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Prime Numerators of the Triangular-Fractional Grid: A shell stratification and a lens for prime-in-progression density

Authors: Huckstead, Jeffery;

Prime Numerators of the Triangular-Fractional Grid: A shell stratification and a lens for prime-in-progression density

Abstract

We study the \emph{prime numerators} of the triangular-fractional grid $a(n,k)=n-1+k/n$. Grouping the reduced fractions by their reduced denominator (their "shell" $s=n/\gcd(k,n)$) gives a stratification of the prime numerators that we did not find in the surveyed literature as a named object. We show that within a fixed shell the numerator has the closed form $\Rs=s^2m-s+t$, so each shell is a union of $\varphi(s)$ arithmetic progressions modulo $s^2$. This reduces the per-shell prime-density question to the prime-number theorem in arithmetic progressions. The naive comparison of per-shell prime density against $1/\varphi(s)$ shows a strong, persistent upward trend (Spearman $\rho_{\mathrm{Sp}}\approx 0.96$--$0.99$); this trend \emph{vanishes} under the cellwise predictor $\widehat\rho_N(s)$, whose coprimality factor $s/\varphi(s)$ is exact because $\varphi(s^2)=s\varphi(s)$. The corrected ratio is flat with no trend, its coefficient of variation "shrinks" with $N$ ($0.138\to0.058$ across $N=250\to2000$), and at $N=1000$ a fraction $0.994$ of shells fall within two binomial standard errors of the prediction. We conclude that the shell stratification is a lens, not a new prime law: the triangular-fractional shell coordinate makes a classical prime-in-progression mechanism visible in a clean, controlled way. We prove a within-shell numerator uniqueness result, record the controls, and fence the claim explicitly. No new prime phenomenon, no Riemann-hypothesis relevance, and no higher-dimensional claim is asserted.

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average