HBP | Main | 1.2 • Helson-Blur: Boundary Guards, Detectors, and Four-Flow Budgets

Helson–Blur: Boundary Guards, Detectors, and Four-Flow Budgets Aleksandar Perišić September 2025 Abstract This note isolates the operative pieces used implicitly in the companion paper Hilbert– Pólya Realizations via Blur. We (i) recall the Helson zeta ambient, (ii) formalize the boundary– guard mechanism via a logarithmic Rouché principle, (iii) package the construction into four summable flows that feed a blurred Cauchy transform, and (iv) give a detector plus a rigidity principle that pins down the model and transfers to ζ . The goal is to provide just enough detail to support the compact arguments in the companion while keeping the exposition self-contained. 1 Helson zetas, blur, and perspective AHelson zeta has Euler product ζχ(s) := Y p1−χ(p)p−s−1, χ :N→Tcompletely multiplicative,|χ(n)|= 1. For background on Helson-type Euler products and structural synthesis see Helson [ 1 ] and Andersson [ 2 ]. This flexible class is large enough to admit controlled deformations and small enough to retain Eulerian bookkeeping across stages. In our use, Helson zetas are proxies: one constructs a stagewise model ζχj that fits ζ with quantitative control on a window, then transfers boundary data and zero counts back to ζ. Blur and boundary collars. Fix a symmetric vertical window K = [ 1 2 + δ, 1] × [ −T, T ] ⋐C+ and a thin collar around ∂K .Blur refers to probing via a positive kernel at scale L≍log ( T + 3); we use either a Gaussian ( ϑ≍L−1 ) or a Paley–Wiener (PW) palette (Λ ≍L ). Blurring moves all comparisons away from pointwise oscillations and onto averaged boundary data, where small Lipschitz controls suffice. 2 Boundary guards and logarithmic Rouché Let F be holomorphic and nonvanishing on a collar of ∂K and H holomorphic there with sup∂K |H|≤ε < log 2. Set G:= FeH. Lemma 2.1 (Logarithmic Rouché).If min∂K |F|≥γ > 0and sup∂K |H| ≤ ε<log 2, then N(G;K)=N(F;K)(same zero count with multiplicity), and moreover |G| ≥ e−εγon ∂K. Idea. On the boundary, |eH− 1 |≤eε− 1 < 1, hence |G−F|<|F| . Classical Rouché (e.g., [ 3 , Ch. 8]) gives equality of zero counts in K , and the lower bound on |G| follows from |eH| ≥ e−ε . 1 Stagewise fit and guards. At stage j we work on Kj = [ 1 2 + δj, 1] × [ −Tj, Tj ]with a boundary mesh Γ j⊂∂Kj . We produce a log-ratio Hj := log ζχj ζ whose boundary norm is controlled by a guard εj<log 2: sup ∂KjHj≤εj(Lipschitz on the collar). Applying Lemma 2.1 with F=ζ,G=ζχjyields the finite-window zero transfer Nζχj;Kj=N(ζ;Kj),min ∂Kj |ζχj|≥e−εjmin ∂Kj |ζ|. 2.1 Helson overlap: freeze old primes, add a new dyadic block What we actually do. Fix a dyadic threshold Xjand let the new block be Bj:= (Xj,2Xj]∩ {primes}. (i) Freeze. Keep previously chosen Euler factors fixed: χj+1 ( p ) = χj ( p )for all p≤Xj . We also keep the branch of log consistent across stages (anchor at, say, s= 2). (ii) Solve only on the fresh block. Choose the new phases {χj+1 ( p ) }p∈Bj⊂T to (a) hit the boundary fit on the enlarged mesh Γ j+1 ⊃ Γ j , (b) respect the guard on Γ j , and (c) meet the moment constraints defined in §7. (iii) Overlap of windows/meshes. Enforce Kj⋐Kj+1 with a thin collar and Γ j⊂ Γ j+1 . This overlap makes the old guard persist automatically. Because old primes are frozen, the stage update is supported on Bj : for any s (in particular s∈Kj), Hj+1(s)−Hj(s) = log ζχj+1 (s) ζχj(s)=X p∈Bj log 1−χj(p)p−s 1−χj+1(p)p−s!.(1) Numerical domain for the logarithm bound. On {Re s≥1 + 1 Lj}we have |p−s|≤p−(1+1/Lj)≤2−(1+1/Lj)<1 2, so the estimate |log (1 −z ) | ≤ 2 |z| (valid for |z| ≤ 1 2 ) applies to each factor with z = χ ( p ) p−s . Therefore Hj+1(s)−Hj(s)≤2X p∈Bj |p−s| ≪ X p∈Bj p−Re s≪τj, uniformly for sin this half-plane (in particular for s∈Kj). Hence the change is exactly accounted for by the prime-tail/block flow τj. In particular: • Past guarantees persist: since Γ j⊂ Γ j+1 and old factors are frozen, the guard on Γ j survives into stage j+1. • Zero transfer is stable: with the guard in place on ∂Kj ,Lemma 2.1 re-applies and preserves the zero count on Kj. • Summability: only the new dyadic block contributes to (1) , making the cumulative effect summable when τjis scheduled as in §10. Reader’s one-liner. At each stage we do not touch old primes ; we only “paint” the new primes Xj<p≤ 2 Xj so that the boundary fit holds on an overlapping (larger) mesh, keeping the guard and zero count we already earned while pushing the fit outward. 2 3 Blur probes, prime-power channel, and the testing menu We work on the prime-power channel −ζ′ ζ(s) = X n≥1 Λ(n) ns=X pX k≥1 Λ(pk) pks , and probe it with even blur kernels f : R→R of unit mass RRf = 1. Writing b f for the Fourier transform and calibrating by subtracting a smooth bulk M (the archimedean main term), the prime fluctuation is P◦:= X pX k≥1 Λ(pk)f(klog p)−M. Definition 3.1 (Admissible blur).An even kernel fwith Rf= 1 is admissible if either (a) Band-limited (Paley–Wiener). b f is supported in [ − Λ , Λ] for some Λ ≥ 1and ∥b f∥∞≤ 1; or (b) Rapid decay (Schwartz). f∈ S(R)and R|u|m|f(u)|du<∞for all m≥0. Lemma 3.2 (Archimedean control and localization).Let f be admissible and let Kj⊂ {Re s≥ 1 2 + δj} , and let Xj denote the dyadic base of the prime-supply block Bj = ( Xj, 2 Xj ]. In the smoothed explicit formula the archimedean term satisfies |A∞| ≪    1+Λ2,in (a), Pm≤m0R|u|m|f(u)|du, in (b) for a fixed m0, and the cross-kleakage obeys X k=ℓX p Λ(pk)Λ(pℓ)f(klog p)f(ℓlog p)≪(e−cΛ,in (a), (log Xj)−A,in (b) for any fixed A > 0, after taking Λ(resp. the decay moments) large enough relative to the targeted budgets. Remark 3.3 (Plug-in menu).Typical choices are: Gaussians; Paley–Wiener band-limited windows; Slepian/DPSS windows on finite intervals (near-optimal time–band concentration); and Vaaler/Beurling extremal majorants/minorants. All are admissible and only change harmless constants. Detector dichotomy (what the probes see). • If a zero sits off the critical line by δ = 0, phase-aligned scans produce exponential-in-variance growth in the zero-side contribution of the explicit formula. • Under RH the response stays polynomial in the scale; a multiscale sweep separates these regimes cleanly (see §7for the use inside the program). Testing upgrades and window construction Bandwidth convention. Write B for the effective spectral bandwidth of f : B = Λ in the PW case and B≍ϑ−1for the Gaussian family. The probe family and sampling geometry we actually use are organized as follows. Items (U5) and (U6) are optional numerics/robustness diagnostics (not needed for logical closure). 3 (U1) Phase-scanned Gaussians on dense grids. Fix scales σj↓ 1 + and grow the integer mode range 1 ≤k≤Kj↑ ∞ . For each σj , scan a phase α on a grid with step ≪B−1 (i.e. ≪ Λ −1 in the PW case and ≪ϑ for Gaussians), yielding a dense family of linear functionals on the prime-phase field and guaranteeing alignment with any off-line zero. (U2) Paley–Wiener probes. When desirable, replace Gaussian f by f∈PWΛ (Fourier support [ − Λ , Λ]) to eliminate aliasing; the prime-band projector then remains stable under modest deblurring. (U3) Prime-power channel only. Work with −ζ′/ζ so blur acts tower-locally on teeth klog p , avoiding composite leakage present on ζitself. (U4) Slepian/DPSS on finite windows. For numerics on bounded u -intervals, use time–band optimizers to minimize leakage into the smooth sector while keeping good concentration. (U5) Multiscale sweep (optional). Run σ on a geometric ladder; at each scale sweep the phase/modes ( α, k ). This separates polynomial archimedean growth from exponential signals due to off-line zeros. (U6) Stochastic dithering (optional). Randomly jitter ( σ, α )within each block and aggregate via median-of-means to suppress coherent leakage and certify many independent small linear readings. (U7) Moment packing. In each block and for the middle object χ, enforce simultaneously max 1≤k≤KjX p≤xj χ(p)k−1 pσj≤ηj,X j ηj<∞, together with zero-count locking on the window. This dense battery of constraints is what the rigidity ultimately consumes (§7). Lemma 3.4 (Deblurring principle).Let f∈PWΛ be band-limited, and let g be holomorphic on Re s≥1. Assume that for a lattice L ⊂ ∂K with mesh ≪Λ−1one has (f∗(g1−g2))L= 0 and sup ∂K |f∗(g1−g2)|≤ε, where ∗ denotes boundary convolution (in the boundary parameter) and g1, g2 are boundary traces of holomorphic functions. Then g1−g2 extends holomorphically across K with interior bound ≪ε . In particular, dense blur matching on the boundary forces an interior discrepancy that is as small as the boundary tolerance. Idea. Paley–Wiener gives f as the boundary trace of an entire function of exponential type Λ. Sampling on a mesh ≪ Λ −1 yields stable reconstruction (frame bounds). The Poisson lift from (2) transfers the boundary bound to the interior with constant 1. Where each blur fits. For logical closure and the detector we default to Gaussians (clean stability, explicit variance control). For alias-free prime-band work and deblurring robustness we switch to Paley–Wiener. For bounded numerical blocks we prefer DPSS windows. Any admissible blur (Def. 3.1) integrates seamlessly with the boundary guards and the four-flow bookkeeping; only constants in the archimedean/localization bounds change. 4 4 Four flows and a BP2 bound Let Lj≍log ( Tj + 3) and take a Gaussian blur with ϑj≍L−1 j (PW variants are analogous). We isolate four nonnegative stagewise quantities—the flows explicitly defined in Definition 4.1: •αj: linear residual of the boundary collocation fit, •βj: quadratic (nonlinear) remainder on the window, •ηj :combined moment defect at a point σj↓ 1 + with a truncation xj (defined precisely in §7), •τj: prime tail beyond a dyadic supply threshold xj↑ ∞. Let m(χj) ϑj denote the blurred Cauchy/Nevanlinna transform associated with the calibrated phase data of ζχj, and write D(χj) ϑj(K) := sup z∈K−Im m(χj) ϑj(z)+. Definition 4.1 (Four-flow budgets: explicit formulas).Fix a window Kj = [ 1 2 + δj, 1] × [ −Tj, Tj ] with boundary mesh Γ j⊂∂Kj , a blur scale ϑj≍L−1 j , a moment point σj = 1 + 1 Lj , and a truncation threshold xj< Xj. Let Hj(s) := log ζχj(s) ζ(s), Bj:= (Xj,2Xj]∩ {primes}. Define the four budgets: αj:= max s∈Γj |Hj(s)|(boundary collocation residual), βj:= sup s∈KjX p∈BjX k≥2 1 kp−kℜs≤X p∈Bj p−2ℜs 1−p−ℜsℜs=1 2+δj (nonlinear log–Euler remainder) ηjas in (4) (combined moment defect), τj:= X p>xj p−σj(prime tail beyond the truncation). All four are nonnegative and finite since ℜs≥1 2+δjon Kj. Proposition 4.2 (Model four-flow bound).For each compact K⋐C+ there exists CK> 0such that D(χj) ϑj(K)≤CKαj+βj+ηj+τj. In particular, if the four flows are summable and the guard εj<log 2, then D(χj) ϑj ( K ) → 0as j→ ∞. Proof sketch. Linear collocation on Γ j controls the boundary discrepancy by αj , while Taylor/Poisson on the collar upgrades boundary → interior with loss ≪βj on K . The truncated first moment and its mean-square defect jointly bound the prime channel on [1 + 1 /Lj, 1+2 /Lj ] by ηj, and the dyadic tail contributes τjby (1). Combine with (3). Remark 4.3 (What the flows measure). αj and βj are purely “geometric” (boundary/interior fit); ηj and τj are “arithmetic” (moment and tail). The dyadic prime supply on blocks Bj = ( Xj, 2 Xj ] makes τjsummable; the combined moment defect ηjis designed to force triviality of χ. Remark 4.4 (Cutoffs).We use Xj for the dyadic block base in Bj = ( Xj, 2 Xj ](prime-supply window), while xj denotes the truncation threshold in the moment/tail controls. These are chosen independently, with xj< Xj for rigidity (so new primes cannot alter the truncated constraints), and xj↑ ∞ ensuring τj=Pp>xjp−σjis summable. 5 5 Boundary calibration, Poissonization, and stability Calibration of the boundary phase. Let Θ (ζ) : R→R be a calibrated argument of ζ ( 1 2 + iλ ), chosen continuous on each zero-free interval and increasing by π times the multiplicity when crossing a zero ordinate. Write dµ(ζ):= 1 πdΘ(ζ), understood in the distributional sense. With the standard archimedean normalization (absorbed into the additive real constant in the Herglotz form), the calibrated boundary measure dµ(ζ) is purely atomic on the critical line, with mass equal to the multiplicity at each ordinate. Poissonized Herglotz transform (exact identity). Let {ψϑ}ϑ>0 be any admissible even blur family with Rψϑ= 1 and set dµ(ζ) ϑ:= ψϑ∗dµ(ζ). For z=x+iy ∈C+define m(ζ) ϑ(z) := ZR1 λ−z−λ 1+λ2dµ(ζ) ϑ(λ)+c0, c0∈R. Then the imaginary part is a Poisson convolution: Im m(ζ) ϑ(x+iy) = Py∗(ψϑ∗dµ(ζ))(x), Py(t) := 1 π y t2+y2.(2) In particular Im m(ζ) ϑ≥ 0on C+ whenever dµ(ζ) is a positive measure, and the blur simply mollifies the boundary data before Poisson lifting. TV–Lipschitz stability for the interior imaginary part. Let ν1, ν2 be finite signed measures on R and form mϑ [ νℓ ]as above with dµ(ζ) replaced by νℓ . Using (2) and that Py≥ 0, RPy= 1, we have for every compact K⋐C+, sup z∈KIm mϑ[ν1](z)−Im mϑ[ν2](z)≤ ∥ψϑ∗(ν1−ν2)∥TV ≤ ∥ν1−ν2∥TV.(3) Consequently, if the calibrated boundary phases of two objects differ by a measure of total variation ≤ε on R , then their interior imaginary parts under the same blur differ by at most ε on any fixed compact K⋐C+. Remark 5.1 (Model → target transfer as a corollary).With ν1 = dµ(ζ) and ν2 = dµ(χj) the guard supplies ∥ν1−ν2∥TV ≪εj . Applying (3) recovers the estimate in Lemma 6.1 with an explicit constant (indeed, 1). Remark 5.2 (Parameter map at a glance).Our default scales are: Gaussian blur of variance L−2 (so ϑ≍L−1 ), or Paley–Wiener with bandwidth Λ ≍L ; window height T∼ecL ; mesh spacing h∼ε/Llog j on zero-free collars. All constants in (2) – (3) are scale-free, shifting only by harmless factors under the admissible choices in §3. 6 Reduction to ζ Define the calibrated phase difference measure on the boundary line by Hj = log ( ζχj/ζ )and let νjbe its (real) boundary increment distribution. The guard yields ∥νj∥≪εj. Lemma 6.1 (Model-to-target transfer).For any compact K⋐C+and ϑj≍L−1 j, sup z∈KIm m(ζ) ϑj(z)−Im m(χj) ϑj(z)≤εj. Consequently, D(ζ) ϑj(K)≤D(χj) ϑj(K)+εj. Proof. This is (3) with ν1=dµ(ζ)and ν2=dµ(χj)and the guard bound ∥ν1−ν2∥TV ≤εj. 6 7 Detector and rigidity We use a Gaussian (or PW) probe against −ζ′/ζ . If a zero lies off the critical line by a distance δ > 0, the probe produces an exponential response exp ( 1 2τ2δ2 )(Gaussian; PW gives exp ( c Λ δ )). Under the stagewise Fit+Zero+Guard, the prime/archimedean side grows only polynomially unless the first-moment channel misbehaves. Lemma 7.1 (Detector dichotomy: quantitative).Fix y≍L−1 . For a Gaussian blur of variance τ2≍L−2, a zero at s0=1 2+δ+iγ contributes to −ℑ m(ζ) ϑ(x+iy)an amount ≫expc τ−2δ2(after phase alignment). Conversely, if all zeros intersecting the probed band satisfy ℜs = 1 2 (i.e. there is no off-line mass in that band), then the total zero-side contribution from |γ|≤T≍ec′L is O ( LC ). For f∈PWΛ the bounds read exp(cΛδ)versus O(ΛC). Idea. Insert the zero term into the smoothed explicit formula and evaluate against the Gaussian/PW kernel; the off-line displacement yields the stated exponential. Under RH, Poisson summation on the critical line gives at most polynomial growth in the bandwidth/variance. Combined moment defect (what we enforce) Fix decreasing σj↓ 1 + , increasing cutoffs xj↑ ∞ with xj< Xj (so new primes cannot influence the truncations), and for each j a finite mesh Σ j⊂ [1 + 1 /Lj, 1+2 /Lj ]with mesh size ≫L−2 j . Define the combined moment defect ηj:= max σ∈ΣjX p≤xj χ(p)−1 pσ | {z } truncated first moment on a mesh + X p≤xj |1−χ(p)|2 p!1/2 | {z } quadratic (mean-square) defect .(4) In the stage schedule (§10) we enforce ηj≤bjwith bj∈ℓ1. Proposition 7.2 (Rigidity from truncated mean-square control).Assume there exist xj↑ ∞ such that ∞ X j=1 X p≤xj |1−χ(p)|2 p!1/2 <∞. Then χ(p) = 1 for every prime p; i.e. χ≡1. Proof. Fix a prime p0. Take jso large that xj≥p0. Then |1−χ(p0)|2 p0 ≤X p≤xj |1−χ(p)|2 p−−−→ j→∞ 0, hence | 1 −χ ( p0 ) | → 0. Since |χ ( p0 ) | = 1, this forces χ ( p0 ) = 1. As p0 was arbitrary, χ≡ 1. Remark 7.3 (Why the first-moment mesh is still included).The quadratic term alone yields triviality, as shown. We keep the truncated first moment on a mesh in (4) because it interfaces naturally with the detector: it prevents thin phase conspiracies inside dyadic blocks and makes the exponential-versus-polynomial dichotomy numerically rigid. 7 8 BP2 and the Herglotz/HP bridge Fix a positive, even blur kernel family {ψϑ}ϑ>0 with RRψ1 = 1 and scaling ψϑ ( t ) = ϑ ψ1 ( ϑt ) (Gaussian or PW are admissible). For the target ζ , define the blurred boundary measure dµ(ζ) ϑ:= ψϑ∗dµ(ζ)and the blurred Cauchy/Nevanlinna transform m(ζ) ϑ(z) = ZR1 λ−z−λ 1+λ2dµ(ζ) ϑ(λ)+c0, z ∈C+, c0∈R. Definition 8.1 (BP2).Let K⋐C+ be fixed. Along any blur scale with parameter ϑ = ϑ ( L ) ≍ L−1and L→ ∞ (e.g., Gaussian variance L−2or PW bandwidth Λ≍L), set D(ζ) ϑ(K) := sup z∈K−Im m(ζ) ϑ(z)+. We say that (BP2) holds on Kif D(ζ) ϑ(L)(K)→0as L→ ∞. Guard budget and BP2 trigger (lightweight closure) Let γj := min∂Kj|ζ| be the moat on the stage boundary (nonzero under the guard/collar assumptions). Define the adaptive guard εj:= min n1 2log 2, γj2−jo. Lemma 8.2 (Summable guard).Let Uj be a boundary collar of ∂Kj free of zeros/poles and set Llog j:= sup Uj(log ζ)′(s)+(log ζχj)′(s). By Cauchy’s estimates on the zero-free collar, Llog j<∞. With the mesh spacing hj=εj 2Llog j , we have maxΓj|Hj|≤εj/ 2, hence by the Lipschitz bound on Uj , sup∂Kj|Hj|≤εj<log 2as claimed. Proof. Since εj≤γj 2 −j , summability is immediate: Pjεj/γj≤Pj 2 −j<∞ . On the collar Uj , Cauchy’s estimates (hence a mean-value Lipschitz bound) give |Hj ( s ) −Hj ( t ) |≤Llog j|s−t| for s, t connected by a boundary arc contained in Uj . Choosing a mesh Γ j⊂∂Kj with spacing hj = εj/ (2 Llog j )ensures maxΓj|Hj|≤εj/ 2, hence by the Lipschitz bound sup∂Kj|Hj|≤εj . Finally, εj≤1 2log 2 yields sup∂Kj|log ζχj−log ζ|<log 2. Lemma 8.3 (Existence of a summable stage schedule).There exist increasing windows Kj = [ 1 2 + δj, 1] × [ −Tj, Tj ] ⋐C+ with zero-free collars Uj , overlapping meshes Γ j⊂∂Kj (with Γ j⊂ Γ j+1 ), blur parameters Lj≍log ( Tj + 3) and ϑj≍L−1 j , dyadic prime blocks Bj = ( Xj, 2 Xj ] with Xj↑ ∞ (and old primes frozen), moment points σj = 1 + 1 Lj , and tail cutoffs xj↑ ∞ with xj< Xj, together with boundary guards εj:= minn1 2log 2, γj2−jo, γj:= min ∂Kj |ζ|>0, such that the four flows and the guard satisfy αj+βj+ηj+τj+εj∈ℓ1(hence each is summable). Consequently, with the overlap protocol (freezing old primes and enlarging Kj,Γj), the zero transfer on Kjpersists stage by stage and the model-to-target comparison remains controlled. 8 Proof sketch (diagonal construction). Fix a target decay bj := 2 −j−3 and ensure each budget αj, βj, ηj, τj, εj≤bj. (1) Zero-free collars and guard. Because the zeros of ζ are discrete, shrink the boundary slightly so each ∂Kj admits a thin zero-free collar Uj ; set γj := min∂Kj|ζ|> 0. Choose εj = min{1 2log 2 , γj 2 −j} . By Lemma 8.2, with spacing hj = εj/ (2 Llog j )(with Llog j as in Lemma 8.2) we get sup∂Kj|Hj|≤εjand Pjεj/γj≤Pj2−j<∞. (2) Blur/scale. Take Tjincreasing so that Lj≍log(Tj+ 3) ≥jand ϑj≍L−1 j. (3) Linear boundary fit ( αj ). Let mj := | Γ j| with hj as above. Choose the dyadic block size so that |Bj|≥C0mjlog(2 + mj), then solve the boundary collocation on the fresh variables {χj+1 ( p ) }p∈Bj (old primes frozen). Standard least-squares/condition estimates on zero-free collars yield αj≤bj. (4) Quadratic remainder ( βj ). The nonlinear drift on the window is O ( h2 j )once the linear fit is in place (Taylor on the collar plus Poissonization). With hj≍εj/Llog j and Lj≥j , choose Tj mildly faster if needed so that βj≤bj. (5) Combined moment defect ( ηj ). Reserve a tiny slice of degrees of freedom inside Bj to ensure the constraints in (4) hold with xj< Xj: max σ∈ΣjX p≤xj χ(p)−1 pσ≤bj,X p≤xj |1−χ(p)|2 p≤b2 j. Because xj< Xj , these constraints touch only frozen primes; feasibility forces their phases to be trivial, in line with Proposition 7.2. The tiny pilot adjustments used for the boundary fit are orthogonal to these truncations and do not spoil the guard. (6) Tail (τj). Choose xjso large that τj:= X p>xj p−σj≤bjwith σj= 1 + 1 Lj. For example xj:= exp(L2 j)gives τj≪exp(−cLj)≤bjfor large j. (7) Overlap and freezing. Enforce Kj⋐Kj+1 ,Γ j⊂ Γ j+1 , and freeze χj+1 ( p ) = χj ( p ) for p≤Xj . Then past guards/zero counts persist by logarithmic Rouché, and the only new contribution to Hj+1 −Hjcomes from primes in Bj, which is controlled by τj. Summing bjgives Pjbj<∞, hence each budget is summable. Corollary 8.4 (Four flows + guard ⇒ BP2).Fix a compact K⋐C+ . If the four flows satisfy αj + βj + ηj + τj∈ℓ1 and the guard εj∈ℓ1 (with the choice above), then along a blur scale ϑj≍L−1 j, D(ζ) ϑj(K)−−−→ j→∞ 0, i.e. (BP2) holds on K. Remark 8.5 (Minimal stop rule, recorded once).To keep the mesh small while staying in the small-target regime, one convenient “once-and-for-all” stop rule at stage jis: |Bj|≥C0mjlog(2 + mj), mj:= |Γj|, hj=εj 2Llog j , which is always reachable by enlarging the dyadic Xj . This choice guarantees the collocation residuals fit within εj/ 2and leaves a small pilot to stabilize the moment constraints; the boundary guard then upgrades mesh→boundary→interior as above. 9