The Binary Totient Series

Finite extension and cofinal estimates

Extending a computation supplies more finite witnesses. To reach a cofinal statement, an argument must instead produce a witness beyond an arbitrary threshold. The examples below specify which of these extensions is needed; the largest computed scale is not asserted to be a limit of the method.

1. Finite certificate tables. Results such as the diagonal certificates at 28 explicit values of $t$ through $t=64$, now a strict subset of the aggregate band $t\le 82$ (certifiedKill_diagonal_all_imported), the certificates for $h\in [1,8]$ at fixed depth 16 (certifiedKill_all_small), and the exclusions for $h\in [1,16]$ at exponent 14 (totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen) are each a finite list of independently verified rows, not a single argument instantiated at a free parameter. The existing lists do not establish further rows; an extension needs either additional computations or an argument for a family of parameters.

2. A bounded window inside a uniform theorem. The rough-integer bound of Proposition 6.79 and the positivity theorem of Proposition 6.84 hold for every $a\ge 8$, subject to their stated bounds on the other parameters. In the sign theorem, for example, $J+a+6<2\cdot 2^a$. Their proofs are therefore not merely enumerations of a fixed set of exponents. The missing conclusion is a suitable residue inequality at arbitrarily large scales, not extension of the range of $a$ in the positivity theorem. No complexity lower bound follows from these examples.

3. Parameter dependence in an approximation estimate. For reduced approximants $p_K/q_K$ to $S$, the conditions $0<|S-p_K/q_K|\le {\epsilon }_K$ and $q_K{\epsilon }_K\to 0$ imply irrationality. A growing denominator alone does not control this product. In particular, the exact reduced-denominator formula of Proposition 6.8 concerns a specified finite Möbius sum; it does not estimate the complementary tail. Likewise, a bound proved with a constant depending on a fixed auxiliary parameter is not automatically uniform when that parameter grows. Both the dependence of the constant and the error estimate must be retained in any passage to an unbounded family.

The missing arithmetic inputs include the stated constant-saving exponential-sum estimate and an upper-endpoint residue inequality. For the positivity theorem, the unresolved step is exclusion of that residue interval, not positivity at more exponents.