Proposition 16 ([Nondegeneration]). For the quantiles in, with ε chosen as in Lemma 14 ([Aligned annular barriers]), _ m→∞ q_m>0. The conclusion holds along all rates in the chosen countable set.

Suppose otherwise. Choose approximate record lows m→∞ such that b_m:=q_m→0 and q_s≥c b_m (1≤s≤m, s in the chosen rate set) for a fixed c>0 . The bounded-rate lower bounds ensure m→∞ . For each fixed i , the quantile at m2^ -i therefore gives the high-probability barrier event in units b_m . Extract jointly with all these backward offsets. Lemmas Lemma 14 ([Aligned annular barriers]) and Lemma 15 ([Densities controlling displacement]) apply in the comparison region. The first inequality of and Lemma 12 ([Densities and traffic]), on the other hand, supply with positive probability one graph's probability flow ν_m between the two macroscopically separated ports, confined in a fixed compact set, with ∑_e j_ ν_m (e)^2≤Cb_m. Retain the event indicator, choose a graph on a positive-probability subevent, and work pathwise in a further coupling. We establish two properties of the projected flows before using locality. Compactness of the flows. For a path P , let J_h(P) be the maximum number of disjoint subintervals making a projected advance of at least h . Choose N large enough that CR_N<ch/2 in , and then take m large. Use a fixed larger box union U containing the compact confinement. By ,, and Cauchy--Schwarz, ∫J_h(P)\,ν_m( P)≤C/h. Thus the paths are tight after reparameterization. One explicit choice of parameters is useful. At every dyadic h=2^ -l place clock atoms at successive completed moves of distance h , each with mass 2^ -l /(1+J_h(P)) . Add the original traversal time of total mass one, invert this clock with pauses at its atoms, and rescale its total duration to one. On sets where every J_ 2^ -l has a fixed bound, these parameterized paths are equicontinuous: an oscillation of several times 2^ -l must cross a whole atom interval of a fixed positive duration. The bounds and a countable union bound make such sets have arbitrarily large flow probability. The finitely many initial resolutions omitted at an individual scale can be included by enlarging its bound. A subsequence therefore converges to a probability law ν on continuous paths. Their endpoints remain separated. An absolutely continuous measure of arclength. For at most J disjoint path intervals compactly confined in an open box union U , the same argument gives ∫∑_ k=1 ^ J |π_m(P(t_k))-π_m(P(s_k))|\,ν_m( P) ≤C√ (U^ +CR_N ) +CJR_N+o_m(1). The supremum may be taken over rational intervals with closure in the prescribed open condition. It is lower semicontinuous under uniform convergence. Pass first to the path limit, next let N→∞ , and finally let J→∞ . These interval sums exhaust variation on open sets. With a larger U containing all paths, they first show that ν -almost every path is rectifiable with integrable length. Its average arclength measure Λ consequently satisfies 0<Λ()<∞, Λ(U)≤C√(U). The positivity follows from the separated endpoints. Open box covers now show Λ . We next convert a tangent of a limiting path into a local event. Choose a sufficiently fine finite grid W of unoriented directions, closed under quarter-turn and reflection in a coordinate axis. For z∈ , w∈ W , and s>0 , consider the tube T(z,w,s)= z+tw+lw^⊥: |t|<2s, |l|<s/16 . A strict crossing test in this tube has ports strictly beyond the two lines t=-s and t=s , and has confinement and ports with positive clearance from the tube boundary. It is computed on a complete local prescription. Accept all countably many selectors with these properties, and require that the prescription be readable within a fixed-factor enlargement of the tube, with positive margins. Lemma 5 ([Spatial and value selectors]) makes existence of such a test independent of the particular selector library after arbitrarily small relaxation of the margins. At Λ -almost every point there is a direction w∈ W for which, at every sufficiently small dyadic scale s , one of these tests on the graph supplying ν_m has finite limiting /b_m . Indeed, arclength parameterization gives a unit tangent at almost every parameter of almost every rectifiable path. The fine direction grid puts that tangent inside one of the tubes with strict slack at all sufficiently small scales. Almost every path is in the support of ν . The existence of the strict crossing subpath is an open path event, so it has positive ν -probability at each such fixed scale. Restrict ν_m to this event, take the crossing portions, and normalize. Its traffic still has squared norm O(b_m) , with a constant allowed to depend on the scale and the event probability. The complete-selector sandwich gives the asserted finite limiting test value. The statement can be imposed simultaneously at the countably many dyadic scales. Let I^ p _w(z) , respectively I^ d _w(z) , be the indicator of existence of these finite- /b_m tests at every sufficiently small dyadic scale, on the primal or dual graph. These are jointly measurable point-germ readouts: the tests and their fixed-factor buffers eventually lie in every neighborhood of z . The preceding paragraph and give a positive-area set of witnesses on our positive-probability event. Lemma 13 ([Determinism of point germs]) and Remark therefore imply that one of these indicators has deterministic value one almost everywhere in ordinary charts. Reflection exchanges the graphs; rotations within the finite direction grid then give the other graph's indicator in the perpendicular direction. The null sets can be discarded simultaneously, since only finitely many directions and two graphs are involved. Choose a point with both perpendicular indicators equal to one, and then a sufficiently small common dyadic scale. We obtain primal and dual strict crossing families with =O(b_m) in perpendicular tubes, as illustrated in Figure. Every path of one family meets every path of the other upstairs by the transverse-tube rule in Proposition 2 ([Topological comparison]). Such an intersection uses an edge and its dual. Choose traffic minimizers j_m,j_m^* by Lemma 12 ([Densities and traffic]). Their unsigned pairing satisfies 1≤∑_e j_m(e)j_m^*(e^*) ≤(∑_e j_m(e)^2)^ 1/2 (∑_ e^* j_m^*(e^*)^2)^ 1/2 =O(b_m), contradicting b_m→0 . This proves .