The Binary Totient Series

The original quantified residue condition

The threshold quantifier excludes every possible starting point for a rational tail period.

The quantified residue condition.

$$S\notin \mathbb{Q}⟺\mathrm{\forall }h\ge 1\mathrm{\forall }{N}_0\mathrm{\exists }N\ge N_0\mathrm{\exists }L,\mathcal{C}(h,N,L).$$

Here $\mathcal{C}(h,N,L)$ is the finite residue test defined earlier; the universal quantifiers over $h$ and $N_0$ belong to the displayed assertion, not to that finite test. Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_certificate_supply The equivalence is proved; the quantified existence assertion is not proved.

The reverse direction is by contradiction against the tail-period law: if $S$ were rational, the eventual-period theorem (Erdos249257.TotientTailPeriodKiller.eventual_period_of_not_irrational, unconditional) supplies a period $h>0$ and pre-period $N_0$ with $R_{N+h}-R_N\in \mathbb{Z}$ for all $N\ge N_0$; the certificate hypothesis instantiated at that $(h,N_0)$ produces an $N\ge N_0$ where $R_{N+h}-R_N\notin \mathbb{Z}$, a contradiction via nonintegrality from a certificate (Erdos249257.TotientTailPeriodKiller.tail_diff_notMem_int_of_certifiedKill).

For the forward direction, irrationality gives pointwise non-integrality of every positive tail shift and certificate completeness supplies a witness already at $N=N_0$.

For fixed $h$, the required certificate may choose both $N\ge N_0$ and $L$. It is not required at every $N$ or every depth. Its error radius is $N+h+L+2$, so for fixed $h,N$ it grows only linearly in $L$, whereas the modulus is $2^L$. The missing assertion is that an admissible residue lies outside both endpoint intervals beyond every threshold, for every positive $h$. The reformulations below specify other ways to choose such witnesses; none establishes this assertion for the actual totient sequence.