A Source-Compatible Attractor Reduction for the 3x+1 Problem

We present a source-compatible attractor framework for the 3x + 1 problem with a fullyprinted finite transition verification. The primary initial states are the odd sourcesx0 = 3u, u odd, rather than arbitrary odd integers treated independently. Every odd integer has a canonicalsource carrier; therefore ordinary descent is replaced by source-compatible descent. Theproof architecture has three parts. First, exact affine identities are established for finitevaluation words. Second, an internal source-depth theorem proves the origin of the coefficientlayers 3^5,3^6,3^7,3^8... a source cylinder with 3-adic exponent α, observed after d stabilizedaccelerated odd steps, has source-depth indexq(α,d) = α+d+3, so the primitive layer 3^8 is reached at the threshold α+d = 5. This derivation is included inthe manuscript and is not used as an external citation. Third, the finite transition verification,printed at the point where it is used, reduces every non-closing source-compatible brancheither to an explicitly contractive word, or to the residual languageL− =(1+112+211+2112+2211)^N.The residual language is excluded by a mixed-radix obstruction: every transition betweentwo residual generators forces a nonzero lifting digit, whereas an ordinary positive integerhas an ultimately zero expansion in the associated mixed radix system. The finite tables areprinted in the body of the manuscript, and an exact Python verifier is included as an auditaid