Sparse and Arithmetic Transforms of the Binary Totient Series

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evidence/: Five retained integer multipliers, exact gap and infinite-tail certificates, and finite reduced-denominator checks for eleven prime-sieve sums.

gap_multipliers: A sufficient irrationality criterion with exact integral forms, a residue nonvanishing test, and a uniform bound on the complete infinite remainder.

logarithmic_products: No integer Euler product has a polynomial logarithmic defect for the totient series, and finite rational logarithmic transforms have unbounded denominators.

prime_sieve: A cyclotomic factor survives in the reduced denominator for every finite prime sieve, and its square outgrows the full positive remainder.

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The target is irrationality of $S=A(1/2)$, where $A(x)=\sum_{n\ge1}\varphi(n)x^n$ and $\varphi(1)=1$. The parent sketch records the target, its dated partial status search, and the tail identities used here. This account gives author-recorded arguments, without independent review or a claim to resolve Problem 249.

The first conclusion is a precise obstruction for one growing family. For any squarefree $Q$, set $M_Q=\sum_{d\mid Q}\mu(d)2^d/(2^d-1)^2$ and let $D_Q$ be its reduced denominator. The prime-sieve argument proves $M_Q>S$ and $D_Q(M_Q-S)\to\infty$ as $Q\to\infty$ through squarefree integers. Thus the literal sieve approximants cannot give small integral forms by clearing their reduced denominators. This is a lower bound on the actual denominator and actual error, not a comparison with an unnecessarily large common denominator.

Two further product obstructions exclude an integer Euler product whose logarithmic derivative differs from $A$ by a polynomial, and a fixed rational prefactor times the exponential of a nonzero finite rational combination of $L(x^d)$, where $L(x)=\sum\varphi(n)x^n/n$. The latter has unbounded Taylor denominators.

The next construction is a sequence of integer polynomial multipliers whose product with $A$ has gaps near a selected degree. The exact integral form and full-tail bound give a sufficient criterion: top coefficient two, the stated gaps of size $k$, and height $H=o(2^k/N^2)$. Its nonvanishing test is proved. The [[research/leads/totient_binary_series/integer_interpolation/_index|integer interpolation theorem]] supplies an integer solution at every order; the required estimate for a growing family remains open. A formal truncation lemma shows exactly when a single sparse integer series and its product with $A$ supply these multipliers. Applied to Euler's product and the sigma series, the same argument recovers the known base-two irrationality result with a quadratic coefficient bound. The exact evidence checks five concrete multipliers through $N=900$. They demonstrate finite feasibility and supply height upper bounds, without proving that criterion.

The mathematical boundary is essential. An obstruction to these sieve approximants does not exclude combinations of approximants, rational corrections, growing multipliers, Padé approximation, or other sparse products. Formal integer Euler products comprise all constant-one integer series; this observation does not make every product method equivalent to a particular gap criterion.

These results support a specific research choice: stop using the literal finite prime-sieve sums as small denominator-cleared errors, and investigate the affine convolution equations for structured integer multipliers. They leave combinations of sieve sums and other analytic transformations open.