Key Role of Dimensional Analysis Homogeneity in Proving Riemann Hypothesis and Providing Explanations on the Closely Related Gram Points
Bibliographic record
Abstract
Riemann zeta function is the famous complex number infinite series consisting of a real and an imaginary part. Non-trivial zeros and Gram points are best seen as mathematically derived entities of this function when its variable Sigma has a value of $\frac{1}{2}$. The presence [but not the actual locations] of the complete set of infinite non-trivial zeros is characterized by the criterion that the sum total of the simultaneous real and imaginary parts in Riemann zeta function equates to zero. In an identical manner this slightly altered criterion for the presence [but not the actual locations] of the complete set of infinite Gram points is that this 'sum total' now refer to the lesser requirement that only the individual imaginary part in Riemann zeta function equates to zero. The key role played by Dimensional analysis homogeneity to rigorously prove Riemann conjecture/hypothesis has been fully outlined in our landmark research paper published earlier on Page 9 - 21 in the preceding Volume 8, Number 3, June 2016 issue of this journal. Those resulting methodology previously employed by us are now mathematically used in an analogical procedure to delineate its role in successfully supplying crucial explanations for Gram points. In this research article, we use the notation \{Non-critical lines\}-Gram points to signify those 'near-identical' (virtual) Gram points when Sigma value is not $\frac{1}{2}$.
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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 | 0.004 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.012 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".