Cofactor restrictions on infinite totient-plus-one inverse chains

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_proof: Compose the exact inverse maps for fixed cofactors, linearize one block at its negative fixed point, and compare 2-adic valuations. Necessary prime-residue conditions force a cofactor sequence bounded by 9 to become constant.

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This elementary deduction is recorded by Plasma AI for Problem 409. An infinite basin would contain an infinite increasing odd inverse chain. The claim excludes an eventually repeated cofactor pattern, including the fixed cofactor $3$ used by finite prime-tuple constructions. It also excludes all chains that eventually have largest-prime cofactor at most $9$.

These exclusions do not prove any exceptional prime's basin finite. Arbitrary nonperiodic sequences of bounded cofactors, and sequences of unbounded cofactors, remain uncontrolled. The research sketch identifies this missing comparison and explains why finite chains from varying roots do not provide one infinite chain from a fixed root.

The proof uses exact affine identities and divisibility, with a four-row necessary-transition table for cofactors $3,5,7,9$. It uses no conjecture about primes and no computational enumeration. The bound applies to each fixed starting integer and cofactor block; it is not a bound uniform over all starting integers or all blocks.

Standing. Tier 0, author-recorded and unconditional, without independent review or formalization. No solution of Problem 409 is claimed. The evidence is the displayed proof, with no executable evidence entry point needed.