Each paper uses the language of regularity or convergence, and each measures a different object: Kaplan et al. measure a single number (test loss), Kaushik et al. measure weights, and Huh et al. measure representations (how models arrange data internally). That makes the trio ideal for practice. The words line up and the measurements do not, and the exercise is to see exactly where.
The three papers, three levels of the network
All three are about convergence, but each measures a different object. Keep the three apart before connecting them.
A regularity in a single aggregate number (loss) says nothing by itself about internal structure. A shared subspace in weights is a claim about parameters. Aligned similarity structure is a claim about behaviour on inputs. “Convergence” means a different thing at each level.
Kaplan, McCandlish, Henighan, Brown, Chess, Child, Gray, Radford, Wu, Amodei · 2020 · arXiv preprint (OpenAI). Highly cited; the allocation advice was later revised by Hoffmann et al. 2022.
What the paper claims (paraphrased from the paper)
Cross-entropy test loss of Transformer language models falls as a smooth power law in non-embedding parameters N, dataset size D and compute C, over many orders of magnitude, with architectural shape mattering little within a wide range.
Method:
Train many decoder-only Transformers (about 103 to 109 non-embedding parameters) on WebText2 and fit power laws such as L(N)=(Nc/N)αN with αN≈0.076.
Evidence type: Empirical: controlled training runs and curve fitting within one lab, one data distribution and one architecture family.
Limitations & falsifier
No theory for why power laws or these exponents, which the authors acknowledge.
Loss is a single scalar. It says nothing about what is happening inside the network.
One data distribution and one architecture family; the exact constants are setup-specific.
Compute-optimal allocation later revised by Chinchilla.
What would falsify it:
Clear, reproducible bends or breaks in the curves inside the measured range, or exponents that change drastically with minor setup changes.
Kaushik, Chaudhari, Vaidya, Chellappa, Yuille · 2025 · arXiv preprint (December 2025). Not yet independently replicated as far as this curriculum knows.
What the paper claims (paraphrased from the paper)
Networks trained on diverse tasks occupy remarkably similar low-dimensional subspaces of weight space: after spectral decomposition, a few principal directions capture most of the variance across many models of the same architecture.
Method:
Mode-wise spectral analysis (principal-component-style decomposition) of weight matrices from over 1,100 models, including about 500 Mistral-7B LoRA adapters, about 500 Vision Transformers and 50 fine-tuned LLaMA-3-8B models, largely sourced from Hugging Face. Plus reconstruction tests: project held-out models onto the shared subspace and measure retained performance.
Evidence type: Empirical: large-scale analysis of existing models, with compression/reconstruction experiments.
Limitations & falsifier
Subspaces are architecture-specific: shared within an architecture, not across architectures.
Many models are adapters or fine-tunes of a shared pretrained base (e.g. LoRAs on one Mistral-7B), so overlap may partly be inherited from that base.
Models collected from a public hub are not a random sample of all trained models.
The authors’ limitations section notes that the shared directions are not yet interpretable and that the method needs many pretrained task models.
What would falsify it:
Independently trained-from-scratch models (different seeds, no shared base) whose weights do not share a low-dimensional subspace beyond what random matrices with the same spectra would show.
Representations: how models arrange data internally
Huh, Cheung, Wang, Isola · 2024 · Published at ICML 2024 (position paper). Argument plus supporting measurements.
What the paper claims (paraphrased from the paper)
Representations in different neural networks are becoming more aligned, across architectures and even across modalities (vision and language), as models grow more capable. The authors hypothesise convergence toward a shared statistical model of reality.
Method:
Survey prior convergence results, then measure representational alignment with a mutual nearest-neighbour metric: do two models agree on which data points are close to which? This is done for 78 vision models and for vision-language pairs on image–caption data.
Evidence type: Empirical measurements plus a theoretical argument (selective pressures toward convergence). The headline "Platonic" claim is a hypothesis.
Limitations & falsifier
Alignment depends on the chosen similarity metric; the paper discusses this under its own limitations.
Measured alignment values are well below perfect; "converging" is a trend, not an arrival.
Information unique to one modality may never align. The authors list this among their counterexamples and limitations.
Correlation with competence, not a causal demonstration.
What would falsify it:
Alignment trends that vanish or reverse under other reasonable similarity metrics, or capable models that remain clearly unaligned on held-out data.
Background theory the stages draw on
A Neural Scaling Law from the Dimension of the Data Manifold(Sharma & Kaplan, 2020) Proposes that scaling exponents relate to the intrinsic dimension of the data. A **theory** proposal that links "low-dimensional structure in data" to the shape of loss curves.
Explaining Neural Scaling Laws(Bahri, Dyer, Kaplan, Lee & Sharma, 2021) Identifies four scaling regimes (variance- vs resolution-limited, in model or data size) and connects some exponents to kernel spectra. A reminder that "the" scaling law may have different mechanisms underneath.
Are Emergent Abilities of Large Language Models a Mirage?(Schaeffer, Miranda & Koyejo, 2023) The metric lesson: what looks like a sharp pattern can be produced by the measuring instrument. Applies directly to similarity metrics.
⌖Cartographer
Maps every possible connection, generously, and labels each one.
Find the recurring concepts, candidate shared mathematical structures and research opportunities across the three papers. Generous but labelled: every link gets a tentative connection level and a note on what evidence would raise it.
Worked example
Worked example, written by the curriculum authors from the papers’ abstracts and key sections (checked 2026-10-30). It is a teaching example, not the output of an AI research run, and no model was called to produce it.
Recurring concept: "convergence" or "regularity" with scale. It shows up at three levels of the same system:
Loss (Kaplan 2020): a scalar summary falls as a power law in N, D, C[empirical].
Weights (Kaushik 2025): many models of one architecture share a few dominant spectral directions [empirical, preprint].
Representations (Huh 2024): models increasingly agree on which inputs are similar [empirical trend], possibly heading toward a shared model of reality [theory / hypothesis].
Candidate shared structure: low dimensionality. Sharma & Kaplan (2020) propose that scaling exponents reflect the intrinsic dimension of the data manifold [theory]. Kaushik et al. find low-dimensional weight subspaces. Huh et al. argue that representations converge on the statistics of the data. A tempting through-line: the data has low-dimensional structure, and everything else inherits it. Tentatively Level 2, since "dimension" is a formal concept in two of the papers.
Candidate shared structure: spectra. Bahri et al. (2021) link some scaling exponents to kernel spectra. Kaushik et al. use spectral decomposition of weights. Both describe a quantity by its ranked eigen-directions. Tentatively Level 2.
Research opportunities 1. Does the dimension of a model family's shared weight subspace track the intrinsic dimension of its training data? 2. Do models that are more representationally aligned (Huh) also share more weight-subspace directions (Kaushik)? 3. Does alignment rise as a power law in scale, with an exponent, like loss?
Checklist
Your work (your own reading; unverified until you cite a page)
AI assist · copy the prompt, run it yourself, paste the reply below
⚔Adversary
Attacks the map: definitions, confounds, missing evidence.
Attack the map as an unsympathetic, fair reviewer. Expose definitional differences, confounds and missing evidence; downgrade every link to the level the evidence actually supports.
Worked example
Worked example, written by the curriculum authors from the papers’ abstracts and key sections (checked 2026-10-30). It is a teaching example, not the output of an AI research run, and no model was called to produce it.
1. "Universal" and "converge" mean three different things. Kaplan's regularity is a curve shape for one aggregate number. Kaushik's universality is shared variance directions among weights within one architecture. Huh's convergence is rising agreement on nearest neighbours across architectures and modalities. Same words, three operational definitions. Downgrade the "all three" link to Level 1.
2. Loss says nothing about internals. Two very different networks can reach the same loss. A smooth L(N) curve is compatible with internal convergence and also with internal divergence. Kaplan cannot be evidence for structural convergence.
3. Shared starting points (confound). Hundreds of the analysed weight sets are LoRA adapters on one Mistral-7B base, and the LLaMA set is fine-tuned models. Fine-tunes stay near their base, so a shared subspace may be inherited, not discovered. Ask how much survives among models trained from scratch with different seeds. The paper reports some from-scratch experiments; their weight relative to the headline counts matters.
4. Selection. Models downloaded from a public hub "without filtering" are still a convenience sample: popular bases, popular tasks.
5. Metric dependence (the Mirage lesson). Representational alignment depends on the similarity measure. A mutual nearest-neighbour score could rise with scale partly because of what it rewards. Before any trend is called "convergence", check it under other reasonable metrics.
6. No mechanism. No paper in the trio, or in the background, has shown that data-manifold dimension causes weight-subspace dimension or representational alignment. The low-dimensionality through-line is a story, currently unfalsified because it is untested.
Strongest surviving link: Kaushik ↔ Sharma & Kaplan, at Level 2 at most. Both make precise claims about low-dimensional structure, and the link is testable. Most tempting link that fails: "scaling produces convergence at every level". It reads as one finding, but it is three findings with three definitions.
Checklist
Your work (your own reading; unverified until you cite a page)
AI assist · copy the prompt, run it yourself, paste the reply below
✎Editor
Writes the strongest explanation that survives the attack.
Write the most interesting explanation that survives the Adversary, no more and no less. Give a verdict line, a ~250-word explainer for sophisticated non-specialists, and experiments that would move the verdict.
Worked example
Worked example, written by the curriculum authors from the papers’ abstracts and key sections (checked 2026-10-30). It is a teaching example, not the output of an AI research run, and no model was called to produce it.
Verdict: The three papers document regularities at three different levels of a neural network. Together they justify a Level 1–2 connection (shared vocabulary, some shared low-dimensional mathematics), not a shared mechanism.
Explainer (≈250 words). As neural networks have grown, researchers keep finding order where they might expect chaos. In 2020, Jared Kaplan and colleagues at OpenAI showed that a language model's test loss falls along a remarkably straight line on a log-log plot as models, datasets and compute grow, a power law spanning several orders of magnitude [empirical]. In 2024, Minyoung Huh and colleagues at MIT reported that models of different designs, even vision and language models, increasingly agree on which inputs are similar as they become more capable. They proposed, as a hypothesis, that models are converging on a shared statistical picture of the world. In late 2025, a preprint by Prakhar Kaushik and colleagues analysed more than a thousand trained models and found that within a given architecture, the learned weights crowd into a few shared directions [empirical, preprint].
It is tempting to read these as one discovery. They are not, at least not yet. Each measures a different thing: a single error score, the raw parameters, or the geometry of internal representations. Each uses its own definition of "convergence". Many of the weight-space models were fine-tuned from a handful of shared base models, which could explain some of their overlap. What links the papers is a family resemblance and a shared mathematical vocabulary of low-dimensional structure. The interesting question is whether that vocabulary points to a real common cause, and it is a question that can now be tested.
Distinguishing experiments 1. Train families of models from scratch with different seeds and no shared base; test whether shared weight subspaces persist beyond a random-matrix baseline. 2. For one model family, measure both weight-subspace overlap and representational alignment; check whether they rise together with scale. 3. Vary a dataset's intrinsic dimension in a controlled way; test whether the scaling exponent (Sharma & Kaplan) and the shared-subspace dimension (Kaushik) shift together. That would be a Level 3b test.
Checklist
Your work (your own reading; unverified until you cite a page)
AI assist · copy the prompt, run it yourself, paste the reply below
Which connections survive?
What the Cartographer proposed, and the level that holds up after the Adversary.
Between
Proposed link
Proposed
Justified
Note
Kaplan ↔ Huh
Both report smooth trends with scale: loss falls, alignment rises. "Bigger models are more similar to each other" and "bigger models have lower loss" may be two views of models approximating the same data distribution better.
≅Level 2 · Shared math
≈Level 1 · Analogy
Both are monotone trends with scale, but there is no shared equation and no derivation linking loss to alignment. A trend in the same direction is Level 1.
Kaushik ↔ Huh
Both say networks "converge": one in weight space, one in representation space.
≅Level 2 · Shared math
≈Level 1 · Analogy
Different objects (parameters vs activations), different metrics (spectral variance vs nearest-neighbour agreement), and a different scope (within an architecture vs across architectures and modalities). Shared vocabulary, not shared math.
Kaushik ↔ Sharma & Kaplan
Low-dimensional structure in weights and low intrinsic dimension of data could be related: if data lives on a low-dimensional manifold, weights fit to it might too.
≡Level 3 · Shared mechanism
≅Level 2 · Shared math
Both use "low-dimensional" in a precise linear-algebra or geometry sense, so some formal kinship is real (Level 2 at a stretch). No study has tested whether data-manifold dimension predicts weight-subspace dimension. That test would be Level 3b.
All three
Scaling produces regularity at the level of loss, weights and representations.
≡Level 3 · Shared mechanism
≈Level 1 · Analogy
Three different regularities are each measured somewhere. No mechanism connects them yet. The defensible claim is a research agenda, not a finding.
Editor’s verdict
Three regularities, three objects, three definitions of "convergence": a Level 1–2 connection (shared vocabulary and some shared low-dimensional mathematics) and a testable research agenda, not a discovered mechanism.
Experiments that would tell the readings apart
Train families of models from scratch (no shared base) and test whether shared weight subspaces exceed a random-matrix baseline.
Measure weight-subspace overlap and representational alignment on the same model family across scale, and test whether they co-vary.
Manipulate a dataset’s intrinsic dimension and test whether the scaling exponent and the shared-subspace dimension shift together (a Level 3b predictive test).