Wieferich co-occurrence along exponential sequences: a weighted tail estimate and its application to (star')

Fix an integer r >= 2 and write b_e = r^e - a' for a' <= A coprime to r. A recent preprint (hexagon:2610.00630, Zeng) reduces the abc conjecture, within a lattice-theoretic framework, to a single remaining hypothesis (star'). We show that on the k = 2 slice the density-zero requirement on the exceptional set attached to (star') follows from one summability condition K_{r,a'} < infinity, where K_{r,a'} = sum over q in C_{r,a'} of s_q log q (1/D_3(q) + 1/e_0(q)), C_{r,a'} is a three-filter congruence set, D_3(q) = ord_{q^3}(r), and e_0(q) is the least solution of q^3 | r^e - a'. The main result is unconditional: splitting the governing weight Xi_r by the block d = ord_q(r), the mass constraint (prod q)^2 | r^d - 1 and Stieltjes partial summation give Xi_r^{(2)} = sum_{s_q = 2} 2 log q / (d_q q) < infinity for every base r, with explicit bound O(1 + log log r); no abc, no GRH, and no count or density of the Wieferich primes W_r. The exact threshold of the first-moment route is Phi_r(E) = o(E), equivalently A_r(t) = sum_{d <= t} M_d / d = o(t); the deep-layer condition Xi_r^{(3)} < infinity sits strictly between Murata's finiteness conjecture P_{6a} and the deep reciprocal sum, and the barrier analysis (one claimed radical barrier withdrawn as a bookkeeping artifact and replaced by an exact identity) shows it cannot be reached by block/radical/partial-summation methods. A Katz double-zero hypothesis (D4) yields Xi_r^{(3)} < infinity and P_{6a} almost surely (Borel-Cantelli); in the Haar ensemble (D4) is free with constant 1, while the diagonal embedding iota(Z) has measure zero, so the missing input is precisely pointwise rigidity of iota(r), equivalently the single-variable statement m(q) -> infinity for the least nontrivial Teichmuller representative m(q) modulo q^3. The bad-set hypothesis (H-j) is supported by computation to 10^6 (five hits, zero growth), while the same data correct the twin-Wieferich null hypothesis to the Teichmuller rate (q-1)/q^2. We prove (H-j) is not a GRH statement, locate it strictly between Katz's 2015 equidistribution conjecture and Murata's 1991 conjectures, record ancillary unconditional results (sum_p log p / (d_p (p-1)) < infinity via Erdos-Murty; the exact identity C_r = S_W + S_ord + LTE_2; strict one-way comparisons), and specify what any new tool must look like (S1-S7). We prove no conjecture: the exact threshold, Xi_r^{(3)}, P_{6a}, P_4, P_1 and (H-j) remain open.