Finite-prime reduction of the totient phase quotient

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_proof: Encode fixed prime divisibility as a least-preimage-ratio weight, count it using the common prefix, and transfer prime-deletion bounds to all phases and to the finite coefficient.

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This conditional deduction is recorded by Plasma AI. It constructs phase profiles for fixed finite-prime observables and transfers a large-prime bound to the finite arithmetic coefficient. It leaves the constancy question open: the profiles may depend on the phase, and only their indicated weighted combination needs to become constant.

The proof uses the structural and counting premises of L65 to count divisibility of the least preimage, with every witness union and global minimum retained. The finite-coefficient regularity from L43 and the unit jumps of these counts give continuous profiles on the circle. Both inputs stand at tier 0 and remain conditional on the unrefereed totients manuscript.

The prime-deletion estimate and the comparison of the mean of $v/\ell(v)$ with $V'/V$ occur in the unrefereed, non-arXiv two-page note An explicit bound for Erdős problem 417, posted by the account LeanMeanBean on 10 October 2026. The proof here spells out the needed estimates and combines them with the manuscript's coefficient construction. The new conclusion is the phase-uniform truncation and the finite-prime criterion, not a new source for the note's numerical bound. Neither the note's numerical sum nor the manuscript's underlying proofs are certified by this deduction.

Standing. Tier 0, author-recorded, with no independent review or formalization. This is a conditional reduction, not a solution of Problem 417. Its research route records the unresolved cancellation among the phase profiles. There is no computational evidence entry point: the evidence is the displayed argument.

The reduction is complete under its premises. The prime truncation error has the explicit upper bound $1/(Y-1)$; evaluating the profiles still requires the manuscript's cut limit. A proof of phase independence, or a phase gap exceeding the truncation and cut errors, remains a separate mathematical obligation.