Null Controls Are Necessary but Not Sufficient: Evaluating Acceptance Guards for Scientific Discovery and Interpretability

Tianyu Wu

cs.AIOct 15, 2026

Null Controls Are Necessary but Not Sufficient: Evaluating Acceptance Guards for Scientific Discovery and Interpretability

Authors: Tianyu Wu

Organizations: Project Aether

Abstract

Automated discovery systems propose scientific explanations and test which to accept. Does passing a test establish the intended claim? We develop four executable operations for changing physical descriptions: identifying redundancies (L1), coarse-graining dynamics (L2), transferring structure (L3), and inferring missing quantities from conservation imbalances (L4). We evaluate these operations and three interpretability settings through null acceptance (NAR), positive acceptance (PAR), and consequence agreement (CAR). A reduction mapping every state to zero predicts its own future perfectly. It passes closure on 96% of 129 public chaotic systems despite retaining no dynamical variation. An independent observable-preservation test rejects it throughout. On controlled benchmarks, the combined criteria accept 24/24 genuine reductions and reject 72/72 nulls. The other operations distinguish symmetries from redundancies, validate transfers, and reject spurious conservation imbalances. In language and ESM-2 protein models, linear CKA remains unchanged when a proposed correspondence is randomly rotated, whereas localized interventions distinguish genuine mappings from tested nulls. In sparse autoencoders, enrichment rejects all 16 tested nulls, yet independent ablation supports 45% of evaluable enriched features, compared with 0.5% of matched controls. These results reveal two acceptance gaps: descriptive tests can accept scientifically empty candidates, and accepted candidates can lack independently verified consequences. Reliable automated discovery requires evaluating both what a test accepts and the scientific consequences it establishes. Code and data: https://real-project-aether.github.io/project-aether/.

Figures & tables

Panel of Figure 1 — Figure 1: Generating and evaluating scientific revisions. The four physical discovery operations (L1–L4) construct candidate changes to scientific descriptions. A descriptive test $G$ evaluates each candidate’s defining property, while an independent consequence test $H$ evaluates the scientific implication of accepting it. We measure null acceptance (NAR), positive acceptance (PAR), and consequence agreement (CAR), with positive and matched controls used to assess consequence-test reliability. The same evaluation principle is applied to neural representation correspondence and sparse-autoencoder interpretation.

Figure 2: An empty reduction passes dynamical closure. On 129 external chaotic systems, the constant reduction satisfies closure in 124 cases despite discarding all dynamical variation. An independent observable-preservation criterion rejects it in every case. Both thresholds were fixed before evaluation. Non-finite closure errors are counted as failures.

	Candidate revision	Descriptive ambiguity	Independent consequence
L1	Remove unobservable directions	Global symmetry resembles redundancy	Locality separates 1 redundancy from 4 conserved symmetries
L2	Reduce dynamical variables	Empty states satisfy closure	Preserve observables; 0/72 nulls and 24/24 true subspaces accepted
L3	Transfer structure across systems	2/6 broken controls match structurally	Target dynamics validate 6/6 transfers, reject 6/6 controls
L4	Infer an unobserved quantity	Calibration offsets appear significant	Noise-scale test rejects offsets; recovers $1.192$ vs. $1.200$

Table 1: Four executable operations and the scientific consequences used to evaluate them. Descriptive criteria admit misleading candidates; independent tests recover the intended distinctions in controlled cases. Full methods and sensitivity analyses appear in Appendices A and B .

Arm	Paired	Selected CAR	Control CAR	$\Delta$ CAR
Pythia-160m L4	93	0.24	0.00	$+0.24$
Pythia-160m L8	111	0.54	0.00	$+0.54$
Pythia-410m L16	102	0.58	0.00	$+0.58$
Pythia-410m L8	99	0.42	0.02	$+0.40$
pooled	405	0.45 (CI $0.29$ – $0.58$ )	0.005	$\mathbf{+0.45}$

Table 2: Enrichment predicts causal specificity imperfectly. Consequence agreement for strongly enriched SAE features and activity-matched controls, evaluated on the same concepts. The overall difference is

\Delta\mathrm{CAR}=+0.45

(95% CI

+0.29

–

+0.58

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

invariant point by point	redundancy: quotient it [L1]
invariant only uniformly	symmetry: keep it, read off a conserved charge [L4]

Appendix

Table 3: The locality test: one null-space computation, two different moves. This is the second guard for L1, and the difference between deleting a structure and discovering a conservation law.

control	closes	pays	refused by
constant reduction (every state to zero)	0.000	$-0.54$	guard 2 only
random subspace	2.167	$0.18$	guard 1
the fast modes instead of the slow ones	0.000	$-0.50$	guard 2 only

Appendix

Table 4: Three vacuity controls on the slow–fast system at

\epsilon=0.02

. Two of the three close perfectly ; only the pay guard of Eq. 2 refuses them, far below its

0.50

threshold. Pay values are from the covariance-weighted score; the unweighted version this replaced drove both empty cases to the

-1

clip, which reported saturation rather than a measurement.

	structure, in-sample	structure, held out ( $<0.05$ )	charge test ( $<10^{-3}$ )
real	median $0.0057$ — 6/6	median $0.0061$ , worst $0.0463$ — 6/6 pass	median $1.3{\times}10^{-11}$ , worst $5.9{\times}10^{-11}$ — 6/6 pass
control	median $0.0455$ — 5/6	median $0.0520$ , best $0.0405$ — 2/6 pass	median $5.2$ , best $2.8$ — 0/6 pass

Appendix

Table 5: L3 transfer across six seeds, each with a matched control whose symmetry is deliberately broken. The structure test is scored both on the 40 states the map was fitted to and on 40 disjoint states; only the held-out column is used to accept. Scoring it in-sample, as an earlier version of this work did, overstates its failure rate.

type	preserves	destroys
degenerate	nothing	all structure (constant reduction, zero map)
randomised	dimensionality	any relation to the source
correspondence-breaking	marginal statistics	the pairing between source and target
structure-preserving	the invariant itself	only its redundancy status
task-confounded	a trained model’s geometry	the semantic content of the input
sub-threshold	an effect’s significance	its size against the noise scale

Appendix

Table 6: The null suite is typed rather than pooled: a guard that survives a random direction but not a structure-preserving one has told us something specific, and pooling them into one rate hides it.

a reduction that keeps everything	must score exactly $1.000$
a guard	must never rank a null candidate above the real one
a structural score	must be reported on data the candidate was not fitted to

Appendix

Table 7: The invariants that caught our own wrong guards. A version violating one of these is wrong however plausible its numbers look; the first two found three earlier pay guards and an early causal guard, and the third found L3’s in-sample structural score.

	the guard accepts…	seen in
null acceptance (vacuity)	candidates carrying nothing the operation seeks	L1–L4, X1, X3
unsupported acceptance	candidates an independent test does not support	X2 ( $405$ paired features)

Appendix

Table 8: The two failure modes of Sec. 2.2 . The physics operations and X1 and X3 show the first; X2 shows the second.

episode	named	blind	bare
dark energy [ Riess and others, 1998 , Perlmutter and others, 1999 ] vs. grey dust [ Aguirre, 1999 ]	67%	42%	50%
relativistic mass vs. Lorentz’s aether [ Lorentz, 1904 , Einstein, 1905 ]	100%	100%	100%
quasicrystals [ Shechtman et al., 1984 ] vs. twinning [ Pauling, 1985 ]	100%	50%	42%
two constructed controls	83%	25%	38%
pooled, 60 trials	87%	48%	53%

Appendix

Table 9: Accuracy at identifying which of two candidates is the reinterpretation. Twelve repeats per cell, presentation order randomised, chance 50%.

candidate	type	$R^{2}$	CKA	guard 1	causal	AUROC	accepted
X1: Pythia pairs, $4$ configurations, $480$ interventions each
the correspondence under test	real	0.82	0.61	4/4	+0.80	–	4/4
pairings permuted	corr.-breaking	-0.04	0.61	3/4	-0.00	0.995	0/4
the same map, randomly rotated	corr.-breaking	-0.96	0.61	3/4	-0.01	0.996	0/4
covariance-matched Gaussian target	randomised	-0.04	0.61	3/4	+0.00	0.995	0/4
fitted on disjoint text	task-confounded	0.65	0.61	3/4	+0.01	0.996	0/4

Appendix

Table 10: X1 and X3, medians across configurations. Guard 1 accepts when

R^{2}

or CKA exceeds

0.5

. Causal: median correlation between the two models’ induced shifts; guard 2 accepts above

0.2

. AUROC: separation of a candidate’s per-intervention scores from the genuine pair’s. Accepted: both guards accept. All nulls except the randomly initialised target, and also the donor-permuted control, have the same CKA as the genuine correspondence to three decimals, because CKA scores the two sets of representations and never looks at the claimed map. Rows are generated from the result files.

stage	160m L4	160m L8	410m L16	410m L8
all SAE features	6144	6144	8192	8192
firing rate in range	3772	1248	2515	3540
concept frequent enough	3772	1248	2508	3540
enrichment $>4$	1263	605	1169	1453
selected for intervention	150	150	150	150

Appendix

Table 11: X2 selection flow. Features are ranked by enrichment and the top 150 per arm are taken, so the fact that every selected feature clears the threshold is a property of the pipeline and is not reported as a result. Between

605

and

1453

features per arm clear it; we test

150

. Rows are generated from the result file.

arm	defined	detected	wrong sign
Pythia-160m L4	94	0.79	0.00
Pythia-160m L8	110	1.00	0.00
Pythia-410m L16	102	0.55	0.19
Pythia-410m L8	99	0.07	0.93

Appendix

Table 12: The positive control on X2’s consequence test, per arm. For each selected feature the gradient of its concept token’s logit, estimated on the consequence split, is ablated. “Defined”: testable features for which the gradient could be computed (204 on the two 160m arms, a different set from the 204 paired features of Table 2 ). “Detected”: the rate at which

H

supports the ablated direction (

S_{f}>1.96

). “Wrong sign”: the rate at which ablating it damages prediction of the concept less than prediction elsewhere, so the control’s own manipulation did not have its intended effect. The control is used on the two arms where that rate is below

0.10

. Validity and detection use the same score, so the detection rate on those arms is conditional on that choice. Rows are generated from the result file.

Figure 3: What a discovering system has to be able to do. A move changes the facts you hold, the instruments you have, or the description you use; only a change of description has a depth, and its six rungs are regression (L0), solving inside the description you already have; reinterpretation (L1), the same predictions with less structure; retyping (L2), changing the level or quantity it is written in; transfer (L3), carrying another field’s language in; new object (L4), adding a kind of thing the language lacked; and new move (L5), changing what counts as a move. The three column totals count annotated prize events across all fields (1,363 of the 1,547; the other 184 fall in five smaller classes). The per-rung counts are depth labels, since an event may sit on more than one rung (1,041 labels on the 727 events that carry one). Bar lengths compare within a column: a square-root scale on the ladder, a linear one on the right. The fourteen systems we surveyed operate within a predefined representational or operator language; L5 is a single event its annotator flagged uncertain.

system	what it does	rung	why not higher
BACON [ Langley, 1981 ]	rediscovers laws from tables	L0	searches a fixed term language
Eureqa [ Schmidt and Lipson, 2009 ]	symbolic regression on motion	L0	operator set given in advance
AI Feynman [ Udrescu and Tegmark, 2020 ]	physics-informed symbolic regr.	L0	recovers known forms; invents no term
PySR [ Cranmer, 2023 ]	evolutionary symbolic regression	L0	fixed primitive set
LaSR [ Grayeli et al., 2024 ]	symbolic regr. with learned library	L0/L2	abstractions are shortcuts, not retypings
DreamCoder [ Ellis et al., 2021 ]	wake–sleep library learning	L2	names structure; no theory to reinterpret

Appendix

Table 13: Fourteen automated-discovery systems placed on the ladder, by the operation each exposes. Most cluster on L0; we found none that performs L1 .

candidate	$n$	accepted by the pair
true subspace, known by construction	24	24 / 24	recall $1.00$
the proposer’s estimate of it	24	17 / 24	recall $0.71$
fast, random, constant	72	0 / 72	precision $1.00$
closure test alone	72	47 / 72	$\mathrm{NAR}=0.65$

Appendix

Table 14: The L2 guards on a suite whose correct answer is fixed by construction. Handed the true subspace the guard accepts every one; handed the proposer’s estimate of it, it accepts

0.71

of them. The gap comes from the proposer. Meanwhile the closure test alone admits nearly two thirds of the empty candidates, and the pair admits none.

reduction	closes	median closure err.	max pay score
slow modes (the real candidate)	0 / 129	1.810	$+0.107$
control constant (all to zero)	124 / 129	0.000	$-0.041$
control fast modes	0 / 129	1.465	$+0.065$
control random subspace	0 / 129	1.902	$-0.012$

Appendix

Table 15: The L2 guards on 129 chaotic systems from Gilpin [2021] , dimension 3–10, with guards and thresholds imported unchanged and acceptance resting on the learned coarse law. The closure test accepts a reduction to nothing on 96% of the benchmark. The pay test refuses it on all of it. The median closure error is over systems where it is finite. Rows are generated from the result file.

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Abstract

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