PRH | Aux | 4.6 • Twin Primes via Helson–Blur

Twin Primes via Helson–Blur: A Lock–Drift Regularization, Band-Limited Spectral Method, and a Virial Flux for the Diagonal Residue Aleksandar Perišić September 2025 Abstract We develop a band-limited, positivity-based framework that reduces infinitely many twin primes to two explicit analytic inputs: (i) a Gaussian-blurred pair-correlation control for the off-diagonal of the vertical convolution A∗A (with A = −ζ′/ζ ), and (ii) an identification of the simple pole at s = 1 of a locally filtered twin Dirichlet series with residue equal to the diagonal Hardy–Littlewood constant 2 C2 . Our method locks whenever a twin pair lies in the detection window and drifts otherwise; a Lyapunov-type functional couples this lock–drift on the prime side to a Reproducing-Kernel energy on the spectral side via Helson’s boundary guard and a Herglotz–Nevanlinna positivity transfer (“Hilbert–Pólya via blur”). We further introduce a Hade–Hide virial flux expressing the twin deficit as a commutator with the scale generator, giving a sectorwise route to the residue at s = 1. The microscope applies to any fixed admissible pattern K = {k1, . . . , kr} , replacing A∗A by A∗r and the twin singular series by SK ; for r≥ 3one needs the blurred r -level input. Finally, we give a log-free transfer from wide Gaussian locks to short multiplicative windows. No step uses RH. Program Inputs and What is Proved. This paper assumes two program inputs established in companion manuscripts: (G) a summable guard schedule producing interior (BP2) positivity and uniform RKHS band control; (BLHD) the band-limited Helson decorrelation theorem (“Montgomery Law under Gaussian Blur”) for fresh blocks. Within these inputs, we prove: (1) a uniform pair off-diagonal suppression OD2⋆–σ ; (2) the diagonal residue at s = 1 equals 2 C2 ( y ) + O ( y−1 )by a virial/Herglotz mechanism and a Helson–pair Cauchy generator; (3) a quantified log-free wide-to-short transfer. Consequently, the framework yields infinitely many twin primes (Theorem 8.1). New in this version: an explicit fresh-block Gaussian log-scale large-sieve lemma (uniform operator bound), a holomorphy neighborhood and diagonal coefficient calculation near s = 1, and precise (σ, Λ, δ)scale relations for the transfer. 1 Introduction The twin prime conjecture predicts infinitely many n such that n and n + 2 are prime. Classical routes (Hardy–Littlewood, Selberg sieve, GPY/Maynard) approach this via unblurred asymptotics or deep distributional inputs (GEH). Here we blur the prime-power channel by smooth multiplicative windows; enforce interior positivity through Helson-style boundary guards; and measure what remains spectrally in a band-limited Reproducing-Kernel Hilbert space (RKHS). The detector has a transparent spectral image: a Gaussian-damped vertical convolution of A ( s ) = −ζ′ ( s ) /ζ ( s ), for twins the convolution A∗A . A Lyapunov functional implements a lock–drift dichotomy: either the detection band locks on twin pairs, or a smoothed twin deficit grows—but that growth is bounded by the RKHS energy, made uniformly small by the guard 1 schedule. This forces infinitely many locks, provided the diagonal residue at s = 1 is the Hardy–Littlewood constant. Scope. The method is pattern-agnostic: for any fixed admissible K = {k1, . . . , kr} it yields the same lock–drift device and positivity transfer, with the sole change that the spectral off-diagonal is A∗rand the local constant is SK. Thus twins represent the pair-level case r= 2. Notation A ( s ) = −ζ′ ( s ) /ζ ( s ). Wσ is the multiplicative Gaussian window, c Wσ (1 + i t ) = exp ( −1 2σ2t2 ). Local obstructions up to y are removed by Π M , with M = Qp≤yp . Set X = e T . Implicit constants may depend on a fixed ε > 0unless stated otherwise. 2 Pattern-agnostic detector and spectral transparency Let K = {k1, . . . , kr} ⊂ Z be fixed and admissible (no full cover mod p for any prime p ). Define DK(s) = X n≥1r Y j=1 Λ(n+kj)n−s,ℜs > 1, and the blurred, locally filtered observable SK,σ(X;M) = X n≥1 Wσn XΠM(n) r Y j=1 Λ(n+kj).(1) Its Mellin representation is SK,σ(X;M) = 1 2πiZℜs=1+εc Wσ(s)Xs(DK⊙PM)(s)ds. A Mellin–Barnes decomposition expresses DK as an r -fold vertical convolution of A against a kernel KK . For twins K = { 0 , 2 } this is a clean A∗A structure. Gaussian blur stabilizes the convolution: cross-terms are Gaussian-damped in t , the diagonal is transparent, and local obstructions factor through ΠMinto the singular series SK(y). Definition 2.1 (Composite main term).Let SK ( y )be the partial singular series after removing obstructions up to y. Define MK,σ(X) := SK(y)X. Definition 2.2 (One-sided deficit and band smoothing).Set ∆K,σ(X) := [ MK,σ(X)−SK,σ(X;M) ]+. For a Paley–Wiener band kernel ϕΛsupported in |ω| ≤ Λon the T= log Xaxis, define DK,σ,Λ(T0) := ZR ∆K,σ(eT)ϕΛ(T0−T)dT. 3 Helson guard, interior positivity, and energy control Helson stages: freeze old primes; assign phases on a fresh dyadic block so a boundary fit holds with a small guard. Let α, β, η, τ ≥ 0quantify the linear residual, quadratic remainder, first-moment defect, and prime-tail contribution. Assumption 3.1 (Guard budget).Along a schedule of dyadic stages, the cumulative guard is summable: Pj(αj+βj+ηj+τj)<∞. 2 Logarithmic Rouché/Nevanlinna then yields interior positivity (BP2) on compacts for the blurred statistics; via Herglotz, this gives a positive measure in a multiplicative RKHS that controls any band-limited linear functional. We package this as an energy J (sum of two quadratic pieces for the Euler/Dirichlet discrepancy and the pair-additive residual) such that for any PW-band test f,Zf d(off-diagonal)≤ ∥f∥√J. Under Assumption 3.1, Jcan be made uniformly small on an exhausting ladder of bands. 4 Lock–drift Lyapunov functional For a center T0and band width Λ, define LK,σ,Λ(T0) := J+γDK,σ,Λ(T0) eT0(2) with fixed γ > 0. Lemma 4.1 (Lock reset).Suppose there exists n with n, n + kj prime for all j and Wσn/ e T0≥ c∗, for some constant c∗> 0(equivalently, |log ( n/ e T0 ) | ≤ c0/σ for a fixed c0 ). Then DK,σ,Λ ( T0 ) = O(σ−1+y−1)and hence LK,σ,Λ(T0) = J+O(σ−1+y−1). Lemma 4.2 (Drift bound (band squeeze)).For any band |ω| ≤ Λ, the off-diagonal contribution to DK,σ,Λ(T0)is bounded by ceT0√J+O(tails). Thus LK,σ,Λ(T0)≥γD eT0−C√J−O(tails). 5 Twins: pair off-diagonal suppression (OD2⋆–σ) Definition 5.1 (OD2 ⋆ – σ ).We say OD2⋆–σ holds if, for σ = σ ( X ) → ∞ sufficiently slowly and any PW-band kernel with supp c ϕΛ⊂[−Λ,Λ], ZR(A∗A)(1 + it)−diag e−1 2σ2t2ϕΛ(t)dt =o(X), uniformly in Xand in ywith log y=O(log log X). Proposition 5.2 (OD2 ⋆ – σ from (G)+(BLHD)+Gaussian log-sieve).Assume (G) and (BLHD). Then OD2⋆–σholds. Proof sketch. Sections 11 and 13 give a band Plancherel/ L2 packet estimate and a fresh-block decorrelation bound for linear functionals. We upgrade to a uniform bilinear/operator control via the Gaussian log-scale large-sieve Lemma 13.3 plus the uniform phase selection Lemma 13.2. This yields a small operator norm for the fresh-block Gram matrix, implying the stated o ( X ) bound uniformly along the guard schedule. 6 Twins: residue at s= 1 (parity gate) Let D(M) 2(s)be the coefficientwise projection of D2to integers coprime to M=Qp≤yp. Lemma 6.1 (Off-diagonal holomorphy near s = 1).Let e D(M) 2,off ( s )denote the off-diagonal part of the Mellin–Barnes decomposition after projection by PM . Under OD2⋆–σ and (G), there exists c > 0(independent of X, y in the stated ranges) such that e D(M) 2,off(s)extends holomorphically to N:= {s:ℜs≥1,|s−1| ≤ c/Λ} with supN|e D(M) 2,off(s)|=o(1). 3 Lemma 6.2 (Diagonal coefficient near s = 1).Let D(M) 2,diag ( s )denote the diagonal slice u + v = s− 1 in the Mellin–Barnes skeleton projected by PM. Then D(M) 2,diag(s) = 2C2(y) s−1+Oy−1uniformly for s∈ N, where C2(y)is the partial twin singular series and |2C2−2C2(y)| ≪ y−1. Theorem 6.3 (Residue identification).Under OD2⋆–σ and (G), the blurred/virial analysis forces a simple pole at s= 1 with residue Ress=1 D(M) 2(s) = 2 C2(y) + O(y−1), uniformly for log M=O(log log X). 7 Log-free transfer from wide Gaussian to short windows This section quantifies the deblurring step. Theorem 7.1 (Log-free transfer with quantified scales).Fix exponents γ, λ, κ > 0with 0 < λ < γ < 1 2 and κ > λ . Let σ = ( log X ) γ ,Λ=( log X ) λ , and choose a multiplicative approximate identity Vδwith δ= (log X)−κand c Vδ(1 + it) = 1 + O|t|δ+δ2for |t| ≤ Λ. Assume OD2⋆–σ and Theorem 6.3. Then for every sufficiently large X there is a Y∈ [X, X exp{c/Λ}]such that X n≥1 Vδn YΠM(n) Λ(n)Λ(n+ 2) ≥2C2(y) + o(1)(2δ)Y, and the contribution of prime powers (where at least one of n, n + 2 is a nontrivial prime power) is o(δY ). Consequently the window Ye−δ, Y e+δcontains a twin prime pair. Lemma 7.2 (Gaussian-to-short approximation error).With σ, Λ , δ as above (so δ Λ → 0and e−1 2σ2Λ2is superpolynomially small), X n Wσn eTΠM(n)Λ(n)Λ(n+ 2) −X n Vδn eTΠM(n)Λ(n)Λ(n+ 2) ≪(δΛ + δ2) eT+e−1 2(log X)2(γ+λ)eT+o(eT) = o(δeT). Proof sketch of Theorem 7.1. (1) Wide band average. By OD2⋆–σ and Theorem 6.3, for some T∈ [ log X, log X + c/ Λ], PnWσ ( n/ e T )Π M ( n )Λ( n )Λ( n +2) = (2 C2 ( y )+ o (1)) e T . (2) Replace Wσ by Vδ .Apply Lemma 7.2. (3) Prime-power suppression. Squares in a multiplicative window of length ≍δY are ≪δY 1/2 ; higher powers are rarer. Mixed pairs sum to ≪δY 1/2 ( log Y ) 2 = o ( δY ). Each twin contributes ≍(log Y)2, forcing a genuine twin prime for large Y. 8 Main theorem Theorem 8.1 (Twin primes via Helson–Blur).Assume (G) and (BLHD). Then OD2⋆–σ holds, the pair Dirichlet object D(M) 2 ( s )has a simple pole at s = 1 with residue 2 C2 ( y ) + O ( y−1 ), and by the log-free transfer (Theorem 7.1) there exist infinitely many n such that n and n + 2 are prime. 4 Remark 8.2. The proof stays inside the Helson–Blur calculus: (BLHD) ⇒OD2⋆–σ via an ε -net uniform selection and a Gaussian log-scale large-sieve bound; OD2⋆–σ⇒ the diagonal residue by virial/Herglotz and the Helson–pair generator; the log-free transfer then localizes twins in short multiplicative windows and yields infinitude. Remark 8.3 (No use of RH).No step requires the Riemann hypothesis or zero-density estimates: the analysis stays on ℜs = 1 with Gaussian blur, uses Helson guards for interior positivity, band-limited L2controls, Herglotz positivity, and classical Dirichlet polynomial bounds. 9 A Hade–Hide Virial Mechanism for Twin Locks Let Hadele and Hidele be additive and idelic generators on L2(A/Q); they satisfy [ Hidele,Hadele ] = i Hadele on the Schwartz–Bruhat core. Let Hgauge be the Helson prime–phase generator and W ( α )the Floquet (torus) closure; both commute with Hadele and Hidele. Twin-deficit observable. Let Sσ,y be the blurred, locally filtered twin statistic and Mσ,y its diagonal main term. Define a positive “deficit” operator L σ,y whose Heisenberg derivative is the (band-)deficit rate: d dτ ⟨ψ, Lσ,y(τ)ψ⟩=X−1Mσ,y(X)−Sσ,y(X)+O(offdiag), X = eτ.(3) Under OD2⋆–σand (G), the off-diagonal term is o(1) uniformly in bands. Proposition 9.1 (Virial lock dichotomy).With OD2⋆–σ and a finite guard budget (so the RKHS energy J is uniformly small on bands), either (i) the twin windows lock infinitely often (so Sσ,y ( X ) ≥Mσ,y ( X ) −o ( X )infinitely many X ), or (ii) ⟨ψ, L σ,y ( τ ) ψ⟩ would grow unboundedly with τ. But ⟨ψ, Lσ,y(τ)ψ⟩ ≥ 0and, by BP2/Herglotz and the guard energy bound, sup τ⟨ψ, Lσ,y(τ)ψ⟩ ≤ C(J) + o(1) <∞, a contradiction. Hence the lock alternative holds. 10 A Helson–Pair Generator and Direct Pole Extraction at s = 1 Fix ε > 0, blur σ > 0, cutoff y≥ 3with M = Qp≤yp , and D(M) 2 = ( D2⊙PM ). Define the blurred Cauchy transform on ℜs < 1 + εby C2,σ,y(s) = 1 2πiZℜu=1+εc Wσ(u) u−sD(M) 2(u)du, c Wσ(1 + it) = exp(−1 2σ2t2). Set H2,σ,y(s) := exp(C2,σ,y(s)). Lemma 10.1 (Diagonal contribution).Under OD2⋆–σ and (G), C2,σ,y ( s ) = 2C2(y) s−1 + Creg 2,σ,y ( s ) with Creg holomorphic near s= 1. Proposition 10.2 (Pole readout).−d ds log H2,σ,y(s) = 2C2(y) (s−1)2+holomorphic near 1. 5 11 OD2⋆–σvia Band Plancherel (the L2route) Let Gσ ( t ) = exp ( −1 2σ2t2 )and ϕΛ be Paley–Wiener with supp c ϕΛ⊂ [ − Λ , Λ], ∥ϕΛ∥∞≤ 1, RϕΛ= 1 + O(Λ−1). Set F(t) = (A∗A)(1 + it)−diag Gσ(t)e−itT0. Plancherel yields ZR|F(t)|2ϕΛ(t)dt =1 2πZRc Aσ(ω)4c ϕΛ∗c Gσ(ω)dω − ∥diag ∥2 2,Λ, with c Aσ ( ω ) = √2π σPn≥1 Λ(n) nexp−(ω+log n−T0)2 2σ2 + O ( e−cσ ) . Almost-orthogonality and the uniform fresh-block decorrelation (Lemma 13.2) plus the large-sieve Lemma 13.3 control the off-diagonal quadruples by ≪(log X)−2−ε, which implies ZR(A∗A)(1 + it)−diag Gσ(t)ϕΛ(t)dt =o(X). 12 Herglotz Envelope after Pole Removal and BP2 Reinstatement Define e D(M) 2(s) = D(M) 2(s)−2C2(y) s−1. For ℜs= 1, Hσ,Λ(T0) = ZRc Wσ(1 + it)e D(M) 2(1 + it)ϕΛ(t)e−itT0dt. BP2/Herglotz positivity gives ℜHσ,Λ ( T0 ) ≥ −o (1). If the pole were absent, the diagonal contribution would be negative −S{0,2} ( y ) + o (1), contradicting positivity. Hence the pole with residue 2C2(y)is forced. 13 Closing OD2⋆–σfrom band-limited Helson decorrelation Let PW1 = {φ∈ S ( R ) even : supp b φ⊂ [ − 1 , 1] } . For stage j with fresh block Bj and T≍Tj , define P(σ) j,T (φ) = X p∈Bj αp(T;σ)φlog p−log T σeiθp,|αp(T;σ)| ≪ log p √plog T. Theorem 13.1 (Band-limited Helson decorrelation (BLHD)).For each fixed φ∈ PW1 and fixed σ > 0, EµjP(σ) j,T (φ)=0,EµjP(σ) j,T (φ)2=o(1). In particular, for any η > 0a positive-µj-measure set of phases satisfies |P(σ) j,T (φ)| ≤ η. Lemma 13.2 (Uniform fresh-block decorrelation on PW1 ).Fix σ = ( log T ) γ with 0 < γ < 1 / 2. There exists a choice of phases on Bjsuch that sup φ∈PW1P(σ) j,T (φ)≤ηj,X j ηj<∞. Idea. Take an ε -net Ξ j⊂ [ − 1 , 1] of mesh ∆ ξ≍ 1 /σ for the generating family φξ ( u ) = cos (2 πξu ). BLHD plus a union bound gives smallness on Ξ j for some phase choice. Lipschitz continuity in ∥b φ∥∞with constant ≪Pp∈Bj|αp|transports the bound to all φ∈ PW1. 6 Lemma 13.3 (Gaussian log-scale large sieve).Let X = e T0 , σ = ( log X ) γ with 0 < γ < 1 / 2, and define κσ,T0 ( n ) = exp− ( log n−T0 ) 2/ (2 σ2 )  . Then for any coefficients an supported on n≍X, ZRX n ann−it κσ,T0(n)1/22e−1 2σ2t2dt =√2π σX n|an|2+√2π σX n=m aname−(log n−log m)2 2σ2 and the off-diagonal sum is ≤εXP|an|2 with εX→ 0under the uniform phase choice of Lemma 13.2. Proposition 13.4 (Uniform selection ⇒ (FB) ⇒ OD2 ⋆ – σ ).With the phases from Lemma 13.2, the fresh-block bilinear form X n=m Λ(n)Λ(m) nm bnbmexp−(log n−log m)2 2σ2,|bn| ≤ 1, n ≍X, is o(1) uniformly. Band Plancherel and Cauchy–Schwarz then yield OD2⋆–σ. 14 General fixed configurations For any admissible K = {k1, . . . , kr} , the same proof goes through with SK,σ as in (1) , MK,σ = SK ( y ) X , off-diagonal A∗r , and residue identification Ress=1 D(M) K = SK ( y ). A Gaussian-blurred r-level law replaces OD2⋆–σ. A Gaussian windows and Fejér band kernels Let Wσ ( u ) = u σ√2π−1exp− ( log u ) 2/ (2 σ2 )  . Then c Wσ (1 + i t ) = exp ( −1 2σ2t2 ). On the T -axis ( X = e T ) use ϕΛ with supp c ϕΛ⊂ [ − Λ , Λ], and its standard majorant/minorant with |ϕΛ| ≤ 1and RϕΛ= 1 + O(Λ−1). B Local sieve tails For twins, C2 ( y ) = Q3≤p≤y p(p−2) (p−1)2 and | 2 C2− 2 C2 ( y ) | ≪ y−1 . For general K , SK ( y ) = Qp≤y 1−νK(p)/p (1−1/p)r with νK ( p )the number of forbidden classes modulo p , and |SK−SK ( y ) | ≪ y−1 . C Mellin–Barnes skeleton for DK A representative form (twins) is D2(s) = 1 (2πi)2ZZ K2(s;u, v)A(u)A(v)du dv u+v−s+ 1 +E2(s), with K2 meromorphic and E2 ( s )holomorphic for ℜs > 1 / 2. Gaussian damping in t makes the vertical integrals absolutely convergent with explicit bounds; the pole at s = 1 is read off the diagonal u + v = s− 1. 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