Erdős #51: an unconditional exact-ratio-2 family (limsup n_a/a ≥ 2), quantitative obstruction lemmas, and a certified record table of minimal-preimage ratios to 3.06×10^10

For every integer k≥1 whose binary expansion contains a bit i such that the Fermat number F_i=2^(2^i)+1 is composite, the smallest n with φ(n)=2^k is exactly 2^(k+1), i.e. n_a/a=2 for a=2^k. Since F_5=641·6700417 is composite, this holds for the infinite set of k with bit 5 set; hence there are infinitely many totient values a with n_a/a exactly 2, and limsup over totient values of n_a/a is ≥ 2. Complement: if every bit i of k has F_i prime, then n_{2^k}=∏_{i∈B(k)}F_i ∈ (2^k, 2^(k+1)), ratio <2. Unconditionally n_{2^k}=2^(k+1) for every 32≤k<2^33.

Obstruction pack (all proved): (a) if a is a totient value with n_a/a≥K then every n with φ(n)=a satisfies ∏_{p|n}p/(p−1)≥K; (b) fully elementary: ω_odd(n)≥(K²−4)/8 and v₂(a)≥(K²−4)/8, equivalently f(a)/a≤2√(2v₂(a)+1) for every totient value a — in particular a≡2 (mod 4) forces f(a)/a≤2√3, and any family witnessing #51 must have v₂(a)→∞; (c) with Mertens (asymptotic) and Rosser–Schoenfeld (explicit): ω_odd(n) ≥ exp((e^{−γ}−o(1))K), explicitly ≥ e^(0.5526K)/(0.5526K)−1 for K>10.226; (d) f(a)/a≤(e^γ+o(1))loglog a (known bound, proof included for completeness); combining (c) with v₂(a)≤log₂a recovers (d) with constant 1.8094 in place of e^γ=1.7811.

Certified computational record table: for every totient value a ≤ A_max = 30,641,761,281 the exact minimal preimage f(a) is determined (sieve over n ≤ 1.94×10^11 plus an elementary completeness lemma guaranteeing all preimages of such a lie ≤ N). There are 3,903,102,222 totient values ≤ A_max; exactly 31,043 of them have f(a)/a ≥ 2 (complete census; a further 3,408,149 have 1.9 ≤ ratio < 2); the maximum ratio is 2.043447, attained at a=387383296=2^16·23·257 with f(a)=791597265=3·5·17·47·257²; no totient value a ≤ 3.06×10^10 has f(a)/a ≥ 2.05. The running-record positions are exactly 2, 8, 128, 5888, 2037248, 387383296, reproducing OEIS A393265 independently and certifying the list's completeness for a ≤ 3.06×10^10. Census structure: 31,029 of the 31,043 have ODD minimal preimage (all built from the Fermat kit 3·5·17(·257) with Sophie-Germain-type carriers, e.g. every top-16 record has f(a)=3·5·17·47·347·q or 3·5·17·47·257²); the 14 even ones include a=2^32, 2^33, 2^34 with ratio exactly 2 — the sieve independently exhibits Theorem 1's family in range; minimum v₂(a) over the census is 8.

Novelty assessment (honest scoping): the classification of φ-fibers of 2^k is classical (AMM Problem E3361, 1991; Guy's UPNT B36/B39 territory; OEIS A002181 comment by T. D. Noe), and the record POSITIONS were published in OEIS A393265 (J. McCranie, Feb 2026, terms to 3.9×10^8 with an uncertified scan to 1.2×10^11). Apparently new: the unconditional infinitude of exact-ratio-2 totient values and the resulting limsup≥2 stated for Erdős #51; the boxed inequality f(a)/a≤2√(2v₂(a)+1); the certified (provably complete) table and ratio≥2 census with reproducible code. None of these appear on the erdosproblems.com page/thread, in OEIS, or in web/arXiv searches (log: novelty.md).

Context/caveat claim: these results do NOT decide #51 and are consistent with both answers. The problem demands n_a/a→∞ along an infinite family; our infinite family has constant ratio 2, and the certified data (max 2.0434 over 3×10^10 values, record growth 1.5→2.0434 over ten orders of magnitude) together with the obstruction pack (ratio K forces v₂(a)≳e^{0.55K}) support Tao's heuristic that the answer is likely negative. The one rigorous positive takeaway is limsup≥2; the natural next wall is proving infinitely many totient values with ratio ≥2+δ for some δ>0 (Sophie-Germain-type kits; conditional on Dickson this looks feasible, unconditionally it meets the #203-type shifted-prime obstruction exactly as Kovač predicted).