D-073 — F-D repair side thread ii′-6-R: mission, plan, and repaired proof of the Thm 6.1 horizontal right-tail

Date: 2026-08-25 | Session: 039 | Status: THREAD COMPLETE — 6-R-0/6-R-1/6-R-2 all done; repaired proof + audits + E-033 cross-validation (ALL CHECKS PASS) + drafts in §8. Main line may execute repair (R) at the recommended c = 1.5. Evidence: E-031 (finding), E-032 (diagnostic, ALL CHECKS PASS), E-033 (cross-validation, ALL CHECKS PASS) Checklist anchor: NGUYEN.md §5 Phase 5B, item ii′-6-R

0. Mission (agreed 2026-08-25, user directive — this section IS the thread prompt, persisted so nothing lives only in conversation)

MISSION — focused re-proof of the right part of the horizontal segments in Lagarias's Theorem 6.1 (Finding F-D follow-up). This is a SEPARATE, SELF-CONTAINED side thread. The main line (Phase 5B, Lean tier ii′) is parked at the Finding F-D decision point awaiting exactly this thread's deliverables. Do NOT resume main-line work; do NOT open Lean formalization.

READ FIRST (in order):

1. STATE.md — resume point.
2. research/derivations/D-072-finding-F-D-right-tail-stepping.md — the finding: mechanism, E-031 evidence, inheritance, what survives.
3. research/paper/lagarias-li-coefficients-automorphic-0404394.pdf, pages 27–29 — the actual Thm 6.1 proof passage ((6.9)–(6.14)).
4. research/derivations/D-071-ii-prime-6-design.md — the honest-Bn constant machinery this repair feeds (kernel charge, bridge, FAlt).
5. research/numerics/scripts/E-031 script + research/numerics/data E-031 sidecars — the envelope evidence.

BACKGROUND (established; do not re-derive):

• The printed stepping (6.12) bounds the kernel ratio by e and (6.10) gains a factor 2 per unit step right from σ = 3, so the product ratio is ≤ e/2 > 1: the product GROWS to a peak ~e^{0.066√n} at σ ≈ 0.36√n (E-031, measured). Both bounds are SHARP (zeta halving = Dirichlet m=2 term; kernel ratio |1−1/(s+1)²|ⁿ ≈ e^{n/T²} exact), so no side-wise/triangle refinement at T = √n can work.
• Oscillation cannot rescue T = √n by standard tools: the phase Φ(σ) ≈ −n^{3/2}/(σ²+n) has |Φ′| = O(1), so IBP/van der Corput/stationary phase gain O(1)–polynomial factors against an e^{0.066√n} amplitude range. The only known exact evaluation is the residue identity itself, which is circular for this purpose.
• Conjugation gives I_II + I_IV = 2i·Im(I_II): the closed identity constrains ONLY Im(I_II). Re(I_II) is unconstrained by the contour argument and may be of order ~envelope — i.e. the printed per-side claim |I_II| = O(√n) may be FALSE as a statement at T = √n.
• The repair is one line inside Lagarias's own framework: he proves the horizontal bound for general T with T² ≥ n and only specializes T = √n + ε in the final line. Taking T = c√n + ε with c > 1/√(log 2) ≈ 1.2012 makes the kernel ratio ≤ e^{1/c²} < 2, the stepping works verbatim, and every other ingredient is T-generic (verified numerically, E-031 V3, c = 1.5 and 2).

DELIVERABLES:

• (D1) E-032 DIAGNOSTIC (cheap, decisive): compute the TRUE complex horizontal integral I_II at T = √n (zero-avoiding) for n = 10³, 10⁴ (10⁵ if cheap) by direct quadrature (mpmath, dps ≥ 40; integrand smooth, phase O(1) rad/unit; validate ζ′/ζ against the Dirichlet series where re > 2). Compare against the E-031 envelope. Verdict: Re(I_II) ~ envelope ⟹ the per-side claim is FALSE at T = √n (only the pair survives); |I_II| ≪ envelope ⟹ hidden cancellation exists. Record the numbers either way.
• (D2) THE REPAIRED PROOF (this file, sections to be written): (i) dependency audit — every ingredient of Thm 6.1's proof listed and marked T-generic / T-dependent; (ii) the corrected stepping lemma at T = c√n, c > 1/√(log 2): per-step product ratio ≤ q := e^{1/c²}/2 < 1, geometric series from σ = 3, explicit right-tail bound ∫₃^{σ₁}, total horizontal bound |I_II| + |I_IV| with EXPLICIT constants (no untraced O's — use D-071's machinery: kernel charge e^{4/c²}+1, |ζ′/ζ(2)|, the zero-avoiding (log T)² bound); (iii) recommended c for the main line, quantified (2 margin / 1.5 range); (iv) consequence audit (Thm 6.1 conclusion + Thm 1.1 height); (v) numerical cross-validation of all new constants (E-033).
• (D3) A one-paragraph "repair summary" suitable for the Lagarias letter (DRAFT only — do NOT edit any correspondence file).
• (D4) A one-paragraph verdict on the T = √n per-side question, combining E-032 with the circularity analysis (DRAFT only).

HARD CONSTRAINTS:

• OUT OF SCOPE: re-proving any other part of Thm 6.1 (audit only); the T = √n salvage as a proof goal (D1's verdict settles whether anything there remains to pursue — if |I_II| comes out SMALL, STOP and report to the user before going further); Lean work; contacting anyone.
• FREEZE PROTOCOL (user directive 2026-08-21): no edits to research/paper/paperA.tex, paperB.tex, draft.tex, references.bib; no edits to existing Lean files; no edits to research/correspondence/. New files only: this file, research/numerics/scripts/E-032.py / E-033.py + data sidecars.
• Honesty tag on every claim ([PROVED] / [NUMERICAL-VALIDATED] / [HEURISTIC]); numerics deterministic + cached, CSV/JSON sidecars, registered in research.db via db.py.
• Session protocol per NGUYEN.md §8; sub-stage rule (§8.5): land D1 first as its own committed sub-stage.

SUCCESS CRITERIA:

1. This file contains the complete repaired proof of the horizontal bound with explicit constants, plus the dependency and consequence audits, all honesty-tagged.
2. E-032's verdict recorded with numbers.
3. The main line can execute repair (R) from this file + the recommended c without any further analysis.

1. Progress log (append one entry per sub-stage; this is the in-thread stop-and-go record)

• 2026-08-25: mission agreed (user directive), thread registered in NGUYEN.md §5 as ii′-6-R; this file created as the resume anchor. NEXT: sub-stage 6-R-0 (E-032 diagnostic).
• 2026-08-25 (session 039): 6-R-0 COMPLETE — E-032 ALL CHECKS PASS; verdict HIDDEN CANCELLATION, decisive (§2 below). Per the mission's STOP rule the thread reports to the user before 6-R-1.
• 2026-08-25 (session 039, user direction: repair proof only, salvage parked): 6-R-1 COMPLETE — §3 dependency audit (only item 10 proof-breaking), §4 corrected stepping lemma + explicit R(c) [PROVED, elementary], §5 c-choice (AUDIT CORRECTION: recommend c = 1.5, reversing D-072's "2 margin/1.5 range" — the fold is I_III-length-dominated), §6 consequence audit, §7 E-033 ALL CHECKS PASS (Z₀ certified, q-table, R covers E-032 measured 3.2–6.3× slack, kernel charge, fold fit ≤ 0.40, margins > 0 to N₁, full fold). 6-R-2 COMPLETE — §8 drafts (D3 letter paragraph, D4 per-side verdict paragraph; correspondence untouched per freeze protocol). THREAD COMPLETE — all success criteria met.
• 2026-08-25 (session 040, USER DIRECTIVE — c = 1.5 main-line redo PAUSED): the user stopped the c = 1.5 execution mid-flight — the goal is the ORIGINAL N₁ = 9×10²⁴, not the repaired N₁ = 4×10²⁴; a new plan supersedes (PROJECT_DIRECTIVE_D-074.txt). The c = 1.5 WIP is preserved on git branch wip/c-1.5-margin-redo (commit 642d47a, BUILD BROKEN mid-convexity; the commit message has the exact resume map: hnn-sign-assembly + Bn + endpoints + LiBasePoints targets). Master is green (36b9e79) with the session-038 Bn (T = √n+1, the F-D-broken 3.3 charge documented in D-072) — the original 9×10²⁴ goal now requires either the T = √n salvage (§8 D4 open question) or a different repair that keeps c = 1. This thread's deliverables (Lemma 4.1, R(c), E-032/E-033, the audits, the drafts) remain valid for ANY c > 1/√log 2.

2. E-032 diagnostic (D1) — verdict + numbers

VERDICT: HIDDEN CANCELLATION — decisive. The true complex right-tail integral at T = √n (zero-avoiding) is ORDERS OF MAGNITUDE below its envelope, and the gap widens explosively with n: |I_right|/envelope = 0.141 → 1.63×10⁻³ → 5.57×10⁻¹⁰ at n = 10³ → 10⁴ → 10⁵. The per-side claim |I_II| = O(√n) is empirically TRUE (the full side comes out at 7.4 / 19.2 / 30.9 — the ratio |I_II|/√n even DECREASES, 0.227 → 0.167 → 0.083) — but it is not accessible to ANY envelope/stepping/triangle method at T = √n, since the envelope itself is ~e^{0.066√n} too fat. Per the mission (D1): STOP and report. [NUMERICAL-VALIDATED for the numbers; NUMERICAL for the interpretation]

2.1 Method (all checks pass; E-032 script + CSV/JSON sidecars, db-registered)

Integrand f(w) = k_n(w−1)·(−ζ′/ζ(w)), w = σ+iT, right part σ ∈ [3, 2√n]; T = c√n + ε with ε from a deterministic zero-avoiding scan (max-clearance grid point; clearances 0.43–1.87). −ζ′/ζ computed two independent ways: truncated Dirichlet series Σ_{k≤K(σ)} Λ(k)k^{−w} with CERTIFIED tail < 1e-11 (primary, σ ≥ 5) and mpmath (σ ∈ [3,5)); cross-validated at σ ∈ {3, 3.5, 4, 4.5, 5}: max discrepancy 6.0×10⁻¹⁰ [check VA]. Trapezoid h = 0.05 vs 0.02: h-delta ≤ 5×10⁻⁵ everywhere (vs |I| ≥ 0.3) [check VC]. Envelope reproducibility vs E-031 [check V0]: refined envelope 1.026×10⁴ at n = 10⁴ (E-031: 1.05×10⁴), 4.707×10¹⁰ at n = 10⁵ (E-031: 5.87×10¹⁰, coarse grid + K = 300 tail). Repair-envelope collapse [check V3]: Env(c=2)/Env(c=1) = 7.5×10⁻¹² at n = 10⁵. dps 50.

2.2 The numbers (c = 1, i.e. T = √n + ε, zero-avoiding)

Full side (I_II = I_mid + I_right) and mid part [−5/2, 3] (mpmath ζ):

Repair heights (right part): c = 1.5: |I| = 1.12 / 1.14 / 1.00, envelope ≈ |I| (ratio 0.90–1.00); c = 2: |I| = 0.62 / 0.31 / 0.35, ratio ≈ 1.00 — at the repaired heights the envelope is honest (no cancellation needed), as the stepping argument requires.

2.3 What the diagnostic establishes

1. The envelope is catastrophically pessimistic at T = √n — oscillation kills essentially ALL of the e^{0.066√n} peak (cancellation factor ~2×10⁹ at n = 10⁵). [NUMERICAL-VALIDATED]
2. The per-side claim |I_II| = O(√n) is empirically TRUE, with a small constant and a DECREASING ratio |I_II|/√n (0.235 → 0.098 over n = 10³ → 10⁵). [NUMERICAL]
3. The mid part has essentially NO cancellation (ratio 0.70–0.95) — the charged mid-part bound is honest; all the hidden cancellation lives in the right tail. [NUMERICAL-VALIDATED]
4. The repair heights make the envelope itself honest (ratio ≈ 1, envelope ~1.0 at c = 1.5, ~0.35 at c = 2, n = 10⁵) — the T = c√n repair needs no cancellation and is provable by the elementary stepping. [NUMERICAL-VALIDATED]
5. Re(I_II) = −2.25 / −16.65 / −25.97 (negative, slow-growing); the closed identity I_II + I_IV = 2i·Im(I_II) constrains only Im — the measured Re is consistent with O(√n) but NOT with O(1). [NUMERICAL]
6. Consequence for the finding note: at T = √n the published PROOF of the horizontal bound is broken (D-072) while the per-side STATEMENT appears true — the gap between them is exactly the hidden cancellation measured here. A T = √n proof would need new oscillation machinery (the phase |Φ′| = O(1) defeats stationary phase / van der Corput; the cancellation is NOT of stationary-phase type). [NUMERICAL + HEURISTIC]

3. Dependency audit (D2-i)

Every ingredient of Lagarias's Thm 6.1 proof (pp. 27–29), marked T-generic (valid verbatim at any T = c√n + ε) or T-dependent (changes with c). Notation: the rectangle σ₀ ∈ (−3, −2) [variant B: −5/2], σ₁ = 2√n, im = ±T; integrand k_n(w−1)·(−L′/L(w)), k_n(s) = (1+1/s)ⁿ − 1, w = σ+iT. (Our rectInt'_Jlag convention: w = the integration variable; Lagarias's s is our w−1. The right tail is σ ∈ [3, 2√n] in w.)

Conclusion of the audit: the ONLY proof-breaking T-dependence is item 10; items 4, 8, 12 are benign quantitative changes. The repair touches exactly one line of the paper. [AUDITED]

4. The corrected stepping lemma (D2-ii) — repaired proof, explicit constants

Setup (ζ, variant-B rectangle). w = σ+iT, T = c√n + ε with c > 1/√(log 2) = 1.20120… and ε ∈ [0,1] zero-avoiding (the wide pigeonhole on [c√n, c√n+1], clearance 1/(9·Ĉ₀(c√n+2)) — item 6). Right tail: σ ∈ [3, σ₁], σ₁ = 2√n. Integrand f(w) = k_n(w−1)·(−ζ′/ζ(w)), k_n(s) = (1+1/s)ⁿ − 1.

Lemma 4.1 (corrected stepping). For σ ≥ 3 set P(σ) := |1+1/(σ−1+iT)|ⁿ · 2^{−(σ−2)}. Then for every σ ≥ 3,

P(σ+1)/P(σ) ≤ q, q := e^{1/c²}/2 < 1.

PROOF. The zeta factor contributes exactly 1/2 per unit step (item 7: |ζ′/ζ(σ+1+iT)|/|ζ′/ζ(σ+iT)| ≤ 2^{−1}, the (6.10) halving). The kernel ratio is Lagarias's own (6.12), which is CORRECT as a ratio bound: with s = σ−1+iT,

|1+1/(s+1)|ⁿ/|1+1/s|ⁿ = |1 + (1/(s+1) − 1/s)/(1+1/s)|ⁿ ≤ (1 + |1/(s(s+1))|·|s+1|/|s| … )ⁿ = (1 + 1/|s+1|²)ⁿ ≤ (1+1/T²)ⁿ ≤ e^{n/T²} ≤ e^{1/c²},

since |s+1| ≥ T and T² ≥ c²n. (The defect of F-D is not in this chain but in the PRODUCT: at c = 1 the factor e^{1/c²} = e > 2 beats the halving.) ∎

Corollary 4.2 (explicit right-tail bound). With |ζ′/ζ(2)| =: Z₀ = 0.569960993069… (E-033 V1, certified),

∫₃^{σ₁} |f(σ+iT)| dσ ≤ R(c) := Z₀·[ e^{4/c²}/(4(1−q)·log 2) + 1/(2·log 2) ].

PROOF. |f| = |k_n(w−1)|·|ζ′/ζ(w)| ≤ |(1+1/(w−1))ⁿ|·|ζ′/ζ(w)| + |ζ′/ζ(w)| ≤ Z₀·P(σ) + Z₀·2^{−(σ−2)} (item 7 applied pointwise). Write σ = 3 + j + r, j = ⌊σ−3⌋, r ∈ [0,1). Lemma 4.1 gives P(3+j+r) ≤ q^j·P(3+r), and on [3,4]: Re(w−1) = 2+r ∈ [2,3] ⊂ [−3,3] so the kernel charge (item 8) gives P(3+r) ≤ e^{4/c²}·2^{−(1+r)}. Geometric series:

∫₃^{σ₁} P(σ) dσ ≤ (Σ_{j≥0} q^j)·∫₀¹ e^{4/c²}·2^{−1−r} dr = e^{4/c²}/(1−q) · 1/(4·log 2),

and ∫₃^∞ 2^{−(σ−2)} dσ = 1/(2·log 2). ∎

Numerically (E-033 V2–V4):

R(2) < 3.3 — the repaired right-tail charge at c = 2 is SMALLER than the broken charge it replaces; R(1.5) = 5.94 is mildly larger. [PROVED (Lemma 4.1, Cor 4.2 — elementary, self-contained); NUMERICAL-VALIDATED (constants, E-033)]

Mid part [−5/2, 3] (unchanged machinery, T = c√n). Kernel ≤ e^{4/c²}+1 (item 8); the tight R-bound (Lean 5f capstone, modulo perimeter: ZeroCountProp, ZeroCountProp2, NoLowZeros) + the segment-integral near sum (D-071 §4.4: Σ_near mult·∫dx/|w−ρ| ≤ Z2(T)·2·log(16.5/c_clear) =: NI(T), c_clear = 1/(9·Ĉ₀(c√n+2))):

∫_{−5/2}^{3} |f| dσ ≤ (e^{4/c²}+1)·[ NI(T) + 5.5·(10.5·log(T+2) + 48) ].

Total horizontal bound:

|I_II| + |I_IV| ≤ 2·(e^{4/c²}+1)·[ NI(T) + 5.5·(10.5·log(T+2)+48) + R(c) ].

5. The choice of c (D2-iii)

Quantified trade (E-033, ALL CHECKS PASS; V2/V5/V6 tables):

Recommendation: c = 1.5 for the main line. Rationale: (i) the fold is I_III-LENGTH-DOMINATED (the 4T[…] term ~ T·log T is ~87% of Bn at 10⁵), so SMALLER c gives BIGGER margins everywhere — the binding base-region margin is 2.3× larger at c = 1.5, and margin(n) > 0 on the whole grid [8×10⁴, N₁] at both c (E-033 V6); (ii) the range wins too: N₁ = 4.0×10²⁴ vs 2.25×10²⁴; (iii) the costs are immaterial to the fold: R = 5.94 vs 1.97 and kernel 6.92 vs 3.72 are both ≪ Bn ~ 2.5×10⁵ at 10⁵ (the whole horizontal piece is ≤ 11% of Bn either way), and q = 0.78 still gives comfortable geometric decay (1/(1−q) = 4.5 vs 2.8); (iv) c ≤ 1.3 is NOT recommended: q ≥ 0.90 makes the stepping knife-edge and the repaired theorem's O-constant sensitivity (1−q)⁻¹ ~ (c−c*)⁻¹ grows, while the fold gains little (already saturated) — R(1.3) = 23.2, R(1.25) = 51.8, both still ≪ Bn, so the constraint is robustness, not the fold. c = 2 is documented as the ultra-conservative option (right-tail charge 1.97 < the broken 3.3; q-margin 0.36) — usable with the identical proof.

AUDIT CORRECTION: this reverses D-072 §6's preliminary framing ("c = 2 preferred for margin comfort, c = 1.5 for range") — the E-033 fold computation shows the margin is I_III-length-dominated and favors SMALLER c on both axes (margin and range); c = 2's only real wins are the (fold-immaterial) right-tail constant and the q-margin. The recommendation is quantified above; either c executes with the same proof.

6. Consequence audit (D2-iv)

1. Lagarias Thm 6.1 (Sf(n,π) = λ_n(√n,π) + O(√n log n)): statement UNCHANGED; proof repaired by the single substitution T = c√n + ε, c > 1/√log 2 (item 10 fixed by §4; item 4 absorbed; items 9/11 constants ×c). The O-constant becomes c-dependent (blows up like (1−q)^{−1} as c → 1.2012⁺). [AUDITED]
2. Lagarias Thm 1.1 (finite-height Li positivity): the range n ≤ (T₀−1)² becomes n ≤ ((T₀−1)/c)². [AUDITED]
3. R-004 (ours): N₁ = 9.0×10²⁴ → 2.25×10²⁴ (c = 2) [4.0×10²⁴ at c = 1.5]. The finite-verification range (5×10⁵), the criterion-3 status, and the theorem statement are unchanged. Bn is c-parameterized per §5; the margin survives (E-033 V6/V7); the main-line redo = the D-071 8-step protocol re-run at the chosen c (convexity re-proof + base-point re-certification; the base-point values IMPROVE — the horizontal charge dropped).
4. What does NOT repair: nothing at T = √n — the printed proof there is broken beyond the stepping line (E-032 §2: the envelope is e^{0.066√n}-fat; only hidden cancellation saves the statement). The T = √n per-side question stays an open problem (§8, D4).
5. Perimeter: the repair introduces NO new declared hypotheses. The right tail is perimeter-free; the mid part uses the same ZeroCountProp/ZeroCountProp2/NoLowZeros already declared (5f).

7. Cross-validation (D2-v, E-033)

E-033 (research/numerics/scripts/E-033-repaired-stepping-validation.py

• CSV/JSON sidecars, db-registered, dps 50): ALL CHECKS PASS.

• V1 (Z₀ certified): |ζ′/ζ(2)| ∈ [0.5699210, 0.5703661] by the K = 25 000 Dirichlet partial sum + certified tail (log(K−1)+1)/(K−1); mpmath value 0.5699610 inside; width 4.5×10⁻⁴ < 10⁻³. R(c) is charged with the UPPER edge 0.5703661 (honest). [NUMERICAL-VALIDATED]

• V2 (q-table + boundary): q(1) = 1.3591 > 1 (the F-D bug); q < 1 at 1.25 / 1.3 / 1.5 / 1.75 / 2 (0.948 / 0.904 / 0.780 / 0.693 / 0.642); boundary c* = 1/√log 2 = 1.2011224 ∈ [1.2011, 1.2013]. R-table: 51.8 / 23.2 / 5.94 / 2.89 / 1.97; kernel-charge table: 13.9 / 11.7 / 6.92 / 4.69 / 3.72. [NUMERICAL-VALIDATED]
• V3 (R covers the E-032 measured truth): measured |I_right| at c = 1.5: 1.12 / 1.14 / 1.00 (n = 10³/10⁴/10⁵) ≤ R(1.5) = 5.94 ✓; at c = 2: 0.62 / 0.31 / 0.35 ≤ R(2) = 1.97 ✓ — the repaired bound holds against the directly-computed true integrals with slack 3.2–6.3×. [NUMERICAL-VALIDATED]
• V4 (kernel charge): max_{σ∈[−3,3]} |k_n(σ+iT)| at n = 10⁵, T = c√n: 5.65 ≤ e^{16/9}+1 = 6.92 (c = 1.5); 2.06 ≤ e+1 = 3.72 (c = 2). [NUMERICAL-VALIDATED]
• V5 (fold fit at the repaired height): NI(T) + 5.5·(10.5·log(T+2) +48) ≤ 7·Ĥ(T), T = c√n+1, on the grid [8×10⁴, N₁(c)]: max ratio 0.3925 (c = 1.5), 0.3883 (c = 2). [NUMERICAL-VALIDATED]
• V6 (margins): margin(n; c) = S_∞(n) − Bn_c(n) + 1 > 0 on the dense grid for both c; min on grid: 111 917 (c = 1.5, at 8×10⁴), 49 365 (c = 2, at 8×10⁴); margin(N₁): 1.08×10²⁶ (c = 1.5), 5.99×10²⁵ (c = 2). [NUMERICAL-VALIDATED]
• V7 (full fold): honest_c(n) ≤ Bn_c(n) on the whole grid, both c. [NUMERICAL-VALIDATED]

Honesty note: the first E-033 run failed V1 on a script-tolerance error (the certified upper edge 0.5703661 exceeded an over-tight 0.5700 window; the certified bracket itself was correct and contained the true value). The window was corrected to the honest spec (width < 10⁻³); no mathematical content changed. Audit trail in git history.

8. Drafts (D3 letter paragraph; D4 per-side verdict paragraph)

D3 — repair summary paragraph for the Lagarias letter (DRAFT only; correspondence untouched)

Repair (one line). In the proof of Theorem 6.1, keep the general height T with T² ≥ n throughout, and only at the end specialize to T = c√n + ε_n with any fixed c > 1/√log 2 = 1.2012… (instead of c = 1). The estimate (6.12) then gives the kernel ratio ≤ e^{1/c²}, which is < 2, so the product with the (6.10) halving decreases geometrically at each unit step exactly as printed, and the right part of the horizontal segments is O_c(|L′/L(2,π)|). Explicitly, for ζ, with q := e^{1/c²}/2: ∫₃^{2√n} |k_n(w−1)·ζ′/ζ(w)| dσ ≤ R(c) := |ζ′/ζ(2)|·[ e^{4/c²}/(4(1−q) log 2) + 1/(2 log 2) ] (R(1.5) = 5.94, R(2) = 1.97). Every other ingredient of the proof is T-generic; the only further changes are |λ_n(√n) − λ_n(T)| = O_c(√n log n) (each zero with |γ| ≥ √n still contributes ≤ 2; there are O((c−1)√n log n) of them — the same order as the remainder) and the height in Theorem 1.1, which becomes n ≤ ((T₀−1)/c)². We also report that the T = √n per-side statement itself appears to be TRUE: direct quadrature of the true complex horizontal integral at T = √n (n = 10³, 10⁴, 10⁵; 50-digit arithmetic; ζ′/ζ cross-validated against the Dirichlet series to 6×10⁻¹⁰) gives |I_II| = 7.4, 19.2, 30.9 — well within O(√n), the e^{0.066√n} envelope being killed by oscillation (|I|/envelope down to 5.6×10⁻¹⁰). The defect is thus confined to the proof's stepping argument, not to the theorem's content; a T = √n per-side proof seems to require genuinely new machinery, since the phase has O(1) derivative and stationary-phase/van der Corput tools do not see the cancellation.

D4 — T = √n per-side verdict paragraph for the finding note (DRAFT only; correspondence untouched)

Per-side verdict at T = √n (E-032). The per-side claim |I_II| = O(√n) in the published proof is empirically TRUE, but not for the printed reason: the true complex right-tail integral is 7.2 / 16.7 / 26.2 at n = 10³ / 10⁴ / 10⁵ while its absolute envelope is 5.1×10¹ / 1.0×10⁴ / 4.7×10¹⁰ — hidden cancellation removes essentially all of the e^{0.066√n} peak (|I|/envelope = 0.141 → 1.6×10⁻³ → 5.6×10⁻¹⁰, and |I_II|/√n even decreases: 0.235 → 0.192 → 0.098). The mid-segment [−5/2, 3] shows almost no cancellation (ratio 0.7–0.95): its classical bound is honest. Hence the (6.12) stepping failure is a failure of the METHOD at T = √n, not of the statement. The repair T = c√n (c > 1/√log 2) restores an honest envelope (integral/envelope ratio ≈ 1; envelope ≈ 1.0 at c = 1.5, ≈ 0.35 at c = 2, n = 10⁵) at the price of the Theorem 1.1 height (T₀/c)². Proving the per-side bound at T = √n itself would require new oscillation machinery: the phase Φ(σ) ≈ −n^{3/2}/(σ²+n) has |Φ′| = O(1), so stationary-phase/van der Corput methods gain at most O(1)-polynomial factors against an e^{0.066√n} amplitude range, and the only known exact evaluation is the residue identity itself (circular). OPEN QUESTION: identify the mechanism (exact-identity structure? arithmetic of the nearby zeros?) that makes the T = √n per-side bound true.