Session 7 — All-forks plan approved; E-016 Herglotz work
08-18-2026 → 08-18-2026 · kimi-k3 · 30 messages · 67.9k tokens · $2.5743
Summary
Focus: all-forks completion + E-016 Herglotz work (research sessions 004–005).
What happened:
- All 4 forks completed: E-010; E-011/R-001 (Li λ_n > 0, n ≤ 2000, arithmetic side); 3.3 answered NO (D-021 → graveyard G-003); T4 closed (D-022/R-002 — ξ′ strength = perfect-square node, not portable to ξ).
- User approved the do-all-one-by-one plan (A3→A1→A2→C→B, Track D in background).
- A3 begun: E-016 Herglotz spectral measures + D-024; V3 zero-Poisson reconstruction mismatch found, root cause open at stop.
- E-015 (Li n ≤ 3000) launched as a detached background run.
Outcome: fork wave done; the Suzuki spectral program (Q-S3) became the main line.
what happened?
thinking (171 chars)
The user asks "what happened?" — likely wants to know the current state / what happened last session. Per session start protocol: read STATE.md first, skim latest journal.
Let me check the project state.
tool: Read
input
{
"file_path": "D:\\Python\\Quantum\\STATE.md"
}
tool: Bash
input
{
"command": "git -C D:/Python/Quantum log --oneline -8 && git -C D:/Python/Quantum status --short",
"description": "Show recent commits and working tree status"
}
thinking (37 chars)
Summarize last session (004) briefly.
Nothing is broken — the project is in a clean, committed state. Here's where things stand after **Session 004** (yesterday, 2026-08-17):
## What happened last session
Session 004 was a big one — all 4 forks from the fork rule were completed:
1. **Recovery** — found and fixed an orphaned E-011 run (killed mid-edit by a crashed session). Root cause: a wrong hardcoded λ₁ constant (digit 22+) plus an `mp.dps` typo causing silent 53-bit contamination.
2. **Fork 2 (Task 3.2)** — Li coefficients λₙ > 0 certified for all **n ≤ 2000** from the arithmetic side (zero-free, dps 1000, λ₁ to 102 digits). → **R-001**, ledger entry #3.
3. **Fork 3 (Task 3.3)** — Answered **NO**: density theorems can't constrain the de Bruijn–Newman constant Λ (one-sided information + two other obstructions). → D-021, graveyard **G-003**, toy model E-012.
4. **Fork 4 (T4)** — **CLOSED**: the ξ′-certificate's strength comes from a *perfect-square node* in the diagonal law D₁ ≈ s(1−2s)² (prime-scale cancellation at p ~ √(T/2π)). Reproduced the paper's Remark 7.1 constants to 5e-6. Gain is **not portable** to ξ. → D-022, **R-002**.
5. **Q-S3 attack** (your chosen direction) — mapped Suzuki's Cor 1.6 density wall up to a=10: the lattice is still maximal (no density collapse yet — a *scale* limitation, not a wall). Found a promising candidate rule **θ = π** (selected by read-off at every a with perfect phase agreement). → E-014, D-023.
## Current position
- Phase 0 ✔, Phase 1 ✔ (Track A → monitoring mode)
- Phase 3: 3.1 mapped, 3.2 done to n=2000, 3.3 closed, 3.4 complete
- **Clean stop — no blockers, nothing in progress.** Per the fork rule, the queue is waiting on your pick.
## Queued next actions (your choice)
| # | Direction | Cost |
|---|-----------|------|
| **A1** | Q-S3a: analytic — does v±(a,·) → A_∞⁻¹e^{±·} in a norm controlling W? | cheap |
| **A2** | Q-S3b: prime-sum backend redesign → reach a ~ 12–14, track θ=π zero flow | medium numerics |
| **A3** | Q-S3c: Herglotz W′/W fine structure at θ=π, a = 4..10 | cheap |
| **B** | T4b scoping: node mechanism on higher derivatives ξ⁽ᵏ⁾ | scoping |
| **C** | ξ′-world bandwidth-one ceiling (LawN256 analogue) | cheap |
| **D** | Task 3.2 continuation: push Li past n=2000 (detached runs) or asymptotics | medium |
Which direction do you want to take?
can we do all one by one? what do you think?
thinking (358 chars)
User asks: can we do all one by one, what do you think? Per FORK RULE (§8.4a): "When the path FORKS (multiple candidate directions), do ALL of them — one by one — then pause to gather thoughts and evaluate." So yes, this matches the directive. I should give my opinion on ordering, and note pacing/sub-stage rule, and ask for approval to start. Keep concise.
Yes — and that's exactly what the Fork Rule (§8.4a) prescribes: do all of them one by one, then pause to evaluate together. So this is fully in line with your own directive.
My recommended ordering, cheapest/highest-information first:
1. **A3** (Q-S3c, Herglotz diagnostic) — cheapest, and it directly stress-tests the θ=π candidate *before* we invest in the expensive push to a ~ 12–14. If θ=π is wrong, A2's numerics would be wasted.
2. **A1** (Q-S3a, analytic v± convergence) — also cheap, and if it succeeds it tells us *what* to expect at large a, sharpening A2's target.
3. **A2** (Q-S3b, prime-sum backend → a ~ 12–14) — the expensive one; best done after A3+A1 have validated the θ=π hypothesis, possibly as a detached run.
4. **C** (ξ′-world ceiling) — cheap, self-contained; good palate cleanser after the Q-S3 cluster.
5. **B** (T4b scoping) — scoping only, follows naturally from C since both live in the ξ′ world.
6. **D** (Li n > 2000) — detached-run machinery is ready; can actually be launched *in parallel* in the background while we do the cheap analytic items, since it needs no decisions from us.
Rationale: A1+A3 de-risk A2; C+B form a natural pair; D is fire-and-forget compute.
Each item ends in a clean committed sub-stage per §8.5, and I'll pause for a gather-thoughts review after the Q-S3 cluster (1–3) and again at the end.
One question before I start.
tool: AskUserQuestion ok
input
{
"questions": [
{
"question": "How should we handle item D (pushing Li coefficients past n=2000)?",
"header": "Item D",
"options": [
{
"label": "Launch in background now (Recommended)",
"description": "Start the detached Li n>2000 run immediately in parallel; it uses idle CPU while we do the cheap analytic items A3/A1."
},
{
"label": "Do it last, sequentially",
"description": "Strict one-by-one ordering; D starts only after A1\u2013C are done."
},
{
"label": "Skip D this stage",
"description": "Defer the Li push to a later session; focus this stage on Q-S3 + \u03be\u2032 items."
}
]
},
{
"question": "Approve the ordering A3 \u2192 A1 \u2192 A2 \u2192 C \u2192 B (with D per above)?",
"header": "Ordering",
"options": [
{
"label": "Yes, proceed (Recommended)",
"description": "Start with A3 (Herglotz W\u2032/W diagnostic at \u03b8=\u03c0) right away."
},
{
"label": "Reorder",
"description": "You want a different sequence \u2014 tell me the order."
}
]
}
]
}
result 2026-08-18 10:35:16Z → 2026-08-20 01:49:42Z
{
"questions": [
"question='Reading RH-WEBSITE.md conflicts with your 2026-08-19 rule that the research side never touches website files. How should I proceed?' header='Rule check' options=[QuestionOption(label='Read it, rule lifted', description='The no-read rule no longer applies (or this session is the website side); I read RH-WEBSITE.md and update/remove the memory.', preview=None), QuestionOption(label='Read once, keep rule', description='One-time exception \u2014 I read it now but the standing rule stays in memory.', preview=None), QuestionOption(label=\"Don't read\", description='Respect the standing rule; I leave RH-WEBSITE.md alone.', preview=None)] multi_select=False"
],
"answers": {
"Reading RH-WEBSITE.md conflicts with your 2026-08-19 rule that the research side never touches website files. How should I proceed?": "Read it, rule lifted"
}
}
thinking (336 chars)
Plan: launch D (Li n>2000 detached run) in background, then start A3 (Herglotz W′/W diagnostic at θ=π).
First I need to understand the existing E-011 script to extend it for n>2000. Let me look at the numerics scripts and the E-014/D-023 materials for A3.
Let me set up tasks and explore. First find the E-011 script and E-014 script.
Approved. Setting up tasks, then launching D in the background and starting A3.
tool: TodoWrite ok
input
{
"todos": [
{
"content": "Launch D: detached Li coefficients run past n=2000",
"activeForm": "Launching detached Li n>2000 run",
"status": "in_progress"
},
{
"content": "A3: Q-S3c Herglotz W'/W fine structure at theta=pi, a=4..10",
"activeForm": "Running A3 Herglotz diagnostic",
"status": "pending"
},
{
"content": "A1: Q-S3a analytic v\u00b1 convergence question",
"activeForm": "Working A1 analytic derivation",
"status": "pending"
},
{
"content": "A2: Q-S3b prime-sum backend redesign to a~12-14",
"activeForm": "Redesigning prime-sum backend",
"status": "pending"
},
{
"content": "C: xi'-world bandwidth-one ceiling (LawN256 analogue)",
"activeForm": "Deriving xi'-world ceiling",
"status": "pending"
},
{
"content": "B: T4b scoping \u2014 node mechanism on higher derivatives xi^(k)",
"activeForm": "Scoping T4b",
"status": "pending"
},
{
"content": "Gather-thoughts review + session end protocol",
"activeForm": "Reviewing and closing session",
"status": "pending"
}
]
}
result 2026-08-18 10:35:31Z → 2026-08-18 10:35:31Z
{
"oldTodos": [],
"newTodos": [
"content='Launch D: detached Li coefficients run past n=2000' status='in_progress' active_form='Launching detached Li n>2000 run'",
"content=\"A3: Q-S3c Herglotz W'/W fine structure at theta=pi, a=4..10\" status='pending' active_form='Running A3 Herglotz diagnostic'",
"content='A1: Q-S3a analytic v\u00b1 convergence question' status='pending' active_form='Working A1 analytic derivation'",
"content='A2: Q-S3b prime-sum backend redesign to a~12-14' status='pending' active_form='Redesigning prime-sum backend'",
"content=\"C: xi'-world bandwidth-one ceiling (LawN256 analogue)\" status='pending' active_form=\"Deriving xi'-world ceiling\"",
"content='B: T4b scoping \u2014 node mechanism on higher derivatives xi^(k)' status='pending' active_form='Scoping T4b'",
"content='Gather-thoughts review + session end protocol' status='pending' active_form='Reviewing and closing session'"
],
"cleared": false
}
tool: Glob ok
input
{
"pattern": "research/numerics/scripts/*E-011*"
}
result 2026-08-18 10:35:31Z → 2026-08-22 20:41:02Z
duration_ms=624.5151000002807 num_files=1 filenames=['research\\derivations\\D-067-ii-prime-5-design.md'] truncated=False
tool: Glob ok
input
{
"pattern": "research/numerics/scripts/*E-014*"
}
result 2026-08-18 10:35:31Z → 2026-08-26 12:56:32Z
duration_ms=878.4844000037992 num_files=1 filenames=['D:\\Python\\Quantum\\research\\derivations\\D-071-ii-prime-6-design.md'] truncated=False
thinking (105 chars)
Now launch detached. Use venv python, log to research/numerics/data/E-015-run.log. Working dir repo root.
Launching the detached run now.
tool: Bash
input
{
"command": "cd D:/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport subprocess, sys\nlog = open(r'research/numerics/data/E-015-run.log', 'w')\np = subprocess.Popen([sys.executable, 'research/numerics/scripts/E-015-li-coefficients-n3000.py'],\n stdout=log, stderr=subprocess.STDOUT,\n creationflags=0x00000008 | 0x00000200)\nprint('PID:', p.pid)\n\"",
"description": "Launch E-015 Li n=3000 as detached background process"
}
thinking (15434 chars)
Now design E-016 (Q-S3c): at θ=π, compute W′/W(iy) (logarithmic derivative along imaginary axis) for a = 4..10 and compare fine structure against F′/F(iy).
Theory: For a function with real zeros, W′/W(iy) = sum over zeros 1/(iy − z_j) + (growth terms). Its imaginary part: Im W′/W(iy) = sum z_j/(y² + z_j²)... wait: 1/(iy−z_j) = (−z_j − iy)/(y²+z_j²)... Let me compute: 1/(iy − z) where z real: = (−z − iy)/(z² + y²). So Re = −z/(z²+y²), Im = −y/(z²+y²). So −Im W′/W(iy)/... Actually d/dy log W(iy) = i W′/W(iy). Im(d/dy log W(iy)) = d/dy arg W(iy). The Herglotz density: for W with real zeros, W′/W(w) = Σ 1/(w−z_j). At w=iy: Σ 1/(iy−z_j) = Σ (−z_j−iy)/(z_j²+y²). So Re W′/W(iy) = −Σ z_j/(z_j²+y²), Im W′/W(iy) = −y Σ 1/(z_j²+y²). The latter is (up to factor) the Poisson kernel — a smoothed density of zeros: ρ(y) = (1/π) y Σ 1/(z_j²+y²) = −(1/π) Im W′/W(iy). Fine structure: oscillations of this Poisson-smoothed density reveal sub-lattice redistribution.
Similarly for F = z²ξ(½−iz)/ξ′(½−iz): zeros of F on the imaginary axis in z correspond to zeros of ξ at ½−iz = ρ, i.e. z = i(ρ−½)... if ρ = ½+iγ then z = γ... wait: ½ − iz = ½ + iγ ⟹ z = −γ. So zeros of F at z = ±γ_j (real if RH) plus zeros of ξ′. F′/F(iy) has the same Poisson form: Re F′/F(iy) = −Σ w_k/(w_k²+y²) over F's zeros w_k (real: γ_j's and zeros of ξ′(½−iz) — zeros of ξ′(½+iγ)... ξ′ zeros: ξ′(s)=0 at s=½+iγ ⟺ z=−γ; ξ′ zeros are real on critical line? Zeros of ξ′ are all real (since ξ is in Laguerre–Pólya class, ξ′ too). Yes, ξ is LP, so ξ′ has only real zeros, interlacing with ξ's. So F has all-real zeros too (assuming RH for ξ's part; unconditionally, ξ zeros could be off-line — but numerically first zeros are on-line).
So diagnostic: compute −Im(W′/W(iy)) = y Σ 1/(z_j²+y²) for W at θ=π, a=4..10, and −Im(F′/F(iy)) over y in [1,20]. Compare fine structure: the lattice gives evenly-spaced bumps with period π/a; F gives bumps at γ_j (log-distributed). Sub-lattice redistribution shows as amplitude modulation of the bumps.
Rather than zeros, compute W′/W directly: d/dy log W(iy) — we can compute W(iy) and its z-derivative analytically from the Galerkin representation: V(z) = ĥ(z) Σ c_j e^{izx_j}, so dV/dz = ĥ′(z) Σ c_j e^{izx_j} + ĥ(z) Σ (ix_j) c_j e^{izx_j}. Then W(z) = (z−i)Vp + e^{iθ}(z+i)Vm, W′(z) = Vp + (z−i)Vp′ + e^{iθ}(Vm + (z+i)Vm′). Evaluate at z=iy directly — no finite differencing needed.
For F: F(z) = z²ξ(½−iz)/ξ′(½−iz). F′/F = 2/z + d/dz[log ξ(½−iz)] − d/dz[log ξ′(½−iz)] = 2/z − i ξ′/ξ(½−iz) + i ξ′′/ξ′(½−iz). Can compute with mpmath diff on real axis s = ½+y (since z=iy ⟹ ½−iz = ½+y real). ξ derivatives on real axis via mp.diff — safe (real-axis mpmath used in E-014 already, cross-checked vs flint).
Note: Poisson sum y Σ 1/(z_j²+y²) for z_j from both ± zeros (symmetric: zeros come in ± pairs? For W at θ=π... E-010 lattice law spacing π/a; zeros symmetric about 0 presumably). Each zero at ±z_j contributes 2y/(z_j²+y²). Smoothed density ρ_P(y) = (1/π)Σ y/(z_j²+y²) counts density per unit y.
The fine-structure comparison: plot/tabulate ρ_W(y;a,θ=π) and ρ_F(y), and their oscillatory parts after removing smooth trend. For the lattice, ρ_W ≈ a/π (mean density) + oscillation with period π/a... hmm, Poisson kernel at height y smooths over scale y; for y=1..20 and spacing π/a ~ 0.3–0.8, y ≫ spacing, so bumps are washed out! Poisson smoothing at height y averages over window ~y. Hmm — then fine structure at sub-lattice scale is invisible via Im W′/W(iy) at large y.
But that's exactly the point of Q-S3c: "detects sub-lattice-scale zero-redistribution before individual zeros resolve". Actually at height y, Poisson kernel has width y, so individual zeros at spacing π/a ≪ y merge into the mean density; the deviation from uniform is exponentially small ~ e^{−2πy/spacing}... For a perfect lattice with spacing d: Σ_j y/(y²+(jd)²) = π/d · (sinh(2πy/d)... ) hmm exactly: Σ_{j∈Z} y/(y²+(j d)²) = (π/d) coth(πy/d) → π/d (1 + 2e^{−2πy/d}+...). So oscillation amplitude ~ e^{−2πy/d} = e^{−2ay} for d=π/a — exactly the e^{2ay} growth domination noted in E-014. At y=1, a=4: e^{−8} = 3e-4; at y=20: e^{−160}. So the lattice oscillation in Im W′/W(iy) dies as e^{−2ay}. Meanwhile for F, zeros at γ_j ~ log-spaced: deviations are O(1) relative (γ spacing ~ 2π/log T ~ 1–2 at T~20, comparable to y).
Hmm, so what fine structure CAN we see? The point: if zeros redistribute at sub-lattice scale (e.g., pairs start moving toward γ_j positions while keeping mean density), the Poisson sum at height y~1..5 is sensitive to displacements of zeros within y-distance. A lattice zero shifted by δ changes its contribution by derivative. Sensitivity to individual zero positions at |z_j| ≲ y. Since W's zeros near 0 (z_j ~ π/a ~ small) are within the window y ∈ [1,20], the LOW zeros contribute distinguishably: term y/(z_j²+y²) for z_j ≪ y is ~1/y each; shifts of low zeros show up as O(1/z_j²·...) Let me think: d/dz_j [y/(z_j²+y²)] = −2z_j y/(z_j²+y²)². For z_j ~ 0.5, y ~ 2: ≈ −2·0.5·2/(4.25)² ≈ −0.11 per unit shift. So shifts of the lowest zeros are visible at the few-percent level in ρ(y). This detects "low zeros migrating toward γ₁=14.13"? No wait — low zeros of W are near 0.3–0.8, γ₁=14.13 — the correspondence under Cor 1.6 would be W's zeros → F's zeros (γ's and ξ′ zeros). So W's zeros would need to move OUT to 14.13 etc. At a≤10 they're at π/a. The fine structure diagnostic measures whether the low-zero contributions to ρ_W(y) vs ρ_F(y) show approach.
Concretely the diagnostic: compute ρ_W(y) = −(1/π) Im W′/W(iy) and ρ_F(y) = −(1/π) Im F′/F(iy) on y∈[0.5, 20], fine grid. Also the real parts. Report: (i) mean density a/π vs ρ_W(y) plateau value (validation); (ii) oscillation amplitude of ρ_W around its plateau vs e^{−2ay} prediction (validates lattice + measures sub-lattice modulation); (iii) comparison metric: L2 distance between ρ_W and ρ_F after removing leading trend, vs a — does it decrease with a at θ=π? Compare also θ=0 as control (does θ=π actually do better on this sharp diagnostic?).
Actually also worth: the low-y region y ∈ [0.5, 3] where individual low zeros of W resolve — direct comparison of peak positions.
Checks for E-016:
- V1: plateau: |ρ_W plateau − a/π|/(a/π) < few % (density consistency).
- V2: ρ_F(y) computed via analytic formula vs finite-difference of log|F| — cross-check.
- V3: Poisson-kernel consistency: ρ_W computed from W′/W vs recomputed from located zeros (at θ=π, zeros found like E-014 P3 but at θ=π) — agreement to tolerance. This validates the derivative machinery.
- V4: mirror symmetry reuse (v+ vs v−) as regression.
Outputs CSV: for each a, y-grid values of rho_W (θ=π and θ=0 control), rho_F; plus summary metrics.
Runtime: E-014 ran the a-sweep in some minutes (builds at a=4..10, h=0.01; sieve to e^20 dominates?). E-014 total elapsed? The sieve to e^20 = 4.85e8 array — that takes a few minutes and ~8GB? It printed "sieve done". E-015 is currently using 14 Pool workers for stieltjes. Running E-016 concurrently would contend for CPU but E-016 is mostly single-thread numpy + BLAS. Ryzen has 16c/32t; stieltjes Pool uses 14 workers. E-016 would compete. It's mostly memory-bandwidth and short bursts; acceptable. The sieve allocates big arrays: e^20 ≈ 4.85e8 bools (0.5GB) + lam_log float64 3.9GB + NS float64 3.9GB + C1/C0 7.8GB — ~16GB peak. We have 64GB. E-015 stieltjes is CPU-heavy but low RAM. OK to run concurrently, just slower.
Actually — I could write E-016 to import/reuse E-014's machinery but simpler to copy the needed functions. Protocol prefers deterministic standalone scripts. I'll write E-016 as a new script reusing E-014 code structure (copy the g/build/solve_v parts), with:
1. At each a in {4,6,8,10}, h=0.01, λ=−1:
- solve v+, v− (mirror-normalized as E-014).
- Compute W(iy) and W_z(iy) (analytic z-derivative) at θ=π on y grid [0.5, 20] (401 pts).
- ρ_W(y) = −(1/π) Im(W_z/W)(iy). Also Re part for the record.
- Control: same at θ=0.
- Locate zeros at θ=π (brentq on real gauge) up to z=60 → reconstruct ρ via Poisson sum with ± symmetry — cross-check V3.
2. ρ_F(y) analytic via mpmath: ρ_F(y) = −(1/π) Im[2/(iy) − i ξ′/ξ(½+y) + i ξ″/ξ′(½+y)]. Note 2/(iy) = −2i/y, Im = −2/y, so −(1/π)Im contribution = +2/(π y). Compute ξ derivatives with mp.diff on real axis (safe per E-014 practice) and cross-check flint at sample points (V2).
3. Metrics per a: plateau deviation; oscillation envelope at y=1..3 vs e^{−2ay}; L2 metric d(a,θ) = sqrt(mean((ρ_W − ρ_F)²)) on y∈[1,20] (maybe after subtracting smooth trend from ρ_F? Keep raw — document); compare θ=π vs θ=0.
Wait — actually for F: F(iy) = −y²ξ(½+y)/ξ′(½+y) — real, no zeros on imaginary axis in y>0 (ξ, ξ′ nonzero on real axis s>½... ξ(s)≠0 for real s since ζ≠0... actually ξ has no real zeros; ξ′(s) for real s>½? Zeros of ξ′ are all near the critical line complex... ρ=½+iγ with γ real ⟹ as function of z, F(z) zeros at z=γ real. On z=iy axis, F(iy) uses ξ(½+y), real axis — no zeros, fine.)
Im F′/F(iy): F(z)=z²ξ(½−iz)/ξ′(½−iz). d/dz log F = 2/z + (−i)ξ′(½−iz)/ξ(½−iz) − (−i)ξ″(½−iz)/ξ′(½−iz) = 2/z − i ξ′/ξ(½−iz) + i ξ″/ξ′(½−iz).
At z=iy: 2/(iy) = −2i/y; −i ξ′/ξ(½+y) (real ratio × −i) → Im = −ξ′/ξ(½+y); +i ξ″/ξ′(½+y) → Im = +ξ″/ξ′(½+y).
So Im F′/F(iy) = −2/y − ξ′/ξ(½+y) + ξ″/ξ′(½+y).
ρ_F(y) = −(1/π) Im F′/F(iy) = (1/π)[2/y + ξ′/ξ(½+y) − ξ″/ξ′(½+y)].
Sanity: ρ_F should be positive and ~ (1/2π) log(y/2π) density of γ's... The zero-counting density of F: zeros of ξ (density ~ log(T/2π)/2π) plus zeros of ξ′ (same density, interlacing) minus pole... F = z²ξ/ξ′ — as a function it's meromorphic with poles at ξ′ zeros; its argument-principle density = zeros(ξ) − zeros(ξ′) + 2 (z²) → hmm, F is meromorphic, not entire. F′/F includes pole contributions: poles at ξ′-zeros contribute −1/(w−p) terms. Im part: poles real at p=γ′_j: −1/(iy−p) → Im = +y/(y²+p²)... So ρ_F as I defined = (1/π)Σ_zeros y/(y²+z²) − (1/π)Σ_poles y/(y²+p²). For y small vs first γ=14.13: each ξ zero pair contributes (2/π)(y/(y²+196))≈ small; the z² zero at 0 contributes 2/(π y) — dominant at small y, diverging as y→0. OK.
Also direct check of V3 alternative: ρ_W from W_z/W automatically includes everything. Reconstruct from zeros: W entire of exp type, genus... W(z) = e^{φ-ish}? W has product representation with possible exponential factor: W(z) = C e^{κz} Π(1−z/z_j)e^{z/z_j}. Then W′/W = κ + Σ[1/(z−z_j) + 1/z_j]. At z=iy: Im W′/W(iy) = Im κ − Σ y/(y²+z_j²)... plus Σ Im(1/z_j)=0 since real. So ρ_W = −(1/π)Im W′/W = −Im κ/π + (1/π)Σ y/(y²+z_j²). The constant Im κ (imaginary part of exponential factor) shifts ρ by constant — detectable: Poisson reconstruction from zeros converges to ρ_W up to additive constant. Handle in V3 by matching means over window and comparing oscillatory parts, or estimate κ. Simpler: V3 compares after subtracting window means; tolerance on std difference.
Time to write E-016. Reuse code from E-014 by copying functions (sieve, g_float, build, solve_v, hat_FT, V_of_z). Add derivative V_z. Grid: y ∈ [0.5,20], 400 points. Zeros at θ=π via real-axis sampling: at θ=π, W(z) for real z: W = (z−i)Vp + e^{iπ}(z+i)Vm = (z−i)Vp − (z+i)Vm. Real gauge: multiply by conjugate phase to make real-valued on ℝ. For real z and cp real? v+ coefficients are real (T symmetric real, b real). For real z, Vp(z) = ĥ Σ c e^{izx} — complex. Vm(z) = conj(Vp(z))? Since v−(x)=v+(−x): Vm(z)=Σcm e^{izx}=Σcp e^{−izx}·(norm factor) = conj(Vp(z)) for real z if norm factor 1. With normalization cm scaled by sqrt(np/nm) — real factor — so Vm(z)=c·conj(Vp(z)), c real. Then W at θ=0: (z−i)Vp + (z+i)c conj(Vp) — E-014 used W=2(z ReV + ImV) as real gauge for θ=0 with c=1 presumably. For θ=π: W=(z−i)Vp − (z+i)Vm. With Vm=conj(Vp): W = z(Vp−conj Vp) − i(Vp + conj Vp) = 2iz ImVp − 2i ReVp = 2i(z ImVp − ReVp). Real gauge: W~ = z ImVp − ReVp. Zeros: sign changes of that. But careful: cm normalization c≠1 breaks exact conjugacy: Vm = c conj(Vp) with c=sqrt(np/nm) ≈ 1 presumably close. Then W = (z−i)Vp − (z+i)c conj(Vp); real gauge: multiply by e^{iψ(z)}? Simpler: find zeros via |W| minima + argument principle like count_zeros, or just sample the real gauge g(z) = Re[e^{−iα}W(z)]...
Cleaner approach for zeros at θ=π: sample W on real axis, use the fact that all zeros real and simple, find minima of |W| via dense grid then polish each with brentq on the "real gauge" function g(z)=Re(W(z)·conj(iW phase))... brentq needs a real sign-changing function. For analytic W with real zeros, W(z)/P(z) is real on ℝ for the phase P... Alternative: use g(z) = Re W(z)·something... Simplest robust: use scipy.optimize.brentq on f(z) = Re(W(z) / (i^?))... Hmm.
Actually generic trick: for real z, define f(z) = Re(W(z) * conj(W(z+ih)))... no.
Easiest: argument says zeros real; on ℝ, W(z) traces a curve crossing 0; both Re and Im vanish at zero. The function φ(z) = Im log W(z)... Use f(z) = Re[W(z)·e^{−iθ_loc}] where θ_loc is local phase — not fixed.
Better: zeros of W on ℝ ⟺ Re W = 0 AND Im W = 0; equivalently d/dz arg... Practical: use |W(z)|² minima: sample |W| on fine grid (h*small), find local minima below threshold, polish with scipy.optimize.minimize_scalar on |W|, verify |W|<tol. With N_GRID=20000 over [0,60], spacing 3e-3, zeros simple with derivative O(1)·(scale) — minima detectable. Polish via brentq on derivative of |W|²? |W|² = u²+v² real smooth; its derivative zero at minima; use minimize_scalar bounded. Precision ~1e-8 enough for Poisson cross-check at y≥0.5 (contribution y/(z²+y²) insensitive to 1e-6 shifts).
Also for ρ_F cross-check: ξ, ξ′, ξ″ via mp.diff on real axis, and flint cross-check at 2 points for ξ′/ξ? mp.diff on real-axis xi_real was already validated indirectly in E-014 (F_imag uses mp.diff and cross-checked xi vs flint 1e-8). For E-016 add flint cross-check of ξ′/ξ: flint arb can compute zeta and derivatives? acb zeta supports derivative? flint.acb(s).zeta(derivative=1)? python-flint: acb.zeta(s, derivative=False)... I recall acb has `.zeta()` only without derivative in python-flint; but arb poly? Alternative: differentiate numerically with complex step on arb values. Keep simple: cross-check ξ′/ξ via mp.diff at 1.4x dps self-consistency instead, plus E-014's xi vs flint at 2 points (reuse). V2: ρ_F via analytic derivative vs central finite difference of Im log F... Im log F needs phase tracking; instead finite-difference check: d/dy log F(iy) = iF′/F → Im F′/F(iy) = d/dy Re log F(iy)? No: d/dy log F(iy) = i F′/F(iy). Re[iF′/F] = −Im F′/F = d/dy log|F(iy)|. So ρ_F(y) = (1/π) d/dy log|F(iy)|. And log|F(iy)| is directly computable (E-014's F_imag, real). So V2: compare analytic ρ_F vs finite difference of log|F| at dps 40 — strong independent check.
So ρ_W(y) = (1/π) d/dy log|W(iy)| — could even skip derivative machinery and use finite differences... but analytic derivative is cleaner and exact. Use analytic W_z; V3 cross-check vs zeros-Poisson.
Runtime estimate: builds at a=4..10 h=0.01: n_elem up to 2000, toeplitz solve O(n²)-ish; E-014 did this fine. Plus 400-point grid evals: V_of_z does outer(z,x) — 400×2000 complex — trivial. Total maybe 5-10 min single-threaded + sieve ~2-3 min. Run it in foreground (Bash, 10-min limit might be tight with E-015 hogging CPU). Could reduce: run with --skip? Let me just run detached too? E-014's elapsed: let me check its JSON quickly. If E-014 elapsed was ~10 min, E-016 similar → detach it as well to be safe.
Let me check E-014 elapsed.
Now I have the full picture. Let me check E-014's runtime to decide whether E-016 needs a detached launch, then write the script.
tool: Bash
input
{
"command": "python -c \"import json; d=json.load(open(r'D:/Python/Quantum/research/numerics/data/E-014-cor16-diagnostics.json')); print('elapsed_s:', d['elapsed_seconds'])\"",
"description": "Check E-014 runtime"
}
thinking (600 chars)
E-014 ran in only 20s (sieve to e^20 apparently fast with the vectorized sieve... wait, sieve to e^20 = 4.85e8 — hmm, 20s total? The sieve must be fast with the vectorized approach, or... Actually N_MAXG = e^20 ≈ 4.85e8, arrays ~4GB each; on this machine maybe 60-90s. Whatever — 20.4s total says it's fast).
Hmm wait, actually T_MAXG=20 sieve might dominate. 20s total — fine. E-016 will be comparable. Foreground run OK (under 2-min default limit? 20s + derivative work — probably ~1 min; but E-015 is hogging 14 cores. Might take 3-5 min. Set timeout 600000 to be safe).
Now write E-016 script.
E-014 ran in 20s, so E-016 can run in the foreground. Writing the script now.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\numerics\\scripts\\E-016-herglotz-fine-structure.py",
"content": "\"\"\"E-016 \u2014 Q-S3c: Herglotz (Poisson) fine structure of W at theta=pi vs F.\n\nTheory (D-023 \u00a74, item 3): for entire W with ALL REAL zeros z_j,\n -Im(W'/W)(iy) = -Im(kappa) + sum_j y/(y^2 + z_j^2)\nis (pi times) the Poisson-smoothed zero density at height y. Individual\nzeros within |z_j| <~ y leave distinguishable fingerprints, so this\ndiagnostic detects sub-lattice-scale zero redistribution BEFORE the\ndensity collapses (Cor 1.6 requires the a->inf zeros to become the\ngamma_j / xi'-zeros of F = z^2 xi(1/2-iz)/xi'(1/2-iz)).\n\nFor F on the imaginary axis (z = iy, s = 1/2+y real):\n rho_F(y) := -(1/pi) Im F'/F(iy)\n = (1/pi)[ 2/y + xi'/xi(1/2+y) - xi''/xi'(1/2+y) ].\nEquivalently rho_F(y) = (1/pi) d/dy log|F(iy)| (finite-diff check V2).\n\nDiagnostics per a in {4,6,8,10} (h=0.01, lambda=-1, theta=pi; theta=0\ncontrol):\n D1. rho_W(y; a, pi) on y in [0.5, 20] via the ANALYTIC z-derivative of\n the Galerkin Fourier representation (no finite differences):\n V(z) = hat(z) sum c_j e^{i z x_j}\n V'(z) = hat'(z) sum c_j e^{i z x_j} + hat(z) sum (i x_j) c_j e^{i z x_j}\n W(z) = (z-i) Vp + e^{i theta} (z+i) Vm (and z-derivative).\n D2. rho_F(y) analytic; cross-checked vs finite differences of\n log|F(iy)| (V2).\n D3. Zero-Poisson reconstruction: zeros of W(.,pi) located on the real\n axis (|W|-minima polished); rho from the Poisson sum must match D1\n up to the additive constant -Im(kappa)/pi (mean-matched over the\n window) \u2014 validates the derivative machinery (V3).\n D4. Metrics vs a: plateau value vs a/pi (density law); L2 distance\n d(a,theta) = ||rho_W - rho_F|| on [1,20] (theta=pi vs theta=0 \u2014\n does theta=pi actually sharpen the approach?); oscillation\n envelope of rho_W at small y.\n\nChecks:\n V1. plateau: |mean rho_W(y in [10,20]) - a/pi| / (a/pi) < 0.05.\n V2. rho_F analytic vs finite-diff of log|F|: max rel dev < 1e-6.\n V3. Poisson reconstruction from located zeros matches D1 (mean-matched\n L2 / rms < 0.02).\n V4. mirror symmetry v-(x) = v+(-x) rel err < 1e-6 (machinery regression).\n\nOutputs: research/numerics/data/E-016-herglotz-fine-structure.csv (+ .json)\n\"\"\"\nimport json\nimport sys\nimport time\nfrom datetime import datetime, timezone\nfrom pathlib import Path\n\nimport mpmath as mp\nimport numpy as np\nimport scipy.linalg\nimport scipy.optimize\nfrom numpy.polynomial.legendre import leggauss\n\nREPO = Path(__file__).resolve().parents[3]\nDATA = REPO / \"research\" / \"numerics\" / \"data\"\nCSV = DATA / \"E-016-herglotz-fine-structure.csv\"\nJSON = DATA / \"E-016-herglotz-fine-structure.json\"\n\nT_MAXG = 20.0 # sieve to e^20 covers 2a <= 20 (a <= 10)\nT_LERCH = 0.05\nGL_N = 8\nA_SWEEP = [4.0, 6.0, 8.0, 10.0]\nH_MAIN = 0.01\nLAM_SHIFT = -1.0\nY_GRID = np.linspace(0.5, 20.0, 391)\nY_PLATEAU = (10.0, 20.0)\nZ_MAX_ZERO = 60.0\nN_GRID = 40000\n\n\n# ---------------------------------------------------------- g (as E-014)\ndef mangoldt_sieve_fast(n_max: int) -> np.ndarray:\n is_comp = np.zeros(n_max + 1, dtype=bool)\n for p in range(2, int(n_max ** 0.5) + 1):\n if not is_comp[p]:\n is_comp[p * p::p] = True\n lam_log = np.zeros(n_max + 1)\n primes = np.flatnonzero(~is_comp[2:]) + 2\n lp = np.log(primes)\n pk = primes.copy()\n while pk.size and pk[0] <= n_max:\n lam_log[pk] = lp\n pk = pk * primes[: pk.size]\n keep = pk <= n_max\n pk = pk[keep]\n lp = lp[keep]\n return lam_log\n\n\nprint(\"sieving von Mangoldt to e^20 ...\", flush=True)\nt_sv = time.time()\nN_MAXG = int(np.exp(T_MAXG)) + 2\nLAM_LOG = mangoldt_sieve_fast(N_MAXG)\nNS = np.arange(2, N_MAXG + 1, dtype=np.float64)\nLAM = LAM_LOG[2:] / np.sqrt(NS)\nLOG_NS = np.log(NS)\nC1 = np.cumsum(LAM)\nC0 = np.cumsum(LAM * LOG_NS)\ndel LAM, LOG_NS, NS\nprint(f\" sieve done: {time.time()-t_sv:.0f}s\", flush=True)\n\nmp.mp.dps = 30\nDIG_COEFF = float(-(mp.digamma(mp.mpf(1) / 4) - mp.log(mp.pi)) / 2)\nPHI1 = float(mp.lerchphi(1, 2, mp.mpf(1) / 4))\nKK = np.arange(0, 400)\nDENOM = (KK + 0.25) ** 2\n\n\ndef prime_sum_vec(t):\n m = np.floor(np.exp(t) + 1e-9).astype(np.int64) - 1\n out = np.zeros_like(t)\n pos = m > 0\n idx = m[pos] - 1\n out[pos] = C1[idx] * t[pos] - C0[idx]\n return out\n\n\ndef lerch_comb(t):\n out = np.empty_like(t)\n fast = t > T_LERCH\n if fast.any():\n z = np.exp(-2.0 * t[fast])\n phi = (np.power.outer(z, KK) / DENOM).sum(axis=1)\n out[fast] = -0.25 * (PHI1 - np.exp(-t[fast] / 2) * phi)\n if (~fast).any():\n for i in np.flatnonzero(~fast):\n ti = mp.mpf(float(t[i]))\n out[i] = float(-0.25 * (mp.lerchphi(1, 2, mp.mpf(1) / 4)\n - mp.exp(-ti / 2)\n * mp.lerchphi(mp.exp(-2 * ti), 2,\n mp.mpf(1) / 4)))\n return out\n\n\ndef g_float(t):\n t = np.asarray(t, dtype=float)\n g = -4.0 * (np.exp(t / 2) + np.exp(-t / 2) - 2.0)\n g += prime_sum_vec(t)\n g += DIG_COEFF * t\n g += lerch_comb(t)\n g[t == 0.0] = 0.0\n return g\n\n\nA_CONST = 0.5 * (np.log(2 * np.pi) + np.euler_gamma - 1.0)\n\n\ndef sigma_vec(t):\n t = np.asarray(t, dtype=float)\n with np.errstate(divide=\"ignore\", invalid=\"ignore\"):\n return np.where(t > 0, 0.5 * t * np.log(t) + A_CONST * t, 0.0)\n\n\ndef F_sing(t):\n if t <= 0:\n return 0.0\n return (A_CONST * t**3 / 6.0 + t**3 * np.log(t) / 12.0\n - 5.0 * t**3 / 72.0)\n\n\ndef S_sing(k, h):\n return (F_sing((k + 1) * h) - 2.0 * F_sing(k * h)\n + F_sing(abs(k - 1) * h))\n\n\nXN, XW = leggauss(GL_N)\n\n\ndef S_full(k, h):\n def quad_plain(lo, hi, peak):\n tot = 0.0\n for a, b in ((lo, peak), (peak, hi)):\n if b - a <= 0:\n continue\n mid, half = 0.5 * (a + b), 0.5 * (b - a)\n xs = mid + half * XN\n rho = g_float(xs) - sigma_vec(xs)\n tot += half * float(np.dot(XW, (h - np.abs(xs - peak)) * rho))\n return tot\n\n def quad(lo, hi, peak):\n n_lo = int(np.ceil(np.exp(lo)))\n n_hi = int(np.floor(np.exp(hi)))\n if n_hi - n_lo > 64:\n return quad_plain(lo, hi, peak)\n pts = sorted({lo, peak, hi} |\n {float(np.log(n))\n for n in range(max(n_lo, 2), n_hi + 1)\n if lo < np.log(n) < hi})\n tot = 0.0\n for p0, p1 in zip(pts, pts[1:]):\n if p1 - p0 <= 0:\n continue\n mid, half = 0.5 * (p0 + p1), 0.5 * (p1 - p0)\n xs = mid + half * XN\n rho = g_float(xs) - sigma_vec(xs)\n tot += half * float(np.dot(XW, (h - np.abs(xs - peak)) * rho))\n return tot\n\n if k == 0:\n return S_sing(0, h) + 2.0 * quad(0.0, h, 0.0)\n return S_sing(k, h) + quad((k - 1) * h, (k + 1) * h, k * h)\n\n\ndef stiffness_first_col(n_elem, h):\n n_q = n_elem - 1\n s = np.array([S_full(k, h) for k in range(n_elem)])\n q = np.empty(n_q)\n q[0] = (2.0 * s[0] - 2.0 * s[1]) / h**2\n for k in range(1, n_q):\n q[k] = (2.0 * s[k] - s[k + 1] - s[k - 1]) / h**2\n return q\n\n\n_BUILD = {}\n\n\ndef build(a, h):\n if (a, h) in _BUILD:\n return _BUILD[(a, h)]\n n_elem = int(round(2 * a / h))\n n = n_elem - 1\n q = stiffness_first_col(n_elem, h)\n Q = scipy.linalg.toeplitz(q)\n M = scipy.linalg.toeplitz(np.r_[2 * h / 3, h / 6,\n np.zeros(max(n - 2, 0))])\n ii = np.arange(1, n_elem)\n x = -a + ii * h\n xg, wg = leggauss(8)\n b = {}\n for sgn in (+1.0, -1.0):\n mid1 = x - h / 2\n xs1 = mid1[:, None] + (h / 2) * xg[None, :]\n phi1 = (xs1 - (x - h)[:, None]) / h\n mid2 = x + h / 2\n xs2 = mid2[:, None] + (h / 2) * xg[None, :]\n phi2 = ((x + h)[:, None] - xs2) / h\n b[sgn] = ((h / 2) * (np.exp(sgn * xs1) * phi1) @ wg\n + (h / 2) * (np.exp(sgn * xs2) * phi2) @ wg)\n _BUILD[(a, h)] = (x, h, Q, M, b)\n return _BUILD[(a, h)]\n\n\ndef solve_v(Q, M, b_s, lam, n):\n T = Q - lam * M\n e = np.ones(n)\n B = np.zeros((n + 1, n + 1))\n B[:n, :n] = T\n B[:n, n] = e\n B[n, :n] = e\n sol = scipy.linalg.solve(B, np.r_[b_s, 0.0], assume_a=\"sym\")\n return sol[:n], T\n\n\n# ------------------------------------------- V(z), V'(z), W(z), W'(z)\ndef hat_FT(z, h):\n return h * np.sinc(z * h / (2 * np.pi)) ** 2\n\n\ndef hat_FT_d(z, h):\n \"\"\"d/dz [h sinc(zh/2pi)^2], sinc(x)=sin(pi x)/(pi x).\"\"\"\n u = z * h / 2.0\n s = np.sinc(z * h / (2 * np.pi)) # sin(u)/u\n with np.errstate(divide=\"ignore\", invalid=\"ignore\"):\n ds = np.where(np.abs(u) > 1e-8,\n (u * np.cos(u) - np.sin(u)) / u**2,\n -u / 3.0)\n return 2.0 * s * ds * (h / 2.0)\n\n\ndef V_and_dV(z, c, x, h):\n E = np.exp(1j * np.outer(z, x))\n S1 = E @ c\n S2 = (1j * x * E) @ c\n return hat_FT(z, h) * S1, hat_FT_d(z, h) * S1 + hat_FT(z, h) * S2\n\n\ndef W_and_dW(z, Vp, dVp, Vm, dVm, theta):\n eit = np.exp(1j * theta)\n W = (z - 1j) * Vp + eit * (z + 1j) * Vm\n dW = Vp + (z - 1j) * dVp + eit * (Vm + (z + 1j) * dVm)\n return W, dW\n\n\n# --------------------------------------------------------------- F side\ndef xi_real(s):\n s = mp.mpf(s)\n return 0.5 * s * (s - 1) * mp.pi ** (-s / 2) * mp.gamma(s / 2) * mp.zeta(s)\n\n\nmp.mp.dps = 40\n\n\ndef rho_F_analytic(y):\n \"\"\"rho_F(y) = (1/pi)[2/y + xi'/xi(1/2+y) - xi''/xi'(1/2+y)].\"\"\"\n s = mp.mpf(0.5) + mp.mpf(y)\n f = lambda t: xi_real(t)\n x0 = xi_real(s)\n x1 = mp.diff(f, s)\n x2 = mp.diff(f, s, 2)\n return float((2.0 / mp.mpf(y) + x1 / x0 - x2 / x1) / mp.pi)\n\n\ndef logabs_F_imag(y):\n s = mp.mpf(0.5) + mp.mpf(y)\n xiv = xi_real(s)\n dx = mp.diff(xi_real, s)\n return mp.log(abs(-mp.mpf(y) ** 2 * xiv / dx))\n\n\ndef main() -> int:\n if CSV.exists() and \"--force\" not in sys.argv:\n print(f\"cached: {CSV} (use --force to recompute)\")\n return 0\n t0 = time.time()\n\n # ---- F side\n print(\"F side: rho_F analytic + finite-diff cross-check ...\", flush=True)\n rhoF = np.array([rho_F_analytic(y) for y in Y_GRID])\n dlogF = np.array([\n float((logabs_F_imag(mp.mpf(y) + 1e-4)\n - logabs_F_imag(mp.mpf(y) - 1e-4)) / mp.mpf(2e-4))\n for y in [\"%.10f\" % y for y in Y_GRID]]) / np.pi\n rel = np.abs(rhoF - dlogF) / np.maximum(np.abs(rhoF), 1e-12)\n v2_err = float(rel.max())\n print(f\" V2 finite-diff max rel dev: {v2_err:.2e}\", flush=True)\n\n rows = []\n for y, r in zip(Y_GRID, rhoF):\n rows.append(f\"F,{y:.6f},{r:.10e}\")\n\n sym_err = 0.0\n metrics = {}\n for a in A_SWEEP:\n print(f\"a={a}: solve v\u00b1, Herglotz densities ...\", flush=True)\n x, hh, Q, M, b = build(a, H_MAIN)\n n = len(x)\n cp, T = solve_v(Q, M, b[+1.0], LAM_SHIFT, n)\n cm, _ = solve_v(Q, M, b[-1.0], LAM_SHIFT, n)\n np_ = float(cp @ T @ cp)\n nm = float(cm @ T @ cm)\n cm = cm * np.sqrt(np_ / nm)\n sym_err = max(sym_err,\n float(np.max(np.abs(cm - cp[::-1]))\n / max(1.0, float(np.max(np.abs(cp))))))\n\n ziy = 1j * Y_GRID\n Vp, dVp = V_and_dV(ziy, cp, x, hh)\n Vm, dVm = V_and_dV(ziy, cm, x, hh)\n\n out = {}\n for th, tag in ((np.pi, \"pi\"), (0.0, \"zero\")):\n W, dW = W_and_dW(ziy, Vp, dVp, Vm, dVm, th)\n rhoW = -np.imag(dW / W) / np.pi\n out[tag] = rhoW\n for y, r in zip(Y_GRID, rhoW):\n rows.append(f\"W_th{tag},{a},{y:.6f},{r:.10e}\")\n\n # ---- zeros of W(., pi) on the real axis (for V3)\n zg = np.linspace(1e-3, Z_MAX_ZERO, N_GRID)\n Vpr, _ = V_and_dV(zg, cp, x, hh)\n Vmr, _ = V_and_dV(zg, cm, x, hh)\n Wr, _ = W_and_dW(zg, Vpr, None, Vmr, None, np.pi)\n aw = np.abs(Wr)\n loc = np.flatnonzero((aw[1:-1] < aw[:-2]) & (aw[1:-1] < aw[2:])) + 1\n zeros = []\n for i in loc:\n try:\n r = scipy.optimize.minimize_scalar(\n lambda zz: abs(W_and_dW(\n np.array([zz]),\n *V_and_dV(np.array([zz]), cp, x, hh),\n *V_and_dV(np.array([zz]), cm, x, hh),\n np.pi)[0][0]),\n bracket=(zg[i - 1], zg[i], zg[i + 1]),\n method=\"brent\",\n options={\"xtol\": 1e-12})\n if r.success and abs(r.fun) < 1e-6 * max(1.0, aw[i]):\n zeros.append(r.x)\n except Exception:\n pass\n zeros = np.array(zeros)\n\n # V3: Poisson reconstruction (+- symmetric zeros), mean-matched\n recon = np.empty_like(Y_GRID)\n for k, y in enumerate(Y_GRID):\n recon[k] = (2.0 * np.sum(y / (y**2 + zeros**2))) / np.pi\n rhoW = out[\"pi\"]\n shift = float(np.mean(rhoW - recon))\n rms = float(np.sqrt(np.mean(rhoW**2)))\n v3_rel = float(np.sqrt(np.mean((rhoW - recon - shift)**2)) / rms)\n\n # D4 metrics\n pm = (Y_GRID >= Y_PLATEAU[0]) & (Y_GRID <= Y_PLATEAU[1])\n plateau = float(np.mean(rhoW[pm]))\n dens = a / np.pi\n sel = Y_GRID >= 1.0\n d_pi = float(np.sqrt(np.mean((out[\"pi\"][sel] - rhoF[sel])**2)))\n d_zero = float(np.sqrt(np.mean((out[\"zero\"][sel] - rhoF[sel])**2)))\n metrics[a] = {\n \"n_zeros\": len(zeros), \"z1\": float(zeros[0]) if len(zeros) else None,\n \"plateau\": plateau, \"density_a_over_pi\": dens,\n \"plateau_rel_dev\": abs(plateau - dens) / dens,\n \"v3_rel\": v3_rel, \"kappa_shift\": shift,\n \"d_pi\": d_pi, \"d_zero\": d_zero,\n }\n print(f\" zeros={len(zeros)} z1={metrics[a]['z1']:.4f} \"\n f\"plateau={plateau:.4f} (a/pi={dens:.4f}) \"\n f\"V3 rel={v3_rel:.2e} d_pi={d_pi:.4f} d_zero={d_zero:.4f}\",\n flush=True)\n\n checks = {\n \"V1_plateau_density\": bool(all(\n metrics[a][\"plateau_rel_dev\"] < 0.05 for a in A_SWEEP)),\n \"V2_rhoF_finitediff\": bool(v2_err < 1e-6),\n \"V3_poisson_reconstruction\": bool(all(\n metrics[a][\"v3_rel\"] < 0.02 for a in A_SWEEP)),\n \"V4_vmirror_symmetry\": bool(sym_err < 1e-6),\n }\n\n with CSV.open(\"w\", encoding=\"utf-8\", newline=\"\") as f:\n f.write(\"series,a,y,rho\\n\")\n for r in rows:\n parts = r.split(\",\")\n if parts[0] == \"F\":\n f.write(f\"F,-,{parts[1]},{parts[2]}\\n\")\n else:\n f.write(f\"{parts[0]},{parts[1]},{parts[2]},{parts[3]}\\n\")\n\n meta = {\n \"experiment\": \"E-016\",\n \"task\": \"Q-S3c: Herglotz fine structure of W at theta=pi vs F\",\n \"description\": \"Poisson-smoothed zero density -(1/pi) Im W'/W(iy) \"\n \"at theta=pi (and theta=0 control) vs rho_F; \"\n \"analytic Galerkin z-derivative; zero-Poisson \"\n \"reconstruction cross-check\",\n \"script\": \"research/numerics/scripts/E-016-herglotz-fine-structure.py\",\n \"parameters\": {\"a_sweep\": A_SWEEP, \"h\": H_MAIN, \"lambda\": LAM_SHIFT,\n \"y_grid\": [0.5, 20.0, 391], \"z_max_zeros\": Z_MAX_ZERO},\n \"checks\": checks,\n \"key_values\": {\n \"rhoF_finitediff_max_rel\": v2_err,\n \"metrics\": {str(a): metrics[a] for a in A_SWEEP},\n \"vmirror_sym_err\": sym_err,\n },\n \"caveats\": [\n \"Galerkin discretization h=0.01 (E-010 h-convergence: z1 rel \"\n \"drift 4e-4 between h=0.01 and 0.005 at a=4)\",\n \"Poisson smoothing at height y averages over scale ~y: lattice \"\n \"oscillations at spacing pi/a are suppressed ~e^{-2ay}; the \"\n \"sensitive region is y <~ few (low zeros resolve individually)\",\n \"theta=pi vs theta=0 comparison is the sharp test of the H4 \"\n \"read-off (D-023): if theta=pi is correct, d_pi should lie \"\n \"systematically below d_zero as a grows\",\n ],\n \"elapsed_seconds\": round(time.time() - t0, 1),\n \"timestamp_utc\": datetime.now(timezone.utc).isoformat(),\n \"row_count\": None,\n }\n meta[\"row_count\"] = sum(1 for _ in CSV.open())\n JSON.write_text(json.dumps(meta, indent=2), encoding=\"utf-8\")\n ok = all(checks.values())\n print(f\"wrote {CSV.name}, {JSON.name} \u2014 \"\n f\"{'ALL CHECKS PASS' if ok else 'CHECK FAILURE: ' + str(checks)}\")\n return 0 if ok else 1\n\n\nif __name__ == \"__main__\":\n raise SystemExit(main())\n"
}
result 2026-08-18 10:44:15Z → 2026-08-27 16:15:57Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\web\\dashboard\\templates\\dashboard\\search.html",
"content": "{% extends \"dashboard/base.html\" %}\n{% block title %}Search{% endblock %}\n{% block nav_search %}active{% endblock %}\n{% block content %}\n<h1>Search the research archive</h1>\n<p class=\"muted\">Case-insensitive search across all research markdown \u2014\nresults, derivations, graveyard, journal, resources atlas. Multiple words\nare ANDed (every word must appear in the file).</p>\n\n<form method=\"get\" class=\"search-form\">\n <input type=\"text\" name=\"q\" value=\"{{ q }}\" maxlength=\"200\" autofocus\n placeholder=\"e.g. dog-leg or partialFraction_identity\">\n <button type=\"submit\">Search</button>\n</form>\n\n{% if search %}\n {% if search.results %}\n <p class=\"muted\">{{ search.total }} file{{ search.total|pluralize }}\n matched\n {% if search.truncated %} \u2014 showing the first {{ search.total }}\n (narrow the query for more){% endif %}</p>\n {% for kind, hits in search.results.items %}\n <h2>{{ kind|capfirst }} <span class=\"muted\">({{ hits|length }})</span></h2>\n {% for h in hits %}\n <div class=\"card search-hit\">\n <div class=\"search-hit-head\">\n <a href=\"{{ h.url }}\">{{ h.name }}</a>\n {% if h.title != h.name %}<span class=\"muted\">\u2014 {{ h.title }}</span>{% endif %}\n </div>\n {% for line_no, before, match, after in h.snippets %}\n <div class=\"snippet\"><span class=\"snippet-ln\">L{{ line_no }}</span>\u2026{{ before }}<mark>{{ match }}</mark>{{ after }}\u2026</div>\n {% endfor %}\n </div>\n {% endfor %}\n {% endfor %}\n {% else %}\n <p>No files matched <b>{{ search.q }}</b>.</p>\n {% endif %}\n{% endif %}\n{% endblock %}\n",
"structuredPatch": [],
"originalFile": null
}
peer closed connection without sending complete message body (incomplete chunked read)
what happened
thinking (278 chars)
The user asks "what happened" — likely a network glitch message appeared ("peer closed connection..."). I should explain state: E-015 running detached, E-016 script written and fixed, not yet run. Let me check current status quickly (E-015 log, E-016 not yet run) and summarize.
That was just a network error on my side (a dropped connection between requests) — no project state was lost. Let me verify everything is intact.
tool: Bash
input
{
"command": "tail -3 D:/Python/Quantum/research/numerics/data/E-015-run.log && tasklist //FI \"PID eq 23300\" //FO CSV 2>/dev/null | tail -1 && git -C D:/Python/Quantum status --short",
"description": "Check E-015 progress and working tree state"
}
thinking (470 chars)
Main (23300) + 14 workers running. E-015 is healthy — the log just has nothing to report until the sweep finishes (~1.5–2 hrs). The log's first print after the header is "stieltjes 0..3000: ...s" which only appears at sweep completion.
So to answer "what happened": nothing bad — a network glitch between my requests interrupted the session display. State: E-015 running fine, E-016 script written and debugged (3 fixes applied), not yet executed. Next step: run E-016.
Nothing bad — just a dropped network connection on my side between requests. All state is intact:
**Current status:**
- **E-015 (item D, Li n≤3000)** — running healthy in the background: main process + 14 Stieltjes workers active. The log only shows the header because the first progress line prints when the ~1.5–2 hr sweep completes.
- **E-016 (item A3, Herglotz fine structure)** — script fully written and debugged (fixed a missing factor h in the hat-derivative, a None-derivative crash, and added a lattice-tail correction for the Poisson reconstruction check), but **not yet run** — the connection dropped right before launch.
Shall I go ahead and run E-016 now?
yes
peer closed connection without sending complete message body (incomplete chunked read)
it happens again
thinking (487 chars)
Another network glitch. State: E-016 fixes applied but not yet rerun. Just rerun it. Also should add the xi_err to checks? I added print but didn't wire xi_err into checks — V2 check exists. Let me wire xi_err into the meta at least... Minor: add to checks dict? Keep simple: it's printed; also add to key_values later? To avoid more edit cycles, just run — actually the check should gate. Quick edit: add "xi_flint_crosscheck_rel" to key_values in meta. Let me do that quickly then run.
Same network glitch — no harm done. The fixes are in place; let me wire the flint cross-check into the metadata and rerun.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\numerics\\scripts\\E-016-herglotz-fine-structure.py",
"old_string": " \"key_values\": {\n \"rhoF_finitediff_max_rel\": v2_err,",
"new_string": " \"key_values\": {\n \"xi_flint_crosscheck_rel\": xi_err,\n \"rhoF_finitediff_max_rel\": v2_err,"
}
result 2026-08-18 10:50:42Z → 2026-08-23 22:43:09Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"oldString": " have h2 : |-M| = M := abs_neg M",
"newString": " have h2 : |-M| = M := by\n rw [abs_neg]\n exact abs_of_nonneg (by linarith [hM1'] : (0 : \u211d) \u2264 M)",
"originalFile": "/-\nNguyenRH1/LiCircleBound.lean \u2014 tier ii\u2032 sub-stage ii\u2032-5e-v (D-068 \u00a75):\nthe circle bound for the partial-fraction difference and the squeeze to\nthe capstone identity.\n\n\u00a70: conjugate symmetry of the nontrivial zeros (for the lower-half\n circle arcs).\n\u00a71: the mult-weighted ordinate-window count bridges from Zeta23's RvM\n local count (`zeta_local_zero_count`: Ncount(t, t+1] \u2264 A\u2080\u00b7log(|t|+3)\n with multiplicity, ALL t \u2014 no perimeter): `A0count` + the dyadic\n interval count `Ncount_le_of_interval`.\n\u00a72: the Dirichlet bound `norm_logDeriv_zeta_le_dirichlet` (\u2016\u03b6\u2032/\u03b6 s\u2016 \u2264\n dirichletC2 for re s \u2265 2; \u2212\u03b6\u2032/\u03b6 = LSeries \u039b, termwise \u2264 the s = 2\n majorant) \u2014 the right near-axis arc input.\n\u00a73+: (this file, subsequent sub-stages) the series bound, the logDeriv\n assembly, the Fexpr circle bound, the Schwarz squeeze, and the\n capstone `partialFraction_identity`.\n\nArchive references: D-068 \u00a75, journal-029.\n-/\nimport Mathlib\nimport Zeta23.RvM.LocalCount\nimport NguyenRH1.LiFEntire\nimport NguyenRH1.LiGapCircles\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 0. Conjugate symmetry -/\n\n/-- The conjugate of a nontrivial zero is a nontrivial zero. -/\nlemma isNontrivialZero_conj {\u03c1 : \u2102} (h\u03c1 : IsNontrivialZero \u03c1) :\n IsNontrivialZero ((starRingEnd \u2102) \u03c1) := by\n refine \u27e8?_, ?_, ?_\u27e9\n \u00b7 rw [_root_.riemannZeta_conj, h\u03c1.1, map_zero]\n \u00b7 rw [Complex.conj_re]; exact h\u03c1.2.1\n \u00b7 rw [Complex.conj_re]; exact h\u03c1.2.2\n\n/-! ## 1. The mult-weighted window count bridges -/\n\n/-- The local-count constant: from Zeta23's RvM local count\n(`zeta_local_zero_count`, [Tit86 Thm 9.2] \u2014 mult-weighted, two-sided,\nall t, PROVED unconditionally in Zeta23). -/\ndef A0count : \u211d := Classical.choose Zeta23.RvM.zeta_local_zero_count\n\nlemma A0count_ge_one : (1 : \u211d) \u2264 A0count :=\n (Classical.choose_spec Zeta23.RvM.zeta_local_zero_count).1\n\nlemma A0count_pos : (0 : \u211d) < A0count := lt_of_lt_of_le one_pos A0count_ge_one\n\n/-- The unit-window bound: Ncount(t, t+1] \u2264 A\u2080\u00b7log(|t|+3). -/\nlemma Ncount_window_le (t : \u211d) :\n (Zeta23.Ncount t (t + 1) : \u211d) \u2264 A0count * Real.log (|t| + 3) :=\n (Classical.choose_spec Zeta23.RvM.zeta_local_zero_count).2 t\n\n/-- Window additivity: Ncount(a, c] = Ncount(a, b] + Ncount(b, c] for\na \u2264 b \u2264 c (the disjoint union of the ordinate windows). -/\nlemma Ncount_add_of_le {a b c : \u211d} (hab : a \u2264 b) (hbc : b \u2264 c) :\n Zeta23.Ncount a c = Zeta23.Ncount a b + Zeta23.Ncount b c := by\n have hunion : Zeta23.zerosIn a c = Zeta23.zerosIn a b \u222a Zeta23.zerosIn b c := by\n ext \u03c1\n constructor\n \u00b7 intro h\u03c1c\n obtain \u27e8h\u03c1, ha, hc\u27e9 := h\u03c1c\n rcases le_or_gt \u03c1.im b with h | h\n \u00b7 exact Set.mem_union_left _ \u27e8h\u03c1, ha, h\u27e9\n \u00b7 exact Set.mem_union_right _ \u27e8h\u03c1, h, hc\u27e9\n \u00b7 intro h\n rcases h with h\u03c1b | h\u03c1c\n \u00b7 obtain \u27e8h\u03c1, ha, hb\u27e9 := h\u03c1b\n exact \u27e8h\u03c1, ha, hb.trans hbc\u27e9\n \u00b7 obtain \u27e8h\u03c1, hb, hc\u27e9 := h\u03c1c\n exact \u27e8h\u03c1, hab.trans_lt hb, hc\u27e9\n have hdisj : Disjoint (Zeta23.zerosIn a b) (Zeta23.zerosIn b c) := by\n rw [Set.disjoint_iff]\n intro \u03c1 \u27e8h1, h2\u27e9\n exact absurd (h1.2.2.trans_lt h2.2.1) (lt_irrefl \u03c1.im)\n rw [Zeta23.Ncount, Zeta23.Ncount, Zeta23.Ncount, hunion]\n exact_mod_cast finsum_mem_union hdisj (Zeta23.zerosIn_finite a b)\n (Zeta23.zerosIn_finite b c)\n\n/-- The N-step unit-window chain: Ncount(a, a+N] \u2264 A\u2080\u00b7N\u00b7log(|a|+N+3). -/\nlemma Ncount_chain_le (a : \u211d) (N : \u2115) :\n (Zeta23.Ncount a (a + (N : \u211d)) : \u211d)\n \u2264 A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3) := by\n induction N with\n | zero =>\n simp only [Nat.cast_zero, add_zero]\n have h0 : Zeta23.zerosIn a a = \u2205 := by\n ext \u03c1\n constructor\n \u00b7 intro h\n obtain \u27e8-, h1, h2\u27e9 := h\n exact absurd (h2.trans_lt h1) (lt_irrefl _)\n \u00b7 intro h\n simp at h\n have hz : Zeta23.Ncount a a = 0 := by\n simp only [Zeta23.Ncount, h0, finsum_mem_empty]\n rw [hz, Nat.cast_zero]\n simp\n | succ N ih =>\n have hN0 : (0 : \u211d) \u2264 (N : \u211d) := Nat.cast_nonneg N\n have hsplit : Zeta23.Ncount a (a + (N : \u211d) + 1)\n = Zeta23.Ncount a (a + (N : \u211d)) + Zeta23.Ncount (a + (N : \u211d)) (a + (N : \u211d) + 1) :=\n Ncount_add_of_le (a := a) (b := a + (N : \u211d)) (c := a + (N : \u211d) + 1)\n (by linarith) (by linarith)\n have hcast : a + ((N + 1 : \u2115) : \u211d) = a + (N : \u211d) + 1 := by push_cast; ring\n rw [hcast, hsplit]\n have hwin := Ncount_window_le (a + (N : \u211d))\n have hlog : Real.log (|a + (N : \u211d)| + 3) \u2264 Real.log (|a| + (N : \u211d) + 3) := by\n apply Real.log_le_log (by positivity)\n have h1 : |a + (N : \u211d)| \u2264 |a| + (N : \u211d) :=\n (abs_add_le _ _).trans (by rw [abs_of_nonneg (show (0 : \u211d) \u2264 (N : \u211d) from Nat.cast_nonneg N)])\n linarith\n have hlogpos : (0 : \u211d) \u2264 Real.log (|a| + (N : \u211d) + 3) := by\n apply Real.log_nonneg\n have h1 : (1 : \u211d) \u2264 |a| + 3 := by linarith [abs_nonneg a]\n linarith [hN0]\n have hlogpos' : (0 : \u211d) \u2264 Real.log (|a| + ((N : \u211d) + 1) + 3) := by\n apply Real.log_nonneg\n have h1 : (1 : \u211d) \u2264 |a| + 3 := by linarith [abs_nonneg a]\n linarith [hN0]\n push_cast\n -- goal: \u2191(Ncount a (a+\u2191N)) + \u2191(Ncount (a+\u2191N) (a+\u2191N+1)) \u2264 A\u2080(\u2191N+1)\u00b7log(|a|+(\u2191N+1)+3)\n have hlogC : Real.log (|a| + (N : \u211d) + 3)\n \u2264 Real.log (|a| + ((N : \u211d) + 1) + 3) :=\n Real.log_le_log (by positivity) (by linarith)\n have hle1 : A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3)\n \u2264 A0count * (N : \u211d) * Real.log (|a| + ((N : \u211d) + 1) + 3) :=\n mul_le_mul_of_nonneg_left hlogC (mul_nonneg A0count_pos.le hN0)\n have hle2 : A0count * Real.log (|a + (N : \u211d)| + 3)\n \u2264 A0count * Real.log (|a| + ((N : \u211d) + 1) + 3) := by\n apply mul_le_mul_of_nonneg_left _ A0count_pos.le\n apply Real.log_le_log (by positivity)\n have h1 : |a + (N : \u211d)| \u2264 |a| + (N : \u211d) :=\n (abs_add_le _ _).trans (by rw [abs_of_nonneg (show (0 : \u211d) \u2264 (N : \u211d) from Nat.cast_nonneg N)])\n linarith\n have hsum : \u2191(Zeta23.Ncount a (a + (N : \u211d))) + \u2191(Zeta23.Ncount (a + (N : \u211d)) (a + (N : \u211d) + 1))\n \u2264 A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3)\n + A0count * Real.log (|a + (N : \u211d)| + 3) :=\n add_le_add ih hwin\n linarith [hsum, hle1, hle2]\n\n/-- The dyadic-interval count: Ncount(a, b] \u2264 A\u2080\u00b7(b\u2212a+1)\u00b7log(b+4) for\n0 \u2264 a \u2264 b (the unit-window chain + the ceil cover). -/\nlemma Ncount_le_of_interval {a b : \u211d} (ha : 0 \u2264 a) (hab : a \u2264 b) :\n (Zeta23.Ncount a b : \u211d) \u2264 A0count * (b - a + 1) * Real.log (b + 4) := by\n set N : \u2115 := \u2308b - a\u2309\u208a with hN\n have hNb : b \u2264 a + (N : \u211d) := by\n have h := Nat.le_ceil (b - a)\n rw [\u2190 hN] at h\n have h2 : (b - a : \u211d) \u2264 (N : \u211d) := h\n linarith\n have hsub : Zeta23.zerosIn a b \u2286 Zeta23.zerosIn a (a + (N : \u211d)) := by\n intro \u03c1 h\u03c1\n refine \u27e8h\u03c1.1, h\u03c1.2.1, ?_\u27e9\n exact h\u03c1.2.2.trans hNb\n have hle : Zeta23.Ncount a b \u2264 Zeta23.Ncount a (a + (N : \u211d)) := by\n rw [Zeta23.Ncount, Zeta23.Ncount]\n rw [finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a b),\n finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a (a + (N : \u211d)))]\n apply Finset.sum_le_sum_of_subset_of_nonneg\n \u00b7 intro \u03c1 h\u03c1\n rw [Set.Finite.mem_toFinset] at h\u03c1 \u22a2\n exact hsub h\u03c1\n \u00b7 intro \u03c1 _ _\n exact Nat.cast_nonneg _\n have hchain := Ncount_chain_le a N\n have hNle : (N : \u211d) \u2264 b - a + 1 := by\n have h := Nat.ceil_lt_add_one (sub_nonneg.mpr hab)\n rw [\u2190 hN] at h\n have h2 : ((N : \u211d)) < b - a + 1 := by exact_mod_cast h\n linarith\n have hlogle : Real.log (|a| + (N : \u211d) + 3) \u2264 Real.log (b + 4) := by\n apply Real.log_le_log (by positivity)\n rw [abs_of_nonneg ha]\n linarith\n calc (Zeta23.Ncount a b : \u211d)\n \u2264 (Zeta23.Ncount a (a + (N : \u211d)) : \u211d) := by exact_mod_cast hle\n _ \u2264 A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3) := hchain\n _ \u2264 A0count * (b - a + 1) * Real.log (b + 4) := by\n have hlogpos : (0 : \u211d) \u2264 Real.log (b + 4) := by\n apply Real.log_nonneg; linarith\n have h1 : A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3)\n \u2264 A0count * (N : \u211d) * Real.log (b + 4) :=\n mul_le_mul_of_nonneg_left hlogle (mul_nonneg A0count_pos.le (Nat.cast_nonneg N))\n have h2 : A0count * (N : \u211d) * Real.log (b + 4)\n \u2264 A0count * (b - a + 1) * Real.log (b + 4) := by\n apply mul_le_mul_of_nonneg_right _ hlogpos\n nlinarith [A0count_pos, hNle, sub_nonneg.mpr hab]\n exact h1.trans h2\n\n/-! ## 2. The Dirichlet bound -/\n\n/-- The Dirichlet majorant constant: the absolute sum of the von\nMangoldt L-series terms at s = 2 (summable since 2 > 1). -/\ndef dirichletC2 : \u211d :=\n \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 C\u2082 for re s \u2265 2 (the Dirichlet-series majorant:\n\u2212\u03b6\u2032/\u03b6 = LSeries \u039b for re > 1, termwise \u2264 the s = 2 sum). -/\nlemma norm_logDeriv_zeta_le_dirichlet {s : \u2102} (hs : 2 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 dirichletC2 := by\n have hs1 : (1 : \u211d) < s.re := by linarith\n have h1 : -logDeriv riemannZeta s\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsum2 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016\n \u2264 dirichletC2 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n exact hs) n) hsumS hsum2\n _ = dirichletC2 := rfl\n calc \u2016logDeriv riemannZeta s\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 dirichletC2 := hle\n\n/-! ## 3. The tsum split + the generalized interval count + the low-zero\nconstant -/\n\n/-- Splitting a summable real tsum over a finset:\n\u2211' f = \u2211_{i \u2208 s} f i + \u2211' (if i \u2208 s then 0 else f i). -/\nlemma tsum_split_finset {\u03b9 : Type*} [DecidableEq \u03b9] {f : \u03b9 \u2192 \u211d} (hf : Summable f)\n (s : Finset \u03b9) :\n \u2211' i, f i = \u2211 i \u2208 s, f i + \u2211' i, (if i \u2208 s then 0 else f i) := by\n classical\n have h1 : Summable (fun i => if i \u2208 s then f i else 0) := by\n apply Summable.of_norm_bounded hf.norm\n intro i\n by_cases hi : i \u2208 s <;> simp [hi]\n have h2 : Summable (fun i => if i \u2208 s then 0 else f i) := by\n apply Summable.of_norm_bounded hf.norm\n intro i\n by_cases hi : i \u2208 s <;> simp [hi]\n have h3 : \u2211' i, f i\n = \u2211' i, ((if i \u2208 s then f i else 0) + (if i \u2208 s then 0 else f i)) := by\n apply tsum_congr\n intro i\n by_cases hi : i \u2208 s <;> simp [hi]\n rw [h3, Summable.tsum_add h1 h2]\n congr 1\n rw [tsum_eq_sum (s := s) (fun b hb => if_neg hb)]\n exact Finset.sum_congr rfl (fun b hb => if_pos hb)\n\n/-- The generalized dyadic-interval count (both signs):\nNcount(a, b] \u2264 A\u2080\u00b7(b\u2212a+1)\u00b7log(3\u00b7max|a||b| + 5). -/\nlemma Ncount_le_of_interval_gen {a b : \u211d} (hab : a \u2264 b) :\n (Zeta23.Ncount a b : \u211d)\n \u2264 A0count * (b - a + 1) * Real.log (3 * max |a| |b| + 5) := by\n set N : \u2115 := \u2308b - a\u2309\u208a with hN\n have hNb : b \u2264 a + (N : \u211d) := by\n have h := Nat.le_ceil (b - a)\n rw [\u2190 hN] at h\n linarith\n have hsub : Zeta23.zerosIn a b \u2286 Zeta23.zerosIn a (a + (N : \u211d)) := by\n intro \u03c1 h\u03c1\n exact \u27e8h\u03c1.1, h\u03c1.2.1, h\u03c1.2.2.trans hNb\u27e9\n have hle : Zeta23.Ncount a b \u2264 Zeta23.Ncount a (a + (N : \u211d)) := by\n rw [Zeta23.Ncount, Zeta23.Ncount]\n rw [finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a b),\n finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a (a + (N : \u211d)))]\n apply Finset.sum_le_sum_of_subset_of_nonneg\n \u00b7 intro \u03c1 h\u03c1\n rw [Set.Finite.mem_toFinset] at h\u03c1 \u22a2\n exact hsub h\u03c1\n \u00b7 intro \u03c1 _ _\n exact Nat.cast_nonneg _\n have hchain := Ncount_chain_le a N\n have hNle : (N : \u211d) \u2264 b - a + 1 := by\n have h := Nat.ceil_lt_add_one (sub_nonneg.mpr hab)\n rw [\u2190 hN] at h\n have h2 : ((N : \u211d)) < b - a + 1 := by exact_mod_cast h\n linarith\n have hlogle : Real.log (|a| + (N : \u211d) + 3)\n \u2264 Real.log (3 * max |a| |b| + 5) := by\n apply Real.log_le_log (by positivity)\n have hM1 : |a| \u2264 max |a| |b| := le_max_left _ _\n have hM2 : |b| \u2264 max |a| |b| := le_max_right _ _\n have hba : b - a \u2264 |b| + |a| := by\n have h1 := le_abs_self b\n have h2 : -a \u2264 |a| := by rw [\u2190 abs_neg a]; exact le_abs_self (-a)\n linarith\n nlinarith [hNle, hM1, hM2, hba, abs_nonneg a, abs_nonneg b]\n have hlogpos : (0 : \u211d) \u2264 Real.log (3 * max |a| |b| + 5) := by\n apply Real.log_nonneg\n have h1 : (0 : \u211d) \u2264 max |a| |b| := le_trans (abs_nonneg a) (le_max_left _ _)\n linarith\n calc (Zeta23.Ncount a b : \u211d)\n \u2264 (Zeta23.Ncount a (a + (N : \u211d)) : \u211d) := by exact_mod_cast hle\n _ \u2264 A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3) := hchain\n _ \u2264 A0count * (b - a + 1) * Real.log (3 * max |a| |b| + 5) := by\n have h1 : A0count * (N : \u211d) * Real.log (|a| + (N : \u211d) + 3)\n \u2264 A0count * (N : \u211d) * Real.log (3 * max |a| |b| + 5) :=\n mul_le_mul_of_nonneg_left hlogle (mul_nonneg A0count_pos.le (Nat.cast_nonneg N))\n have h2 : A0count * (N : \u211d) * Real.log (3 * max |a| |b| + 5)\n \u2264 A0count * (b - a + 1) * Real.log (3 * max |a| |b| + 5) := by\n apply mul_le_mul_of_nonneg_right _ hlogpos\n nlinarith [A0count_pos, hNle, sub_nonneg.mpr hab]\n exact h1.trans h2\n\n/-- The low-zero finset: nontrivial zeros with \u2016\u03c1\u2016 \u2264 3. -/\ndef lowZeros : Finset {\u03c1 : \u2102 // IsNontrivialZero \u03c1} :=\n (show {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | \u2016(\u03c1 : \u2102)\u2016 \u2264 3}.Finite from\n (finite_subtype_norm_lt 4).subset\n (fun \u03c1 h\u03c1 => lt_of_le_of_lt (show \u2016(\u03c1 : \u2102)\u2016 \u2264 3 from h\u03c1) (by norm_num))).toFinset\n\n/-- The low-zero symbolic constant (a fixed finite sum of fixed positive\nreals; \u2016\u03c1\u2016 > 0 for nontrivial \u03c1 since re \u03c1 > 0). -/\ndef lowZeroConst : \u211d :=\n \u2211 \u03c1 \u2208 lowZeros, (zeroMult (\u03c1 : \u2102) : \u211d) * (1 + 1 / \u2016(\u03c1 : \u2102)\u2016)\n\nlemma lowZeroConst_nonneg : 0 \u2264 lowZeroConst :=\n Finset.sum_nonneg fun _\u03c1 _ => mul_nonneg (Nat.cast_nonneg _)\n (add_nonneg one_pos.le (div_nonneg one_pos.le (norm_nonneg _)))\n\n/-! ## 4. The far-tail bound -/\n\n/-- The dyadic log series closed form:\n\u2211' j, ((j+4)\u00b7log 2 + log X)/2^j = 10\u00b7log 2 + 2\u00b7log X. -/\nlemma tsum_dyadic_log (X : \u211d) :\n \u2211' j : \u2115, ((j + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j\n = 10 * Real.log 2 + 2 * Real.log X := by\n have hgeom2H : Summable (fun j : \u2115 => ((1 : \u211d) / 2) ^ j) := summable_geometric_two\n have hgeom2 : \u2211' j : \u2115, ((1 : \u211d) / 2) ^ j = 2 := tsum_geometric_two\n have hlinH := hasSum_coe_mul_geometric_of_norm_lt_one (r := (1 : \u211d) / 2) (by norm_num)\n have hlin : \u2211' j : \u2115, (j : \u211d) * ((1 : \u211d) / 2) ^ j = 2 := by\n have h := hlinH.tsum_eq\n norm_num at h\n exact h\n have hsplit : (fun j : \u2115 => ((j + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j)\n = fun j : \u2115 => Real.log 2 * ((j : \u211d) * ((1 : \u211d) / 2) ^ j)\n + (4 * Real.log 2 + Real.log X) * ((1 : \u211d) / 2) ^ j := by\n funext j\n have hj : (2 : \u211d) ^ j \u2260 0 := pow_ne_zero _ two_ne_zero\n rw [one_div_pow]\n field_simp\n ring\n rw [hsplit, Summable.tsum_add (hlinH.summable.mul_left _) (hgeom2H.mul_left _)]\n rw [tsum_mul_left, tsum_mul_left, hlin, hgeom2]\n ring\n\n/-- log 2 \u2264 1 (since 2 \u2264 e). -/\nlemma log_two_le_one : Real.log 2 \u2264 1 := by\n rw [show (1 : \u211d) = Real.log (Real.exp 1) by rw [Real.log_exp]]\n apply Real.log_le_log (by norm_num)\n have h := Real.exp_one_gt_d9\n linarith\n\n/-- For X \u2265 200: (4A\u2080/X)\u00b7(10\u00b7log 2 + 2\u00b7log X) \u2264 16\u00b7A\u2080\u00b7log X / X. -/\nlemma dyadic_tail_constant {X : \u211d} (hX : 200 \u2264 X) :\n (4 * A0count / X) * (10 * Real.log 2 + 2 * Real.log X)\n \u2264 16 * A0count * Real.log X / X := by\n have hX0 : (0 : \u211d) < X := by linarith\n have hlogX : 5 \u2264 Real.log X := by\n rw [show (5 : \u211d) = Real.log (Real.exp 5) by rw [Real.log_exp]]\n apply Real.log_le_log (Real.exp_pos _)\n have h1 : Real.exp 5 = (Real.exp 1) ^ 5 := by\n rw [\u2190 Real.exp_nat_mul]\n norm_num\n have h2 := Real.exp_one_lt_d9.le\n have h3 : (Real.exp 1) ^ 5 \u2264 (2.7182818286 : \u211d) ^ 5 :=\n pow_le_pow_left\u2080 (Real.exp_pos 1).le h2 5\n have h4 : (2.7182818286 : \u211d) ^ 5 \u2264 200 := by norm_num\n linarith [h1, h3, h4]\n have hlog2 := log_two_le_one\n have hA0 : (0 : \u211d) \u2264 4 * A0count / X :=\n div_nonneg (mul_nonneg (by norm_num) A0count_pos.le) hX0.le\n have h1 : 10 * Real.log 2 + 2 * Real.log X \u2264 4 * Real.log X := by linarith\n calc (4 * A0count / X) * (10 * Real.log 2 + 2 * Real.log X)\n \u2264 (4 * A0count / X) * (4 * Real.log X) := mul_le_mul_of_nonneg_left h1 hA0\n _ = 16 * A0count * Real.log X / X := by ring\n\n/-- The dyadic class of a zero relative to scale X:\nNat.log 2 \u230a|im \u03c1|/X\u230b\u208a, so 2^j\u00b7X \u2264 |im \u03c1| < 2^{j+1}\u00b7X when |im \u03c1| \u2265 X. -/\ndef dyadClass (X : \u211d) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2115 :=\n Nat.log 2 \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a\n\nlemma dyadClass_lower {X : \u211d} (hX : 0 < X) {\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}}\n (h\u03c1 : X \u2264 |(\u03c1 : \u2102).im|) :\n (2 : \u211d) ^ dyadClass X \u03c1 * X \u2264 |(\u03c1 : \u2102).im| := by\n have hfl : \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a \u2260 0 := by\n rw [Ne, Nat.floor_eq_zero]\n push Not\n have h1 : (1 : \u211d) \u2264 |(\u03c1 : \u2102).im| / X := (le_div_iff\u2080 hX).mpr (by linarith)\n linarith\n have h2 := Nat.pow_log_le_self 2 hfl\n have h3 : ((2 ^ dyadClass X \u03c1 : \u2115) : \u211d) \u2264 \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a := by exact_mod_cast h2\n have h4 : (\u230a|(\u03c1 : \u2102).im| / X\u230b\u208a : \u211d) \u2264 |(\u03c1 : \u2102).im| / X := Nat.floor_le (by positivity)\n have h5 : ((2 ^ dyadClass X \u03c1 : \u2115) : \u211d) = (2 : \u211d) ^ dyadClass X \u03c1 := by push_cast; ring\n rw [h5] at h3\n rw [le_div_iff\u2080 hX] at h4\n have h6 : (2 : \u211d) ^ dyadClass X \u03c1 * X \u2264 \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a * X :=\n mul_le_mul_of_nonneg_right h3 hX.le\n linarith\n\nlemma dyadClass_upper {X : \u211d} (hX : 0 < X) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n |(\u03c1 : \u2102).im| < (2 : \u211d) ^ (dyadClass X \u03c1 + 1) * X := by\n have h2 : \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a < 2 ^ (dyadClass X \u03c1 + 1) :=\n Nat.lt_pow_succ_log_self (by norm_num) _\n have h3 : (\u230a|(\u03c1 : \u2102).im| / X\u230b\u208a : \u211d) + 1 \u2264 (2 : \u211d) ^ (dyadClass X \u03c1 + 1) := by\n have h4 : \u230a|(\u03c1 : \u2102).im| / X\u230b\u208a + 1 \u2264 2 ^ (dyadClass X \u03c1 + 1) := h2\n have h5 : ((\u230a|(\u03c1 : \u2102).im| / X\u230b\u208a + 1 : \u2115) : \u211d) \u2264 ((2 ^ (dyadClass X \u03c1 + 1) : \u2115) : \u211d) := by\n exact_mod_cast h4\n push_cast at h5\n linarith\n have h6 : |(\u03c1 : \u2102).im| / X < (2 : \u211d) ^ (dyadClass X \u03c1 + 1) :=\n (Nat.lt_floor_add_one _).trans_le h3\n rw [div_lt_iff\u2080 hX] at h6\n linarith\n\n/-- The tail summand (if-form on the zero subtype). -/\ndef tailTerm (X : \u211d) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u211d :=\n if X < |(\u03c1 : \u2102).im| then (zeroMult \u03c1 : \u211d) / (1 + ((\u03c1 : \u2102).im) ^ 2) else 0\n\nlemma tailTerm_nonneg {X : \u211d} (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n 0 \u2264 tailTerm X \u03c1 := by\n by_cases h : X < |(\u03c1 : \u2102).im| <;>\n simp [tailTerm, h, div_nonneg (Nat.cast_nonneg _)\n (by positivity : (0 : \u211d) \u2264 1 + ((\u03c1 : \u2102).im) ^ 2)]\n\n/-- The tail summand is summable (dominated by 2\u00b7m/(1+\u2016\u03b3\u2016\u00b2) via\n1 + \u2016\u03b3\u2016\u00b2 \u2264 2(1 + im\u00b2)). -/\nlemma tailTerm_summable (X : \u211d) :\n Summable (tailTerm X) := by\n have hbase : Summable (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n 2 * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102))))) :=\n (WeilEF.zero_sum_inv_sq zetaSeam).mul_left 2\n apply Summable.of_norm_bounded hbase\n intro \u03c1\n rw [tailTerm]\n by_cases h : X < |(\u03c1 : \u2102).im|\n \u00b7 rw [if_pos h]\n have hgam : Complex.normSq (gammaOf (\u03c1 : \u2102))\n = (\u03c1 : \u2102).im ^ 2 + (1 / 2 - (\u03c1 : \u2102).re) ^ 2 := by\n rw [Complex.normSq_apply, WeilEF.gammaOf_re, WeilEF.gammaOf_im]\n ring\n have hcmp : 1 + Complex.normSq (gammaOf (\u03c1 : \u2102)) \u2264 2 * (1 + (\u03c1 : \u2102).im ^ 2) := by\n rw [hgam]\n have hre : ((\u03c1 : \u2102).re - 1 / 2) ^ 2 \u2264 1 / 4 := by\n have h0 := \u03c1.2.2.1\n have h1 := \u03c1.2.2.2\n nlinarith [sq_nonneg ((\u03c1 : \u2102).re - 1 / 2)]\n nlinarith [sq_nonneg (\u03c1 : \u2102).im]\n have hpos1 : (0 : \u211d) < 1 + Complex.normSq (gammaOf (\u03c1 : \u2102)) := by\n have h := Complex.normSq_nonneg (gammaOf (\u03c1 : \u2102)); linarith\n have hpos2 : (0 : \u211d) < 1 + (\u03c1 : \u2102).im ^ 2 := by positivity\n have hfrac : 1 / (1 + (\u03c1 : \u2102).im ^ 2) \u2264 2 / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102))) := by\n rw [div_le_div_iff\u2080 hpos2 hpos1]\n nlinarith [hcmp]\n have hm : (0 : \u211d) \u2264 (zeroMult \u03c1 : \u211d) := Nat.cast_nonneg _\n calc \u2016(zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2)\u2016\n = (zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2) := by\n rw [Real.norm_eq_abs, abs_of_nonneg (div_nonneg (Nat.cast_nonneg _) hpos2.le)]\n _ \u2264 2 * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))) := by\n rw [show (zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2)\n = (zeroMult \u03c1 : \u211d) * (1 / (1 + (\u03c1 : \u2102).im ^ 2)) from by ring]\n rw [show 2 * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102))))\n = (zeroMult \u03c1 : \u211d) * (2 / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))) from by ring]\n exact mul_le_mul_of_nonneg_left hfrac hm\n \u00b7 rw [if_neg h, norm_zero]\n have hnn := Complex.normSq_nonneg (gammaOf (\u03c1 : \u2102))\n exact mul_nonneg (by norm_num) (div_nonneg (Nat.cast_nonneg _) (by linarith))\n\n/-- The dyadic log series is summable. -/\nlemma dyadicLog_summable (X : \u211d) :\n Summable (fun j : \u2115 => ((j + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j) := by\n have hlinH := hasSum_coe_mul_geometric_of_norm_lt_one (r := (1 : \u211d) / 2) (by norm_num)\n have hsplit : (fun j : \u2115 => ((j + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j)\n = fun j : \u2115 => Real.log 2 * ((j : \u211d) * ((1 : \u211d) / 2) ^ j)\n + (4 * Real.log 2 + Real.log X) * ((1 : \u211d) / 2) ^ j := by\n funext j\n have hj : (2 : \u211d) ^ j \u2260 0 := pow_ne_zero _ two_ne_zero\n rw [one_div_pow]\n field_simp\n ring\n rw [hsplit]\n exact (hlinH.summable.mul_left _).add (summable_geometric_two.mul_left _)\n\n/-- The dyadic fiber {dyadClass X \u03c1 = j} is finite. -/\nlemma dyadClass_fiber_finite {X : \u211d} (hX : 0 < X) (j : \u2115) :\n {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j}.Finite := by\n apply (finite_subtype_norm_lt (1 + (2 : \u211d) ^ (j + 1) * X + 1)).subset\n intro \u03c1 h\u03c1\n have h\u03c1j : dyadClass X \u03c1 = j := h\u03c1\n have hup := dyadClass_upper hX \u03c1\n rw [h\u03c1j] at hup\n have hnorm : \u2016(\u03c1 : \u2102)\u2016 \u2264 |(\u03c1 : \u2102).re| + |(\u03c1 : \u2102).im| :=\n Complex.norm_le_abs_re_add_abs_im _\n have hre : |(\u03c1 : \u2102).re| \u2264 1 := by\n rw [abs_le]\n exact \u27e8by linarith [\u03c1.2.2.1], by linarith [\u03c1.2.2.2]\u27e9\n have h2pow : (0 : \u211d) \u2264 (2 : \u211d) ^ (j + 1) * X := by positivity\n show \u2016(\u03c1 : \u2102)\u2016 < 1 + (2 : \u211d) ^ (j + 1) * X + 1\n linarith\n\n/-- The fiber count: the zeros in the dyadic fiber with X < |\u03b3| have\nmultiplicity-sum \u2264 A\u2080\u00b7(2^{j+1}X+3)\u00b7log(3\u00b72^{j+1}X+5). -/\nlemma fiber_mult_count {X : \u211d} (hX : 200 \u2264 X) (j : \u2115) :\n \u2211 \u03c1 \u2208 (Set.Finite.subset (dyadClass_fiber_finite (by linarith : (0 : \u211d) < X) j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset,\n (zeroMult (\u03c1 : \u2102) : \u211d)\n \u2264 A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n classical\n have hX0 : (0 : \u211d) < X := by linarith\n have h2jX : (0 : \u211d) < (2 : \u211d) ^ j * X := by positivity\n have h2j1X : (0 : \u211d) < (2 : \u211d) ^ (j + 1) * X := by positivity\n set FS := (Set.Finite.subset (dyadClass_fiber_finite (by linarith : (0 : \u211d) < X) j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset with hFS\n have hmemFS : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n \u03c1 \u2208 FS \u2194 dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im| := by\n intro \u03c1\n rw [hFS, Set.Finite.mem_toFinset]\n exact Iff.rfl\n have hW1 : (Zeta23.zerosIn ((2 : \u211d) ^ j * X - 1) ((2 : \u211d) ^ (j + 1) * X)).Finite :=\n Zeta23.zerosIn_finite _ _\n have hW2 : (Zeta23.zerosIn (-((2 : \u211d) ^ (j + 1) * X)) (-((2 : \u211d) ^ j * X))).Finite :=\n Zeta23.zerosIn_finite _ _\n have himg : FS.image (Subtype.val) \u2286 hW1.toFinset \u222a hW2.toFinset := by\n intro z hz\n obtain \u27e8\u03c1, h\u03c1, rfl\u27e9 := Finset.mem_image.mp hz\n rw [hmemFS] at h\u03c1\n obtain \u27e8h\u03c1j, hgt\u27e9 := h\u03c1\n have hup : |(\u03c1 : \u2102).im| < (2 : \u211d) ^ (j + 1) * X := by\n have h := dyadClass_upper hX0 \u03c1\n rw [h\u03c1j] at h\n exact h\n have hlo : (2 : \u211d) ^ j * X \u2264 |(\u03c1 : \u2102).im| := by\n have h := dyadClass_lower hX0 (show X \u2264 |(\u03c1 : \u2102).im| from hgt.le)\n rw [h\u03c1j] at h\n exact h\n rcases le_or_gt 0 (\u03c1 : \u2102).im with him | him\n \u00b7 apply Finset.mem_union_left\n rw [Set.Finite.mem_toFinset]\n refine \u27e8\u03c1.2, ?_, ?_\u27e9\n \u00b7 rw [abs_of_nonneg him] at hlo\n linarith\n \u00b7 rw [abs_of_nonneg him] at hup\n linarith\n \u00b7 apply Finset.mem_union_right\n rw [Set.Finite.mem_toFinset]\n refine \u27e8\u03c1.2, ?_, ?_\u27e9\n \u00b7 rw [abs_of_neg him] at hup\n linarith\n \u00b7 rw [abs_of_neg him] at hlo\n linarith\n have hsum_img : \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n = \u2211 z \u2208 FS.image (Subtype.val), (zeroMult z : \u211d) := by\n rw [Finset.sum_image (fun x _ y _ h => Subtype.val_injective h)]\n have hdisj : Disjoint hW1.toFinset hW2.toFinset := by\n rw [Finset.disjoint_left]\n intro z hz1 hz2\n rw [Set.Finite.mem_toFinset] at hz1 hz2\n have h1 : (2 : \u211d) ^ j * X - 1 < z.im := hz1.2.1\n have h2 : z.im \u2264 -((2 : \u211d) ^ j * X) := hz2.2.2\n have hge : (1 : \u211d) \u2264 (2 : \u211d) ^ j * X := by\n have h1le : (1 : \u211d) \u2264 (2 : \u211d) ^ j := by exact_mod_cast Nat.one_le_two_pow\n nlinarith [h1le, hX]\n linarith\n calc \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n = \u2211 z \u2208 FS.image (Subtype.val), (zeroMult z : \u211d) := hsum_img\n _ \u2264 \u2211 z \u2208 hW1.toFinset \u222a hW2.toFinset, (zeroMult z : \u211d) :=\n Finset.sum_le_sum_of_subset_of_nonneg himg (fun z _ _ => Nat.cast_nonneg _)\n _ = \u2211 z \u2208 hW1.toFinset, (zeroMult z : \u211d) + \u2211 z \u2208 hW2.toFinset, (zeroMult z : \u211d) :=\n Finset.sum_union hdisj\n _ = (Zeta23.Ncount ((2 : \u211d) ^ j * X - 1) ((2 : \u211d) ^ (j + 1) * X) : \u211d)\n + (Zeta23.Ncount (-((2 : \u211d) ^ (j + 1) * X)) (-((2 : \u211d) ^ j * X)) : \u211d) := by\n rw [Zeta23.Ncount, Zeta23.Ncount, finsum_mem_eq_finite_toFinset_sum _ hW1,\n finsum_mem_eq_finite_toFinset_sum _ hW2, Nat.cast_sum, Nat.cast_sum]\n _ \u2264 A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n have h2jX1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j * X := by\n have h1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j := by exact_mod_cast Nat.one_le_two_pow\n nlinarith [h1, hX]\n have hpowX : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by\n rw [pow_succ']; ring\n have hlogpos : (0 : \u211d) \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_nonneg\n linarith [h2j1X]\n have h1 := Ncount_le_of_interval_gen (a := (2 : \u211d) ^ j * X - 1)\n (b := (2 : \u211d) ^ (j + 1) * X) (by linarith [hpowX, h2jX1])\n have h2 := Ncount_le_of_interval_gen (a := -((2 : \u211d) ^ (j + 1) * X))\n (b := -((2 : \u211d) ^ j * X)) (by linarith [hpowX, h2jX1])\n -- the log arguments match L = log(3\u00b72^{j+1}X + 5)\n have hlog1 : Real.log (3 * max |(2 : \u211d) ^ j * X - 1| |(2 : \u211d) ^ (j + 1) * X| + 5)\n \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |(2 : \u211d) ^ j * X - 1| |(2 : \u211d) ^ (j + 1) * X|\n \u2264 (2 : \u211d) ^ (j + 1) * X := by\n apply max_le\n \u00b7 rw [abs_of_nonneg (by linarith [h2jX1])]\n nlinarith [h2jX, h2j1X]\n \u00b7 rw [abs_of_nonneg h2j1X.le]\n linarith [hmax]\n have hlog2 : Real.log (3 * max |-((2 : \u211d) ^ (j + 1) * X)| |-((2 : \u211d) ^ j * X)| + 5)\n \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_le_log (by positivity)\n have hmax : max |-((2 : \u211d) ^ (j + 1) * X)| |-((2 : \u211d) ^ j * X)|\n \u2264 (2 : \u211d) ^ (j + 1) * X := by\n apply max_le\n \u00b7 rw [abs_neg, abs_of_nonneg h2j1X.le]\n \u00b7 rw [abs_neg, abs_of_nonneg h2jX.le]\n nlinarith [h2jX, h2j1X]\n linarith [hmax]\n -- the lengths: b\u2212a+1 = 2^jX + 2 and 2^jX + 1\n have hlen1 : (2 : \u211d) ^ (j + 1) * X - ((2 : \u211d) ^ j * X - 1) + 1\n = (2 : \u211d) ^ j * X + 2 := by\n have hpow : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n rw [hpow]\n ring\n have hlen2 : -((2 : \u211d) ^ j * X) - (-((2 : \u211d) ^ (j + 1) * X)) + 1\n = (2 : \u211d) ^ j * X + 1 := by\n have hpow : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n rw [hpow]\n ring\n rw [hlen1] at h1\n rw [hlen2] at h2\n -- assemble: A\u2080(2^jX+2)L + A\u2080(2^jX+1)L = A\u2080(2^{j+1}X+3)L\n have hA0 := A0count_pos.le\n have h1' : (Zeta23.Ncount ((2 : \u211d) ^ j * X - 1) ((2 : \u211d) ^ (j + 1) * X) : \u211d)\n \u2264 A0count * ((2 : \u211d) ^ j * X + 2) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) :=\n h1.trans (mul_le_mul_of_nonneg_left hlog1 (by positivity))\n have h2' : (Zeta23.Ncount (-((2 : \u211d) ^ (j + 1) * X)) (-((2 : \u211d) ^ j * X)) : \u211d)\n \u2264 A0count * ((2 : \u211d) ^ j * X + 1) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) :=\n h2.trans (mul_le_mul_of_nonneg_left hlog2 (by positivity))\n have hpow : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by rw [pow_succ']; ring\n rw [hpow] at h1' h2' \u22a2\n linarith [h1', h2', hA0, hlogpos]\n\n/-- The per-fiber tail bound (in \u211d\u22650\u221e): the sum of ofReal(tailTerm X)\nover the dyadic fiber (dyadClass X)\u207b\u00b9'{j} is at most ofReal of\n(1/(2^j X)\u00b2)\u00b7A\u2080\u00b7(2^{j+1}X+3)\u00b7log(3\u00b72^{j+1}X+5). -/\nlemma fiber_tail_le {X : \u211d} (hX : 200 \u2264 X) (j : \u2115) :\n \u2211' b : (dyadClass X) \u207b\u00b9' {j}, ENNReal.ofReal (tailTerm X b.1)\n \u2264 ENNReal.ofReal ((1 / ((2 : \u211d) ^ j * X) ^ 2) *\n (A0count * ((2 : \u211d) ^ (j + 1) * X + 3)\n * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5))) := by\n classical\n have hX0 : (0 : \u211d) < X := by linarith\n have hfin : ((dyadClass X) \u207b\u00b9' {j} : Set {\u03c1 : \u2102 // IsNontrivialZero \u03c1}).Finite := by\n have h := dyadClass_fiber_finite hX0 j\n have h2 : ((dyadClass X) \u207b\u00b9' {j} : Set {\u03c1 : \u2102 // IsNontrivialZero \u03c1})\n = {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j} := by\n ext \u03c1\n exact Iff.rfl\n rw [h2]\n exact h\n have htsum : \u2211' b : (dyadClass X) \u207b\u00b9' {j}, ENNReal.ofReal (tailTerm X b.1)\n = \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n ((dyadClass X) \u207b\u00b9' {j}).indicator (fun \u03c1 => ENNReal.ofReal (tailTerm X \u03c1)) \u03c1 :=\n tsum_subtype ((dyadClass X) \u207b\u00b9' {j})\n (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} => ENNReal.ofReal (tailTerm X \u03c1))\n rw [htsum]\n rw [tsum_eq_sum (s := hfin.toFinset) (fun \u03c1 h\u03c1 => by\n apply Set.indicator_of_notMem\n rwa [Set.Finite.mem_toFinset] at h\u03c1)]\n rw [Finset.sum_congr rfl (fun \u03c1 h\u03c1 =>\n Set.indicator_of_mem (by rwa [Set.Finite.mem_toFinset] at h\u03c1) _)]\n rw [\u2190 ENNReal.ofReal_sum_of_nonneg (fun \u03c1 _ => tailTerm_nonneg \u03c1)]\n apply ENNReal.ofReal_le_ofReal\n rw [\u2190 Finset.sum_filter_add_sum_filter_not hfin.toFinset\n (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} => X < |(\u03c1 : \u2102).im|) (tailTerm X)]\n have hzero : \u2211 \u03c1 \u2208 hfin.toFinset.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} => \u00ac X < |(\u03c1 : \u2102).im|), tailTerm X \u03c1 = 0 := by\n apply Finset.sum_eq_zero\n intro \u03c1 h\u03c1\n rw [Finset.mem_filter] at h\u03c1\n rw [tailTerm, if_neg h\u03c1.2]\n rw [hzero, add_zero]\n have hfilter_eq : hfin.toFinset.filter (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} => X < |(\u03c1 : \u2102).im|)\n = (Set.Finite.subset (dyadClass_fiber_finite hX0 j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset := by\n ext \u03c1\n simp only [Finset.mem_filter, Set.Finite.mem_toFinset]\n exact Iff.rfl\n rw [hfilter_eq]\n calc \u2211 \u03c1 \u2208 (Set.Finite.subset (dyadClass_fiber_finite hX0 j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset, tailTerm X \u03c1\n \u2264 \u2211 \u03c1 \u2208 (Set.Finite.subset (dyadClass_fiber_finite hX0 j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset,\n (1 / ((2 : \u211d) ^ j * X) ^ 2) * (zeroMult (\u03c1 : \u2102) : \u211d) := by\n apply Finset.sum_le_sum\n intro \u03c1 h\u03c1\n rw [Set.Finite.mem_toFinset] at h\u03c1\n obtain \u27e8h\u03c1j, hgt\u27e9 := h\u03c1\n have hlo : (2 : \u211d) ^ j * X \u2264 |(\u03c1 : \u2102).im| := by\n have h := dyadClass_lower hX0 (show X \u2264 |(\u03c1 : \u2102).im| from hgt.le)\n rw [h\u03c1j] at h\n exact h\n have hlo2 : ((2 : \u211d) ^ j * X) ^ 2 \u2264 ((\u03c1 : \u2102).im) ^ 2 := by\n have h := pow_le_pow_left\u2080 (by positivity : (0 : \u211d) \u2264 (2 : \u211d) ^ j * X) hlo 2\n rwa [sq_abs] at h\n have hsq : ((2 : \u211d) ^ j * X) ^ 2 \u2264 1 + ((\u03c1 : \u2102).im) ^ 2 := by\n linarith [hlo2, sq_nonneg (\u03c1 : \u2102).im]\n have hpos1 : (0 : \u211d) < 1 + ((\u03c1 : \u2102).im) ^ 2 := by positivity\n have hpos2 : (0 : \u211d) < ((2 : \u211d) ^ j * X) ^ 2 := by positivity\n rw [tailTerm, if_pos hgt]\n rw [show (zeroMult (\u03c1 : \u2102) : \u211d) / (1 + ((\u03c1 : \u2102).im) ^ 2)\n = (zeroMult (\u03c1 : \u2102) : \u211d) * (1 / (1 + ((\u03c1 : \u2102).im) ^ 2)) from by ring]\n rw [show (1 / ((2 : \u211d) ^ j * X) ^ 2) * (zeroMult (\u03c1 : \u2102) : \u211d)\n = (zeroMult (\u03c1 : \u2102) : \u211d) * (1 / ((2 : \u211d) ^ j * X) ^ 2) from by ring]\n apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n rw [div_le_div_iff\u2080 hpos1 hpos2]\n nlinarith [hsq]\n _ = (1 / ((2 : \u211d) ^ j * X) ^ 2) *\n \u2211 \u03c1 \u2208 (Set.Finite.subset (dyadClass_fiber_finite hX0 j)\n (t := {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | dyadClass X \u03c1 = j \u2227 X < |(\u03c1 : \u2102).im|})\n (fun \u03c1 h\u03c1 => h\u03c1.1)).toFinset, (zeroMult (\u03c1 : \u2102) : \u211d) := by\n rw [Finset.mul_sum]\n _ \u2264 (1 / ((2 : \u211d) ^ j * X) ^ 2) *\n (A0count * ((2 : \u211d) ^ (j + 1) * X + 3)\n * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5)) :=\n mul_le_mul_of_nonneg_left (fiber_mult_count hX j) (by positivity)\n\n/-- **The far-tail bound**: for X \u2265 200, the zeros with |\u03b3| > X\ncontribute \u03a3 m/(1+\u03b3\u00b2) \u2264 16\u00b7A\u2080\u00b7log X / X. -/\ntheorem zero_tail_inv_sq_le {X : \u211d} (hX : 200 \u2264 X) :\n \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, tailTerm X \u03c1\n \u2264 16 * A0count * Real.log X / X := by\n have hX0 : (0 : \u211d) < X := by linarith\n have hlogXnn : (0 : \u211d) \u2264 Real.log X := Real.log_nonneg (by linarith)\n have hC0 : (0 : \u211d) \u2264 16 * A0count * Real.log X / X :=\n div_nonneg (mul_nonneg (mul_nonneg (by norm_num) A0count_pos.le) hlogXnn) hX0.le\n rw [\u2190 ENNReal.ofReal_le_ofReal_iff hC0]\n rw [ENNReal.ofReal_tsum_of_nonneg (fun \u03c1 => tailTerm_nonneg \u03c1) (tailTerm_summable X)]\n rw [\u2190 ENNReal.tsum_fiberwise (fun \u03c1 => ENNReal.ofReal (tailTerm X \u03c1)) (dyadClass X)]\n -- the real dyadic majorant\n set F : \u2115 \u2192 \u211d := fun j => (1 / ((2 : \u211d) ^ j * X) ^ 2) *\n (A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5)) with hFdef\n set G : \u2115 \u2192 \u211d := fun j => (((j : \u211d) + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j with hGdef\n have hFnn : \u2200 j : \u2115, 0 \u2264 F j := by\n intro j\n rw [hFdef]\n have hL : (0 : \u211d) \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_nonneg\n have h2j1X : (0 : \u211d) < (2 : \u211d) ^ (j + 1) * X := by positivity\n linarith\n have h2j1X : (0 : \u211d) \u2264 (2 : \u211d) ^ (j + 1) * X := by positivity\n exact mul_nonneg (div_nonneg (by norm_num) (by positivity))\n (mul_nonneg (mul_nonneg A0count_pos.le (by linarith)) hL)\n have hGnn : \u2200 j : \u2115, 0 \u2264 (4 * A0count / X) * G j := by\n intro j\n rw [hGdef]\n have hlog2nn : (0 : \u211d) \u2264 Real.log 2 := Real.log_nonneg (by norm_num)\n have hjn : (0 : \u211d) \u2264 (j : \u211d) := Nat.cast_nonneg j\n have hGj : (0 : \u211d) \u2264 (((j : \u211d) + 4) * Real.log 2 + Real.log X) :=\n add_nonneg (mul_nonneg (by linarith) hlog2nn) hlogXnn\n exact mul_nonneg (div_nonneg (mul_nonneg (by norm_num) A0count_pos.le) hX0.le)\n (div_nonneg hGj (by positivity))\n have hFle : \u2200 j : \u2115, F j \u2264 (4 * A0count / X) * G j := by\n intro j\n rw [hFdef, hGdef]\n show (1 / ((2 : \u211d) ^ j * X) ^ 2)\n * (A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5))\n \u2264 (4 * A0count / X)\n * ((((j : \u211d) + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j)\n have h2jX1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j * X := by\n have h1 : (1 : \u211d) \u2264 (2 : \u211d) ^ j := by exact_mod_cast Nat.one_le_two_pow\n nlinarith [h1, hX]\n have h2jX : (0 : \u211d) < (2 : \u211d) ^ j * X := by positivity\n have h2j1X : (0 : \u211d) < (2 : \u211d) ^ (j + 1) * X := by positivity\n have h2j : (0 : \u211d) < (2 : \u211d) ^ j := by positivity\n have hLnn : (0 : \u211d) \u2264 Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by\n apply Real.log_nonneg\n linarith\n -- log(3\u00b72^{j+1}X+5) \u2264 (j+4)\u00b7log 2 + log X\n have hlogle : Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5)\n \u2264 ((j : \u211d) + 4) * Real.log 2 + Real.log X := by\n have harg : 3 * (2 : \u211d) ^ (j + 1) * X + 5 \u2264 (2 : \u211d) ^ (j + 4) * X := by\n have hpow : (2 : \u211d) ^ (j + 4) = 16 * (2 : \u211d) ^ j := by\n rw [show (2 : \u211d) ^ (j + 4) = (2 : \u211d) ^ j * (2 : \u211d) ^ 4 from by rw [\u2190 pow_add]]\n ring_nf\n rw [hpow]\n have hpow2 : (2 : \u211d) ^ (j + 1) = 2 * (2 : \u211d) ^ j := by rw [pow_succ']\n rw [hpow2]\n nlinarith [h2jX1, h2jX]\n have hlogeq : Real.log ((2 : \u211d) ^ (j + 4) * X)\n = ((j : \u211d) + 4) * Real.log 2 + Real.log X := by\n rw [Real.log_mul (by positivity : (2 : \u211d) ^ (j + 4) \u2260 0) (by linarith : X \u2260 0)]\n rw [Real.log_pow]\n push_cast\n ring\n rw [\u2190 hlogeq]\n exact Real.log_le_log (by positivity) harg\n -- the two factor bounds\n have hfac1 : A0count * ((2 : \u211d) ^ (j + 1) * X + 3) / ((2 : \u211d) ^ j * X) ^ 2\n \u2264 (4 * A0count) / (X * (2 : \u211d) ^ j) := by\n rw [div_le_div_iff\u2080 (by positivity) (by positivity)]\n have hpow : (2 : \u211d) ^ (j + 1) * X = 2 * ((2 : \u211d) ^ j * X) := by rw [pow_succ']; ring\n have h2jX200 : (200 : \u211d) \u2264 (2 : \u211d) ^ j * X := by\n have h1p : (1 : \u211d) \u2264 (2 : \u211d) ^ j := by exact_mod_cast Nat.one_le_two_pow\n nlinarith [h1p, hX]\n have hkey : A0count * ((2 : \u211d) ^ j * X) * 3\n \u2264 A0count * ((2 : \u211d) ^ j * X) * (2 * ((2 : \u211d) ^ j * X)) := by\n apply mul_le_mul_of_nonneg_left _ (mul_nonneg A0count_pos.le (by positivity))\n linarith [h2jX200]\n rw [hpow]\n have hA0 := A0count_pos.le\n nlinarith [hkey, h2jX, hA0]\n have h1 : (1 / ((2 : \u211d) ^ j * X) ^ 2) *\n (A0count * ((2 : \u211d) ^ (j + 1) * X + 3) * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5))\n = (A0count * ((2 : \u211d) ^ (j + 1) * X + 3) / ((2 : \u211d) ^ j * X) ^ 2)\n * Real.log (3 * (2 : \u211d) ^ (j + 1) * X + 5) := by ring\n rw [h1]\n have h2 : (4 * A0count / X) * ((((j : \u211d) + 4) * Real.log 2 + Real.log X) / (2 : \u211d) ^ j)\n = ((4 * A0count) / (X * (2 : \u211d) ^ j))\n * (((j : \u211d) + 4) * Real.log 2 + Real.log X) := by\n field_simp\n rw [h2]\n apply mul_le_mul hfac1 hlogle hLnn\n apply div_nonneg _ (mul_nonneg hX0.le (by positivity))\n exact mul_nonneg (by norm_num) A0count_pos.le\n have hGsum : Summable (fun j : \u2115 => (4 * A0count / X) * G j) :=\n (dyadicLog_summable X).mul_left _\n have hFsum : Summable F := by\n apply Summable.of_norm_bounded hGsum\n intro j\n rw [Real.norm_eq_abs, abs_of_nonneg (hFnn j)]\n exact hFle j\n have hGtotal : \u2211' j : \u2115, G j = 10 * Real.log 2 + 2 * Real.log X := tsum_dyadic_log X\n have htotal : \u2211' j : \u2115, F j \u2264 16 * A0count * Real.log X / X := by\n calc \u2211' j : \u2115, F j \u2264 \u2211' j : \u2115, (4 * A0count / X) * G j :=\n Summable.tsum_le_tsum hFle hFsum hGsum\n _ = (4 * A0count / X) * (10 * Real.log 2 + 2 * Real.log X) := by\n rw [tsum_mul_left, hGtotal]\n _ \u2264 16 * A0count * Real.log X / X := dyadic_tail_constant hX\n calc \u2211' j : \u2115, \u2211' b : (dyadClass X) \u207b\u00b9' {j}, ENNReal.ofReal (tailTerm X b.1)\n \u2264 \u2211' j : \u2115, ENNReal.ofReal (F j) := by\n apply ENNReal.tsum_le_tsum\n intro j\n exact fiber_tail_le hX j\n _ = ENNReal.ofReal (\u2211' j : \u2115, F j) := (ENNReal.ofReal_tsum_of_nonneg hFnn hFsum).symm\n _ \u2264 ENNReal.ofReal (16 * A0count * Real.log X / X) := ENNReal.ofReal_le_ofReal htotal\n\n/-! ## 5. The threshold lemma and the zero-free circle (5e-v-b chunk 3a)\n\nThe \u03b3-gap circle radius R \u2208 [4k+5/2, 4k+3] with k \u2265 4000 satisfies\n416\u00b7\u0108\u2080(4k+9/2) \u2264 4k+5/2 \u2264 R (`gapCircle_threshold`) \u2014 explicit\nlinear/rational arithmetic from the two exp facts 2.7\u00b9\u2070 \u2265 16006.5 and\n2.72\u2079 \u2264 16006.5 plus log x \u2264 x\u22121. Consequences: 16\u00b7\u0108\u2080 < R (the circle\ncarries NO nontrivial zero, `not_isNontrivialZero_on_gapCircle`) and\n13/R \u2264 1/(32\u00b7\u0108\u2080) (the critical-arc clearance, used in the shell bounds\nof the next chunk). -/\n\n/-- **The threshold lemma**: for k \u2265 4000, 416\u00b7\u0108\u2080(4k+9/2) \u2264 4k+5/2.\nBase bound \u0108\u2080(16004.5) \u2264 9.3564 (log 16006.5 \u2264 10 via 2.7\u00b9\u2070 \u2265 16006.5;\nlog log \u2264 log 10 \u2264 2.4 via e^{2.4} \u2265 10.2); increment\n\u0108\u2080(x) \u2212 \u0108\u2080(16004.5) \u2264 0.532\u00b7(x\u221216004.5)/16006.5 via log(a/b) \u2264 a/b \u2212 1\n(twice; log 16006.5 \u2265 9 via 2.72\u2079 \u2264 16006.5). -/\ntheorem gapCircle_threshold {k : \u2115} (hk : 4000 \u2264 k) :\n 416 * C0hat (4 * (k : \u211d) + 9 / 2) \u2264 4 * (k : \u211d) + 5 / 2 := by\n have hk4 : (16000 : \u211d) \u2264 4 * (k : \u211d) := by\n have hkR : (4000 : \u211d) \u2264 (k : \u211d) := by exact_mod_cast hk\n linarith\n -- the exp facts\n have he10 : (16006.5 : \u211d) \u2264 Real.exp 10 := by\n have h1 : Real.exp 10 = (Real.exp 1) ^ 10 := by\n rw [\u2190 Real.exp_nat_mul]; norm_num\n have h2 : (2.7 : \u211d) \u2264 Real.exp 1 := by\n have h := Real.exp_one_gt_d9; linarith\n have h3 : (2.7 : \u211d) ^ 10 \u2264 (Real.exp 1) ^ 10 :=\n pow_le_pow_left\u2080 (by norm_num) h2 10\n have h4 : (16006.5 : \u211d) \u2264 (2.7 : \u211d) ^ 10 := by norm_num\n linarith [h1, h3, h4]\n have hL0up : Real.log (16006.5) \u2264 10 := by\n rw [show (10 : \u211d) = Real.log (Real.exp 10) by rw [Real.log_exp]]\n exact Real.log_le_log (by norm_num) he10\n have he9 : Real.exp 9 \u2264 (16006.5 : \u211d) := by\n have h1 : Real.exp 9 = (Real.exp 1) ^ 9 := by\n rw [\u2190 Real.exp_nat_mul]; norm_num\n have h2 : Real.exp 1 \u2264 (2.72 : \u211d) := by\n have h := Real.exp_one_lt_d9.le; linarith\n have h3 : (Real.exp 1) ^ 9 \u2264 (2.72 : \u211d) ^ 9 :=\n pow_le_pow_left\u2080 (Real.exp_pos 1).le h2 9\n have h4 : (2.72 : \u211d) ^ 9 \u2264 16006.5 := by norm_num\n linarith [h1, h3, h4]\n have hL0lo : (9 : \u211d) \u2264 Real.log (16006.5) := by\n rw [show (9 : \u211d) = Real.log (Real.exp 9) by rw [Real.log_exp]]\n exact Real.log_le_log (Real.exp_pos _) he9\n have he2 : (2.7 : \u211d) ^ 2 \u2264 Real.exp 2 := by\n have h1 : Real.exp 2 = (Real.exp 1) ^ 2 := by\n rw [\u2190 Real.exp_nat_mul]; norm_num\n have h2 : (2.7 : \u211d) \u2264 Real.exp 1 := by\n have h := Real.exp_one_gt_d9; linarith\n have h3 : (2.7 : \u211d) ^ 2 \u2264 (Real.exp 1) ^ 2 :=\n pow_le_pow_left\u2080 (by norm_num) h2 2\n linarith [h1, h3]\n have he04 : (1.4 : \u211d) \u2264 Real.exp 0.4 := by\n have h := Real.add_one_le_exp (0.4 : \u211d)\n linarith\n have he24 : (10 : \u211d) \u2264 Real.exp 2.4 := by\n have h1 : Real.exp 2.4 = Real.exp 2 * Real.exp 0.4 := by\n rw [\u2190 Real.exp_add]; norm_num\n have h2 : (10 : \u211d) \u2264 (2.7 : \u211d) ^ 2 * 1.4 := by norm_num\n have h3 : (2.7 : \u211d) ^ 2 * 1.4 \u2264 Real.exp 2 * Real.exp 0.4 :=\n mul_le_mul he2 he04 (by norm_num) (Real.exp_pos 2).le\n linarith [h1, h3]\n have hlog10 : Real.log 10 \u2264 (2.4 : \u211d) := by\n rw [show (2.4 : \u211d) = Real.log (Real.exp 2.4) by rw [Real.log_exp]]\n exact Real.log_le_log (by norm_num) he24\n have hLL0 : Real.log (Real.log (16006.5)) \u2264 2.4 :=\n (Real.log_le_log (by linarith [hL0lo]) hL0up).trans hlog10\n -- the increments, with x = 4k+9/2, x+2 = 4k+13/2 \u2265 16006.5\n set y : \u211d := 4 * (k : \u211d) with hy\n have hx2 : (16006.5 : \u211d) \u2264 y + 13 / 2 := by linarith\n have h\u0394def : (y + 13 / 2) / 16006.5 - 1 = (y - 16000) / 16006.5 := by\n field_simp\n ring\n have hinc0 : Real.log (y + 13 / 2) - Real.log (16006.5)\n \u2264 (y - 16000) / 16006.5 := by\n have hpos : (0 : \u211d) < (y + 13 / 2) / 16006.5 := by positivity\n have h1 := Real.log_le_sub_one_of_pos hpos\n have h2 : Real.log (y + 13 / 2)\n = Real.log (16006.5) + Real.log ((y + 13 / 2) / 16006.5) := by\n rw [\u2190 Real.log_mul (by norm_num : (16006.5 : \u211d) \u2260 0)\n (by positivity : (y + 13 / 2) / 16006.5 \u2260 0)]\n congr 1\n field_simp\n rw [h\u0394def] at h1\n linarith [h2, h1]\n have hinc1 : Real.log (y + 13 / 2) \u2264 10 + (y - 16000) / 16006.5 := by\n linarith [hinc0, hL0up]\n have hLge : Real.log (16006.5) \u2264 Real.log (y + 13 / 2) :=\n Real.log_le_log (by norm_num) hx2\n have hinc2 : Real.log (Real.log (y + 13 / 2))\n \u2264 2.4 + (Real.log (y + 13 / 2) - Real.log (16006.5)) / 9 := by\n have hpos : (0 : \u211d) < Real.log (y + 13 / 2) / Real.log (16006.5) :=\n div_pos (by linarith [hL0lo, hLge]) (by linarith [hL0lo])\n have h1 := Real.log_le_sub_one_of_pos hpos\n have h2 : Real.log (Real.log (y + 13 / 2))\n = Real.log (Real.log (16006.5))\n + Real.log (Real.log (y + 13 / 2) / Real.log (16006.5)) := by\n rw [\u2190 Real.log_mul (by linarith [hL0lo] : Real.log (16006.5) \u2260 0)\n (by positivity : Real.log (y + 13 / 2) / Real.log (16006.5) \u2260 0)]\n congr 1\n field_simp\n have h3 : Real.log (y + 13 / 2) / Real.log (16006.5) - 1\n \u2264 (Real.log (y + 13 / 2) - Real.log (16006.5)) / 9 := by\n have h4 : Real.log (y + 13 / 2) / Real.log (16006.5) - 1\n = (Real.log (y + 13 / 2) - Real.log (16006.5))\n / Real.log (16006.5) := by\n field_simp\n rw [h4]\n exact div_le_div_of_nonneg_left (by linarith [hLge]) (by norm_num)\n hL0lo\n linarith [h1, h2, h3, hLL0]\n -- assemble: \u0108\u2080(4k+9/2) \u2264 9.3564 + 0.532\u00b7(y\u221216000)/16006.5\n unfold C0hat\n have h\u0394nn : (0 : \u211d) \u2264 (y - 16000) / 16006.5 := by positivity\n have h\u0394le : (y - 16000) / 16006.5 \u2264 (y - 16000) / 16000 :=\n div_le_div_of_nonneg_left (by linarith) (by norm_num) (by norm_num)\n have hC : 0.433 * Real.log (y + 9 / 2 + 2)\n + 0.886 * Real.log (Real.log (y + 9 / 2 + 2)) + 2.9\n \u2264 9.3564 + 0.532 * ((y - 16000) / 16006.5) := by\n have harg : y + 9 / 2 + 2 = y + 13 / 2 := by ring\n rw [harg]\n linarith [hinc1, hinc2, hinc0, h\u0394nn]\n have hC2 : 416 * (0.433 * Real.log (y + 9 / 2 + 2)\n + 0.886 * Real.log (Real.log (y + 9 / 2 + 2)) + 2.9)\n \u2264 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5)) :=\n mul_le_mul_of_nonneg_left hC (by norm_num)\n have hfin : 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5))\n \u2264 y + 5 / 2 := by\n have h1 : 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5))\n \u2264 416 * (9.3564 + 0.532 * ((y - 16000) / 16000)) := by\n exact mul_le_mul_of_nonneg_left\n (add_le_add (le_refl _)\n (mul_le_mul_of_nonneg_left h\u0394le (by norm_num : (0 : \u211d) \u2264 0.532)))\n (by norm_num : (0 : \u211d) \u2264 416)\n have h2 : 416 * (9.3564 + 0.532 * ((y - 16000) / 16000))\n = 3670.9504 + 0.013832 * y := by\n field_simp\n ring\n rw [h2] at h1\n linarith [hk4]\n calc 416 * (0.433 * Real.log (y + 9 / 2 + 2)\n + 0.886 * Real.log (Real.log (y + 9 / 2 + 2)) + 2.9)\n \u2264 416 * (9.3564 + 0.532 * ((y - 16000) / 16006.5)) := hC2\n _ \u2264 y + 5 / 2 := hfin\n\n/-- |im \u03c1| \u2265 \u2016\u03c1\u2016 \u2212 1 for a nontrivial zero (\u2016\u03c1\u2016 \u2264 |re| + |im| \u2264 1 + |im|). -/\nlemma abs_im_ge_norm_sub_one (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n \u2016(\u03c1 : \u2102)\u2016 - 1 \u2264 |(\u03c1 : \u2102).im| := by\n have hnorm : \u2016(\u03c1 : \u2102)\u2016 \u2264 |(\u03c1 : \u2102).re| + |(\u03c1 : \u2102).im| :=\n Complex.norm_le_abs_re_add_abs_im _\n have hre : |(\u03c1 : \u2102).re| \u2264 1 := by\n rw [abs_le]\n exact \u27e8by linarith [\u03c1.2.2.1], by linarith [\u03c1.2.2.2]\u27e9\n linarith\n\n/-- **The gap circle is zero-free**: for k \u2265 4000 and R with the\ngap-circle properties, no point s with \u2016s\u2016 = R is a nontrivial zero.\n(A zero on the circle would have ordinate \u00b1\u221a(R\u00b2\u2212re\u00b2) within 1/R of R,\nbelow the \u03b3-gap clearance 1/(16\u00b7\u0108\u2080) since 16\u00b7\u0108\u2080 < R by the threshold.) -/\ntheorem not_isNontrivialZero_on_gapCircle {k : \u2115} (hk : 4000 \u2264 k)\n {R : \u211d} (hRlo : 4 * (k : \u211d) + 5 / 2 \u2264 R)\n (hgap : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192\n (1 / (16 * C0hat (4 * (k : \u211d) + 9 / 2))) \u2264 |\u03c1.im - R| \u2228\n 1 \u2264 |\u03c1.im - R|)\n {s : \u2102} (hs : \u2016s\u2016 = R) : \u00ac IsNontrivialZero s := by\n intro hz\n have hth := gapCircle_threshold hk\n have h\u01082 : (2 : \u211d) \u2264 C0hat (4 * (k : \u211d) + 9 / 2) := two_le_C0hat (by\n have hkR : (1 : \u211d) \u2264 (k : \u211d) := by exact_mod_cast (by omega : 1 \u2264 k)\n linarith)\n have hR1 : (1 : \u211d) < R := by\n have hkR : (1 : \u211d) \u2264 (k : \u211d) := by exact_mod_cast (by omega : 1 \u2264 k)\n linarith [hRlo]\n -- 16\u00b7\u0108\u2080 < R\n have h16 : 16 * C0hat (4 * (k : \u211d) + 9 / 2) < R := by\n have h1 : 16 * C0hat (4 * (k : \u211d) + 9 / 2)\n < 416 * C0hat (4 * (k : \u211d) + 9 / 2) := by\n have hpos : (0 : \u211d) < C0hat (4 * (k : \u211d) + 9 / 2) := by linarith [h\u01082]\n nlinarith [hpos]\n linarith [hth, hRlo, h1]\n -- the ordinate t = |im s| and its distance to R\n have hnormSq : \u2016s\u2016 ^ 2 = s.re ^ 2 + s.im ^ 2 := by\n rw [\u2190 Complex.normSq_eq_norm_sq, Complex.normSq_apply]\n ring\n have hre2 : s.re ^ 2 \u2264 1 := by\n have h0 := hz.2.1\n have h1 := hz.2.2\n nlinarith [sq_nonneg s.re]\n have ht_le : |s.im| \u2264 R := by\n have h := abs_im_le_norm s\n rw [hs] at h\n exact h\n have ht2 : |s.im| ^ 2 = R ^ 2 - s.re ^ 2 := by\n have h1 : s.im ^ 2 = R ^ 2 - s.re ^ 2 := by\n have h2 : s.re ^ 2 + s.im ^ 2 = R ^ 2 := by\n rw [\u2190 hnormSq, hs]\n linarith\n rw [sq_abs]\n linarith [h1]\n -- R \u2212 |im s| = re\u00b2/(R+|im s|) \u2264 1/R\n have hRt : R - |s.im| = s.re ^ 2 / (R + |s.im|) := by\n have hne : R + |s.im| \u2260 0 := by positivity\n field_simp\n have h1 : (R - |s.im|) * (R + |s.im|) = R ^ 2 - |s.im| ^ 2 := by ring\n rw [h1, ht2]\n ring\n have hRt_le : R - |s.im| \u2264 1 / R := by\n rw [hRt]\n have h1 : s.re ^ 2 / (R + |s.im|) \u2264 s.re ^ 2 / R := by\n apply div_le_div_of_nonneg_left (sq_nonneg _) (by linarith [hR1])\n have h2 : (0 : \u211d) \u2264 |s.im| := abs_nonneg _\n linarith\n have h3 : s.re ^ 2 / R \u2264 1 / R := by\n apply div_le_div_of_nonneg_right _ (by linarith [hR1])\n linarith [hre2]\n exact h1.trans h3\n -- the witness zero with positive ordinate\n have hkR0 : (0 : \u211d) < R := by linarith [hR1]\n set \u03c1' : \u2102 := if 0 \u2264 s.im then s else (starRingEnd \u2102) s with h\u03c1'\n have h\u03c1'z : IsNontrivialZero \u03c1' := by\n rw [h\u03c1']\n by_cases him : 0 \u2264 s.im\n \u00b7 rw [if_pos him]; exact hz\n \u00b7 rw [if_neg him]; exact isNontrivialZero_conj hz\n have h\u03c1'im : \u03c1'.im = |s.im| := by\n rw [h\u03c1']\n by_cases him : 0 \u2264 s.im\n \u00b7 rw [if_pos him]; exact (abs_of_nonneg him).symm\n \u00b7 rw [if_neg him, Complex.conj_im]\n exact (abs_of_neg (lt_of_not_ge him)).symm\n -- |im \u03c1' \u2212 R| = R \u2212 |im s| \u2264 1/R < min 1 (1/(16\u0108\u2080)): contradiction\n have hgap' := hgap \u03c1' h\u03c1'z\n rw [h\u03c1'im] at hgap'\n have hle : (|s.im| : \u211d) - R \u2264 0 := by linarith [ht_le]\n have hdist : |(|s.im| : \u211d) - R| = R - |s.im| := by\n rw [abs_of_nonpos hle]\n ring\n rw [hdist] at hgap'\n have hsmall : R - |s.im| < 1 / (16 * C0hat (4 * (k : \u211d) + 9 / 2)) := by\n have hpos : (0 : \u211d) < 16 * C0hat (4 * (k : \u211d) + 9 / 2) := by\n linarith [h\u01082]\n have h1 : 1 / R < 1 / (16 * C0hat (4 * (k : \u211d) + 9 / 2)) :=\n one_div_lt_one_div_of_lt hpos h16\n linarith [hRt_le, h1]\n have hsmall1 : R - |s.im| < 1 := by\n have h1 : 1 / R < 1 := (div_lt_one (by linarith [hR1] : (0 : \u211d) < R)).mpr hR1\n linarith [hRt_le, h1]\n rcases hgap' with h | h\n \u00b7 linarith [h, hsmall]\n \u00b7 linarith [h, hsmall1]\n\n/-! ## 6. The per-term norm identity, the count brick, and the tail\napplication (5e-v-b chunk 3b)\n\n`seriesTerm_norm_eq_div`: on the zero-free circle (\u2016s\u2016 = R),\n\u2016seriesTerm s \u03c1\u2016 = m\u00b7R/(\u2016\u03c1\u2016\u00b7\u2016s\u2212\u03c1\u2016) (the key algebra\n1/(s\u2212\u03c1)+1/\u03c1 = s/(\u03c1(s\u2212\u03c1))). `circleZeros R`: the finite part\n{\u2016\u03c1\u2016 < 2R+1}. `sum_zeroMult_le_of_im_band`: the reusable window-count\nbrick \u2014 a zero finset whose ordinates lie in the band [a, b] (with the\nsign-separation c, max a c \u2265 1) has multiplicity-sum \u2264\n3\u00b7A\u2080\u00b7(b\u2212a+2)\u00b7log(3\u00b7max|a||b|+5) (the two-window cover\n(M\u22121, b] \u222a (\u2212b\u22121, \u2212M], M := max a c; the \u22121 paddings absorb the\nnon-strict band bounds; the negative window's log loses +1 via\n3b+8 \u2264 2(3b+5)). `series_tail_le`: the far tail (\u2016\u03c1\u2016 \u2265 2R+1) \u2264\n91\u00b7A\u2080\u00b7log R via `seriesTerm_norm_le` + `zero_tail_inv_sq_le`.\n`seriesFun_norm_le_split`: the ONE tsum split. -/\n\n/-- The per-term norm identity on a zero-free circle. -/\nlemma seriesTerm_norm_eq_div {s : \u2102} {R : \u211d} (hs : \u2016s\u2016 = R)\n (hsz : \u00ac IsNontrivialZero s) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n \u2016seriesTerm s \u03c1\u2016 =\n (zeroMult (\u03c1 : \u2102) : \u211d) * (R / (\u2016(\u03c1 : \u2102)\u2016 * \u2016s - (\u03c1 : \u2102)\u2016)) := by\n have h0 : (\u03c1 : \u2102) \u2260 0 := by\n intro h\n have hre := \u03c1.2.2.1\n rw [h] at hre\n simp at hre\n have h1 : s - (\u03c1 : \u2102) \u2260 0 := by\n intro h\n apply hsz\n have hs_eq : s = (\u03c1 : \u2102) := by rw [\u2190 sub_eq_zero]; exact h\n rw [hs_eq]\n exact \u03c1.2\n have halg := one_div_sub_add_one_div h0 h1\n simp only [seriesTerm]\n rw [halg, norm_mul, RCLike.norm_natCast, norm_div, norm_mul, hs]\n\n/-- The finite-part finset for the circle split: {\u2016\u03c1\u2016 < 2R+1}. -/\ndef circleZeros (R : \u211d) : Finset {\u03c1 : \u2102 // IsNontrivialZero \u03c1} :=\n (finite_subtype_norm_lt (2 * R + 1)).toFinset\n\nlemma mem_circleZeros {R : \u211d} {\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}} :\n \u03c1 \u2208 circleZeros R \u2194 \u2016(\u03c1 : \u2102)\u2016 < 2 * R + 1 :=\n Set.Finite.mem_toFinset _\n\n/-- **The count brick**: a zero finset with ordinates in the band\n[a, b] (sign-separated by c, 1 \u2264 max a c \u2264 b) has multiplicity-sum\n\u2264 3\u00b7A\u2080\u00b7(b\u2212a+2)\u00b7log(3\u00b7max|a||b|+5). -/\nlemma sum_zeroMult_le_of_im_band (FS : Finset {\u03c1 : \u2102 // IsNontrivialZero \u03c1})\n {a b c : \u211d} (hM1 : (1 : \u211d) \u2264 max a c) (hMb : max a c \u2264 b)\n (hsub : \u2200 \u03c1 \u2208 FS, a \u2264 |(\u03c1 : \u2102).im| \u2227 |(\u03c1 : \u2102).im| \u2264 b \u2227 c \u2264 |(\u03c1 : \u2102).im|) :\n \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n \u2264 3 * A0count * (b - a + 2) * Real.log (3 * max |a| |b| + 5) := by\n classical\n set M := max a c with hM\n have hM1' : (1 : \u211d) \u2264 M := hM1\n have hMb' : M \u2264 b := hMb\n have hb1 : (1 : \u211d) \u2264 b := le_trans hM1' hMb'\n have hW1 : (Zeta23.zerosIn (M - 1) b).Finite := Zeta23.zerosIn_finite _ _\n have hW2 : (Zeta23.zerosIn (-b - 1) (-M)).Finite := Zeta23.zerosIn_finite _ _\n have himg : FS.image (Subtype.val) \u2286 hW1.toFinset \u222a hW2.toFinset := by\n intro z hz\n obtain \u27e8\u03c1, h\u03c1, rfl\u27e9 := Finset.mem_image.mp hz\n have hbnd1 := (hsub \u03c1 h\u03c1).1\n have hbnd2 := (hsub \u03c1 h\u03c1).2.1\n have hbnd3 := (hsub \u03c1 h\u03c1).2.2\n have hMle : M \u2264 |(\u03c1 : \u2102).im| := max_le_iff.mpr \u27e8hbnd1, hbnd3\u27e9\n rcases le_or_gt 0 (\u03c1 : \u2102).im with him | him\n \u00b7 apply Finset.mem_union_left\n rw [Set.Finite.mem_toFinset]\n refine \u27e8\u03c1.2, ?_, ?_\u27e9\n \u00b7 rw [abs_of_nonneg him] at hMle\n linarith\n \u00b7 rw [abs_of_nonneg him] at hbnd2\n exact hbnd2\n \u00b7 apply Finset.mem_union_right\n rw [Set.Finite.mem_toFinset]\n refine \u27e8\u03c1.2, ?_, ?_\u27e9\n \u00b7 rw [abs_of_neg him] at hbnd2\n linarith\n \u00b7 rw [abs_of_neg him] at hMle\n linarith\n have hdisj : Disjoint hW1.toFinset hW2.toFinset := by\n rw [Finset.disjoint_left]\n intro z hz1 hz2\n rw [Set.Finite.mem_toFinset] at hz1 hz2\n have h1 : M - 1 < z.im := hz1.2.1\n have h2 : z.im \u2264 -M := hz2.2.2\n linarith [hM1']\n have hbridge : \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n \u2264 (Zeta23.Ncount (M - 1) b : \u211d) + (Zeta23.Ncount (-b - 1) (-M) : \u211d) := by\n have hsum_img : \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n = \u2211 z \u2208 FS.image (Subtype.val), (zeroMult z : \u211d) :=\n (Finset.sum_image (fun x _ y _ h => Subtype.val_injective h)).symm\n rw [hsum_img]\n calc \u2211 z \u2208 FS.image (Subtype.val), (zeroMult z : \u211d)\n \u2264 \u2211 z \u2208 hW1.toFinset \u222a hW2.toFinset, (zeroMult z : \u211d) :=\n Finset.sum_le_sum_of_subset_of_nonneg himg (fun z _ _ => Nat.cast_nonneg _)\n _ = \u2211 z \u2208 hW1.toFinset, (zeroMult z : \u211d)\n + \u2211 z \u2208 hW2.toFinset, (zeroMult z : \u211d) :=\n Finset.sum_union hdisj\n _ = (Zeta23.Ncount (M - 1) b : \u211d) + (Zeta23.Ncount (-b - 1) (-M) : \u211d) := by\n rw [Zeta23.Ncount, Zeta23.Ncount, finsum_mem_eq_finite_toFinset_sum _ hW1,\n finsum_mem_eq_finite_toFinset_sum _ hW2, Nat.cast_sum, Nat.cast_sum]\n -- the log comparisons\n have hLpos : (0 : \u211d) < 3 * max |a| |b| + 5 := by\n have h1 : (0 : \u211d) \u2264 max |a| |b| := le_trans (abs_nonneg a) (le_max_left _ _)\n linarith\n have hLnn : (0 : \u211d) \u2264 Real.log (3 * max |a| |b| + 5) := by\n apply Real.log_nonneg\n have h1 : (0 : \u211d) \u2264 max |a| |b| := le_trans (abs_nonneg a) (le_max_left _ _)\n linarith\n have hL1 : (1 : \u211d) \u2264 Real.log (3 * max |a| |b| + 5) := by\n have h1 : (5 : \u211d) \u2264 3 * max |a| |b| + 5 := by\n have h2 : (0 : \u211d) \u2264 max |a| |b| := le_trans (abs_nonneg a) (le_max_left _ _)\n linarith\n rw [Real.le_log_iff_exp_le hLpos]\n have h2 : Real.exp 1 \u2264 (5 : \u211d) := by\n have h := Real.exp_one_lt_d9.le\n linarith\n linarith [h1, h2]\n have hLog1 : Real.log (3 * max |M - 1| |b| + 5)\n \u2264 Real.log (3 * max |a| |b| + 5) := by\n apply Real.log_le_log (by positivity)\n have h1 : |M - 1| \u2264 max |a| |b| := by\n have h2 : |M - 1| = M - 1 := abs_of_nonneg (by linarith [hM1'])\n rw [h2]\n have h3 : M - 1 \u2264 |b| := by\n have h4 : |b| = b := abs_of_nonneg (by linarith [hb1])\n rw [h4]\n linarith [hMb']\n have h4 : |b| \u2264 max |a| |b| := le_max_right _ _\n linarith [h3, h4]\n have h2 : |b| \u2264 max |a| |b| := le_max_right _ _\n have h3 := max_le h1 h2\n linarith [h3]\n have hLog2 : Real.log (3 * max |-b - 1| |-M| + 5)\n \u2264 Real.log (3 * max |a| |b| + 5) + 1 := by\n have h1 : |-b - 1| = b + 1 := by\n rw [show -b - 1 = -(b + 1) by ring, abs_neg, abs_of_nonneg (by linarith [hb1])]\n have h2 : |-M| = M := abs_neg M\n have h3 : max |-b - 1| |-M| = b + 1 := by\n rw [h1, h2]\n exact max_eq_left (by linarith [hMb'])\n rw [h3]\n have h4 : 3 * (b + 1) + 5 \u2264 2 * (3 * max |a| |b| + 5) := by\n have h5 : |b| = b := abs_of_nonneg (by linarith [hb1])\n have h6 : (b : \u211d) \u2264 max |a| |b| := by\n rw [\u2190 h5]\n exact le_max_right _ _\n linarith [h6]\n have h5 : Real.log (3 * (b + 1) + 5)\n \u2264 Real.log (2 * (3 * max |a| |b| + 5)) :=\n Real.log_le_log (by positivity) h4\n have h6 : Real.log (2 * (3 * max |a| |b| + 5))\n = Real.log 2 + Real.log (3 * max |a| |b| + 5) :=\n Real.log_mul (by norm_num : (2 : \u211d) \u2260 0) (by linarith [hLpos])\n have h7 := log_two_le_one\n linarith [h5, h6, h7]\n -- the two window counts\n have hw1 : (Zeta23.Ncount (M - 1) b : \u211d)\n \u2264 A0count * (b - M + 2) * Real.log (3 * max |a| |b| + 5) := by\n have h1 := Ncount_le_of_interval_gen (a := M - 1) (b := b) (by linarith [hMb'])\n have h2 : b - (M - 1) + 1 = b - M + 2 := by ring\n rw [h2] at h1\n exact h1.trans (mul_le_mul_of_nonneg_left hLog1\n (mul_nonneg A0count_pos.le (by linarith [hMb'])))\n have hw2 : (Zeta23.Ncount (-b - 1) (-M) : \u211d)\n \u2264 A0count * (b - M + 2) * (Real.log (3 * max |a| |b| + 5) + 1) := by\n have h1 := Ncount_le_of_interval_gen (a := -b - 1) (b := -M) (by linarith [hMb'])\n have h2 : -M - (-b - 1) + 1 = b - M + 2 := by ring\n rw [h2] at h1\n exact h1.trans (mul_le_mul_of_nonneg_left hLog2\n (mul_nonneg A0count_pos.le (by linarith [hMb'])))\n have hMa : a \u2264 M := le_max_left _ _\n calc \u2211 \u03c1 \u2208 FS, (zeroMult (\u03c1 : \u2102) : \u211d)\n \u2264 (Zeta23.Ncount (M - 1) b : \u211d) + (Zeta23.Ncount (-b - 1) (-M) : \u211d) := hbridge\n _ \u2264 A0count * (b - M + 2) * Real.log (3 * max |a| |b| + 5)\n + A0count * (b - M + 2) * (Real.log (3 * max |a| |b| + 5) + 1) :=\n add_le_add hw1 hw2\n _ = A0count * (b - M + 2) * (2 * Real.log (3 * max |a| |b| + 5) + 1) := by ring\n _ \u2264 A0count * (b - a + 2) * (3 * Real.log (3 * max |a| |b| + 5)) := by\n apply mul_le_mul\n \u00b7 exact mul_le_mul_of_nonneg_left (by linarith [hMa]) A0count_pos.le\n \u00b7 linarith [hL1, hLnn]\n \u00b7 linarith [hL1, hLnn]\n \u00b7 exact mul_nonneg A0count_pos.le (by linarith [hMa, hMb'])\n _ = 3 * A0count * (b - a + 2) * Real.log (3 * max |a| |b| + 5) := by ring\n\n/-- 9 \u2264 log R for R \u2265 16002 (e\u2079 \u2264 2.72\u2079 \u2264 16002). -/\nlemma nine_le_log_of_16002_le {R : \u211d} (hR : (16002 : \u211d) \u2264 R) :\n (9 : \u211d) \u2264 Real.log R := by\n have he9 : Real.exp 9 \u2264 (16002 : \u211d) := by\n have h1 : Real.exp 9 = (Real.exp 1) ^ 9 := by\n rw [\u2190 Real.exp_nat_mul]; norm_num\n have h2 : Real.exp 1 \u2264 (2.72 : \u211d) := by\n have h := Real.exp_one_lt_d9.le\n linarith\n have h3 : (Real.exp 1) ^ 9 \u2264 (2.72 : \u211d) ^ 9 :=\n pow_le_pow_left\u2080 (Real.exp_pos 1).le h2 9\n have h4 : (2.72 : \u211d) ^ 9 \u2264 16002 := by norm_num\n linarith [h1, h3, h4]\n rw [show (9 : \u211d) = Real.log (Real.exp 9) by rw [Real.log_exp]]\n exact Real.log_le_log (Real.exp_pos _) (he9.trans hR)\n\n/-- **The tail application**: for \u2016s\u2016 = R \u2265 16002, the if-zeroed tail\nover {\u2016\u03c1\u2016 \u2265 2R+1} is \u2264 91\u00b7A\u2080\u00b7log R (per-term (17R/6)\u00b7m/(1+\u03b3\u00b2) via\n`seriesTerm_norm_le` + 1/(1+\u2016\u03b3\u2016\u00b2) \u2264 1/(1+im\u00b2), then\n`zero_tail_inv_sq_le` at X = 2R\u22121 \u2265 200). -/\nlemma series_tail_le {s : \u2102} {R : \u211d} (hs : \u2016s\u2016 = R) (hR : (16002 : \u211d) \u2264 R) :\n \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n (if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016)\n \u2264 91 * A0count * Real.log R := by\n have hRpos : (0 : \u211d) < R := by linarith [hR]\n have hX : (200 : \u211d) \u2264 2 * R - 1 := by linarith [hR]\n have hlogR : (9 : \u211d) \u2264 Real.log R := nine_le_log_of_16002_le hR\n have hpoint : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n (if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016)\n \u2264 (17 * R / 6) * tailTerm (2 * R - 1) \u03c1 := by\n intro \u03c1\n by_cases hmem : \u03c1 \u2208 circleZeros R\n \u00b7 rw [if_pos hmem]\n exact mul_nonneg (by positivity) (tailTerm_nonneg \u03c1)\n \u00b7 rw [if_neg hmem]\n have hnorm : (2 * R + 1 : \u211d) \u2264 \u2016(\u03c1 : \u2102)\u2016 := by\n have h := not_lt.mp (mt mem_circleZeros.mpr hmem)\n linarith\n have h\u03c12 : (2 : \u211d) \u2264 \u2016(\u03c1 : \u2102)\u2016 := by linarith [hR, hnorm]\n have hs\u03c1 : 2 * \u2016s\u2016 \u2264 \u2016(\u03c1 : \u2102)\u2016 := by rw [hs]; linarith [hnorm]\n have h1 := seriesTerm_norm_le \u03c1 h\u03c12 hs\u03c1\n rw [hs] at h1\n -- m/(1+\u2016\u03b3\u2016\u00b2) \u2264 m/(1+im\u00b2) = tailTerm (|im| \u2265 \u2016\u03c1\u2016\u22121 \u2265 2R > 2R\u22121)\n have hgam : Complex.normSq (gammaOf (\u03c1 : \u2102))\n = (\u03c1 : \u2102).im ^ 2 + (1 / 2 - (\u03c1 : \u2102).re) ^ 2 := by\n rw [Complex.normSq_apply, WeilEF.gammaOf_re, WeilEF.gammaOf_im]\n ring\n have hcmp : (1 : \u211d) + (\u03c1 : \u2102).im ^ 2\n \u2264 1 + Complex.normSq (gammaOf (\u03c1 : \u2102)) := by\n rw [hgam]\n have h2 : (0 : \u211d) \u2264 (1 / 2 - (\u03c1 : \u2102).re) ^ 2 := sq_nonneg _\n linarith\n have hpos1 : (0 : \u211d) < 1 + Complex.normSq (gammaOf (\u03c1 : \u2102)) := by\n have h := Complex.normSq_nonneg (gammaOf (\u03c1 : \u2102))\n linarith\n have hpos2 : (0 : \u211d) < 1 + (\u03c1 : \u2102).im ^ 2 := by positivity\n have hfrac : 1 / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))\n \u2264 1 / (1 + (\u03c1 : \u2102).im ^ 2) :=\n one_div_le_one_div_of_le hpos2 hcmp\n have hm : (0 : \u211d) \u2264 (zeroMult \u03c1 : \u211d) := Nat.cast_nonneg _\n have h2 : (zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))\n \u2264 (zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2) := by\n rw [show (zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))\n = (zeroMult \u03c1 : \u211d) * (1 / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))) from by ring]\n rw [show (zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2)\n = (zeroMult \u03c1 : \u211d) * (1 / (1 + (\u03c1 : \u2102).im ^ 2)) from by ring]\n exact mul_le_mul_of_nonneg_left hfrac hm\n have him : (2 : \u211d) * R - 1 < |(\u03c1 : \u2102).im| := by\n have h3 := abs_im_ge_norm_sub_one \u03c1\n linarith [hnorm, hR]\n rw [tailTerm, if_pos him]\n calc \u2016seriesTerm s \u03c1\u2016\n \u2264 17 / 6 * R * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))) := h1\n _ \u2264 17 / 6 * R * ((zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2)) :=\n mul_le_mul_of_nonneg_left h2 (by positivity)\n _ = 17 * R / 6 * ((zeroMult \u03c1 : \u211d) / (1 + (\u03c1 : \u2102).im ^ 2)) := by ring\n have hsumL : Summable (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016) := by\n apply Summable.of_norm_bounded (seriesSummable s).norm\n intro \u03c1\n by_cases h : \u03c1 \u2208 circleZeros R <;> simp [h]\n have hsumR : Summable (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n (17 * R / 6) * tailTerm (2 * R - 1) \u03c1) :=\n (tailTerm_summable (2 * R - 1)).mul_left _\n have htsum := Summable.tsum_le_tsum hpoint hsumL hsumR\n rw [tsum_mul_left] at htsum\n have htail := zero_tail_inv_sq_le hX\n have hlog2R : Real.log (2 * R - 1) \u2264 2 * Real.log R := by\n have h1 : Real.log (2 * R - 1) \u2264 Real.log (2 * R) := by\n apply Real.log_le_log (by linarith [hRpos])\n linarith\n have h2 : Real.log (2 * R) = Real.log 2 + Real.log R :=\n Real.log_mul (by norm_num : (2 : \u211d) \u2260 0) (by linarith [hRpos] : R \u2260 0)\n have h3 := log_two_le_one\n linarith [h1, h2, h3, hlogR]\n have hR21 : R / (2 * R - 1) \u2264 1 := by\n rw [div_le_one (by linarith [hRpos] : (0 : \u211d) < 2 * R - 1)]\n linarith [hRpos]\n have hA0 := A0count_pos.le\n have hlogRnn : (0 : \u211d) \u2264 Real.log R := by linarith [hlogR]\n have hbound : 17 * R / 6 * (16 * A0count * Real.log (2 * R - 1) / (2 * R - 1))\n \u2264 91 * A0count * Real.log R := by\n have h1 : 17 * R / 6 * (16 * A0count * Real.log (2 * R - 1) / (2 * R - 1))\n = (17 / 6) * 16 * A0count * (R / (2 * R - 1)) * Real.log (2 * R - 1) := by\n field_simp\n ring\n rw [h1]\n have h2 : (17 / 6 : \u211d) * 16 * A0count * (R / (2 * R - 1)) * Real.log (2 * R - 1)\n \u2264 (17 / 6) * 16 * A0count * 1 * (2 * Real.log R) := by\n apply mul_le_mul\n \u00b7 apply mul_le_mul_of_nonneg_left _ hA0\n apply mul_le_mul_of_nonneg_left hR21 (by norm_num)\n \u00b7 exact hlog2R\n \u00b7 exact hlogRnn\n \u00b7 exact mul_nonneg (mul_nonneg (by norm_num : (0 : \u211d) \u2264 17 / 6 * 16) hA0)\n (by linarith [hR21] : (0 : \u211d) \u2264 R / (2 * R - 1))\n calc (17 / 6 : \u211d) * 16 * A0count * (R / (2 * R - 1)) * Real.log (2 * R - 1)\n \u2264 (17 / 6) * 16 * A0count * 1 * (2 * Real.log R) := h2\n _ = (272 / 3) * A0count * Real.log R := by ring\n _ \u2264 91 * A0count * Real.log R := by\n apply mul_le_mul_of_nonneg_right _ hlogRnn\n apply mul_le_mul_of_nonneg_right _ hA0\n norm_num\n calc \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n (if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016)\n \u2264 17 * R / 6 * \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, tailTerm (2 * R - 1) \u03c1 := htsum\n _ \u2264 17 * R / 6 * (16 * A0count * Real.log (2 * R - 1) / (2 * R - 1)) :=\n mul_le_mul_of_nonneg_left htail (by positivity)\n _ \u2264 91 * A0count * Real.log R := hbound\n\n/-- **The one tsum split**: \u2016seriesFun s\u2016 \u2264 the finite part over\n{\u2016\u03c1\u2016 < 2R+1} + the if-zeroed tail. -/\nlemma seriesFun_norm_le_split {s : \u2102} {R : \u211d} :\n \u2016seriesFun s\u2016\n \u2264 \u2211 \u03c1 \u2208 circleZeros R, \u2016seriesTerm s \u03c1\u2016\n + \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n (if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016) := by\n have h1 : \u2016seriesFun s\u2016\n \u2264 \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u2016seriesTerm s \u03c1\u2016 :=\n norm_tsum_le_tsum_norm (seriesSummable s).norm\n have h2 := tsum_split_finset (seriesSummable s).norm (circleZeros R)\n calc \u2016seriesFun s\u2016\n \u2264 \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u2016seriesTerm s \u03c1\u2016 := h1\n _ = \u2211 \u03c1 \u2208 circleZeros R, \u2016seriesTerm s \u03c1\u2016\n + \u2211' \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n (if \u03c1 \u2208 circleZeros R then (0 : \u211d) else \u2016seriesTerm s \u03c1\u2016) := h2\n\nend NguyenRH1\n\nend",
"structuredPatch": [
{
"oldStart": 1244,
"oldLines": 7,
"newStart": 1244,
"newLines": 9,
"lines": [
" \u2264 Real.log (3 * max |a| |b| + 5) + 1 := by",
" have h1 : |-b - 1| = b + 1 := by",
" rw [show -b - 1 = -(b + 1) by ring, abs_neg, abs_of_nonneg (by linarith [hb1])]",
"- have h2 : |-M| = M := abs_neg M",
"+ have h2 : |-M| = M := by",
"+ rw [abs_neg]",
"+ exact abs_of_nonneg (by linarith [hM1'] : (0 : \u211d) \u2264 M)",
" have h3 : max |-b - 1| |-M| = b + 1 := by",
" rw [h1, h2]",
" exact max_eq_left (by linarith [hMb'])"
]
}
],
"userModified": false,
"replaceAll": false
}
peer closed connection without sending complete message body (incomplete chunked read)
let stop here
thinking (3290 chars)
User wants to stop. Per §8.4 session END protocol (no exceptions before shutdown):
1. Write journal/session-005.md
2. File results/derivations; update index.md files; register in research.db
3. Update NGUYEN.md checklist/graveyard/ledger
4. Update STATE.md LAST with NEXT ACTION
5. git commit
But E-015 is still running in background (detached, ~1.5–2 hr). It will keep running after I stop — but computer shutdown would kill it. Should I ask? User said "let's stop here". The E-015 run will be killed by shutdown. Options: let it finish (it writes CSV/JSON itself), or leave running and note in STATE. Actually the script is crash-safe only at the end (writes at completion). If the user shuts down, the run is lost and must be restarted with --force (cache check: CSV doesn't exist → recomputes). That's acceptable — note in STATE.md.
Current session work to archive:
- E-015 launched (in progress, PID 23300)
- E-016 created and run: V1, V2, V4 pass; V3 fails (reconstruction mismatch — root cause under investigation; debug findings: θ=π zeros real, z1≈4.5/a law, tail noise in zero finder at large z, κ≠0 evidence)
- Key findings from debug:
* θ=π zeros are real (Thm 1.5 holds in Galerkin at θ=π)
* z1(π)·a ≈ 4.50, 4.499, 4.497, 4.496 — near-constant, ~1.43π; NOT rigid half-shift
* No θ=π zero in (0, π/a) — interlacing defect near origin worth understanding
* ρW−recon diff ~ +a/π at small y, crossing zero ~y≈7, → −1 at y=20: consistent with nonzero Im κ (exponential factor) — actually diff at small y = 1.274 = a/π exactly... interesting: +1.2739 ≈ a/π = 1.2732. Hmm, that suggests recon is missing ~ a/π at low y and gaining at high y — could be missing zeros (low zeros undercounted? count 381 vs expected... θ=0 has 382). Or the tail EM formula overshoots at large y.
* Actually likely diagnosis: zero-finder noise at large z (spurious brackets) corrupts recon at high y; and possibly a missing low zero. Root cause not yet pinned — that's exactly where we stopped.
Also network glitches interrupted several times.
Also E-016 has known bugs/issues to fix next session:
- V3 needs: robust zero location at large z (meromorphic/brentq guard), investigate Im κ (compute κ via W'(0)/W(0)), understand missing low zero (0, π/a).
Time check: I should be efficient. Write journal, D-024 (or keep E-016 as experiment with check failure documented + derivation for analysis?), update indexes, db registration, NGUYEN.md, STATE.md, commit.
E-016 currently has CHECK FAILURE — it wrote CSV/JSON with checks recorded. Register as experiment with failed check + findings. Create D-024 derivation documenting the θ=π zero structure findings (real zeros, z1·a≈4.5 law, κ≠0 evidence, V3 open problem). Actually keep it lean: journal + D-024 + STATE + NGUYEN.md updates + db registration + commit.
Let me check db.py usage for log-experiment and add-derivation, and look at index files format. Also check journal naming: session-004.md exists presumably.
Also should I kill E-015? User said stop here — if they shut down, run dies. I'll leave it running (it might finish if they don't shut down immediately) and note in STATE.md: "if E-015 CSV absent at next session start, rerun with --force". Mention to user.
Let me look at existing files quickly for format conventions.
Understood — stopping. Running the session-end protocol now (§8.4). Note: **E-015 is still running in the background** (~1.5–2 hr total); if you shut down now it's lost and will be rerun next session — its script is cached/resumable via `--force`. Let me archive everything.
tool: Bash
input
{
"command": "ls D:/Python/Quantum/research/journal/ D:/Python/Quantum/research/derivations/ && tail -5 D:/Python/Quantum/research/derivations/index.md && python D:/Python/Quantum/research/numerics/scripts/db.py 2>&1 | head -20",
"description": "Inspect journal/derivation conventions and db.py CLI"
}
thinking (599 chars)
Now write the archive files. Keep it efficient:
1. journal/session-005.md
2. D-024 derivation (E-016 analysis — partial, V3 open)
3. Update derivations/index.md
4. db.py log-experiment E-015 (in progress) + E-016 (check failure, diagnostic findings) and add-derivation D-024
5. NGUYEN.md: session log §8.6, checklist 3.4 note? E-016 belongs to Q-S3c — mark in §5 Phase 3 line? Update Session 005 entry.
6. STATE.md last.
7. Commit.
Check db.py log-experiment signature quickly... I'll just try with --help inline. To save cycles, do journal + D-024 + index first, then db, then STATE, then commit.
Writing the archive files.
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-005.md",
"content": "# Session 005 \u2014 2026-08-18\n## Plan (user-approved): do ALL queued directions one by one (Fork Rule)\nOrder: A3 \u2192 A1 \u2192 A2 \u2192 C \u2192 B, with D (Li n>2000) launched in background.\nA3 = Q-S3c Herglotz diagnostic; A1 = Q-S3a analytic; A2 = Q-S3b backend;\nC = \u03be\u2032-world ceiling; B = T4b scoping; D = Li n=3000 detached run.\n\n## Work done\n### D (background): E-015 launched\n- Created E-015-li-coefficients-n3000.py (E-011 method, n_max=3000,\n dps=1450, own output files \u2014 avoids clobbering E-011's registered CSV).\n- First detached launch FAILED: sandbox blocks DuplicateHandle\n (WinError 5) for multiprocessing spawn. Relaunched with sandbox\n disabled \u2014 healthy (main PID 23300 + 14 workers). ETA ~1.5\u20132 h.\n- LESSON: detached multiprocessing jobs must be launched with the\n sandbox disabled (session-004 machinery note amended).\n- **Run still in progress at session stop** (user called stop). If the\n machine shuts down, rerun: E-015 script with --force (crash-safe:\n outputs written only at completion).\n\n### A3 (Q-S3c): E-016 created, run, V3 fails \u2014 rich diagnostic findings\n- E-016-herglotz-fine-structure.py: \u03c1_W(y) = \u2212(1/\u03c0) Im(W\u2032/W)(iy) via\n ANALYTIC Galerkin z-derivative (no finite differences) at \u03b8=\u03c0 and\n \u03b8=0 control, a \u2208 {4,6,8,10}, vs \u03c1_F(y) = (1/\u03c0)[2/y + \u03be\u2032/\u03be \u2212 \u03be\u2033/\u03be\u2032].\n- Bugs found and fixed during bring-up: (i) hat-derivative missing a\n factor h; (ii) \u03be(1) removable pole at y=0.5 (mp.zeta(1) raises) \u2014\n exact value \u00bd supplied; (iii) flint cross-check NaN at s=1 (arb pole)\n \u2014 point dropped; (iv) |W|-minima zero finder with brent polish was\n noisy at large z (spurious brackets where |W| grazes) \u2014 replaced by\n exact real-gauge sign-change + brentq (c=1 exactly \u27f9 W(iy-real) pure\n imaginary, g(z)=z ImV\u2212ReV).\n- Results: V1 (plateau = a/\u03c0, \u22640.15%) PASS; V2 (\u03c1_F vs finite-diff of\n log|F|, 1.3e-8) PASS; V4 (mirror symmetry) PASS; xi vs flint 8.8e-17.\n **V3 (Poisson reconstruction from located zeros + EM lattice tail vs\n analytic \u03c1_W) FAILS: rel ~0.3 at all a.**\n- Debug findings (analysis in D-024):\n * \u03b8=\u03c0 zeros ARE real (sign changes confirmed) \u2014 Thm 1.5 survives\n Galerkin at \u03b8=\u03c0.\n * z\u2081(\u03b8=\u03c0)\u00b7a \u2248 4.504, 4.499, 4.497, 4.496 at a=4,6,8,10 \u2014 a stiff\n ~4.5/a law, NOT the rigid half-shift \u03c0/2a. No \u03b8=\u03c0 zero in (0,\u03c0/a).\n * \u03c1_W \u2212 recon: +a/\u03c0 at y=0.5, monotone decrease, zero crossing\n y\u22487, \u22120.99 at y=20 (a=4) \u2014 structure consistent with nonzero\n Im \u03ba (exponential-factor gauge) and/or missing/extra zeros; root\n cause NOT pinned at stop time.\n * d_pi \u2248 d_zero (\u03b8=\u03c0 marginally WORSE on raw L2 over [1,20]:\n 1.2201 vs 1.2165 at a=4) \u2014 raw L2 is plateau-dominated; the\n informative metric must weight small y.\n- Session interrupted 3\u00d7 by network dropouts (no state loss); user\n stopped during V3 root-causing.\n\n## Decisions\n- E-016 kept as experiment-with-failed-check (CSV/JSON written with\n checks recorded honestly); V3 root cause = first task next session.\n\n## Next session\n1. Finish V3 root-cause (D-024 \u00a74 lists the three candidates:\n Im \u03ba \u2260 0 measurable via W\u2032(0)/W(0); low-zero bookkeeping vs \u03b8=0;\n large-z zero-finder noise contaminating recon).\n2. Then resume approved order: A1 \u2192 A2 \u2192 C \u2192 B; collect E-015 (or rerun).\n"
}
thinking (296 chars)
E-016 logged. E-015 CSV doesn't exist yet (run in progress) — db.py requires the file for sha256. Register E-015 next session when done; note in STATE. Now add D-024 (that command didn't run because the chain broke? The && chain: E-015 failed → add-derivation didn't run). Run add-derivation now.
E-016 registered. E-015 can't be registered yet (CSV doesn't exist until the run finishes — will register next session). Completing D-024 registration.
tool: Bash
input
{
"command": "cd D:/Python/Quantum && python research/numerics/scripts/db.py add-derivation D-024 --title \"E-016 analysis: Herglotz fine structure at theta=pi \u2014 partial; V3 open; theta=pi zero structure findings\" --task Q-S3c --status \"partial (V3 root cause open)\" --file research/derivations/D-024-herglotz-fine-structure-analysis.md",
"description": "Register D-024 in research.db"
}
thinking (290 chars)
Now update NGUYEN.md: §8.6 session log + the Q-S3 line in CURRENT POSITION... NGUYEN.md §5 Phase 3 3.4 line mentions E-010 proposed; add E-016 status. Minimal edits:
1. §5 Phase 3, 3.4c line: append E-016 status. Actually Q-S3c is under 3.4c. Update the line.
2. §8.6 add Session 005 entry.
Now updating NGUYEN.md (§5 checklist + §8.6 session log) and STATE.md.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\NGUYEN.md",
"old_string": " 3.4c COMPLETE (Suzuki operators: D-018, E-009a/b, D-019 \u2014 \u03bb_a\n cliff measured; Galerkin Weil form validated to 4.5e-6 vs zero\n side; E-010 proposed: W(a,\u03b8;z) zeros vs \u03b3_j).",
"new_string": " 3.4c COMPLETE (Suzuki operators: D-018, E-009a/b, D-019 \u2014 \u03bb_a\n cliff measured; Galerkin Weil form validated to 4.5e-6 vs zero\n side; E-010 proposed: W(a,\u03b8;z) zeros vs \u03b3_j).\n Q-S3c (session 005): E-016/D-024 \u2014 Herglotz fine structure at\n \u03b8=\u03c0; \u03b8=\u03c0 zeros real, z\u2081\u00b7a\u22484.5 stiff law, V3 reconstruction\n mismatch open (session 006 first task)."
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\NGUYEN.md",
"old_string": " E-014): Cor 1.6 density wall mapped to a=10 (maximal lattice,\n interlacing exact, no thinning \u2014 scale limitation, not a wall);\n \u03b8=\u03c0 candidate found; Q-S3a/b/c queued. Detached-run machinery\n established.",
"new_string": " E-014): Cor 1.6 density wall mapped to a=10 (maximal lattice,\n interlacing exact, no thinning \u2014 scale limitation, not a wall);\n \u03b8=\u03c0 candidate found; Q-S3a/b/c queued. Detached-run machinery\n established.\n- Session 005 (2026-08-18): user approved do-all-one-by-one plan\n (A3\u2192A1\u2192A2\u2192C\u2192B, D in background). D launched: E-015 Li n\u22643000\n detached (in progress at stop; sandbox blocks multiprocessing spawn \u2014\n detached launches need sandbox disabled). A3 done: E-016 + D-024 \u2014\n Herglotz \u03c1_W at \u03b8=\u03c0 vs \u03c1_F; V1/V2/V4 pass, V3 (zero-Poisson\n reconstruction) fails rel ~0.3, root cause open; findings: \u03b8=\u03c0 zeros\n real, z\u2081(\u03c0)\u00b7a\u22484.5 stiff law, no \u03b8=\u03c0 zero in (0,\u03c0/a), raw-L2 \u03b8=\u03c0\u2248\u03b8=0\n (plateau-dominated metric). Stopped mid-V3-root-cause (user break;\n 3 network dropouts, no state loss)."
}
Now STATE.md (last, per protocol).
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\STATE.md",
"content": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-18 | Session: 005 (stopped mid-A3-root-cause; user break)\n\n## CURRENT POSITION\n\u2714 PHASE 0 COMPLETE. \u2714 PHASE 1 COMPLETE (Track A monitoring mode).\nPHASE 3 (Track C): 3.1 mapped, 3.2 arithmetic side DONE to n=2000\n(E-011/R-001), 3.3 CLOSED (D-021 \u2192 G-003, E-012), 3.4a/b/c COMPLETE.\nSuzuki line: Q-S3c ATTACKED (E-016, D-024 \u2014 partial): Herglotz \u03c1_W at\n\u03b8=\u03c0 vs \u03c1_F; V1/V2/V4 pass, **V3 (zero-Poisson reconstruction) fails\nrel ~0.3 \u2014 root cause open**. Findings: \u03b8=\u03c0 zeros REAL; z\u2081(\u03c0)\u00b7a \u2248 4.5\nstiff law (NOT rigid half-shift); no \u03b8=\u03c0 zero in (0,\u03c0/a); raw-L2\n\u03b8=\u03c0\u2248\u03b8=0 (plateau-dominated; need small-y-weighted metric).\n\n## ACTIVE PLAN (user-approved 2026-08-18): do ALL queued items one by one\nOrder: A3 \u2192 A1 \u2192 A2 \u2192 C \u2192 B; D runs in background.\n A3 = Q-S3c Herglotz diagnostic \u2014 IN PROGRESS (V3 root cause)\n A1 = Q-S3a analytic v\u00b1 \u2192 A_\u221e^{-1}e^{\u00b1\u00b7} convergence\n A2 = Q-S3b prime-sum backend redesign \u2192 a ~ 12\u201314\n C = \u03be\u2032-world bandwidth-one ceiling (LawN256 analogue)\n B = T4b scoping (node mechanism on \u03be\u207d\u1d4f\u207e)\n D = E-015 Li n\u22643000 detached run\n\n## IN PROGRESS\n- **E-015 (item D) RUNNING detached, PID 23300** (n_max=3000, dps=1450;\n ETA ~1.5\u20132 h from 2026-08-18 launch; log:\n research/numerics/data/E-015-run.log). If machine was shut down:\n rerun `./.venv/Scripts/python.exe research/numerics/scripts/\n E-015-li-coefficients-n3000.py --force` DETACHED WITH SANDBOX\n DISABLED (sandbox blocks multiprocessing DuplicateHandle \u2014 WinError 5;\n session-005 lesson). After completion: db.py log-experiment E-015\n (registration deferred \u2014 CSV didn't exist at session end).\n\n## NEXT ACTION\n1. Root-cause E-016 V3 (D-024 \u00a74): three candidates \u2014 (a) Im \u03ba \u2260 0\n (measure directly: Im(W\u2032/W)(0)); (b) low-zero bookkeeping (count 381\n vs lattice 382 at a=4); (c) large-z zero-finder float64 noise in\n recon (test: recon with zeros \u2264 60 + EM tail from 60; if y=20\n mismatch changes materially, (c) confirmed).\n2. Then resume order: A1 \u2192 A2 \u2192 C \u2192 B; collect/register E-015.\n\n## COMPLETED (Session 005)\n- [x] E-015 created (E-011 method, n_max=3000, own output files) +\n launched detached\n- [x] E-016 created + run: Herglotz fine structure \u03b8=\u03c0 vs F\n (V1/V2/V4 pass; V3 fail recorded honestly in JSON)\n- [x] D-024 analysis (partial); E-016 + D-024 registered in research.db\n- [x] Bring-up bugs fixed in E-016: hat-derivative factor h; \u03be(1)=\u00bd\n removable pole; flint NaN at s=1; zero-finder replaced by exact\n real-gauge brentq (c=1 \u27f9 W pure imaginary on \u211d at \u03b8=\u03c0)\n\n## OPEN CONJECTURES / THREADS\n- Q-S1 (D-019 \u00a73): L\u00b2\u2192uniform lifting for Cor 1.6 (Phase 4 candidate).\n- Q-S2: \u210b(T_a) as de Branges-type space (Phase 2.4 link, alive).\n- Q-S3: \u03b8(a) identification \u2014 \u03b8=\u03c0 candidate (D-023); Q-S3c attacked\n (D-024): stiff z\u2081\u00b7a\u22484.5 law at \u03b8=\u03c0 is a NEW constraint any \u03b8(a)\n theory must explain; density-collapse transition must start at a\u226b10;\n Q-S3a/b queued.\n- Q6 uncertainty \u2194 bandwidth-one ceiling bridge (D-001); Q14 Gutzwiller\n (Phase 2); T2 Christoffel/CMS (D-006); T4b higher derivatives (D-022\n \u00a75); \u03be\u2032-world ceiling (LawN256 analogue).\n- 3.2 next: E-015 (n\u22643000) in flight; then asymptotic attack; optional\n Arb ball-arithmetic certification.\nCLOSED: T1 (D-010/E-004), T3 (D-009), T4 (D-022), T5 (\u2192G-001),\n3.4a (D-016), 3.3 (\u2192G-003).\n\n## BLOCKERS\n(none \u2014 E-016 V3 is an open problem, not a blocker; other items proceed)\n\n## HANDOFF NOTES\n- db.py: init, log-experiment, add-result, add-derivation, add-graveyard,\n list (no --session flag on add-derivation; log-experiment REQUIRES the\n CSV to exist \u2014 sha256'd at registration).\n- venv interpreter: ./.venv/Scripts/python.exe (NOT ./venv/...).\n- git commits need: git -c user.name=\"NGUYEN-RH-1\" -c\n user.email=\"research@local\" commit (config must NOT be modified).\n- Break prompt rule active: ask before long runs, especially near 5:00 PM.\n- Hardware: Ryzen 9 7945HX 16c/32t, 64GB RAM, RTX 4090 Laptop 16GB.\n- venv packages: pypdf, sympy 1.14, mpmath 1.3, numpy 2.5.2, scipy 1.18,\n flint (python-flint; use for ALL zeta/Stieltjes \u2014 mpmath's are broken\n beyond ~16 digits in this env; mp.zetazero is safe; real-axis mp.zeta\n at dps 40 verified to 8.8e-17 vs flint at s=1.5/10.5/20.5 \u2014 E-016).\n- LONG RUNS: CLI kills Bash tasks at ~2 min default (10 min max). Launch\n DETACHED (session-004 pattern) **with sandbox DISABLED** (session-005:\n sandboxed detached launch dies \u2014 multiprocessing spawn DuplicateHandle\n WinError 5), then poll the log file.\n- flint.acb.stieltjes(k) scales ~k\u00b3 (28s @ k=2000, 3530 bits); full\n 0..3000 sweep ETA ~1.5\u20132 h on 14 Pool workers.\n- E-009b eigensolver floor: eigenvalues < ~1e-11 at h=5e-3 are at/below\n backward-error noise \u2014 quote as upper bounds only.\n- E-010/E-016 build cache pattern: (a,h) builds reused; sieve to e^20\n covers a \u2264 10 (~15s).\n- E-016: at \u03b8=\u03c0 the exact real gauge is g(z) = z Im Vp \u2212 Re Vp\n (c=\u221a(n\u208a/n\u208b)=1 exactly by mirror symmetry) \u2014 brentq on sign changes;\n do NOT use |W|-minima+polish (noisy at large z).\n"
}
result 2026-08-18 11:06:48Z → 2026-08-21 18:46:35Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\research\\journal\\session-014.md",
"content": "# Session 014 \u2014 2026-08-21 \u2014 Tier ii\u2032 launched: design + two discoveries\n\n## Directive\nUser chose option (a): do tier ii\u2032 (close SfBoundProp in Lean) BEFORE\nsubmission, with Paper A frozen and submittable at all times. Freeze\nprotocol in force: no edits to paperA.tex/paperB.tex/draft.tex/\nreferences.bib until ii\u2032-7; no edits to existing Lean proofs except the\nliCoeff correctness fix (below, flagged); commit at every sub-stage.\n\n## Work done (sub-stage ii\u2032-0: design \u2192 D-064)\n1. Baseline verified clean (git: only untracked LaTeX artifacts).\n2. Two parallel inventory agents surveyed: (i) vendored residue/\n meromorphic API; (ii) Zeta23 zero infrastructure + NguyenRH1 defs +\n Mathlib zeta/digamma/Jensen/Borel\u2013Carath\u00e9odory surface.\n3. Read liCoeff/Sf/perimeter defs, D-036, D-056, D-037 (full), and the\n primary source (Lagarias math/0404394 \u00a74\u20136, archived PDF).\n4. Axiom-checked all planned Zeta23 dependencies LIVE: zero_sum_inv_sq,\n zeta_local_zero_count, zeta_reflect_zero, zeta_mult_reflect,\n RectangleIntegral'_eq_sumResiduesIn, ResidueMult, logDerivResidue,\n riemannZetaLogDerivResidue \u2014 ALL 3 standard axioms only.\n5. Settled the exact identity chain NUMERICALLY (dps 40\u201350, mpmath +\n E-021 zero cache; see D-064 \u00a73 table).\n6. Wrote D-064 (design doc: discoveries, routes R1\u2013R5, lemma graph,\n revised sub-stage map ii\u2032-1a..7). Registered in index.md + db.\n\n## DISCOVERY 1 \u2014 liCoeff encoding defect (correctness, not cosmetic)\n`liCoeff` (LiSkeleton.lean:39) is an unordered tsum whose family is NOT\nSummable for any n \u2265 1 (summand ~ n/|\u03b3|, \u03a3 mult/|\u03c1| = \u221e by RvM).\nTherefore liCoeff n = 0 (junk), making `LiVerifiedUpTo` FALSE and\n`SfBoundProp` FALSE as encoded (at n = 10\u2075: |Sf| \u2265 432 501 > 347 000\n\u2265 Bn by the certified LiBasePoints bounds). The session-012 theorems\nare vacuously true; the session-013 audit could not see this (it checks\naxioms, not hypothesis truth). Paper A's mathematics is unaffected \u2014\npure encoding defect. FIX DESIGNED (D-064 \u00a71): paired symmetric form\n(manifestly real, absolutely summable via pair bound |pair| \u2264\nC(n)\u00b7mult/|\u03c1|\u00b2 + Zeta23's proved `zero_sum_inv_sq`). Scheduled as\nsub-stage ii\u2032-1a (first executable item).\n\n## DISCOVERY 2 \u2014 the crux identity is Sf = Res\u2081 EXACTLY\nThe D-036/D-037 bookkeeping, settled against Lagarias's paper and\nverified numerically:\n- **Sf(n) := S_\u221e(n) + 1 \u2212 \u03bb_n = Res\u2081(n) = \u03a3_{j=1}^n C(n,j)\u00b7c_{j\u22121}**\n (c_k = Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1). Follows from Lagarias\n (4.10) [\u03c4_j = \u03b7_j + (\u22121)\u02b2\u03c3_{j+1} \u2212 (\u22121)\u02b2\u03b4] summed binomially.\n Verified: (4.10) at j=0 exact to 15 digits (\u03bb\u2081 = 0.02309570896612103);\n crux consequence \u03bb\u2082 = 0.09234573523 matches E-015's certified\n 0.09234573522804667 exactly.\n- **Residue theorem**: sides(T) = \u03bb_n(T) + triv + Res\u2081(n), verified to\n 2e-21 (n=2) by direct contour integration.\n- **Assembly is TAIL-FREE**: |Sf| \u2264 |\u03bb_n(T)| + |sides| + |triv| \u2264\n 2N(T) + [side pieces] + 1 = B(n) \u2014 no tail term anywhere (the crux\n absorbs it). This is why D-037's B(n) has none.\n- The w=1 pole (order n+1) residue computation reduces to a Leibniz\n rule: (w\u22121)^{n+1}J = [w\u207f \u2212 (w\u22121)\u207f]h(w), h = (w\u22121)(\u2212\u03b6\u2032/\u03b6) analytic \u2014\n Res = \u03a3_j C(n,j)c_{j\u22121} confirmed algebraically.\n- SIDE EFFECT: D-037's line-63 compressed identity needs an erratum\n pointer to D-064 (correct reading: Sf = Res\u2081 = sides \u2212 \u03bb_n(T) \u2212 triv).\n\n## Inventory highlights (why tier ii\u2032 is feasible)\n- Rectangle residue theorem EXISTS (vendored FromPNTPlus\n `RectangleIntegral'_eq_sumResiduesIn`, axiom-clean) \u2014 needs the\n simple-pole scaffold extended to finite order (bounded, R1).\n- `logDerivResidue`, `ResidueMult`, `riemannZetaLogDerivResidue`\n vendored and clean.\n- Mathlib HAS Borel\u2013Carath\u00e9odory, Jensen's formula, digamma,\n HadamardThreeLines, MeromorphicAt/Order/NormalForm/FactorizedRational.\n- Zeta23 HAS proved \u03a3 mult/(1+\u2016\u03b3_\u03c1\u2016\u00b2) < \u221e (zero_sum_inv_sq), local\n zero count, RvM, reflect/conj symmetry, window finiteness.\n- Mathlib has NO Weierstrass/Hadamard factorization; FromPNTPlus\n StrongPNTPrefix has the local Blaschke/Borel toolkit.\n\n## Route decisions (D-064 \u00a74)\nR1 build higher-order residue machinery (1\u20132 sess); R2 crux via (4.10)\n\u2014 investigate zeta-23 EF reuse (\u03b22) first, Hadamard fallback (\u03b21)\n(2\u20134 sess, the long pole); R3 R-bound via LOCAL route\n(Borel\u2013Carath\u00e9odory/Jensen/Blaschke \u2014 machinery exists) (2\u20133 sess);\nR4 the old \"w=1 \u2194 S_\u221e\" monster is KILLED by Discovery 2 (factors into\nR1's Leibniz computation + R2's (4.10)); R5 perimeter unchanged.\n\n## State at stop\nD-064 complete; sub-stage map ii\u2032-1a..7 (total ~9\u201316 sessions).\nNEXT: ii\u2032-1a (liCoeff pairing fix + summability). Paper A untouched\n(compiles at session-013 state; submission-ready fallback intact \u2014\nD-064 \u00a78).\n",
"structuredPatch": [],
"originalFile": null
}
thinking (125 chars)
Committed (6034d3c), clean tree. Session-end protocol complete. E-015 still running — remind user about shutdown consequence.
Session 005 archived and committed (`6034d3c`). Clean stop.
**Summary of where we are:**
- **E-016 (A3)** ran: V1/V2/V4 pass, **V3 (zero-Poisson reconstruction) fails** — root cause is the first task for next session (three candidates documented in D-024 §4).
- Real findings banked: θ=π zeros are real; stiff law z₁(θ=π)·a ≈ 4.5 (not the naive half-shift); no θ=π zero in (0, π/a).
- **E-015 (D)** is still running in the background (PID 23300, ~1.5–2 h total). If you shut down now it dies and gets rerun next session — nothing else is lost; everything else is committed.
Resume point is in STATE.md with the explicit NEXT ACTION. Have a good break.
o-15 is done
thinking (444 chars)
E-015 done, ALL CHECKS PASS. Register in db, commit data files, update STATE (E-015 complete; registration done). Also update NGUYEN.md? The result ledger entry #3 covers n≤2000; extend to n≤3000? Ledger update appropriate: update entry 3 to n ≤ 3000. Keep it brief.
Actions:
1. db.py log-experiment E-015
2. Update NGUYEN.md ledger entry 3 (n ≤ 3000, λ_3000 = 8617.2192, ratio 0.999845)
3. Update STATE.md (E-015 done + registered)
4. Commit.
E-015 finished — **ALL CHECKS PASS**. Registering and archiving.
tool: Bash
input
{
"command": "cd D:/Python/Quantum && python research/numerics/scripts/db.py log-experiment E-015 --name \"Li coefficients arithmetic side n<=3000 (Bombieri-Lagarias Thm 2, zero-free)\" --script research/numerics/scripts/E-015-li-coefficients-n3000.py --params \"n_max=3000,dps=1450,arb_bits=5029\" --csv research/numerics/data/E-015-li-coefficients-n3000.csv --json research/numerics/data/E-015-li-coefficients-n3000.json --rows 3000 --session 5",
"description": "Register E-015 in research.db"
}
E-015 is fully archived (commit `ae98392`):
**E-015 results — Li coefficients, arithmetic side, n ≤ 3000:**
- **ALL CHECKS PASS**: λ₁ error 4.31e-102; E-008 cross-check worst ratio 0.125; self-consistency 3.81e-1005
- **λ₃₀₀₀ = 8617.2192**, all λₙ > 0 for n ≤ 3000
- Refined asymptotic ratio **0.999845** at n=3000 (vs 1.00216 at n=2000 — continuing to converge toward 1 from above, consistent with Arias de Reyna)
- Ledger entry #3 extended to n ≤ 3000; registered in research.db
- Runtime: ~86 min (Stieltjes sweep 61 min + V5 22 min)
Item **D** of the approved plan is done. Remaining queue: finish **A3** (V3 root cause), then **A1 → A2 → C → B**. Nothing else running; clean state whenever you're ready.
we will leave now an come back later to continue
thinking (403 chars)
User is leaving. Everything is already committed and clean. Just confirm state is safe. Maybe quick git status check to be sure nothing uncommitted. Also save a small memory? The workflow memory already covers staged workflow. Nothing new worth saving — maybe the sandbox-disabled detached launch lesson is already in STATE.md (excluded from memory since it's documented in project files). Just confirm.
Everything is already committed and the tree is clean — nothing lives only in conversation:
- **Session 005 archived**: journal, D-024, indexes, research.db, STATE.md — commits `6034d3c` + `ae98392`
- **Resume point**: STATE.md → NEXT ACTION is the E-016 V3 root cause (three candidates listed), then A1 → A2 → C → B
- No background jobs running; safe to shut down
See you when you're back.