Present but Hard to Read: Probing Latent Reasoning with Minimal Pairs

Aur Shalev Merin

cs.LGOct 20, 2026

Present but Hard to Read: Probing Latent Reasoning with Minimal Pairs

Authors: Aur Shalev Merin

Organizations: Independent researcher

Abstract

Probing studies of latent reasoning often read a model's state where the answer begins and take a failed probe as evidence that a value is absent. We test this with a digit copied from the question, in the looped model Ouro-1.4B and the standard transformer Qwen 2.5-1.5B. Comparing prompts that differ only in that digit shows it is present at the last prompt token, in a code shared across questions (readout about 0.9). A probe that sees one prompt at a time, the usual setting, barely reads it: 0.36 in Ouro and 0.19 in Qwen even after 5,249 training questions. The digit's change is small rather than crowded: random noise lands on equally busy directions, while a random word in its place moves the state far more and is readable. Since every varied token leaves a trace, a successful probe is ambiguous too. We give a protocol, including a synthetic null for generic mixing, for testing computed values against copies and mixtures of the inputs. In pre-registered studies of two-step problems it finds a small signal for the unwritten intermediate a+b just before Ouro writes the answer's last digit, and its sign follows the operation, though mixing alone produces much of that sign-following part. Holding that value at another problem's from the first loop on does not move the written digit toward the other problem's, while the same patch of the answer does: the value is readable but not used.

Figures & tables

varied token	minimal-pair probe	standard probe	trace
a number the problem needs	0.92–0.97	0.15–0.23	5–7%
a number in an irrelevant sentence	0.96–1.00	0.14–0.30	3–6%
a colour word in that sentence	0.99–1.00	0.19–0.40	4–6%

Table 1: Ouro, the last prompt token (chance 0.10); ranges over mid 1 and the ends of loops 1–4. The trace is the spread of a family’s ten variants as a share of the spread between questions.

Figure 1: Reading a copied token in Ouro (chance 0.10). Left: the last prompt token, by loop. Middle: word problems after loop 4, by position; “+k” is k tokens after the digit. Right: the last prompt token, for a number the problem needs, a number in an added, irrelevant sentence, and a colour word in that sentence. In the right panel, solid lines show the minimal-pair probe and dotted lines the standard probe, each the mean over two insertion points; the right panel’s needed number comes with the added sentence, so it differs slightly from Table 1 ’s first row.

	Ouro-1.4B (looped)	Qwen2.5-1.5B (standard)
minimal-pair probe	0.92–0.97	0.75–0.98 (mean 0.88)
standard probe	0.15–0.23	0.12–0.14
structured decoder	0.17–0.21	0.12–0.14
MLP probe	0.11–0.17	0.10–0.11
gain, half to all templates	+0.015	+0.004
Gaussian ceiling of the decoder	0.75–0.92	0.53–0.73

Table 2: The last prompt token in a looped and a standard model, 77 distinct questions (chance 0.10). Ouro: end-of-loop states, and mid 1 where Appendix B gives it; Qwen: after layers 7, 14, 21 and 28, the pre-registered range. The gain is the standard probe’s from half to all training templates.

Figure 2: Single-prompt probes trained on more and more distinct questions (last prompt token; best over Qwen’s layers 7–28 or Ouro’s loop ends; 1,000 fixed test questions). Full-state: logistic regression on all dimensions,

C

from

10^{-4}

to 10; PCA-256: the standard probe. Dotted: minimal pairs on 150 (Qwen) and 100 (Ouro) of the test questions. The best state is picked on the test set, which favours the single-prompt probe.

change	input distance	overlap / shuffled	signal/noise in subspace	standard probe	minimal-pair probe
a different digit	0.70	0.68–1.19	0.0009–0.0037	0.12–0.14	0.75–0.98
a random word	1.42	0.50–1.06	0.011–0.205	0.24–0.74	0.95–1.00
random noise	1.46	0.75–1.24	0.0001–0.0010	0.11–0.15	0.76–0.99

Table 3: Qwen2.5-1.5B, last prompt token, layers 7–28 (pre-registered, except the input distance, a descriptive check). Input distance: mean distance between the ten variants’ input embeddings. Overlap / shuffled: each change’s overlap over that of its own shuffled-label subspace. Signal/noise: inside the change’s own 9-dimensional subspace. The same ten words and ten noise vectors are used in every family.

	Study 1	Study 2 (new templates)
$a+b$ at P1	+0.205 [0.172, 0.236]	+0.219 [0.194, 0.244]
margin over best control	+0.118 [0.094, 0.141] ( $a-b$ )	+0.075 [0.052, 0.098] (answer $+\,2c$ )
bar for the margin	0.15	0.025 (set after Study 1)
label	UNCLEAR	passes; fails Study 1’s 0.15
$a+b$ at P0	$-0.016$	$-0.007$
bare addition at P0	sum 0.71 vs ceiling 0.10 (no controls)	$a+b$ +0.372 vs $a-b$ +0.463: mixing

Table 4: Pre-registered results, after loop 4: excess over the copy ceiling. Study 1 was scored mod 10 against

a-b

and

a+c

; Study 2 mod 5 against

a-b

a+c

and two mixtures of the answer with

c

Figure 3: Study 2, after loop 4, mod 5: excess of the minimal-pair probe over each label’s copy ceiling. Left: bare addition at the last prompt token. Right: word problems at P1, just before the answer’s units digit.

needed vs flipped	Study 1, loop 4	Study 2, loop 4	Study 1, loop 3	Study 2, loop 3
$a+b$ vs $a-b$	+0.087 [0.069, 0.105]	+0.130 [0.105, 0.154]	+0.173	+0.161
$a-c$ vs $a+c$	+0.122 [0.093, 0.152]	+0.065 [0.038, 0.092]	+0.064	+0.044
$b-c$ vs $b+c$	+0.052 [0.027, 0.079]	+0.042 [0.015, 0.069]	$-0.032$	$-0.041$

Table 5: Sign-flipped twins at P1 (excess over each label’s copy ceiling; difference with 97.5% interval over families). Only the first row has a plus as the needed sign.

Figure 4: Every mod-5 combination of the operands’ units digits at P1, after loop 4: excess over its own copy ceiling, with 97.5% intervals over families. Each colour is one line through the answer (answer

+\,k\,\times

one copied number); the dark bar is that line’s two-number member (

a+b

a-c

b-c

). Grey: combinations on no answer line. The answer digit and the copies are not shown.

$I=+0.157$ [0.141, 0.172]: FOLLOWS; fixed preference $F=+0.032$ [0.012, 0.053]
	plus problems (need $a+b$ )	minus problems (need $a-b$ )
excess of $a+b$	+0.207	+0.078
excess of $a-b$	+0.082	+0.267
needed minus flipped	$D_{+}=+0.124$	$D_{-}=+0.189$

Table 6: Study 3, operation swap (pre-registered): P1 after loop 4, excess over the copy ceiling (mod 5), with a 97.5% interval over twin families.

pairs	patched after	answer	$a+b$	$a-b$	random	statistic	label
		hit rate by patched subspace
same $c$	loop 3	0.902	0.017	0.032	0.001	$-0.015$ [ $-0.027$ , $-0.004$ ]	NOT-USED
same $c$	loop 2	0.331	0.007	0.006	0.002	+0.001 [ $-0.005$ , +0.007]	NOT-USED
equal answer	loop 3	–	0.003	0.002	0.000	+0.001 [ $-0.003$ , +0.007]	NOT-USED
equal answer	loop 2	–	0.003	0.003	0.001	+0.001 [ $-0.002$ , +0.003]	NOT-USED

Table 7: Study 3, patching at P1 on plus problems (pre-registered; the loop-2 point and the equal-answer pairs were added before analysis), 2,000 pairs per row. A hit is the digit the patch implies. Same-

c

pairs move the answer with

a+b

; equal-answer pairs hold it fixed and have no answer patch. Statistic: hit(

a+b

) minus the larger of hit(

a-b

) and hit(random), with a 99.375% interval.

test	answer patch hit	same- $c$ pairs	equal-answer pairs
patch at end of loop 1	0.009	UNSCORABLE	UNSCORABLE
clamp at ends of loops 1–3	0.914	$-0.016$ [ $-0.034$ , +0.001]	+0.005 [ $-0.001$ , +0.012]
answer ablated, end of loop 3	–	+0.005 [ $-0.011$ , +0.015]	$-0.001$ [ $-0.016$ , +0.013]

Table 8: Patching

a+b

at P1 without a formed answer to override it (pre-registered), 2,000 pairs of each kind. Statistic: hit(

a+b

) minus the larger of hit(

a-b

) and hit(random), with a 99.17% interval (six tests). In the ablation test the patch is restricted to directions outside the answer subspace; that test has no positive control, and the ablation alone changes the digit in about 68% of pairs.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Answer-only GSM8K accuracy by loop, with 95% Wilson intervals.

state	overlap	shuffled	signal/noise in subspace	structured decoder	Gaussian ceiling	standard probe 50% / 100%	no PCA	MLP
mid 1	11.7	10.7	0.005	0.17	0.75	0.14 / 0.15	0.16	0.11
end 1	7.2	9.1	0.014	0.17	0.90	0.18 / 0.19	0.18	0.13
end 2	6.5	7.9	0.023	0.21	0.92	0.21 / 0.23	0.24	0.17
end 3	9.2	10.1	0.011	0.20	0.88	0.17 / 0.18	0.19	0.17
end 4	9.4	11.3	0.010	0.18	0.85	0.17 / 0.19	0.18	0.12

Appendix

Table 9: Ouro, last prompt token, held-out templates. Overlap: context variance per dimension inside the digit subspace over its average per dimension; shuffled: the same for a subspace fitted with the digit labels shuffled within each family. Structured decoder: a mass-mean probe with a shrunk covariance, built from the minimal-pair digit means and the training prompts, applied to raw single prompts. Gaussian ceiling: the same decoder on samples from its own fitted model. Standard probe trained on 50% and 100% of the training templates. No PCA: the standard probe on all 2,048 dimensions,

C

from

10^{-4}

to 10. MLP: one hidden layer of 256 units on the 256 principal components.

Analysis	Sec.	Status	Outcome
Answer-only accuracy by loop (GSM8K; Huginn check)	A	Pre-registered	passed: +16.5 points over four loops
Re-score with the final scorer	A	Re-run, no test	reproduces the numbers; loop 2–4 gain +3.0 [ $-1.5$ , +7.5]
Gain on GSM-Symbolic variants	A	Pre-registered	UNCLEAR: +6.4 points, below the 8.25 bar
Intermediate value at the last prompt token	3.1	Pre-registered	unscorable: sanity check at the majority rate
Copied digits by position; bare arithmetic	3.1	Exploratory	digit read only on its own token; bare operands 0.96–1.00
Minimal pairs; irrelevant number; colour word; value check	3.1	Exploratory, rule in code before data	present at 0.92–1.00 in a shared code; not a magnitude

Appendix

Table 15

Language Model Probing

Language model probing measures what information and capabilities models encode and how those signals relate to their behavior.

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Abstract

Language Model Probing

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