How reproducible is the mathematics found inside trained models?
Deep networks trained on very different data and objectives keep converging on a small vocabulary of mathematical structures — linear feature directions, low-dimensional manifolds, belief-state simplices, concept lattices, attractors, collapse geometries. The evidence is spread across machine learning, computational neuroscience and statistical physics and described in incompatible language, so how much of it holds up cannot be answered from the literature as it stands.
The map is the instrument for answering it. Every entry names a structure, names the model it was found in, points at the paper that reported it, and is read to primary source before it is written.
Reproducibility and equivalence.
The question splits in two, and each half decomposes into things a corpus can actually count. Every axis below is computed from the records, never asserted by a curator.
Reproducibility — did it happen more than once?
- The same structure reported in more than one paper.
- Reported by more than one research group — papers that share no author.
- The overall frequency with which a structure appears in the corpus at all.
Equivalence — does it hold across kinds of model?
- The same structure across modalities — language, vision, audio, protein, control.
- Across model families, trained independently by different labs.
- Across architecture classes — transformer, CNN, RNN, diffusion, GNN.
Two by-products. The corpus is a source of hypotheses about where structures come from — the empty cells show where to look. And it is an instrument for analysing the field: which methods dominate, which structures are the mastodons, whether models are linear or linear methods are simply used more often.
The map as an object.
A domain knowledge base in the form of a graph of claims, decomposed into structures, models, methods, families and hypotheses.
A continuously updated map of empirically reported mathematical structure in trained models.
What gets into the map.
A paper enters the corpus if and only if all five conditions hold. They are checked in order, and the first failure ends the assessment.
Qualifying evidence: criteria (a), (b), (c)
Slot 03 requires at least one of three kinds of evidence. Together they define what counts as a measurement of structure, and the boundaries between them are what keeps the corpus from drifting.
Standing exclusions
Each of these was decided once, on a real case, and is not reopened per paper. The reason is recorded so the boundary can be argued with rather than guessed at.
| Relative encoding strength | A claim that X is encoded more strongly than Y is about magnitude, not about shape. |
| Method papers with no structural claim | A technique that is never used to establish a structure inside a real model is not a finding. |
| Imposed geometry | Shape that follows from the architecture or the loss by construction. |
| Behaviour and benchmarks | Capability, robustness, pruning-survival and evaluation results with no representational claim attached. |
| Theory with no measurement | Purely analytic results about what a network could represent. These can appear as context on a theory page; they do not become paper records. |
| Position and survey papers | No original experiments, no entry. They may be cited on an existing node as background. |
Two things that look like exclusions and are not. A paper whose full text cannot be obtained is deferred, never rejected on its abstract. And a negative or critical result — a failed replication, a demonstration that a reported structure is an artefact — is not rejected at all: it is recorded as a caveat on the structure it bears on, because that is exactly the evidence a map like this exists to keep.
Three outcomes, and only three
Every screened candidate ends in exactly one of these, dated and logged in public. There is no silent middle.
How far it has been reproduced.
Computed from the authors of the papers and from each model's architecture class, domain and family. The unit here is a node, not a finding: every count below is out of the 66 structures and hypotheses that have at least one paper behind them.
| axis | breadth | replication |
|---|---|---|
| papers with no shared author | — | 31 |
| architecture classes | 38 | 29 |
| domains | 33 | 28 |
| model families | 48 | 30 |
Breadth — two or more values turned up at all, even if one study supplied them. Replication — the values came from papers with no shared author. Only the second is called replication.
Read the denominator: 35 of these 66 nodes rest on a single paper, so the ceiling for every row above is 31.
How a paper becomes an entry.
Four stages, run separately and logged separately, so that discovery bias, reading errors and classification errors stay distinguishable from one another.
What this map does not yet do.
These are the places a reviewer will press, so they are named here first — each one paired with what is being done about it.
- Coverage is skewed. The discovery sweep runs mostly on arXiv, and arXiv is where recent machine learning lives; non-arXiv venues and older work are under-represented. A separate back-fill track for 2000s–2020 results and non-arXiv venues.
- Replication is tracked, not tested. The map records when two papers report the same structure. Nobody here re-runs the experiment, and a paper may also ship code the map does not yet point at. A layer of links to executable replications.
- Curation is single-reader. Inclusion decisions run through one pipeline with one person's audit pass, not two independent screeners with a measured agreement rate. A human-verification loop with a published inter-rater agreement.
- Thin on theory. The structure pages record what was measured, but rarely build the intuition or give an exercise, so the map informs without teaching. A track of worked problems and intuitions.
One schema change is also outstanding: how the evidence was obtained — criteria (a), (b), (c) — is decided during screening but not stored on the finding, the way the Gene Ontology stores evidence codes. Two grades were retired for a related reason: canonical, which graded consensus in the field rather than evidence in the corpus, and replicated, where a hand-set grade duplicated, worse and with drift, what the corpus computes for itself.
The corpus is not public yet.
The corpus is plain text in Git — 2 500+ YAML and Markdown files with a schema and a validator — but the repository is closed while the schema is still moving. Every record is readable through this site: each structure, method, model family and paper has a page carrying the same fields the files do.
The method is written down here so that it can be checked. If a finding is wrong, or the inclusion rule was applied incorrectly, that is a defect worth reporting.
- The classification. A structure filed under the wrong role or type; two entries that are the same object under different names; a class the tree has no slot for.
- The inclusion rule. A paper that passes the five slots and is missing, or one that is here and should not be.
- The replication scheme. The four axes, the thresholds, the author-disjointness test — cases where they read the corpus wrongly.
- A misread paper. A claim attributed to a paper that the paper does not make.
- Missing work. Especially pre-2020, non-arXiv, and results outside language models.
Built and curated by Sabrina Sadiekh.