Neutrosophic MR-Metric Spaces: A Topos-Theoretic Framework with Applications
Bibliographic record
Abstract
This paper introduces and systematically investigates the category of Neutrosophic MR-Metric Spaces (NMR-MS), which generalizes classical metric spaces by incorporating neutrosophic logic to model truth (T), indeterminacy (I), and falsity (F). We define the category NMRMS and construct sheaves of NMR-MS over topological spaces, proving that the category Sh(X, NMRMS) forms an elementary topos. This provides a rich mathematical framework for reasoning about uncertainty, vagueness, and contextual truth in a localized manner. We develop the internal language of this topos as a neutrosophic type theory and establish its soundness and completeness. The framework is applied to diverse fields including manifold theory, dynamic systems, image processing, data fusion, functional analysis, graph theory, differential equations, machine learning, topology optimization, quantum systems, and financial modeling. Our work unifies and extends recent advances in fixed point theory, fractional calculus, and neutrosophic fuzzy metrics within a single, category-theoretic foundation.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".