The Shifted-Prime Singular-Series Law and the Restricted Goldbach Problem

Source abstract

This paper consolidates two companion studies in the author’s restricted-Goldbach and shifted-prime programme into a single, non-redundant treatment, together with a genuinely new synthesis of a third. Part I gives a fully rigorous, self-contained treatment of the restricted binary, ternary, and quaternary Goldbach problem, in which the first prime of a Goldbach pair is constrained to a residue class $a$ ($mod$ $q$). We prove an elementary but decisive local obstruction showing that no main term of size $≍N/φ(q)$ can hold uniformly once $q$ has an odd prime factor, identify the correct local main term for $q=2^{k}$, and give a fully self-contained explicit pointwise minor-arc bound (Vaughan’s identity with balanced parameters), an exact master second-moment bound, and an effective almost-all theorem with a threshold that we prove cannot be replaced by an $X$-independent constant once a matching second-moment lower bound is granted. We give a fully unconditional restricted Chen-type theorem, a ternary prime-anchoring transfer, and positivity of the restricted quaternary singular series. Part II proves, once and citably, an unconditional limit law: for every fixed integer $h≠0$, the odd part ${\mathfrak{S}}_{0}(n):=\prod_{\mathrm{\ell }\mid n,\mathrm{\ell }\gt 2} (\mathrm{\ell }-1)/(\mathrm{\ell }-2)$ of the Hardy–Littlewood singular series, evaluated at shifted primes $n=p+h$, converges in distribution to an explicit random Euler product $Y_{h}=\prod_{\mathrm{\ell }\gt 2,\mathrm{\ell }\nmid h} a_{\mathrm{\ell }}^{B_{\mathrm{\ell }}}$ with independent Bernoulli local factors $P(B_{\mathrm{\ell }}=1)=1/(\mathrm{\ell }-1)$; we establish an entire Mellin transform of order one, convergence of every integer moment, non-atomicity, unbounded support, and a superpolynomial upper tail, strictly amplified relative to generic integers. Part III connects this law to two separate targets: the amplitude of the restricted Goldbach main term of Part I along the shifted-prime sequence $N=p+h$ (unconditional), and, under a stated Hardy–Littlewood-type uniformity hypothesis, the fluctuations of the additive richness R(p) of the prime-generation DAG studied in Part IV. Part IV records the DAG-invariant material in full: the additive richness $R(p)$ and hub count $H(p)$ of the shifted-sum operation $p⊕q:=p+q-1$ satisfy a stable empirical inverse correlation $corr(R,H)≈-0.7935$ over the first 5000 primes; we show, by an independently coded computational study, that the singular-series factor ${\mathfrak{S}}_{0}(p+1)$ – realising $Y_{1}$ at the shift relevant to $p+1$ – is a genuine but only partial mediator of this correlation, reducing its magnitude by about 8%.