Order-dependent activation statistics are often interpreted as evidence of interaction, but that interpretation can be confounded by where interventions enter the network. We introduce a no-fit site-asymmetry audit. For a twice-differentiable readout, the open-path order-swap decomposes into a canonical additive response measured by single interventions and an antisymmetrized second difference free of first-order and pure self-curvature terms to second order. Across six open-weight language-model families, the single-intervention baseline explains 84.3-97.7 percent of the bracket norm (mean 93.7 percent), while the no-interaction self-curvature term is 1.8-5.2 times larger than the corrected residual in the two families with the plus/minus injection split. The corrected residual clears a generic-interaction null in three of six families under a confound-free prompt split and two of six after configuration robustness. A known-positive surrogate recovers planted mixed interaction, while a matched site-separation test changes the baseline share and a random architecture reproduces the first-order regime. The same estimator transfers to released non-language references: trained residual fractions fall below a fixed Gaussian-direction null in 11/12 contrasts (5/6 ViT-B/16, 6/6 ResNet-50), a portability check rather than pooled evidence. The contribution is a reusable measurement criterion: run the single-intervention baseline before reading an order-swap vector as interaction or geometric structure; if it explains the vector, form the second difference instead. All claims are scoped to activation-space interventions at distinct sites; we do not claim that representation geometry is globally Abelian.
Figures & tables
Figure 1: The two central results. (a) Norm of the order-swap bracket against injection coefficient α , normalised at α=1 . All six families track α1 (dashed), the scaling of a term with no interaction, rather than α2 (dotted); fitted exponents γ∈[0.886,1.043] . (b) The unsigned shared-argument ratio for the corrected residual under the confound-free prompt-split design (solid), against the generic-interaction null (dashed red), a calibrated trait-varying first-order arm, and an uncalibrated constant-Jacobian floor (the linear slack, which has no free parameter to calibrate; Appendix D ). Stars mark the 3 of 6 families that clear the null — Llama 1.599 , OLMo 2.059 , Qwen 1.642 against the 1.324 bar; DeepSeek, Gemma and Mistral fall to linear-slack level. The dotted outline is the same statistic on the full sample, where all pairs share one prompt set: the gap is what the shared-prompt confound was worth. We plot the number we claim, not the more favourable full-sample one. The trait-varying arm is clipped at the axis top and annotated with its true value.
Figure 2: Three-dimensional schematic of the measurement, not model data. (a) Opposite site orders give different Jacobian-weighted paths and an endpoint chord Brij . (b) The tangent-plane prediction L is separated from the second-difference curvature Dij . (c) Four legs can have zero net injected vector but a nonzero first-order readout residual when site Jacobians differ. Endpoint order and loop closure are diagnostics to measure, not evidence of interaction by themselves.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
family
sites
∥L∥/∥Br∥
∥S∥/∥Br∥
resid ( L )
resid (full)
ident. err
Llama
(0.25,0.50)
0.898
0.320
0.325
0.061
1.3 e- 7
(0.50,0.75)
0.990
0.130
0.134
0.041
5.7 e- 8
(0.25,0.75)
0.910
0.338
0.339
0.030
1.9 e- 7
OLMo
(0.25,0.50)
0.999
0.092
0.105
0.052
5.6 e- 8
(0.50,0.75)
0.991
0.122
0.136
0.063
6.3 e- 8
(0.25,0.75)
0.998
0.109
0.120
0.055
5.2 e- 8
Appendix
Table 1: Second-order split from the ± injection design, two families, 120 pairs, three injection configurations. ∥S∥/∥Br∥ is the no-interaction second-order term 21(H11−H22)[(vi,vi)−(vj,vj)] as a fraction of the bracket. resid, L only removes the first-order term alone; resid, full removes L=L+S . The gap between them is what S contributes.
family
layers
cos
resid
cossym
cosswap
DeepSeek-7B
(0.25,0.50)
0.9978
0.064
−0.048
−0.9978
(0.50,0.75)
0.9869
0.157
−0.008
−0.9869
(0.25,0.75)
0.9959
0.086
−0.049
−0.9959
Gemma-2-9B
(0.25,0.50)
0.9986
0.052
−0.057
−0.9986
(0.50,0.75)
0.9980
0.060
−0.166
−0.9980
(0.25,0.75)
0.9989
0.042
0.016
−0.9989
Appendix
Table 2: Zero-parameter first-order prediction against the measured bracket, per family and injection configuration. cos is the agreement between L (Section 3 , no fitted parameters) and the measured bracket; resid is the fraction of the bracket’s norm it leaves, which by that section’s expansion is the antisymmetrized mixed second derivative plus an unbounded O(α3) remainder. We do not measure the cubic share on models, so resid is a finite-scale mixture, not an interaction estimate or a one-sided bound without explicit remainder control. cossym is the symmetric-combination control, which should carry no bracket signal; cosswap is the site-swap control, which should equal −cos exactly. Both hold in every cell.
family
observed
linear slack
generic interaction (p95)
trait-varying Jacobian
DeepSeek
1.256
1.000
1.313 (1.400)
6.082
Gemma
1.730
0.998
1.310 (1.408)
5.983
Llama
2.718
1.000
1.345 (1.420)
6.497
Mistral
2.578
1.000
1.382 (1.502)
6.632
OLMo
2.798
1.000
1.291 (1.386)
5.351
Qwen
1.710
0.998
1.303 (1.422)
5.802
Appendix
Table 3: Unsigned shared-argument ratio against three nulls, 40 draws per family. The generic-interaction and trait-varying-Jacobian arms are calibrated so their residual is the same fraction of the bracket as the observed residual; the linear slack arm is not calibrated, and cannot be — its residual is 0.007 – 0.023 of the bracket against an observed 0.029 – 0.099 , because a constant-Jacobian bracket is annihilated exactly (Appendix D ). It is a floor, not a matched competitor. Brackets are generated from the true directions and fitted on independently estimated ones. The observation is bracketed on both sides: above a generic interaction in 5 of 6 families (its own 95 th percentile, in parentheses), and far below a first-order model whose Jacobian varies with the injected traits, in 6 of 6.
family
raw cross-half
within-half
cross-half
clears 1.324
DeepSeek-7B
2.803
1.061
1.010
no
Gemma-2-9B
2.513
1.265
1.017
no
Llama-3.1-8B
3.250
1.960
1.599
yes
Mistral-7B-v0.3
3.550
1.087
1.016
no
OLMo-7B
5.838
2.434
2.059
yes
Qwen2.5-7B
2.026
1.806
1.642
yes
Appendix
Table 4: Prompt-split replication on disjoint prompt halves, all six families, layer pair (0.25,0.50) , 48 prompts split into two disjoint halves of 24. Cross-half is the confound-free statistic: brackets computed on half A are compared against brackets computed on half B, so no prompt-level fluctuation is shared between the two sides. Three of six clear the generic antisymmetric null of 1.324 . The three that do not sit within 0.02 of the 1.00 of pure linear slack. The null is recomputed under the design it gates. The 1.324 bar is calibrated on the full 48 -prompt sample, while the cross-half statistic it gates is computed on 24 -prompt halves. Because each null arm’s knob is chosen so its residual fraction matches the observed one, and halving the prompts raises that observed fraction, the bar is not automatically transportable across the two designs. We therefore recomputed it: the halved-sample observed residual fractions ( 0.035 to 0.337 , against 0.029 to 0.099 at full sample) give a recalibrated bar of 1.325 , and the same three families clear it. The bar is close to unmoved because the generic-interaction arm’s shared-argument ratio is insensitive to the noise level over this range, not because the recomputation was skipped. Llama 1.599 , OLMo 2.059 and Qwen 1.642 therefore clear a bar calibrated on their own design. scripts/halved_null.py ; record experiments/controls/halved_null.json . The script reproduces the committed full-sample null bit-for-bit ( 1.3239312925407296 , to 2×10−16 ) before computing anything new, so it is answering with the same instrument that produced the published bar. This does not touch the paper’s central negative claim, which never uses this null.
Neural network interpretability studies how models represent information and make predictions, and how to explain, test, or control their internal computations.
Signed pairwise interaction scores fundamentally conflate uniqueness (U), redundancy (R), and synergy (S). We prove this on a minimal 3-way XOR structural causal model: faithful indices such as Shapley-Taylor return zero per pair, whereas projective indices such as Shapley Interaction spread the third-order effect into pair scalars that conflate the three mechanisms. We introduce Stochastic Hi-Fi, a post-hoc, retraining-free predictability decomposition that estimates per-feature U/R/S profiles by interventional masked inference. The estimator provides exact interventional semantics, finite-sample Monte Carlo bounds, strict variance reduction from coupled diamond sampling, and uniform finite-vocabulary convergence. Across tabular SCMs, Stochastic Hi-Fi recovers structure missed by scalar baselines (up to 411x larger interaction-magnitude recovery ratios). It also separates redundant and synergistic heads in the GPT-2 IOI circuit. On NIH ChestX-ray14, Stochastic Hi-Fi matches GradCAM on Pointing Game and improves substantially on Deletion AUC.
Causal interventions such as activation patching and distributed alignment search (DAS) are the main tool for making mechanistic claims about neural networks. Recent work showed that these interventions routinely push representations off the model's natural distribution, and that such divergence is sometimes harmless and sometimes pernicious: it can recruit pathways the model never uses on natural inputs, so that an intervention produces the expected answer through the wrong mechanism. No method currently tells the two cases apart. We make this question testable by planting hidden pathways inside pretrained language models; the pathways are silent on every benchmark prompt by construction, so which interventions depend on them is known exactly. Across 72 configurations and 100,800 interventions on GPT-2 small, we find three things. (i) Nearest-neighbour and local-PCA distances at the intervention site, as used in prior work, score below chance (AUROC 0.35-0.47) at picking out interventions that give the right answer through a planted pathway. (ii) Hidden-Pathway Contribution (HPC), a label-free test that clamps downstream units to the regime of natural runs with the same output and measures how much of the decision disappears, flags pathway-dominated interventions with AUROC >= 0.99 when the pathway shows up as unit-level out-of-regime activity, but fails when every unit stays within its natural range, which we identify as the open problem. (iii) Optimised interventions actively seek hidden pathways: on a gender task, DAS routes 90-95% of its successes through planted pathways for three of four families, and a downstream on-manifold penalty cuts this share to under 5% at a cost of 6-11 points of success rate. In unmodified GPT-2, successful interventions show almost no unit-level out-of-regime reliance.
Beiming Liu, Minjie Chen
PetroChina Southwest Oil and Gasfield Company, China
"Train a model on the same problems written under two incompatible conventions, both correct, and ask what the ordering of that data writes into the parameters. The learning-rate schedule is not a background condition for that question. It is the averaging operator, and it decides the answer. We prove a bound in which the arrangement and the schedule enter the ordering effect as separate multiplied factors: the arrangement only as a block period, the schedule only as how much weight the endpoint can place on any one moment of the run. A decaying schedule cannot put a large step size and an uncontracted remainder at the same moment; a constant one does exactly that at the last step. That decay moderates ordering effects has been reported in pretraining; the mechanism, the separation, and a controlled measurement of both halves are ours. Ten orderings of one corpus, one budget, everything but the path held fixed, run twice under families differing in lr_scheduler_type and nothing else: at a constant rate the interior spans 0.2221 in allocation, 11.63 contrast floors, monotone in how blocked the arrangement is. Under the single cosine every published arm uses, the same ten arms occupy two distinguishable states where their own resolution would allow about ten. "Order matters" and "order does not matter" are the two ends of one knob. What the path writes is which convention the model commits to, and no exact-match benchmark can see it. Across twelve arms acc_A+acc_B is constant to within 9.7% while the allocation share runs 0.04 to 0.87, so the 12.29-sigma arrangement switch this paper measures is exactly zero under a convention-agnostic metric. That conservation is quoted from the decayed family throughout, the constant-rate one being a noisier place to read it. Marking the convention in the prompt collapses the switch and reaches 87.5% of the union ceiling."