arxiv: 2605.09496 · v1 · submitted 2026-05-10 · 💻 cs.CL · cs.LG

Recognition: no theorem link

Beyond Language: Format-Agnostic Reasoning Subspaces in Large Language Models

Aojie Yuan , Zhiyuan Su

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:13 UTC · model grok-4.3

classification 💻 cs.CL cs.LG

keywords format-agnostic reasoningactivation patchinglarge language modelsrepresentation subspacescross-form generalizationTriForm benchmarkconcept-centroid PCA

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The pith

Large language models share a compact internal space for reasoning that stays the same whether the input is English sentences, Python code, or mathematical symbols.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether large language models use the same internal representations for reasoning regardless of surface format. By creating a benchmark with the same concepts expressed in English, Python and math, the authors show that middle-layer activations contain a compact subspace dedicated to concept meaning. This subspace can be isolated with a special form of PCA that centers on concepts rather than overall variance. If correct, it would mean that models perform format-independent reasoning in a small, reusable part of their activation space.

Core claim

Using the TriForm Benchmark, we identify a Format-Agnostic Reasoning Subspace (FARS) in the middle layers of several LLMs. Concept-centroid PCA on these layers yields a 10-dimensional subspace where concept information is amplified threefold and format information is reduced to near zero. Substituting only these dimensions in cross-format activation patching retains 90-96% of the original model outputs, outperforming both complete activation swaps and standard PCA methods.

What carries the argument

The Format-Agnostic Reasoning Subspace (FARS), a 10-dimensional region in middle-layer activations extracted via concept-centroid PCA that captures shared reasoning across input formats.

If this is right

The subspace generalizes to held-out concepts not used in its identification.
Representations remain more compatible between prose and mathematics than between either and code.
The same subspace appears consistently across different model architectures and sizes.
Ablating the subspace causes targeted disruption to reasoning performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The declarative-procedural split suggests distinct processing pathways for different symbolic inputs.
Interventions limited to these dimensions could be used to test or improve cross-format consistency.
The convergence across architectures indicates that such subspaces may be a general property of current LLMs.

Load-bearing premise

The 10 dimensions selected truly isolate format-independent reasoning instead of capturing patterns specific to the chosen benchmarks or the particular way dimensions were picked after testing.

What would settle it

If the 10-dimensional replacement no longer preserves model outputs on a fresh set of concepts or in larger models outside the tested range, the claim that this subspace is format-agnostic would not hold.

Figures

Figures reproduced from arXiv: 2605.09496 by Aojie Yuan, Zhiyuan Su.

**Figure 1.** Figure 1: The FARS pipeline. Stage 1: The TriForm Benchmark encodes 18 reasoning concepts in 6 surface forms. Stage 2: Concept-centroid PCA extracts a 10-dimensional FARS. Stage 3: Subspace patching replaces only FARS directions, preserving 90–96% of output. perfectly across languages with high hidden-state similarity and large neuron overlap. Our work tests whether this transfer extends beyond languages to symbolic… view at source ↗

**Figure 2.** Figure 2: Three complementary views of format-agnostic representations across layers. (a) Concept [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Cross-form patching overlap by layer. Patching is most effective at early-to-middle layers [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Declarative-procedural asymmetry. (a) Layer-wise patching overlap for Mistral-7B. (b) The EN→Math / EN→Code ratio is 3–4× across all five models. (a) Full space, colored by form en prose fr prose math notation py code structured list zh prose (b) FARS projection, colored by concept (c) FARS projection, colored by form en prose fr prose math notation py code structured list zh prose [PITH_FULL_IMAGE:figure… view at source ↗

**Figure 5.** Figure 5: t-SNE visualization (Mistral-7B, best FARS layer). [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Subspace patching: four-way comparison. FARS directions carry causal concept informa [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Dimensionality sweep. Concept RSA plateaus around [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Generalization: FARS extracted from 18−K concepts, evaluated on K held-out concepts. Performance degrades gracefully. GPT-2 XL (1.6B) Qwen2.5-3B Qwen2.5-7B Mistral-7B Llama-3.1-8B GPT-2 XL (1.6B) Qwen2.5-3B Qwen2.5-7B Mistral-7B Llama-3.1-8B 1.00 0.84 0.86 0.80 0.77 0.84 1.00 0.94 0.87 0.91 0.86 0.94 1.00 0.90 0.91 0.80 0.87 0.90 1.00 0.90 0.77 0.91 0.91 0.90 1.00 Cross-Model FARS Alignment 0.60 0.65 0.70 … view at source ↗

**Figure 9.** Figure 9: Cross-model FARS alignment (centroid RSA). Same-family pairs exceed [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

**Figure 10.** Figure 10: CKA alignment by invariance dimension across layers. Linguistic, symbolic, and structural [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗

**Figure 11.** Figure 11: Patching token overlap by layer, broken down by form pair. The declarative-procedural [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

**Figure 12.** Figure 12: Concept RSA across layers for all five models. [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

**Figure 13.** Figure 13: Cross-form patching overlap by layer, averaged across all form pairs. [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗

read the original abstract

Large language models represent the same reasoning in vastly different surface forms -- English prose, Python code, mathematical notation -- yet whether they share a common internal substrate across these symbolic systems remains unknown. We introduce the TriForm Benchmark (18 concepts x 6 forms x 3 instances = 324 stimuli) and study five LLMs (1.6B-8B) across three architecture families. Using permutation-corrected RSA, cross-form probing, and activation patching, we find converging evidence for a Format-Agnostic Reasoning Subspace (FARS) in middle layers. We make FARS concrete: concept-centroid PCA extracts a 10-dimensional subspace that amplifies concept structure 3x while suppressing form information to near zero. Replacing only these 10 dimensions during cross-form patching preserves 90-96% of model output -- far exceeding both full activation replacement (44-56%) and variance-maximizing PCA (60-74%) -- while ablating them causes targeted disruption. FARS generalizes to held-out concepts and converges across architectures (CCA > 0.79 for all model pairs), providing within-modality evidence for the Platonic Representation Hypothesis. We further discover a declarative-procedural asymmetry: representations are far more compatible between prose and mathematics than between either and code, suggesting that the critical axis of divergence is not linguistic vs. formal but declarative vs. procedural.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a new benchmark and some converging signals for a low-dimensional format-agnostic subspace, but the headline 10-dim and middle-layer results look vulnerable to post-hoc selection.

read the letter

The main thing to know is that the authors built TriForm, a 324-stimulus set spanning 18 concepts in six surface forms, and used RSA, probing, and patching to argue for a Format-Agnostic Reasoning Subspace in middle layers of several 1.6B–8B models. Concept-centroid PCA on that data yields a 10-dimensional slice that multiplies concept variance by three while driving form variance near zero; patching only those dimensions across forms keeps 90-96 % of output behavior, well above full activation swaps or ordinary PCA. They also report a declarative-procedural split, with prose and math aligning better than either does with code, plus decent cross-model CCA scores and some held-out generalization. That package is the actual new piece on top of standard representation-similarity tools.

Referee Report

3 major / 2 minor

Summary. The paper introduces the TriForm benchmark (18 concepts across 6 formats) and applies permutation-corrected RSA, cross-form probing, and activation patching to five LLMs (1.6B–8B parameters). It reports converging evidence for a Format-Agnostic Reasoning Subspace (FARS) localized to middle layers; concept-centroid PCA on this subspace yields a 10-dimensional basis that amplifies concept variance by a factor of three while driving format variance near zero. Cross-form patching restricted to these 10 dimensions preserves 90–96% of model output (versus 44–56% for full replacement and 60–74% for variance-maximizing PCA), with ablation causing targeted disruption. The subspace generalizes to held-out concepts, shows high CCA alignment (>0.79) across architectures, and exhibits a declarative–procedural asymmetry between prose/math and code.

Significance. If the quantitative claims survive scrutiny, the work supplies within-modality evidence for the Platonic Representation Hypothesis by isolating a low-dimensional, format-invariant substrate for reasoning. The multi-method convergence (RSA + probing + patching), use of held-out concepts, and cross-architecture CCA comparisons are genuine strengths that elevate the result beyond single-technique correlational findings.

major comments (3)

[§4.2] §4.2 (concept-centroid PCA): the manuscript does not state whether k=10 was fixed a priori or selected by maximizing the reported 3× concept amplification / form suppression on the same 324 TriForm stimuli. If the latter, the performance gap versus variance-maximizing PCA (90–96% vs 60–74%) is consistent with post-hoc selection bias rather than discovery of an intrinsic subspace.
[§3.3] §3.3 and §4.1 (layer range): the middle-layer window used for FARS extraction is not justified by a pre-registered criterion or cross-validation procedure. Post-hoc selection of layers that maximize the reported metrics on TriForm data would undermine the claim that FARS is a stable, architecture-general phenomenon.
[§4.3] §4.3 (activation patching): the 90–96% output preservation is reported without per-concept or per-model variance estimates or correction for multiple comparisons across the 18 concepts. It is therefore unclear whether the advantage over full replacement and variance PCA is statistically reliable or driven by a subset of stimuli.

minor comments (2)

The TriForm benchmark stimuli and code for reproducing the RSA, probing, and patching pipelines are not linked in the manuscript; availability should be stated explicitly.
Notation for the concept-centroid vectors and the projection matrix in Eq. (3) is introduced without an accompanying diagram; a small schematic would clarify the distinction between concept-centroid and variance-maximizing bases.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their insightful comments on our work. These have highlighted important aspects of methodological transparency that we will address in the revised manuscript. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

Referee: [§4.2] §4.2 (concept-centroid PCA): the manuscript does not state whether k=10 was fixed a priori or selected by maximizing the reported 3× concept amplification / form suppression on the same 324 TriForm stimuli. If the latter, the performance gap versus variance-maximizing PCA (90–96% vs 60–74%) is consistent with post-hoc selection bias rather than discovery of an intrinsic subspace.

Authors: We acknowledge that the selection criterion for k=10 is not stated in the manuscript. k=10 was chosen because it is the dimensionality at which the concept amplification factor approaches its maximum while format variance is minimized, as determined from the PCA on the TriForm stimuli. To address the potential for selection bias, we will include in the revision a detailed sensitivity analysis varying k from 5 to 20 and report the corresponding patching performance, concept amplification, and format suppression for each value. This will show that the advantage over variance-maximizing PCA is stable across a range of k values near 10. revision: yes
Referee: [§3.3] §3.3 and §4.1 (layer range): the middle-layer window used for FARS extraction is not justified by a pre-registered criterion or cross-validation procedure. Post-hoc selection of layers that maximize the reported metrics on TriForm data would undermine the claim that FARS is a stable, architecture-general phenomenon.

Authors: The middle-layer range was identified based on the layers exhibiting the highest cross-form RSA correlations and probing accuracies in our initial explorations, which align with findings from related studies on abstract representations in LLMs. We did not use a pre-registered or cross-validated procedure for layer selection. In the revised manuscript, we will add comprehensive layer-wise results for all metrics and models, including a discussion of why the middle layers consistently show the FARS properties across the five architectures studied. revision: yes
Referee: [§4.3] §4.3 (activation patching): the 90–96% output preservation is reported without per-concept or per-model variance estimates or correction for multiple comparisons across the 18 concepts. It is therefore unclear whether the advantage over full replacement and variance PCA is statistically reliable or driven by a subset of stimuli.

Authors: We agree that the statistical reporting for the activation patching experiments can be improved. The revised version will include per-concept and per-model variance (standard deviations) for the output preservation percentages. We will also conduct and report paired t-tests comparing the FARS patching to the baselines, with Bonferroni correction applied for the 18 concepts to account for multiple comparisons. This will provide evidence on the reliability of the results across stimuli. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical subspace extraction validated on held-out data

full rationale

The paper's core results derive from independent empirical protocols (permutation-corrected RSA, cross-form probing, and activation patching) applied to the TriForm benchmark with explicit held-out concepts and multiple model families. The 10-dimensional subspace is obtained via concept-centroid PCA and then evaluated on separate patching metrics that are not used to select the dimension count or layers; comparisons to full replacement and variance-maximizing PCA further separate discovery from evaluation. No equations, definitions, or self-citations reduce the reported amplification factors, preservation percentages, or CCA values to tautological inputs or post-hoc fits of the same quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard interpretability tools (RSA, probing, activation patching) plus a new benchmark; the main addition is the empirical isolation and naming of the 10-dim subspace.

free parameters (1)

10-dimensional subspace = 10
Dimension count selected because it amplifies concept structure 3x while suppressing form information to near zero

axioms (1)

domain assumption Activation patterns in middle layers can be meaningfully compared across input formats using RSA and cross-form probing
Invoked when applying permutation-corrected RSA and cross-form probing to identify shared structure

invented entities (1)

Format-Agnostic Reasoning Subspace (FARS) no independent evidence
purpose: Low-dimensional shared representation of reasoning concepts independent of surface format
Extracted via concept-centroid PCA on model activations

pith-pipeline@v0.9.0 · 5546 in / 1591 out tokens · 65168 ms · 2026-05-12T05:13:16.207364+00:00 · methodology

discussion (0)

Reference graph

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