# Verification report

On the annulus exp(-pi/2)<|z|<exp(pi/2), based at1, every outer-boundary Martin kernel with |theta-pi|<=1/40 lies strictly below the Harnack envelope throughout one common punctured neighborhood of1, with gap at least |log z|/25. Four competing normalized kernel gradients and analytic series bounds prove the statement. No curve of equality, hence no Green line of equality, can issue from the base point for any of these boundary kernels.

Hayman-Lingham Problem7.42, pp.173-174, attributed to M.Arsove and G.Johnson: does the simply connected Martin-kernel/Green-line Harnack contact property persist for multiply connected Green domains? A smooth bounded circular annulus is a complete negative answer to this yes/no extension question.

# Detailed originating-system self-audit

1. The domain is bounded, connected and doubly connected with two analytic circles. The base1 is interior. Logarithmic coordinates use a2pi-periodic strip of widthpi; locally they are injective near1.
2. Each Poisson summand cos(u)/(cosh(v-theta+2pi m)-sigma sin(u)) is positive and harmonic. Denominators cannot vanish in the open strip. Exponential tails justify termwise derivatives on compact substrips and uniformity in the compact parameter arc.
3. The Green function has the correct logarithmic pole and zero boundary data. Its boundary derivatives have the signs shown. The factors2pi and the change between angular measure/arclength cancel in the normalized quotient. The four competitors are actual positive harmonic functions equal to1 at the same basepoint.
4. Minimality is proved, not assumed: the circle average of a positive harmonic function is affine in u and has bounded finite endpoint limits; weak limits of the boundary measures on exhausting cylinders give the two-measure Poisson representation. Fourier coefficients solve a second-order ODE and recover the endpoint measures, proving uniqueness. Splitting a point-mass representation proves each kernel is minimal. The Green quotient identifies its Martin endpoint.
5. The base derivatives are (sigma T(theta)/S(theta),-S'(theta)/S(theta)). Reflection theta->-theta changes only the vertical sign, and switching boundary component changes only the horizontal sign. Thus all four rectangle vertices occur.
6. Exact bounds: sech1>1/2; tanh1>1/2; e^(2pi)>400; the noncentral tail is less than8/399<1/25. This gives a>25/104>1/5 and b>21/104>1/5. No cancellation is incorrectly dropped: each signed tail term is bounded below by minus its absolute value.
7. T/S is a positive weighted average of sech terms. On |theta-pi|<=1/40 their arguments have absolute value at least pi-1/40. Hence c_theta<4/39. The identities |S'|<=S and |S''|<=S give |(log S)''|<=2 and, using S'(pi)=0, |d_theta|<=1/20.
8. The support function of the four gradient vertices is a|u|+b|v|. Subtracting the target linear term gives at least19/195|u|+3/20|v|, and hence9/100 times the Euclidean norm. This works in every direction, including negative u and v.
9. All functions have value1 at the base. Compactness of the target arc and uniform C2 bounds give one Taylor remainder constant. Shrinking the radius makes9/100 r-C r^2>=r/25. No specific numerical radius is claimed or needed.
10. H_1 is at least the maximum of the four admissible competitors; the proof does not presume it is harmonic, differentiable, or given by one kernel everywhere. A strict gap on a punctured neighborhood rules out any continuous equality curve issuing from1. The positive-length boundary arc also has positive harmonic measure because its Poisson density is positive.
11. Thirteen exact symbolic/rational checks,42 independent Fourier comparisons,10 Green quotient checks and288 sampled directional gaps support the analytic calculation. Floating-point samples do not prove the quantified continuum theorem. The all-points statement follows from the written uniform bounds. No independent-review or formal-verification claim is made.
12. The exact source question has one yes/no extension clause, unlike bundled multi-part source records. This example answers it negatively in full. We do not classify every multiply connected domain or claim absence of contact at all other boundary points. Classical ingredients and bounded priority search are explicit.

No gap was found in this self-audit. The accepted result is one complete negative answer to AMR-022-7042.

# Bounded literature review,26October2026

Primary sources inspected: Hayman-Lingham1809.07200v2, Problem7.42 and its update; Herron1987, DOI10.2996/kmj/1138037356, complete extracted text with emphasis on Theorems1,4,5; Bear-Smith, An extension of the Schwarz-Pick lemma, author manuscript https://math.hawaii.edu/~wayne/mathpapers/Harnack2.pdf , Sections2-4 including Theorem4.5 and Corollary4.6; Vuorinen1982, DOI10.5186/aasfm.1982.0732, introduction at https://www.acadsci.fi/mathematica/Vol07/vol07pp259-277.pdf .

Queries combined Arsove/Johnson with Harnack, Martin kernel and Green line; exact Problem7.42 with Harnack; annulus/Poisson-kernel Harnack extrema; and Harnack noncontact. The frozen HF record had no existing discussion mention in the fresh667-event collision snapshot. No matching explicit resolution was located. Search-engine date labels for old PDFs were not treated as publication dates.

Herron is directly relevant classical background, not a new result of this work. Bear-Smith concerns strict comparison with hyperbolic distance and rigidity of maps from a simply connected cover; it is not a statement that every fixed normalized kernel contacts the envelope along a Green line. Vuorinen's introduction concerns boundary behavior of the broader class called Harnack functions, which differs from the particular envelope in the question. No complete audit of Vuorinen's paper is claimed. The present proof independently establishes the specific local noncontact arc. The review is bounded and does not certify exhaustive novelty or priority.

All five final PDF pages were directly visually inspected. No clipping, missing symbols, unresolved references or layout defects were found. Built-in compilation and Tectonic succeeded. Name is prominent; affiliation/contact is a footnote; AI assistance appears in prose.