Continuity of the totient phase coefficients

../

_proof: Piecewise continuity of the finite coefficients, removal of interior jumps by unit increments of the two counts, and the common jump factor of G_m at the end of a phase period.

---

This conditional deduction is recorded by Plasma AI. It takes as premises the results of the unrefereed totients manuscript that L65 assumes, and it uses L65's conclusions: the uniform limit $A(f_0;s)$, its positive lower bound, the asymptotic $V'(x)=Z(x)(A(f_0;\theta(x))+o(1))$ with $Z(x)=xG_m/\log x$, and the equivalence between convergence of $V/V'$ and constancy of $q(s)=A(f_0;s)/A(1;s)$. The manuscript states that no regularity of its phase coefficient is asserted or needed; this card supplies that regularity from the same premises.

The proof has three parts. For a fixed cut the witness sets are constant between finitely many phases, so each finite coefficient is piecewise continuous and uniform limits keep one-sided limits. The counts $V$ and $V'$ change by at most one between consecutive integers while the phase moves by $o(1)$, which removes every interior jump. At the end of a period the scale factor $G_m$ jumps by the factor $\lambda/\gamma$, and both coefficients jump back by the same factor, so their quotient joins up.

Significance for Problem 417. Constancy of $q$ cannot come from separate constancy of the two coefficients: $A(1;\cdot)$ is provably nonconstant, and the two coefficients carry the same boundary factor. A limit for $V/V'$ is therefore an exact cancellation between two nonconstant functions. If $q$ is nonconstant, the ratio oscillates continuously with the phase between $\min 1/q$ and $\max 1/q$, and every intermediate value is a limit along an exact-phase sequence. Since $q$ is continuous, the exact missing identity is the vanishing of its nonzero Fourier coefficients, each a limit of an explicit finite-cut integral. Continuity also means that constancy on a dense set of phases would suffice. Because $q$ returns to $q(0)$ at the end of the period, a one-sided comparison is enough as well: if $q(s)\le q(s')$ whenever $0\le s\le s'<1$, then $q(0)\le q(s)\le\lim_{t\to1}q(t)=q(0)$ for every $s$, so $q$ is constant. The same holds if the finite quotients satisfy $q_H(s)\le q_H(s')+o(1)$ uniformly, since they converge uniformly to $q$.

Standing. Tier 0, author-recorded; no independent review or formalization. It remains conditional on the unrefereed manuscript through L65, whose own standing is tier 0. No constancy or nonconstancy of $q$ is established, so Problem 417 remains open. The research route records the remaining phase test. The strict inequality $\lambda>\gamma$ follows from the increase of the coefficients $a_j$, without a numerical calculation. There is no computational evidence entry point.