The Binary Totient Series

Changing the coefficients after a finite prefix

Proposed argument. The quantified residue condition requires certificates beyond every starting index. A finite computation can check many instances, but one fixed inspection bound cannot establish these quantifiers. The following comparison explains this limitation without making any claim about all possible irrationality arguments. Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_certificate_supply

Counterexample. Theorem 2.4 gives an explicit splice construction; its series value, exact reduced denominator and failure of the separation condition are checked in Lean. Choose an integer $P>B$, set $\gamma (n)=\phi (n)$ for $n\le B$, and $\gamma (n)=n-{\mathbf{1}}_{P\mid n}$ for $n>B$. Then $0\le \gamma (n)\le n$, and the binary series $\sum \gamma (n)/2^n$ is rational: it is a finite rational prefix, plus the tail of $\sum _{n}n{2}^{-n}$, minus a geometric sum over the multiples of $P$ beyond $B$. Because $\gamma$ agrees with $\phi$ on every index $\le B$, $\gamma$ passes every finite certificate that only inspects indices $\le B$; in particular it passes or fails every instance of $\mathcal{C}(h,N,L)$ with $N+h+L\le B$ exactly as $\phi$ does. Yet $\sum \gamma (n)/2^n\in \mathbb{Q}$ by construction.

Scope. For each fixed bound $B$, a different rational comparison sequence can agree with $\phi$ on all inspected coefficients. This rules out a deduction from that finite prefix alone. It does not rule out a proof that establishes the required instances for arbitrarily large $B$ with the correct quantifiers. Such a proof would no longer be a single finite inspection. The comparison also does not preserve multiplicativity or every other arithmetic property of $\phi$.

Consequence. The finite-prefix example and the parity example below are distinct constructions. The former preserves a prescribed finite prefix; the latter preserves parity and has separated nonperiodic blocks. Each shows why its own list of assumptions is insufficient. The recorded finite diagonal examples remain useful individual certificates, but their number alone gives no assertion at arbitrarily large scales. Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_all_imported_through_t64