# Verification and attribution report

For deterministic positive variances v_n<=C/n with divergent sum and sum v_n/log(n+1)<infinity, the Gaussian lacunary series sum sqrt(v_n) xi_n z^(2^n) almost surely has almost every boundary point Plessner, every Stolz image there has finite set area, and truncated image areas tend to zero. One full-measure set works for every aperture. The explicit logarithmic family has 0<beta<=1.

# Detailed originating-system self-audit — 26 October 2026

The complete written argument is in proof.md. This is a mathematical self-audit, not independent peer review or formal verification.

- The Gaussian convention is E|xi|^2=1. Consequently the Brownian kernel is (pi t)^{-1}exp(-|z|^2/t); the occupation estimate and time clock use this same convention.
- The unit-time sausage bound is proved by occupation after a stopping time, not assumed from a heuristic about Brownian paths. On a hit by time 1, the next unit interval lies inside [0,2]. Integrating the larger target disk over target centers gives exactly 8pi epsilon^2. The lower bound is 8e^{-9}epsilon^2 log(1/epsilon). Both powers cancel correctly.
- Integer clock crossings use strict positivity of v_n and divergence of T_n. Once v_n<=1/2, crossings of successive levels are distinct. Overshoot at level j-1 is <v_{n_{j-1}}, not v_{n_j}. This yields at least half a unit of variance per disjoint block. The denominator inequality has the correct direction.
- A time T_n equal to an integer is assigned to that integer's block. Since n>=n_j, its disk radius is no larger than n_j^{-1/8}. Overlap of adjacent closed time intervals only enlarges the cover.
- Unit-time Brownian increments have the same sausage area distribution after subtracting B_j. Independence between blocks is unnecessary: finite sum of expected nonnegative areas implies finite sum almost surely.
- Area of tail unions goes to zero by summability of the covering areas. Finiteness of the entire union would not suffice on its own.
- The coefficient bound is polynomial and uniform in theta. In the low-frequency shadowing sum, split at n/2 to control the early terms; bounding every k^{-1/4} by n^{-1/4} would be invalid.
- The random coefficient constant is removed from the disk radii by weakening n^{-1/4} to n^{-1/8} for sufficiently large n. Thus the Brownian covering uses deterministic radii and no illicit dependence assumption.
- Circular Gaussian rotation gives equality in law of the whole discrete sequence, for each theta. Fubini is applied to measurable functions of that sequence; it does not require a Brownian motion family indexed measurably by theta.
- Recurrence alone along arbitrary sampling times is insufficient. The reflection-principle estimate explicitly controls all intervals between the chosen times. The v_n<=C/n hypothesis makes the exceptional probabilities summable.
- Plessner's theorem is used only after radial oscillation excludes an angular limit. The conclusion is almost everywhere, never everywhere.
- The full-measure set works for every aperture because the same disk union controls every deterministic aperture after its own finite threshold. No uncountable intersection of probability-one events is used.
- Finitely many initial shells lie in a compact subdisk. Their images are bounded and therefore have finite set area, even if the mapping has high multiplicity.
- The theorem concerns image area without multiplicity. It does not claim the Dirichlet integral over a cone is finite. Beardon-Carne's area conventions and the Gardiner-Manolaki strengthening quoted by Ivrii concern multiplicity and do not contradict this theorem.
- Exact attained range is distinguished from cluster set. Only exact attained range has area zero here; every truncated image is dense.
- The logarithmic coefficient family uses 0<beta<=1. At beta>1 the variance sum converges; beta=0 fails the sufficient summability condition. Neither case is improperly included, and necessity is not asserted.
- Ivrii's prior negative resolution and key proof method are credited. The proposed new content is the finite-area/vanishing-tail-area strengthening and sufficient variance criterion. Bounded searches found no matching prior statement; absolute priority is not certified.

Conclusion: no gap was found in the complete written proof. The finite checks required for publication are provenance, source/PDF portability, metadata, and rendering checks; stochastic simulation would not establish this infinite almost-sure theorem and is not represented as proof.

# Literature and attribution — 26 October 2026

The frozen record is Hayman-Lingham Problem 5.57 (Baernstein), with the weaker Collingwood area question numbered 5.20. The 2018 source's open status is superseded.

1. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2, https://arxiv.org/abs/1809.07200. The local source spans PDF pages 107–108 for Problem 5.57. It asks about the actual image, not merely its closure.
2. Oleg Ivrii, Non-tangential ranges of holomorphic functions at Plessner points, arXiv:2609.18785v1, 16 September 2026, https://arxiv.org/abs/2609.18785. Theorem 1.1 gives the prior negative answer with variances 1/n, density-zero full Stolz images and area-zero exact nontangential ranges. Sections 3 and 5 provide the shadowing and recurrence framework adapted here. Section 7 uses an exponentially shrinking Wiener sausage to prove density zero. Theorem 1.2 gives full Hausdorff dimension of the exact range at almost every Plessner point. This paper has been read directly; the proposed finite-total-area conclusion is not stated there. Its remark about superlacunary exponents concerns horoballs, a different direction.
3. A. F. Beardon and T. K. Carne, Euclidean and Hyperbolic Lengths of Images of Arcs, arXiv:0711.0219v1, https://arxiv.org/html/0711.0219v1, published in Proceedings of the London Mathematical Society 97 (2008), 183–208, doi:10.1112/plms/pdm060. The introductory area A(f) explicitly counts multiplicity. Its finite-area hypotheses are not the set-area conclusion proved here.

Classical Brownian recurrence, the reflection principle, strong Markov property, and Plessner's theorem are used with their usual hypotheses. The only needed Wiener-sausage estimate is proved in full by an occupation argument. No precise sausage asymptotic is imported without proof.

Bounded web searches on this date included: "Plessner" "finite area"; "Plessner" "finite-area"; "Stolz" "finite area" holomorphic; "Stolz" "finite area" "lacunary"; "lacunary" "Stolz" "log" "area"; "Ivrii" "Stolz" "finite"; and "2609.18785" "finite". Primary-source follow-up found the sources above and no exact match for the variance-summability finite-area theorem. This is not an exhaustive priority certification. Search engines and secondary review pages are discovery aids, not mathematical evidence.

The contribution is a short extension of Ivrii's method, not an independent first solution of the historical question. A publication or HF notice must foreground that attribution. Zero new original source closures are credited to this work; one complete strengthening may be counted.