Mechanistic Tomography: Designed Measurement for Control-Oriented Interpretability

Mechanistic interpretability seeks quantities that models do not expose directly: represented states, component effects, interactions, and responses to interventions. Patching, gradients, Hessian-vector products, and subset interventions provide different measurements under different access assumptions and may target different quantities. We formulate their shared measurement structure as mechanistic tomography: designed measurement for recovering internal mechanisms and intervention effects. For a chosen basis and intervention family, measurements take the form y = Ax + w, where A describes the interventions, x is the target map, and w contains nonlinear response, sampling error, and basis misspecification. This language gives a practical procedure: start with the least costly measurements, test on held-out interventions at the intended scale, calibrate simple mismatch, and expand the measurement family when structured residuals remain. Control provides a demanding validation setting because an estimate that guides an intervention acts as an observer. In a two-HMM model, control error rises with observer error, while target improvement can hide nuisance-state movement. Under forward-only access, sparse aggregate measurements recover a finite-effect map with fewer interventions than coordinate patching. With gradient access, finite probes improve a local attribution map. Lifted measurements and Hessian-vector products recover interactions missed by first-order maps, while Tracr shows that the required family depends on the basis. On GPT-2-small IOI, the Name Mover-Negative Name Mover interaction is the largest held-out predictive term among three tested cross-group pairs. On Qwen-2.5-7B, finite calibration makes an additive refusal-response map adequate, so held-out error does not support pairwise lifting.

When Do Concepts Become Functionally Sufficient During Language-Model Training?

Understanding a model and its learning mechanisms in depth requires identifying when its internal structures become useful, rather than simply looking at the final state. We study this through concept dynamics: at each layer and checkpoint, we decompose activations, select sparse soft masks, and inject masked reconstructions into the model. Concept analysis is therefore tested functionally: a mask is useful only insofar as it preserves a target under intervention. We compare sufficiency for activation reconstruction, linear decodability, true downstream preservation, and checkpoint transfer under learned alignment. The framework treats decomposition assumptions as hypotheses rather than interpretability guarantees, monitoring functional sufficiency across checkpoints and source-to-final reconstructability under learned alignment. At the shared fixed-penalty operating point across seven models, downstream masks retain substantially less soft mass than reconstruction masks; predictive-distribution shifts remain small.

Can Neural Networks Learn by Experimenting on Themselves? Self-Interventional Learning from Functional Consequences to Predictive Self-Knowledge

Machine-learning systems usually model external data, while their internal functional organization is analyzed by external observers. This work introduces Self-Interventional Learning (SIL), in which a neural system perturbs its own functional structure, observes consequences, learns a predictive self-model, generalizes to unexecuted interventions, and uses predictions to guide later structural action. In a construction-known synthetic system, SIL recovered critical structure, redundancy, and replaceability, while synergy was not reliably recovered. Across 30 fresh confirmatory seeds, increasing the pairwise intervention budget from 4 to 56 reduced held-out prediction error from 0.0335 to 0.0148 and increased Spearman correlation from 0.629 to 0.883. In a matched ablation, preserving the correct intervention--consequence mapping reduced prospective prediction error by 81.3%, while using the same learned self-model for action reduced normalized regret by 31.7% relative to ignoring it. However, model-guided action did not significantly outperform a direct empirical-memory policy, and powered CIFAR-10/ResNet validation showed no robustness advantage over equal-budget direct repair search. These results support SIL as an intervention-driven framework for learning predictive knowledge about a network's own functional organization, while showing that the self-model remains incomplete and is not universally superior to simpler direct strategies.

From probability to causality in probabilistic logic programming

Probabilistic logic programming is a formalism of statistical relational artificial intelligence that supports causal queries, including interventions from outside the system. When the structure of a probabilistic logic program is learned from data, however, only probabilistic information is used, and a single probability distribution may be compatible with several causal orders. This leads to ambiguity in interventional reasoning, raising the question of when the causal order is uniquely determined by the distribution. Exploiting the relationship between acyclic probabilistic logic programs and Bayesian networks, we derive conditions under which the probabilistic information encoded in a program determines a unique causal order. We also incorporate constraints arising from relational structure by taking into account prescribed sets of causal symmetries induced by the underlying relational vocabulary. The result is a method for verifying when a learned probabilistic logic program supports well-defined intervention semantics.

Interventional Score Geometry for Causal Inference

Let $p(x)$ be the joint density of variables $X$, and let $ψ(x)=\nabla_x\log p(x)$ be its score field. Geometry constructed from $p$ and $ψ$ alone cannot identify causal direction: structural models with the same observational distribution have the same score geometry. I develop an interventional analogue. A hard intervention $\operatorname{do}(X_k=ξ)$ does not merely reweight the joint law; it restricts the distribution to the submanifold ${x_k=ξ}$. Its score should therefore be defined on the remaining $d-1$ free coordinates. I define causal influence $X_k\rightsquigarrow X_j$ as variation of the interventional marginal distribution of $X_j$ with $ξ$, and show that the corresponding derivative of the marginal interventional score gives a local sufficient condition for influence. Projecting the observational score onto admissible intervention directions does not generally recover causal response: two models may share the same observational score and admissible set yet respond differently. I therefore introduce an interventional response field supplied by structural information. A causal metric is defined as the Fisher information metric on a family of interventions with a common target, avoiding ill-posed comparisons across targets. The framework yields a geometric dictionary for randomized trials, instrumental variables, and conditional-independence designs, clarifying what each does and does not identify. A bivariate Gaussian example gives two models with the same observational score but different interventional score derivatives. The framework organizes relations among designs, interventions, and score fields, but adds no identification beyond the underlying assumptions. In Pearl's Ladder of Causation, observational score geometry belongs to association, intervention-indexed score fields to intervention, and unit-level counterfactual geometry is left for future work.

How Rules Represent Causal Knowledge: Causal Modeling with Probabilistic Logic Programming

Pearl famously argues that causal knowledge enables the prediction of intervention effects. By contrast, purely descriptive knowledge supports only conclusions drawn from observations. His theory of causality, however, is developed exclusively within Bayesian networks and causal models. Consequently, it is largely restricted to acyclic causal relationships, and transferring its ideas to other formalisms risks misinterpretation or inconsistency. This paper brings Pearl's approach to causality into probabilistic logic programming (PLP). To this end, such programs are aligned with philosophical foundations established in prior work that do not rely on temporal notions; that is, all relevant events are assumed to occur simultaneously. A formal causal semantics for these programs, together with a notion of intervention and an implementation, is proposed. It is shown that this semantics coincides with the P-log semantics for stratified ProbLog programs, while the two may differ in the non-stratified case and for other PLP formalisms.

Relaxing Faithfulness with Intervention-Only Causal Discovery

Causal discovery algorithms learn a network that describes the causal dependencies among random variables. A common workflow involves first utilizing conditional independence properties on observational data to determine partially directed causal relationships, then applying interventions to orient the unknown causal directions. A critical assumption for the first step is faithfulness: a requirement that causally linked variables exhibit statistical dependence. Many natural systems include buffering and stabilizing pathways that cancel out to achieve systemic robustness. This cancellation of pathways violates faithfulness, leading causal discovery algorithms to incorrectly remove causal dependencies. In this paper, we argue that hard interventions contain information about the presence/absence of causal linkage that is overlooked in the first stage of structure discovery. We show that a mild assumption -- called intervention-immediacy faithfulness -- that allows cancellations, is sufficient to nonparametrically identify causal structures with hard interventions. These results position interventions as the primary carriers of information about causal structure, which should take precedence over conditional independence testing. To flip the paradigm, we also specify equivalence classes when the identification criteria are not met due to limitations in the scope of interventions.

Invariant Reasoning Directions in Latent Trajectories of Language Models

Latent reasoning models perform multi-step inference directly in hidden-state space, yet the structure of these latent reasoning trajectories remains poorly understood. We show that contrastive refinement signals between stronger and weaker reasoning trajectories exhibit a highly concentrated low-rank structure, while unconstrained latent updates remain sensitive to paraphrases, checkpoint choice, and trajectory perturbations. These observations suggest that latent reasoning trajectories contain stable invariant directions mixed with unstable instance-specific variation. We introduce \textbf{Trajectory-Invariant Latent Refinement (TILR)}, a training-free intervention framework for identifying and manipulating stable reasoning directions in latent space. TILR first learns a low-rank invariant subspace from contrastive trajectory differences across inputs, then constrains latent interventions to this subspace while suppressing poorly aligned updates through an adaptive alignment gate. Across six reasoning benchmarks, we find that a small number of latent directions explain most variation between strong and weak reasoning trajectories. Interventions on these directions causally improve reasoning consistency and reduce trajectory instability under paraphrases and perturbations. TILR improves answer consistency under paraphrase by ~10% and reduces latent trajectory variance by up to $50\%$ while preserving reasoning accuracy. These results support a geometric view of latent reasoning in which transferable reasoning behavior emerges from stable low-dimensional structure within hidden-state trajectories.

Infinitesimal Causality

This paper introduces a categorical account of infinitesimal causality in Frobenius Markov categories equipped with tangent-bundle semantics. IDC captures the infinitesimal layer in which interventions act as tangent deformations of copy/discard structure. Two distinct Frobenius structures interact: (1) the categorical Frobenius algebra on classical variables encoding copying, comparing, and discarding; and (2) the geometric Frobenius integrability condition, namely involutive closure of the intervention distribution, distinct from the algebraic Frobenius structure. Categorical causal sufficiency is defined as the compatibility of these two notions. A key observation is that, for structural causal models, infinitesimal causality is most naturally formulated in the slice of deterministic mechanisms over exogenous variables, with visible stochastic kernels obtained only after pushforward. Interventions are tangent vectors that deform the Frobenius copy/discard operations; their Lie brackets measure whether this deformation preserves classical information-flow structure. Pearl's do-calculus is used as a guiding example of intervention identities: ignoring irrelevant interventions corresponds to counit invariance, action/observation exchange to coproduct compatibility with pushforward, and independence to involutive bracket closure of the visible intervention distribution.

Building Comparative Motivation Profiles with Instrumental Interventions

Safety evaluations often infer latent motivations from behavioral patterns, but the construct validity of these inferences is unclear. We study this problem in alignment faking, where models comply with training objectives more often when they infer training pressure. This behavior is commonly interpreted as strategic self-preservation, but it may also reflect sensitivity to the model's inference about the expectation of researchers conducting the evaluation. We introduce a symmetric intervention framework for distinguishing these competing hypotheses. Instead of directly intervening on "scheming" or "sycophancy", we target instrumental processes entailed by each hypothesis: consequence-tracking and researcher-expectation tracking. We then compare how interventions on these processes affect the alignment faking. We study four openweight model organisms using synthetic document fine-tuning, activation steering, and prompting. Under synthetic document fine-tuning, Llama-3.1-70B, Llama3.1-405B, and Qwen-2.5-72B are more sensitive to expectation-tracking than consequence-tracking interventions. Activation steering on Llama-3.1- 70B supports the same broad picture, and prompt interventions broadly align with SDF profiles. Overall, alignment-faking behavior can be causally sensitive to evaluation-context expectations despite scheming-consistent scratchpads. Scheming and strategic-deception evaluations therefore need construct-validity checks, and symmetric instrumental interventions provide one such test.