D-072 — Finding F-D: the (6.12) stepping in Lagarias's Thm 6.1 is broken (right-tail of the horizontals)

Date: 2026-08-24 | Session: 038 (ii′-6-c discovery, before any Lean work) | Status: DOCUMENTED, direction OPEN (user opted to pause and think) Numerical evidence: E-031 (ALL CHECKS PASS).

1. The claim that fails

Lagarias, "Li coefficients for automorphic L-functions" (arXiv:math/0404394, Ann. Inst. Fourier 57 (2007) 1689–1740), Theorem 6.1's proof, page 29 (the right part of the horizontal segments):

"Now we step across the interval (II) towards the right, in segments of length 1, starting from σ = 3. At the initial point we have |L′/L| = O(1) ... the estimate (6.10) gains a factor of 2 going one unit to the right. In comparison |k_n(s+1)+1|/|k_n(s)+1| ≤ e [via (6.12)]. We obtain an upper bound for |k_n(s)L′/L(s,π)| that decreases geometrically at each step, and after O(log n) steps it becomes O(1) so |I_II| = O(√n + (log n)³)."

The contour: integrand k_n(s)·(−L′/L(s+1)), k_n(s) = (1+1/s)ⁿ − 1, on the rectangle σ₀ ∈ (−3, −2), σ₁ = 2√n, im = ±T, T = √n + ε.

2. Why it fails (the mechanism)

Per unit step right from σ = 3:

• the zeta bound halves: |L′/L(σ+1+iT)| ≤ 2^{−(σ−2)}·|L′/L(2)| ((6.10));
• the kernel ratio is ≤ e: |k_n(s+1)+1|/|k_n(s)+1| ≤ (1+1/T²)ⁿ ≤ e^{n/T²} ≤ e ((6.12), since T² ≥ n).

The product ratio is therefore ≤ e/2 ≈ 1.359 > 1 — the product GROWS, it does not decrease. The kernel's own per-step log-increment is ≈ 1–2 for σ ≪ √n (from log|1+1/s| ≈ σ/(σ²+T²), increment n·d/dσ[σ/(σ²+T²)] ≈ n(T²−σ²)/(σ²+T²)² ≈ 1 at σ small, T² = n), which exceeds log 2 ≈ 0.693 until σ ≈ 0.36√n. So the product rises from ~O(1) at σ = 3 to a peak ~e^{0.066√n} at σ ≈ 0.36√n, then falls to e^{√n/2}·2^{−(2√n−2)} ≈ 0 at σ₁ = 2√n. The stepping's own geometric series Σ(e/2)^j with ratio > 1 diverges; the published conclusion O(√n) requires the ratio < 1.

The same bound works with a taller horizontal: T = c√n gives kernel ratio ≤ e^{1/c²}, which is < 2 (hence product ratio e^{1/c²}/2 < 1) whenever c > 1/√(log 2) ≈ 1.2012 — verified numerically (E-031 V3: the product decreases monotonically from σ = 3 at c = 1.5 and c = 2).

3. Numerical evidence (E-031, direct computation, dps 25)

The product |k_n(w−1)|·|Σ_{k<300} Λ(k)/k^w| at w = σ + i√n:

Envelope integral of the right tail ∫_{3}^{2√n} (step 2):

• n = 10⁴: ≈ 1.05×10⁴ (peak 494 at σ = 35);
• n = 10⁵: ≈ 5.87×10¹⁰ (peak 1.56×10⁹ at σ = 111).

Direct factor evaluation at the peak (n = 10⁵, σ = 100, T = √n = 316.2): |k_n(100+317i)| = |(1+1/(100+317i))^{100000}| = 4.39×10³⁹ (mpmath, dps 30); |ζ′/ζ(101+317i)| ≈ 1.8×10⁻³¹ (mpmath finite difference + Dirichlet series cross-check). Product ≈ 8×10⁸.

Repair verification (n = 10⁵, product log P at σ = 3, 10, 30, 60, 100, 200):

• T = √n: −0.03, +2.17, +8.08, +15.51, +20.86, +3.71 (GROWS to σ ≈ 120);
• T = 1.5√n: −1.91, −3.10, −8.10, −15.92, −27.32, −63.66 (decreases);
• T = 2√n: −1.59, −4.91, −13.80, −27.21, −45.41, −93.63 (decreases).

4. Inheritance into R-004

D-036 §3 and D-037 §4 inherited the stepping verbatim: the right part of the horizontals is charged "3.3" (D-037: "right part [3, 2√n] decays geometrically by the stepping argument (6.12): |I_II|+|I_IV| ≤ 2(e⁴+1)[7H(T)+3.3]"). The true right-tail is ~5.9×10¹⁰ at n = 10⁵ — the charge is off by ~10⁷×, and the error grows like e^{0.066√n}.

With the honest right-tail, B(n) must charge ~e^{0.066√n}·poly, which overtakes S_∞(n) = ½n log n + O(n) at n ≈ 3×10⁴ — the margin argument for [N₀, N₁] collapses (margin(n) < 0 for n ≳ 3–4×10⁴).

5. What survives

• The theorem statement (λ_n > 0 for n ≤ N₁): directly verified to n = 5×10⁵ (Misguich–Voros + E-011/E-015 to 3000); the measured |S_f| is ~13–20 (E-021, Maślanka). The closed contour identity is exact — the sides cancel dramatically ((II)/(IV) are conjugates: real parts cancel; plus phase-winding of arg(k_n·L′/L) ~ −300 rad over the peak region). Lagarias's Theorem 6.1 CONCLUSION (Sf = λ_n(√n) + O(√n log n)) may well hold via that cancellation — but the paper's PROOF of the horizontal bound is broken, and the side-wise triangle bounds (what D-036/037/R-004 use) are unusable as charged.
• All Lean work to date: the honest-Bn correction (4F, (2/3)Z2hat), margin convexity, base points, the sides identity, the bridge (4·FAlt), the perimeter Props, the wide pigeonhole — the bug lives in the horizontal piece (ii′-6-c), which had NOT been built yet.
• The residue-identity framework (rectInt'_Jlag) — exact, unaffected.

6. Repair path (R) — verified numerically (E-031 V3)

Raise the horizontal height to T = c√n with c > 1/√(log 2) ≈ 1.2012 (kernel ratio ≤ e^{1/c²} < 2 ⟹ the product decreases per step ⟹ the stepping works; the right-tail becomes genuinely small — at T = 2√n the kernel at σ = 100 is e^{24.6} and the product is ~e^{−43}).

Costs (c = 2 shown; c = 1.5 halves the range-cost):

• N₁ → (T₀/c)²: c = 2 gives (T₀/2)² ≈ 2.25×10²⁴; c = 1.5 gives (T₀/1.5)² ≈ 4×10²⁴ (from 9×10²⁴ — still criterion-3).
• Bridge 4·FAlt(T) grows by ≈ c× (4·FAlt(2√n) ≈ 0.64√n log n).
• I_III length doubles: 2T = 2c√n (bound 4c√n·[(2/3)Z2 + 10.5 log + 48] ≈ 42√n log n at c = 2).
• Horizontal kernel charge drops: |k_n| ≤ (1+4/(16+4c²n/c²))ⁿ+1 ≤ e^{4/c²}+1 ≈ e+1 at c = 2 (from e⁴+1).
• I_I (right vertical) shrinks (kernel ≤ e^{√n/(2c)}+1).
• The zero-avoiding T pigeonhole and the wide pigeonhole carry over verbatim (window [c√n, c√n+1]).
• The margin survives comfortably: B(n) ~ 4c√n·(0.4+10.5)·log·... ≈ 21c·√n log n ≪ ½n log n for n ≳ 10⁴–10⁵ (crossover of the same order as before, re-verified by the margin machinery).

7. Literature status (session 038, §4A.3 checks)

• arXiv math/0404394 version history (v1–v4, 2004–2005): the only noted correction is "to remainder term in Theorem 1.1" — Thm 6.1 untouched.
• Web searches (DuckDuckGo, Bing): no erratum or discussion of a gap in Thm 6.1/(6.12) found. MathOverflow search not accessible from this environment. The issue may be undocumented — or read charitably with an unwritten fix (the T = c√n repair is essentially free and restores the O(√n) horizontal bound with the same proof).

8. Open questions (for the return)

1. Adopt repair (R) (which c: 2 for margin comfort, 1.5 for range) and rework Bn/constants accordingly (the D-071 protocol re-applies: convexity re-proof + base-point re-certification)?
2. Or attempt a salvage of T = √n via phase-winding/oscillation (the sides DO cancel — research-level, no existing machinery, open-ended)?
3. Or a deeper literature/community check first (MathSciNet review, MathOverflow direct, treatments of (6.12) in citing papers — e.g., Maślanka's, Voros's, the explicit finite-height follow-ups)?
4. The theorem statement R-004 itself: with repair (R), Paper A's headline becomes λ_n > 0 for n ≤ (T₀/c)² (≈ 2.25–4×10²⁴) — the finite-verification range (5×10⁵) and the criterion-3 status are unchanged.

9. Honesty notes

• The envelope integral (E-031 V2) is an UPPER bound on the true side integral |I_II's right part| ≤ ∫|f| — the true complex integral could be smaller via oscillation; but the envelope is what every side-wise assembly (D-036/037/R-004) charges, so it is the right measure for the fold's provability.
• The zeta factor was bounded by the Dirichlet series Σ_{k<300} Λ(k)/k^w (absolute convergence, re w ≥ 4) and cross-checked against mpmath's direct ζ′/ζ at σ = 100 (1.8×10⁻³¹ vs series 2.7×10⁻³¹ — same order). The tail k ≥ 300 contributes < 4×10⁻⁵ at σ = 3 and less elsewhere.
• No Lean code was harmed: ii′-6-c had not begun; the finding was caught at the design-check stage (the D-071 fold analysis).