# Verification and scope report An attained dual Hilbert-transform test characterizes almost-everywhere positive-real-part modulus majorizability. The coefficient pi/2 is sharp and rational squared-polynomial tests suffice. Explicit separated spikes have bounded weak-L1 quasinorm but no majorant, while their symmetric decreasing rearrangement has one; N spikes require center modulus between N/8 and N+1. # Dual tests and spatial obstructions for positive-real-part majorants 26 October 2026. Complete written arguments for the stated results; publication and novelty decisions are separate. No independent reviewer or formal proof assistant is being represented as having checked this work. ## 1. Conventions and the logarithmic representation Let T=R/(2 pi Z), dm=dt/(2 pi), and define the real periodic Hilbert transform H by the Fourier multiplier -i sign(n), with H1=0. It is a bounded isometry on real mean-zero L2 and is skew-adjoint there. In particular H maps L-infinity into L2 on this probability space. Let P denote Poisson extension. If F is holomorphic with Re F>0, the branch L=log F with imaginary part in (-pi/2,pi/2) is well defined. Its bounded harmonic imaginary part is P theta for a real theta in L-infinity with |theta|<=pi/2. Harmonic conjugacy and the L2 Fourier multiplier give Re L = C - P(H theta), C = log |F(0)|. There is no hidden singular harmonic term: P theta determines its harmonic conjugate uniquely up to the real constant C. Thus L belongs to H2 after addition of its constant, and has finite nontangential boundary values a.e. In particular log |F*| = C - H theta a.e. (1) Conversely, for real theta with |theta|<=pi/2 and real C, the harmonic pair L_theta(z) = C - P(H theta)(z) + i P theta(z) is holomorphic. One may verify this on Fourier coefficients, or first for trigonometric polynomials and then by local L2 convergence. If theta is not identically one of the two endpoint constants, the maximum principle gives |P theta|=0, integral g dm=1 }. (2) **Theorem A.** The following are equivalent: 1. f<=|F*| a.e. for a holomorphic F with Re F>0; 2. D(h)=integral h dm>=0. **Clamping step.** If F majorizes f, then G=F+1 has positive real part and |G*|>=max(1,|F*|)>=max(1,f), because |1+w|^2=1+2 Re w+|w|^2 for Re w>=0. This changes the optimal center modulus, so (3) is stated only for the clamped datum. It does not incorrectly identify the optimum for the original, unclamped f. **Necessity.** Apply (1) to a majorant of max(1,f). It gives (3)'s lower bound, since for every normalized nonnegative smooth g, integral h g <= C - integral (H theta)g = C + integral theta Hg <= C + (pi/2)||Hg||_1. It also gives h in L1 (indeed h is bounded above by a nonnegative L2 function). This proves the required necessity without assuming in advance that an arbitrary boundary logarithm is integrable. **Sufficiency and attainment.** Suppose D(h)=C= integral h g - C integral g. (4) Each functional is weak-* continuous because Hg is smooth and integrable. These constraints have the finite intersection property. To see this, consider a finite collection g_1,...,g_k. If its feasible set were empty, separate the compact convex image of B in R^k from the closed upper orthant with corner a_j=integral h g_j-C integral g_j. A separating normal must have all coordinates lambda_j>=0. It would imply sup_{theta in B} integral theta H(sum lambda_j g_j) < integral (h-C) sum lambda_j g_j. The left side is exactly (pi/2)||H(sum lambda_j g_j)||_1, contradicting (2) after normalization of the nonzero nonnegative sum. Thus compactness gives one theta satisfying all constraints. Because h and H theta are integrable, (4) is precisely the distributional inequality h<=C-H theta, hence it holds almost everywhere. Section 1 constructs a positive-real-part F with this majorant and log|F(0)|=C. The endpoint-constant exception is handled there by a positive constant. This proves every assertion of Theorem A. The proof also works for any real C>=D(h); no strict inequality is needed. ## 3. A countable testing version The supremum in (2) is unchanged if its tests are restricted to g_p(e^{it}) = |p(e^{it})|^2 / integral |p|^2 dm, 0 != p in (Q+iQ)[z]. (5) For a smooth nonnegative g, its Fejer means are nonnegative trigonometric polynomials, converge uniformly to g, and their Hilbert transforms converge to Hg in L2 and hence in L1. Fejer–Riesz factorization expresses each as |p|^2 for an analytic polynomial with complex coefficients. Approximating those coefficients by rational complex numbers gives uniform convergence of the squared moduli and L2 convergence of their Hilbert transforms. Normalize the integrals. Pairing with h in L1 passes to the limit by uniform convergence. This proves (5). The tests involve only the measurable datum and explicitly specified trigonometric polynomials, rather than an unknown analytic majorant. No effective stopping rule for arbitrary measurable h is claimed. ## 4. The coefficient pi/2 is sharp For g_r=P_r, the normalized Poisson kernel centered at 1, H g_r(t) = 2r sin t/(1-2r cos t+r^2), ||H g_r||_1 = (2/pi) log((1+r)/(1-r)). (6) The norm follows by integration on (0,pi), where the numerator is positive, followed by symmetry. Let f_0(e^{it})=2/|1-e^{it}|. It is at least 1 and is the modulus of 2/(1-z), a positive-real-part function. For h_0=log f_0, Poisson extension gives integral h_0 g_r dm = log 2 - log(1-r). (7) Replacing pi/2 in (2) by any smaller constant a makes its value at g_r tend to infinity as r tends to 1, since the leading coefficient of log(1/(1-r)) is 1-2a/pi>0. Thus pi/2 cannot be reduced in a criterion valid for all positive-real-part majorants. Also D(h_0)=log 2: the constant test g=1 gives the lower bound, and F=2/(1-z) gives the matching upper bound. For a constant datum f=A>=1, D(log A)=log A, another normalization check. ## 5. An elementary weak-type bound and local mass test Every F with Re F>0 has the Herglotz representation F(z)=i b + integral_T (zeta+z)/(zeta-z) dmu(zeta), mu>=0, mu(T)=Re F(0), b=Im F(0). (8) One can obtain (8) directly: take weak limits of Re F(r e^{it})dm, all of mass Re F(0); use the Poisson formula on smaller disks to identify the limit, then fix the harmonic-conjugate constant. This uses no unverified specialized representation theorem. Write G_nu for the integral in (8) with a finite positive measure nu and no imaginary constant, and M_nu=nu(T). For lambda>0 the bounded analytic function Q(z)=G_nu(z)/(lambda+G_nu(z)) has real part between 0 and 1. At a boundary value w with Re w>=0, Re[w/(lambda+w)] >= 1/2 iff |w|>=lambda. The mean-value identity for Re Q therefore proves m{|G_nu*|>=lambda} <= 2 M_nu/(lambda+M_nu) <= 2 M_nu/lambda. (9) Boundary values exist by Section 1 when M_nu>0; for the zero measure there is nothing to prove. The bounded real part of Q justifies passage of its mean to the boundary. Let J be an arc of angular radius rho<=1 and I its concentric subarc of radius at most rho/2. For t in I and s outside J the circular angular distance is at least rho/2. The boundary Herglotz kernel has modulus |cot((t-s)/2)|<=4/rho. Thus the part of (8) outside J is bounded on I by 4 mu(T)/rho. If f>=A on I, |F*|>=f, and A >= 2 (|b| + 4 mu(T)/rho), (10) then |G_(mu restricted J)*|>=A/2 a.e. on I. Equation (9) gives the local mass requirement mu(J) >= A m(I)/4. (11) Restriction to J and its complement is valid even when mu has atoms at the endpoints: the smaller arc I stays a positive distance from the complement, and the latter integral extends analytically across I. ## 6. A weak-L1 function which has no such majorant For integers n>=20 define t_n=1/n, rho_n=1/(10 n^2), J_n=(t_n-rho_n,t_n+rho_n), I_n=(t_n-pi 2^(-n),t_n+pi 2^(-n)), f(t)=sum_{n>=20} 2^n 1_(I_n)(t). (12) All intervals are viewed as arcs in (-pi,pi]. The J_n are pairwise disjoint: consecutive centers are separated by 1/(n(n+1)), larger than rho_n+rho_(n+1). The I_n lie in the middle half of J_n. For the latter, 20 pi n^2<=2^n holds at n=20 using pi<4, and its ratio increases thereafter. In particular m(I_n)=2^(-n). The datum f is finite at every point if f(0)=0 is specified. Its support has only 0 as an accumulation point of the intervals. For every lambda>0, lambda m{f>lambda} <= 2, (13) as follows by summing the geometric tail of their lengths. Therefore f is in weak L1, and log max(1,f) is integrable. Suppose F were a positive-real-part majorant with the fixed finite constants M=mu(T), b from (8). Since 2^n rho_n tends to infinity, (10) holds for every sufficiently large n. Equation (11) then gives mu(J_n)>=1/4 for all such n. The J_n are disjoint, contradicting finiteness of mu. This is an explicit counterexample to the shortcut “weak L1 is sufficient,” not a contradiction to Theorem A. Theorem A instead implies D(log max(1,f))=infinity. ## 7. Finite versions have a linear center cost despite bounded weak-L1 size Let f_N be the sum in (12) over n=20,...,N+19, and let M(f_N)=inf{|F(0)| : Re F>0, |F*|>=max(1,f_N) a.e.}. Then N/8 <= M(f_N) <= N+1. (14) The upper bound follows by taking a Herglotz transform whose boundary real part is max(1,f_N); its value at zero is the integral of that density, which is at most N+1. For the lower bound suppose B=|F(0)|=20, the elementary inequality 2^n>=100 n^3 holds (check n=20 and use 2(n/(n+1))^3>1 afterwards), so 2^n rho_n>=10n. Among the last ceil(N/2) indices, n>=20+floor(N/2)>N/2, and therefore 2^n rho_n > 5N > 10B >= 2(|b| rho_n + 4M), where rho_n<=1 was used in the last comparison. Thus (11) applies on each of these disjoint arcs, giving M >= ceil(N/2)/4 >= N/8, a contradiction. This proves (14), including the complex value F(0); normalizing F(0) to a positive real number by an impermissible rotation has not been assumed. The weak-L1 quasinorm of all f_N remains at most 2. ## 8. The class is not rearrangement invariant Let f^# be the symmetric decreasing rearrangement of (12). By (13), f^#(e^{it})<=2 pi/|t| for 0<|t|<=pi. Since |1-e^{it}|<=|t|, f^#(e^{it}) <= |2 pi/(1-e^{it})|. The right side is the boundary modulus of 2 pi/(1-z), which has positive real part. Thus f and f^# are equimeasurable, but only the latter admits a positive-real-part majorant. One may choose representatives finite at every point by assigning an arbitrary finite value at 0, without changing this a.e. assertion. Distributional integrability information alone cannot characterize the class. ## 9. Scope and remaining bibliographic work Theorems A and (14), the explicit counterexample (12), and non-invariance under rearrangements have complete proofs above. The dual criterion is a variational characterization with only prescribed real test functions, not a finite decision procedure. Its derivation uses classical logarithmic representations, Hilbert-transform duality, compactness, and separation. The separated-spike construction transfers the local-mass idea from the Riesz-majorant dossier AMR-022-5073, with a new elementary Herglotz weak-type bound in place of a positive potential kernel. No claim of absolute priority is justified by the bounded search to date. The exact historical problem is quoted only with its a.e. interpretation. Further literature comparison must distinguish minorant problems for model spaces, radial Korenblum growth spaces, and coefficient-majorization papers from the present boundary-modulus question. # Source and literature review 26 October 2026. The source is Hayman--Lingham, Research Problems in Function Theory, arXiv:1809.07200v2, Problem 5.74, printed page 112 / PDF page 113. The page was rendered and inspected. Its almost-everywhere qualifier is explicit. The frozen statement.for_research has normalized SHA256 174c2d7609c86ba06b850919608bb81b54e7ce59cb5c855cbeb05d163903f66d. The 2018 source's lack of a progress note does not establish open status in 2026. The main analytic ingredients are classical: bounded harmonic Poisson representation, the L2 Hilbert transform and its skew adjoint, the maximum principle, weak-star compactness, and finite-dimensional convex separation. The logarithmic representation and Herglotz representation are derived in the paper, and the local weak-type estimate follows from an explicit Cayley identity. Dritschel--Rovnyak, The operator Fejer--Riesz theorem, arXiv:0903.3639v1, https://arxiv.org/html/0903.3639v1, Section 1, was read for the scalar statement: a nonnegative trigonometric polynomial factors as the squared modulus of an analytic polynomial. No operator version is needed. Herglotz's 1911 bibliography was verified against the Saxon Academy archive: volume 63, pages 501--511. The original full text was not read; the representation used here is proved directly by weak limits and Poisson extension. Bounded searches included Korenblum / positive real part / majorant; Problem 5.74; Herglotz / solid hull / weak; and Hilbert transform / majorant / duality. Searches returned model-space admissible upper majorants, coefficient majorization and radial Korenblum growth-space questions, none of which supplies the exact boundary-modulus result here. No exhaustive priority certification is asserted. A Holland--Twomey integral-means article was not successfully read in full and is not used as evidence or as a theorem dependency. The neighboring internal dossier AMR-022-5073 concerns Riesz-potential majorants and remains partial. Its almost-everywhere/pointwise distinction must not be transferred to this source, which expressly asks for almost-everywhere domination. The present paper has its own complete analytic arguments. Its counterexample refutes sufficiency of weak L1, not the existence of all possible characterizations. The dual test is an exact variational characterization, not a finite algorithm. Absolute novelty and first historical resolution are not claimed or counted. ## Supplementary checks ```json { "cayley_identity_exact": true, "finite_cost_lengths_checked": 400, "formal_verification_completed": false, "hilbert_sign_and_factor_cases": 3, "infinite_arguments": "../proof_audit.md", "novelty_established_by_checks": false, "poisson_antiderivative_exact": true, "quadrature": [ { "norm_error": "8.75811540203010669327330989556197582050163714e-47", "pairing_error": "8.75811540203010669327330989556197582050163714e-47", "r": "0.2" }, { "norm_error": "0.0", "pairing_error": "0.0", "r": "0.5" }, { "norm_error": "0.0", "pairing_error": "3.50324616081204267730932395822479032820065485e-46", "r": "0.9" }, { "norm_error": "3.50324616081204267730932395822479032820065485e-46", "pairing_error": "0.0", "r": "0.99" } ], "source_sha256": "66e399047b3e768eeebe8baf6a51606e53537a8b95e9b66799db8bfdbdf746ea", "spike_indices_checked": 481, "status": "PASS_EXACT_ALGEBRA_AND_SUPPLEMENTARY_QUADRATURE" } ```