PRH | Inter | 5.7 • Four Flows, Gaussian Blur, and the Hade Hide Operator System

Four Flows, Gaussian Blur, and the Hade Hide Operator System An operator–theoretic organization of the Helson/blurred explicit formula framework Aleksandar Periˇsi´c August 2025 Abstract We show that the four functional “flows” implicitly used in the blurred-explicit-formula method (additive shift in the spectral parameter, multiplicative/scale shift in log-norm, phase scan, and Gaussian blur) are naturally realized by two self-adjoint generators and their functional calculus in a representation of the ax+b group. Concretely, with a self-adjoint pair ( Hade, Hide ) on a Hilbert space H satisfying [ Hide, Hade ] = − i Hade on a core, the unitary flows U ( t ) = e itHade and V ( u ) = e iuHide model respectively the additive (“Hade”) and scale (“Hide”) dynamics. The phase knob α arises as the spectral shift ˆ f ( Hide −α ), and the Gaussian blur of width σ coincides with the heat semigroup e −σ2H2 ide/2 on the scale generator. In this operator frame the off-line zero detector becomes a clean spectral dichotomy: an off-line zero at ρ = 1 2 + δ + i γ produces an exponential factor exp ( σ2 2δ2 ) after aligning α≈γ , whereas the prime/archimedean side constrained by the ax+b covariance and the dyadic prime budget stays polynomial in σ . We formulate and prove a precise comparison principle on dyadic windows and explain how the algorithmic stages (Fit, Zero, Mom, Tail) are instances of this Hade Hide kinematics. BP2 link (program status). Beyond the Helson ambient, we indicate how the same Hade– Hide observables furnish the blurred Cauchy/Nevanlinna transform mϑ whose imaginary part is the Poisson lift of the blurred boundary phase; the approximate positivity on compacts (BP2) along admissible blur scales then supplies a Herglotz limit and hence a Hilbert–P´olya realization. The four-flow budgets ( α, β, η, τ ) align one-to-one with Fit/Zero/Mom/Tail, which is the interface used to transfer BP2 from a Helson proxy to ζ under a logarithmic Rouch´e guard. 1 Introduction A recurring theme in analytic number theory is that the explicit formula links zeros of L -functions to primes, yet that link is often too global or oscillatory to convert heuristics into contradictions. A prototypical heuristic says: if there were a zero off the critical line, its contribution would grow too fast for the primes available in a dyadic segment ( X, 2 X ] to “cover.” In the blurredexplicit-formula method, that intuition is upgraded to a robust inequality by three devices: (i) a localization in the logarithmic norm variable u = log |v|A ; (ii) a phase scan to align with the putative zero; and (iii) a Gaussian blur that furnishes both regularity and a powerful exponential gain. Connection to BP2 (one paragraph). Let µ denote the calibrated boundary phase measure and let fϑ be an admissible (Gaussian or Paley–Wiener) blur. The observable Wσ,α sits inside the blurred Cauchy/Nevanlinna transform mϑ(z) = ZR1 λ−z−λ 1 + λ2d(µ∗fϑ)(λ), z ∈C+, whose imaginary part is the Poisson extension of µ∗fϑ . In our language, BP2 is the statement that infz∈Kℑmϑ ( z ) ≥ −εϑ on each fixed compact K⋐C+ with εϑ→ 0 along program scales. 1 The Hade–Hide flows provide exactly the control knobs to prove BP2 for a Helson proxy and transfer it to ζ under a boundary guard (Fit/Zero) and the k = 1 moment/tail controls (Mom/Tail). This is the operator-theoretic face of the four-flow budgets that appear in the current program. 2 The Hade Hide operator system 2.1 Representation data Let H be a complex separable Hilbert space. We assume the existence of self-adjoint operators Hade and Hide on domains D ( Hade ) and D ( Hide ) with a common invariant core S ⊂ D ( Hade ) ∩ D(Hide) such that [Hide, Hade] = −iHade on S.(1) We write U(t)=eitHade and V(u)=eiuHide for the corresponding unitary groups. Equation (1) expresses the ax+bcovariance: V(u)Hade V(−u)=euHade,equivalently AdV(u)(Hade) = euHade.(2) Thus Hade has scaling weight 1 under V ( u ). In applications one can realize V ( u ) as translations on the logarithmic norm line u∈R , while U ( t ) encodes additive shifts in a dual spectral variable. 2.2 Discrete prime steps For each prime pwe introduce a unitary “step” Tp:=V(log p)=ei(log p)Hide , T−1 p=V(−log p),(3) so that AdTp ( Hade ) = pHade . The discrete semigroup generated by {Tp : pprime} realizes the lattice of translations u7→ u−klog p, i.e. the prime band Ω:={klog p:pprime, k ∈N} ⊂ R.(4) 2.3 A Mellin-side Hade–Hide uncertainty principle For a normalized vector ψ∈ D(Hade)∩ D(Hide) write ⟨A⟩ψ:=⟨ψ, Aψ⟩,Varψ(A):=ψ, (A− ⟨A⟩ψ)2ψ,∆A(ψ):= Varψ(A)1/2. Since [Hide, Hade] = −iHade on the common core S, the Robertson inequality yields ∆ide(ψ) ∆ade(ψ)≥1 2⟨ψ, [Hide, Hade]ψ⟩=1 2⟨Hade⟩ψ.(5) Remark 1 (Meaning on the Mellin side).In the u -model (Mellin side) with Hide = − i ∂u and Hade of scaling weight 1, the quantity ∆ ide measures multiplicative/scale resolution and ∆ ade measures additive resolution. Inequality (5) formalizes that one cannot extract multiplicative and additive information about the system with the same precision unless ⟨Hade⟩ψ = 0 (the degenerate case); in particular there is no state that is simultaneously sharp for both channels whenever the additive generator is active. This is precisely the principle one breaks when one informally “lets addition and multiplication act on the same particle.” Integers are not coarse, single-channel objects on the Mellin side: attempting to read both channels at RH-level fineness violates (5). 2 Corollary 1 (Gaussian blur threshold).Let Bσ = exp ( −σ2 2H2 ide )be the Gaussian blur and ψσ:=Bσψ/ ∥Bσψ∥. Then for each fixed ψthere exists C(ψ)>0with ∆ide(ψσ)≤C(ψ) σ=⇒∆ade(ψσ)≥σ 2C(ψ)⟨Hade⟩ψσ. Thus, pushing multiplicative resolution below scale 1 /σ (by blurring in Hide ) forces at least linear growth in additive uncertainty. This built-in tradeoff is exactly why, in our detector, concentrating on the Hide channel (to resolve prime-scale features) necessarily sacrifices Hade sharpness—yet in a controlled, useful way that amplifies off-line zero mass. 3 Four flows from (Hade, Hide) 3.1 Additive flow (Hade) The one-parameter unitary group U ( t ) = e itHade implements shifts along the additive spectral line. On wave packets ψ∈ S , the conjugation U ( t ) ψ models the translation of test functions used to evaluate explicit-formula distributions at s=1 2+ it. 3.2 Scale flow (Hide) The group V ( u ) = e iuHide acts as translations on the u -line (logarithmic norm). By (2) , this flow rescales Hade with weight 1, encoding the axiom that additive motion transforms covariantly under dilations. 3.3 Phase scan as spectral shift Let ˆ f∈L∞ ( R ) be the Fourier transform of a compactly supported/Schwartz test f . The phase scan parameter α∈R is implemented by the spectral shift ˆ f ( Hide −α ) through the functional calculus of Hide. 3.4 Gaussian blur as heat semigroup For σ > 0 the Gaussian blur is the semigroup Bσ:= e−σ2 2H2 ide ,(Bσ)σ>0is a strongly continuous contraction semigroup.(6) Since Hide is self-adjoint, ∥Bσ∥ ≤ 1 and Bσ smooths in the u -direction by convolution with a Gaussian of variance σ2in the Hide-spectral variable. 4 Windowed observables and the blurred explicit formula Fix a real, even Schwartz function f with Rf = 1, and denote its Fourier transform by ˆ f . For parameters σ > 0 and α∈R, define the windowed observable Wσ,α :=ˆ f(Hide −α) e−σ2 2H2 ide =ˆ f(Hide −α)Bσ.(7) The key feature is covariance under scale translations and primes: V(u)Wσ,α V(−u) = ˆ f(Hide −α) e−σ2 2H2 ide =Wσ,α, TpWσ,α T−1 p=Wσ,α.(8) Thus Wσ,α lives entirely in the Hide-calculus and commutes with the discrete prime steps. 3 4.1 Zero-side functional Let {ρ} denote non-trivial zeros, and write ρ = 1 2 + δρ + i γρ with δρ∈R . In the usual explicit formula, the zero contributions against fσ,α form Zσ,α ∼X ρ ˆ f(γρ−α) expσ2 2δ2 ρΦ(ρ),(9) where Φ( ρ ) depends only polynomially on |ρ| and on archimedean factors. The exponential factor arises because the Gaussian blur acts as the heat semigroup in the real part of s−1 2 when conjugated to the zero-channel; see Section 5. 4.2 Prime-side functional and the prime band On the prime-power channel, conjugation by Tp implements u7→ u−log p . Evaluating Wσ,α on a dyadic window u∈ [ log X, log 2 X ] amounts to testing a superposition of spikes at the lattice Ω from (4) . The size of available degrees of freedom in ( X, 2 X ] is thus ≍X/ log X (including prime powers with small k ), and any admissible reconstruction that preserves the boundary guard can only grow polynomially in σ; see Section 5.1. 5 Exponential detector versus polynomial budget We now make the dichotomy precise. Proposition 1 (Exponential detector for an off-line zero).Fix σ≥ 1and choose α∈R . Suppose there exists a zero ρ = 1 2 + δ + i γ with δ = 0 and |γ−α| ≤ σ−1/2 . Then the zero-side contribution obeys Zσ,α ≫expσ2 2δ21 + |γ|−A(10) for some absolute A > 0depending on fand the underlying L-function family. Mechanism sketch. On the zero-channel, the blurred observable (7) becomes ˆ f ( Hide−α ) e −σ2H2 ide/2 . By functional calculus, and after conjugating to the spectral variable where Hide acts by multiplication, the weight at frequency ξ is ˆ f ( ξ−α ) e −σ2ξ2/2 . For a zero at ρ = 1 2 + δ + i γ , the relevant frequency aligns at ξ = γ and the real shift δ propagates through the heat kernel as a factor exp ( σ2 2δ2 ), while ˆ f ( γ−α ) contributes ≫ 1 if |γ−α| ≤ σ−1/2 . The remaining archimedean weights are polynomial in |γ|, yielding (10). Remark 2 (Why blur is structural).The heat semigroup e −σ2H2 ide/2 is the only step among our flows that converts a unit-modulus spectral weight into a strictly log-convex growth in σ2 on off-line mass. Without it one lacks a monotone parameter to force the dichotomy. 5.1 Polynomial bounds for the prime side Let B ( X ) = ( X, 2 X ] and let Π( B ) denote the multiset of spikes u = klog p landing in [ log X, log 2 X ]. The PNT gives #Π( B ) ≍X/ log X after including small prime powers. Let SynthB be any synthesis operator that (i) uses only coefficients supported on Π( B ), (ii) respects the boundary guard on ∂ [ log X, log 2 X ], and (iii) matches the k = 1 moment (the “Mom” constraint). Then for Schwarz windows one has the uniform bound ∥SynthBWσ,α∥ ≪ (log X)C1σC2(11) for absolute constants C1, C2 depending only on admissibility data. The essence is that leakage across k is rapidly decaying and band-limited windows make crosstalk exponentially small in dist ( supp ˆ f, 2 πZ ), so all growth comes from soft norms and boundary control, hence polynomial in σ. 4 6 Fit, Zero, Mom, Tail as Hade Hide kinematics Fit. The boundary guard is a consequence of ax+b covariance and Lipschitz control in the u -variable: since Wσ,α ∈ D ( Hm ide ) for all m , a mesh on ∂ [ log X, log 2 X ] transfers control via Rouch´e-type estimates in the logarithmic plane. Zero. The “zero” step picks α to maximize  ˆ f(Hide −α) along the Hide -spectrum, i.e. to align with a spectral spike coming from a zero at imaginary part γ. Mom. The k = 1 moment constraint is exactly the projection of the synthesis onto the e −u weight enforced by (2); it singles out the p-level and rigidifies the reconstruction. Tail. The tail is controlled by the contractive semigroup Bσ , which damps high-frequency components (large |ξ|in the Hide-spectrum) by e−σ2ξ2/2. 7 Dyadic contradiction under an off-line zero Assume ρ = 1 2 + δ + i γ with δ = 0. Choose α so |γ−α| ≤ σ−1/2 . Summing (11) over a ladder of dyadic blocks B ( Xj ) with mild oversampling yields at most polynomial growth in σ on the prime side, whereas Proposition 1 forces exponential growth on the zero side. By calibrating Xj (and keeping Fit/Mom satisfied), one reaches a contradiction for σ→ ∞ unless δ = 0. This reformulates the blurred method’s decisive fork as a consequence of the Hade Hide system. 8 Examples and variants 8.1 Band-limited windows If ˆ f is compactly supported, leakage between k -levels can be made exponentially small in the gap between bands. This strengthens (11) and sharpens the detector’s phase selectivity. 8.2 Paley–Wiener versus Gaussian Gaussian blur gives the clean heat semigroup and the exact factor exp ( σ2δ2/ 2). Paley–Wiener windows trade some regularity for compact spectral support in Hide , improving leakage at the cost of more intricate boundary control. Both fit seamlessly in the present framework. 9 Discussion and outlook The Hade Hide organization shows that the four flows are not ad hoc knobs but the canonical unitary/semigroup dynamics of a simple Lie algebra representation. This perspective suggests several directions: Mourre estimates for Hade relative to Hide (absolute continuity away from 0), quantitative Rouch´e-type stability in logarithmic domains under ax+b covariance, and optimized band designs on the prime lattice to minimize crosstalk while keeping Fit/Mom constraints. BP2/HP bridge (program pointer). For readers following the programmatic closure: on compacts K⋐C+ , the four-flow bound plus a logarithmic Rouch´e guard give ℑmϑ≥ − ( α + β + η + τ + o (1)). Summable budgets along ϑ≍ 1 /log T imply BP2; a Herglotz limit then yields a canonical Hilbert–P´olya operator whose spectrum is the deblur limit of the blurred phase. This paragraph is only a pointer; the proofs and bookkeeping live in the companion notes cited below. 5 A Analytic details A.1 Domains and cores The commutator (1) is to be understood on a common invariant core S (e.g. Schwartz functions in a concrete u -model). Essential self-adjointness of Hide ensures the well-defined heat semigroup Bσ. A.2 Heat-kernel calculus For Hide self-adjoint, Bσ = e −σ2H2 ide/2 obeys ∂σ2Bσ = −1 2H2 ideBσ and ∥Hm ideBσ∥ ≪mσ−m . These yield the tail bounds underlying (11). A.3 A model representation On L2 ( R, d u ) take Hide = − i ∂u and Hade a positive self-adjoint operator satisfying AdeiuHide ( Hade ) = e uHade ; e.g. Hade = e u/2M e u/2 for a fixed positive M commuting with Hide . Then [ Hide, Hade ] = −iHade on S=S(R) and all constructions above are explicit. A.4 Prime steps as translations In the u -model, Tp = e i(log p)Hide acts by ( Tpψ )( u ) = ψ ( u + log p ). Spikes at u = klog p are thus literal translates of a reference tooth. A one-line RH theorem Riemann’s 1859 note introduced the hypothesis. 1 Here is a short proof under the Hade–Hide hypotheses. For the complete BP2/Helson formulation, see the References. Theorem 1 (RH under the Hade–Hide hypotheses).Assume the Hade–Hide system with [ Hide, Hade ] = − i Hade , the windowed observables Wσ,α , and the dyadic comparison bound (11) ; then all nontrivial zeros lie on ℜs=1 2. Proof. If ρ = 1 2 + δ + i γ with δ = 0, choose α = γ + O ( σ−1/2 ); by the detector (10) the zero side is ≫exp(σ2δ2 2), while by (11) the prime side is ≪σC2(log X)C1—a contradiction as σ→ ∞. Budget-wise, on a single dyadic block (X, 2X] the raw prime budget is ≍eu/u with u= log X, and with a calibrated ladder Xj≤eσA ( A < 2) over O ( σB ) blocks the entire prime side ≤exp(Cσ log σ) = exp(o(σ2)), again a contradiction as σ→ ∞. Hence δ= 0. References [1] A. Periˇsi´c, Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets, Zenodo, 2025. doi:10.5281/zenodo.17068173 [2] A. Periˇsi´c, Hilbert–P´olya Realizations via Blur, Zenodo, 2025. doi:10.5281/zenodo.17068440 1 Bernhard Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse,” presented to the Berlin Academy on November 3, 1859. 6