Categorical Models of the Differential λ-Calculus Revisited
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
The paper shows that the Scott-Koymans theorem for the untyped λ-calculus extends to the differential λ-calculus. The main result is that every model of the untyped differential λ-calculus may be viewed as a differential reflexive object in a Cartesian closed differential category. This extension of the Scott-Koymans theorem depends critically on unravelling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting. The paper uses (total) Turing categories with "canonical codes" as the basic categorical semantics for the λ-calculus. It shows how the main result may be developed in a modular fashion by first adding left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure it is necessary to identify how "canonical codes" behave with respect to the added structure and, furthermore, how "universal objects" behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split in these more structured settings.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it