Minimal-Pair Probing: Present, Readable, Used

Sources#

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

Summary#

Aur Shalev Merin (independent researcher), Present but Hard to Read: Probing Latent Reasoning with Minimal Pairs (arXiv 2610.28498, 2026-10-20, empirical, pre-registered), is a methods paper about what a linear probe on a model's state can and cannot tell you. Probing studies of latent reasoning usually read the state at the summary position, the last prompt token where the answer begins, and treat a failed probe as evidence that an intermediate value is absent. The paper shows that inference fails in both directions:

A failed probe says little about absence. A digit copied from the question is demonstrably present at the last prompt token, in a code shared across questions. A probe that compares a question's ten variants reads it at about 0.9. A probe that sees one prompt at a time, the usual setting, reads it at 0.12–0.24 with tens of training templates, and at 0.36 in Ouro-1.4B and 0.19 in Qwen2.5-1.5B after 5,249 distinct training questions.
A successful probe says little about computation. Every token varied, including a number in an irrelevant sentence and a colour word, leaves a readable trace at the same position. A probe that reads a "computed" value may be reading the copied inputs it is a function of.
A readable value need not be used. In Ouro, the unwritten intermediate a + b of a two-step problem is readable just before the answer's units digit. Clamping it to another problem's value from the first loop on does not move the written digit, while clamping the answer does in 91% of pairs.

The three findings separate three questions that a single probe accuracy runs together: is the value present, is it readable from a new prompt, and is it used. Each needs its own instrument: minimal pairs, a single-prompt probe with input controls, and an intervention.

Minimal pairs: presence without readability#

The test takes one question per template and one of its multi-digit numbers, and makes ten copies that differ only in that number's last digit (0–9). The copies are identical token for token except one, so any difference in a later state is caused by that digit. Subtracting the family's mean removes what the ten variants share. A probe trained on those centred states, on templates it never saw, reads the digit at 0.92–0.97 in Ouro, while the standard probe (PCA to 256 components, then multinomial logistic regression, scored on held-out templates) reads it at 0.15–0.23, near the 0.10 chance rate (Table 1, ranges over loop states).

Three controls make the code a copy rather than the answer in progress:

The code is shared. A probe trained on needed numbers reads numbers in an added irrelevant sentence ("Unrelated fact: a nearby park has 43 benches.") at 0.78–0.97, and a colour word in that sentence at 0.99–1.00 with the minimal-pair probe.
It is not the answer. The irrelevant number changes the first token of Ouro's answer in at most 2.6% of families.
It is not a magnitude. A linear value component exists (R² 0.57), but the probe's errors are no closer to the true digit than random guesses (mean distance 3.8 against 3.7), and the top one to four principal components read the digit at only 0.21–0.45. It behaves like a code for which token was there, which is what an attention head copying a token predicts.

A pre-registered replication in the standard transformer Qwen2.5-1.5B, with the same families, probes and code, passed its rule. The minimal-pair probe averaged 0.88, the standard probe and the best structured linear decoder (a mass-mean probe with a shrunk covariance) stayed at 0.12–0.14, and an MLP probe did no better. Two further pre-registered studies gave each model 6,249 GSM8K questions with a randomized units digit. At 5,249 training questions the best single-prompt probe reaches 0.19 (Qwen) and 0.36 (Ouro), against 0.91 and 0.96 for minimal pairs on held-out questions. Both curves are still rising (Figure 2), so the claim is a gap at the data sizes tested, not unreadability in principle.

The minimal-pair probe is a test for presence, not a decoder: it needs all ten variants of the test question. The two probes answer different questions, and here the answers come apart.

Why the digit is hard to read: small, not crowded#

The obvious explanation is that the digit's directions are unusually busy. They are not. Within its own 9-dimensional subspace the digit's signal is a few percent of the context variance around it, but a subspace fitted with shuffled labels is just as crowded (ratio 0.83 in Ouro, 1.01 in Qwen). Any change within a question lands on the directions along which questions vary most.

What differs between changes is their size. In Qwen, a pre-registered null swapped the digit for one of ten random words, or added random noise of a token embedding's size. All three changes land on equally busy directions, but the word swap is read by a standard probe at 0.24–0.74 while the digit and the noise read at 0.11–0.15. The ten digits start about half as far apart in input-embedding space as random words do, but that does not explain it: the word and noise variants start equally far apart, and only the word change reaches the last token. A pre-registered linear-response study found the word swap's effect almost entirely nonlinear (its first-order prediction is orthogonal to the actual change and 4.5–15 times too small). What reaches the summary position depends on which token is present, not on how far the input moves.

A methodological side result: causal tracing corrupts inputs with Gaussian noise on token embeddings, and here noise of a token embedding's size barely reaches the summary position while a token swap does. That supports the recommendation to replace tokens rather than add noise.

The author places this outside superposition in the usual sense: the digit shares its directions with variation between contexts, not with a fixed set of other features. The neuroscience analogue cited is information-limiting correlations, where only noise along the signal's own direction limits a decoder.

A protocol for computed values#

Within a minimal-pair family a computed value is a fixed function of the varied digit, so reading it proves nothing. The protocol varies the operands' units digits together and checks a candidate value against five alternatives:

Copy ceiling. Score every label as its excess over the same probe run on one-hot copies of the inputs, which can read any function of a single digit. Add any input-derived quantity the model can see; the strict version adds the carry and borrow.
Prime modulus. Mod 10, many combinations share their parity with a + b or with the answer digit. In a synthetic state holding only copies and the answer, a + b + c mod 10 read 0.14 above its ceiling; mod 5 every label stayed within 0.04.
Every combination. Rank the target against all 31 forms xa + yb + zc mod 5, not a few chosen controls, because periodic number features make some combinations easier to read.
Sign-flipped twins. Compare a + b with a − b. A generic mixture has no reason to prefer the needed sign, and swapping the operation separates the needed sign from a fixed preference.
Mixtures of the output. Compare the target with the other forms on its line through the answer (answer + kc), and calibrate against synthetic states that mix the inputs generically.

The fifth step is where the paper's own results get weaker, and it says so.

What the protocol finds in Ouro#

The test bed is Ouro-1.4B, a pretrained looped model that runs its 24 layers up to four times. Answer-only GSM8K accuracy rises from 6.5% after one loop to 23% after four, but almost all of the gain comes at loop 2, where Ouro improves equally on HellaSwag and MMLU (Appendix A). Accuracy alone therefore cannot say whether the extra loops act as a hidden chain of thought.

Three pre-registered studies used two-step problems whose answer is a + b − c, so the intermediate a + b is never written. Study 3 is the clean one: pre-registered with its bar unchanged, it swaps "gave her B more" for "took B of her", so plus problems need a + b and minus problems need a − b.

The readable sign follows the operation. At P1, the answer token just before its units digit, after loop 4: on plus problems a + b reads +0.207 over its copy ceiling against +0.082 for a − b. On minus problems the mirror image holds (+0.267 against +0.078). The interaction is I = +0.157 [0.141, 0.172], labelled FOLLOWS. A fixed preference for differences is small, F = +0.032. Adding the carry and borrow to the ceiling leaves I = +0.126.
Before the answer starts, nothing is there. At P0, the last prompt token, I is −0.013 once the carry and borrow are included. A follow-up at the period ending the sentence that gives b, where a running total would sit, found no sign-following (I at most +0.018).
Mixing explains about half of it. A pre-registered synthetic null built states with copies of a, b and c, a strong answer code, no a ± b term, and a generic mixture. Those worlds produce I of +0.09 on average and at most +0.14, against +0.157 in Ouro. Pairwise mixing also tops the answer line by as much as Ouro does, so that test does not separate a held value from mixing. And since a + b equals the answer plus c, no probe label can separate "a + b formed first" from "answer formed, c added back". The answer is readable as early as a + b.

Patching says the value is not used there. For two plus problems sharing c, the paper replaces x's component in a 9-dimensional subspace at P1 with partner x′'s, runs the remaining loops, and checks whether Ouro writes x′'s digit. Patching the answer subspace at the end of loop 3 moves the prediction in 90% of pairs. Patching a + b does so in 1.7%, slightly less than the a − b control, and with the answer held fixed by construction (equal-answer pairs) it hits 0.3%. All four tests are NOT-USED.

The obvious objection is that by loop 3 an intact answer overrides a weak a + b. A further pre-registered study removes that escape. Clamping the component to the partner's at the ends of loops 1, 2 and 3 moves the written digit in 91.4% of pairs for the answer and in at most 4% for a + b, within 0.02 of the a − b and random controls. Both clamp tests are NOT-USED, including equal-answer pairs. A single patch at the end of loop 1 is unscorable, because loops 2–4 undo any patch there, even one of the answer (hit 0.009), which fits Ouro re-deriving its state each loop. The paper's reading: at P1, in the subspaces tested, a + b is readable but not used.

Loop dynamics add one shape: within each loop, the second half of the layers makes combinations of the numbers readable, and unneeded ones then fade (a + c, never needed, peaks at +0.46 after loop 2 and falls to +0.09 by loop 4, while a + b peaks after loop 3 and keeps +0.22).

What this does and does not establish#

Scale. Two models of about 1.5B parameters. The computation studies use Ouro alone, on problems of one form (a ± b − c, two-digit numbers), read where the gold answer is fed in.
Only P1 was patched, and only in subspaces spanned by class means. An a + b held at another position (the question sentence, read through attention) or in a code outside those subspaces would be missed. Subspace patches can mislead through dormant directions, but the author notes that risk bears on positive results more than on a null.
Computed values may be encoded differently from copies. The paper did not test whether the presence-readability gap holds for them.
Mixed pre-registration record, reported in full. Of ten pre-registered probing studies, one was unscorable and two UNCLEAR. Study 2's bar was set after Study 1 missed its own. Study 3's loop-2 patch point and equal-answer pairs were added after extraction, before analysis. The §5.2 comparisons and the loop dynamics are post hoc on saved states. Appendix C lists every analysis's status.

The single-author, independent-researcher provenance carries no product conflict. The paper's own transparency (frozen pre-registrations in a public repository, every unscorable or UNCLEAR outcome reported) is the main reason to keep the empirical tier without a downgrade.

Why it matters for monitoring#

Every activation monitor in the corpus is some version of a probe at a position, and this paper prices the three inferences such a monitor makes.

Absence. A single-position probe that reads nothing does not show the information is gone. Invisible Reasoning (Filler-Token Latent Computation)'s filler-span experiment and White-Box Activation Monitoring's pooled probes report that single-position probing "yields nothing". By this paper's result, that null is weak evidence about presence, though it is still evidence about what a deployable single-prompt detector could read.
Computation. A probe that succeeds at the summary position may be reading copies of the inputs. Pre-Reasoning Commitment's end-of-prompt correctness probe (AUROC ≈ 0.76) is a readout of something the prompt already contains. That is consistent with the page's claim that the answer is set before reasoning, but by itself it cannot distinguish a computed commitment from a well-chosen function of the inputs. The copy-ceiling control is the piece that would.
Use. Readability is not causal role. This is the same gap WorkspaceBench faces when it scores readers on intermediate variables without intervening, and the reason the workspace work pairs its lens with swap and ablation tests.

The constructive technique transfers. Subtracting a family mean is the within-prompt version of Model Diffing's finetune-minus-base subtraction, and the same move as matched-twin hard negatives in Retroactive Permission Laundering: remove what the comparison shares, and a small shared signal becomes readable. For Chain-of-Thought Monitorability, the result is a caution on the argument that looped or latent-reasoning models leave activation probing as the fallback channel: in the one looped model tested, the probe-visible intermediate was not the one the model used.

Open Questions#

Does the presence-readability gap hold for computed values, not just copies? The paper tested only copied tokens. Settler: a minimal-pair probe and a single-prompt probe on an intermediate that patching shows is used, at the position where it is used.
Does single-prompt readability of a present value close with scale or data? Both learning curves were still rising at 5,249 questions on ~1.5B models. Settler: the same minimal-pair versus single-prompt comparison on a ≥7B open-weight model, or with a training set an order of magnitude larger.

Connections#

Invisible Reasoning (Filler-Token Latent Computation): the filler-span probe this page's absence result qualifies, and the computation-without-tokens claim that motivates probing for latent intermediates at all
White-Box Activation Monitoring: the probe family whose three inferences (absent, computed, used) this page separates
Pre-Reasoning Commitment: an end-of-prompt probe at the same summary position, whose success the copy-ceiling control would sharpen
WorkspaceBench: a benchmark that scores readers on intermediate variables, where presence and use are not separately measured
Model Diffing: the same subtraction at a different grain, a finetune minus its base rather than a prompt minus its family mean
Retroactive Permission Laundering: matched authorized twins as hard negatives, another minimal-pair construction for a probe
Chain-of-Thought Monitorability: the oversight case for activation probes on latent reasoning, and the caution that a readable intermediate may not be the computation
The Global Workspace in Language Models (J-space): the readout that is paired with causal swaps, which is the use test this paper insists on

Sources#

Present but Hard to Read: Probing Latent Reasoning with Minimal Pairs — Aur Shalev Merin (independent researcher), Present but Hard to Read: Probing Latent Reasoning with Minimal Pairs, arXiv 2610.28498, 2026-10-20, 20 pp, empirical, pre-registered with a public repository of freezes. Sections used: §1 and Abstract (the three findings); §2 (Ouro-1.4B and Qwen2.5-1.5B setup, probe definitions, P0/P1); §3.1 + Table 1 (minimal pairs in Ouro; shared code, not a magnitude); §3.2 + Table 2 + Figure 2 (the Qwen replication; the 5,249-question curves); §3.3 + Table 3 + App. B (small-not-crowded; random-word and noise nulls; linear response); §4 (the protocol); §5.3 + Tables 6–7 (Study 3's operation swap and patching); §5.4 + Table 8 (clamping and the synthetic mixing null); §5.5 (loop dynamics); §6 (limitations); App. A (accuracy by loop). Parse note: PDF-derived (docling 2.126.0 / docling-mlx 0.1.1, confidence_grade: excellent). verify warned table-collapse on the Appendix C status table (raw line ~394). That table also splits across three docling grids with welded cells, and nothing is cited from it; statuses are taken from the §6 prose and §5 text. Table 4 (Studies 1–2) is a genuine docling cell collapse (every row label and value merged into one cell per column) and is cited nowhere; the Study 1–2 figures used come from §5.1–5.2 prose. Tables 7 and 8 were reconciled against pdftotext -layout on the local PDF and parse clean. The Table 1–3 figures quoted are each restated in prose. Figure 2 viewed: it confirms both single-prompt curves still rising at 5,249 questions and the 0.96 / 0.91 minimal-pair lines