D-054 — Task 4.3b: independent-instance adversarial review of T-1/T-2/T-3 + D-052 — findings, verifications, dispositions

Date: 2026-08-19 | Session: 009 | Task: 4.3b Status: complete. Outcome: all three theorem TAGS downgraded; theorem statements survive; four repair tasks opened (4.5a–4.5d).

Method: four independent review instances (fresh-context agents) attacked T-1 (R-004 chain), T-2 (R-003 + D-041/D-051), T-3 (D-048–D-050), and D-052's accounting. Every claimed finding was then RE-VERIFIED by hand against the archive before disposition. Findings that did not survive re-verification are marked REFUTED below.

1. T-1 (R-004, finite-height Li theorem)

Verified findings

• F1. D-039 Binet tail broken for |W| ≳ 35 [VERIFIED]. The split at t = 1 bounds ∫₁^∞ by a CONSTANT (~4e-5, via |W²+t²| ≥ 2U|V| ≥ 5), so the assembly "|r(W)| ≤ 0.097/|W|²" fails once 0.097/|W|² < 8e-5. The BOUND is true (asymptotic |r|·|w|² → 1/12; E-grid max 0.084); the proof is repairable: split the tail at t ~ |V|/2, using |W²+t²| ≥ V²/2 for t ≤ |V|/2 and exponential smallness beyond.
• F2. D-039 recurrence estimate misuses the Cauchy bound [VERIFIED]. The termwise estimate gives |f(u)+u³/3| ≤ 0.45 ABSOLUTE on |u| ≤ 0.2 (via |g(u)| ≤ 3.62·(|u|/0.4)⁴/(1−|u|/0.4)), not ≤ 0.45·|u|⁴. Extracting |u|⁴ costs a factor 1/(0.4⁴·(1−½)) ≈ 78, giving |E(w)| ≤ 1/(3|w|³) + 283/|w|⁴ — too lossy at |w| = 5 to assemble |r| ≤ 1/|w|². Repairable with a sharper Cauchy circle or direct series majorant (true max |g| on |u| ≤ 0.2 is ~0.01).
• F3. D-036 §1 far-sum bound is O(|s| log t), not O(log t) [VERIFIED — the load-bearing one]. The written assembly |s/(ρ(s−ρ))| ≤ |s|/(0.999|γ||t−γ|) + 1/(ab) ≤ ½(a⁻²+b⁻²) yields Σ_far ≲ 0.74|t| log t. The classical O(log t) (Lagarias Lemma 6.1, Davenport Ch. 15) requires the REFERENCE-POINT trick (compare at 2+it; the far-sum difference has bounded numerator (2+it−s) and converges via Σ C₀/(t−γ)²), which is NOT in D-036. The R-bound constants A = 2.0/0.9 are therefore NOT traced as claimed; their status is E-021-measured (15 points, 20–30% margin). Robustness: the R-bound enters B(n) at leading order (coefficient 0.577 + 2A of √n log n) but the margin at N₁ is 2.46×10²⁶ and margin(N₀) = 67 705; the theorem survives A, B₀ inflated 10×. Sensitivity low, but the tracing claim was wrong.
• F4. Margin-convexity proof invalid as written [VERIFIED]. "Every component of B(n) is concave" is asserted; 4T·f(T) pieces are products (e.g. √n·log log n-type terms) whose concavity for all n ≥ 20 is not shown. The fact itself is grid-verified (E-023 V2, 4000 log-points to N₁ + contiguity to 4×10⁵) and a symbolic second-derivative check is routine. → repair task.
• F5. I_I length/constant [VERIFIED, negligible]: contour length 2(√n+1) not 2√n; |ζ′/ζ(2)| ≤ 0.5699 uncertified. Effect < 10⁻⁸⁰. Fold into 4.5a.

Refuted / downgraded findings

• Bridge window (T−1)² vs printed T²/(4log²T): D-037 §6 already discloses the discrepancy, derives the stronger window from the argument (per-zero bound after Lagarias (6.7)), and carries the conservative fallback N₁* = 2.73×10²¹ in the theorem statement. Disposition: keep both windows; re-verify against Lagarias's §6 text at paper write-up (joins H5).
• Finite-verification overlap, N₁ arithmetic, S_∞ lower bound: re-verified OK by the review instance and again here.

T-1 disposition

Theorem STATEMENT stands (on E-021-measured A = 0.9, B₀ = 1.0, with margin dwarfing any plausible constant error). TAG downgraded: R-004 → [PROVED-CONDITIONAL(computational RH-to-T₀; computational finite verification; R-bound constants numerically measured (E-021), analytic tracing incomplete → task 4.5a)]. D-039 status → ERRATUM (bound true, proof broken, repair opened). Repairs:

• 4.5a D-036 §1 reference-point rewrite (O(log t) R-bound with traced constants) + D-039 repair (Binet split at t ~ |V|; sharper recurrence majorant) + I_I housekeeping.
• 4.5b Symbolic concavity proof of B(n)'s components (n ≥ 20).

2. T-2 (R-003 + D-041 reduction)

Verified findings (with severity reassessed)

• F6. Compact-convergence formulation. Suzuki's (1.12) as printed says "uniformly on every compact subset K ⊂ ℂ" while the target F = z²ξ/ξ′ has poles at zeros of ξ′ — literal reading impossible for entire e^{ϕ}W. This is SUZUKI's own abuse of notation (standard: compacts avoiding poles / spherical metric). Our files quoted it without the precision. Disposition: precision note added to R-003 / D-041 (punctured compacts); no content change. [MINOR]
• F7. τ̂± "analytic functionals carried by |x| ≥ a" [VERIFIED as a rigor gap, already flagged in spirit]. A distribution carried by the UNBOUNDED set |x| ≥ a is outside Paley–Wiener–Schwartz; R_τ's representation (z−i)τ̂₊ + e^{iθ}(z+i)τ̂₋ is distributional, and the exact identity W = F(R₀+R_τ) holds as meromorphic functions with R_τ := W/F − R₀. This sharpens (not extends) R-003's flagged "tempered-FT bookkeeping". Note strengthened in R-003. [GAP, known]
• F8. Converse direction (⇐) is a skeleton [VERIFIED — but already disclosed]. D-041 §4 states (⇐) "proof SKELETON with lemmas L3/L4 named", and D-051 shows L4 collapses to (★). The overstatement lives in D-052's T-2 line and tag, which presented the ⟺ as proved "modulo L1 write-up". Correct accounting: (★) ⟹ (1.12) chain and (1.12) ⟹ spec(D) ⊂ ℝ are proved (modulo L0/L1 bookkeeping); spec(D) ⊂ ℝ ⟹ (1.12) is a conditional skeleton. [Tag fix in D-052]
• W-2 paywall honesty: re-verified OK (correctly located, not claimed discharged). W-1 (κ) correctly named unknown.

T-2 disposition

Factorization R-003 keeps [PROVED (modulo tempered-FT bookkeeping)] — the surviving bookkeeping items are F6/F7, both noted. The REDUCTION's honest statement: "Cor 1.6 ⟺ (★)" [PROVED modulo bookkeeping] and "(1.12) ⟹ spec(D) ⊂ ℝ" [PROVED via Suzuki + D1]; the full "spec(D) ⊂ ℝ ⟺ RH" has its ⟹-via-(1.12) leg as skeleton (L3/L4). Criterion-3 ACHIEVED stands on the (★) reduction + spectral framing (novel content vs Suzuki: the explicit F·(R₀+R_τ) form and the deficiency-convergence theorem D1), with W-3 circularity now cited.

3. T-3 (valley theorem, D-048–D-050)

Verified findings

• F9. B1 lower bound: Lemma D's 1/g² term was dropped [VERIFIED — proof broken]. Lemma D gives Σ ≤ 4C₀/g_γ² + …; with g_γ ≥ 4/a the 1/g² term is ≤ C₀a²/4 — an O(a²) contribution, NOT O(a). The written "(1 + 1/g_γ)" bound is inconsistent with Lemma D as stated. Honest worst-case from this method: denominator 4a²(1 + Θ(C₀)) ⟹ a CONSTANT floor ~0.65–0.9, not 1 − 1/(2a). The measured floors ≈ 1.00 reflect true zero spacing (~2π/log γ ≫ 1/a neighbors rare), which worst-case density C₀ cannot see. Repair route: better test functions (v̂ with nodes at the nearest neighbors — the sup in (I1) allows it) or an averaged-density lemma. → task 4.5c.
• F10. B3 sinc² argument factor 2 [VERIFIED]. v̂₀(γ') = 2sin((γ−γ')a)/(γ−γ') ⟹ the profile is sinc²((z−γ)a), not sinc²((z−γ)a/2) as written; the printed envelope is an OVERCLAIM (stronger than proved) for 0 < |z−γ| < π/a. Statement fix required.
• F11. K(γ,γ) ≤ 1 needs m_γ = 1 [VERIFIED]. Under RH alone the bound is ≤ 1/m_γ; simplicity is known only to T₀ (and for the first 10¹³ zeros unconditionally). Statement fix: ≤ 1/m_γ.
• F12. δ_crit "iff" [VERIFIED]: the carrier computation shows a negative direction EXISTS for δ > π/(2a); non-existence for δ ≤ κ is unproved (one test function). Statement fix: "if", with the converse as the open refinement.

Refuted findings

• "v₀ ∉ form domain H₀¹" [REFUTED as fatal; downgraded to a write-up gap]. The form domain via the zero side is {v ∈ L² : Σ m_γ|v̂(γ)|² < ∞}, and for v₀ = e^{∓iγx}1_{(−a,a)} the sum converges (sin² gives Σ O(1/(γ'−γ)²) against log-density — convergent). So v₀ IS admissible. Remaining honest item: the identification of the operator-form domain with the zero-side domain should be written out (joins the (I1) rigor queue). Note also a sign-convention slip: with v̂(z) = ∫v e^{izx}, localization at +γ needs v₀ = e^{−iγx}.
• B2 upper bound K ≤ 2a: re-verified correct (T ≥ I ⟹ T⁻¹ ≤ I; the inverse-inequality flip was handled correctly).

T-3 disposition

Split the theorem: upper bounds (K(γ,γ) ≤ 1/m_γ; plateau K(z,z) ≤ 2a) [PROVED-CONDITIONAL(RH)] stand; lower bounds and the sinc² profile → [SKETCH — measured (E-022 family), proof repair task 4.5c]; detector δ_crit = π/(2a) one-direction proved, converse open, numerical stress test = E-027 (task 4.4, this session). The valley PHENOMENOLOGY (measured) is unaffected.

4. D-052 accounting corrections (all applied to D-052)

• C1. G-001/(A) mislabel in §3: zero-density invisibility is graveyard entry (A); G-001 is the s₂/p substitution symmetry. Fixed.
• C2. Residual value: D-052 used Z/(2π) without citing D-045's supersession of D-042's Z/6. Citation added.
• C3. T-2 tag split per §2 above. Criterion-3 wording now cites W-3.
• C4. T-3's detector moved from criterion-3 support to criterion-4 (it is a conditional detector statement, not an equivalence).
• C5. "Track C complete" narrowed to "the Li route complete" (3.1 Weil-positivity horizon remains resolution-limited, [~]).

5. Net effect on the Phase 4 accounting

• Criterion 1: not claimed (unchanged; paywall W-2 located).
• Criterion 2: not achieved; proved impossible in the certificate framework (D-013) — unchanged.
• Criterion 3: ACHIEVED via the (★)-reduction (R-003) + D1 spectral framing — with the honest scope of §2 (one direction proved, one skeleton). T-3's detector no longer counted here.
• Criterion 4: ACHIEVED — obstruction map, now including the repair map 4.5a–4.5d (below) which is itself documented-obstruction content.
• T-1 remains a new explicit theorem; its tag is downgraded pending 4.5a/4.5b (statement unchanged; margins dwarf the gaps).

6. Repair tasks opened