The Tri-Quarter Framework: Unifying Complex Coordinates with Topological and Reflective Duality across Circles of Any Radius
Bibliographic record
Abstract
In this paper, we introduce the Tri-Quarter Topological Duality Theorem, the foundation of a novel mathematical framework that unifies complex, Cartesian, and polar coordinate systems on the complex plane ℂ while equipping the circle T r of radius r > 0 with a new topological property. Our framework integrates a generalized coordinate system-where real and imaginary components are assigned unique phase pairs-with a structured orientation that elevates T r to an active separator with intrinsic directional properties. We prove that T r , as the boundary zone, exhibits topological duality with the inner zone X −,r ( ||x|| < r ) and outer zone X +,r ( ||x|| > r ), ensuring consistent separation between inner and outer radial directions across T r with a phase pair map encoding additional information. We also introduce the Escher Tri-Quarter Reflective Duality Theorem, proving reflective duality across T r via a circle inversion map that preserves phase pairs while swapping X −,r and X +,r . This approach offers insights into topological separation, orientation, and reflection, facilitating analysis of systems with circular symmetry, with potential applications in fields such as black hole physics, signal processing, and other areas reliant on complex domain partitioning. A case study on quadrant-based transformations demonstrates streamlined directional mappings, geometric elegance, unified classification, and computational efficiency in ℂ . For instance, in the case study, the Tri-Quarter approach consistently reduces the number of comparison conditional checks from a maximum of 7 in the standard method to a maximum of 4, significantly enhancing computational efficiency. A software tool visualizes some of these concepts, with future work aimed at exploring practical implementations.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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".