Mathematical truths are causal (not merely descriptive), satisfying counterfactual and explanatory criteria of causation

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• Mathematics is not a guarantor of truth but a language for describing structure; the mathematical concepts function as lenses, not guarantors of correctness.claim0.789
  Proposition 1, the paper's core methodological stance.
• Linear truth directions in LLMs are reliable primarily in factual recall cases and break down when truth assessment depends on computing and storing intermediate results.claim0.762
  Central empirical conclusion of the paper about the fundamental limits of truth directions.
• We hypothesize that explicitly instructing the model to evaluate the correctness of the given statement may change the geometry of truth directions.hypothesis0.752
  Motivating hypothesis for Section 5's investigation of prompt template effects.
• Does the multi-directional nature of truth imply an underlying nonlinear representation, or is it compatible with linear separability?question0.750
  Theoretical open question about the geometry of truth in LLMs raised in Discussion
• "Thus, beyond all questions of quantity there lie questions of pattern, which are essential for the understanding of Nature."quote0.749
  Epigraph opening §3, anchoring the paper's focus on pattern over mere quantity.
• Even mathematical catastrophes always begin as features consistent with the symmetries of the earlier state and develop smoothly from within the existing wholeness.claim0.745
  Alexander's response to the apparent counter-examples from catastrophe theory: discontinuities are themselves structure-preserving at a deeper level
• Truth directions emerge in earlier layers for factual tasks and later layers for arithmetic tasks.claim0.742
  Core empirical claim about the layer-dependence of truth direction emergence as a function of task type.
• Causal abstraction is not enough for mechanistic interpretability because it becomes vacuous without assumptions about how models encode informationclaim0.741
  Central thesis of the paper