Session 008 — 2026-08-19

Regroup + R7 registration; task 3.8 COMPLETE (R-004 finite-height Li theorem)

Part 1 — Regroup (user-requested) and resources registration

• User asked for a regroup + formal registration of the resources folder.
• Created research/resources/index.md: all 43 notes cataloged in 4 groups (Foundations / Zeros & explicit formulas / RH equivalences / RMT statistics), each mapped to the track/task it feeds.
• NGUYEN.md §2: R7 row added. D-001 Part III: pointer entry.
• Found: research/paper/Detailed_Derivations_...txt is an OLDER DRAFT duplicate (54,034 vs 54,367 bytes, missing intro) of the resources compendium — untracked, flagged to user for deletion (not deleted).
• Committed as session-008 opening (e7b72ad).

Part 2 — Task 3.8 (D-037 → R-004) COMPLETE

Assembled the explicit finite-height Li theorem. Full chain in D-037:

1. Read inputs: D-032/D-033 (design), D-036 (constants), E-021 data, Lagarias math/0404394 PDF §1 (Thm 1.1, bridge (1.16)) and §6 (Thm 6.1 proof, read in full).
2. Coordinate-system audit of D-036 (variant A kernel-plane vs variant B w-plane): resolved — D-036/E-021 consistent in variant B (contour in the ζ′/ζ argument plane, σ₀ = −5/2, trivial zero −2 inside, residue ≤ 1). No bug.
3. Real gap found: D-036's uniform |ζ′/ζ| bound on segment (III) rests on the R-bound verified only for |t| ≥ 10; the line passes within 1/2 of the trivial pole at w = −2. Closed by E-023 S1: M₀ = max_{|t|≤10} |ζ′/ζ(−5/2+it)| = 1.3352 (grid + Lipschitz ×2). Contribution 40·M₀ = 53.4 — negligible.
4. S_∞ elementary lower bound derived (new, D-037 §3): from Lagarias (4.11) via finite differences: S_∞(n) = (1−a₁)n − 1 + Σ_{m odd≥3} [(1−1/m)ⁿ − 1 + n/m] ⟹ S_∞(n) ≥ ½n log n + (C₁−γ/2)n − 4 (n ≥ 20), fully explicit (harmonic bounds). Validated vs E-021 exact values (E-023 V1). Avoids tracing Lagarias §5's O(1) lemmas entirely.
5. Margin assembly (E-023): margin convex for n ≥ 20; crossover 32 871 (traced A=2/B₀=2) / 30 217 (measured 0.9/1.0); adopted N₀ = 5×10⁴ (margin 67 705). Finite verification contiguous: ours to 3000 (ledger #3) + Misguich–Voros to 5×10⁵ (10× overlap).
6. Bridge: n ≤ (T₀−1)² from the proof's actual requirement (√n+1 ≤ T₀); Lagarias's printed (1.16) window T²/(4log²T) is conservative slack — fallback N₁* = 2.726×10²¹ documented.

THEOREM (R-004, ledger #6): λ_n > 0 for all 1 ≤ n ≤ N₁ = (T₀−1)² = 8 999 999 999 994 000 000 000 001 ≈ 9.0×10²⁴, T₀ = 3×10¹² (Platt–Trudgian). Tag [PROVED-CONDITIONAL(computational RH-to-T₀; finite verification; one numeric sub-bound)]. Criterion-3 deliverable; the maximal content of the Li route short of RH (Voros).

Adversarial review highlights (D-037 §7)

• P1–P6 proved/elementary; H1–H5 computational/heuristic, each with sensitivity analysis (theorem insensitive to H2/H3 links) and upgrade paths: D-039 (Binet-form digamma sub-bound proof), E-023b (acb box cover for M₀).
• Sanity gates pass: no claim beyond N₁; no D-H/Epstein analogue.

State of the plan

• Track C (Phase 3): 3.1–3.8 ALL CLOSED. The Li/front-A line has delivered its maximal deliverable.
• Next per plan: 2.10 (E-022 spectral-measure numerics), 2.9-D1 (Krein frame), 2.11 (D-038 transition scale). Optional: D-039, E-023b.

Part 3 — Task 2.10 (E-022/E-022b, D-040) COMPLETE

Spectral-measure convergence numerics (D-035 deliverable D2).

• E-022: Clark/de Branges spectral measures μ_a = Σ (1/K_a(z_k,z_k))δ_{z_k} for a ∈ {4,6,8,10,12}, θ = π. New Schur-Toeplitz bordered-solve machinery (validated 3.6e-14 vs E16.solve_v; V0–V3 all pass). Backend: own cache E-022-backend-cache.npz (E-019's cache is keyed to its t-points — mismatch assertion).
• Coarse weak metric FLAT (d_a ≈ 0.0106); mass conservation (≈184).
• Key finding (E-022b): K_a(z,z) has deep valleys exactly at the true γ's — Δ(a) = 2.16→2.94 growing but decelerating; fit Δ ≈ 3.33 − 4.76/a SATURATES (resid 0.032). Valleys are at γ's, NOT at the model's own lattice zeros. Within-z-bin + shifted-null controls ran (the naive O5 elevation was mostly distributional — artifact caught by null test before claiming).
• Weights-not-positions: atoms within 0.1 of γ carry e^{2.0}–e^{2.8}× more weight; positions stay chance-level (E-019). Mass within ±0.1 of γ's: 63% → 72.5% then flat.
• Interpretation: fixed-θ=π truncation has a saturation ceiling; full consolidation (Δ → ∞) requires the MOVING θ(a) of Cor 1.6 — a new concrete constraint on 2.9-D1. Δ(a) is now the project's sharp, cheap resolvent-convergence probe.
• D-040 filed; db registered (E-022, E-022b, D-040).

Part 4 — Tasks 2.9-D1 (D-041) and 2.11 (D-038) COMPLETE

• Read Suzuki §7.1–7.6 in full (local text). Key identifications: limit operator D = U⁻¹M_{π/2}U (the π/2-extension has spectrum Γ); masses m_γ = 1 (multiplicities); §7.5's expectation is strong resolvent convergence D_{a,θ(a)} → D; Cor 1.6 is ONE-directional ((1.12) ⟹ RH).
• D-041 (2.9-D1): unconditional Krein frame; Theorem D1 (deficiency convergence ⟺ strong resolvent convergence, proof skeleton via Krein resolvent formula); reduction theorem "(1.12) ⟺ spec(D) ⊂ ℝ" with lemma map L0–L4 (L1 unconditional embedding open; L3/L4 WH symbol class assembly); D-040's Δ-constraint formalized as necessary conditions N1–N3. Fixed θ=π family CANNOT converge to D (violates N1/N2 asymptotically, measured); θ(a) flow obligatory, target the π/2-extension.
• D-038 (2.11): consolidation functional R(a) = (aZ/π)e^{−Δ(a)} (validated to 15% against measured residual); onset criterion Δ′(a) > 1/a; margin C(a) = Δ − log a decreasing 0.77→0.46 — fixed-θ consolidation asymptotically impossible; transition a* > 12; the θ(a) flow is the only channel. Valley masses ≈ 1.00 already (a≥4) — the per-γ target mass is attained early; only the plateau residual must outflow.
• New task 2.12 (E-024): θ-flow numerics (scan θ, pin θ↔ψ convention, test saturation-breaking). Open convention item: finite-a θ vs B-side ψ mapping.

Part 5 — Task 2.12 (E-024, D-042) COMPLETE — θ-independence + linear-K law

• E-024: θ-sweep (7 θ × 3 a), all checks pass. θ-INDEPENDENCE: residual/mass/valley stats constant over θ to ≤2.2%; only lattice phase responds (z₁·a: 3.15→4.50, ≈87% of skeleton shift). The θ(a) flow as a consolidation channel is CLOSED by evidence (de Branges prediction confirmed: K is blind to the extension).
• Linear-K law: plateau logK = ln a + 0.64 exact to 0.3% (5 pts); valley floor ≈ 1 (= mass gauge m_γ = 1). Residual → Z/6 ≠ 0: the family equilibrates EXACTLY at the consolidation boundary (margin C(a) → ~0.4). Linear-K is generic Szegő bulk behavior — consolidation needs non-generic superlinear K-growth: the (★) cancellation, absent at a ≤ 12.
• Synthesis (D-042 §3): at fixed λ = −1, NO θ(a) at any a consolidates. Remaining channel: λ → λ_a⁻ (Suzuki's own remark after Cor 1.6; ground-state-dominated regime = sister conjecture (1.2)). → task 2.13 (E-025 λ-flow) opened.

Part 6 — Task 2.13 (E-025, D-043) COMPLETE — floor, λ-response, LOCKING CORRECTION

• E-025 (λ-flow): V2/V4 pass; V1/V3 fail — the failures are the findings.
• Discretization floor: (Q,M) pencil bottom crosses 0 at a ≈ 5.2 (h=0.01); O(h) artifact confirmed by refinement (a=8: −2.9e-2 → −1.6e-2 → −8.3e-3). True λ_a unresolvable for a ≳ 3; D-019 F1's cliff tail = floor (upper-bound tag stands). The λ→λ_a⁻ experiment is machinery-limited (needs structured eigensolvers; not pursued).
• λ-flow response: near-singular T ⟹ Δ deepens (2.16→6.99 at a=4, clean bottom), residual drops 48→11 — channel qualitatively real, but the approached mode at a ≥ 6 is the artifact mode (sister corr(v̂,Ξ) ≈ 0; nothing learned about (1.2)).
• MAJOR CORRECTION: E-019 B2's "γ-proximity chance-level" was a wrong-direction metric (median over ~10³ zeros to z≤300 vs only 30 γ's to z≤77 → trivial 49). Correct direction (γ → nearest W-zero): min-dist ≈ 0.65/a² — quadratic locking, improving through a=12. D-029/D-040 errata filed. Revised consolidation picture: positions lock + weights concentrate; the obstruction is exactly the excess-zero outflow + residual → Z/6 (linear-K equilibrium).
• Next: analytic derivation of the linear-K law (what breaks the equilibrium); then the outflow question.

Part 7 — Linear-K analysis (D-044) COMPLETE

• Framework: windowed-symbol law K_a(z,z) ≈ a·(1/β ⊛ w_a) (Szegő/ Christoffel mode counting). Measured confirmations: linear plateau (0.3%), a-stationary floor (−0.11…+0.23), valley width ~ c/a, valley masses ≈ 1 (mass gauge), residual → Z/6 = Z/(πρ̄) with ρ̄ = 1.9. Valleys = simple zeros of 1/β at the true γ's (symbol poles). The equilibrium (margin → const) is generic bulk behavior.
• The breaking mechanism, quantitative: λ-shift leaves the symbol's poles unchanged ⟹ valley floors PINNED (masses stay 1) while the plateau 1/(β̄−λ) diverges as λ → β_min ⟹ residual dies. E-025's measured response matches the direction. Blocker = discretization floor (D-043 §A): dense refinement infeasible; needs spectral Galerkin for the rough kernel g.
• OPEN: closed form of β. Refuted: |Ξ/Ξ′| family (corr ≤ 0.62). The 6/π constant unexplained (note K_plateau ≈ 6·(a/π) = 6 × PW-kernel diagonal). The flat plateau suggests truncation-bulk dominance — classical symbol calculus needs care (g grows exponentially; ĝ is an analytic functional).

Part 8 — Fork complete: D-045 (symbol) + D-046 (λ-sizing → G-004)

• Fork 1 (D-045): extended K-profile below γ₁ (flat at ≈2a) + plateau prefactor per a (1.95→1.99→~2) → K_plateau → 2a = free Paley–Wiener diagonal (the "6/π" was a finite-a shadow). Valley floors ≈ 1.04 ± 0.06 for all γ (mass gauge m_γ ≈ 0.96). Symbol identified STRUCTURALLY: β_a → 1 + zero-comb (the explicit formula as operator symbol); residual law refined to Z/(2π). Plateau micro-regression: no residual structure (corr ≤ 0.28).
• Fork 2 (D-046): λ-channel sizing. Artifact mode diagnosed (frequency-46 oscillatory, O(h) decay). True λ_a: trustworthy points only a ≤ 1 (0.414 → 2.07e-4 → 1.44e-9, superexponential); at a=2 already < 3e-12 (floor). Feasibility windows DISJOINT (ground state a ≲ 1.3 vs zero resolution a ≳ 4–6) → G-004 (λ-channel closed; revival: λ_a lower bound / symbol-space formulation / spectral panels).
• Fork evaluation queued for user: candidates (1) symbol-space ground-state formulation, (2) 2.9 L1/L3/L4, (3) Phase-4 synthesis, (4) domain sweep.

Part 9 — paper assembly sub-stage 1 + imaginary-frequency probe

• K(iy,iy) probe (ground-state at imaginary frequency): K ≈ 0.8–0.9 × free value e^{2ay}/(2y) — NO ground-state signature at imaginary frequency; the symbol-space quick avenue is exhausted. Direction (1) remains theory-only.
• Proceeded to paper assembly (user delegated). draft.tex: Results section now has the R-004 finite-height theorem as a complete theorem–proof-sketch unit (5 steps, constants, margin values, honesty remark incl. conservative window N₁* = 2.73×10²¹). references.bib: +8 entries (Li, Bombieri–Lagarias, Lagarias, Trudgian, Platt–Trudgian, Voros ×2, Suzuki).
• CAVEAT: no LaTeX toolchain on this machine (pdflatex/xelatex/ tectonic all absent) — the draft is NOT compile-tested. Compile check deferred (user may have a TeX install elsewhere, or add one).
• Remaining paper sub-stages: spectral-program section (R-003 + D-040–D-046), obstruction section (G-001–G-004), numerics appendix (E-tables), Lean appendix placeholder.

Part 10 — L1 sketch + domain sweep (§4A.5)

• L1 upgraded (D-041): injectivity of ℋ(A_a) ↪ ℋ(A∞) has a clean unconditional argument (g∗v = 0 ⟹ ĝ·v̂ = 0 ⟹ v̂ ≡ 0; analytic- functional bookkeeping as flagged in D-026). L3/L4 remain library assembly (Gohberg–Feldman).
• Domain sweep for the residual-equilibrium wall: Christoffel- function universality theory (Levin–Lubinsky line; Eichinger– Lukić–Woracek arXiv:2409.18045 — sine-kernel asymptotics ⟺ unique Lebesgue tangent measure). Our measured picture (plateau 2a + valleys at γ's) matches a limit measure = ac background + atoms; consolidation ⟺ limit measure pure atomic ⟺ tangent measure Dirac. The equilibrium is thereby reframed as a spectral-purity question for the underlying canonical system — the named next opening. Secondary hits: RHP-with-poles/dressing (W-3), prediction theory (same dichotomy), Beurling–Malliavin (λ_a bounds).

Part 11 — E-026 inverse-spectral machinery (built + validated; parked)

• Stieltjes/Lanczos reconstruction (dps 50, full reorthogonalization) of the Jacobi coefficients of the Clark measures, a = 4/8/12, 240 coefficients each. Moment round-trip 3e-14 (correct check; the script's atom-matching V1 was misdesigned — cosmetic, noted).
• Finding: the b_k are truncation-dominated (tail → band edge Z/2, period-2 from the origin gap); the γ-structure is diffuse (ratio to equal-mass reference: 2% std, 14% max early; no clean γ-beats even in a narrow band). The inverse-spectral representation smooths the localized valley signal — the K-diagonal remains the sharp probe.
• Avenue parked with revival notes (D-047): full Krein-string decode, longer sequences, de Branges subspace-chain algorithm.

Part 12 — D-039 (digamma proof) + D-048 (valley theorem skeleton)

• D-039: digamma remainder bound PROVED (Binet second formula + recurrence, explicit Cauchy bounds): |r(w)| ≤ 0.36/|w|² < 1/|w|² for Re w ≥ −3/2, |w| ≥ 5 (numeric check max 0.084). R-004's H2 link upgraded: theorem tag is now [PROVED-CONDITIONAL(computational RH-to-T₀; finite verification)] — only M₀ remains grid-verified.
• D-048: valley theorem skeleton. The variational identity K_a(z,z) = sup_v |v̂(z)|²/(‖v‖² + Q_W^a(v)) makes the phenomenon transparent: floor theorem K(γ,γ) = 1 − O((1+log γ)/a) (the γ-term of the comb dominates); plateau theorem K = 2a(1−o(1)) (comb–smooth cancellation = RvM on bumps); width ~1/a. The Δ ≈ log a + 0.64 law is log(2a/1) (log 2 = 0.693 ✓ within floor drift). The saturating 5-point fit (Δ_∞ = 3.33) was a model artifact — the log law is the backed one; margin Δ − log a → log 2 (const) — consistent with the equilibrium. RH-detector remark: off-line zeros would give anomalous valley signatures — potential zero-free-region criterion (paper discussion item). Bookkeeping B1–B3 named for full rigor.

Part 13 — D-049: valley theorem FULL PROOFS (B1–B3 closed)

• Key identity confirmed from E-009a: Q_W^a(v) = Σ_γ m_γ|v̂(γ)|² exactly (pure comb, no background) under RH.
• B1 floor: K(γ,γ) ≤ 1 EXACT (single-term domination) and ≥ 1 − 1/(2a) − C(1+log γ)/a² (bump test + density sampling). Measured floors 1.04 explained as grid-minimum upward bias — consistent.
• B2 plateau: K ≤ 2a EXACT (T ≥ I under RH, Yoshida direction) and ≥ 2a(1 − C'/a) — the measured approach 0.973 → 0.997 ✓.
• B3 profile: universal lower envelope 1/(sinc²((z−γ)a/2) + 1/(2a)).
• Δ law now theorem: Δ = log 2a + o(1); the early saturating fit formally superseded.
• Adversarial catch during write-up: the Weil form is form-unbounded on L² (log-density weight) — the finite-height tail is NOT L²-controlled; fixed by stating the clean theorems under RH and naming the finite-height tail estimate T2 (PW sampling + subharmonic comparison; not yet written out).

Part 14 — D-050: T2 write-up (valley theorem completed)

• Exact carrier computation: an off-line pair at height γ with deviation δ produces negative form directions iff δ > π/(2a) (threshold exact; pair term −8cosh²(aδ)(δ²−κ²)/(δ²+κ²)², κ = π/2a).
• Quotient analysis: high-frequency directions cannot corrupt K(z,z) at z ≤ Z (numerator suppression 1/(γ₀−z)²).
• Honest inventory: valley lower bounds are unconditional (bump tails directly summable, Lemma T-bump ~3e-7 at a=12); the upper bounds are exactly Weil-positivity-at-level-1/(2a) statements — free under RH, NOT provable from RH-to-T₀ alone, and MEASURED (λ_a > 0 + valleys present). Consequence: the measured valleys are numerical evidence of Weil positivity in the ℋ(T_a) metric to scale 12 — independent of and consistent with the λ_a cliff.

Part 15 — paper valley-theorem section + D-051 (L3/L4 assembly)

• draft.tex: valley-theorem subsection added (plateau/floors/profile theorem + detector remark). Paper now covers R-003, R-004, the spectral program, obstructions, numerics appendix.
• D-051: L3 answered — classical Gohberg–Feldman symbol classes are INAPPLICABLE (poles on ℝ are the content, not a defect; Suzuki's "formal symbol" remark). Correct frame: Krein–Langer screw-function continuation theory; L4 collapses onto (★) itself (as predicted by D-035's W-3). The D1 reduction chain is fully assembled.
• All session-008 queue items now done. Next stage candidate: Phase 4 synthesis (4.1–4.4).

Handoff notes

• E-023 script: research/numerics/scripts/E-023-finite-height-assembly.py; data E-023-assembly.csv/.json; db registered (session 8).
• Platt–Trudgian exact height: "3×10¹²" per title/abstract (arXiv:2004.09765); confirm exact certified height at paper write-up (N₁ insensitive at <1% level).
• N₀ could be pushed under 3000 (fully self-contained verification) only with a fundamentally sharper horizontal-segment treatment (I_II/IV dominate B(n) below n ~ 10⁵: 2(e⁴+1)·7H(T), H = 3C₀² ~ zero-avoidance d_min); not needed for the theorem — literature bridge suffices.
• E-021 n=10⁴ rerun still available cheaply via caches (unchanged).

Part 16 — Phase 4 begun: D-052 (4.1/4.2) + D-053 (4.3a)

• D-052: result inventory after sessions 001–008; best theorems stated in final form and classified: T-1 = R-004 finite-height Li [PROVED-CONDITIONAL]; T-2 = R-003 + D-041 reduction "(1.12) ⟺ spec(D) ⊂ ℝ" [PROVED modulo L1 write-up]; T-3 = valley theorem (D-049/D-050) [PROVED-CONDITIONAL(RH)]. Success-criteria accounting: criterion 3 (new reduction) ACHIEVED via T-2; criterion 4 (obstruction map) ACHIEVED; criteria 1–2 not claimed, paywall located exactly (W-2). Revival conditions collected.
• D-053: Davenport–Heilbronn/Epstein sanity gate run structurally on all three theorems — GATE PASSES. T-1: the bridge (input 3) is the load-bearing discriminator, false for D-H at the first off-line zero; finite verification also fails there (Voros/Misguich sign changes). T-2: formally true with both sides false for D-H — can never certify a false RH-analogue. T-3: conditional form vacuous; unconditional lower bounds cannot even be formulated (indefinite form, no positive T_a^{-1}); detector threshold δ_crit = π/(2a) built for exactly this failure mode. Numerical D-H experiment not run (no D-H screw machinery); prediction logged for if-built. Files uncommitted at session end (carried to 009).