HBP | Question | 4.2 • The Prime-Factorization Crystal: A Hopf-Algebraic Perspective

The Prime-Factorization Crystal: A Hopf-Algebraic Perspective Aleksandar Perišić August 2025 Abstract We present a canonical framework that pairs two “crystals”: a prime–Euler side, where primes and their powers appear as Hopf–algebraic primitives in the two–variable Dirichlet kernel ζ ( s ) −∂slog ζ ( s + σ )  , and a zeta–zero side, where the centered completed factors ξ ( s±σ/ 2) form a lattice whose singular frequencies coincide with the prime–power spectrum. In both settings the prime–power comb {klog p} is recovered by integral projectors, with exponents read off from the maximal harmonic. This yields an explicit, structural recovery of the prime factorization of any integer n —viewing factorization as a resonance between equivalent analytic objects—and unifies Euler products, Hopf algebras, the zero–lattice of ζ , and Weil’s explicit formula in one formalism [ 1 , 16 , 2 , 3 , 8 , 9 , 6 , 5 , 4 ]. The goal is conceptual, not algorithmic speed. Having fixed factorization, we then examine what lies beyond its horizon. We identify the prime–band projector Π 0 and show that RH/ ¬ RH is invisible on a fixed slice; we introduce an almost–prime (blurred) theory and σ –dynamical diagnostics that are kernel–robust and decisive for RH; and we outline extensions that detect off–line "excitations" via positive–level poles/cuts without altering the prime comb. Thus factorization is the entry point; the main contribution is a unified language pinning down where RH lives and how to test it. Motivation: Zeta and Primitive Fibers The Dirichlet series ζ(s) = ∞ X n=1 n−s is group-like under the Dirichlet coproduct ∆ n−s = Pd|nd−s⊗ ( n/d ) −s [ 8 , 9 , 3 ], and log ζ ( s ) = Pp,k≥1p−ks/k exhibits the Euler product [ 1 , 16 , 2 ]. However, each local factor Pk≥1p−ks = (1 −p−s ) −1 bundles all powers of p into a single block, so ζ ( s )alone does not isolate the individual prime-power exponents in the factorization of n. To separate those contributions at the coefficient level, we pass to the mixed kernel K(s, σ) := ζ(s)−ζ′(s+σ) ζ(s+σ)ℜs > 1,ℜ(s+σ)>1, the standard logarithmic derivative kernel, cf. [6, 5]; whose Dirichlet expansion reads K(s, σ) = ∞ X n=1 Dn(σ) ns, Dn(σ) = X d|n Λ(d)d−σ=X p|n (log p) vp(n) X j=1 p−jσ . [ 6 , Ch. 13][ 5 , Ch. 5]. Thus each coefficient Dn ( σ )is primitive: it depends only on the prime-power divisors pj∥n , with natural log p weights and a finite geometric tail up to the true exponent vp ( n ). In this sense the family {Dn ( σ ) } already disentangles each prime’s powers, without further Möbius or convolutional differentiation. Notation (RH). Throughout, “RH” denotes the Riemann Hypothesis: all nontrivial zeros ρ of ζ(s)satisfy ℜρ=1 2. We do not assume RH unless explicitly stated. 1 The Euler–Hopf Crystal (Primitive Form) Definition 1.1 (Euler–Hopf crystal).Fix σ0> 0. For n≥ 1with n = Qppep , define the Euler–Hopf crystal as the tempered distribution CEuler n(σ0) := X pep∥n ep X k=1 (log p)p−kσ0δω−klog p. Origin. This crystal is the σ–Fourier transform of the coefficient family Dn(σ) = X d|n Λ(d)d−σ=X pep∥n (log p) ep X k=1 p−kσ arising from the logarithmic derivative kernel K(s, σ) = ζ(s)−ζ′(s+σ) ζ(s+σ),ℜs > 1,ℜ(s+σ)>1. It is therefore already primitive: its support consists only of the prime–power lines ω = klog p with natural log p weights, and contains no composite log d . This Fourier–Dirichlet viewpoint is standard; see e.g. [2, Ch. 5], [1, §I.3]. Use. Because the Euler–Hopf crystal is primitive from the outset, one reads off the ( p, ep ) data of n directly from its spikes, without any additional convolutional Möbius inversion. On the zero–lattice side, this is the natural basis for matching the primitive frequencies in explicit formulae and autocorrelation identities. Definition 1.2 (Charged Euler–Hopf primitive crystal).Fix σ0>0. For n=Qppep, define CEuler # n(σ0) := X pep∥n ep X k=1 (klog p)p−kσ0δω−klog p (the superscript #indicating “logarithmic derivative weight”). It is related to the uncharged Euler–Hopf crystal by CEuler # n(σ0) = −∂ ∂σ0 CEuler n(σ0). The Zero-Hopf Crystal Definition 1.3 (Zero–lattice Hopf crystal).Let the nontrivial zeros be {ρ = βρ + itρ} , counted with multiplicity, and write tρ:= ℑρ. Define the zero–lattice Hopf algebra Hρ:= Sym(spanC{uρ1,ρ2}ρ1,ρ2), with coproduct ∆ρ(uρ1,ρ2) = uρ1,ρ2⊗1+1⊗uρ1,ρ2, so each generator uρ1,ρ2is primitive. (Hopf primitives vs. group-likes in the sense of [3, 8, 9]). Evaluation is via the exponential character χu(uρ1,ρ2) := eiu (tρ2−tρ1), measuring the frequency difference tρ2−tρ1. 2 Fix an even, vertically rapidly decaying weight H, and define the (signed) zero measure µH:= X ρ H(ρ)δtρ,˜µH(A) := µH(−A) (the reflection of µH about the origin). Evenness of H ensures µH and ˜µH have the same total mass. For an even Schwartz function ϕwith Fourier transform b ϕ, define the two–point projector Pϕ(ω) := ZRb ϕ(u)|AH(u)|2e−iωu du = 2πϕ∗(µH∗˜µH)(ω), where AH ( u ) := PρH ( ρ ) eiutρ . The convolution form shows that Pϕ is supported on the full difference set {tρ2−tρ1}, with amplitudes µH∗˜µH=X ρ1,ρ2 H(ρ1)H(ρ2)δtρ1−tρ2. Definition 1.4 (Primitive zero–lattice crystal).With µH := PρH ( ρ ) δtρ and ˜µH ( A ) := µH ( −A ), define CZero H(ω) := µH∗˜µH=X ρ1,ρ2 H(ρ1)H(ρ2)δω−(tρ2−tρ1)∈ S′(Rω). for the (weighted) zero–difference distribution. Then for any even Schwartz ϕ with Fourier transform b ϕ, Pϕ(ω) = 2πϕ∗ CZero H(ω). Thus CZero His the raw (unsmoothed) zero–difference comb, while Pϕis its ϕ–smoothed probe. Remark (RH not assumed).The support of CZero H lies on the real axis {ω = tρ2−tρ1} irrespective of ℜρj ; horizontal displacements affect only weights and vertical envelopes in later σ –dynamics, not the fact that differences are real. Interpretation. This is the zero–lattice counterpart of the Euler–Hopf crystal: •On the Euler side, spikes occur at ω=klog pwith log pweights. •On the zero–lattice side, spikes occur at ω=tρ2−tρ1with weights H(ρ1)H(ρ2). Both are primitive in their respective Hopf structures and serve as the natural bases for the transfer principle. Definition 1.5. Prime–band (level–0) projection. Define the prime–band projector Π0:S′(R)−→ S′(R) as the distributional projection that retains only the singularities at the prime comb {klog p : k∈Z≥1, p prime } and annihilates all other singular support. Concretely, for any T∈ S′ ( R ) and any test ϕwhose Fourier support isolates a single klog p, ⟨Π0T, ϕ⟩=⟨T, ϕ⟩,while if sing supp ϕ∩ {klog p}=∅,then ⟨Π0T, ϕ⟩= 0. Equivalently, sing supp(Π0T)⊆ {klog p}and Π0is the identity on atoms at klog p. Zero–difference closure (completions). Call a distribution Z∈ S′ ( R )acompletion (or closure) of the zero–difference data if it agrees with CZero Hon the prime band, i.e. Π0Z= Π0CZero H. 3 The set of all completions is the affine space CℓZ:= {Z∈ S′(R):Π0Z= Π0CZero H}=CZero H+ ker Π0, where ker Π 0 = {T∈ S′ ( R ) : Π 0T = 0 } is the invisible sector (“off–band” with respect to primes). Definition 1.6 (Prime–band projector Π 0 ).Let Ω := {klog p : k∈Z≥1, p prime } , a closed discrete subset of R . Choose pairwise disjoint open intervals Uω around each ω∈ Ω, and pick χ∈C∞ c ( R )such that χ≡ 1on Sω∈ΩUω and supp χ⊂V , where V is a slightly larger open union of those intervals. Define the linear, continuous operator Π0:S′(R)→ S′(R),⟨Π0T, φ⟩:= ⟨T, χ φ⟩, φ ∈ S(R). Lemma 1.7 (Basic properties).Π 0 is idempotent (Π 2 0 = Π 0 ), sing supp (Π 0T ) ⊆ Ω, and for every ω0∈ Ωone has Π 0δω0 = δω0 . If T is supported away from Ω(i.e. sing supp T∩ Ω = ∅ ), then Π 0T = 0. Moreover, if ˜χ is any other cutoff with ˜χ≡ 1on SUω , the resulting projector ˜ Π0 satisfies ˜ Π0T− Π 0T∈ker Π 0 for all T ; hence the prime–band part Π 0T is independent of the cutoff modulo the invisible sector ker Π0. Sketch. Multiplication by a fixed C∞ function defines a continuous endomorphism of S′ ( R ). Since χ2 = χ , idempotence follows. The multiplication does not create new singularities and vanishes off V , so sing supp (Π 0T ) ⊆sing supp T∩V⊆ Ω. For δω0 with ω0∈ Ω, χ ( ω0 ) = 1 gives Π 0δω0 = δω0 . If sing supp T avoids Ω, choose χ supported in a neighborhood of Ωdisjoint from sing supp T ; then χT = 0. For two cutoffs χ, ˜χ , the difference ( ˜χ−χ ) T is supported away from Ω, hence in ker Π0. Primes Distributional Equivalence Theorem 1.8 (Explicit–formula crystal identity with smooth part; explicit–formula form in the sense of Weil [ 4 ]; see also [ 6 , 5 ] ).Let σ0> 0, H be even and rapidly decaying on vertical lines, and ϕeven Schwartz. Then, distributionally in ω, Pϕ(ω)=2πX pX k≥1 (klog p)p−kσ0b ϕ(klog p)WH(klog p;σ0)δklog p+ Smooth(ϕ, H;σ0), where WH ( · ; σ0 )is a smooth envelope coming from the vertical weights and the archimedean factor of −ζ′/ζ, and Smooth(ϕ, H;σ0) = ϕ(0)X ρ Hρ+σ0 2Hρ−σ0 2+1 2πZRb ϕ(ξ)W(∞) H(ξ;σ0)dξ, with the archimedean kernel given by the symmetric integral W(∞) H(ξ;σ0) := ZRH1 2+σ0 2+iu+ξ 2H1 2−σ0 2+iu−ξ 2r(u)du, where the function r(u)is r(u) = ℜ "−log π+1 2ψ1 2+σ0 2+iu 2+1 2ψ1 2−σ0 2+iu 2#, ψ =Γ′ Γ. Here: 4 •Symmetry: W(∞) H(ξ;σ0) = W(∞) H(−ξ;σ0)follows from the evenness of Hand r. • Smoothness: if H is rapidly decaying on vertical lines, then W(∞) H is C∞ in both variables (ξ, σ0)and decays rapidly in ξ. Remark (Smooth part is atom-free and invisible to spike extraction).In Theorem 1.8, the term Smooth ( ϕ, H ; σ0 )is independent of ω (as written) and, more generally under admissible deformations of H and ϕ , remains a C∞ function of ω ; in particular sing suppωSmooth = ∅ . Consequently it creates no Dirac masses at ω = klog p and cannot alter the coefficients of any existing spikes. Operationally, any spike–extraction procedure that localizes at a comb point (e.g. residues of the σ –Laplace resolvent at ℓ = −klog p , or pairings with test functions shrinking around ω = klog p ) ignores Smooth . In the rigorous model of Π 0 as multiplication by a cutoff χ equal to 1near the prime comb, χ·Smooth may persist as a smooth background, but it never generates atoms; all δω−klog pcontributions come solely from the prime–power part of the formula. In particular, for any admissible H, ϕ, sing suppωPϕ={klog p:k∈Z≥1, p prime }(primitive Euler support), and for each fixed n, sing suppωCEuler # n(σ0)={klog p:pk∥n}⊂{klog p}. Proof (direct from the Weil explicit formula [4]; cf. [6, 5]). Write the nontrivial zeros as ρ = βρ + itρ and fix σ0> 0. Let H be even and rapidly decaying on vertical lines, and let ϕ be even Schwartz with Fourier transform b ϕ. Set µH:= X ρ H(ρ)δtρ,˜µH(A) := µH(−A),AH(u) := X ρ H(ρ)eiutρ. Then |AH(u)|2=X ρ1,ρ2 H(ρ1)H(ρ2)eiu(tρ2−tρ1)⇐⇒ µH∗˜µH=X ρ1,ρ2 H(ρ1)H(ρ2)δtρ2−tρ1. Consequently Pϕ(ω) = ZRb ϕ(u)|AH(u)|2e−iωu du (integral form) = 2πϕ∗(µH∗˜µH)(ω)(convolution form) = 2πX ρ1,ρ2 H(ρ1)H(ρ2)ϕ ω−(tρ2−tρ1)(expanded sum). (∗) Step 1: Centered explicit formula with vertical shift. Consider the centered completed product η(s, σ) := ξs+σ 2ξs−σ 2, s =1 2+iu. Apply the (weighted) Weil explicit formula to the logarithmic derivative of each factor, with test function in the u –variable given by the even Schwartz kernel u7→ b ϕ ( u ) e−iωu , and with vertical weights H ( 1 2±σ0 2 + iu ). Summing the two applications (the + σ0/ 2and −σ0/ 2parts) yields a centered explicit formula in which the zero sum becomes precisely the left side of (∗), i.e. ZRb ϕ(u)|AH(u)|2e−iωu du =(prime–power side) +(archimedean + diagonal zero terms). 5 Step 2: Prime–power contributions. The prime side of the Weil formula comes from −ζ′/ζ , hence produces the series X pX k≥1 (log p)p−k(1 2+σ0 2+iu)and X pX k≥1 (log p)p−k(1 2−σ0 2+iu). Multiplying (i.e. centering) and integrating against b ϕ ( u ) e−iωu collapses the u –dependence by the identity ZRb ϕ(u)e−iu(ω−klog p)du = 2π ϕ ω−klog p, which (in the distributional limit where ϕ localizes to a point) yields Dirac atoms δω−klog p . The vertical weights contribute a smooth envelope WH ( klog p ; σ0 )depending on H and on the symmetric placement σ=σ0of the two factors. Collecting terms gives the discrete comb 2πX pX k≥1 (klog p)p−kσ0b ϕ(klog p)WH(klog p;σ0)δω−klog p. Step 3: Smooth remainder (archimedean + diagonal). The archimedean part of the Weil formula (gamma factor) and the diagonal zero term produce no point masses in ω ; they enter as a smooth function of ω. After the same weighting by Hand test ϕ, these assemble into Smooth(ϕ, H;σ0) = ϕ(0)X ρ Hρ+σ0 2Hρ−σ0 2+1 2πZRb ϕ(ξ)W(∞) H(ξ;σ0)dξ, with W(∞) H(ξ;σ0) = ZRH1 2+σ0 2+i(u+ξ 2)H1 2−σ0 2+i(u−ξ 2)r(u)du, where r ( u )is the standard archimedean kernel from the explicit formula, (a standard archimedean kernel in the explicit formula [5, §5.15–§5.18][6, Ch. 2]) r(u) = ℜ"−log π+1 2ψ 1 2+σ0 2+iu 2!+1 2ψ 1 2−σ0 2+iu 2!#. Conclusion (singular support). Combining Steps 2–3 with ( ∗ )yields the claimed identity. Since the smooth terms have no point masses, the singular support of Pϕ in ω consists exactly of the prime comb {klog p} with weights as stated. For any fixed integer n , applying the same transform to the finite sum defining CEuler # n ( σ0 )produces atoms only at {klog p : pk∥n} , proving the last assertion. Definition 1.9 (Pern zero-side crystal on the prime band).Define CZero n,σ0,ϕ,H as Pϕ restricted to the finite set {ω = klog p : pk∥n} (i.e. keep only those atoms of Pϕ whose locations match the prime powers dividing n). Proposition 1.10 (Prime-band isomorphism).Fix σ0>0and admissible H, ϕ. Assume b ϕ(klog p)WH(klog p;σ0)= 0 for every klog p∈sing suppωCEuler # n(σ0). Define the diagonal multiplier on the prime band by Mϕ,H,σ0:δω−klog p7−→ b ϕ(klog p)WH(klog p;σ0)δω−klog p, extended linearly to finite (or tempered) prime-combs. Then, modulo ker Π0, CZero n,σ0,ϕ,H =Mϕ,H,σ0CEuler # n(σ0). In particular, the restriction of Mϕ,H,σ0 to the prime band is an automorphism; hence the Euler and zero crystals are canonically equivalent on the prime band (gauge-equivalent via the above diagonal multiplier), modulo ker Π0. 6 Sketch. By Theorem 1.8, both sides have the same singular support {klog p} , and the zero-side coefficients differ from the Euler-side ones by the scalar factor b ϕ ( klog p ) WH ( klog p ; σ0 ). The stated Mϕ,H,σ0 applies exactly these scalars at each atom; nonvanishing makes it invertible on the prime band, yielding the identity modulo ker Π0. Explicit Prime Projector on the Euler–Hopf Crystal Fix σ0>0and n≥1. Using K(s, σ) := ζ(s)−ζ′(s+σ) ζ(s+σ)=X m≥1 Dm(σ) ms(ℜs > 1,ℜ(s+σ)>1), the vertical coefficient–extraction identity gives Dn(σ) = nclim T→∞ 1 2TZT −T ζ(c+it)−ζ′(c+it +σ) ζ(c+it +σ)eit log ndt, c > 1, c +ℜσ > 1. a standard orthogonality/Dirichlet-coefficients device [1, §I.3][2, §1.1]. Set σ=σ0+it. Then Dn(σ0+it) = X pj∥n (log p)p−jσ0e−it j log p. Taking the (distributional) vertical Fourier transform in t, d Dn(ω) := ZRDn(σ0+it)eiωt dt = 2πX pj∥n (log p)p−jσ0δω−jlog p= 2πCEuler n(σ0), which is exactly the (primitive) Euler–Hopf crystal with logarithmic weights. Laplace–peeling on the Euler side (primitive/uncharged). Fix σ0>0and set e Dn(t) := Dn(σ0+it) = X pj∥n (log p)p−jσ0e−it j log p. Its one–sided Laplace transform (for ℜs > 0) is a finite sum of simple fractions L+{e Dn}(s) = Z∞ 0 e−st e Dn(t)dt =X pj∥n (log p)p−jσ0 s+i j log p,Res s=−i j log p= (log p)p−jσ0. Smallest prime. Let α1:= min{β > 0 : s=−iβ is a pole of L+{e Dn}} and set log p1=α1, p1=eα1,Res s=−ilog p1 = (log p1)p−σ0 1. Exponent of p1 without prior knowledge. Cancel all harmonics of p1 by multiplying successively by the linear factors A1(s) := Y m≥1s+i m log p1L+{e Dn}(s), stopping when no factor at s = −i m log p1 remains. The number of cancelled factors equals ep1 . Iterate. Set A(0) ( s ) = L+{e Dn} ( s )and, after extracting p1 and ep1 , let A(1) ( s ) = A1 ( s ). Then repeat: •αj:= min{β > 0 : s=−iβ is a pole of A(j−1)},pj:= eαj. • Cancel all poles at s = −i m log pj (for m = 1 , 2 , . . . ) to obtain A(j) ( s )and record epj as the number of cancellations performed. Because only prime-power poles occur, this process terminates after finitely many steps and recovers the full factorization n=Qpepj j. 7 Explicit Prime Projector on the Zero–Hopf Crystal Let {ρ = βρ + itρ} denote the nontrivial zeros and set CZero H := µH∗˜µH . Throughout we work in the quotient space S′ ( R ) /ker Π 0 (equivalently, we read all identities modulo ker Π 0 ), so only the prime–band component Π 0CZero H is relevant. We work with the centered completed zeta product Gξ(s, σ) := ξs+σ 2ξs−σ 2, ξ(s) = 1 2s(s−1) π−s/2Γ s 2ζ(s) = eA+Bs Y ρ1−s ρes/ρ. (Hadamard product; cf. [6, Ch. 2].) Set s=1 2+iu and fix an even vertical weight H; define the smoothed zero–sum AH(u) := X ρ H(ρ)eiutρ, µH:= X ρ H(ρ)δtρ,˜µH(A) := µH(−A). For any even Schwartz cutoff ϕwith Fourier transform b ϕ, define the two–point projector Pϕ(ω) := ZRb ϕ(u)|AH(u)|2e−iωu du = 2πϕ∗(µH∗˜µH)(ω) = 2πX ρ1,ρ2 H(ρ1)H(ρ2)ϕ ω−(tρ2−tρ1). Thus the primitive frequencies on the zero side are precisely the differences tρ2−tρ1 , paralleling the role of log pon the Euler–Hopf side. To isolate the prime–power spikes, fix σ0> 0, regard σ = σ0 + it as the vertical variable, and take the σ–Fourier transform of the (weighted) centered kernel: CZero n,σ0,ϕ,H(ω) := ZRb ϕ(t)"lim U→∞ 1 2UZU −U log Gξ1 2+iu, σ0+itdu#e−iωt dt, then convolve with Pϕ. By Theorem 1.8, the singular support of CZero n,σ0,ϕ,H is the comb sing suppωCZero n,σ0,ϕ,H ={klog p:pep∥n, 1≤k≤ep}, with calibrated amplitudes (log p)p−kσ0b ϕ(klog p)WH(klog p;σ0), up to the smooth remainder Smooth(ϕ, H;σ0)(no discrete ω–support). Slope–peeling on the zero side. Let Ap,k ( σ )denote the coefficient of δ ( ω−klog p )in CZero n,σ,ϕ,H . Then, for each prime pand k≥1, Ap,k(σ) = (log p)p−kσ b ϕ(klog p)WH(klog p;σ), so in particular (assuming b ϕ(log p)= 0 and ∂σlog WH(log p;σ) = o(1) as σ→ ∞), log p1= lim σ→∞−∂σlog Ap1,1(σ). Multiplicity is read off by counting present harmonics: ep= max{k≥1 : Ap,k(σ)≡ 0}(equivalently, spikes at klog pappear up to k=ep). (Choose ϕ so that b ϕ ( klog p )  = 0 for the relevant k to avoid smoothing them away.) With p1 and e1 obtained this way, subtract the contribution of CZero pe1 1,σ,ϕ,H —i.e., remove the entire block {klog p1: 1 ≤k≤e1}—and iterate to reconstruct npurely from zero–lattice data. 8 Corollary 1.11 (Prime–Factorization Crystal Resonance).The slope-peeling limits on either Hopf side yield a theoretical, explicit factorization algorithm purely in terms of Hopf–algebraic data. Proof. By Theorem 1.8, for each fixed σ > 0the singular support of both CEuler# n ( σ )and CZero n,σ,ϕ,H is the discrete comb {klog p:pep∥n, 1≤k≤ep}. Write Ap,k(σ)for the coefficient of δ(ω−klog p)on either side. Then Ap,k(σ) = (log p)p−kσ b ϕ(klog p)WH(klog p;σ), so (assuming b ϕ(klog p)= 0 and that ∂σlog WH(klog p;σ) = o(1) as σ→ ∞) log p1= lim σ→∞−∂σlog Ap1,1(σ), recovering the smallest prime p1dividing n. Multiplicity is determined by harmonic count: sing suppωC contains klog p1 exactly for 1 ≤ k≤e1, and omits it for k > e1, hence e1= max{k≥1 : klog p1∈sing suppωC }. Removing the block {klog p1 : 1 ≤k≤e1} from the comb and iterating yields all remaining primes and exponents. The same argument applies on either Hopf side, completing the proof. Explicit Zero Projector on the Euler–Hopf Crystal We use only K(s, σ) := ζ(s)−ζ′(s+σ) ζ(s+σ),ℜs > 1,ℜ(s+σ)>1, viewed as a function of σ with s fixed. All transforms below are in the tempered–distribution sense; meromorphic continuation justifies moving beyond the domain of absolute convergence when needed. Zero part in the σ –variable. Fix s = c + iu with c > 1. Remove the known pole at 1and set Es(σ) := −ζ′(s+σ) ζ(s+σ)−1 s+σ−1=X ρ m(ρ) s+σ−ρ, where m(ρ)∈Z≥1is the multiplicity of the zero ρof ζ. Thus K(s, σ) = ζ(s)Es(σ) + 1 s+σ−1, so the σ–poles (and their multiplicities) of Kcoincide with those of Es. Trivial–zero cleaning To avoid a large contribution at the ordinate t = 0 coming from the trivial zeros of ζ, we replace Esby Ent s(σ) := Es(σ)−X m≥1 1 s+σ+ 2m so that only the nontrivial zeros contribute poles in σ. Write σ=σ0+it with σ0>0, and define e Es,σ0(t) := Ent s(σ0+it). 9 obtained from the renormalized atoms e A ( a ) = a on the prime band. Define the canonical function extracted from the lattice by ζcan(s) := exp Z(0,∞) e−sa adµlog(a)!,ℜs > 1, ζcan(s)−−−−−→ ℜs→+∞1. Then log ζcan(s) = X pX m≥1 1 mp−ms,−ζ′ can ζcan (s) = X pX m≥1 (log p)p−ms for ℜs > 1, hence ζcan ( s ) = ζ ( s )there; by analytic continuation and the chosen normalization, ζcan coincides with the Riemann zeta function on C\ { 1 } . In particular, ζ is already built into the lattice: our extraction adds no new arithmetic information and makes only the canonical normalization choice at +∞. Remark (Assumptions audit: the single choice we make).Beyond the zero–difference data and the centered explicit–formula identification of prime–band atoms, the only additional input is the decision to integrate the recovered logarithmic derivative to a function: we set −ζ′ can ζcan (s) = Z(0,∞) e−sa dµlog(a),ℜs > 1, and fix the multiplicative constant by the normalization ζcan(s)→1as ℜs→+∞. Concretely, this means: 1. Normalization/limit exists. The Laplace integral converges absolutely for ℜs > 1, so log ζcan(s) = Re−sa adµlog(a)→0as ℜs→+∞, hence ζcan(s)→1. 2. The extracted shape is a logarithmic derivative. We interpret the recovered measure dµlog as the logarithmic derivative of a function (and not some other nonlinear functional of dµlog). 3. Unit scale k= 1 is forced.Replacing sby ks gives −F′(s) F(s)=Ze−ks a dµlog(a) = −ζ′(ks) ζ(ks), so F ( s ) = ζ ( ks ) 1/k after normalization at + ∞ . This moves each weight at a = log p to p−ks and breaks the calibrated identity e A ( a ) = a . Equivalently, the linear scale is fixed by d ds p−ss=0 =−log p, which enforces k= 1. This is not extra arithmetic content; it is merely selecting a primitive of a given logarithmic derivative with the canonical normalization at + ∞ . All dependence on the probes ( H, ϕ, σ0 ) cancels in e A(a) = a, so µlog and hence ζcan are probe–independent. No RH is used. Interpretation. In this sense, ζ is merely the unit–scale lens through which the lattice is read: the reconstruction is lossless (no information discarded) and faithful (no information added). Choosing k = 1 amounts either to rescaling the logarithm (replacing log by klog ) or—if one insists on the usual log —to redefining the “primes” as pα with α = k = 1, which no longer matches the integer prime alphabet. Thus ζ exhausts the multiplicative content of the lattice on the prime band. What remains undetermined is only the off–band “invisible sector”: the lattice lives in S′ ( R ) /ker Π 0 , and any completion differing by an element of ker Π 0 leaves all Euler–side (prime–band) observables unchanged. 16 Proposition 1.14 (Calibration selects α = 1; multiplicative cover of N is independent of RH).With the zero–difference setup and prime–band extraction as above, choose the logarithmic–derivative shape and unit scale k= 1: −ζ′ can ζcan (s) = Z(0,∞) e−sa dµlog(a), ζcan(s)−−−−−→ ℜs→+∞1. Then: 1. (Multiplicative cover at α = 1) The renormalized atoms satisfy e A ( mlog p ) = mlog p , hence log ζcan(s) = X pX m≥1 1 mp−ms,−ζ′ can ζcan (s) = X pX m≥1 (log p)p−ms, so the spectral letters are precisely {pm} ( α = 1), giving the usual multiplicative cover of N . 2. (What k = 1 means) If one replaces s by ks then −F′/F ( s ) = Re−ksa dµlog ( a )and, after the same normalization, F ( s ) = ζ ( ks ) 1/k . Equivalently, the letters become { ( pα ) m} with α = k . For α∈Z>1 this covers only the α th-power subsemigroup, and for non-integer α it is not a multiplicative cover of N at all. Thus the requirement “cover N ”forces α = k = 1. 3. (No implication for the critical line) This calibration does not constrain zero locations. In our framework, ℜρ=1 2is equivalent to level-0 flatness (no positive-level poles beyond the Euler lattice, as explained later) and is logically independent of the base-level multiplicative calibration that fixes α= 1. Remark (On centering and symmetry).The “shifting we choose” is 0(we center via the symmetric product ξ ( s±σ/ 2)), and “reflection” is 1(we respect the functional–equation symmetry). These fix the lens but do not move zeros onto the critical line; RH is the deeper, dynamical flatness property, not a byproduct of enforcing coverage over N. Commuting Diagram Zero side: Gξ(s,σ)=ξs+σ 2ξs−σ 2Euler side: K(s,σ)=ζ(s)−ζ′(s+σ) ζ(s+σ) zeros + centered product; choose H, ϕ; autocorr. Pϕ  y  yDirichlet coeff. projector Pn average in uwith H; set σ=σ0+it;t–FT in σ Dn(σ); set σ=σ0+it;t–FT in σ,−∂σ0 CZero n,σ0,ϕ,H sing suppΩ =========⇒ CEuler# n(σ0) Legend. Dirichlet coefficient projector Pn A(·, σ):= nclim T→∞ 1 2TZT −T A(c+it, σ)eit log ndt, c > 1, c +ℜσ > 1. Vertical Fourier transform b fσ0(ω) := ZRf(σ0+it)eiωt dt. Zero autocorrelation projector AH(u) := X ρ H(ρ)eiutρ, µH:= X ρ H(ρ)δtρ,˜µH:= X ρ H(ρ)δ−tρ, Pϕ(ω) := ZRb ϕ(u)|AH(u)|2e−iωu du = 2πϕ∗(µH∗˜µH)(ω). Prime band and projector Ω := {klog p:k∈Z≥1, p prime } ⊂ log N>1, Π0:S′(R)→ S′(R)(prime-band projector; identity on atoms in Ω). Prime-band singular support sing suppΩ(T) := sing supp(Π0T)⊂Ω (isomorphic projection on Ω,mod ker Π0). 17 This “transfer” picture echoes trace–formula viewpoints relating primes and spectral data (cf. Connes [13]). Proposition 1.15 (LCM–completion realizes the full prime comb on the prime band).Fix σ0>1. Then, as N→ ∞, DLN(σ)−→ X pX k≥1 (log p)p−kσ =X n≥2 Λ(n) nσ=−ζ′(σ) ζ(σ)(ℜσ > 1), with locally uniform convergence on compact subsets of {ℜσ > 1 } (by absolute convergence and the Weierstrass M–test). Consequently, at the vertical slice σ=σ0+it, d DLN(ω)−−−−→ weak-* 2πX pX k≥1 (log p)p−kσ0δω−klog p= 2πCEuler ∞(σ0) (σ0>1), and (by differentiation in σ0) the same holds for the charged crystal: −∂σ0d DLN(ω)−−−−→ weak-* 2πX pX k≥1 (klog p)p−kσ0δω−klog p= 2πCEuler # ∞(σ0) (σ0>1). Moreover, by Proposition 1.10, the zero–side prime–band component coincides with this limit up to the fixed diagonal multiplier: CZero n,σ0,ϕ,H =Mϕ,H,σ0CEuler # n(σ0) =⇒ CZero ∞,σ0,ϕ,H =Mϕ,H,σ0CEuler # ∞(σ0). Thus the “image on the zero–crystal side” is complete on the prime band in the N→ ∞ LCM–limit. Sketch. For each prime p,vp(LN) = ⌊logpN⌋, so DLN(σ) = X pX k≤logpN (log p)p−kσ. For ℜσ > 1the double series Pp,k≥1 ( log p ) p−kσ converges absolutely; hence the M–test yields locally uniform convergence of DLN ( σ )to −ζ′ ( σ ) /ζ ( σ )on compacta. (For real σ > 1the convergence is also pointwise monotone increasing.) Taking the vertical Fourier transform at σ0 + it sends each term p−k(σ0+it) to a Dirac mass at ω = klog p with weight ( log p ) p−kσ0 , giving weak-* convergence of finite Borel measures on R (and hence convergence in S′ ( R )). Differentiation in σ0is justified termwise on compacta, yielding the charged limit. Remark (Level–0 vs. infinite closure; Abel forcing).The weak-* limit above produces a finite positive measure for σ0> 1; in particular it is tempered. For 0 < σ0≤ 1, the total mass is infinite, so one may either (i) work modulo ker Π 0 with compactly supported prime–band tests, for which local finiteness suffices, or (ii) enforce convergence by Abel regularization: µ(ε) σ0:= X p,k≥1 (log p)p−k(σ0+ε)δω−klog p=⇒µσ0:= lim ε↓0µ(ε) σ0 in the sense of distributions on the prime band. All prime–band statements (and the isomorphism with the zero side) remain valid under this Abel limit. 18 Example: n= 12 (Primitive Form) For n= 12 = 22·31and σ0= 1, the charged Euler–Hopf crystal (from Definition 1.2) is CEuler# 12 (1) = (log 2)1 2δω−log 2 +1 4δω−2 log 2+ (log 3)1 3δω−log 3. This contains only the prime–power locations {log 2 , 2 log 2 ,log 3 } , each weighted by ( log p ) p−kσ0 . Prime power pk∥12 Frequency ω=klog pAmplitude (log p)p−k 21log 2 (log 2)/2 222 log 2 (log 2)/4 31log 3 (log 3)/3 By contrast, the full divisor comb X d|12 d−1δω−log d includes all divisor frequencies { 0 ,log 2 ,log 3 , 2 log 2 ,log 6 ,log 12 } and arises by convolving the primitive prime blocks of 2and 3. Only the primitive crystal isolates the pure prime–power spectrum, which is the singular support appearing on the Euler side of the explicit formula. (For reference, the atomic prime blocks are 2with e2 = 2: locations { 0 ,log 2 , 2 log 2 } with weights {1,1/2,1/4}, and 3with e3= 1: locations {0,log 3}with weights {1,1/3}) The zero–lattice Hopf crystal (with admissible ϕ, H ) yields the same primitive support {klog p : pep∥n, 1 ≤k≤ep} . It does not include composite frequencies such as log 6or log 12; those appear only in the Euler divisor crystal Pd|nd−σ0δ(ω−log d). Figure 1: Euler vs Zero crystal for n= 12 Why Hopf? Structural Payoffs and Level–0 Flatness Why Hopf. In a Hopf algebra ( H, m, ∆ , S, ε ),group–like elements g satisfy ∆ g = g⊗g and primitives x satisfy ∆ x = x⊗ 1+1 ⊗x . On the Euler side, the primitives are the prime powers p−ks and ζ ( s )is group–like; on the zero side (after centering), the primitives are the zero–differences tρ2−tρ1 , while ξ ( s±σ/ 2) plays the group–like role. The tensor Hopf Hs⊗Hσ separates primes and exponents; the zero–Hopf compresses them into a single difference coordinate. The explicit formula then matches their singular supports functorially (exp/log interchanges group–likes and primitives [3, 8, 9]). 19 Lemma 1.16 (Coefficient characters are automatic).Let HE be the Dirichlet–convolution Hopf algebra with coproduct ∆ n−s = Pd|nd−s⊗ ( n/d ) −s . For any Dirichlet series F ( s ) = Pna ( n ) n−s , the coefficient functionals P: n [ F ] = a ( n )satisfy P: mn =P: m∗ P: n (convolution on Hom ( HE,C )), so multiplicativity at coefficients is structural—no re-derivation is needed per kernel. RH as Level–0 Flatness (Laplace–Hopf View) Euler side (resolvent at level 0). Let HEbe the Dirichlet–convolution Hopf algebra with basis en and ασ ( en ) := n−σen ( σ≥ 0). Its generator is the Hopf derivation δ := d dσ σ=0 ασ , so δ(en) = −(log n)en. Define the Laplace–Hopf resolvent LE(ℓ) := Z∞ 0 e−ℓσ ασdσ = (ℓ−δ)−1,(ℜℓ > 0), hence LE(ℓ)epk=1 ℓ+klog pepkand LE(0+)epk=1 klog pepk(level 0). The appearance of exponential tilts from off–line zeros aligns with the de Bruijn–Newman paradigm [10, 11, 12]. Zero side (centered kernel, same eigenvalues). For the centered kernel Gξ ( s, σ ) = ξ ( s + σ/ 2) ξ ( s−σ/ 2), after applying the projector Pϕ and subtracting the explicit smooth (archimedean + diagonal) terms, the comb part decays like e−σ k log p in the σ –direction. Its Laplace–Hopf resolvent for the n–projected crystal is LZ,n(ℓ, ω) = X pep∥n ep X k=1 (log p)b ϕ(klog p)f WH(klog p;ℓ) ℓ+klog pδω−klog p+(no ω–spikes from the smooth part) with f WH(·;ℓ)the σ–Laplace transform of the vertical envelope. Lemma 1.17 (Matching level–0 poles under RH).Assume RH. Then, for each prime p|n and 1≤k≤ep, both resolvents LE(ℓ)=(ℓ−δ)−1and LZ,n(ℓ, ·) have a simple pole at ℓ = −klog p , and there are no other poles with ℜℓ > 0. Moreover, provided b ϕ(klog p)f WH(klog p;−klog p)= 0, Res ℓ=−klog pLE(ℓ)epk=epk, Res ℓ=−klog pLZ,n(ℓ, ω) = (log p)b ϕ(klog p)f WH(klog p;−klog p)δω−klog p, so the pole locations coincide, and the residues correspond under the level–0 transfer (up to the scalar envelope factor). Proof (sketch). On the Euler side, δ ( epk ) = − ( klog p ) epk , so ( ℓ−δ ) −1epk = 1 ℓ+klog pepk has a simple pole at ℓ=−klog p. On the zero side, after removing the explicit smooth terms and applying Pϕ , the comb contributes terms of the form (log p)b ϕ(klog p)f WH(klog p;ℓ) ℓ+klog pδω−klog p, hence the same pole locations; provided b ϕ ( klog p ) f WH ( klog p ; −klog p )  = 0, the residues are nonzero. Under RH there are no off–line zeros ρ = β + iγ with β = 1 2 , so no factors e±(β−1 2)σ appear in the centered kernel; consequently the Laplace transform has no additional poles in the open right half–plane (the potential extra poles would be at ℓ=±(β−1 2)). 20 Corollary 1.18 (Level–0 resolvent recovers the primitive crystal under RH).Assume RH. Then, as ℓ→0+, LE(ℓ)=(ℓ−δ)−1and LZ,n(ℓ, ·) recover the same prime–power support: {ω=klog p:pep∥n, 1≤k≤ep}. More precisely, LE(ℓ)epk−−−→ ℓ→0+ 1 klog pepk, and LZ,n(ℓ, ω)−−−→ ℓ→0+X pep∥n ep X k=1 (log p)b ϕ(klog p)f WH(klog p; 0) klog pδω−klog p, so (for admissible H, ϕ with b ϕ ( klog p ) f WH ( klog p ; 0)  = 0) the spikes at klog p are present exactly for 1 ≤k≤ep . Hence the prime–power factorization data ( p, ep )are read off on both sides from level 0. Proof (sketch). By Lemma 1.17 and RH, the only poles with ℜℓ > 0occur at ℓ = −klog p . Evaluating at ℓ→ 0 + gives the stated limits on each side; the zero-side smooth terms contribute no ω–atoms. Nonvanishing of b ϕ(klog p)f WH(klog p; 0) ensures the spikes are detected. We now phrase level–0 flatness as the exact RH obstruction. Proposition 1.19 (RH ⇔ level–0 flatness of the zero–side resolvent).After removing the explicit smooth terms, the meromorphic continuation in ℓ of LZ,n ( ℓ, · )has no poles with ℜℓ > 0beyond the Euler-type poles {−klog p} if and only if all nontrivial zeros satisfy ℜρ = 1 2 . Equivalently, any off–line zero ρ = β + iγ with β = 1 2 produces additional poles at ℓ = ± ( β−1 2 )in the zero–side resolvent, obstructing evaluation at level 0. Sketch. Write the nontrivial zeros as ρ = 1 2 + βρ−1 2 + itρ = 1 2 + δρ + itρ and consider the centered product η ( s, σ ) = ξ ( s + σ/ 2) ξ ( s−σ/ 2) with s = 1 2 + iu . After removing the known smooth (archimedean + diagonal) terms, the σ –dependence of the prime part is e−(klog p)σ , so its σ–Laplace transform has poles only at ℓ=−klog p < 0. Now look at the zero contribution. The logarithmic derivative of η contributes, for each zero ρ , terms of the form 1 s+σ/2−ρ+1 s−σ/2−(1 −ρ)and conjugates. Setting s = 1 2 + iu and averaging in u against an even weight H , the u –dependence cancels, and one obtains a σ –factor e+δρσ + e−δρσ (see the vertical–Fourier computation preceding Eq. (energy/tilt) in the text). Hence the σ–Laplace transform contributes 1 ℓ−δρ +1 ℓ+δρ , i.e. simple poles at ℓ=±δρ. If RH holds, then every δρ = 0, so the zero part contributes only a constant in σ (which is precisely part of the subtracted smooth term), and there are no poles with ℜℓ > 0beyond the Euler-type set {−klog p} . Conversely, if some zero is off the critical line with δρ> 0, then a pole appears at ℓ = δρ> 0in the zero–side resolvent, obstructing level 0evaluation. This proves the stated equivalence. 21 Reading: Euler’s crystal is intrinsically “level–0.” RH asserts the zero–lattice crystal is, after known smooth subtractions, equally level–0—no spectral mass at higher levels—so factorization is fully captured on the base level in both Hopf pictures. In that sense, factorization remains faithful on both sides, there would be no additional work beyond "level-0" under RH. A sharp invariant. Define the flatness index FZ:= sup nℜℓ:ℓis a pole of LZ,n(ℓ, ·)not of Euler type (−klog p),ℜℓ > 0o. Then RH ⇐⇒ FZ= 0. Quantitative knobs. “Almost flat” ( FZ≤η ) is equivalent to a zero-free strip ℜρ−1 2≤η . Conversely, any off–line zero with β−1 2=δ > 0forces a pole at ℓ=δ. Interpretation and caution. The prime band is the only part of the zero–difference distribution that the Euler side can actually detect. To make this explicit, note: • The prime side probes only Π 0CZero H : anything in ker Π 0 lies in the “invisible sector” and is undetectable by prime–band tests (Euler combs, factorization functionals, etc.). Thus primes determine CZero Honly modulo ker Π0. • Our level–0 transfer and all peeling procedures operate entirely within Π 0CZero H and remain valid for every completion Z∈ CℓZ. •The Riemann Hypothesis is exactly the statement that, after the known smooth subtractions, no extra discrete spectrum lies outside the prime band: RH ⇐⇒ sing supp(Z)\ {klog p}has no discrete atoms for (the canonical) Z=CZero H. Equivalently, positive–level poles of the zero–side resolvent vanish: FZ= 0 (flatness). • Why we impose level–0 restrictions on the prime side. Because completions differ by elements of ker Π 0 , a naive “zero–side property” may not be visible to primes. By restricting to level–0 (prime–band local) properties, we ensure they depend only on Π 0CZero H and therefore do transfer faithfully. • Full closure: This is an additional object, which one can close by imposing some arbitrary property that may help reveal the projection more clearly, though in principle RH need not hold for such a closure. However, this delicate “dance” between unintended closure and projection may already be occurring, which is precisely why total rigor—and using factorization as the primary test—remains essential in our modeling. • Isomorphism and the “apparent” resolution of RH. For any admissible ( ϕ, H ) and after the standard smooth subtractions, one has, for each fixed σ0> 0, the static prime–band identity CZero ∞,σ0,ϕ,H =Mϕ,H,σ0CEuler # ∞(σ0)in S′(R)/ker Π0. This calibrates the comb at the slice σ = σ0 : it preserves the discrete support {klog p} and rescales amplitudes diagonally. Crucially, it is not a statement about the σ –dynamics. In particular, it neither detects nor rules out positive–level leakage in σ(additional Laplace poles with ℜℓ > 0caused by off–line zeros), because fixing σ0 and projecting with Π 0 cannot see exponential factors e±(ℜρ−1 2)σ . Hence the prime–band isomorphism, by itself, does not imply that the zero side “sits at level 0,” nor does it resolve RH; additional control in σis required. 22 Finally, the identity holds only modulo ker Π 0 : different completions Z∈ CℓZ agree on the prime band but may differ off–band, so properties depending on the invisible sector cannot be inferred from this isomorphism. If any argument hinges on off–band data of a completion—whether a particular choice or the completion space in general—the prime–band projector cannot see it. What it does supply is a sharp localization of the open issue: it pinpoints the obstruction to σ –flatness (positive–level leakage) and, independently of the eventual method of resolution, furnishes a unified dynamical testbed in which to probe it. A Theoretical testbed: Pushing prime properties through the Hopf transfer. We regard the Euler–side atomic measure µE as the carrier of prime–side properties P, and ask which such properties survive verbatim under the Hopf transfer to the zero–side crystal. Formally, a prime–side property Pis any statement about the prime–power lattice encoded by µE that is local on the frequency axis—that is, testable by finitely many linear functionals against compactly supported test functions in ω— and invariant under the level–0 weighting. Examples include: • Isolation (microlocal detectability). For each spike ω0 = klog p there exists an even Schwartz ϕ supported in a small neighborhood of ω0 such that ⟨µE, ϕ⟩  = 0 but ⟨µE, ϕ(· − η)⟩= 0 for all sufficiently small η= 0. (Support–only.) • Discrete difference constraints. For fixed h , properties of the translated difference DhµE := µE ( · + h ) −µE ( · )(e.g. exact vanishing on certain intervals). (Support + finite linear relations.) • Prime–power ratios (envelope–free). For each p and k≥ 1, ratios that cancel the archimedean envelope, e.g. Rp,k := (k+ 1) ⟨µE, ϕk+1⟩ k⟨µE, ϕk⟩, ϕjisolating jlog p, which satisfy Rp,k ≡1on the Euler side. (Support + envelope cancellation.) •Quadratic positivity on the lattice. Positivity of finite sums X (p,k),(q,ℓ) ap,k aq,ℓ ⟨µE, ψ(p,k);(q,ℓ)⟩, where each kernel ψ(p,k);(q,ℓ) is supported near ω = ( klog p ) − ( ℓlog q ). Such statements can be phrased as Gram–matrix positivity conditions over the prime–power lattice. (Support + finite bilinear relations.) We say Pis •level–0supp readable if it depends only on the support of µE(the set {klog p}), and • level–0 full readable if it depends on both the support and on which spikes carry nonzero mass at fixed σ0 (via envelope–free linear combinations such as those above that detect multiplicity or vanishing). 23 Transfer principle. For any prime–side property Pthat is level–0 full readable, the Euler–side statement P(µE) transfers verbatim to the zero side, P(µZ), because the transfer map T0 preserves the zero–lattice and annihilates all positive–level envelopes. Factorization and RH. The factorization functional Pfac(µ) := support(µ),which spikes are nonzero is level–0full readable on the Euler side. Hence: RH holds ⇐⇒ Pfac(µE)transfers to Pfac(µZ), i.e. the complete prime–power factorization of n (via support + maximal k ) is visible on the zero side if and only if the Riemann Hypothesis is true. Realistic outlook. This reframes RH as the single obstruction to transferring full factorization: one may test RH by checking whether P fac (or any other level–0 full property) remains invariant under T0 . Failure of invariance for any such Pwould disprove RH, while the absence of counterexamples across infinitely many structural tests sharpens our confidence in it. This does not preclude the existence of a level–0 full functional powerful enough to prove RH; rather, it shows that the Hopf–transfer testbed is valuable in its own right. Sensitivity of level–0 flatness to off–line zeros The level–0 resolvent is an unusually sensitive detector of off–line zeros. Any single nontrivial zero ρ = β + iγ with β = 1 2 injects an exponential factor e±(β−1 2)σ into the centered kernel, whose Laplace transform produces poles at ℓ=±(β−1 2). These poles are independent of γ and sit strictly to the right of the Euler-type lattice {−klog p} in the ℓ–plane whenever β > 1 2. Thus: • Immediate effect of a single offender. Even one zero with β = 0 . 500 1 would produce a pair of poles at ℓ = ± 10 −4 , instantly visible in principle, far separated from the nearest Euler pole (which for p = 2 is at ℓ≈ − 0 . 693). The detection threshold is not governed by height |γ|but solely by the horizontal deviation |β−1 2|. • Accumulated effect of many offenders. Infinitely many zeros with β > 1 / 2bounded away from 1 / 2by δ > 0would yield an infinite set of poles at ℓ = δ , rendering the level–0 evaluation meaningless. Even if δ decays with height, any supremum δmax > 0shows up as FZ=δmax in the flatness index. • Empirical meaning of high–up searches. The verification that all zeros with |γ| ≤ 10 12 lie on the line certifies that no pole with ℓ > 0has yet appeared from an off–line zero in that range. While this does not preclude tiny δ above 10 12 , it constrains any such deviation to be below the resolution of existing tests and ensures flatness up to FZ≲δbound. In short, the horizontal error of any off–line zero is mapped directly to the real part of a new resolvent pole, without dilution by height. This makes level–0 flatness a sharp and non–heuristic obstruction: it is as sensitive to β−1 2as one could hope for in an analytic test. 24 Energy per Zero via Parseval (pair model) Fix σ0>1 2and consider a symmetric pair of zeros with respect to the critical line, ρ±=1 2±δ+it, δ := β−1 2, t ∈R, so that δ = 0 corresponds to an on–line pair. The contribution of this pair to the logarithmic derivative of the completed zeta is Eδ,t(s) := 1 s−1 2+δ+it+1 s−1 2−δ+it. Prime–frequency picture (vertical Fourier slice). Place s = σ0 + iτ and take the vertical Fourier transform in τ: b Eδ,t(ω) := ZREδ,t(σ0+iτ)eiωτ dτ. A standard contour computation yields, for σ0>1 2+|δ|, b Eδ,t(ω)=2π e−(σ0−1 2−δ) (ω−t)1{ω>t}+ 2π e−(σ0−1 2+δ) (t−ω)1{ω<t}. (This asymmetric exponential tilt (when δ = 0) is the same positive-level signature discussed in the de Bruijn–Newman program [11, 12].) Thus each zero produces a Laplace kernel centered at the prime frequency ω = t , with unequal decay rates to the right/left when δ= 0. For an on–line zero (δ= 0) the decay is symmetric. Energy per zero (Parseval). Define the (local) energy carried by this pair at the vertical level σ0by Eδ,t(σ0) := 1 (2π)2ZRb Eδ,t(ω)2dω. Using the explicit kernel above, Z∞ t (2π)2e−2(σ0−1 2−δ)(ω−t)dω =(2π)2 2(σ0−1 2−δ),Zt −∞ (2π)2e−2(σ0−1 2+δ)(t−ω)dω =(2π)2 2(σ0−1 2+δ), and therefore Eδ,t(σ0) = π σ0−1 2−δ+π σ0−1 2+δ, σ0>1 2+|δ|. Consequences. •On–line balance. For δ= 0 (RH case) the energy is symmetric and minimal: E0,t(σ0) = 2π σ0−1 2 . Prime–side oscillations at frequency ω=tare purely level–0(no exponential tilt). • Off–line tilt. For δ = 0 one side decays more slowly (larger energy cost), and the prime–frequency kernel acquires an exponential modulation: b Eδ,t(ω) = e+δ(ω−t)·2πe−(σ0−1 2)(ω−t)1{ω>t}+e−δ(ω−t)·2πe−(σ0−1 2)(t−ω)1{ω<t}. Equivalently, in the time variable conjugate to ω this is a sinusoid with exponential envelope e±δ t—the tilted (positive–level) mode detected by the Laplace–Hopf resolvent. • Compensation. Removing this pair eliminates exactly the energy Eδ,t ( σ0 )at frequency t ; to preserve Parseval balance, the prime side must supply an equal amount, which manifests as the exponential scaffold observed by the peeling procedure. When δ = 0 the scaffold is absent (level–0flatness). 25 B. Energy growth: flatness index (secondary diagnostic) Define the kernel energy and flatness index EH,ϕ(σ) := ∥RH,ϕ(σ, ·)∥L2 ω,FZ:= lim sup σ→∞ 1 σlog EH,ϕ(σ). Then RH ⇐⇒ FZ = 0 for all admissible ( H, ϕ ); if some off–line zero has β−1 2 = δ > 0then FZ≥δ. Equivalently, for every ε > 0, RH ⇐⇒ lim σ→∞ e−εσ EH,ϕ(σ)=0 for all admissible (H, ϕ). C. Local ridge (frequency–localized) test Fix an ordinate t⋆ and an even bump χ∈C∞ c ( R )with χ (0) = 1. Demodulate and Laplace in σ : Ψ(t⋆) H,ϕ(ℓ) := Z∞ 0 e−ℓσZRχ(ω−t⋆)RH,ϕ(σ, ω)dω dσ. Then Ψ (t⋆) H,ϕ has simple poles precisely at ℓ = ± ( βj−1 2 )where {βj + it⋆} are the zeros on that ordinate (sum of residues gives the total multiplicity). Under RH, Ψ (t⋆) H,ϕ is holomorphic for ℜℓ > 0. D. Asymmetry (left–right) test at a frequency Tilt from an off–line pair 1 2±δ + it⋆ creates asymmetric decay to the right/left of ω = t⋆ . For any even χas above define AH,ϕ(σ;t⋆) := ZRχ(ω−t⋆) [RH,ϕ(σ, ω)−RH,ϕ(σ, 2t⋆−ω)] dω. Then under RH, AH,ϕ ( σ ; t⋆ ) ≡ 0. If an off–line pair with shift δ > 0is present at t⋆ , one has AH,ϕ(σ;t⋆)≍sinh(δσ)(after the standard smooth subtractions). E. Probe–invariance (robustness checks) Signals in ker Π0must be probe-invariant: • (Nullifier choice.) Use families of even ϕ with b ϕ|Ω≡ 0(and varying high-order vanishing) and even vertical weights H; genuine kernel features persist across choices. • (Envelope cancellation.) Compare ∂σ –logs or Laplace residues; these kill smooth multipliers coming from archimedean/vertical envelopes and isolate the ℓ–poles. F. Practical blueprint (how to look in ker Π0) 1. Prime-band nullification: choose ϕ with b ϕ (Ω) = 0 (to any desired order) and an even, rapidly decaying H; form RH,ϕ = (I−Π0)Pϕ. 2. Pole scan: compute LH,ϕ ( ℓ, ω )and search for poles with ℜℓ > 0; a single pole at ℓ = δ > 0 falsifies RH and gives an off–line zero at ρ=1 2+δ+i ω. 3. Energy bound: estimate FZ via the growth rate of EH,ϕ ( σ ); a bound FZ≤η yields a zero–free strip |ℜρ−1 2| ≤ η. 4. Localization: around a suspected t⋆ , use Ψ (t⋆) H,ϕ to extract individual shifts and multiplicities from residues. 32 5. Robustness: repeat across several ( H, ϕ )to confirm probe-invariance of any detected feature. Lemma 1.33 (Prime-only σ –tests are RH–blind).Let T ( σ, ω )be any of our zero–side distributions and Π 0 the prime-band projector in ω . For any linear operator S built from ∂σ , Lσ , and bω , we have S Π 0 = Π 0S . Hence S (Π 0T )depends only on the band data and is independent of horizontal zero positions. In particular, positive–level poles caused by off–line zeros lie in (I−Π0)ST and are annihilated by Π0. Summary. The kernel sector RH,ϕ ∈ker Π 0 is where RH lives. Fixed-slice prime–band data are RH–blind; what matters here is the σ –dynamics: absence (resp. presence) of positive–level poles in ℓ certifies (resp. falsifies) RH, and quantitative growth of ∥RH,ϕ∥ bounds horizontal zero deviations. 2 Beyond the kernel: what to add, why it helps, and the theory we need 2.1 An extended prime–band calculus Definition 2.1 (Multiplicative blur Bτ and σ –dynamics).Let {ϕτ}τ≥0 be even Schwartz kernels on the a –line with ϕ0 = δ0 and ϕτ1∗ϕτ2 = ϕτ1+τ2 . Define the multiplicative blur on the log–scale by BτT:= ϕτ∗T, T ∈ S′(R). For the vertical parameter σ , let D be any of: ∂σ , finite differences in σ , or the one–sided Laplace transform Lσ[f](ℓ) = R∞ 0e−ℓσf(σ)dσ. Definition 2.2 (Extended observables: prime and kernel channels).Let T ( σ, · )be a zero–side distribution (e.g. Pϕ(·)at σ). Fix ψ∈ S(R)with b ψ|Ω≡0, where Ω = {klog p}. Define (prime channel) Π0BτT(σ0,·),(kernel channel) QψDBτT(·), where Qψ is convolution in ω by ψ . We work in the smallest closed subspace of S′ ( R )stable under Bτ,D, and convolution by ψ. Remark (Relation to primes; “blur  =rescale”).The blur Bτ acts on the logarithmic Dirichlet measure dµlog(a) = PpPk≥1(log p)δa−klog pby Bτµlog =X pX k≥1 (log p)ϕτ(· − klog p). Equivalently, each atom at a = klog p is replaced by a bump centered at a . Heuristically, writing a = klog p + ε , this averages contributions of the form p−(k+ε/ log p)s . Thus Bτ locally perturbs exponents around each k , but it does not replace the global scale k = 1 by an arbitrary function: it is an averaging (convolution) on the a–line, not a deterministic rescaling s7→ f(s). 2.2 Why this helps for RH (vs. prime–only tests) • Height–agnostic detection. If some zero satisfies ℜρ = 1 2 + δ > 0, then kernel–channel energies like Eτ,ψ(σ) := Qψ(BτPϕ)(σ, ·)L2 ω grow like eδσ as σ→ ∞ (rate independent of ℑρ ). No individual prime spike needs to “hit” the zero. 33 • Zero–free regions from bounds. Uniform subexponential bounds for Eτ,ψ ( σ )(in τ, ψ ) imply explicit zero–free strips without locating zeros. • Noise–robust aggregation. Bτ averages nearby klog p contributions, stabilizing estimates while preserving the RH–relevant σ–signature in the kernel channel. • Separation of concerns. Π 0 keeps exact prime arithmetic (reconstruction), while QψD captures horizontal displacements ℜρ−1 2that are invisible to Π0. 2.3 A program: theories we likely need (T1) Multiplicative heat flow on the log–line. A semigroup {Bτ} with generator G such that for Lτ(s) := Z∞ 0 e−sa d(µlog ∗ϕτ)(a)we have ∂τLτ=GsLτ, and −ζ′ τ/ζτ = Lτ defines a blurred zeta family. (This sits under the almost* framework umbrella explained further.) (T2) Explicit–formula stability under blur. A centered explicit formula for BτPϕ : singular support remains on Ω, with a smooth multiplier depending on τ ; RH–sensitive content moves to the kernel channel. (T3) Kernel–channel equivalences. For admissible ψwith b ψ|Ω= 0 and any τ≥0, ∃δ > 0 : Eτ,ψ(σ)≍eδσ⇐⇒ max ρℜρ−1 2=δ, with only residues/envelopes depending on ψ, Bτ. (T4) Height–free zero–free criteria. Inequalities for Eτ,ψ ( σ )(convexity/monotonicity in σ or τ) that yield ℜρ≤1 2+εwithout resolving ℑρ. (T5) Kernel universality and stability. RH/ ¬ RH dichotomy in the kernel channel is unchanged under a broad class of blurs ϕτ and notch filters ψ ; i.e. the closure is not “fine–tuned.” (T6) Model calibration. Exact computations for a Hadamard model with one off–line pair to verify Eτ,ψ(σ)∼eδσ and to calibrate constants. (T7) Extension to L –functions. The same calculus for primitive L –functions (conductor–normalized log–scale), so the kernel channel is equivalent to GRH for the family. Remark (“ pf(t) ” vs. blur).Allowing a global rescaling s7→ ks (with k = 1) is equivalent to replacing log by klog , or to redefining “primes” as pα with α = k . By contrast, the blur Bτ is alocal averaging in the a = log pk variable: each atom δa−klog p is replaced by ϕτ ( · − klog p ). Heuristically this integrates p−(k+ε/ log p)s over small ε , but the mean scale remains k = 1. Thus blur preserves the prime alphabet and its multiplicative cover of N , while creating a stable channel (QψD) where horizontal zero displacements leave a σ–exponential signature. Remark (Practical choice of blur and a stochastic viewpoint).Let ϕ be a probability kernel on R(so Rϕ= 1) and let Φ(s) := ZRe−sε ϕ(dε) (ℜs > 0) be its Laplace transform. Define the blurred logarithmic derivative by 34 −ζ′ ϕ ζϕ (s) := Φ(s)−ζ′ ζ(s),ℜs > 1, and fix the multiplicative constant by the normalization ζϕ ( s ) → 1as ℜs→ + ∞ . If ϕ is small/centered (an approximate identity), then near s= 0 Φ(s)=1−E[ε]s+1 2E[ε2]s2+O(s3), so in particular, if E [ ε ]=0we have Φ( s ) = 1 + O ( s2 ); the blur is then a gentle, invertible deformation on {ℜs≥1 + δ}(for fixed δ > 0). Stochastic interpretation. Let {εp,k}p,k be i.i.d. with law ϕ , independent across primes p and exponents k≥1. Consider the random prime–power series Lϕ(s) := X pX k≥1 (log p)p−ks e−sεp,k ,ℜs > 1. Then E [ Lϕ ( s )] = Φ( s ) PpPk≥1 ( log p ) p−ks = Φ( s ) −ζ′/ζ ( s ). Defining the random analytic function ζrand,ϕ by −d ds log ζrand,ϕ(s) = Lϕ(s), ζrand,ϕ(s)→1as ℜs→+∞, we have Eh−ζ′ rand,ϕ/ζrand,ϕi=−ζ′ ϕ/ζϕ. Moreover, for fixed swith ℜs > 1, Var Lϕ(s)=Φ(2ℜs)− |Φ(s)|2X pX k≥1 (log p)2p−2kℜs, so fluctuations are square–summable and small when ϕ is tight (e.g. Var ( ε ) ≪ 1). This formalism identifies blurring with a multiplicative random modulation of prime–power weights, whose mean effect is exactly the deterministic multiplier Φ(s). Definition 2.3 (Almost–prime theory).Fix an even, centered probability kernel ϕ on R (so Rϕ = 1 and Rε ϕ ( dε )=0) with Laplace transform Φ( s ) = RRe−sε ϕ ( dε ). The ϕ -blurred prime log–measure is the convolution dµ(ϕ) log := ϕ∗dµlog =X pX k≥1 (log p) (ϕ∗δklog p), i.e. each atom at a=klog pis replaced by the translated copy ϕ(· − klog p). The associated blurred logarithmic derivative is −ζ′ ϕ ζϕ (s) := Φ(s)−ζ′ ζ(s),ℜs > 1, with normalization ζϕ(s)→1as ℜs→+∞. We call the study of dµ(ϕ) log , ζϕ the almost–prime theory over ϕ kernel. The classical prime theory is recovered in the limit ϕ⇒δ0(no blur). Remark (Blur on ζvs. on −ζ′/ζ).Let µlog := X pX k≥1 (log p)δa−klog p, µint := X n≥1 δa−log n. For any blur family BτT:= ϕτ∗Ton the a–line with Laplace factor Φτ(s) = RRe−saϕτ(a)da, Z∞ 0 e−sa dBτµlog(a)=Φτ(s)−ζ′ ζ(s),Z∞ 0 e−sa dBτµint(a)=Φτ(s)ζ(s). 35 Thus blurring is the same operator (multiplication by Φ τ ( s )in s ), but the structure it acts on differs: supp(µlog) = G p {klog p:k∈N}(disjoint towers), supp(µint) = {log n:n∈N}(additive semigroup generated by {log p}). Consequently, Bτ is "tower-local” for −ζ′/ζ (each δa−klog p is blurred independently), but not for ζ : blurring δa−log n smears the entire composite log n , not individual prime exponents vp ( n ). This is why our tower decomposition and prime calibration are formulated on the −ζ′/ζ (prime–power) channel rather than on ζitself. Normalization note. The atom at a = 0 (from n = 1) in µint becomes ϕτ centered at 0, contributing a global factor Φ τ ( s )in front of ζ ( s ). One may subtract δ0 (work with ζ− 1) if desired; it does not affect prime–band statements. Normalization (safe primitive). For any s0 with ℜs0> 1fix the branch by log ζϕ ( s0 ) := 0 and define log ζϕ(s) = −Zs s0 Φ(w)ζ′(w) ζ(w)dw, (ℜs > 1), which uniquely determines ζϕ and matches the normalization ζϕ ( s ) → 1as ℜs→ + ∞ .If ϕ is supported in ( −ρ, ρ )with ρ < log 2(so dµ(ϕ) log has no support near a = 0), then one may also write log ζϕ(s) = Z(0,∞) e−sa adµ(ϕ) log (a). Basics of the almost* theory To support the almost* viewpoint (unifying “almost–prime” and “almost–integer” from the multiplicative and additive sides), we split the geometries and give each its own blur. Addition lives on the integer axis, multiplication on the log–axis; the two channels are probed by Dirichlet/Euler functionals and meet on a controlled overlap band. Setup: two channels and bandlimited blur Let H:= Hadd |{z} additive line Rt ⊕ Hmult |{z} log line Ru , u = log n. Fix even, positive–definite, bandlimited kernels ϕadd ∈PWΛadd (Rt), ϕmult ∈PWΛmult (Ru), where PWΛ denotes a Paley–Wiener band ( b f supported in [ − Λ , Λ]). These profiles are the blurs on the two lines. Two–channel integer atoms For n≥1define the integer atom |n⟩:= ϕadd(· − n), ϕmult(· − log n)∈ Hadd ⊕ Hmult. Before any probe, n is known only up to the chosen additive blur on the t –axis and the chosen multiplicative blur on the u–axis. 36 Actions: addition vs. multiplication Addition by m∈Ztranslates the additive channel: Tm:f(t)7→ f(t−m),|n⟩ 7→ ϕadd(· − (n+m)), ϕmult(· − log n). Multiplication by k∈Ntranslates the log channel: Sk:g(a)7→ g(a−log k),|n⟩ 7→ ϕadd(· − n), ϕmult(· − (log n+ log k)). Probes (measurements) • Dirichlet/additive probe: linear functionals on Hadd (Fourier tests, congruence/CRT questions, additive convolution). • Euler/multiplicative probe: linear functionals on Hmult (Mellin tests, prime/prime–power structure, multiplicative convolution). The overlap band and “intermediate” probes Set the overlap bandlimit Λ∧:= min{Λadd,Λmult},Hoverlap := PWΛ∧. Let Π ∧ be the orthogonal projection onto PWΛ∧ on either line (Π ∧ commutes with convolutions by kernels whose Fourier support lies in [−ΛΛ,ΛΛ] , so intermediate probes do not introduce aliasing.). An intermediate probe with mixing parameter r∈[0,1] acts on |n⟩∧,r := rΠ∧ϕadd(· − n),(1 −r) Π∧ϕmult(· − log n), so all coupling automatically lives in the common band PWΛ∧(no aliasing). Why this is useful •Blur/band is structural: inequalities and energies live natively in Hadd ⊕ Hmult; nothing is retrofitted. • Clean separation of arithmetic: additive facts (convolutions, CRT, residues) live in the add channel; multiplicative facts (Euler product, prime powers at klog p ) live in the log channel. •Riesz/independence principle (RH–testable): The blurred prime–power family {ϕmult (· − klog p)}p,k is Riesz in PWΛΛ on growing windows; equivalently, no late prime layer is δ –approximable by earlier layers. Failure (extra dependence beyond power-tower identities) forces a positive-level pole ℓ > 0in the σ-resolvent, i.e. an off-line zero. • Energies are RKHS norms: each channel carries a reproducing–kernel norm; the joint squeeze is the norm in the direct sum. Additive vs. multiplicative blur on the log line Let µlog := X pX k≥1 (log p)δa−klog p, µint := X n≥1 δa−log n. 37 For any even L1 kernel ψ on Ru with Laplace/Mellin multiplier Ψ( s ) := RRe−suψ ( u ) du one has ZRe−su d ψ∗µlog(u) = Ψ(s)−ζ′ ζ(s),ZRe−su d ψ∗µint(u) = Ψ(s)ζ(s) (ℜs > 1). Interpretation. The operator (convolution on u ) is the same, but the structures differ: µlog is a disjoint union of prime towers {klog p}(tower–local blur), while µint sits on {log n}and mixes primes inside each composite (global blur). Accordingly, “tower calculus” is carried out on −ζ′/ζ (prime–power channel), not on ζitself. One–box definition Definition 2.4 (Two–channel integer atom).Fix even, positive–definite, bandlimited kernels ϕadd ∈PWΛadd (Rt)and ϕmult ∈PWΛmult (Ru). For n≥1, |n⟩=ϕadd(· − n), ϕmult(· − log n)∈ Hadd ⊕ Hmult. Addition by m acts as t7→ t + m on Hadd ; multiplication by k acts as u7→ u−log k on Hmult . Dirichlet/Euler functionals probe the respective channels; intermediate probes operate on PWΛ∧ with mixing parameter r∈[0,1]. A continuous split between Dirichlet and Euler We work in the Dirichlet–convolution Hopf algebra. For a Dirichlet series F ( s ) = Pn≥1a ( n ) n−s with a(1) = 1, write m:= log∗aso that a= exp∗(m), where log∗,exp∗ are with respect to Dirichlet convolution ( m (1) = 0, a (1) = 1). For ζ one has m(pk)=1/k. Definition 2.5 (r–interpolating split Tr).For r∈[0,1] set Er:= exp∗(r m), D1−r:= exp∗((1 −r)m), and define Tr[F] := Er, D1−r, F(s) = Er(s)·D1−r(s). Endpoints: r = 0 gives ( E0, D1 )=( δ1, F )(pure Dirichlet); r = 1 gives ( E1, D0 )=( F, δ1 )(pure Euler). Zeta explicitly (for ℜs > 1, principal branch). Using m(pk)=1/k, Er(s) = Y p (1 −p−s)−r=ζ(s)r, D1−r(s) = X n≥1 g1−r(n)n−s=ζ(s)1−r, with multiplicative coefficients g1−r(pk) = (1 −r) + k−1 k!=Γ(1 −r+k) Γ(1 −r)k!. Hence Er ( s ) D1−r ( s ) = ζ ( s )for all r∈ [0 , 1]: the split reallocates prime–power cumulants without changing the function. Composition law (dyadics and closure). Applying Tq after Tr moves a fraction q of the remaining mass to the Euler side, so the Euler share updates by r⊕q:= r+ (1 −r)q= 1 −(1 −r)(1 −q),Tq◦ Tr=Tr⊕q. Iterating r = 1 2 generates the dyadics and converges monotonically to the Euler presentation; conversely, dyadic descent converges to the Dirichlet presentation. 38 Clipped universe [1 , N ].Replacing m by mN := m· 1 {n≤N} yields Er,N = exp∗ ( rmN )and D1−r,N = exp∗ ((1 −r ) mN )whose product, after discarding terms > N , equals the truncated sum Pn≤Nn−s. This gives a finite–window version of the same split. Remark (Two–nature metaphor).With Tr and the two–channel atom |n⟩ , an integer has two coherent “natures”: additive (on the t –axis) and multiplicative (on the a = log n axis). The parameter r acts as a polarizer, continuously rotating the presentation while leaving the underlying function ζunchanged. Structurally, the channels carry different group laws (Cauchy vs. Dirichlet convolution) and live in different Fourier geometries. Any transfer between them (Mellin/explicit–formula, prime–band projection Π 0 ) preserves the discrete prime–comb singular support but necessarily leaves a C∞ remainder (archimedean and diagonal terms) together with kernel pieces invisible on the prime band. In particular, there is no global identification of addition with multiplication; they agree only after projection to the overlap band (or asymptotically, in the PNT sense). The “smooth residue” one must drag along when passing from one side to the other is therefore not an artefact of method, but a consequence of their fundamentally different algebraic and microlocal natures. Unified RH/Non–RH Modeling Even if the Riemann Hypothesis turns out to be true, the complete spectral imagery suggests a richer dynamical interpretation: the zero–lattice system is then in its ground state, with all spectral mass confined to the critical line (level–0 flatness). However, there is no structural obstruction to exciting the system—introducing additional horizontal displacement of zeros—while keeping all prime–side structures intact at base level. From this viewpoint, RH describes the unperturbed configuration, and non–RH corresponds to controlled excited states where new spectral modes appear at positive levels. The framework below encodes both cases in a single model, allowing for purely discrete “orbital” excitations or continuous “band” excitations, and quantifies their prime–side signatures. Ground vs. excited state imagery. Left: RH (ground state) — all nontrivial zeros lie on ℜs = 1 2 , producing only level–0 poles in the prime–side Laplace index. Middle: discrete excitation — a finite number of horizontal shifts δj> 0create parallel excited lines with isolated poles at ℓ = δj . Right: continuous excitation — a band of shifts fills the strip 1 2<ℜs≤1 2 + ∆, producing a real–axis cut ℓ∈ [0 , ∆] on the prime side. In all cases, the level–0 core (prime comb) is preserved; excitation is detected only in the positive–level sector. We now introduce a general framework in which the Riemann Hypothesis (RH) appears as the ground state of the system, and any departure from RH corresponds to an excitation that shifts zeros horizontally off the critical line. The model preserves all prime–side structures at level 0 while making the off–line contribution explicitly measurable. Definition 2.6 (Base core and excitation operator).Let the level–0 core be the completed zeta with all zeros projected to the critical line: ξ0(s) := ξ(0) Y ρ⋆1−s ρ⋆, ρ⋆=1 2+itρ. An excitation is specified by an even finite signed Borel measure ν on horizontal displacements δ∈R(evenness enforces the functional equation symmetry). Define the excitation factor Hν(s) := Y ρ⋆ exp(ZRlog1−s ρ⋆+δ1−s 1−ρ⋆−δ 1−s ρ⋆1−s 1−ρ⋆dν(δ)). 39 The general state is ξν(s) := ξ0(s)Hν(s), ζν(s) := ξν(s) 1 2s(s−1) π−s/2Γ(s/2). Remark (Excitation factor). •ν= 0 (no excitation) ⇒RH exactly: all zeros on ℜs=1 2. • Any nonzero ν redistributes paired zeros horizontally while preserving the functional equation and trivial zeros. The "amount" and "pattern" of non–RH is encoded by ν. Definition 2.7 (Discrete vs. continuous excitation). • Discrete (orbital) excitation: ν = PJ j=1 ajδδj + δ−δj with δj> 0produces parallel excited lines ℜs=1 2±δj; prime–side signature: isolated poles at ℓ=δj. • Continuous (band) excitation: ν = h ( δ ) dδ with even h∈L1 ( R )produces a continuous excited strip 1 2<ℜs≤1 2+ ∆; prime–side signature: branch cut on ℓ∈[0,∆]. Proposition 2.8 (Prime–side detection).Let ην ( s, σ ) = ξν ( s + σ/ 2) ξν ( s−σ/ 2) and project to the prime band as in Theorem 1.8. Then the σ–Laplace transform Lν(ℓ, ω) = Z∞ 0 e−ℓσ (prime-band part of log ην)dσ, where ℓ is the Laplace dual of σ (the “level” index) and ω is the Fourier dual of the vertical variable t, satisfies: •Under RH (ν= 0)Lνhas no poles or cuts for ℜℓ > 0:level–0 flatness. •Discrete ν: isolated poles at ℓ=δj(excited orbits). •Continuous ν: real–axis cut {ℓ > 0 : h(ℓ)= 0}(excited band). Definition 2.9 (Excitation energy).For σ0>1 2define the pair energy density Eν(t;σ0) = ZRπ σ0−1 2−δ+π σ0−1 2+δdνt(δ), where νt is the restriction of ν to the pair centered at t . RH corresponds to ν≡ 0and minimal energy. Corollary 2.10 (Stable core and excited envelope).Every ξνsplits as ξν(s) = ξ0(s) |{z} RH core ·Hν(s) |{z} excited envelope , and the prime–side projector Π 0 detects only the core at level 0. Excitation appears solely through positive–level poles or cuts of Lν. Ground and prime states. We distinguish two reference configurations: • Ground (level–0) state: all nontrivial zeros lie on the critical line ℜs = 1 2 , producing only level–0 poles on the prime–side Laplace index. • Prime state: the prime–side spectrum is exactly the unperturbed Euler–type comb, with no additional positive–level structures. 40 In the unperturbed model, RH ⇐⇒ ground state =prime state, so that the spectral configuration of zeros and the prime–side distribution are in perfect correspondence. This framework, however, allows one to excite and test the system in a controlled manner on either side: • perturb zeros by adding, shifting, or deleting them (off–line excursions, new excited lines, or bands); • perturb primes by adding, shifting (in log p ), or deleting them (altering the comb structure). Such perturbations break the RH equivalence, but in a specified and measurable way, enabling one to probe the internal coupling between the zero–lattice and prime spectrum. This enriches the investigation beyond the binary RH/non–RH dilemma, opening the possibility of reading finer structural information about the zeta system from targeted excitations. Operational cases: •Single over–prime line: one δ1>0, one pole at ℓ=δ1. •Finite ladder: equally spaced δj=jα, poles at α, 2α, . . . . • Q–ladder: discretely spaced δj∈Q+ (not necessarily in arithmetic progression), producing a set of isolated poles at rational abscissae; the distribution of poles reflects the chosen rational pattern. •Band: continuous hon [0,∆], cut on ℓ∈[0,∆]. This construction unifies RH and non–RH cases: the base level–0 core encodes the stable RH structure; any nonzero excitation ν corresponds to an excited state whose spectral signature is fully visible in the positive–level index ℓ . Instead of resisting the rich framework in which primes and zeros coexist, we turn it to our advantage: learning from it by observing the system together with its full richness, rather than trying to enforce a narrow set of properties the system may or may not support. Unified RH/Non–RH Modeling (Sharp and Almost–Prime) A. Blur operator and its commutation with the pipeline Assume throughout that the Laplace transform Φof the blur kernel ϕ is entire and nonvanishing on a right half–plane: Φ(s) = ZRe−sε ϕ(dε),Zϕ= 1,∃s∗<1 : Φ(s)= 0 for all ℜs≥s∗. Let ϕbe even and centered. Define the blur/convolution operator on ω–distributions by Bϕ[T] := ϕ∗T. On the Euler side this corresponds to the multiplier −ζ′ ϕ ζϕ (s) := Φ(s)−ζ′ ζ(s), ζϕ(s)→1as ℜs→+∞, i.e. the almost–prime theory (with kernel ϕ). 41