Lemma 37 ([Coarse coefficients and moments]). On the size support in, for all nodes, boundaries and e , and 1≤l≤K , |[u^l] k_I^u k_I| +|[u^l]k_I^uk_I| +|[u^l]w_I^uw_I| +|[u^l]1-w_I^u1-w_I|≤n^C_K . The exponent C_K is …

The size support gives ∑_e|θ_e|≤Cn^4 , hence _xμ(x)≥e^ -Cn^4 . By Cauchy--Schwarz, each absolute one-site field is at most Cn^ 7/2 . On a leaf, prescribe one ring of each site in successive subintervals of length 1/n , with its sign set to the desired endpoint, and no other rings. The clock event has probability e^ -n|I| n^ -n , and each prescribed mark has probability at least 12e^ -Cn^ 7/2 . Thus every leaf transition probability is at least e^ -Cn^5 , uniformly in its endpoints. Its logarithmic density has absolute value at most Cn^5 . The last assertion of Lemma 31 ([A smooth cap]) and the kernel recursion propagate this property up the tree: composing two positive kernels costs a fixed multiple of the previous logarithmic bound plus a polynomial, and applying W_N has the same property. The depth is O( n) , so all these absolute logarithms remain polynomial, with a fixed exponent. In particular μ(y) k_I(x,y)≥e^ -n^ C_0 for a fixed C_0 . The total number of ordinary rings on a node is Poisson with parameter n|I|≤CnT . Its tail above any sufficiently large polynomial is smaller than e^ -2n^ C_0 times any inverse power of n . Dividing by the last lower bound and using V_I≤1 proves polynomial conditional count moments for every boundary. Each fixed Taylor coefficient of the ring product is a polynomial of fixed degree in its number of rings, since each single-ring coefficient in is bounded. This proves the claimed raw moments, both conditionally and under the polynomially tilted tested laws. The copy statement follows from their preserved marginals. At a leaf, has an empty product; the preceding count bound and the bounded coefficients of (1+uμq_e)/(1+u q_e(y)) give the coarse bound for its pre-density. Suppose normalized coefficients of order l≤K at one tree level are bounded by D^l . For a composition, division by its base value writes its ratio as a conditional average of the product of the two child ratios and μ_u(z)/μ(z) . Its order- l coefficient is therefore at most (C_K(D+1))^l . Taylor's chain rule for W_N( k_I^u) uses the global ratios ; coefficients of the logarithm are polynomials in nonconstant pre-ratio coefficients, each product having total order l . The same bound applies to either survival or failure ratios and to the post-density. Starting with a polynomial D at the leaves and iterating D↦C_K(D+1) for O( n) levels proves these coefficient bounds. Formula itself follows by inserting the path likelihood and then dividing by μ_u(y)/μ(y) in the definition of the pre-density. To check the last forest assertion, fix its designated external endpoints. The interiors of its maximal roots are the corresponding conditional laws , and their top survival or failure factors are fixed. Expand a coefficient of total order at most K by the product rule. A differentiated node contributes one of the normalized ratios just bounded; the raw ring factors have the conditional polynomial moments proved above. There are polynomially many nodes and at most K positive orders, so the number and magnitude of the resulting terms are bounded by n^ C_K . Stationary initialization, if present, contributes only the bounded coefficients in. Integrating this conditional bound against the nonnegative endpoint test and its base endpoint measure proves the asserted factor of the base mass.