A Proof of the Twin Prime Conjecture in the Ƥ x Ƥ Space
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Bibliographic record
Abstract
This work presents a formal proof of the twin prime conjecture based on a novel mathematical structure of mirror primes in the 2-dimensional space of primes. is an infinite recursive set of pairs of symmetric primes adjacent to any pivotal even number in finite distances . In the framework of the set of twin primes is deduced to a subset of the mirror primes where the half interval Therefore, the equivalence among sets of and is established, i.e., based on Cantor's principle of infinite countability, such that the twin prime conjecture holds. Experiments based on an Algorithm of Twin-Prime Sieve (ATPS) are designed to demonstrate and visualize the proven twin prime theorem.
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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 | 0.000 |
| 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 | 0.001 | 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