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Record W3183622905 · doi:10.1080/00029890.2022.2115779

Extremal Values of Pi

2022· preprint· en· W3183622905 on OpenAlex
Nikhil Henry Bukowski Sahoo

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.

aboutThe title or abstract carries a Canadian signal from the geographic lexicon.
no affNo Canadian affiliation: this work is invisible to an affiliation-only frame.
No Canadian affiliation. An affiliation-only frame, the usual design, would never have seen this work. It is one of the works that make the case for inverting the frame.

Bibliographic record

VenueAmerican Mathematical Monthly · 2022
Typepreprint
Languageen
FieldMathematics
TopicMathematics and Applications
Canadian institutionsnot available
Fundersnot available
KeywordsPiEuclidean geometryQuarter (Canadian coin)Norm (philosophy)MathematicsEuclidean distanceValue (mathematics)CombinatoricsSymmetry (geometry)GeometryLawStatisticsPolitical scienceGeography

Abstract

fetched live from OpenAlex

We give an introduction to the classical results of Stanisław Gołąb, on the values that pi may attain in arbitrary normed planes, including a classification of the extremal values. This fascinating topic is accessible to undergraduates, but introductory accounts tend to elide the fussy details of classifying the extremal values; we instead give the necessary background to understand the full proofs, with limited outside references. We then reprove a result due to J. Duncan, D. Luecking, and C. McGregor, which states that any norm with quarter-turn symmetry has pi-value at least π. We also classify when this lower bound is obtained. Finally, we define Radon norms and note an analogy that we hope will help motivate further results.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.001
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMeta-epidemiology (narrow), Insufficient payload (model declined to judge)
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Other · Consensus signal: none
Teacher disagreement score0.734
Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.001
Meta-epidemiology (narrow)0.0010.001
Meta-epidemiology (broad)0.0020.001
Bibliometrics0.0000.000
Science and technology studies0.0000.001
Scholarly communication0.0000.000
Open science0.0010.002
Research integrity0.0000.001
Insufficient payload (model declined to judge)0.0030.000

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.

Opus teacher head0.057
GPT teacher head0.350
Teacher spread0.293 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it