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Updated 08-20-2026

D-056 — Repair 4.5a: reference-point R-bound (traced A = 10.5, B₀ = 48) + digamma remainder repair (supersedes D-039) + I_I housekeeping

[Title erratum 2026-08-19 (session 010, D-060): the title previously read "A = 6.0, B₀ = 40" — stale intermediate values; the proved constants of this file are A = 10.5, B₀ = 48 (§ below, E-028).] Date: 2026-08-19 | Session: 009 | Task: 4.5a Status: complete. Supersedes D-036 §1 (erratum D-054 F3) and D-039 (erratum D-054 F1/F2). Support computation E-028 (ALL CHECKS PASS).

1. The digamma remainder, repaired (replaces D-039)

Lemma D. For Re w ∈ [−5/4, 1], |Im w| ≥ 5: ψ(w) = log w − 1/(2w) + r(w), |r(w)| ≤ 1.2/|w|². (True size ~ (1/12)/|w|² asymptotically; the constant is application-insensitive — only the 1/|w|² decay enters.)

Proof. Binet piece (W = w + 2, so U := Re W ∈ [3/4, 3], V := Im W, |V| ≥ 5). Binet's second formula: r(W) = −2 ∫₀^∞ t / ((t² + W²)(e^{2πt} − 1)) dt. |t²+W²|² = (t² + U² − V²)² + 4U²V²t². Split [0,∞) at 1 and |V|/2: - [0,1]: |t²+W²| ≥ |W|² − 1 ≥ 0.96|W|² (|W| ≥ 5.05); ∫₀¹ t/(e^{2πt}−1)dt < Γ(2)ζ(2)/(2π)² = 1/24. Piece ≤ 2/(24·0.96|W|²) = 0.0868/|W|². - [1, |V|/2]: |t²+W²| ≥ V² − U² − t² ≥ 3V²/4 − 9 ≥ 0.39V² (U ≤ 3, |V| ≥ 5). Same integral: piece ≤ 2/(24·0.39V²) = 0.214/V² ≤ 0.214/(0.648|W|²) = 0.330/|W|² (V² ≥ |W|² − 9 ≥ 0.648|W|²). - [|V|/2,∞): |t²+W²| ≥ 2U|V|t ≥ 1.5|V|t; the t cancels and ∫_{|V|/2}^∞ dt/(e^{2πt}−1) ≤ e^{−π|V|}/(2π(1−e^{−2π})) — ≤ 3×10⁻⁸, absorbed in the rounding of the middle piece. Total: |r(W)| ≤ 0.417/|W|² (U ∈ [3/4, 3], |V| ≥ 5).

Recurrence piece. ψ(w) = ψ(w+2) − 1/w − 1/(w+1) gives r(w) = r(w+2) + E(w) with, writing u = 1/w, E(w) = f(u) := log(1+2u) − u/(2(1+2u)) − u/2 − u/(1+u). f is analytic on |u| ≤ 1/5 (nearest singularity u = −1/2) with f(u) = −u³/3 + Σ_{k≥4} c_k u^k, c_k = (−1)^k[−2^k/k + 2^{k−2} + 1], |c_k| ≤ 2^k/k + 2^{k−2} + 1. Hence for |u| ≤ 1/5: |f(u) + u³/3| ≤ Σ_{k≥4} (2^k/k + 2^{k−2} + 1)/5^k ≤ 0.010667 + 0.010667 + 0.002 = 0.0233. g(u) := f(u) + u³/3 vanishes to order 4, so by the maximum principle applied to g(u)/u⁴ (analytic on |u| ≤ 1/5): |g(u)| ≤ 0.0233·(5|u|)⁴ = 14.6·|u|⁴. Therefore |E(w)| ≤ 1/(3|w|³) + 14.6/|w|⁴.

Assembly. |w|²/|W|² = ((U−2)² + V²)/(U² + V²) ≤ 1.04 on the range (decreasing in U on [3/4,3] and in V on [5,∞); max at U = 3/4, V = 5: 26.5625/25.5625 = 1.0391). Thus for |w| ≥ 5: |r(w)| ≤ 0.417·1.04/|w|² + 1/(3·5|w|²) + 14.6/(25|w|²) = (0.4337 + 0.0667 + 0.584)/|w|² = 1.084/|w|² ≤ 1.2/|w|². ∎

2. The R-bound by the reference-point method (replaces D-036 §1)

Theorem R. For s = σ + it, σ ∈ [−5/2, 3], |t| ≥ 10: R(s) := ζ′/ζ(s) − Σ_{|γ−t|<1} 1/(s−ρ) satisfies |R(s)| ≤ A log(|t|+2) + B₀, A = 10.5, B₀ = 48.0, assembled below as 5.95 log(t+2) + 12.16 log log(t+2) + 47.8, then folded via log log u ≤ (1/e) log u (max at u = e^e).

Proof. Reference point s₀ = 2 + it. Partial fractions (ξ′/ξ(s) = B + Σ_ρ(1/(s−ρ) + 1/ρ); ζ′/ζ = ξ′/ξ − 1/s − 1/(s−1)

  1. |ζ′/ζ(s₀)| ≤ −ζ′/ζ(2) =: Z₂ ≤ 0.5700 (absolute convergence; certified: partial sum to 10⁶ + tail (log N+1)/N = 1.5e-5, E-028 V5).
  2. Near at s₀: |s₀−ρ| ≥ 2 − β ≥ 1 ⟹ Σ_{|γ−t|<1} 1/|s₀−ρ| ≤ C₀(t).
  3. Far difference: |s₀−s| = |2−σ| ≤ 4.5; |s−ρ| ≥ |t−γ|; |s₀−ρ| ≥ √(1+(t−γ)²) ≥ (1+|t−γ|)/√2. Each term ≤ 4.5√2/(|t−γ|(1+|t−γ|)). Group by unit intervals (Σ_{k≥1} 1/(k(k+1)) = 1 exactly): left side ≤ C₀(t), right side k ≤ t ≤ C₀(2t) ≤ C₀(t) + 0.58; the k > t tail ≤ C₀(2t)/(t+1) + 0.433/t ≤ 0.53 at t = 10 (decreasing). Total ≤ 4.5√2·[2C₀(t) + 0.58 + 0.53] = 12.73·C₀(t) + 7.06.
  4. |1/s − 1/s₀| ≤ 4.5/|t|² ≤ 0.045.
  5. |1/(s−1) − 1/(s₀−1)| ≤ 4.5/|t|² ≤ 0.045 (|s−1| ≥ |t|, etc.).
  6. ψ-difference: ½|log(s/s₀)| ≤ 0.23 at |t| = 10 (decreasing; |arg(s/s₀)| ≤ 4.5/|t|, |log|s/s₀|| ≤ 0.02); ½|1/(2s) − 1/(2s₀)| ≤ 0.012; ½(|r(s/2)| + |r(s₀/2)|) ≤ ½·1.2·(4/|s|² + 4/|s₀|²) ≤ 4.8/|t|² ≤ 0.048 (Lemma D). Total ≤ 0.29. Assemble: |R(s)| ≤ 0.57 + C₀(t) + 12.73 C₀(t) + 7.06 + 0.09 + 0.29 = 13.73·C₀(t) + 7.97 ≤ 13.73·(0.433 log(t+2) + 0.886 log log(t+2) + 2.9) + 7.97 = 5.95 log(t+2) + 12.16 log log(t+2) + 39.8 + 7.97 ≤ (5.95 + 12.16/e) log(t+2) + 47.8 ≤ 10.5 log(|t|+2) + 48. ∎

Remarks.

3. Sensitivity / theorem survival (R-004 with traced constants)

B(n) with A = 10.5, B₀ = 48 (honest C₀ incl. the log log term): leading coefficient of √n log n becomes 4·(0.1443 + 10.5/2) ≈ 21.6; B₀ term 4T·B₀ = 192√n; horizontals 2(e⁴+1)·7·H with H = 3C₀² + 10.5 log + 48. E-028 exact (ALL CHECKS PASS):

4. I_I housekeeping (D-054 F5)

5. Status of the R-004 input inventory after this repair