Intrinsic Structure: Spectral Identifiability for Mechanistic Interpretability

• URL: http://arxiv.org/abs/2608.10172v1
• Date: Mon, 10 Aug 2026 19:42:01 GMT
• Title: Intrinsic Structure: Spectral Identifiability for Mechanistic Interpretability
• Abstract summary: We prove the first identifiability theorem for a mechanistic-interpretability primitive.<n>The spectrum is an identifiable, model-intrinsic fingerprint with a stated error bar, not a legible decomposition.
• Score: 51.56484100374058
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Mechanistic interpretability explains models by identifying circuits inside them, but has no way to tell whether a circuit is a property of the model or an artifact of the method that found it. Sparse autoencoders illustrate the problem: different seeds and widths recover materially different features from the same activations, and no theory says whether that variability is incidental or structural. We put dictionary learning for interpretability on an identifiability footing. Treating the forward pass as a controlled dynamical system with depth as time and lifting it with the Koopman operator yields a finite linear realisation whose \emph{spectrum} is a coordinate-free property of the model. We prove the spectrum is recoverable from $M$ calibration samples at rate $M^{-1/2}$ up to permutation - to our knowledge the first identifiability theorem for a mechanistic-interpretability primitive, with a matching minimax lower bound, a median-of-means variant for heavy-tailed activations, and a dissociation theorem: whenever the realisation is non-normal, the directions carrying activation variance and the directions carrying information across depth cannot coincide. The identifiable object and the legible object are not the same object. On GPT-2 small, Gemma-2-2B and Qwen3-8B-Base the spectrum converges everywhere and attains the predicted exponent on Qwen3-8B-Base ($0.506 \pm 0.031$); shortfalls collapse onto one curve against each cell's sample threshold. Koopman modes beat random directions but lose to principal components on indirect-object identification, with the gap decaying $4.1\times$ in depth-distance, as the theorem predicts. The Koopman spectrum is an identifiable, model-intrinsic fingerprint with a stated error bar, not a legible decomposition.

Related papers

• How Many Samples Are Needed to Determine Causal Direction? Sharp Minimax Bounds for Bivariate LiNGAM [7.7753818665096945]
  We study how many observations are needed to determine the causal direction between two linearly related variables.<n>We establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty.
  arXiv  Detail & Related papers  (2026-08-16T16:20:19Z)
• Is Spurious Correlation Removal Always Learnable? [56.28155520961125]
  Invariant learning can fail even when the structure is statistically identifiable.<n>We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive, there exist emphamplable multi-dimensional environments.<n>Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
  arXiv  Detail & Related papers  (2026-06-11T05:49:43Z)
• Self-Attention as Transport: Limits of Symmetric Spectral Diagnostics [0.0]
  We show that uniform causal attention satisfies an $n$-independent floor $ge 1/5$ with worst cut at $tast/n approx 0.32$, while window attention pierces the floor as $O(w/n)$.<n>The resulting two-axis diagnostic yields a falsifiable polarity prediction.
  arXiv  Detail & Related papers  (2026-05-06T13:25:13Z)
• Operationally induced preferred basis in unitary quantum mechanics [0.0]
  The preferred-basis problem and the definite-outcome aspect of the measurement problem persist even if the detector is modeled unitarily.<n>A change of mathematical type constitutes the core of the 'cut': a structurally necessary interface from group-based kinematics to set-based counting.
  arXiv  Detail & Related papers  (2026-01-26T17:22:03Z)
• Phase-space entropy at acquisition reflects downstream learnability [54.4100065023873]
  We propose an acquisition-level scalar $S_mathcal B$ based on instrument-resolved phase space.<n>We show theoretically that (S_mathcal B) correctly identifies the phase-space coherence of periodic sampling.<n>$|S_mathcal B|$ consistently ranks sampling geometries and predicts downstream reconstruction/recognition difficulty emphwithout training.
  arXiv  Detail & Related papers  (2025-12-22T10:03:51Z)
• Transformer Injectivity & Geometric Robustness - Analytic Margins and Bi-Lipschitz Uniformity of Sequence-Level Hidden States [0.0]
  We show that the map from discrete prompts to last-token hidden states is generically injective on finite prompt sets.<n>We study behavior across layers, sequence lengths, model scales, and 8- and 4-bit activation quantization.
  arXiv  Detail & Related papers  (2025-11-17T19:39:15Z)
• Robustness of Nonlinear Representation Learning [60.15898117103069]
  We study the problem of unsupervised representation learning in slightly misspecified settings.<n>We show that the mixing can be identified up to linear transformations and small errors.<n>Those results are a step towards identifiability results for unsupervised representation learning for real-world data.
  arXiv  Detail & Related papers  (2025-03-19T15:57:03Z)
• Quantum stochastic phase-space theorems lead to hidden causal loops in a model for measurement consistent with macroscopic realism, Bell nonlocality and no-signaling [0.0]
  We show how quantum measurement and nonlocality can be explained consistently with macroscopic realism and no-signaling.<n>We analyze a measurement of $hatx$ on a system prepared in a superposition of eigenstates.
  arXiv  Detail & Related papers  (2022-05-12T13:10:20Z)
• Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements [62.997667081978825]
  characterisation of the brain grey matter cytoarchitecture with quantitative sensitivity to soma density and volume remains an unsolved challenge in dMRI. We propose a new forward model, specifically a new system of equations, requiring a few relatively sparse b-shells. We then apply modern tools from Bayesian analysis known as likelihood-free inference (LFI) to invert our proposed model.
  arXiv  Detail & Related papers  (2021-11-15T09:08:27Z)
• Analytic Signal Phase in $N-D$ by Linear Symmetry Tensor--fingerprint modeling [69.35569554213679]
  We show that the Analytic Signal phase, and its gradient have a hitherto unstudied discontinuity in $2-D $ and higher dimensions. This shortcoming can result in severe artifacts whereas the problem does not exist in $1-D $ signals. We suggest the use of Linear Symmetry phase, relying on more than one set of Gabor filters, but with a negligible computational add-on.
  arXiv  Detail & Related papers  (2020-05-16T21:17:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.