Tail uniqueness and cofactor factorization for the totient coefficient

../

_proof: Compare the common prefixes at the cuts H and P of one value, identify the tail prime tuple of every witness and the least preimage, and transfer the exceptional tuple count to the mass of the data and to the phase quotient.

---

This conditional deduction is recorded by Plasma AI. It applies the common-prefix proposition of the unrefereed totients manuscript at two cuts at once: at the manuscript's cut $H$, whose tail totient $d$ enters the finite coefficient, and at the shorter cut $P=\lfloor\log\log H\rfloor$, whose common prefix contains every retained tail prime of the first. Outside the two exceptional sets the second prefix pins down the tail prime tuple of every witness of $d$, forbids its smallest prime from dividing the cofactor, and splits the least preimage. The manuscript is an explicit premise, including its assertion that every preimage of a non-exceptional value has the common prefix.

Significance for Problem 417. The finite coefficient of the manuscript sums over distinct tail totients, each weighted by the probability of a union over its witnesses and by its global least preimage. On factorized tail totients the union has one member, the cofactor-to-value map is replaced by distinct values $e=\varphi(a)$, and $g(d)=g(e)\prod_h(1-1/Q_h)$ with the product within $2/Q_P$ of one. The second part of the statement therefore expresses the phase quotient $q$ of Problem 417 through the reciprocal least-preimage ratio $g(e)$ of the cofactor totient alone. This removes the union, the duplication of odd $a$ and $2a$, and the coprimality question at $Q_P$ from the comparison of phases; it does not compare two phases. The research route uses it to locate the phase dependence in the cofactor fiber and records why the manuscript's error rates leave constancy undecided.

Dependency roles. The second part consumes L65 for the comparison of $V'(x)$ with the mass of the weight $1/r$ and for the identification of $q(\theta(x))$ with $V'(x)/V(x)$ up to $o(1)$. The first part uses only the manuscript.

The proof uses the statements of Propositions 3.3 and 5.3 at the two cuts, the band conditions of Equation (2.5), and the prime count for the largest prime from the proof of Proposition 6.1. It does not use Lemma 6.4 or Proposition 6.5, so it says nothing directly about the finite coefficient $A_H$; the passage to the coefficient's probability measure is on the research route, with the extra reading of those proofs that it needs.

Standing. Tier 0, author-recorded, with no independent review or formalization. It remains conditional on the unrefereed manuscript, and through L65 on the same premises. No constancy or nonconstancy of $q$ is established, and Problem 417 remains open. The exceptional mass is bounded only through the manuscript's rates at the two cuts. At the cut $P$ the coverage rate of Proposition 3.3 is $e^{-c\lfloor\log\log P\rfloor}$ up to smaller terms, with an unspecified constant $c$, so the bound tends to zero extremely slowly. There is no computational evidence entry point: the evidence is the displayed deduction.