Lemma 1 ([Finite pin inputs]). For the finite picture above the following assertions hold. For every finite E and nonempty single-time T disjoint from E , ℙ_E(T⊆ B_E) ≤(2r(E,T))^ |T| . Almost surely under ℙ_P , g_P( a):=ℚ_a (G_P)>0 , and …

The statements are the finite-pin forms of . We give the arguments to specify the scope of the extra operations and the measure used after exposure. Cut at all pin times and at zero, and let V be the disjoint union of the slot layers at these cuts. The maps between successive cuts are independent admissible bijections. Their symbols in disjoint input and output variables multiply to give a stable polynomial _ b ∏_ i∈ V (x_i+y_ π(i) ) . Delete a nonpin q by setting x_q=y_q=t and extracting the coefficient of t . If its predecessor and successor are i,j , then (x_i+t)(t+y_j) is replaced by x_i+y_j . A fixed point instead contributes 2t , hence a factor two. Successive deletion therefore gives the nonzero stable polynomial _ b [2^ n_P ∏_ i∈P (x_i+y_ σ(i) )]. Differentiate in the input variables indexed by P T , set all input variables to zero, and divide by Z_P . The result is _P∏_ i∈T y_ σ(i) , the stable generating polynomial of the subset σ(T) . By and the arithmetic--geometric mean inequality, ℙ_P σ(T)=T ≤∏_ j∈T ℙ_P j∈σ(T) ≤r(E,T)^ |T| . Here ∑_ j∈T ℙ_P j∈σ(T) =|T|r(E,T) by bijectivity. The event σ(T)=T is precisely T⊆ B_E : iterating the next-pin permutation from T then meets no pin in E . Finally, 2^ n_P ≤2^ n_E ≤2^ |T| 2^ n_P and Z_E≥ Z_P , proving (i). For (ii), first fix a proposed set A of initial exposed labels. Use two independent Poisson clock families on every edge, each with the original rate: accept the first family's clock when an endpoint contains an A -label and the second when neither does. This realizes the base process. Given the past, the superposed clock has twice the total original rate, each edge is accepted with its original relative rate, and the rejection probability is 1/2 . The A -label paths use only the first family. Given these paths, the second family restricted to the hole intervals remains fresh. Revealing the A -label images at a uniform permutation leaves a uniform residual bijection. At a Bernoulli transposition its outcome is revealed exactly when an exposed label is involved; other outcomes remain independent. This constructs ℚ_ a . More explicitly, let μ_A be the base law of the paths and endpoint images of the proposed labels. Let F_A mean that those paths avoid P and close on A , and let c_A count their cycles on this event. For every bounded measurable function Φ of the exposure and remaining picture, _PΦ = Z_P^ -1 ∑_A∫_ F_A 2^ c_A( a) _ ℚ_ a [\1_ G_P Φ]\, μ_A(d a). Each full picture occurs in exactly one branch, namely its set of labels on cycles missing P . The exposure marginal on that branch is therefore Z_P^ -1 2^ c_A g_P\,μ_A restricted to F_A . It assigns zero mass to g_P=0 , and division by g_P proves (ii). In particular the exposure marginal includes the all-hit probability g_P ; it is not the unweighted law of the proposed histories. For (iii), inserting one transposition joins two cycles or splits one. It increases the number of cycles missing the fixed pins by at most one, and so multiplies the weight by at most two, regardless of the other instantaneous operations. The Poisson insertion formula for j ordered distinct marks, followed by division by Z_E , gives. Independent fresh clocks have their ordinary factorial measure, proving the last assertion.