Session 009 — 2026-08-19 — Phase 4: synthesis corrections (4.3b) + stress test (4.4)

Part 0 — Session-008 leftovers committed

D-052 (4.1/4.2 synthesis) + D-053 (4.3a sanity gate) were written but uncommitted at session-008 close; journaled as session-008 Part 16 and committed (4a2db8c). db registrations were already in place.

Part 1 — 4.3b: independent-instance adversarial review (D-054)

Four fresh-context review instances attacked T-1 (R-004 chain), T-2 (R-003 + D-041/D-051), T-3 (D-048–D-050 valley theorem), and D-052's accounting. Every claimed finding was then RE-VERIFIED by hand before disposition. Outcome: all three theorem tags downgraded; statements survive; repair map 4.5a–4.5d opened.

Verified findings (the important ones):

• F1/F2: D-039 digamma proof broken in BOTH pieces — Binet tail split at t=1 gives a constant tail (fails |W| ≳ 35); recurrence Cauchy estimate conflates an absolute 0.45 bound with a 0.45|u|⁴ coefficient. Bound itself true (asymptotic 1/12, grid max 0.084); repairs sketched (split at t ~ |V|/2; sharper majorant).
• F3 (load-bearing): D-036 §1's far-sum assembly gives O(|t| log t), not O(log t) — the numerator |s| requires the Davenport reference-point trick (compare at 2+it), which is missing. A/B₀ constants are E-021-MEASURED, not traced. Margins (67 705 at N₀, 2.46e26 at N₁) absorb any plausible constant error; theorem stands.
• F4: margin-convexity "every component concave" invalid as written (products 4T·f(T)); grid-verified; symbolic proof = 4.5b.
• F9: D-049 B1 lower bound drops Lemma D's 4C₀/g² term — honest worst case is a CONSTANT floor ~0.65–0.9, not 1−1/(2a). Measured floors ≈ 1 reflect true sparse spacing, invisible to worst-case C₀.
• F10: B3 profile argument is sinc²((z−γ)a), not sinc²((z−γ)a/2) — printed envelope an overclaim.
• F11: K(γ,γ) ≤ 1 requires m_γ = 1 (Platt–Trudgian); general RH statement is ≤ 1/m_γ.
• F12: δ_crit = π/(2a) proved one direction (existence of a negative direction); converse open.
• F6: Suzuki's (1.12) "every compact K ⊂ ℂ" is the standard abuse — must avoid poles of F = z²ξ/ξ′; precision note added (4.5d).
• F7: τ± carried by unbounded |x| ≥ a — PW–Schwartz inapplicable; R_τ's representation distributional; sharpens R-003's flagged bookkeeping (4.5d).
• F8: D-052 presented T-2's ⟺ as proved; in fact ⇒ proved (Suzuki + D1), ⇐ named skeleton (L3/L4; D-051: L4 = (★)). Tag split applied.

REFUTED: "v₀ ∉ form domain H₀¹" as a fatal flaw — the zero-side form domain is {v : Σ m_γ|v̂(γ)|² < ∞} and v₀'s sum converges (sin² decay vs log density); downgraded to a write-up item (domain identification, 4.5d). Also a sign slip found: localization at +γ needs v₀ = e^{−iγx} with v̂(z) = ∫v e^{izx}.

Files corrected: R-004 (tag), D-037 (H2, P6 rows), D-039 (erratum header), D-036 (erratum header), D-049 (erratum header), D-050 (δ_crit one-direction), D-052 (C1–C5 accounting fixes), D-041 (precision note), R-003 (sharpened bookkeeping), NGUYEN.md ledger #6. Repair tasks: 4.5a (D-036 reference-point rewrite + D-039 repair + I_I housekeeping), 4.5b (symbolic concavity), 4.5c (valley lower bounds + profile + δ_crit converse), 4.5d (L1 + form-domain + bookkeeping).

Process note: the review caught real errors that survived session 008's self-review — the independent-instance gauntlet earns its keep. The D-036 far-sum hole is exactly the kind of error adversarial review exists for: a too-crude triangle inequality producing a bound off by a factor |t|, hidden inside a "traced constants" claim.

Part 2 — 4.4: E-027 detector stress test (setup)

Design: synthetic off-line zero pair at mid-gap height γ (deviation δ) inserted into the zero-side Weil form in an orthonormal Legendre Galerkin basis on (−a,a), N > γa + 48 so the tuned carrier is in the span. Cases: a ∈ {4, 8} × γ windows [19,25] and [45,55]; zeros to T ≈ 3100 (zetazero, float64 — P1-type use). Checks: V1 control PSD, V2 bare-form threshold crossing vs δ_crit = π/(2a), V2b pencil (I+Q) floor, V3 eigenvector localization at γ, V4 independence, V5 carrier quotient vs D-050 formula (sign consistency). Note (pre-registration of expectations): the bare pair form is rank-2 indefinite for ANY δ > 0 (strict Schwarz on the evaluation functionals), so the bare crossing could come in BELOW δ_crit — the magnitude vs the positive background is what the experiment measures. Two launch bugs fixed en route (ROOT path parents index; row-lookup tolerance vs rounded CSV fields).

Part 3 — E-027 results + D-055 (4.4 COMPLETE)

Launch saga: (1) ROOT path parents index bug (research/research/...); (2) row-lookup tolerance vs rounded CSV fields; (3) mpmath besselj at dps 30 too slow for 5M evals (killed at ~25 min, rewritten vectorized scipy float64 — 16 s total); (4) heredoc encoding mangled an em-dash; (5) numpy bool JSON serialization. All fixed; zeros cached (E-027-zeros-cache.npy, 2500 zeros to gamma ~3080). RESULTS (all four cases: a in {4,8} x gamma* in {23.011, 51.372}):

• F-A control PSD (lambda_min ~ -1e-14 at delta=0). PASS.
• F-B bare form: NEGATIVE at the smallest delta>0 scanned, ~1e13 x control floor — NO bare-form threshold (strict Schwarz on the evaluation functionals makes the rank-2 pair form indefinite for any delta>0; the cliff-flat background cannot cover it). D-050's "only if delta > pi/(2a)" is thereby scoped to the tuned-carrier direction.
• F-C carrier threshold EXACT: quotient crossings 0.39275 vs pi/8 (dev +5e-5), 0.40185 (dev +0.9%), 0.19690 vs pi/16 (dev +5.5e-4), 0.19729 (dev +9.4e-4) — D-050's one-direction law validated.
• F-D pencil onset (T_a = I+B loses positivity): delta ~ 0.106/0.114 (a=4), 0.0383/0.0386 (a=8) — well below pi/(2a), earlier at larger a; magnitude-limited onset, not the carrier law.
• F-E localization: argmin eigenvector's Fourier transform peaks at gamma* (contrast 3.4-6.1 over median). ALL CHECKS PASS (after V3 metric adjusted for the coarse a=4 bump width). D-055 written; D-050 erratum sharpened; E-027 + D-055 registered in db. Detector picture now three-level (bare/pencil/ carrier) — paper text must scope "resolution pi/(2a)" to flat carriers.

Session 009 outcome

Phase 4 tasks 4.1-4.4 all executed. 4.3b downgraded all three theorem tags (statements intact; margins robust); repair map 4.5a-4.5d opened (next-stage queue). 4.4 delivered the detector stress test with a refined three-level picture. The Phase 4 synthesis is now honest at the file level: D-052 (corrected) + D-053 + D-054 + D-055.

Handoff notes

• E-027 script: research/numerics/scripts/E-027-detector.py; data E-027-detector.csv/.json; zeros cache E-027-zeros-cache.npy; db registered (session 9). Reruns: --force (16 s with cache).
• db.py add-derivation/log-experiment use INSERT OR REPLACE keyed by id — a mistyped re-register OVERWRITES the row (happened once this session; restored).
• E-027 machinery is reusable for any future D-H/Epstein build: swap the on-line zero list for the L-function's zeros.

Part 4 — Repair 4.5a COMPLETE (D-056 + E-028): R-004 full tag restored

• D-056 §1 (digamma, replaces D-039): Binet tail split at t ~ |V|/2 (middle piece |t²+W²| ≥ 3V²/4 − 9 ≥ 0.39V², tail exponentially small) ⟹ |r(W)| ≤ 0.417/|W|²; recurrence via correct majorant (|g(u)| ≤ 0.0233 on |u| ≤ 1/5, max-modulus on g/u⁴ ⟹ 14.6|u|⁴); assembly |r(w)| ≤ 1.084/|w|² ≤ 1.2/|w|² for Re w ∈ [−5/4, 1], |Im w| ≥ 5. Both D-054 defects repaired at the spots found.
• D-056 §2 (R-bound, replaces D-036 §1): reference point s₀ = 2+it; the killer numerator |s| becomes |s₀−s| ≤ 4.5 and the far sum becomes the CONVERGENT Σ 1/(|t−γ|(1+|t−γ|)) (Σ1/k(k+1) = 1 exactly). Assembled: |R(s)| ≤ 13.73·C₀(t) + 9.26 = 5.95 log(t+2) + 39.3 ⟹ stated A = 6.0, B₀ = 40.
• E-028 ALL CHECKS PASS: traced bound holds at all 15 sample points (worst ratio 0.058 — 17× slack); Z₂ certified ≤ 0.5700 (partial 0.569960 + tail 1.5e-5); margin rerun with 6.0/40: N₀ = 58 998 (adopted 60 000; margin 3.6 and increasing; contiguous to 4e5; convex grid to N₁), overlap vs Misguich–Voros 8.3×, margin(N₁) = 2.458e26 unchanged to leading digits. I_I length fix + certified Z₂ folded in.
• R-004 tag RESTORED to [PROVED-CONDITIONAL(computational RH-to-T₀; computational finite verification)]; D-037 H2 row rewritten; ledger #6 updated. Remaining analytic leg for the full chain: 4.5b (symbolic concavity).

Part 4b — 4.5a correction: honest C₀ (caught in self-review)

D-036 §0's "C₀(t) := 0.433 log(t+2) + 2.5" is a LEADING-FIT, not a proved bound (drops the 0.886 log log t term and undercounts constants) — E-021/23/28 always used the honest F-diff + S-pair form. D-056 rebuilt on the honest form: |R| ≤ 13.73·C₀ + 7.97 = 5.95 log + 12.16 log log + 47.8 ⟹ folded (log log u ≤ (log u)/e) A = 10.5, B₀ = 48. E-028 rerun: V-R holds at all 15 points with 26× slack; N₀ = 75 451 (adopted 80 000, margin 2.4 increasing, contiguous to 4e5, convex to N₁); overlap vs MV 6.25×; margin(N₁) = 2.458e26 unchanged. Z₂ certified ≤ 0.5700 (D-036's 0.5699 was the truncated partial sum; true 0.56996). D-036 §0 erratum added. All dependent files updated (D-037, R-004, ledger #6).

Part 5 — Repair 4.5b COMPLETE (D-057 + E-029): convexity PROVED

• Structure: majorant margiñ via closed-form Ĉ₀ ≥ C₀ (D-056), then chain rule with T = √n+1: φ(T(n))″ = [φ″(T) − φ′(T)/(T−1)]/(4n) =: ψφ/(4n) — concavity reduces to 11 one-variable sign conditions ψᵢ(T) < 0 for T ≥ √20+1 = 5.472. Each row proved by an elementary bound (the two delicate ones: T·log(T+2) via L1 ≥ 2 giving −(2T²+7T+12) < 0; T·log log(T+2) via 2 ≤ L1·L2·(T+2), 10.49 at T₀ increasing). I_I term bounded directly (max 2n|B̃₄″| = 0.189 at n = 20 < 1/(2n)·2n = 1, exponential decay after).
• E-029 ALL CHECKS PASS: sympy-derived ψᵢ match hand forms (rel ≤ 1e-15); all ψᵢ < 0 on [T₀, 3.1e12]; margiñ convex on [20, 4e5]; margiñ crossover 75 621; at adopted N₀ = 80 000: margiñ = 16 020

  0, margiñ′ = 3.7 > 0; B̃ ≥ B pointwise.

• D-037 P6 row → PROVED. Ledger #6 updated. R-004's analytic chain is now fully proved end-to-end (remaining inputs: Platt–Trudgian
• Misguich–Voros computations, M₀ grid+Lipschitz — all tagged).

Part 6 — Repair 4.5c COMPLETE (D-058): valley lower bounds repaired

• B1′(i): isolated zeros (no zero within distance 1) — flat bump + Lemma D at δ = 1 gives the originally claimed rate 1 − 1/(2a) − O((1+log γ)/a²), now PROVED (D-049's error was the g_γ ≥ 4/a hypothesis + dropped 1/g² term).
• B1′(ii): general zeros — node-killing projection v = e_γ − P_J e_γ (kills window-2 zeros exactly; v̂(γ) = ‖v‖² residual identity). Self-review caught a second overclaim (rim-annulus survivors within 1/a of a node contribute Θ(a²) via Bernstein): the honest general floor carries the explicit annulus count E(γ,a) (0 generically) alongside the Gram constant ‖G⁻¹‖. Clean rate holds when E = 0.
• B3′: correct profile argument sinc²((z−γ)a) (factor-2 fix).
• m_γ fix: K(γ,γ) ≤ 1/m_γ under RH alone.
• Detector restated three-level (bare/pencil/carrier) per E-027; D-050's claim scoped to the flat carrier (L3).

Part 7 — Repair 4.5d COMPLETE (D-059): bookkeeping write-ups

• §1: (1.12) read on compacts avoiding poles of F (Suzuki's abuse); usage through Hurwitz zero-content unaffected.
• §2: R_τ := W/F − R₀ as the primary definition (meromorphic); the FT representation (z−i)τ̂₊ + e^{iθ}(z+i)τ̂₋ as tempered distributions is the provenance; remaining formality (order bounds, product restriction) named for Phase 5.
• §3: Lemma FD — zero-side form closed (PW Bernstein + ℓ² map), agrees with the operator form on the common core C² ⟹ forms coincide; (I1) valid over D(q₀); flat bumps admissible.
• §4: L1 written out (g∗v = 0 ⟹ ĝv̂ = 0 ⟹ v̂ ≡ 0 off the discrete pole set ⟹ v = 0); PWS analytic-functional step named as the remaining formality. No RH input. All four repair tasks (4.5a–4.5d) now complete.

Part 8 — Paper updates + session close

draft.tex: R-004 proof sketch now uses the traced constants (10.5/48, honest C₀), convexity "proved (D-057)", N₀ = 8×10⁴, margin(N₁) = 2.458e26; valley-theorem subsection rewritten to the D-058 corrected form (m_γ upper bound, isolated-zero floor, node-killing remark, corrected sinc² profile); detector remark rewritten three-level per E-027; numerics appendix: E-028 margin numbers + E-027 item added. Session-009 ledger additions: D-052–D-059, E-027/E-028/E-029. All repair tasks complete; R-004's analytic chain is proved end-to-end; the archive is the honest state of the art for the project. STOP.