A Fresh Attack on the Binary Totient Series

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dilation_designs: A signed combination of prime dilations that cancels the first K levels of the shift tail for every function uses at least K+1 distinct primes, so its Chinese-remainder modulus exceeds K^(K+1). When that modulus is at most the averaging scale, the cancelled levels form a vanishing fraction of a logarithmic window.

evidence/: Exact finite checks of kernel shadows, retained fixed-prime robust vectors, complete smooth-orbit rounding, sparse comparisons, dilation designs and residue-lifting inequalities, with explicit limits on their coverage.

kernel_rigidity: On the short support [J,2J] the shadow u of every vector is fixed by its window coefficients at any complementary pair. Fixed-prime robust vectors have u=0 for sufficiently large even J. At suitable powers of two the kernel shadows fill 2^J Z x 2Z, giving two quantitative implications between odd cost and inhomogeneous approximation of S.

partial_multiplicativity: Comparisons with dyadic binary sums preserve all prime-power values and every totient multiplication law at finitely many fixed primes. They give an exponential cost bound for robust odd kernels without restricting the length of their supports.

perturbation_barrier: Sparse changes bounded by the square root except at one place, preserving growing powers of two, give the binary series any real value. The construction preserves empirical distributions of o(log N) windows up to o(1), delimiting arguments that use only these shared properties.

proof_sketch: Exact base-two target and complete-tail criterion, quantitative kernel shadows, rational comparisons preserving all prime-power values and fixed prime laws, scoped construction obstructions, and the remaining estimates.

residue_lifting: Residue truncations approach the totient sum exponentially fast, but a zero block occurs only after the first altered coefficient. Rescaling there amplifies the positive truncation error to at least half the modulus.

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This folder concerns Problem 249, the irrationality of $\sum_{n\ge1}\varphi(n)2^{-n}$. Its original results are Plasma AI's, author-recorded and without independent review. It uses specified results of the existing lead at their recorded standing. The editable proof sketch carries the exact target, held implications, limitations and open steps. There is no proposed solution here.

The kernel page computes the possible shadows on $[J,2J]$ at suitable powers of two and gives quantitative implications between odd-vector cost and inhomogeneous approximation of $2^{J-2}S$. It does not prove the required asymptotic approximation estimate.

The strongest comparison is L220: positive integer functions can preserve every prime-power value and the exact totient multiplication laws at any fixed finite set of primes, remain arbitrarily close in relative size, and have any binary sum in an interval about $S$. The [[research/totient_binary_new_attack/partial_multiplicativity|application to robust kernels]] uses a dyadic rational in that interval to bound odd kernel costs on supports of arbitrary length. A growing prime set or additional rough-composite relations remains outside its strongest bound.

The [[research/totient_binary_new_attack/perturbation_barrier|sparse comparison]], based on L218, preserves empirical distributions of $o(\log N)$ windows up to $o(1)$, while allowing any real binary sum. The [[research/totient_binary_new_attack/dilation_designs|dilation design bound]], based on L219, limits formal level cancellation when the Chinese-remainder modulus fits the averaging scale. The [[research/totient_binary_new_attack/residue_lifting|residue-lifting note]] identifies the positive error that prevents a direct transfer from bounded-residue zero blocks. Each obstruction is restricted to its stated mechanism.

All three claim cards remain at tier 0. The finite evidence checks lattice examples, retained robust vectors, complete-orbit rounding bounds, small dilation designs and the residue-lifting inequalities. It supplies no unbounded family of successful irrationality certificates.

A related comparison mechanism appears in the prime binary-series research: its increasing integer sequences preserve almost all prime positions and nested sublogarithmic congruences while their sums fill a one-sided interval. Here the preserved properties are prime-power values and fixed-prime multiplication laws. The two constructions test different coefficient classes; neither proves a transfer between the irrationality questions, and neither comparison preserves all defining arithmetic information.