FREE PATH RINGS AND THEIR GEOMETRIC REALIZATION AS CYCLE COMPLEXES. EXTENSION OF J.J. ROTMAN'S FREE GROUP THEOREM
Bibliographic record
Abstract
This paper introduces free path groups and free path rings. A free ring is a Herstein ring on a free group, which is useful in simplified presentation of maximal nucleus clusters (MNCs) in triangulated foreground regions of video frames. A free group (denoted G(β, +)) is a nonempty set G with a binary operation + and a basis β ⊆ G so that every member a ∈ G can be written as a linear combination of the basis elements. A free path group G which is a J.H.C. Whitehead homotopy group free on a collection of continuous maps h : [0, 1] → X (called paths) in a path cycle on a space X. Each path cycle is a sequence of paths with no end path. A free path ring is a Herstein ring on a free path group. The main results in this paper are (1) Every path cycle is realizable as a free path ring (Extension of J.J. Rotman's free group theorem), (2) Every 1-cycle cell complex is realizable as a path cycle and (3) Every 1-cycle is realizable as a free ring (this reverbates back to W. Dyke's 1882 work on the derivation of free groups from circuits on simply connected polygons). Inspired by Lupton-Oprea-Scoville digital fundamental groups, a number of conjectures about the geometric realization of free path rings as structures in digital images and as vector fields are given.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".