Kendiukhov (2026) runs an autonomous executor-brainstormer loop that proposed, tested, and refined 141 geometric/topological hypotheses across 52 iterations about gene-embedding representations in scGPT and Geneformer, with explicit null controls and disjoint gene-pool splits. Persistent homology is statistically significant (p<0.05) in 11/12 transformer layers in the weakest domain and 12/12 in the other two; manifold-aware distance metrics outperform Euclidean distance for identifying regulatory gene pairs; graph community partitions track known transcription-factor-target relationships; canonical correlation analysis between scGPT and Geneformer's independently- trained gene-embedding spaces yields a canonical correlation of 0.80 and 72% gene-retrieval accuracy (though none of 19 tested methods reliably recover exact gene-level correspondence); robust signal concentrates in immune tissue under stringent nulls.

Kendiukhov exports a fixed operator from scGPT's frozen attention value-projection weights (no forward-pass activations) and compresses it into a latent space whose effective dimensionality plateaus at ~8-10 [kendiukhov-2026-scgpt-hematopoietic-manifold] The manifold's geodesic distances correlate with an independent hematopoietic developmental ontology (erythroid rho=0.768 p=0.0017, granulocytic rho=0.568, trunk rho=0.611; internal rho=0.835) [kendiukhov-2026-scgpt-hematopoietic-manifold] It validates on a strict non-overlap external panel (Tabula Sapiens, 564,253 cells) and zero-shot transfer to an immune panel (trustworthiness 0.993, blocked-permutation p=0.0005) [kendiukhov-2026-scgpt-hematopoietic-manifold] A lightweight readout on the extracted manifold beats scVI, Palantir, DPT, CellTypist, and PCA on pseudotime ordering (|rho|=0.439 vs 0.331) with ~1,000x fewer parameters [kendiukhov-2026-scgpt-hematopoietic-manifold]

Kendiukhov (2026) runs an autonomous executor-brainstormer loop that proposed, tested, and refined 141 geometric/topological hypotheses across 52 iterations about gene-embedding representations in scGPT and Geneformer, with explicit null controls and disjoint gene-pool splits. Persistent homology is statistically significant (p<0.05) in 11/12 transformer layers in the weakest domain and 12/12 in the other two; manifold-aware distance metrics outperform Euclidean distance for identifying regulatory gene pairs; graph community partitions track known transcription-factor-target relationships; canonical correlation analysis between scGPT and Geneformer's independently- trained gene-embedding spaces yields a canonical correlation of 0.80 and 72% gene-retrieval accuracy (though none of 19 tested methods reliably recover exact gene-level correspondence); robust signal concentrates in immune tissue under stringent nulls.

Conneau et al. (2020) train five same-architecture masked LMs from scratch, one per language (en/fr/de/ru/zh), with no shared parameters, vocabulary or parallel data [conneau-etal-2020] Orthogonal Procrustes mapping between language pairs recovers substantial cross-lingual structure at subword, word and sentence granularity, improving at higher layers [conneau-etal-2020] CKA between paired parallel-sentence representations shows monolingual models are more similar to each other than to a random encoder but less than bilingually co-trained models (e.g. en-fr 0.58/0.59 vs 0.69 vs 0.46) [conneau-etal-2020] CKA similarity correlates with alignment-based sentence-retrieval performance at >0.9 Pearson, evidencing universal latent symmetries across independently-trained models [conneau-etal-2020]

Dar replaces vec2vec's adversarial unpaired embedding translation with a three-stage linear pipeline: k-means cluster-landmark matching, Quadratic-Assignment-Problem permutation solving, and ICP-style orthogonal Procrustes refinement [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] Evaluated on the four text encoders vec2vec used (gtr, e5, stella, granite) over the 12 directed pairs, it matches or exceeds vec2vec's top-1 accuracy and average rank on nearly every pair [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] It never collapses on the pairs where adversarial vec2vec was weakest, runs in under 10 minutes on CPU versus 1-7 GPU-days, and shows <=0.01 run-to-run standard deviation [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] This reduces vec2vec's implicit geometric claim to a literal distance-preserving orthogonal rotation between two embedding geometries, recoverable without paired data [dar-2025-mini-vec2vec-linear-universal-geometry-alignment]

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Dar replaces vec2vec's adversarial unpaired embedding translation with a three-stage linear pipeline: k-means cluster-landmark matching, Quadratic-Assignment-Problem permutation solving, and ICP-style orthogonal Procrustes refinement [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] Evaluated on the four text encoders vec2vec used (gtr, e5, stella, granite) over the 12 directed pairs, it matches or exceeds vec2vec's top-1 accuracy and average rank on nearly every pair [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] It never collapses on the pairs where adversarial vec2vec was weakest, runs in under 10 minutes on CPU versus 1-7 GPU-days, and shows <=0.01 run-to-run standard deviation [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] This reduces vec2vec's implicit geometric claim to a literal distance-preserving orthogonal rotation between two embedding geometries, recoverable without paired data [dar-2025-mini-vec2vec-linear-universal-geometry-alignment]

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Dar replaces vec2vec's adversarial unpaired embedding translation with a three-stage linear pipeline: k-means cluster-landmark matching, Quadratic-Assignment-Problem permutation solving, and ICP-style orthogonal Procrustes refinement [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] Evaluated on the four text encoders vec2vec used (gtr, e5, stella, granite) over the 12 directed pairs, it matches or exceeds vec2vec's top-1 accuracy and average rank on nearly every pair [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] It never collapses on the pairs where adversarial vec2vec was weakest, runs in under 10 minutes on CPU versus 1-7 GPU-days, and shows <=0.01 run-to-run standard deviation [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] This reduces vec2vec's implicit geometric claim to a literal distance-preserving orthogonal rotation between two embedding geometries, recoverable without paired data [dar-2025-mini-vec2vec-linear-universal-geometry-alignment]

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Dar replaces vec2vec's adversarial unpaired embedding translation with a three-stage linear pipeline: k-means cluster-landmark matching, Quadratic-Assignment-Problem permutation solving, and ICP-style orthogonal Procrustes refinement [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] Evaluated on the four text encoders vec2vec used (gtr, e5, stella, granite) over the 12 directed pairs, it matches or exceeds vec2vec's top-1 accuracy and average rank on nearly every pair [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] It never collapses on the pairs where adversarial vec2vec was weakest, runs in under 10 minutes on CPU versus 1-7 GPU-days, and shows <=0.01 run-to-run standard deviation [dar-2025-mini-vec2vec-linear-universal-geometry-alignment] This reduces vec2vec's implicit geometric claim to a literal distance-preserving orthogonal rotation between two embedding geometries, recoverable without paired data [dar-2025-mini-vec2vec-linear-universal-geometry-alignment]

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Lee et al. train a message-passing GNN on ~5,000 molecules labeled with 138 human odor descriptors (cross-validated AUROC 0.89), taking the 256-dim penultimate layer as a Principal Odor Map [lee-etal-2023-principal-odor-map] POM pairwise distances correlate R=0.73 with an independently-measured human perceptual odor map, versus R=-0.12 for a Morgan-fingerprint embedding [lee-etal-2023-principal-odor-map] Same-label molecules cluster tighter in the POM (cluster density 0.51 vs 0.68 for fingerprints) [lee-etal-2023-principal-odor-map] On 320 novel odorants rated by a trained panel, the POM model beats the median panelist for 53% of molecules and on 58% of odor descriptors [lee-etal-2023-principal-odor-map] The evidence is an RSA-style distance correlation to perceptual ground truth; no representation-space intervention is performed [lee-etal-2023-principal-odor-map]

Craig et al. compute Centered Kernel Alignment between AI weather models and find forecast skill correlates with cross-model representational alignment [craig-etal-2026-the-physics-of-ai-weather-models] GraphCast and Aurora represent the atmosphere similarly despite differing architectures and capacity [craig-etal-2026-the-physics-of-ai-weather-models] They propose a "particle description" in which latent variables move under gradient flow toward a minimum of a learned free-energy functional [craig-etal-2026-the-physics-of-ai-weather-models] Consistent with this, processor layers shift from large-spatial-scale changes early to small-spatial-scale changes with depth [craig-etal-2026-the-physics-of-ai-weather-models]

Huang et al. fit an ordinary-least-squares linear map T between paired sentence-representations of a source and target real LLM [huang-etal-2025-cross-model-platonic-transferability] Transferring a concept steering vector through T succeeds behaviorally in 212/216 cases, while no-map and random-map controls fail to beat baseline in 23/35 cases [huang-etal-2025-cross-model-platonic-transferability] Maps fit for different concepts are themselves highly similar (SSIM 0.87-0.95) versus a random-matrix baseline (0.05-0.13) [huang-etal-2025-cross-model-platonic-transferability] This cross-concept map consistency is a quantified geometric statistic about the transformation itself, beyond downstream task accuracy [huang-etal-2025-cross-model-platonic-transferability]

Lan, Torr, Meek, Khakzar, Krueger & Barez (2024) introduce "Analogous Feature Universality": even if independently-trained SAEs learn different individual feature directions, the subspaces spanned by those features may still be similar under rotation-invariant transformations. They first pair SAE features across model pairs by activation correlation, then measure representational similarity of the paired feature spaces via SVCCA (SVD-truncated CCA, mean canonical correlation) and RSA (Pearson-correlation inner similarity, Spearman outer similarity, Euclidean-distance RDMs), applied to SAE decoder weight matrices. Across four main model pairs (Pythia-70m vs Pythia-160m; Gemma-1-2B vs Gemma-2-2B; Gemma-2-2B vs Gemma-2-9B; Llama-3-8B-Instruct vs Llama-3.1-8B), similarity is high at matched middle layers and far exceeds a random-feature-pairing null: e.g. Gemma-2-2B layer 11 vs Gemma-2-9B layer 21 reaches SVCCA=0.70 (random mean 0.009) and RSA=0.195 (random mean 4.38e-4); Llama-3 vs Llama-3.1 (both layer 25) reaches SVCCA=0.3 with 66% mean activation correlation among paired features, retaining 7% of feature pairs after significance filtering; almost all post-layer-0 comparisons yield p-values between 0.00 and 0.01. A second baseline (Appendix D) trains SAEs on a randomized-weight Pythia-70m and finds very low SVCCA/RSA scores with only 1% of feature pairs surviving filtering, confirming the middle-layer similarity is not an artifact of the pairing/scoring procedure itself. Section 4.3 further decomposes similarity by semantic concept category (e.g. "Emotions", "Country"): concept- specific subspaces pass a paired-vs-shuffled-pairing significance test in nearly every category tested (the one reported exception is "Country" for the Gemma-1-2B/Gemma-2-2B pair), meaning some semantically coherent subspaces are more reliably cross-model-similar than the feature space taken as a whole. No causal intervention (feature transplant, steering-vector transfer) is performed; the paper explicitly frames cross-model steering-vector transfer as an unrealized implication of the hypothesis rather than a tested claim. A base-vs-fine-tuned control (Appendix C, Gemma-1-2B vs its Instruct variant, same layer 12) shows even higher similarity (SVCCA=0.84, RSA=0.25) than the cross-architecture pairs, as expected for models sharing a base checkpoint. Son, Rathore, Rufail, Simon, Zhang, Dave, Blondin, Zhu & O'Brien (2025) independently replicate this finding for the identical Gemma-2-2B vs Gemma-2-9B pair with independently-trained SAEs: "we aligned the resulting monosemantic features via activation correlation, and compared the matched feature spaces with metrics such as SVCCA and RSA," reporting paired SVCCA peaking at 0.73 (vs. Lan et al.'s 0.70 for the same pair) and RSA peaking at 0.22 (vs. 0.195), with random-pairing baselines again far lower (0.005-0.034) -- a close, genuinely independent replication of the same model pair and method, not a duplicate submission. They additionally extend the analysis beyond single tokens to multi-token/phrase-level semantic subspaces (e.g. concatenated concept phrases like "happy child"), finding the same convergence pattern holds at the phrase level.

Xu et al. probe LLM (mainly LLaMA3-70B) concepts via an in-context reverse-dictionary task, characterizing each context's representation by its pairwise-similarity matrix [xu-etal-2025-emergent-conceptual-representations] RSA alignment across contexts rises from 0.800 at 1 demonstration to 0.970 at 24, converging toward a single context-independent relational structure [xu-etal-2025-emergent-conceptual-representations] Alignment with the 120-demonstration structure correlates with reverse-dictionary accuracy at rho=0.976, and cross-model alignment across 67 LLMs predicts task performance (rho=0.870), all correlational with no intervention [xu-etal-2025-emergent-conceptual-representations] The convergent structure aligns via RSA with human similarity judgments (SimLex-999 rho=0.776), THINGS odd-one-out, and voxel-wise fMRI activity in LOC/FFA/PPA and other regions [xu-etal-2025-emergent-conceptual-representations]

Puri, Berend, Lapuschkin & Samek (2025/2026) build a "Concept Atlas" from a Gemma Scope sparse autoencoder trained on Gemma-2-2B's residual stream (layer 20), then fit an orthogonal Procrustes map from three distinct "subject" models' activation spaces (Llama-3.1-8B base, Llama-3.1-8B-Instruct, DeepSeek-R1-Distill-Llama-3.1-8B-Instruct) into this atlas. Translation quality is quantified via AUROC (0.82-0.86) and average precision (0.39-0.49, versus a random baseline of 0.046) across five subject-model layers, and via near-perfect mean-reciprocal-rank retrieval on 454 independently-validated concept features. Atlas concept directions mapped back into a subject model and added (norm-preserving) to its residual stream at multiple layers simultaneously demonstrably steer generation toward the target concept, though this steering effect is reported qualitatively rather than with an inline quantified success rate.

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Wu, Yu, Yogatama, Lu & Kim (2024) test whether LMs route semantically-equivalent inputs through a shared intermediate representation anchored by the dominant pretraining language, across Llama-2-7B/70B, Llama-3-8B/70B, and Baichuan-2. Baseline-corrected cosine similarity of intermediate hidden states for English-Chinese translation pairs is elevated at middle layers relative to a non-translation-pair control; logit-lens projections show intermediate representations of non-English and non-textual (code, arithmetic, vision, audio) inputs decode to the dominant language's semantically-equivalent tokens. Causal activation patching -- replacing an intermediate representation with the dominant-language- equivalent representation -- predictably shifts downstream output across all tested data types.

Yuan & Søgaard (2025) apply supervised and unsupervised (adversarial + iterative refinement) Procrustes alignment, adapted from cross- lingual word-embedding literature, to final-hidden-layer Othello-move representations from seven differently-architected real models (GPT-2-small, BART-base, T5-base, Flan-T5-XL/3B, LoRA-fine-tuned LLaMA-2-7B, LoRA-fine-tuned Mistral-7B, Qwen2.5-7B). Cross-model cosine similarity after alignment reaches up to 93.1% (supervised, GPT-2 <-> BART) and 97.2% (unsupervised, BART <-> Mistral); PCA visualization of the per-game move trajectory shows convergent geometric structure across models; layer-wise similarity heatmaps peak along the diagonal at corresponding depths; and a "latent move projection" analysis finds the nearest-neighbor tile embedding to any given tile is consistently its actual spatial neighbor on the physical board -- a genuine spatial/geometric isomorphism claim beyond simple linear decodability. Clears scope on criterion (a): a measured, quantified cross-model shared-manifold structure in real pretrained-and-fine-tuned models, going beyond the original linear- probing Othello-GPT results already in this map. No causal intervention is performed (purely representational/alignment analysis). Cross-model convergence of this kind is platonic- representation-adjacent; see [[platonic-representation]].

Huang et al. fit an ordinary-least-squares linear map T between paired sentence-representations of a source and target real LLM [huang-etal-2025-cross-model-platonic-transferability] Transferring a concept steering vector through T succeeds behaviorally in 212/216 cases, while no-map and random-map controls fail to beat baseline in 23/35 cases [huang-etal-2025-cross-model-platonic-transferability] Maps fit for different concepts are themselves highly similar (SSIM 0.87-0.95) versus a random-matrix baseline (0.05-0.13) [huang-etal-2025-cross-model-platonic-transferability] This cross-concept map consistency is a quantified geometric statistic about the transformation itself, beyond downstream task accuracy [huang-etal-2025-cross-model-platonic-transferability]

Xu et al. probe LLM (mainly LLaMA3-70B) concepts via an in-context reverse-dictionary task, characterizing each context's representation by its pairwise-similarity matrix [xu-etal-2025-emergent-conceptual-representations] RSA alignment across contexts rises from 0.800 at 1 demonstration to 0.970 at 24, converging toward a single context-independent relational structure [xu-etal-2025-emergent-conceptual-representations] Alignment with the 120-demonstration structure correlates with reverse-dictionary accuracy at rho=0.976, and cross-model alignment across 67 LLMs predicts task performance (rho=0.870), all correlational with no intervention [xu-etal-2025-emergent-conceptual-representations] The convergent structure aligns via RSA with human similarity judgments (SimLex-999 rho=0.776), THINGS odd-one-out, and voxel-wise fMRI activity in LOC/FFA/PPA and other regions [xu-etal-2025-emergent-conceptual-representations]

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Sarkar (2026) constructs k-nearest-neighbor graphs over 1,000 Flickr30k image-caption pairs separately in DINOv2 (vision) and all-MiniLM-L6-v2 (text) embedding spaces, computes each graph's normalized Laplacian eigenbasis (a manifold-respecting analogue of PCA), and fits a functional map C between the two truncated spectral bases via regularized least squares with a Laplacian-commutativity penalty. Three diagnostics decompose cross-modal geometric compatibility: the normalized eigenvalue-spectrum distance between the two encoders is just 0.043 (both follow the same concave growth profile from ~0.03 to ~0.66), indicating the two independently trained manifolds capture a comparable amount of structure at each scale -- consistent with the Platonic Representation Hypothesis. However, the functional map C exhibits near-zero mean diagonal dominance (<0.05, vs. >0.7 typical of near-isometric shape correspondence) and a large orthogonality deviation (70.15, vs. <0.1 typical), meaning the i-th eigenvector of one manifold does not correspond to any single eigenvector of the other -- energy scatters into off-diagonal bands (rows ~15, ~30) rather than concentrating on the diagonal. This decoupling, termed the "spectral complexity-orientation gap," persists across anchor budgets (|S|=5 to 500), spectral truncations (k_s=10 to 100), and a second text encoder (mpnet), ruling out estimation artifacts. Practically, the functional map substantially underperforms ambient-space baselines on cross-modal retrieval: at |S|=500 anchors, Procrustes achieves 55.5% i2t Recall@1 versus only 4.3% for the functional map (12.9x gap, which widens rather than narrows with more anchor supervision); a composability test (composing a DINOv2-to-MiniLM map with a MiniLM-to-mpnet map) shows the composed map is 4.3x worse than a direct map, consistent with multiplicative error compounding of two already-weak individual maps rather than a failure of the composability mechanism itself. The paper argues this is a genuine structural property of independently-trained neural representations rather than a limitation of the functional-map computation, since varying anchor budget, spectral truncation, or text encoder choice does not close the gap.

Sarkar (2026) constructs k-nearest-neighbor graphs over 1,000 Flickr30k image-caption pairs separately in DINOv2 (vision) and all-MiniLM-L6-v2 (text) embedding spaces, computes each graph's normalized Laplacian eigenbasis (a manifold-respecting analogue of PCA), and fits a functional map C between the two truncated spectral bases via regularized least squares with a Laplacian-commutativity penalty. Three diagnostics decompose cross-modal geometric compatibility: the normalized eigenvalue-spectrum distance between the two encoders is just 0.043 (both follow the same concave growth profile from ~0.03 to ~0.66), indicating the two independently trained manifolds capture a comparable amount of structure at each scale -- consistent with the Platonic Representation Hypothesis. However, the functional map C exhibits near-zero mean diagonal dominance (<0.05, vs. >0.7 typical of near-isometric shape correspondence) and a large orthogonality deviation (70.15, vs. <0.1 typical), meaning the i-th eigenvector of one manifold does not correspond to any single eigenvector of the other -- energy scatters into off-diagonal bands (rows ~15, ~30) rather than concentrating on the diagonal. This decoupling, termed the "spectral complexity-orientation gap," persists across anchor budgets (|S|=5 to 500), spectral truncations (k_s=10 to 100), and a second text encoder (mpnet), ruling out estimation artifacts. Practically, the functional map substantially underperforms ambient-space baselines on cross-modal retrieval: at |S|=500 anchors, Procrustes achieves 55.5% i2t Recall@1 versus only 4.3% for the functional map (12.9x gap, which widens rather than narrows with more anchor supervision); a composability test (composing a DINOv2-to-MiniLM map with a MiniLM-to-mpnet map) shows the composed map is 4.3x worse than a direct map, consistent with multiplicative error compounding of two already-weak individual maps rather than a failure of the composability mechanism itself. The paper argues this is a genuine structural property of independently-trained neural representations rather than a limitation of the functional-map computation, since varying anchor budget, spectral truncation, or text encoder choice does not close the gap.

Sarkar (2026) constructs k-nearest-neighbor graphs over 1,000 Flickr30k image-caption pairs separately in DINOv2 (vision) and all-MiniLM-L6-v2 (text) embedding spaces, computes each graph's normalized Laplacian eigenbasis (a manifold-respecting analogue of PCA), and fits a functional map C between the two truncated spectral bases via regularized least squares with a Laplacian-commutativity penalty. Three diagnostics decompose cross-modal geometric compatibility: the normalized eigenvalue-spectrum distance between the two encoders is just 0.043 (both follow the same concave growth profile from ~0.03 to ~0.66), indicating the two independently trained manifolds capture a comparable amount of structure at each scale -- consistent with the Platonic Representation Hypothesis. However, the functional map C exhibits near-zero mean diagonal dominance (<0.05, vs. >0.7 typical of near-isometric shape correspondence) and a large orthogonality deviation (70.15, vs. <0.1 typical), meaning the i-th eigenvector of one manifold does not correspond to any single eigenvector of the other -- energy scatters into off-diagonal bands (rows ~15, ~30) rather than concentrating on the diagonal. This decoupling, termed the "spectral complexity-orientation gap," persists across anchor budgets (|S|=5 to 500), spectral truncations (k_s=10 to 100), and a second text encoder (mpnet), ruling out estimation artifacts. Practically, the functional map substantially underperforms ambient-space baselines on cross-modal retrieval: at |S|=500 anchors, Procrustes achieves 55.5% i2t Recall@1 versus only 4.3% for the functional map (12.9x gap, which widens rather than narrows with more anchor supervision); a composability test (composing a DINOv2-to-MiniLM map with a MiniLM-to-mpnet map) shows the composed map is 4.3x worse than a direct map, consistent with multiplicative error compounding of two already-weak individual maps rather than a failure of the composability mechanism itself. The paper argues this is a genuine structural property of independently-trained neural representations rather than a limitation of the functional-map computation, since varying anchor budget, spectral truncation, or text encoder choice does not close the gap.

Sarkar (2026) constructs k-nearest-neighbor graphs over 1,000 Flickr30k image-caption pairs separately in DINOv2 (vision) and all-MiniLM-L6-v2 (text) embedding spaces, computes each graph's normalized Laplacian eigenbasis (a manifold-respecting analogue of PCA), and fits a functional map C between the two truncated spectral bases via regularized least squares with a Laplacian-commutativity penalty. Three diagnostics decompose cross-modal geometric compatibility: the normalized eigenvalue-spectrum distance between the two encoders is just 0.043 (both follow the same concave growth profile from ~0.03 to ~0.66), indicating the two independently trained manifolds capture a comparable amount of structure at each scale -- consistent with the Platonic Representation Hypothesis. However, the functional map C exhibits near-zero mean diagonal dominance (<0.05, vs. >0.7 typical of near-isometric shape correspondence) and a large orthogonality deviation (70.15, vs. <0.1 typical), meaning the i-th eigenvector of one manifold does not correspond to any single eigenvector of the other -- energy scatters into off-diagonal bands (rows ~15, ~30) rather than concentrating on the diagonal. This decoupling, termed the "spectral complexity-orientation gap," persists across anchor budgets (|S|=5 to 500), spectral truncations (k_s=10 to 100), and a second text encoder (mpnet), ruling out estimation artifacts. Practically, the functional map substantially underperforms ambient-space baselines on cross-modal retrieval: at |S|=500 anchors, Procrustes achieves 55.5% i2t Recall@1 versus only 4.3% for the functional map (12.9x gap, which widens rather than narrows with more anchor supervision); a composability test (composing a DINOv2-to-MiniLM map with a MiniLM-to-mpnet map) shows the composed map is 4.3x worse than a direct map, consistent with multiplicative error compounding of two already-weak individual maps rather than a failure of the composability mechanism itself. The paper argues this is a genuine structural property of independently-trained neural representations rather than a limitation of the functional-map computation, since varying anchor budget, spectral truncation, or text encoder choice does not close the gap.

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Ryskina, Tuckute, Fung, Malkin & Fedorenko (2025) use an fMRI dataset (Pereira et al. 2018) in which the same concepts are presented as sentences, word clouds, and images, defining a "meaning consistency" metric that identifies brain voxels responding similarly to a concept regardless of presentation modality. Using representational similarity analysis (RSA) to compare the geometry of these regions' response patterns against 15 language and vision-language models' own representational geometry, both language-only and language-vision models predict brain signal better in meaning-consistent regions, even in areas with low language-selectivity -- a genuine shared-geometry finding (RSA-based, not merely correlation-strength) linking model representational structure to a cross-modal property of brain organization.

Zhang et al. train pairs of ViT-S/16 I-JEPA encoders fully independently on different views of the same scenes (smallNORB, nuScenes multi-camera, ImageNet augmentation views) [zhang-etal-2026-social-jepa] They define geometric isomorphism as an invertible linear map z2 approximately W z1, fit by closed-form ridge regression and quantified by MSE, R-squared, linear CKA, distance-structure consistency and neighborhood overlap [zhang-etal-2026-social-jepa] Best case (smallNORB) reaches MSE 0.036, R-squared 0.891, DSC 0.872, a multiply-corroborated approximate linear isometry between independently-learned spaces [zhang-etal-2026-social-jepa] They prove the JEPA loss is invariant under GL(d) reparameterization of the encoder, giving a theoretical reason to expect this convergence class [zhang-etal-2026-social-jepa] The fitted map transfers a linear probe zero-shot with no gradient steps and enables teacher-student representation migration at 0.28x the FLOPs of training from scratch [zhang-etal-2026-social-jepa]

Jha, Zhang, Shmatikov & Morris (2026) propose the Strong Platonic Representation Hypothesis -- that neural networks trained with the same objective and modality, but different data and architectures, converge to a universal latent space such that a translation between their representations can be learned without any pairwise correspondence -- and test it constructively with vec2vec: input adapters (A1, A2) map each of two embedding spaces into a shared latent, a shared backbone T processes the latent, and output adapters (B1, B2) map back out, trained adversarially (GAN-style discriminators on both the final outputs and the shared latents) with a generator loss combining reconstruction, cycle-consistency, and Vector Space Preservation (VSP) -- an explicit term matching the pairwise inner-product (Gram) matrix of a batch of embeddings before and after translation. Across six independently-trained text embedding models (gtr, 110M/T5/2021; clip, 151M/CLIP/2021; e5, 109M/BERT/2022; gte, 109M/BERT/2023; stella, 109M/BERT/2023; granite, 278M/RoBERTa/2024, multilingual) trained on entirely disjoint, unpaired subsets of Natural Questions text (1 million sequences per side, no shared documents), vec2vec achieves cosine similarity up to 0.92 (stella to gte) and top-1 retrieval accuracy up to 100% (rank as low as 1.00) on a 65536-text held-out evaluation, versus a naive identity-mapping baseline (near-zero cosine similarity, 0% top-1 on cross-backbone pairs) and an optimal-assignment baseline using Hungarian/Sinkhorn/ Gromov-Wasserstein solvers (0% top-1, rank near-random ~4096-4209 on cross-backbone pairs) -- both baselines perform at essentially chance on cross-backbone pairs while vec2vec does not. A component ablation (gte to gtr) directly ties this success to the explicit geometry- preservation term rather than to model capacity generally: the full method reaches cosine 0.75/top-1 0.91/rank 2.64, but removing only the VSP loss drops this to cosine 0.58/top-1 0.00/rank 4196.64 -- a collapse to nearly the same failure regime as removing the latent GAN entirely (cosine 0.49/top-1 0.00/rank 3897.09) or the cycle-consistency loss (cosine 0.50/top-1 0.00/rank 3941.36). The translation further extends, with reduced but still far-above-baseline performance, to out-of-distribution domains (TweetTopic, MIMIC-III clinical text) and to CLIP's multimodal text tower, and the translated embeddings retain enough source information to support downstream attribute-inference (top-1 accuracy up to 0.90 on a 10-way clinical-code task) and document-reconstruction attacks (LLM-judged information extraction for up to 80% of a 50-email test set) -- so the same convergence that enables unsupervised translation is also demonstrated as a concrete privacy vulnerability of embedding-based systems.

Lan, Torr, Meek, Khakzar, Krueger & Barez (2024) introduce "Analogous Feature Universality": even if independently-trained SAEs learn different individual feature directions, the subspaces spanned by those features may still be similar under rotation-invariant transformations. They first pair SAE features across model pairs by activation correlation, then measure representational similarity of the paired feature spaces via SVCCA (SVD-truncated CCA, mean canonical correlation) and RSA (Pearson-correlation inner similarity, Spearman outer similarity, Euclidean-distance RDMs), applied to SAE decoder weight matrices. Across four main model pairs (Pythia-70m vs Pythia-160m; Gemma-1-2B vs Gemma-2-2B; Gemma-2-2B vs Gemma-2-9B; Llama-3-8B-Instruct vs Llama-3.1-8B), similarity is high at matched middle layers and far exceeds a random-feature-pairing null: e.g. Gemma-2-2B layer 11 vs Gemma-2-9B layer 21 reaches SVCCA=0.70 (random mean 0.009) and RSA=0.195 (random mean 4.38e-4); Llama-3 vs Llama-3.1 (both layer 25) reaches SVCCA=0.3 with 66% mean activation correlation among paired features, retaining 7% of feature pairs after significance filtering; almost all post-layer-0 comparisons yield p-values between 0.00 and 0.01. A second baseline (Appendix D) trains SAEs on a randomized-weight Pythia-70m and finds very low SVCCA/RSA scores with only 1% of feature pairs surviving filtering, confirming the middle-layer similarity is not an artifact of the pairing/scoring procedure itself. Section 4.3 further decomposes similarity by semantic concept category (e.g. "Emotions", "Country"): concept- specific subspaces pass a paired-vs-shuffled-pairing significance test in nearly every category tested (the one reported exception is "Country" for the Gemma-1-2B/Gemma-2-2B pair), meaning some semantically coherent subspaces are more reliably cross-model-similar than the feature space taken as a whole. No causal intervention (feature transplant, steering-vector transfer) is performed; the paper explicitly frames cross-model steering-vector transfer as an unrealized implication of the hypothesis rather than a tested claim. A base-vs-fine-tuned control (Appendix C, Gemma-1-2B vs its Instruct variant, same layer 12) shows even higher similarity (SVCCA=0.84, RSA=0.25) than the cross-architecture pairs, as expected for models sharing a base checkpoint. Son, Rathore, Rufail, Simon, Zhang, Dave, Blondin, Zhu & O'Brien (2025) independently replicate this finding for the identical Gemma-2-2B vs Gemma-2-9B pair with independently-trained SAEs: "we aligned the resulting monosemantic features via activation correlation, and compared the matched feature spaces with metrics such as SVCCA and RSA," reporting paired SVCCA peaking at 0.73 (vs. Lan et al.'s 0.70 for the same pair) and RSA peaking at 0.22 (vs. 0.195), with random-pairing baselines again far lower (0.005-0.034) -- a close, genuinely independent replication of the same model pair and method, not a duplicate submission. They additionally extend the analysis beyond single tokens to multi-token/phrase-level semantic subspaces (e.g. concatenated concept phrases like "happy child"), finding the same convergence pattern holds at the phrase level.

Lan, Torr, Meek, Khakzar, Krueger & Barez (2024) introduce "Analogous Feature Universality": even if independently-trained SAEs learn different individual feature directions, the subspaces spanned by those features may still be similar under rotation-invariant transformations. They first pair SAE features across model pairs by activation correlation, then measure representational similarity of the paired feature spaces via SVCCA (SVD-truncated CCA, mean canonical correlation) and RSA (Pearson-correlation inner similarity, Spearman outer similarity, Euclidean-distance RDMs), applied to SAE decoder weight matrices. Across four main model pairs (Pythia-70m vs Pythia-160m; Gemma-1-2B vs Gemma-2-2B; Gemma-2-2B vs Gemma-2-9B; Llama-3-8B-Instruct vs Llama-3.1-8B), similarity is high at matched middle layers and far exceeds a random-feature-pairing null: e.g. Gemma-2-2B layer 11 vs Gemma-2-9B layer 21 reaches SVCCA=0.70 (random mean 0.009) and RSA=0.195 (random mean 4.38e-4); Llama-3 vs Llama-3.1 (both layer 25) reaches SVCCA=0.3 with 66% mean activation correlation among paired features, retaining 7% of feature pairs after significance filtering; almost all post-layer-0 comparisons yield p-values between 0.00 and 0.01. A second baseline (Appendix D) trains SAEs on a randomized-weight Pythia-70m and finds very low SVCCA/RSA scores with only 1% of feature pairs surviving filtering, confirming the middle-layer similarity is not an artifact of the pairing/scoring procedure itself. Section 4.3 further decomposes similarity by semantic concept category (e.g. "Emotions", "Country"): concept- specific subspaces pass a paired-vs-shuffled-pairing significance test in nearly every category tested (the one reported exception is "Country" for the Gemma-1-2B/Gemma-2-2B pair), meaning some semantically coherent subspaces are more reliably cross-model-similar than the feature space taken as a whole. No causal intervention (feature transplant, steering-vector transfer) is performed; the paper explicitly frames cross-model steering-vector transfer as an unrealized implication of the hypothesis rather than a tested claim. A base-vs-fine-tuned control (Appendix C, Gemma-1-2B vs its Instruct variant, same layer 12) shows even higher similarity (SVCCA=0.84, RSA=0.25) than the cross-architecture pairs, as expected for models sharing a base checkpoint. Son, Rathore, Rufail, Simon, Zhang, Dave, Blondin, Zhu & O'Brien (2025) independently replicate this finding for the identical Gemma-2-2B vs Gemma-2-9B pair with independently-trained SAEs: "we aligned the resulting monosemantic features via activation correlation, and compared the matched feature spaces with metrics such as SVCCA and RSA," reporting paired SVCCA peaking at 0.73 (vs. Lan et al.'s 0.70 for the same pair) and RSA peaking at 0.22 (vs. 0.195), with random-pairing baselines again far lower (0.005-0.034) -- a close, genuinely independent replication of the same model pair and method, not a duplicate submission. They additionally extend the analysis beyond single tokens to multi-token/phrase-level semantic subspaces (e.g. concatenated concept phrases like "happy child"), finding the same convergence pattern holds at the phrase level.

Puri, Berend, Lapuschkin & Samek (2025/2026) build a "Concept Atlas" from a Gemma Scope sparse autoencoder trained on Gemma-2-2B's residual stream (layer 20), then fit an orthogonal Procrustes map from three distinct "subject" models' activation spaces (Llama-3.1-8B base, Llama-3.1-8B-Instruct, DeepSeek-R1-Distill-Llama-3.1-8B-Instruct) into this atlas. Translation quality is quantified via AUROC (0.82-0.86) and average precision (0.39-0.49, versus a random baseline of 0.046) across five subject-model layers, and via near-perfect mean-reciprocal-rank retrieval on 454 independently-validated concept features. Atlas concept directions mapped back into a subject model and added (norm-preserving) to its residual stream at multiple layers simultaneously demonstrably steer generation toward the target concept, though this steering effect is reported qualitatively rather than with an inline quantified success rate.

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Ryskina, Tuckute, Fung, Malkin & Fedorenko (2025) use an fMRI dataset (Pereira et al. 2018) in which the same concepts are presented as sentences, word clouds, and images, defining a "meaning consistency" metric that identifies brain voxels responding similarly to a concept regardless of presentation modality. Using representational similarity analysis (RSA) to compare the geometry of these regions' response patterns against 15 language and vision-language models' own representational geometry, both language-only and language-vision models predict brain signal better in meaning-consistent regions, even in areas with low language-selectivity -- a genuine shared-geometry finding (RSA-based, not merely correlation-strength) linking model representational structure to a cross-modal property of brain organization.

Wang et al. fit a linear map from monolingual English BERT to multilingual BERT using contextual word alignments from only 10,000 Europarl sentence pairs per language [wang-etal-2019-clbt] They compare closed-form orthogonal Procrustes against an unconstrained gradient-descent linear map; the unconstrained map is marginally better and a nonlinear map adds nothing [wang-etal-2019-clbt] On zero-shot cross-lingual dependency parsing (17 languages), CLBT beats the static-embedding SOTA (avg LAS 60.63->63.54) and matches XLM despite using 10K versus millions of parallel sentences [wang-etal-2019-clbt] The alignment claim rests on downstream transfer performance rather than an independent similarity statistic (only a qualitative t-SNE case study is shown) [wang-etal-2019-clbt]

Ryskina, Tuckute, Fung, Malkin & Fedorenko (2025) use an fMRI dataset (Pereira et al. 2018) in which the same concepts are presented as sentences, word clouds, and images, defining a "meaning consistency" metric that identifies brain voxels responding similarly to a concept regardless of presentation modality. Using representational similarity analysis (RSA) to compare the geometry of these regions' response patterns against 15 language and vision-language models' own representational geometry, both language-only and language-vision models predict brain signal better in meaning-consistent regions, even in areas with low language-selectivity -- a genuine shared-geometry finding (RSA-based, not merely correlation-strength) linking model representational structure to a cross-modal property of brain organization.

Yuan & Søgaard (2025) apply supervised and unsupervised (adversarial + iterative refinement) Procrustes alignment, adapted from cross- lingual word-embedding literature, to final-hidden-layer Othello-move representations from seven differently-architected real models (GPT-2-small, BART-base, T5-base, Flan-T5-XL/3B, LoRA-fine-tuned LLaMA-2-7B, LoRA-fine-tuned Mistral-7B, Qwen2.5-7B). Cross-model cosine similarity after alignment reaches up to 93.1% (supervised, GPT-2 <-> BART) and 97.2% (unsupervised, BART <-> Mistral); PCA visualization of the per-game move trajectory shows convergent geometric structure across models; layer-wise similarity heatmaps peak along the diagonal at corresponding depths; and a "latent move projection" analysis finds the nearest-neighbor tile embedding to any given tile is consistently its actual spatial neighbor on the physical board -- a genuine spatial/geometric isomorphism claim beyond simple linear decodability. Clears scope on criterion (a): a measured, quantified cross-model shared-manifold structure in real pretrained-and-fine-tuned models, going beyond the original linear- probing Othello-GPT results already in this map. No causal intervention is performed (purely representational/alignment analysis). Cross-model convergence of this kind is platonic- representation-adjacent; see [[platonic-representation]].

Ryskina, Tuckute, Fung, Malkin & Fedorenko (2025) use an fMRI dataset (Pereira et al. 2018) in which the same concepts are presented as sentences, word clouds, and images, defining a "meaning consistency" metric that identifies brain voxels responding similarly to a concept regardless of presentation modality. Using representational similarity analysis (RSA) to compare the geometry of these regions' response patterns against 15 language and vision-language models' own representational geometry, both language-only and language-vision models predict brain signal better in meaning-consistent regions, even in areas with low language-selectivity -- a genuine shared-geometry finding (RSA-based, not merely correlation-strength) linking model representational structure to a cross-modal property of brain organization.

Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022) define the relative representation of an input x as its vector of cosine similarities to a fixed set of anchor points from the same domain: r_x = (sim(e_x, e_a1), ..., sim(e_x, e_a|A|)), with sim typically cosine similarity S_C(a,b) = ab/(||a|| ||b||) (vector quantization is explored as an alternative in an appendix). This construction is invariant to any rotation, reflection, or positive rescaling applied uniformly to the encoder's output space -- exactly the transformations that make two independently-trained encoders' raw coordinates incomparable -- with translation invariance recoverable separately via a normalization layer (e.g. InstanceNorm). Diagnostically, across ~2,000 independently-trained Cora GCN classifiers varying seed, epoch count, layer depth, dropout, activation, optimizer, learning rate, and embedder type, the Pearson correlation between each model's relative- representation cosine similarity to a fixed reference model and its own classification accuracy is 0.955 (after filtering models with best validation accuracy below 0.5) -- models that reach similar accuracy converge, in relative coordinates, to highly similar representations. Constructively, decoders/classifiers trained on one encoder's relative representations are spliced with zero fine-tuning onto a different, independently-trained encoder's relative representations of the same domain ("zero-shot stitching"), and recover most non-stitched performance while absolute-representation stitching performs at or near chance throughout: image reconstruction MSE (Table 3, 5 seeds) drops from an Absolute-Stitch baseline of 97.79/120.54/86.74/97.17 (MNIST/F-MNIST/CIFAR-10/CIFAR-100, AE) to a Relative-Stitch 2.83/6.37/5.39/18.03; cross-lingual text classification F1 (Table 4, RoBERTa, Amazon Reviews coarse, English decoder) drops from 91.54 (English, non-stitched) to 43.67/54.41/48.72 (Spanish/French/Japanese, Absolute-Stitch) but only to 90.06/82.78/ 78.49 with Relative-Stitch (translated anchors); cross-architecture image classification (Table 6, CIFAR-100/ImageNet1k, ViT-Base decoder + ViT-Small-ResNet50 encoder direction) goes from Absolute-Stitch 6.21/0.07 to Relative-Stitch 81.42/44.72 (and the reverse direction 4.69/0.08 to 84.46/62.21). The paper explicitly summarizes: "Stitching with absolute representations yields performances comparable to random guessing across the board." Typical anchor-set sizes are 300 (word embeddings, Cora) or 500 (cross-architecture vision, matching the larger ViT-Base's 768 dimensions); an ablation (Figure 6/Appendix A.2) shows performance improves monotonically with more anchors for frozen encoders but is less stable for end-to-end-trained ones (Cora), which the authors attribute to greater susceptibility to model collapse. This is among the earliest constructive demonstrations of representational convergence across independently-trained networks: unlike vec2vec/mini-vec2vec, which *learn* a translation map between two embedding spaces, this shows that for a sufficiently invariant choice of coordinates (cosine similarity to a shared anchor set), no learned map is required at all -- only a shared anchor domain, no paired correspondence, and no training.

Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022) define the relative representation of an input x as its vector of cosine similarities to a fixed set of anchor points from the same domain: r_x = (sim(e_x, e_a1), ..., sim(e_x, e_a|A|)), with sim typically cosine similarity S_C(a,b) = ab/(||a|| ||b||) (vector quantization is explored as an alternative in an appendix). This construction is invariant to any rotation, reflection, or positive rescaling applied uniformly to the encoder's output space -- exactly the transformations that make two independently-trained encoders' raw coordinates incomparable -- with translation invariance recoverable separately via a normalization layer (e.g. InstanceNorm). Diagnostically, across ~2,000 independently-trained Cora GCN classifiers varying seed, epoch count, layer depth, dropout, activation, optimizer, learning rate, and embedder type, the Pearson correlation between each model's relative- representation cosine similarity to a fixed reference model and its own classification accuracy is 0.955 (after filtering models with best validation accuracy below 0.5) -- models that reach similar accuracy converge, in relative coordinates, to highly similar representations. Constructively, decoders/classifiers trained on one encoder's relative representations are spliced with zero fine-tuning onto a different, independently-trained encoder's relative representations of the same domain ("zero-shot stitching"), and recover most non-stitched performance while absolute-representation stitching performs at or near chance throughout: image reconstruction MSE (Table 3, 5 seeds) drops from an Absolute-Stitch baseline of 97.79/120.54/86.74/97.17 (MNIST/F-MNIST/CIFAR-10/CIFAR-100, AE) to a Relative-Stitch 2.83/6.37/5.39/18.03; cross-lingual text classification F1 (Table 4, RoBERTa, Amazon Reviews coarse, English decoder) drops from 91.54 (English, non-stitched) to 43.67/54.41/48.72 (Spanish/French/Japanese, Absolute-Stitch) but only to 90.06/82.78/ 78.49 with Relative-Stitch (translated anchors); cross-architecture image classification (Table 6, CIFAR-100/ImageNet1k, ViT-Base decoder + ViT-Small-ResNet50 encoder direction) goes from Absolute-Stitch 6.21/0.07 to Relative-Stitch 81.42/44.72 (and the reverse direction 4.69/0.08 to 84.46/62.21). The paper explicitly summarizes: "Stitching with absolute representations yields performances comparable to random guessing across the board." Typical anchor-set sizes are 300 (word embeddings, Cora) or 500 (cross-architecture vision, matching the larger ViT-Base's 768 dimensions); an ablation (Figure 6/Appendix A.2) shows performance improves monotonically with more anchors for frozen encoders but is less stable for end-to-end-trained ones (Cora), which the authors attribute to greater susceptibility to model collapse. This is among the earliest constructive demonstrations of representational convergence across independently-trained networks: unlike vec2vec/mini-vec2vec, which *learn* a translation map between two embedding spaces, this shows that for a sufficiently invariant choice of coordinates (cosine similarity to a shared anchor set), no learned map is required at all -- only a shared anchor domain, no paired correspondence, and no training.

Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022) define the relative representation of an input x as its vector of cosine similarities to a fixed set of anchor points from the same domain: r_x = (sim(e_x, e_a1), ..., sim(e_x, e_a|A|)), with sim typically cosine similarity S_C(a,b) = ab/(||a|| ||b||) (vector quantization is explored as an alternative in an appendix). This construction is invariant to any rotation, reflection, or positive rescaling applied uniformly to the encoder's output space -- exactly the transformations that make two independently-trained encoders' raw coordinates incomparable -- with translation invariance recoverable separately via a normalization layer (e.g. InstanceNorm). Diagnostically, across ~2,000 independently-trained Cora GCN classifiers varying seed, epoch count, layer depth, dropout, activation, optimizer, learning rate, and embedder type, the Pearson correlation between each model's relative- representation cosine similarity to a fixed reference model and its own classification accuracy is 0.955 (after filtering models with best validation accuracy below 0.5) -- models that reach similar accuracy converge, in relative coordinates, to highly similar representations. Constructively, decoders/classifiers trained on one encoder's relative representations are spliced with zero fine-tuning onto a different, independently-trained encoder's relative representations of the same domain ("zero-shot stitching"), and recover most non-stitched performance while absolute-representation stitching performs at or near chance throughout: image reconstruction MSE (Table 3, 5 seeds) drops from an Absolute-Stitch baseline of 97.79/120.54/86.74/97.17 (MNIST/F-MNIST/CIFAR-10/CIFAR-100, AE) to a Relative-Stitch 2.83/6.37/5.39/18.03; cross-lingual text classification F1 (Table 4, RoBERTa, Amazon Reviews coarse, English decoder) drops from 91.54 (English, non-stitched) to 43.67/54.41/48.72 (Spanish/French/Japanese, Absolute-Stitch) but only to 90.06/82.78/ 78.49 with Relative-Stitch (translated anchors); cross-architecture image classification (Table 6, CIFAR-100/ImageNet1k, ViT-Base decoder + ViT-Small-ResNet50 encoder direction) goes from Absolute-Stitch 6.21/0.07 to Relative-Stitch 81.42/44.72 (and the reverse direction 4.69/0.08 to 84.46/62.21). The paper explicitly summarizes: "Stitching with absolute representations yields performances comparable to random guessing across the board." Typical anchor-set sizes are 300 (word embeddings, Cora) or 500 (cross-architecture vision, matching the larger ViT-Base's 768 dimensions); an ablation (Figure 6/Appendix A.2) shows performance improves monotonically with more anchors for frozen encoders but is less stable for end-to-end-trained ones (Cora), which the authors attribute to greater susceptibility to model collapse. This is among the earliest constructive demonstrations of representational convergence across independently-trained networks: unlike vec2vec/mini-vec2vec, which *learn* a translation map between two embedding spaces, this shows that for a sufficiently invariant choice of coordinates (cosine similarity to a shared anchor set), no learned map is required at all -- only a shared anchor domain, no paired correspondence, and no training.

Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022) define the relative representation of an input x as its vector of cosine similarities to a fixed set of anchor points from the same domain: r_x = (sim(e_x, e_a1), ..., sim(e_x, e_a|A|)), with sim typically cosine similarity S_C(a,b) = ab/(||a|| ||b||) (vector quantization is explored as an alternative in an appendix). This construction is invariant to any rotation, reflection, or positive rescaling applied uniformly to the encoder's output space -- exactly the transformations that make two independently-trained encoders' raw coordinates incomparable -- with translation invariance recoverable separately via a normalization layer (e.g. InstanceNorm). Diagnostically, across ~2,000 independently-trained Cora GCN classifiers varying seed, epoch count, layer depth, dropout, activation, optimizer, learning rate, and embedder type, the Pearson correlation between each model's relative- representation cosine similarity to a fixed reference model and its own classification accuracy is 0.955 (after filtering models with best validation accuracy below 0.5) -- models that reach similar accuracy converge, in relative coordinates, to highly similar representations. Constructively, decoders/classifiers trained on one encoder's relative representations are spliced with zero fine-tuning onto a different, independently-trained encoder's relative representations of the same domain ("zero-shot stitching"), and recover most non-stitched performance while absolute-representation stitching performs at or near chance throughout: image reconstruction MSE (Table 3, 5 seeds) drops from an Absolute-Stitch baseline of 97.79/120.54/86.74/97.17 (MNIST/F-MNIST/CIFAR-10/CIFAR-100, AE) to a Relative-Stitch 2.83/6.37/5.39/18.03; cross-lingual text classification F1 (Table 4, RoBERTa, Amazon Reviews coarse, English decoder) drops from 91.54 (English, non-stitched) to 43.67/54.41/48.72 (Spanish/French/Japanese, Absolute-Stitch) but only to 90.06/82.78/ 78.49 with Relative-Stitch (translated anchors); cross-architecture image classification (Table 6, CIFAR-100/ImageNet1k, ViT-Base decoder + ViT-Small-ResNet50 encoder direction) goes from Absolute-Stitch 6.21/0.07 to Relative-Stitch 81.42/44.72 (and the reverse direction 4.69/0.08 to 84.46/62.21). The paper explicitly summarizes: "Stitching with absolute representations yields performances comparable to random guessing across the board." Typical anchor-set sizes are 300 (word embeddings, Cora) or 500 (cross-architecture vision, matching the larger ViT-Base's 768 dimensions); an ablation (Figure 6/Appendix A.2) shows performance improves monotonically with more anchors for frozen encoders but is less stable for end-to-end-trained ones (Cora), which the authors attribute to greater susceptibility to model collapse. This is among the earliest constructive demonstrations of representational convergence across independently-trained networks: unlike vec2vec/mini-vec2vec, which *learn* a translation map between two embedding spaces, this shows that for a sufficiently invariant choice of coordinates (cosine similarity to a shared anchor set), no learned map is required at all -- only a shared anchor domain, no paired correspondence, and no training.

Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022) define the relative representation of an input x as its vector of cosine similarities to a fixed set of anchor points from the same domain: r_x = (sim(e_x, e_a1), ..., sim(e_x, e_a|A|)), with sim typically cosine similarity S_C(a,b) = ab/(||a|| ||b||) (vector quantization is explored as an alternative in an appendix). This construction is invariant to any rotation, reflection, or positive rescaling applied uniformly to the encoder's output space -- exactly the transformations that make two independently-trained encoders' raw coordinates incomparable -- with translation invariance recoverable separately via a normalization layer (e.g. InstanceNorm). Diagnostically, across ~2,000 independently-trained Cora GCN classifiers varying seed, epoch count, layer depth, dropout, activation, optimizer, learning rate, and embedder type, the Pearson correlation between each model's relative- representation cosine similarity to a fixed reference model and its own classification accuracy is 0.955 (after filtering models with best validation accuracy below 0.5) -- models that reach similar accuracy converge, in relative coordinates, to highly similar representations. Constructively, decoders/classifiers trained on one encoder's relative representations are spliced with zero fine-tuning onto a different, independently-trained encoder's relative representations of the same domain ("zero-shot stitching"), and recover most non-stitched performance while absolute-representation stitching performs at or near chance throughout: image reconstruction MSE (Table 3, 5 seeds) drops from an Absolute-Stitch baseline of 97.79/120.54/86.74/97.17 (MNIST/F-MNIST/CIFAR-10/CIFAR-100, AE) to a Relative-Stitch 2.83/6.37/5.39/18.03; cross-lingual text classification F1 (Table 4, RoBERTa, Amazon Reviews coarse, English decoder) drops from 91.54 (English, non-stitched) to 43.67/54.41/48.72 (Spanish/French/Japanese, Absolute-Stitch) but only to 90.06/82.78/ 78.49 with Relative-Stitch (translated anchors); cross-architecture image classification (Table 6, CIFAR-100/ImageNet1k, ViT-Base decoder + ViT-Small-ResNet50 encoder direction) goes from Absolute-Stitch 6.21/0.07 to Relative-Stitch 81.42/44.72 (and the reverse direction 4.69/0.08 to 84.46/62.21). The paper explicitly summarizes: "Stitching with absolute representations yields performances comparable to random guessing across the board." Typical anchor-set sizes are 300 (word embeddings, Cora) or 500 (cross-architecture vision, matching the larger ViT-Base's 768 dimensions); an ablation (Figure 6/Appendix A.2) shows performance improves monotonically with more anchors for frozen encoders but is less stable for end-to-end-trained ones (Cora), which the authors attribute to greater susceptibility to model collapse. This is among the earliest constructive demonstrations of representational convergence across independently-trained networks: unlike vec2vec/mini-vec2vec, which *learn* a translation map between two embedding spaces, this shows that for a sufficiently invariant choice of coordinates (cosine similarity to a shared anchor set), no learned map is required at all -- only a shared anchor domain, no paired correspondence, and no training.

Kudugunta et al. train one shared 375M Transformer NMT model on over 25B pairs across 103 languages, then compare per-language representations with SVCCA [kudugunta-etal-2019] For X->En, encoder representations of different source languages become MORE similar moving up the encoder; for En->X, the same English input's encoder representations DIVERGE by target language [kudugunta-etal-2019] Embedding-layer representations are highly stable under per-language-pair fine-tuning (mean canonical correlation >0.98), with later layers more sensitive [kudugunta-etal-2019] Encoder representations visually cluster by language family in Laplacian-eigenmap embeddings, but this is qualitative with no clustering statistic; no activation intervention is performed [kudugunta-etal-2019]

Yin et al. compare a pixel-based (I3D) and pose-based (ST-GCN) sign-recognition model on ASL Citizen, testing whether they encode phonological structure rather than shortcuts [yin-etal-2026-sign-language-phonological-perception] On minimal pairs, ST-GCN wins on 81.08% (vs I3D 69.50%), stronger on handshape contrasts while I3D is stronger on location [yin-etal-2026-sign-language-phonological-perception] RSA correlates ST-GCN's handshape similarity with human perceptual confusion (r=0.49) and an articulatory Handshape-Distance metric (r=0.55), versus I3D 0.31/0.20 [yin-etal-2026-sign-language-phonological-perception] The evidence is an RSA distance-correlation to perceptual/articulatory ground truth with no causal intervention; full text was not fully available for deeper verification [yin-etal-2026-sign-language-phonological-perception]

Yin et al. compare a pixel-based (I3D) and pose-based (ST-GCN) sign-recognition model on ASL Citizen, testing whether they encode phonological structure rather than shortcuts [yin-etal-2026-sign-language-phonological-perception] On minimal pairs, ST-GCN wins on 81.08% (vs I3D 69.50%), stronger on handshape contrasts while I3D is stronger on location [yin-etal-2026-sign-language-phonological-perception] RSA correlates ST-GCN's handshape similarity with human perceptual confusion (r=0.49) and an articulatory Handshape-Distance metric (r=0.55), versus I3D 0.31/0.20 [yin-etal-2026-sign-language-phonological-perception] The evidence is an RSA distance-correlation to perceptual/articulatory ground truth with no causal intervention; full text was not fully available for deeper verification [yin-etal-2026-sign-language-phonological-perception]

Puri, Berend, Lapuschkin & Samek (2025/2026) build a "Concept Atlas" from a Gemma Scope sparse autoencoder trained on Gemma-2-2B's residual stream (layer 20), then fit an orthogonal Procrustes map from three distinct "subject" models' activation spaces (Llama-3.1-8B base, Llama-3.1-8B-Instruct, DeepSeek-R1-Distill-Llama-3.1-8B-Instruct) into this atlas. Translation quality is quantified via AUROC (0.82-0.86) and average precision (0.39-0.49, versus a random baseline of 0.046) across five subject-model layers, and via near-perfect mean-reciprocal-rank retrieval on 454 independently-validated concept features. Atlas concept directions mapped back into a subject model and added (norm-preserving) to its residual stream at multiple layers simultaneously demonstrably steer generation toward the target concept, though this steering effect is reported qualitatively rather than with an inline quantified success rate.

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Yuan & Søgaard (2025) apply supervised and unsupervised (adversarial + iterative refinement) Procrustes alignment, adapted from cross- lingual word-embedding literature, to final-hidden-layer Othello-move representations from seven differently-architected real models (GPT-2-small, BART-base, T5-base, Flan-T5-XL/3B, LoRA-fine-tuned LLaMA-2-7B, LoRA-fine-tuned Mistral-7B, Qwen2.5-7B). Cross-model cosine similarity after alignment reaches up to 93.1% (supervised, GPT-2 <-> BART) and 97.2% (unsupervised, BART <-> Mistral); PCA visualization of the per-game move trajectory shows convergent geometric structure across models; layer-wise similarity heatmaps peak along the diagonal at corresponding depths; and a "latent move projection" analysis finds the nearest-neighbor tile embedding to any given tile is consistently its actual spatial neighbor on the physical board -- a genuine spatial/geometric isomorphism claim beyond simple linear decodability. Clears scope on criterion (a): a measured, quantified cross-model shared-manifold structure in real pretrained-and-fine-tuned models, going beyond the original linear- probing Othello-GPT results already in this map. No causal intervention is performed (purely representational/alignment analysis). Cross-model convergence of this kind is platonic- representation-adjacent; see [[platonic-representation]].

Huh et al. (2024) measure mutual nearest-neighbor kernel alignment between ~20 named open-weight language models (LLaMA/LLaMA-3, BLOOM, OpenLLaMA, Mistral/Mixtral, Gemma, OLMo, spanning 560M to 70B parameters) and a range of vision models (DINOv2, MAE, CLIP, ImageNet-21K-supervised ViTs), using paired Wikipedia image-caption data to bridge the two modalities. They find alignment between a language model and vision models rises linearly with the language model's own language-modeling competence (lower bits-per-byte), and symmetrically with the vision model's competence; separately, among 78 vision-only models, alignment within a competence bucket rises as the bucket's average downstream (VTAB) performance rises. Critically, a language model's alignment score to a strong vision encoder (DINOv2) itself predicts that language model's own downstream task performance (Hellaswag common-sense reasoning shows a linear relationship; GSM8K math shows an emergence-like threshold effect) — alignment with other modalities correlates with, and may indicate, general competence, not just modality-specific skill. This is the primary cross-model, cross-modality empirical support cited for the [[platonic-representation]] hypothesis: unlike every other Observation in this map, its "structure" is not a shape found within one model's representation space, but a convergence relationship measured *between* many independently-trained models and even across data modalities.

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

Yuan & Søgaard (2025) apply supervised and unsupervised (adversarial + iterative refinement) Procrustes alignment, adapted from cross- lingual word-embedding literature, to final-hidden-layer Othello-move representations from seven differently-architected real models (GPT-2-small, BART-base, T5-base, Flan-T5-XL/3B, LoRA-fine-tuned LLaMA-2-7B, LoRA-fine-tuned Mistral-7B, Qwen2.5-7B). Cross-model cosine similarity after alignment reaches up to 93.1% (supervised, GPT-2 <-> BART) and 97.2% (unsupervised, BART <-> Mistral); PCA visualization of the per-game move trajectory shows convergent geometric structure across models; layer-wise similarity heatmaps peak along the diagonal at corresponding depths; and a "latent move projection" analysis finds the nearest-neighbor tile embedding to any given tile is consistently its actual spatial neighbor on the physical board -- a genuine spatial/geometric isomorphism claim beyond simple linear decodability. Clears scope on criterion (a): a measured, quantified cross-model shared-manifold structure in real pretrained-and-fine-tuned models, going beyond the original linear- probing Othello-GPT results already in this map. No causal intervention is performed (purely representational/alignment analysis). Cross-model convergence of this kind is platonic- representation-adjacent; see [[platonic-representation]].