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Formalizing Latent Thoughts: Four Axioms of Thought Representation in LLMs

Fahd Seddik, Fatemeh Fard

arXiv:2606.27378v1cs.CLcs.LG

TL;DR

Existing evaluations largely judge continuous thought representations by downstream accuracy, leaving their intrinsic functional quality unclear. This paper proposes four direct axiomatic measures and finds that evaluated representations preserve task identity but collapse per-question identity, with input embeddings matching them across measured axes.

  • Problem

    Existing evaluations mainly use downstream accuracy and lack formal, intrinsic criteria for assessing continuous thought representation quality independently of the decoder or prompt.

  • Method

    The framework defines Causality, Minimality, Separability, and Stability, measuring each directly without retraining across candidate representations and open-weight LLMs.

  • Results

    No evaluated candidate satisfies all four axioms; representations retain coarse task identity but lose per-question identity, while input embeddings match them on every measured axis.

  • Takeaways & Limitations

    The four axioms provide explicit, decomposable diagnostic targets for comparing representations and identifying binding constraints beyond aggregate accuracy.

  • Takeaways & Limitations

    The audit covers 23 BBEH reasoning tasks and five open-weight English-language LLMs, and lexical invariance is unmeasured because it is trivial for the evaluated candidates.

Abstract

from arXiv · show

We introduce an axiomatic evaluation framework for latent thought representations in LLMs, comprising metrics that are independent of downstream benchmark scores and reveal representational failures that benchmark accuracy masks. Existing evaluations conflate representation quality with model capacity. Therefore, failures cannot be attributed to the representation rather than to the model that processes it. We formalize four functional axioms (Causality, Minimality, Separability, and Stability) and define a quantitative measure for each, computed directly on the representation independently of downstream accuracy. We audit open-weight LLMs across 23 reasoning tasks (e.g., Spatial Reasoning, Factual QA). We find that no candidate satisfies all four axioms simultaneously, that the representations distinguish task type reliably but cannot distinguish between two questions within the same task, and that the representations encode little information beyond what is already present in the input embedding. The failure is consistent across dense, reasoning-distilled, and RL-trained model families, indicating that the gap is structural rather than a property of model size or training procedure.

1 Introduction

The paper introduces an axiomatic framework for evaluating latent thought representations independently of downstream accuracy and audits candidates, finding systematic failures in representing individual questions despite retaining task identity.

  • Motivation: The framework addresses the lack of formal functional requirements and intrinsic evaluation that would separate representation quality from model capacity and downstream accuracy.It evaluates what an LLM encodes about a problem rather than only the reasoning quality of a subsequent chain.
  • Axiomatic formalization: The framework defines four functional properties—Causality, Minimality, Separability, and Stability—that characterize thought representations independently of their form.It applies to vectors, tensors, or sets of vectors, and the axioms are proved logically consistent, independent, and complete.
  • Empirical audit: No evaluated candidate satisfies all four axioms, retaining coarse task identity while losing per-question identity and matching the input embedding across every measured axis.The audit on BBEH exposes fine-grained representational collapse that downstream accuracy can mask.
  • Intrinsic evaluation protocol: Each axiom receives an intrinsic measure computed on the source LLM without retraining and decoupled from downstream task accuracy.The measures are KL substitution error, Minimality Gap, same- and cross-task discriminator accuracy, and Distributional Consistency Score.
  • Contributions: The four measures provide quantifiable optimization targets and diagnostics for attributing downstream accuracy changes to specific representational properties.This positions thought representations as objects of study rather than heuristic proxies such as step counts, token budgets, or explicit-CoT imitation.

2 Related work

Prior work shows that LLM hidden states encode rich signals about reasoning, logic, and task difficulty, motivating latent representations of model intent. Continuous reasoning methods explore soft conceptual spaces but face degeneracy, limited task competitiveness, and layer-dependent tradeoffs between signal and noise.

  • Latent representations: Hidden states predict chain-of-thought success, encode sparse logical features, and correlate with task difficulty before token generation [1] [48], [34], [35].These findings motivate representations that capture model intent without relying on explicit outputs.
  • Latent representations: COCONUT [31] and CODI [63] use hidden states to represent or compress multiple reasoning paths into continuous representations.
  • Continuous reasoning methods: Soft-token methods explore continuous concept spaces [81] [64] but require constraints against degenerate greedy behavior [73] [9] and remain competitive with textual CoT on only a narrow task subset [77].
  • Continuous reasoning methods: Layer-wise analyses suggest last-layer activations support next-token prediction, whereas mid-depth representations better balance signal preservation and noise compression [65] [62].

3 Formalizing thought

This section defines Functional Thought as a latent state mediating inputs and semantic outputs, grounded in semantic equivalence and geometry. It formalizes four axioms and metrics for evaluating thought representations directly.

  • 3.1 Semantic equivalence and geometric space: Semantic equivalence is defined by a mapping Φ from output sequences to a semantic manifold, with equivalent sequences sharing the same semantic image.The manifold is equipped with a metric that places partially overlapping meanings closer than conceptually disjoint meanings.
  • 3.2 Thought as a latent functional state: Functional Thought is a latent representation mediating the transformation from inputs to semantic outputs, rather than a communicable linguistic artifact such as Chain-of-Thought.It is intended to capture sufficient statistics of the output distribution and is defined independently of human interpretability.
  • 3.2 Thought as a latent functional state: The thought generator g maps inputs into a thought space while preserving distinctions only when inputs produce semantically distinct outputs, thereby compressing the input space.The construction induces an input equivalence relation based on semantic outputs.
  • 3.3 Quantifying the axioms of Functional Thought: The framework evaluates Functional Thought through four axiomatic properties, formalized with information theory and probability and paired with quantitative metrics.Consistency is proved using an idealized one-hot semantic bottleneck, while independence follows from counter-models that violate one axiom while preserving the other three.
  • 3.3 Quantifying the axioms of Functional Thought: Causality requires substituting T for a reasoning prefix while preserving the predictive distribution over the answer suffix.It is quantified by divergence after replacing prefix token embeddings with the projected representation; lower divergence indicates stronger causal encapsulation.
  • 3.3 Quantifying the axioms of Functional Thought: Minimality requires T to compress inputs while retaining maximal relevance to the output distribution and filtering nuisance information unrelated to generation.Its intractable information-bottleneck objective is approximated by a cross-entropy surrogate, where larger ΔIB indicates greater relevance and minimality.
  • 3.3 Quantifying the axioms of Functional Thought: Separability requires functional discrimination of semantic content in latent space, using learned alignment between T and outputs with same-task and cross-task negatives.The discriminator is implemented as a trainable linear projection into the embedding space of a frozen LLM backbone.

4 Experimental setup

The audit evaluates two last-token hidden-state candidates across 23 BBEH tasks and five diverse open-weight LLMs, using beam-search outputs and frozen-backbone probes to assess the axioms.

  • Candidates: The study analyzes Last Input Token representations from all layers and from the final layer, capturing pre-generation context before the model emits y.Prior work shows linear probes can recover non-trivial information about upcoming generation from these activations [48].
  • Table layout: Input Embedding is the prompt baseline, while Output Embedding provides Exact and Pooled upper-bound references; failure to outperform Input Embedding indicates no added information beyond the prompt.Output-based encodings are not references for Minimality or Stability because they incur their own penalties.
  • Models: The audit covers 23 BBEH tasks [41] and five open-weight models spanning dense, sparse-MoE, reasoning-distilled, and RL-trained paradigms.The models are Llama-3.1-8B-Instruct, Llama-3.3-70B-Instruct, DeepSeek-R1-Distill-Qwen-32B, Skywork-OR1-32B [32], and GPT-OSS-20B [53].
  • Generation: Eight distinct beam-search sequences of up to 8192 tokens provide an empirical slice of high-probability outputs for evaluating each axiom.Beam search reduces sampling variance, guarantees distinct candidates, and exposes an empirical distribution over reasoning paths [24].
  • Probes: Probes use a frozen Llama-3.2-1B backbone with trainable projections into token-embedding space and a trained discriminator head.The frozen backbone supplies a shared decoding surface, while projections learn features specific to each candidate and source-model pair.

5 Results

Across four axioms, no candidate thought representation consistently outperforms the Input Embedding reference across LLMs, and benchmark accuracy does not reveal these representational failures. Representations distinguish tasks but generally fail to distinguish questions within the same task, while iterative thinking degrades with step count.

  • Causality: Every thought representation yields causality KL substantially below the information-free Random Vector baseline, while Output Embedding and Input Embedding are comparable on this axis.The Output Embedding causality score is within bootstrap noise of the Input Embedding reference.
  • Minimality: Minimality results are mixed: LIT usually falls below Input Embedding, soft-thinking is at or above it, and Latent Thinking is nearly equivalent.No candidate consistently encodes more output-relevant compression than the prompt already provides.
  • Separability: Cross-task accuracy is near saturation for all candidates, but every candidate except Output Embedding remains near random on within-task question discrimination.Output Embedding is the only family meaningfully above random in within-task discrimination, with Exact outperforming Pooled.
  • Stability: On Stability, candidates substantially beat Random Vector when beam outputs form multiple semantic classes, with Output Embedding highest and iterative families declining as steps increase.This analysis covers the four LLMs producing non-singleton beam clusters for a sizeable fraction of questions.
  • Overall results: No candidate consistently exceeds the Input Embedding reference on any axis when averaged across LLMs, and no candidate satisfies all four axioms.Per-LLM advantages do not generalize, and iterative thinking variants degrade as step count grows.

6 Conclusion · A Extended Related Work

The paper introduces a direct, four-measure axiomatic evaluation framework that reveals per-question representational collapse across diverse LLM paradigms. Related work situates this framework among latent reasoning, latent world-model, and sentence-level representation methods.

  • 6 Conclusion: The framework evaluates candidate thought representations directly on source LLMs without retraining, using four measures across dense, sparse-MoE, reasoning-distilled, and RL-trained paradigms.Its protocol exposes per-question identity collapse that downstream task accuracy does not reveal.
  • 6 Conclusion: The four axioms provide independent optimization targets and diagnostic readouts, enabling attribution of downstream accuracy changes to specific representational properties.Audits can identify the binding constraint before retraining and support comparisons among competing representations.
  • 6 Conclusion: Stability’s lexical-invariance sub-property is not measured because all evaluated candidates produce paraphrase-identical vectors by construction.A protocol for this sub-property awaits candidate constructions with non-trivial paraphrase variation.
  • 6 Conclusion: The evaluation protocol costs more than a single accuracy benchmark because it requires LLM generations and an additional short probe-training step.This measurement cost is an explicit limitation of the proposed audit.
  • A Extended Related Work: Related surveys organize latent and continuous reasoning methods into token-wise horizontal approaches and layer-wise vertical approaches, while theoretical work characterizes fundamental exploration–exploitation issues.Chen et al. [13] provide the two-axis taxonomy; the supplied passage introduces Zou et al. ’s theoretical characterization but truncates its description.
  • A Extended Related Work: Latent-world-model research extracts structured propositions from hidden states, providing empirical grounding for Separability as linearly decodable semantic-world information.Feng et al. [26] demonstrate this extraction using propositional probes.
  • A Extended Related Work: Sentence-level representation methods model meaning in continuous embedding spaces rather than token vocabularies, including Large Concept Models [68] with SONAR [22] and SONAR-LLM [20].The supplied passage describes these approaches as extending sentence-embedding-space language modeling and reasoning.

B Formal Analysis of the Axiomatic System … C.6 Stability Sub-Properties and DCS Diagnostics

The axiomatic system is logically consistent, independent, and complete only in an idealized setting, while empirical diagnostics operationalize its evaluation and expose limitations of stability and uncertainty decoding.

  • B.1 Consistency (Existence): The four axioms are logically consistent in an idealized semantic-bottleneck model, but arbitrary frozen models face a Pareto frontier between causal exactness and semantic stability.The construction satisfies Causality, Minimality, Separability, and Stability simultaneously; empirical representations instead provide ε-optimal approximations.
  • B.2 Independence (Non-Redundancy): Each axiom is logically independent: counter-models violate Causality, Minimality, Separability, or Stability while preserving the other properties.The constructions use fixed-head coordinate permutations, nuisance information, XOR geometry, and output-dependent lexical features to isolate each failure.
  • B.3 Completeness (Sufficiency): Strict satisfaction of the four axioms establishes a bijective functional isomorphism between reachable representations and semantic classes, becoming approximate under empirical stability tolerances and finite separability margins.Stability and Minimality make the mapping well-defined, Separability makes it injective, Minimality makes it surjective, and Causality preserves downstream behavior.
  • C.1 LLM Data Generation: Representations are extracted from every decoding layer and step, using the last prefill token as the primary thought representation across source LLMs.Models run in native bfloat16 precision, with per-model layer counts varying across the evaluated architectures.
  • C.2 Soft Thinking and Latent Thinking Generation: Soft, noisy, and recurrent latent-thinking procedures are evaluated at iterative steps 1, 16, 32, 64, and 128, with stochastic extraction made deterministic by fixed seeds.Soft Thinking mixes token embeddings, Gumbel noise uses temperature τ = 1.0, and Latent Thinking follows a COCONUT-like recurrent protocol.
  • C.3 Causality Evaluation Protocol: Causality is evaluated without training by replacing each beam’s reasoning prefix with a tiled thought representation and comparing suffix distributions under teacher forcing.The protocol uses K = 8 beams, excludes beams shorter than 51 tokens, and controls position effects by tiling candidates to 128 positions.
  • C.4 Minimality Probe Architecture and Training: Minimality and Separability use fixed probe and discriminator architectures, including LayerNorm, learned projections to 2048 dimensions, and a frozen LLaMA-3.2-1B backbone.Configurations were fixed across source LLMs and candidate representations, with no held-out-test hyperparameter tuning.
  • C.6 Stability Sub-Properties and DCS Diagnostics: Singleton-per-question representations satisfy lexical invariance by construction, whereas GPT-OSS-20B produces non-singleton semantic clusters on only 1.0% of questions versus 16–46% for four dense LLMs.The MoE model’s routing may re-converge diverging beams; threshold-sweep rankings remain stable, while near-random DCS scores only argue against linear encoding, not nonlinear encoding.

C.7 Reproducibility and Code Release … D.4 Causality with the Output-Reconstruction Projection

The evaluation is reproducible through a public, pinned pipeline and reports uncertainty with problem-level bootstrap procedures. Detailed analyses show that causality depends substantially on the projection objective, while representation-level results and distributional diagnostics characterize the four-axiom measurements.

  • C.7 Reproducibility and Code Release: The repository is publicly released under the MIT license as a Hydra project with one runnable script per phase, shared configuration, and pinned random seeds.Source LLMs and the text embedder are public HuggingFace assets, enabling phase-level re-execution without additional parameter passing.
  • C.7 Reproducibility and Code Release: Python 3.12 and CUDA 12.6 dependencies are pinned and resolved with uv sync, while flash-attention is optional for reproducing the reported numbers.Most compute used H100 SXM 80 GB GPUs; a small subset of probe-training and evaluation runs used A100 40 GB GPUs without configuration changes.
  • D.1 Bootstrap Confidence Intervals: Problem-level bootstrap resampling preserves within-problem beam correlations across Minimality, Causality, and discriminator-based DCS measurements.Causality uses clustered valid beams, while DCS contributes one per-problem score; paired Causality advantages keep each candidate-reference pair together.
  • D.1 Bootstrap Confidence Intervals: Causality’s paired advantage treats negative values as evidence that a representation conveys information beyond the evaluation backbone’s projection, whereas nonnegative values indicate no detectable advantage.Pairing cancels per-problem shifts in the KL scale and yields a sign-interpretable statistic with uncertainty from paired bootstrap differences.
  • D.2 Detailed per-axiom results: Tables 9–11 report per-source-LLM Separability, Causality, and Minimality metrics, while Figure 4 reports Stability AUROC and DCS diagnostics.Minimality candidate comparisons are meaningful within source LLM because the residual’s absolute scale shifts with an omitted LLM-dependent constant.
  • D.3 Distributional Views of Causality: Causality distributions reveal whether lower mean KL reflects tighter per-problem behavior or heavy upper tails, while ICC separates between-problem from within-problem dispersion.Figure 5 uses per-beam KL CDFs, and Figure 6 assesses whether problem-level differences support the cluster-bootstrap design.
  • D.4 Causality with the Output-Reconstruction Projection: Soft thinking’s KL drops by roughly half when using the output-reconstruction projection, while the exact output embedding worsens and the random-vector reference is unchanged.This representation-specific effect shows that projection choice materially contributes to high causality KL for source-model residual-stream representations.

D.5 Information-Bottleneck Decomposition for the Minimality Metric

The minimality metric ΔIB is an information-bottleneck Lagrangian surrogate at β = 2, estimated through probe cross-entropies and comparable across representations up to a fixed offset. This equivalence requires T to be a deterministic function of X; candidates derived from generated outputs violate that assumption and require a correction.

  • Chain-rule decomposition: At β = 2, ΔIB decomposes the information-bottleneck Lagrangian into output information and conditional input information terms via the mutual-information chain rule.When T is deterministic from X, I(Y; T | X) = 0, yielding L(2) = −I(T; Y) + I(X; T | Y).
  • Barber–Agakov surrogates: Probe cross-entropies provide Barber–Agakov estimates of the two mutual-information terms, becoming tight with a sufficiently expressive probe and matching the true conditional.The estimates use H(Y | T) ≈ CE(Y | T) and H(X | Y, T) ≈ CE(X | Y, T), with probes conditioned on Y for the conditional term.
  • Bounding direction and ranking preservation: Comparisons of ΔIB across representations differ from −L(2) only by a representation-independent constant determined by the dataset and unconditional baseline probe.This offset is C = H(Y) − CE(X | Y), while shared candidate bias can cancel within a fixed probe class and source LLM.
  • Random Vector cross-entropy interpretation: The Random Vector anchor produces unconditional-language-model-level CE(Y | T), confirming that the metric detects when a representation contains no information about the output.Because random vectors carry no information about Y, the probe cannot optimize a predictive signal from T.
  • When the chain-rule assumption fails: For exact and pooled output embeddings, T depends on Y as well as X, so ΔIB no longer estimates −L(2) even up to a T-independent constant.These candidates violate I(Y; T | X) = 0, introducing a correction term that does not collapse into a representation-independent offset.

D.6 Length Sensitivity of the Causality Metric

Length-sensitivity tests show that causality KL is not meaningfully driven by prompt or output length for informative candidates, and its coarse ranking is stable across evaluation windows. However, substitution length can inflate absolute KL and reorder candidates, motivating common-length tiling for subsequent comparisons.

  • No candidate shows meaningful coupling between causality KL and input length, ruling out prompt size as the source of the main-text ordering.
  • Output length couples with KL only for the Random Vector lower-bound reference; informative candidates show near-zero coupling, indicating KL reflects representational fit rather than generation length.
  • Causality-KL rankings remain coarsely stable across 10, 25, 50, 100, and 200-token windows, although fine-grained orderings shift at longer windows and separation weakens at the shortest.
  • Tiling candidates to common substitution lengths reveals that Random Vector and, less strongly, Exact Output Embedding accumulate higher KL as length increases, making absolute values incomparable across native lengths.
  • All subsequent causality and minimality comparisons therefore feed T representations at a common substitution length, reserving varying-length comparisons for the diagnostic.

D.7 Geometric Analysis of Representational Collapse

Direct geometric diagnostics show that same-task discrimination fails because latent thought representations collapse within tasks, not because probes lack capacity. Across LLMs, low within-task dimensionality, noise drift, or both prevents a representation from simultaneously achieving high task purity and rich within-task variation.

  • Geometric diagnostics: The analysis uses k-NN task purity and participation ratio (PR) to test task clustering and within-task spread directly from flattened thought vectors.Purity is compared with a random-neighbour baseline of approximately 0.042, while PR exceeds 1 only when within-task vectors span multiple directions; on this split, the ceiling is approximately 19.
  • Geometric results: The top-right region of the (PR, k-NN purity) plane is empty for every LLM, showing that no candidate combines rich within-task geometry with strong task clustering.LIT, one-step Latent Thinking, Input Embedding, and Output Embedding generally occupy a moderate-PR, high-purity region; on GPT-OSS-20B, both LIT variants instead lose purity while the embedding candidates remain in their usual region.
  • Geometric results: Same-task discrimination fails because within-task geometry is collapsed—too narrow, too noise-drifted, or both—across every tested LLM, independently of probe capacity.Every candidate apart from Output Embedding has either low PR or low purity in each panel, and all candidates fail same-task discrimination in Table 9.
  • Geometric results: Adding latent-thinking steps cuts purity by roughly half on each LLM and also reduces PR on the larger Llama and distilled models.Latent Thinking starts near LIT in the high-purity region at one step, but longer trajectories move away from that geometry.
  • Geometric results: The anisotropy-adjusted similarity diagnostic ranges from near 0 for Random Vector to near 0.52 for Input Embedding, with LIT and Output Embedding between roughly 0.18 and 0.45.All three geometric diagnostics converge on the same failure pattern without using a trained discriminator.

D.8 Probe Capacity Ablation · D.9 Relationship to Downstream Task Accuracy · E Dataset and Benchmark Details

Across 23 BBEH tasks and five LLMs, within-task separability of thought representations is unrelated to downstream accuracy, while increasing probe capacity does not remove the same-task failure. Together, these results attribute the collapse to the representations rather than discriminator weakness or task difficulty.

  • D.8 Probe Capacity Ablation: The same-task failure persists across three strictly increasing probe architectures, indicating that it is a property of the representation rather than the discriminator.The comparison uses Llama-3.3-70B and evaluates latent-thinking and last-input-token candidates.
  • D.8 Probe Capacity Ablation: The probe ablation increases capacity from a single linear projection to a two-layer MLP with LayerNorm, then additionally unfreezes the 1B model’s last two transformer blocks.Each step strictly enlarges the trainable-parameter budget, with the deepest tier carrying roughly an order of magnitude more trainable parameters.
  • D.8 Probe Capacity Ablation: Last Input Token and Soft Thinking at 128 steps represent the main same-task failure mode, whereas Output Embedding (Exact) serves as a positive control derived from the generated answer.The positive control should be the first representation to improve if probe capacity can overcome the random-guess plateau.
  • D.8 Probe Capacity Ablation: Output Embedding (Exact) reaches 0.726 test accuracy with the Baseline probe, but falls to 0.617 with Deep and 0.642 with Deep+UF2 as capacity increases.The decline is attributed to overfitting despite an order-of-magnitude increase in trainable parameters, while BCE worsens toward the random-guess plateau.
  • D.9 Relationship to Downstream Task Accuracy: ρ = 0.10 (p = 0.31, n = 115): thought-representation separability shows no significant pooled correlation with downstream BBEH pass@1.No individual LLM reaches significance, supporting the conclusion that separability collapse is not driven by task difficulty.
  • D.9 Relationship to Downstream Task Accuracy: The separability collapse therefore reflects a representational failure that downstream task accuracy does not register, rather than merely the difficulty of the underlying tasks.This interpretation follows the absence of a significant accuracy relationship and the convergence of geometric and increasing-capacity probe tests.

E.1 BBEH Task Coverage … F Broader Impacts

The evaluation spans 23 BBEH tasks and compares source LLM outputs, parsing, accuracy, length, and beam-embedding similarity. Results show substantial variation in output behavior and accuracy, while embedding similarity primarily tracks response format rather than answer content; the work is diagnostic and introduces no deployed capability.

  • E.1 BBEH Task Coverage: The evaluation covers all 23 BBEH tasks, using 4,520 problems and eight beams per problem for 36,160 beam–problem pairs.Problems are further partitioned into stratified probe and discriminator splits within each task.
  • E.1 BBEH Task Coverage: The BBEH setup acknowledges originating benchmarks, including Kazemi et al. [40], Nie et al. [52], Kıcıman et al. [42], and Tyen et al. [69].Additional task origins include Kazemi et al. [39], Sánchez et al. [60], Hessel et al. [36], Zhang et al. [80], Yamada et al. [75], Fatemi et al. [25], White et al. [72], and Shah et al. [61].
  • E.2 BBEH Answer Extraction: Per-beam answer-parsing failure varies by more than an order of magnitude across LLMs, and the fraction of problems with at least one parsed beam follows the same ordering.The evaluation uses Kazemi et al.’s [41] official extractor with task-specific regular expressions; recoverable problems quantify headroom for a beam-level verifier.
  • E.3 Output-Length Statistics: At the median, GPT-OSS-20B, Skywork-OR1-32B, and DS-R1-Qwen-32B produce outputs roughly an order of magnitude longer than both instruction-tuned Llama checkpoints.Their interquartile bands reach the ten-thousand-character range, while hard combinatorial tasks remain longest and short answer-format tasks remain shortest across columns.
  • E.5 Qualitative Beam Outputs and Embedder Similarity: The qualitative beam comparisons use held-out test examples and the Llama-Embed-Nemotron-8B space shared by the Pooled and Exact Output Embedding candidates.One high-similarity pair from Time Arithmetic follows the same arithmetic chain and answer, differing only in the closing wording; the low-similarity Causal Understanding pair contrasts long-form reasoning with a one-line verdict.
  • E.5 Qualitative Beam Outputs and Embedder Similarity: The embedder separates beams primarily by response format and only secondarily by answer content.In the Causal Understanding example, beams within a mode have cos ≥0.98, cross-mode similarity falls to [0.06, 0.10], and the two short modes reach cos ≈0.80 despite contradictory verdicts.
  • F Broader Impacts: The work is a diagnostic protocol for representations inside frozen LLMs, producing no new generative capability, dataset, or deployed system.It is intended to support better reasoning models and understanding of functional thought representations, and the authors identify no negative societal effects.

G Earlier Formulations

Earlier formulations exposed methodological failures in discriminator-based DCS, causality projection, and minimality measurement. Qualitative and quantitative probes likewise showed that thought representations can discard instance-specific information and fail same-task separability.

  • G.1 Discriminator-Based DCS Evaluation Protocol: Discriminator-based DCS collapsed to the random baseline because a cross-question-trained discriminator returned near-0.5 scores for within-question beam pairs across all representation families and source LLMs.The protocol compared embedding- and answer-based equivalence matrices using eight beams per problem, but provided no useful within-question gradient signal.
  • G.2 Causality with the Discriminator-Trained Projection: The earlier causality results used a discriminator-trained projection, which was later replaced by output reconstruction after a projection-swap pilot motivated the corrected protocol.Table 22 is retained to show the earlier projection’s effect cell by cell against current results; its training procedure expanded representations to a common fixed length.
  • G.3 Cross-Entropy Proxy for Minimality: The initial minimality proxy defined ∆CE = CE(X | T)−CE(Y | T), but it matches the information-bottleneck Lagrangian only at trade-off weight β = 1.These limitations motivated the corrected ∆IB formulation adopted in Section 3 and derived in Section D.5.
  • G.3 Cross-Entropy Proxy for Minimality: A high ∆CE is ambiguous: it can reflect input compression with output retention or simultaneous loss of information about both input and output.The input and output components are reported separately in Tables 23 and 24, with Random Vector providing a shared input-reconstruction anchor of CE(X | RV) ≈ 1.87 ± 0.04.
  • G.3 Cross-Entropy Proxy for Minimality: On Llama-3.1-8B, Input Embedding already achieved a ∆CE in the high 0.9 range, while only Output Embedding (Exact) clearly exceeded it.Most thinking candidates were at or below the input-alone gap, so the proxy did not show latent thinking producing a more compressed, output-relevant summary than the prompt.
  • G.4 Qualitative Probe Reconstruction: For a BoardGame QA example, the Last Input Token representation produced the same boilerplate prediction for all 10 sampled problems, discarding problem-specific content.The probe reported ∆CE = 0.18 and CE(X|T) = 1.63, and generated an unrelated multiple-choice template rather than instance-specific reconstructions.
  • G.4 Qualitative Probe Reconstruction: Because the input probe could not distinguish the 10 BoardGame QA problems from T, the same-task discriminator likewise could not distinguish them.This qualitative failure confirms the collapse of same-task separability in the thought representation.

G.5 Discriminator-Based DCS Results

Discriminator-based DCS saturates near chance on within-question pairs, making correlations noisy despite close agreement between embedding variants. Its bimodal similarity structure and threshold-stable ranking indicate genuine paraphrase versus cross-answer separation, but not reliable discriminator scores.

  • Similarity structure: Cosine similarities are bimodal across models, separating paraphrase-equivalent pairs near 1 from genuine cross-answer pairs below 0.5.The right mode is large, while the left mode is smaller.
  • Cross-model results: Values near 0.5 across representations reflect discriminator saturation on within-question pairs, with Llama-3.3-70B-Instruct correlations collapsing to noise despite small absolute disagreement.Eparse and Eemb agree closely across models, so results are reported under Eemb; Eemb also remains defined when BBEH answer extraction fails.
  • Threshold sensitivity: Across the full threshold sweep on Llama-3.1-8B-Instruct, rankings remain stable and the chance-floor cluster does not reshuffle.This confirms that saturation is intrinsic to discriminator-based scoring rather than caused by the binarisation threshold; the τ=0.90 row reproduces Table 26 cell-for-cell.
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