Neutrosophic MR-Metric Spaces: A Topos-Theoretic Framework with Applications
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
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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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.010 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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