimated reference points and probe scores for the nested baselines

O_{1}\subset O_{2}\subset O_{3}\subset O_{4}

, reported as mean

\pm

sample standard deviation (SD) across five seeds.

Dataset	Floor (95% CI)	Probe $R^{2}$ (7B / 70B)	$U_{H}$ (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

Appendix

Table 5: Spelling floors for the space and time datasets of Gurnee and Tegmark (2024) . The floor is the held-out

R^{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;

U_{H}

uses a ceiling of 1, which can only lower a positive

U_{H}

(Appendix K.1 ).

Floor features	Decoder	Floor acc. (95% CI)	Linear floor acc.	Chance acc.	Occupied acc.	Probe acc. (95% CI)	Linear probe acc.	$U_{H}$	$U_{H}$ linear probe
Black/White/Empty
$O_{1}$ : 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
$O_{2}$ : 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
$O_{3}$ : placement	GBM	0.80 (0.80–0.81)	0.76	0.46	0.64	0.89 (0.89–0.90)	0.74	0.45	$-0.30$
Mine/Theirs/Empty
$O_{1}$ : 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

Appendix

Table 6: Floors, probe accuracy and headroom-normalized decodability for the Black/White/Empty and Mine/Theirs/Empty targets under nested floor feature sets

O_{1}\subset O_{2}\subset 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.

U_{H}

is the fraction of the headroom recovered by the probe.

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

Appendix

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.

Train $\to$ Test	Decoder	Floor acc. (95% CI)	Linear acc.	Published probe acc. (13B / 70B)
larger_than + smaller_than $\to$ sp_en_trans	GBM	0.49 (0.44–0.55)	0.50	0.97 / 0.97
cities + neg_cities $\to$ neg_sp_en_trans	Shallow GBM	0.50 (0.45–0.55)	0.50	0.96 / 0.99
cities $\to$ neg_cities 1 1 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- $R^{2}=0.035$ , linear pseudo- $R^{2}=0.001$ ); 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 ( $0.35$ , below the $0.50$ chance level), while the near-prior linear decoder stays at chance ( $0.47$ ).	Shallow GBM	0.35 (0.32–0.37)	0.47	0.75 / 0.37
larger_than $\to$ smaller_than	Linear	0.00 (0.00–0.00)	0.00	0.07 / 0.55

Appendix

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.

Attribute	Evaluation	Floor acc. (95% CI)	Chance acc.	Published probe acc. (reading / control)	$U_{H}$ (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 $\to$ Llama-2-Chat	0.88 (0.86–0.90)	0.25	–	–

Appendix

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

\to

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).

U_{H}

uses a ceiling of 1 and is unstable here because the headroom is small.

Condition	ID		OOD
Target	$\log S_{1}$	$\tau^{2}$	$\log S_{1}$	$\tau^{2}$
Estimated floor $\hat{\rho}_{O}^{2}$	$0.01\pm 0.01$	$-0.00\pm 0.00$	$0.09\pm 0.01$	$-0.00\pm 0.00$
Estimated ceiling $\hat{\rho}_{I}^{2}$	$0.63\pm 0.02$	$0.66\pm 0.01$	$0.41\pm 0.02$	$0.36\pm 0.03$
Estimated headroom $\hat{\Delta}=\hat{\rho}_{I}^{2}-\hat{\rho}_{O}^{2}$	$0.62\pm 0.01$	$0.66\pm 0.01$	$0.33\pm 0.02$	$0.36\pm 0.03$
Hidden probe $R^{2}_{H}$	$0.60\pm 0.04$	$0.64\pm 0.02$	$0.40\pm 0.02$	$0.34\pm 0.02$
Probe on $[O;H]$ , $R^{2}_{O,H}$	$0.59\pm 0.04$	$0.64\pm 0.02$	$0.40\pm 0.02$	$0.34\pm 0.03$

Appendix

Table 10: Meta-analysis transformer (xl, last layer, final position): normalized headroom and conditional probing with baseline

B=O

. Every probe, floor and ceiling is the linear decoder ; the floor and ceiling are estimated from

O

and

I

. Mean

\pm

SD over five seeds.

Target	Black/White/Empty	Mine/Theirs/Empty
Layer	5	6
Estimated floor (rung $O_{3}$ )	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)

Appendix

Table 11: Othello-GPT: normalized headroom and conditional probing with baseline

B=O

(rung

O_{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

;

U_{H}

and

U_{H\mid O}

are measured from the estimated floor.

	$\log S_{1}$ $R^{2}$		$\tau^{2}$ $R^{2}$		$\mathrm{MSE}_{\rm model}/\mathrm{MSE}_{\rm oracle}$		MSE
	ID	OOD	ID	OOD	ID	OOD	OOD/ID
xs	0.28 $\pm$ 0.20	0.30 $\pm$ 0.04	0.22 $\pm$ 0.20	0.24 $\pm$ 0.03	1.05 $\pm$ 0.01	5.05 $\pm$ 2.25	28.3
s	0.60 $\pm$ 0.04	0.29 $\pm$ 0.03	0.54 $\pm$ 0.03	0.26 $\pm$ 0.02	1.02 $\pm$ 0.00	2.44 $\pm$ 0.37	14.2
m	0.60 $\pm$ 0.03	0.34 $\pm$ 0.03	0.60 $\pm$ 0.04	0.28 $\pm$ 0.03	1.03 $\pm$ 0.00	2.58 $\pm$ 0.73	14.9
l	0.61 $\pm$ 0.02	0.35 $\pm$ 0.05	0.58 $\pm$ 0.05	0.29 $\pm$ 0.05	1.03 $\pm$ 0.01	2.23 $\pm$ 0.59	12.9
xl	0.64 $\pm$ 0.05	0.40 $\pm$ 0.02	0.64 $\pm$ 0.02	0.34 $\pm$ 0.02	1.03 $\pm$ 0.00	2.08 $\pm$ 0.51	12.0

Appendix

Table 12: In-distribution (ID) vs. out-of-distribution (OOD) probe

R^{2}

and prediction MSE at the final checkpoint (mean

\pm

sample standard deviation across five seeds). MSE is measured at the final position; the oracle is the conditional-Bayes estimator

S_{2}/(S_{1}+1)

, which uses the true

\tau^{2}

Target	Floor	Ceiling	Headroom	Representation probe	$U_{H}$
Neuroscience
Force $x$	$0.11\pm 0.02$	$0.99\pm 0.01$	$0.88\pm 0.03$	$0.75\pm 0.01$	$0.72\pm 0.02$
Force $y$	$0.09\pm 0.02$	$1.00\pm 0.00$	$0.91\pm 0.02$	$0.57\pm 0.02$	$0.53\pm 0.03$
Chemistry
HOMO	$0.39\pm 0.00$	$0.63\pm 0.00$	$0.24\pm 0.00$	$0.56\pm 0.01$	$0.69\pm 0.07$
LUMO	$0.50\pm 0.02$	$0.69\pm 0.01$	$0.19\pm 0.01$	$0.66\pm 0.01$	$0.82\pm 0.09$

Appendix

Table 13: Headroom-normalized decodability across domains (mean

\pm

sample standard deviation across five seeds).

U_{H}=(R_{H}^{2}-\rho_{O}^{2})/(\rho_{I}^{2}-\rho_{O}^{2})

is the fraction of headroom recovered by the representation probe. Neuroscience uses the

O_{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

O_{4}

physicochemical floor and a probe on the LoRA-fine-tuned TabPFN.

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