Categorical models of the differential λ-calculus
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
Abstract The paper shows how the Scott–Koymans theorem for the untyped λ -calculus can be extended 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 unraveling 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 develops the main result in a modular fashion by showing how to add 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” must behave with respect to the added structure and, furthermore, how “universal objects” must behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split while preserving the differential structure of the setting. This paper is the full version of a conference paper and includes the proofs which were omitted from that version due to page-length restrictions.
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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 | 0.000 | 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 | 0.000 |
| 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