Detection Is Not Enough: Reasoning Traces as Testable Program Hypotheses

Vaidehi Hegde

cs.SEOct 21, 2026

Detection Is Not Enough: Reasoning Traces as Testable Program Hypotheses

Authors: Vaidehi Hegde

Organizations: Indian Institute of Technology (Indian School of Mines) Dhanbad

Abstract

When a language model is deployed, no reference solution exists, so the reasoning trace it writes is the only evidence for judging its answer. The reference-free substitutes each fail differently: self-assessment degrades the answer it was meant to protect, sampling agreement measures stability rather than correctness, and perturbation studies read the expected answer off a human-written template, grading a model rather than a trace. We ask how far a single trace can be verified with no reference solution, no judge model and no weight update, and find, for a frozen LLaMA-2-7B-chat on GSM8K, that detection is not what limits such a pipeline. Checking a trace against itself reaches an AUROC of 0.741 and stops there for a structural reason: two thirds of wrong traces state no false equation, having executed a mistaken plan correctly, and on wrong-plan failures the detector flags a wrong trace less often than a right one. A candidate-derived metamorphic probe exposes many of these failures: we abstract the trace into a program over the question's quantities, perturb one quantity, execute the program, and ask the model the rewritten question once, so every trace supplies its own test. Valid perturbations exist for 937 of 1,319 test questions (71.0%); among the 847 that also have a parseable first trace, supported traces are correct 60.6% of the time against a 31.6% base rate, although the probe also refutes 45.5% of correct traces. No tested policy yields a statistically significant improvement in accuracy. A flag earns its retry only when detector precision exceeds D/(R+D), where R and D are the rates at which a retry repairs a wrong trace and damages a correct one; our repair rate is so low that, at the same retry budget, even a perfect detector would recover an estimated 66 of 1,319 questions. Repair, not detection, is the open problem.

Figures & tables

Figure 1: Overview of the pipeline. A frozen model answers a question with a trace, which is parsed into a chain of arithmetic claims. The static detector (orange) decides from the trace alone whether to spend a retry. The metamorphic probe (blue) reads the trace as a program, predicts the answer to a perturbed question, and compares that prediction with the model’s own answer. No component uses a reference answer or a judge model.

question	finding	section
How far does checking a trace against itself go?	It stalls at AUROC 0.741 because 67.1% of failures contain no false equation; on wrong-plan failures the detector fires less often than on correct traces.	§ 4
Can the missing evidence be obtained without a judge?	Partly. 937 of 1,319 questions (71.0%) admit a valid probe. On the 847 of these with a parseable first trace, the probe refutes 81.4% of wrong-plan and misread-question failures, and supported traces are correct 60.6% of the time against 31.6% in that subset.	§ 5
Does better detection improve accuracy?	Not in our tests: no policy improves accuracy significantly. A flag pays only above precision $D/(R+D)$ , and at the measured repair rate of 12.3% a perfect detector on the same budget would gain an estimated five points.	§ 6

Table 1: Summary of findings.

symbol	meaning
$Q$ , $\bm{q}=(q_{0},\dots,q_{k})$	question and its numeric quantities
$M$ , $\tau=M(Q)$ , $\hat{y}(\tau)$	frozen model, its trace, the trace’s final answer
$\mathcal{E}(\tau)=(e_{1},\dots,e_{m})$	equation chain; each claim $e_{j}$ is $\ell_{j}=c_{j}$
$\alpha$ , $P_{\tau}$	abstraction map and the program it extracts ( Definition 2 )
$Q^{\prime}$ , $\bm{q}^{\prime}$	perturbed question and its quantities
$p_{\mathrm{wrong}}(\tau)$	static detector’s estimate that $\tau$ is wrong

Table 2: Notation.

approach	no reference	no trained verifier	no judge LM	one trace	extra generations
Outcome / process verifiers [ 5 , 10 ]	$\checkmark$	$\times$	$\times$	$\checkmark$	0
Self-correction [ 7 , 12 ]	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\geq 1$
Self-consistency [ 21 ]	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$k$
Template perturbation [ 13 , 27 ]	$\times$	$\checkmark$	$\checkmark$	$\times$	many
Self-verification [ 23 ]	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\geq 1$
VeryTrace [ 25 ]	$\checkmark$	$\checkmark$	$\sim$	$\checkmark$	$\geq 1$

Table 3: Verification approaches for a single deployed answer.

\checkmark

yes,

\times

no,

\sim

partly. “No reference” means the expected answer comes from neither a label nor a template; “one trace” means the verdict concerns the specific trace rather than the model or an answer distribution. The table compares properties, not performance.

	out of fold	development (400)	test (1,319)
detection AUROC	0.781	0.770	0.741
type-head accuracy	0.598	–	0.590

Table 4: Results of the static detector. Out-of-fold estimates use GroupKFold by question on the training split.

Figure 2: Distribution of failure types over the 1,068 training-split pass-1 failures. The two largest classes (red) have sound arithmetic and leave no symbolic sign of the error. Definitions and per-class recall of the type head are in Table 8 .

Figure 3: A wrong-plan failure from the GSM8K test set. Every equation in the trace is true, so the static detector clears it (

p_{\mathrm{wrong}}=0.581

, below the cut of 0.683). The abstraction of Section 5.1 turns the trace into the program shown, with question slots

q_{0}{=}2

q_{1}{=}3

q_{2}{=}4

q_{3}{=}16

in the parser’s ordering; the first step, multiplying the eggs by the number eaten, is the wrong plan. Under both perturbations the answer the program predicts differs from the model’s answer, so both probes refute the trace. The true answers are shown for illustration only; the probe never uses them.

Figure 4: Coverage of the metamorphic probe on the test set, and its verdicts. Detection results use the 847 testable questions with a parseable pass-1 answer; accuracy results use all 1,319 questions, with abstentions falling back to the non-probe policy.

verdict	correct	wrong	total	$P(\text{correct})$	diagnostic	value
supported	146	95	241	60.6%	refutation rate, wrong traces	83.6%
refuted	122	484	606	20.1%	refutation rate, correct traces	45.5%
all	268	579	847	31.6%	likelihood ratio, support	3.32
					likelihood ratio, refutation	1.84
					odds ratio (Fisher $p=5.6\times 10^{-29}$ )	6.1

Table 5: Results of the metamorphic probe on the 847 testable pass-1 traces. Left: verdicts by correctness. Right: the verdicts read as diagnostic tests.

Figure 5: Rate at which each verifier calls a trace wrong, by failure type, on the 579 wrong testable traces. Dashed lines give each verifier’s rate on the 268 correct traces; a verifier is informative on a class only where its bar rises above its own line. The probe does so on every class (all per-class Fisher

p<0.002

, Appendix E ); the static detector falls below its line on WP .

policy	correct	accuracy	$p$
pass-1	367	27.82%	–
replace flagged	375	28.43%	0.50
replace flagged, with metamorphic veto	374	28.35%	0.55
select, symbolic	379	28.73%	0.21
select, metamorphic	382	28.96%	0.091
select, metamorphic + back-substitution	382	28.96%	0.096

Table 6: End-to-end accuracy of every policy on the 1,319 test questions, with exact McNemar tests against pass-1.

Figure 6: The break-even condition. (a) Precision a detector needs before retries help, as a function of the repair rate, at the measured damage rates. (b) Net gain on the 540-flag test budget for three repair rates. Moving from the observed operating point to perfect precision gains 58 questions; doubling

R

at the current precision gains about the same.

change	$\Delta$ (questions)	accuracy (points)
none (observed)	$+8$	$+0.6$
no damage ( $D=0$ )	$+58$	$+4.4$
perfect detector ( $\pi=1$ )	$+66$	$+5.0$
doubled repair rate ( $R=24.5\%$ )	$+66$	$+5.0$
repair rate $R=50\%$	$+187$	$+14.1$
perfect detector, unlimited budget	$+117$	$+8.9$

Table 7: Counterfactual gains on the 540-flag test budget from ( 1 ), with all other quantities at their measured values. These are first-order estimates, not experiments.

Figure 7: Where the first error lies in pass-1 failures. Only failures whose first false equation comes after the first step (green) contain a correct prefix that a repair could preserve. The 17.0% and 15.9% split the 32.9% of failures with a locatable false equation by whether that equation is the first step (51.6%).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

type	what went wrong	train $n$	share	recall	test $n$
WP	wrong plan, sound arithmetic	254	23.8%	69.2%	180
SM	misread the question, solved a different problem	253	23.7%	38.2%	185
CE	computation error inside a good plan	136	12.7%	42.5%	83
ST	a needed intermediate never computed	114	10.7%	62.8%	56
AL	sets up symbols, never solves them	94	8.8%	87.1%	28
UNCL	no rule fired confidently	88	8.2%	51.7%	35

Appendix

Table 8: Failure types over the 1,068 training-split pass-1 failures, with the type head’s out-of-fold recall (GroupKFold by question, 2,568 traces) and the count of each type among the 579 wrong testable test traces of Fig. 5 .

budget	flagged	$p_{\mathrm{wrong}}$ cut	precision	expected net
10%	150	0.90	97.3%	$+20.4$
20%	300	0.83	95.7%	$+36.3$
40%	600	0.683	90.7%	$+44.7$
50%	750	0.60	88.0%	$+37.4$
70%	1,050	0.48	82.1%	$-5.0$
100%	1,500	0.13	71.2%	$-158.5$

Appendix

Table 9: Budget-to-threshold calibration, measured out of fold on the 1,500 training-split pass-1 traces, as recorded in the calibration log. Expected net is

\mathrm{TP}\cdot R-\mathrm{FP}\cdot D

with the training-run rates

R=16.1\%

and

D=76.5\%

Figure 8: Precision of the static detector’s flags as a function of the flag budget. Below its break-even line, retrying flagged answers loses more correct answers than it repairs.

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Abstract

Language Model Reasoning Evaluation

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