Proof Sketch for a Fresh Attack on the Binary Totient Series

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Target and standing

The target is Problem 249 in its exact form: is

$$S=\sum_{n\ge1}\frac{\varphi(n)}{2^n}=1.3676308019850\ldots$$

irrational? Other bases and modified coefficient sequences are different questions. The problem page records the target as open. This sketch contains no proof or disproof. Its original arguments are Plasma AI's, author-recorded and without independent review. The claims L218, L219 and L220 are at tier 0. The existing lead and its subordinate results retain their recorded standing; a review of that lead does not cover the arguments here or its later dilation construction.

Complete tails and the sufficient kernel criterion

For $n\ge0$ and $J\ge1$ put

$$\Phi_n=\sum_{m=1}^n\varphi(m)2^{n-m},\qquad T_n=2^nS-\Phi_n=\sum_{j\ge1}\frac{\varphi(n+j)}{2^j},\qquad U_J(n)=\sum_{j=1}^J\varphi(n+j)2^{J-j}.$$

Then $0<T_n\le n+2$ and $2^JT_n=U_J(n)+T_{n+J}$. If $S=a/b$ with integers $a$ and $b\ge1$, every $bT_n$ is an integer. For a finitely supported integer vector $z$ on $n\ge B\ge4$, define

$$u=\sum_nz_n2^n,\qquad v=\sum_nz_n\Phi_n,\qquad C_J(z)=\sum_n|z_n|(n+J+2).$$

If $\sum_nz_nU_J(n)=0$, the complete identity is

$$uS-v=2^{-J}\sum_nz_nT_{n+J},\qquad |uS-v|\le 2^{-J}C_J(z),\qquad 2^B\mid u. \tag{1}$$

Call $z$ odd when $v\equiv2\pmod4$. Write a proposed denominator as $b=2^s q$, with $q$ odd. If $s+1<B$, rationality would give $v_2(au-bv)=s+1$, hence $|uS-v|\ge2/q$. Thus

$$s+1<B,\qquad qC_J(z)<2^{J+1} \tag{2}$$

exclude that denominator. This is the valuation-gap argument of [[research/leads/totient_binary_series/integer_interpolation/two_depth_carries|Proposition 6 and Corollary 7 of the two-depth page]], with the full tail retained. Consequently an odd family with $B\to\infty$ and $C_J(z)=o(2^J)$ proves irrationality. On $[J,2J]$, $J\ge4$, the simpler condition $bC_J(z)<2^{J+1}$ suffices: oddness implies $C_J(z)\ge2J+2$, which ensures the cutoff condition in (2).

The author-recorded [[research/leads/totient_binary_series/integer_interpolation/phase_localization|arithmetic localization theorem]] proves a converse on a subsequence of powers of two: irrationality is equivalent to the existence of such cheap odd vectors on $[J,2J]$. That inherited argument uses the fixed-modulus prime number theorem at its disclosed external standing. The equivalence does not supply the vectors.

What the short-support construction establishes

The kernel page proves the following statements, with its hypotheses explicit.

For $J=2^m$ sufficiently large, the shadows $(u,v)$ of integer kernel vectors on $[J,2J]$ fill $2^J\mathbb Z\times2\mathbb Z$. At an individual $J$, the sufficient column-generation condition can be checked directly. Put

$$\gamma_J=\min_{1\le k<2^{J-1}} \left\|k2^{J-2}S-\tfrac12\right\|, \qquad \kappa_J=\min_{\substack{z\text{ odd}\\\sum z_nU_J(n)=0}}C_J(z),$$

where $\|\cdot\|$ is distance to the nearest integer and the minimum for $\kappa_J$ uses $[J,2J]$. Under the auxiliary-prime hypotheses of its Lemma 4, Corollary 6 gives

$$\kappa_J\le160J^6(1+2^J\gamma_J). \tag{3}$$

Conversely, an odd vector of cost $C<2^{J+1}$ yields a nonzero integer $k$ with

$$|k|\le\frac{2^JC}{2J+2},\qquad \left\|k2^{J-2}S-\tfrac12\right\|\le2^{-J-2}C. \tag{4}$$

These are two implications with different multiplier ranges; they are not an equivalence at the same approximation threshold. In particular, $\gamma_J=o(J^{-6})$ on an unbounded subsequence of powers of two would make (3) satisfy the criterion (2). No such estimate is proved.

The complementary-pair identity on the same page shows that vectors whose cancellation is robust under multiplication laws at any fixed finite set of primes have $u=0$ for sufficiently large even $J$. Fixing every prime-power value does not remove this eventual obstruction. The precise robustness class and sufficient thresholds are in that page. Its finite LLL computations find bases containing cheap odd vectors for $J\in\{16,32,64,128\}$. They do not compute minima or establish an asymptotic construction.

Fixed-prime laws on supports of any length

The claim L220 constructs positive integer functions $f$ whose binary sums fill an open interval about $S$. For every fixed $y\ge2$, modulus $M$ and $0<\varepsilon<1$, these functions preserve all prime-power values, congruence to $\varphi$ modulo $M$, relative error at most $\varepsilon$, and both exact totient multiplication laws at every prime $p\le y$. Only $O(X/\log X)$ values up to $X$ change. Complete contributions from all smooth multiples of a changed rough integer are included in the rounding weights. This is partial multiplicativity, not full multiplicativity.

The interval contains a dyadic rational. Applying the kernel identity to that comparison gives the [[research/totient_binary_new_attack/partial_multiplicativity|support-independent cost bound]]: for each fixed $y$ and $\varepsilon$, there is $B_0$ such that every odd kernel vector supported on $n\ge B_0$, whose cancellation holds for all functions with those fixed-prime laws and prime-power values, satisfies

$$C_J(z)\ge\frac{2^{J+1}}{1+\varepsilon}. \tag{5}$$

There is no upper restriction on support length. Along any such family with lower cutoff tending to infinity, $\liminf C_J(z)/2^J\ge2$. Thus simply widening the support does not turn fixed-prime robust relations into the required $o(2^J)$ family.

When $y$ grows, choose distinct primes $r_0,t_0>y$ and let $Q=r_0t_0\prod_{p\le y}p$. The same page gives the explicit weaker bound $C_J(z)\ge2^{J-Q-3}$, with no cutoff requirement. If $Q=o(J)$, it excludes cost $2^{o(J)}$, but it does not exclude every $o(2^J)$ cost. No uniform version of (5) as $y\to\infty$ is established.

Other comparisons and their limits

The sparse rounding claim L218 gives any prescribed real binary sum while changing $\varphi$ only sparsely, by multiples of $2^{\lfloor\log_2n/4\rfloor}$, with additive error at most $\sqrt n$ except at one place. The comparison note proves that empirical distributions of windows of length $o(\log N)$ agree up to $o(1)$ in total variation. Arguments using only these shared properties cannot distinguish rational sums. This does not exclude quantitative error rates, selected exceptional windows, arithmetic laws not preserved, or global arguments. In particular it proves no universal requirement that every irrationality proof use logarithmic windows.

The dilation claim L219 shows that nonempty signed prime designs cancelling levels $1,\ldots,K$ for every function, with primes greater than $K$ and compatible shifts, need at least $K+1$ distinct primes. If their modulus is at most $x$, this forces $K\le(1+o(1))\ln x/\ln\ln x$. The design note records the resulting limit on formal cancellation when the tail cutoff is of order $\log x$, as in the retained averaged-error method. A six-prime example improves the three-level cube's vertex count, without supplying the missing phase estimate.

The [[research/totient_binary_new_attack/residue_lifting|residue-lifting obstruction]] explains a separate failed transfer. With $R_m=\sum_n(\varphi(n)\bmod m)2^{-n}$ one has $0<S-R_m\le(m+2)2^{-m}$, but whenever a nonempty zero block starts at $N+1$, $2^N(S-R_m)\ge m/2$. Thus the positive difference cannot be ignored in a small-tail contradiction at the zero block. Irrationality of bounded residue series alone does not handle the full series.

The [[research/leads/totient_binary_series/window_cost|termwise window bounds]] concern whole-progression divisibility constructions. They do not bound selected representatives or every possible tail method. For a square-divisible prime cube with primes greater than $K$ and pairwise distinct positive tail indices, the normalized weights give a kernel vector at depth $K$ with $2^K$ nonzero integer coefficients. Its cost is at least $2^K(K+3)$, so this direct cost bound is insufficient. This observation requires distinct indices; it is not a prohibition on all cube constructions or signed phase estimates.

The coprime-pair identity

$$S-\tfrac12=Z(\tfrac12,\tfrac12),\qquad Z(x,y)=\sum_{\substack{a,c\ge1\\\gcd(a,c)=1}}x^ay^c =xy+Z(xy,y)+Z(x,xy)$$

is another exact representation. Splitting coprime pairs into $a=c=1$, $a<c$ and $a>c$ proves the functional equation. No theorem establishing irrationality of this specialization from that equation is supplied. The Möbius and cross-base representations likewise provide identities without a denominator-cleared error estimate that reaches the target.

Open interfaces and conditions for reopening

One way to finish is a family in (2) with $B\to\infty$ and $C_J(z)=o(2^J)$. On the short support, (3) offers the sufficient target $\gamma_J=o(J^{-6})$ along powers of two. Both remain open.

An alternative is a phase saving: for each fixed positive integer $q$, find $\delta_q>0$ and arbitrarily large $X$ such that

$$\left|\frac1X\sum_{N=1}^X e(qT_N)\right|\le1-\delta_q, \qquad e(t)=\exp(2\pi i t).$$

Rationality with denominator $q$ makes every summand one. A family of multipliers containing a multiple of each candidate denominator also suffices. The normalized dilation construction provides a different sufficient phase target with a complete averaged error bound; its required saving on the same progression and scales is still unproved.

The fixed-prime cost obstruction can be escaped by a genuinely growing prime set, by further relations among rough composite arguments, or by cancellation that is not robust in its comparison class. None of these has supplied a cheap odd family here. A fully multiplicative comparison with approximate prime-power values is also unresolved; the fixed-prime comparison is not such a construction. New work on that question must control the contributions from all affected multiples.

The finite evidence tests the stated examples and bounds. No finite computation, author check or change of coordinates here closes the irrationality gap.