The note separates three layers: (i) current literature status, (ii) rigorous reductions andobstruction theorems derived in the thread, and (iii) conditional routes and dead ends. Thecentral conclusion is negative but sharp: many natural corridors — classical Apéry clearing,the EMN simple-pole and polynomial higher-pole programs, fixed-base q-difference compression,several modular/holonomic packetizations, and a broad discrete moment/Hankel family— fail for precise structural reasons. The revised version also records five structural ingredientsthat were missing from the earlier draft: a four-jet modular rigidity theorem for the X1(5)Picard-Fuchs engine, a noncharacteristic edge-splitting theorem for higher poles, injectivity/isomorphism theorems for the EMN collapse operators, a boundary-annihilator theoremfor the edge packet, and formal torsion-support / multiplicative-complexity obstructions onthe modular q-side.