A conditional phase coefficient for the distinct-preimage ratio

../

_proof: A conditional extension of the manuscript's prime count, followed by a uniform tail argument and an exact-phase criterion for convergence.

---

This conditional deduction is recorded by Plasma AI. Here $V(x)$ counts distinct totients at most $x$, and $V'(x)$ counts distinct values of $\varphi(n)$ with $n\le x$. The functions $f_k$ and the phase are those of the unrefereed totients manuscript. The manuscript is an explicit premise, including its assertions about all preimages of almost every totient. The checked fixed-dilation result held with that source does not certify these stronger premises.

The proof extends the manuscript's largest-prime count to $f_0(r)=1/r$. It controls the lower prime cutoff, obtains uniform convergence without assuming continuity, and supplies the uniform tail bound needed for the infinite sum. The expression $A(f_0;s)/A(1;s)$ is a limiting weighted average of $d/\ell(d)$ over tail totients, with witnesses deduplicated by their totient value.

Standing. Tier 0, author-recorded; no independent review or formalization of this deduction. It remains conditional on an unrefereed manuscript. No constancy or nonconstancy is established, so Problem 417 remains open. The research route names the remaining structural test. There is no computational evidence entry point: the evidence is the displayed deduction from the specified source premises.