# Verification report

For every compact E in [0,1], viewed in the real line L in the plane, the 2-modulus of all rectifiable curves joining E to L minus E is infinity when cap(B_2,E)>0, and zero when cap(B_2,E)=0. No positive-length or regularity assumption is added. Separated condenser tests, critical Sobolev trace rigidity, finite Mazur combinations and zero-trace approximation prove the positive-capacity case, including null-length positive-capacity sets.

Hayman-Lingham Problem7.58, printed178 of arXiv1809.07200v2, attributed to A.A.Gonchar and communicated by M.Vuorinen. Does positive conformal2-capacity of a compact E in [0,1] force infinite 2-modulus between E and its real-line complement? The answer is affirmative for every source set.

# Originating-system adversarial audit, 26 October 2026

1. The source asks for 2-modulus, not reciprocal extremal length. All plane
curves connecting E to its line complement are used; no separation of the
two final plates or regularity of E is assumed.
2. F_j is compact and at distance at least 1/j from E. G_j is open and
connected through any ball about a point of E. Its boundary is contained
in F_j union the outer circle. Hence both types of condenser curves are
controlled. The compact plate lies strictly inside G_j.
3. rho_0 has squared L^2 norm 2pi and integral at least one on every
E-to-radius-three curve. Squaring rho+rho_0 gives the uniform bound
2||rho||_2^2+4pi. No modulus continuity for touching plates is assumed.
4. The cited classical equality applies to arbitrary compact plates and
domains, not just smooth curves. Smooth plateau tests follow by doubling,
truncation and mollification; a factor-four bound suffices, avoiding an
unproved claim that mollification preserves values at a limiting plate.
5. Supports lie in B_3, so Poincare controls L^2. Weak compactness alone
would not preserve pointwise values; finite convex combinations of tails
retain exact plateaus and zeros and converge strongly by Mazur.
6. The trace identification uses L^2 limits on each F_m and on E. Their
countable union covers the line up to the two outer endpoints, which do
not affect the trace. The trace outside the disk is zero.
7. The moment formula has sign -h|E|, and support [-h,1] has |x|<=1.
The L^1/L^2-square identity is specific to a difference of indicators.
The logarithmic divergence establishes |E|=0, not capacity zero.
8. The essential extra step uses zero-trace density. The explicit vertical
cutoff estimate is valid for almost every Sobolev slice; its contribution
tends to zero by absolute continuity of the gradient integral. Subsequent
mollification leaves a strip disjoint from the line.
9. Each v_k is a finite sum with a nonempty plateau neighborhood of E.
Each w_k has compact support away from the line. Their difference is
therefore one near E even when E has length zero and positive capacity.
Convergence is in full H^1, which controls the derivative of the final
cutoff in B_2. This is the contradiction in the critical Cantor-set case.
10. For zero capacity, localized small-energy plateau functions exist by
Poincare and multiplication by a fixed cutoff for each n. No uniform
bound on the cutoff derivatives in n is needed: the energy is made small
after the cutoff is fixed. The series of nonnegative gradient magnitudes
has L^2 norm at most delta. Every fixed endpoint outside the closed E
eventually leaves all supports. Tonelli and the chain rule apply to all
rectifiable curves, not only almost every curve.
11. No computation certifies the general theorem. The proof is analytic;
supporting checks are reserved for the moment, strip and energy formulas.
Neither independent review nor proof-assistant verification occurred.
12. Classical trace, capacity and critical rigidity tools are credited.
The bounded literature search found no matching resolution of the exact
Gonchar question; it does not certify absolute priority.

The complete written argument passes this self-audit. It answers the
positive-capacity source question and also proves the zero-capacity case.
These are one dichotomy, one manuscript and one frozen-record closure.

# Bounded prior-art review, 26 October 2026

The exact source is Hayman-Lingham2018v2, Problem7.58. The current review
used primary sources Nasser-Vuorinen2026 (DOI10.1007/s10958-025-08141-0),
Bourgain-Brezis-Mironescu2015 (DOI10.4171/JEMS/551), and inspected parts
of Shlyk1994 (MathNet im859). See source_audit.md for the precise reading
scope; full-book or full-citation-chain review is not claimed.

Queries combined Gonchar/Vuorinen with capacity, complement, real line,
infinite modulus and Sobolev; exact Problem7.58 with solved/update;
compact subset of the real axis with modulus; extremal distance with
real-line complement; and condensers with touching plates. A secondary
search result for the 2019 book mentions unclear terminology but was
not used as primary evidence or as a mathematical premise. Our density,
curve and capacity definitions remove that ambiguity explicitly.

The fresh public HF668-event snapshot had no mention of AMR-022-7058.
No matching published resolution was located by these bounded searches.
This is not an exhaustive priority certification. Critical trace rigidity,
zero-trace density, weak compactness, Mazur's lemma, and the capacity-modulus
identity are classical. The contribution claimed is their explicit
application, with the null-length capacitary step supplied, to the precise
Gonchar question and its zero-capacity counterpart. No novelty is claimed
for the tools or for the general zero-capacity exceptional-curve principle.

All four final PDF pages were directly visually inspected. No clipping, missing symbols, unresolved references or layout defects were found. Built-in compilation and Tectonic succeeded after renaming a conflicting custom capacity macro. The name is prominent; affiliation and contact are a footnote; AI assistance appears in prose.