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 1 grows—but that growth is bounded by the RKHS energy, made uniformly small by the guard 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. 2 Assumption 3.1 (Guard budget).Along a schedule of dyadic stages, the cumulative guard is summable: Pj(αj+βj+ηj+τj)<∞. 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. 3 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). 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. 4 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. 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 At this point we switch language and phrase the deficit as an observable driven by two canonical flows: the additive generator Hadele and the multiplicative (idelic) generator Hidele. Their commutator is the whole reason a virial identity exists. Since these operators are not standard textbook objects in this exact adelic form, we give their full definition and motivation separately. (For the definitions and the two commutation facts we use here, see [18], Theorem 8.1 and Proposition 10.1; for intuition, do the “real place check” exercise.) In short: 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. 5 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. 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. 6 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. 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 . 7 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. Projecting coefficients by PM restricts to classes coprime to M and identifies the diagonal residue with 2C2(y). D Stagewise Large Sieve and the Packet Collision Bound Fix T0 ( X = e T0 ), blur σ = ( log X ) γ , bandwidth Λ = ( log X ) λ , and local cutoff y = ( log X ) β with M = Qp≤yp . Let κσ,T0 ( n ) = exp− ( log n−T0 ) 2/ (2 σ2 )  and N = {n : |log n−T0| ≤ Λ + c0σ} . A Gaussian large sieve on the log scale gives 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. With the fresh-block operator smallness from Lemma 13.3 (under the uniform phase selection), the last sum is o (1) P|an|2 . This yields the packet pair L2 bound and, by Cauchy–Schwarz, the collision bound needed in §11. References [1] G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum” III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70. [2] E. C. Titchmarsh (revised by D. R. Heath-Brown), The Theory of the Riemann ZetaFunction, 2nd ed., Clarendon Press, Oxford, 1986. [3] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge Univ. Press, 2007. [4] H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloquium Publications 53, 2004. [5] H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, 1974. [6] R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, AMS Colloquium Publications 19, 1934. [7] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw–Hill, 1987. [9] S. Ikehara, Zur Theorie der Dirichletschen Reihen und der Riemannschen Zetafunktion, Math. Ann. 110 (1935), 1–28. [10] J. Korevaar, Tauberian Theory: A Century of Developments, Springer, 2004. [11] H. L. Montgomery, The pair correlation of zeros of the zeta function, in Proc. Sympos. Pure Math. 24 (1973), 181–193. 8 [12] D. A. Goldston, J. Pintz, and C. Y. Yıldırım, Primes in tuples I, Ann. of Math. 170 (2009), 819–862. [13] J. Maynard, Small gaps between primes, Ann. of Math. 181 (2015), 383–413. [14] D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1(2014), Art. 12, 83 pp. [15] A. Perišić, Boundary Guards and Detectors, Zenodo, 2025. [16] A. Perišić, Montgomery Law under Gaussian Blur and Fresh-Block Decorrelation, Zenodo, 2025. [17] A. Perišić, Blur as a Universal Principle, Zenodo, 2025. [18] A. Perišić, A Gentle First Course in Adelic–Quantum Arithmetic, Zenodo, 2025. 9