# Prior-art check: Singh Lemmas 5, 20, 48, 51, 67

Date: 2026-11-05. Statements taken from paper.tex (not edited). Argon not consulted (cost $0.00).
Links marked [verified] were opened or returned with matching title, authors and abstract during this check.

| Lemma | Verdict | Closest prior work | Recommended wording |
|---|---|---|---|
| SL5 (sine-process third point near a close pair is like x^4; numerical values of s E[sum xi^-6 given gap s]; E[R3^2 given s<eps] diverges like 1/eps) | B | 3-point sine-kernel determinant: Dyson 1962, Mehta, Random Matrices (standard; not reopened here). Tracy-Widom, "Correlation functions, cluster functions, and spacing distributions for random matrices" (1998) http://www.math.ucdavis.edu/~tracy/selectedPapers/1990s/CV59.pdf [verified exists]. No paper found computing this conditional inverse sixth moment or the 1/eps divergence. | "Singh Lemma 5 (numerical). The x^4 repulsion is the standard consequence of the 3-point sine-kernel determinant (Dyson; Mehta). The constants and the divergence of the conditional second moment of R3, and the repaired forms, are new." |
| SL20 (on RH, 2k-th moments of a short Dirichlet polynomial over a positive-proportion set of zeros are << p^-1 C^k k! sigma^2k #E) | A (essentially) | Soundararajan, "Moments of the Riemann zeta function", Ann. of Math. 170 (2009), Lemma 3 (continuous k! sigma^2k bound), https://arxiv.org/abs/math/0612106 [verified]. Discrete version over zeros via the Landau-Gonek formula: Milinovich, "Upper bounds for the moments of zeta'(rho)", Bull. LMS 42 (2010), Sec. 4 (Lemmas 4.1 and following), https://arxiv.org/abs/0806.0786 [verified]. Gonek, "An explicit formula of Landau and its applications" (1993) is the source of Lemma 4.1 (cited there; not opened). | Do not claim as new. "Lemma 20 (cf. Soundararajan 2009, Lemma 3; Milinovich 2010, Sec. 4). Restricting to E costs only the trivial factor p^-1." If kept as a Singh Lemma, label it "a restatement". |
| SL48 (support bookkeeping 2delta+4beta<2 for RS; Fourier tail of 1/x beyond K; band-limited part misses O(1/(K^2 beta)) of A_K) | B | Rudnick-Sarnak, "Zeros of principal L-functions and random matrix theory", Duke Math. J. 81 (1996) 269-322, https://doi.org/10.1215/S0012-7094-96-08115-6 [verified]; support condition sum|u_j|<2 is their main theorem. Fourier facts on 1/x are textbook. No paper found stating this specific far-field obstruction. | "Singh Lemma 48. Using the Rudnick-Sarnak support theorem and elementary Fourier analysis of 1/x, we isolate the high-frequency obstruction to coarse DC." Credit RS for (Q1); (Q2)-(Q3) are his. |
| SL51 (numerical A^3 m_A in {0.54,0.66,0.72,0.80}; fixed-support majorants saturate, rate 3 needs A ~ 1/eps) | B | Beurling-Selberg / extremal majorant method for small gaps: Carneiro, Chandee, Littmann, Milinovich, "Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function", J. reine angew. Math. 725 (2017) 143-182, https://arxiv.org/abs/1406.5462 [verified]; Chirre, Goncalves, de Laat, "Pair correlation estimates for the zeros of the zeta function via semidefinite programming", Adv. Math. 361 (2020) 106926, https://arxiv.org/abs/1810.08843 [verified]; Goldston-Gonek-Ozluk-Snyder, Proc. LMS (2000) (seen in Goldston's list, not opened). These give N(T,beta) bounds at fixed support; none found states the A^-3 saturation scaling or the A ~ 1/eps requirement. | "Singh Lemma 51 (numerical), using the extremal-majorant framework of Carneiro-Chandee-Littmann-Milinovich and Chirre-Goncalves-de Laat. The saturation constants and the support-versus-rate barrier are new." |
| SL67 (Type II character-variance bound E(theta,C) implies pair-correlation extension H(delta,C), delta=4theta/(1-2theta); threshold C<~5 at delta=50/1093) | B (medium-low confidence; full texts of Goldston-Vaughan, Goldston-Montgomery, Friedlander-Goldston and Hooley I not reachable) | Read in full: Heath-Brown, "Prime numbers in short intervals and a generalized Vaughan identity", Canad. J. Math. 34 (1982) 1365-1377, https://doi.org/10.4153/CJM-1982-095-9 (Lemma 1, the identity giving Type I sums (4) and bilinear Type II sums (5); no zeros, no pair correlation, no theta to delta conversion). Reached only as abstract or metadata: Goldston-Montgomery 1987, https://doi.org/10.1007/978-1-4612-4816-3_10 (RH: pair correlation conjecture equivalent to J(X,theta) ~ (3/2) X^2 theta log(1/theta); this is primes in short intervals, not a character-variance Type II input); Goldston-Vaughan 1997, https://doi.org/10.1017/cbo9780511526091.010; Friedlander-Goldston 1999, https://doi.org/10.1515/9783110285581.841; Hooley, BDH I, Crelle 274/275 (1975) 206-223, https://doi.org/10.1515/crll.1975.274-275.206. The relation delta=4theta/(1-2theta) was not found in any source reached. | "Singh Lemma 67 (conditional). The decomposition is Heath-Brown's identity [HB82, Lemma 1]; the link between pair correlation and prime variances goes back to Goldston-Montgomery [GM87] (cf. Goldston-Vaughan [GV97], Hooley's BDH series). The hypothesis E(theta,C), the conversion delta=4theta/(1-2theta) and the threshold are ours." Author must check that a character-variance bound (q-aspect) actually controls the zeta form factor F(alpha), since GM87 uses short-interval variance instead. |

## Notes
- No reference above was invented; items marked "not opened" are well-known but their exact theorem numbers were not checked here.
- SL20 is the only lemma where the same statement (up to the trivial restriction to E) is already in print.