An analytically quasiconformal homeomorphism distorts quadrilateral moduli by at most K

For this ring clause, an end path is a proper continuous $\gamma :(0,1)\to \tilde{A}$ whose two tails approach different exhaustion ends. It is locally rectifiable if all compact parameter restrictions are rectifiable; define ${\mathrm{\ell }}_{\rho }(\gamma )={\sup }_{0<a<b<1}{\mathrm{\ell }}_{\rho }(\gamma {\mathrm{\mid }}_{[a,b]})$, setting this to infinity if any such restriction is nonrectifiable. Define $\lambda$ by the same supremum of ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2\mathrm{/}A(\rho )$ over finite positive-area Borel densities and $\mu =1\mathrm{/}\lambda$, including zero/infinite reciprocal conventions. This local end-path convention allows nonlanding tails, punctures and infinity; it requires no finite global Euclidean length. For a globally rectifiable compact physical-boundary path, atomless arc-length measure makes this improper integral equal its ordinary integral. Compact-trace paths of infinite Euclidean length form a zero-modulus family: for traces in a fixed ball, a constant density on that ball has infinite improper length and arbitrarily small area after scaling; take the countable union over balls. Removing this family preserves modulus by the vanishing-cost barrier argument. Thus the finite-boundary family values agree with the compact-path convention even though end paths allow merely local rectifiability. Scaling densities gives the admissible-area characterization exactly as in the compact convention Extremal length and the curve-family modulus of a path family.

Write the spherical complement of $\tilde{A}$ as disjoint compact connected sets $K_0,K_1$. Each is nonseparating: the connected domain lies in one component of the open complement of $K_0$; any other component U would lie in the closed $K_1$, while its nonempty boundary lies in $K_0$, contradicting disjointness. For a finite-chart nonseparating continuum K, cover it by finitely many small disks centered on K. Their union is connected because each component meets the covered connected K. Slightly perturb radii within the positive cover margin to avoid the finitely many tangencies/triple intersections. Exposed boundary arcs then form disjoint finite degree-two cycles, hence Jordan curves. Connectedness gives one outer cycle; the others bound holes. Fill those holes to obtain a closed Jordan neighborhood. These neighborhoods can be nested: choose the next disks in the preceding interior, and filling cannot cross the preceding outer boundary. They shrink to K, since each outside point has a path to infinity avoiding K at positive distance, which small enough disk unions miss, placing that point in their exterior. Normalize each $K_i$ in a chart avoiding the other and choose disjoint nested neighborhoods. The closed cores between their Jordan boundaries are compact, nested and exhaust $\tilde{A}$. Their complement in the domain has exactly two connected collars: any point in a neighborhood minus K can follow a path avoiding K to its first outer-boundary hit, and all hits connect through the thin inner boundary collar, disjoint from K. Thus each proper tail eventually lies in one consistent collar, and there are exactly two ends. A homeomorphism is proper as a map of these locally compact domains and carries this exhaustion and its two collar components to an exhaustion and two collar components of A. Therefore f induces an end bijection and maps their end-path families bijectively. No extension to individual wild boundary points is asserted.

On a countable compact-neighborhood exhaustion of the source use the local barriers of [F3]. Extend each by zero and multiply by positive constants so their $L^2$ norms have finite sum. Their sum G belongs to $L^2$. Every path with a bad compact restriction has infinite G integral; the family Z of such end paths has modulus zero, since G/m is admissible there with area tending to zero. Outside Z, each image compact restriction is AC and has the weighted speed identity. For a nonnegative Borel target density $\tau$, the local chain rule gives ${\mathrm{\ell }}_{\tau }(f\circ \gamma {\mathrm{\mid }}_{[a,b]})\le {\mathrm{\ell }}_{(\tau \circ f)\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}}(\gamma {\mathrm{\mid }}_{[a,b]})$. Increasing compact parameter intervals to (0,1) proves the same inequality for the improper lengths, including infinite values. The lower-area inequality and $\mathrm{\parallel }Df{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}^2\le K{J}_f$, exhausted over compact source sets, give pullback area at most $KA(\tau )$. If $\tau$ is target admissible, the pullback is admissible outside Z; add G/m to handle Z and let its $L^2$ norm tend to zero. Taking infima gives $\mu (\mathrm{\Gamma }(\tilde{A}))\le K\mu (\mathrm{\Gamma }(A))$. This is only one direction.

Apply step 3.1 separately to the independently K-QC inverse in [F2]. Its end correspondence is the inverse of that in step 2.1, so it gives $\mu (\mathrm{\Gamma }(A))\le K\mu (\mathrm{\Gamma }(\tilde{A}))$. Together the two inequalities give the claimed annular bounds. Reciprocals give the extremal-length bounds even when one value is zero or infinite. The finite-round numerical constants follow from [F1]. This argument uses exhaustion for individual lengths and density areas, not an unproved continuity of modulus under arbitrary ring-core exhaustion.

The quadrilateral assertion is [F1], and f carries its joining family onto the full image family because it is a homeomorphism on a neighborhood of its compact closure. Area equality on all relatively compact Borel sets is [F2]; applying it to both f and its independently regular inverse proves their null-set preservation. This establishes the entire area and inverse-null clause and completes all three assertions without using any circular-dilatation or MRMT supplier.