← Session 37 — Paper A revisions: certificate retraction + margin fix
thinking (9936 chars)
The user raises three edits plus asks me to verify first. Key instruction: "Always verify first before make changes. My suggestion might not be right."
Claims to verify:
**Claim 1:** Rosser's Theorem 19 directly gives |N(T) − P(T)| ≤ 0.137 log T + 0.443 log log T + 1.588 for the appropriate range, where P(T) = (T/2π)log(T/2πe) + 7/8. And N(T) = F(T) + S(T) with that F is "not exact" — the exact relation has an additional remainder (Trudgian's Corollary 1 form: |N(T) − P(T)| ≤ |S(T)| + 0.2/T₀).
Let me think about the mathematics carefully.
The Riemann–von Mangoldt formula: N(T) = (T/2π)log(T/2πe) + 7/8 + S(T) + O(1/T), where S(T) = (1/π)arg ζ(1/2 + iT) (when T is not a zero ordinate). The classical exact statement:
N(T) = (θ(T)/π) + 1 + S(T), where θ(T) is the Riemann–Siegel theta function. And θ(T)/π + 1 = (T/2π)log(T/2πe) + 7/8 + O(1/T). Specifically the expansion: θ(T) = (T/2)log(T/2πe) − π/8 + 1/(48T) + 7/(5760T³) + ... So N(T) = (T/2π)log(T/2πe) + 7/8 + S(T) + [1/(48πT) + ...].
So indeed N(T) = P(T) + S(T) + r(T) with r(T) = O(1/T) — the identity N(T) = F(T) + S(T) with F = P is NOT exact; it has a small remainder. Trudgian-II's Corollary 1 (which I read in the PDF): "For T ≥ T₀ ≥ e: |N(T) − (T/2π)log(T/2πe) − 7/8| ≤ 0.111 log T + 0.275 log log T + 2.450 + 0.2/T₀." — yes! That's exactly the +0.2/T₀ remainder form. So the user is RIGHT that N = F + S is not exact with F = P; there's the arctan/theta remainder bounded by ~0.2/T (or 1/(48πT)-ish leading term... wait: θ(T) expansion: θ(T) = (T/2)ln(T/2π) − T/2 − π/8 + 1/(48T) + 7/(5760T³) − ... So θ(T)/π + 1 = (T/2π)ln(T/2πe) + 7/8 + 1/(48πT) + ... and N(T) = θ(T)/π + 1 + S(T). So N(T) − P(T) − S(T) = 1/(48πT) + O(1/T³) ≈ 0.0066/T. Trudgian's 0.2/T₀ is a safe explicit bound for this remainder. So the exact relation: N(T) = P(T) + S(T) + O(1/T) with the O-term ≤ 0.2/T explicitly (for T ≥ e say).
Now — is Rosser's Theorem 19 the direct |N(T) − P(T)| bound? The user cites Alex Kontorovich's PNT+ formalization blueprint (PrimeNumberTheoremAnd project) as stating Rosser's theorem in this form. Let me verify: Rosser 1941 "Explicit bounds for some functions of prime numbers" — Theorem 19. I need to check this. Rosser's paper does contain explicit bounds on N(T). I recall Rosser 1941 Theorem 19: "|N(T) − (T/2π)log(T/2πe) − 7/8| ≤ 0.137 log T + 0.443 log log T + 1.588 for T ≥ 1467"? Hmm — actually I believe Rosser's paper has the S(T) bound as Theorem... and the N(T) bound as a corollary or separate theorem. The user says Theorem 19. Kontorovich's PNTAnd blueprint (the Lean formalization project for PNT) states "Rosser's theorem" as the |N(T) − P(T)| bound — I should verify by fetching the Kontorovich blueprint page (link provided by the user) and/or MaRDI portal link.
The user gave two links:
1. MaRDI Portal: https://portal.mardi4nfdi.de/wiki/Item%3AQ2436775 — about Trudgian-II.
2. Alex Kontorovich: https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html — the zeta chapter of the PNTAnd blueprint.
I should fetch the Kontorovich blueprint to see exactly how they state Rosser's theorem (theorem number, constants, range). And check whether it's Theorem 19 of Rosser 1941.
**Claim 2 (notation collision Λ_n):** TRUE — I introduced Λ_n for Keiper normalization in Step 5, while footnote 3 uses Λ_n for Voros's deformed sequence. The footnote says "those concern a deformed sequence Λ_n, not the ordinary λ_n". Indeed a collision. Fix: rename the Keiper-normalized coefficient to κ_n (kappa) in Step 5, and keep Λ_n for Voros's deformed sequence in footnote 3 (or rename footnote's to Λ_n^V). The user suggests κ_n for Johansson/Keiper: λ_n = nκ_n. Good — verify Voros's notation: does Voros 2017 actually use Λ_n for his deformed sequence? Session 042's footnote text says "a deformed sequence Λ_n" — presumably following Voros's paper notation. I'll keep Λ_n for Voros (as the footnote already has) and switch Johansson's to κ_n.
Also the bib note for johansson2015 now says "the Keiper coefficients $\Lambda_n$ to $n = 10^5$" — should become "$\kappa_n$"? The bib note should match the paper's notation: change to "$\kappa_n$". Hmm — but Johansson himself calls them λ_n. The note could say "the Keiper coefficients ($\kappa_n$ in the text's notation)". Simpler: "rigorous interval-arithmetic computation of the Keiper coefficients to $n = 10^5$ ($\kappa_n$ in this paper's notation)". Or just "the Keiper coefficients $\kappa_n$". I'll match paper notation with a parenthetical.
**Claim 3 (F vs P notation collision):** TRUE — the paper uses F(T) for the RvM main term in the new Step-2 sentence AND F as the "honest zero-count majorant" (FAlt). Also the Appendix B derivation note uses N(T) = F(T) + S(T). Fix: use P(T) for the classical main term, keep F for the majorant.
**Verification needed:**
1. Is Rosser 1941 Theorem 19 the |N(T) − P(T)| bound with those constants and T ≥ 1467? Verify via Kontorovich blueprint + another source if possible.
2. What does Trudgian-II's Corollary 1 say — I already read it: "|N(T) − T/2π log(T/2πe) − 7/8| ≤ 0.111 log T + 0.275 log log T + 2.450 + 0.2/T₀ for T ≥ T₀ ≥ e". ✓ (I have this from the PDF page 2.)
3. Is the exact N(T) = θ(T)/π + 1 + S(T)? Yes, standard (Backlund). And θ(T)/π + 1 = P(T) + O(1/T). The user is right that N = P + S is not exact.
4. Voros's deformed sequence notation Λ_n — check Voros 2017 paper? We have arXiv 1703.02844 reference; not downloaded. The footnote text was written in session 042 saying "deformed sequence Λ_n". Whether Voros uses Λ or another letter — for the paper's purposes, the footnote just needs to not collide; if Voros uses different notation, even easier. I could verify via arXiv PDF. Let me download it and check quickly.
Also the user's MZeroProp strengthening suggestion: "the archive should provide the actual interval subdivision, rigorous derivative/Lipschitz bound, precision, source code, output certificate, and hash" — that's an archive/documentation task, not a paper edit (the paper could add a sentence pointing to the E-023 artifacts). The user frames it as "not necessarily a flaw — deserves unusually explicit documentation". I could add a short parenthetical in the MZeroProp note: the certificate's components (script E-023, grid step, Lipschitz constant source, ×2 margin, output CSV/JSON in the archive). Do I have the E-023 details? E-023 = research/numerics/scripts/E-023-finite-height-assembly.py + E-023-assembly.csv/.json. The M₀ certificate: "grid evaluation with sampled-Lipschitz margin of safety ×2". What exactly was done? From NGUYEN.md: "D-036 |t|<10 gap found and closed (M₀ = 1.3352)". E-023 computed ‖ζ′/ζ(−5/2 + it)‖ on a grid over |t| ≤ 10 with mpmath, and a Lipschitz bound between grid points with factor ×2 safety. Details: I'd need to check E-023 script/json for the grid step and Lipschitz value. For the paper, a sentence: "(E-023: grid step h = ..., certified Lipschitz constant L = ..., margin ×2; script + CSV/JSON certificate in the project archive)". Let me check the E-023 JSON sidecar for those parameters.
Plan:
1. Verify Rosser Theorem 19 via the Kontorovich blueprint (fetch the provided link) + MaRDI link if needed.
2. Verify Voros's deformed-sequence notation (download arXiv 1703.02844, grep).
3. Check E-023 JSON for the M₀ certificate parameters (grid step, Lipschitz).
4. Make edits:
a. Step 2: replace "the Riemann--von Mangoldt formula $N(T) = F(T) + S(T)$ with Rosser's explicit bound |S(T)| ≤ ..." with "Rosser's explicit Riemann--von Mangoldt error bound $|N(T) - P(T)| \le 0.137\log T + 0.443\log\log T + 1.588$ ($T \ge 1467$, \cite[Thm~19]{rosser1941}), $P(T) = \frac{T}{2\pi}\log\frac{T}{2\pi e} + \frac78$, together with the finite-range bounds ..." — wait, but the finite-range verifications |S| ≤ 1, 2 are for S(T), not for N − P. Hmm — the small-T coverage: for T < 1467, we need |N(T) − P(T)| bounds too. N(T) − P(T) = S(T) + O(1/T), so |N − P| ≤ |S| + 0.2/T ≤ 1 + 0.2/T (T ≤ 280) etc. The additive constants in C0hat/Z2hat absorb this. In the paper's derivation note I'll phrase: "the additive constants absorb the ranges below Rosser's threshold via the finite-range verifications |S(T)| ≤ 1 (T ≤ 280), |S(T)| ≤ 2 (T ≤ 6.8×10⁶) \cite[Eq.~(1.2)]{trudgian2014ii} and the $O(1/T)$ theta remainder \cite[Cor.~1]{trudgian2014ii}". Trudgian-II Corollary 1 gives the |N − P| ≤ |S-bound| + 0.2/T₀ form — good to cite for the remainder.
b. Derivation note after table: rewrite to use |N(T) − P(T)| bound directly: "the two window forms apply the bound at the window endpoints and take differences; the global form uses |N(T)| ≤ P(T) + R(T) directly; the additive constants and the enlargement of F absorb ...".
c. Rename F(T) → P(T) in the RvM contexts (Step 2 sentence, derivation note, table ZeroCountGlobalProp row "global zero count N(T) ≤ F(T)" → "N(T) ≤ \widehat N(T)"? hmm — the Lean Prop is N-count ≤ FAlt(T). The table row says "global zero count $N(T) \le F(T)$, $T \ge 2$" — F here = the formal majorant (FAlt). Keep F there (it's the majorant, consistent with Step 2's "F is the honest zero-count majorant"). The classical main term becomes P everywhere. So: Step 2 RvM sentence uses P; derivation note uses P; F remains the majorant. Clean.
d. κ_n rename in Step 5 (Λ_n → κ_n), keep λ_n = nκ_n, Λ_{100000} → κ_{100000} = 4.6258…, λ_{100000} = 10⁵κ_{100000}. Also bib note. Also check: does anything else use Λ_n? The footnote 3 (Voros deformed Λ_n) stays.
e. MZeroProp sentence: add certificate specifics (grid step, Lipschitz, ×2; script/CSV/JSON in archive) after checking E-023 sidecar.
5. Recompile, verify, read pages, commit.
Also — the user says "After those, I would stop iterative internal rewriting and move to an independent expert audit". Noted — I'll do these edits and stop.
Let me start verification. First fetch the Kontorovich blueprint zeta chapter and check Rosser theorem statement.
Verifying before editing, per your instruction. Three things to check: (1) Rosser 1941 Thm 19 form and range via the Kontorovich blueprint you linked; (2) whether Voros's deformed sequence actually uses $\Lambda_n$ (arXiv:1703.02844); (3) the E-023 certificate parameters for the MZeroProp documentation note.
tool: WebFetch ok
input
{
"url": "https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html",
"prompt": "Find any statement of Rosser's theorem about N(T), the zero-counting function: exact statement, constants (0.137, 0.443, 1.588?), the main-term formula P(T), the valid range (T >= 1467?), and whether it cites Rosser 1941 Theorem 19 or another theorem number. Also any statement relating N(T) to S(T) and a remainder term."
}
result 2026-08-27 03:05:04Z → 2026-08-27 03:06:38Z
bytes=187140 code=200 code_text='OK' result='## Rosser\'s theorem on N(T) — Theorem 8.3.1\n\n**Exact statement (Theorem 8.3.1, titled "Rosser 1941, Theorem 19"):** For T ≥ 1467,\n\n> "|N(T) − (T/2π) log(T/2πe) − 7/8| ≤ 0.137 log T + 0.443 log log T + 1.588"\n\n- **Constants:** 0.137, 0.443, **1.588** (yes, 1.588)\n- **Valid range:** T ≥ 1467 (yes — not T ≥ 2)\n- **Main term P(T):** (T/2π)·log(T/2πe) + 7/8\n- **Citation:** Rosser **1941, Theorem 19** (bibliography ref. [51], "rosser1941"). The page explicitly corrects earlier blueprint text, noting that "Rosser–Schoenfeld (1962) has no such theorem, and Rosser\'s range is \\(T\\ge 1467\\)".\n- **Lean declaration:** `RS.theorem_19`\n- Section 8.3 is headed "The zeta function bounds of Rosser and Schoenfeld," but the theorem itself is attributed solely to Rosser 1941.\n\n## Related statements on the page\n\n- **Definition 8.1.5 (Riemann–von Mangoldt estimate):** the general template |N(T) − (T/2π) log(T/2πe) − 7/8| ≤ b₁ log T + b₂ log log T + b₃, stated for T ≥ 2 (Lean: `riemannZeta.RvM`, `riemannZeta.Riemann_vonMangoldt_bound`).\n\n- **Lemma 8.5.7 (Backlund\'s bound):** same shape with constants "(b_1, b_2, b_3) = (0.137, 0.443, 6.1)", valid for all T ≥ 2, with main term written (T/2π)log(T/2π) − T/2π + 7/8. The page explains Backlund\'s original used constant 5.225 against the simpler main term (T/2π)log(T/2πe), and absorbing the 7/8 offset gives 5.225 + 7/8 = 6.1.\n\n## N(T) and S(T)\n\n**No statement relating N(T) to the argument function S(T) plus a remainder term appears in this content.** The Riemann–von Mangoldt estimates above (with the 7/8 built into the main term) are the only forms given; S(T) is never mentioned.' duration_ms=94233.32879999999 url='https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html'
tool: Bash
input
{
"command": "cd /tmp && curl -sL -o voros2017.pdf \"https://arxiv.org/pdf/1703.02844\" && file voros2017.pdf && cat /d/Python/Quantum/research/numerics/data/E-023-assembly.json 2>/dev/null | head -50",
"description": "Download Voros 2017 arXiv PDF and show E-023 JSON sidecar",
"timeout": 60000
}
raw content_json (database row)
[{"type":"thinking","thinking":"The user raises three edits plus asks me to verify first. Key instruction: \"Always verify first before make changes. My suggestion might not be right.\"\n\nClaims to verify:\n\n**Claim 1:** Rosser's Theorem 19 directly gives |N(T) − P(T)| ≤ 0.137 log T + 0.443 log log T + 1.588 for the appropriate range, where P(T) = (T/2π)log(T/2πe) + 7/8. And N(T) = F(T) + S(T) with that F is \"not exact\" — the exact relation has an additional remainder (Trudgian's Corollary 1 form: |N(T) − P(T)| ≤ |S(T)| + 0.2/T₀).\n\nLet me think about the mathematics carefully.\n\nThe Riemann–von Mangoldt formula: N(T) = (T/2π)log(T/2πe) + 7/8 + S(T) + O(1/T), where S(T) = (1/π)arg ζ(1/2 + iT) (when T is not a zero ordinate). The classical exact statement: \n\nN(T) = (θ(T)/π) + 1 + S(T), where θ(T) is the Riemann–Siegel theta function. And θ(T)/π + 1 = (T/2π)log(T/2πe) + 7/8 + O(1/T). Specifically the expansion: θ(T) = (T/2)log(T/2πe) − π/8 + 1/(48T) + 7/(5760T³) + ... So N(T) = (T/2π)log(T/2πe) + 7/8 + S(T) + [1/(48πT) + ...].\n\nSo indeed N(T) = P(T) + S(T) + r(T) with r(T) = O(1/T) — the identity N(T) = F(T) + S(T) with F = P is NOT exact; it has a small remainder. Trudgian-II's Corollary 1 (which I read in the PDF): \"For T ≥ T₀ ≥ e: |N(T) − (T/2π)log(T/2πe) − 7/8| ≤ 0.111 log T + 0.275 log log T + 2.450 + 0.2/T₀.\" — yes! That's exactly the +0.2/T₀ remainder form. So the user is RIGHT that N = F + S is not exact with F = P; there's the arctan/theta remainder bounded by ~0.2/T (or 1/(48πT)-ish leading term... wait: θ(T) expansion: θ(T) = (T/2)ln(T/2π) − T/2 − π/8 + 1/(48T) + 7/(5760T³) − ... So θ(T)/π + 1 = (T/2π)ln(T/2πe) + 7/8 + 1/(48πT) + ... and N(T) = θ(T)/π + 1 + S(T). So N(T) − P(T) − S(T) = 1/(48πT) + O(1/T³) ≈ 0.0066/T. Trudgian's 0.2/T₀ is a safe explicit bound for this remainder. So the exact relation: N(T) = P(T) + S(T) + O(1/T) with the O-term ≤ 0.2/T explicitly (for T ≥ e say).\n\nNow — is Rosser's Theorem 19 the direct |N(T) − P(T)| bound? The user cites Alex Kontorovich's PNT+ formalization blueprint (PrimeNumberTheoremAnd project) as stating Rosser's theorem in this form. Let me verify: Rosser 1941 \"Explicit bounds for some functions of prime numbers\" — Theorem 19. I need to check this. Rosser's paper does contain explicit bounds on N(T). I recall Rosser 1941 Theorem 19: \"|N(T) − (T/2π)log(T/2πe) − 7/8| ≤ 0.137 log T + 0.443 log log T + 1.588 for T ≥ 1467\"? Hmm — actually I believe Rosser's paper has the S(T) bound as Theorem... and the N(T) bound as a corollary or separate theorem. The user says Theorem 19. Kontorovich's PNTAnd blueprint (the Lean formalization project for PNT) states \"Rosser's theorem\" as the |N(T) − P(T)| bound — I should verify by fetching the Kontorovich blueprint page (link provided by the user) and/or MaRDI portal link.\n\nThe user gave two links:\n1. MaRDI Portal: https://portal.mardi4nfdi.de/wiki/Item%3AQ2436775 — about Trudgian-II.\n2. Alex Kontorovich: https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html — the zeta chapter of the PNTAnd blueprint.\n\nI should fetch the Kontorovich blueprint to see exactly how they state Rosser's theorem (theorem number, constants, range). And check whether it's Theorem 19 of Rosser 1941.\n\n**Claim 2 (notation collision Λ_n):** TRUE — I introduced Λ_n for Keiper normalization in Step 5, while footnote 3 uses Λ_n for Voros's deformed sequence. The footnote says \"those concern a deformed sequence Λ_n, not the ordinary λ_n\". Indeed a collision. Fix: rename the Keiper-normalized coefficient to κ_n (kappa) in Step 5, and keep Λ_n for Voros's deformed sequence in footnote 3 (or rename footnote's to Λ_n^V). The user suggests κ_n for Johansson/Keiper: λ_n = nκ_n. Good — verify Voros's notation: does Voros 2017 actually use Λ_n for his deformed sequence? Session 042's footnote text says \"a deformed sequence Λ_n\" — presumably following Voros's paper notation. I'll keep Λ_n for Voros (as the footnote already has) and switch Johansson's to κ_n.\n\nAlso the bib note for johansson2015 now says \"the Keiper coefficients $\\Lambda_n$ to $n = 10^5$\" — should become \"$\\kappa_n$\"? The bib note should match the paper's notation: change to \"$\\kappa_n$\". Hmm — but Johansson himself calls them λ_n. The note could say \"the Keiper coefficients ($\\kappa_n$ in the text's notation)\". Simpler: \"rigorous interval-arithmetic computation of the Keiper coefficients to $n = 10^5$ ($\\kappa_n$ in this paper's notation)\". Or just \"the Keiper coefficients $\\kappa_n$\". I'll match paper notation with a parenthetical.\n\n**Claim 3 (F vs P notation collision):** TRUE — the paper uses F(T) for the RvM main term in the new Step-2 sentence AND F as the \"honest zero-count majorant\" (FAlt). Also the Appendix B derivation note uses N(T) = F(T) + S(T). Fix: use P(T) for the classical main term, keep F for the majorant.\n\n**Verification needed:**\n1. Is Rosser 1941 Theorem 19 the |N(T) − P(T)| bound with those constants and T ≥ 1467? Verify via Kontorovich blueprint + another source if possible.\n2. What does Trudgian-II's Corollary 1 say — I already read it: \"|N(T) − T/2π log(T/2πe) − 7/8| ≤ 0.111 log T + 0.275 log log T + 2.450 + 0.2/T₀ for T ≥ T₀ ≥ e\". ✓ (I have this from the PDF page 2.)\n3. Is the exact N(T) = θ(T)/π + 1 + S(T)? Yes, standard (Backlund). And θ(T)/π + 1 = P(T) + O(1/T). The user is right that N = P + S is not exact.\n4. Voros's deformed sequence notation Λ_n — check Voros 2017 paper? We have arXiv 1703.02844 reference; not downloaded. The footnote text was written in session 042 saying \"deformed sequence Λ_n\". Whether Voros uses Λ or another letter — for the paper's purposes, the footnote just needs to not collide; if Voros uses different notation, even easier. I could verify via arXiv PDF. Let me download it and check quickly.\n\nAlso the user's MZeroProp strengthening suggestion: \"the archive should provide the actual interval subdivision, rigorous derivative/Lipschitz bound, precision, source code, output certificate, and hash\" — that's an archive/documentation task, not a paper edit (the paper could add a sentence pointing to the E-023 artifacts). The user frames it as \"not necessarily a flaw — deserves unusually explicit documentation\". I could add a short parenthetical in the MZeroProp note: the certificate's components (script E-023, grid step, Lipschitz constant source, ×2 margin, output CSV/JSON in the archive). Do I have the E-023 details? E-023 = research/numerics/scripts/E-023-finite-height-assembly.py + E-023-assembly.csv/.json. The M₀ certificate: \"grid evaluation with sampled-Lipschitz margin of safety ×2\". What exactly was done? From NGUYEN.md: \"D-036 |t|<10 gap found and closed (M₀ = 1.3352)\". E-023 computed ‖ζ′/ζ(−5/2 + it)‖ on a grid over |t| ≤ 10 with mpmath, and a Lipschitz bound between grid points with factor ×2 safety. Details: I'd need to check E-023 script/json for the grid step and Lipschitz value. For the paper, a sentence: \"(E-023: grid step h = ..., certified Lipschitz constant L = ..., margin ×2; script + CSV/JSON certificate in the project archive)\". Let me check the E-023 JSON sidecar for those parameters.\n\nPlan:\n1. Verify Rosser Theorem 19 via the Kontorovich blueprint (fetch the provided link) + MaRDI link if needed.\n2. Verify Voros's deformed-sequence notation (download arXiv 1703.02844, grep).\n3. Check E-023 JSON for the M₀ certificate parameters (grid step, Lipschitz).\n4. Make edits:\n a. Step 2: replace \"the Riemann--von Mangoldt formula $N(T) = F(T) + S(T)$ with Rosser's explicit bound |S(T)| ≤ ...\" with \"Rosser's explicit Riemann--von Mangoldt error bound $|N(T) - P(T)| \\le 0.137\\log T + 0.443\\log\\log T + 1.588$ ($T \\ge 1467$, \\cite[Thm~19]{rosser1941}), $P(T) = \\frac{T}{2\\pi}\\log\\frac{T}{2\\pi e} + \\frac78$, together with the finite-range bounds ...\" — wait, but the finite-range verifications |S| ≤ 1, 2 are for S(T), not for N − P. Hmm — the small-T coverage: for T < 1467, we need |N(T) − P(T)| bounds too. N(T) − P(T) = S(T) + O(1/T), so |N − P| ≤ |S| + 0.2/T ≤ 1 + 0.2/T (T ≤ 280) etc. The additive constants in C0hat/Z2hat absorb this. In the paper's derivation note I'll phrase: \"the additive constants absorb the ranges below Rosser's threshold via the finite-range verifications |S(T)| ≤ 1 (T ≤ 280), |S(T)| ≤ 2 (T ≤ 6.8×10⁶) \\cite[Eq.~(1.2)]{trudgian2014ii} and the $O(1/T)$ theta remainder \\cite[Cor.~1]{trudgian2014ii}\". Trudgian-II Corollary 1 gives the |N − P| ≤ |S-bound| + 0.2/T₀ form — good to cite for the remainder.\n b. Derivation note after table: rewrite to use |N(T) − P(T)| bound directly: \"the two window forms apply the bound at the window endpoints and take differences; the global form uses |N(T)| ≤ P(T) + R(T) directly; the additive constants and the enlargement of F absorb ...\".\n c. Rename F(T) → P(T) in the RvM contexts (Step 2 sentence, derivation note, table ZeroCountGlobalProp row \"global zero count N(T) ≤ F(T)\" → \"N(T) ≤ \\widehat N(T)\"? hmm — the Lean Prop is N-count ≤ FAlt(T). The table row says \"global zero count $N(T) \\le F(T)$, $T \\ge 2$\" — F here = the formal majorant (FAlt). Keep F there (it's the majorant, consistent with Step 2's \"F is the honest zero-count majorant\"). The classical main term becomes P everywhere. So: Step 2 RvM sentence uses P; derivation note uses P; F remains the majorant. Clean.\n d. κ_n rename in Step 5 (Λ_n → κ_n), keep λ_n = nκ_n, Λ_{100000} → κ_{100000} = 4.6258…, λ_{100000} = 10⁵κ_{100000}. Also bib note. Also check: does anything else use Λ_n? The footnote 3 (Voros deformed Λ_n) stays.\n e. MZeroProp sentence: add certificate specifics (grid step, Lipschitz, ×2; script/CSV/JSON in archive) after checking E-023 sidecar.\n \n5. Recompile, verify, read pages, commit.\n\nAlso — the user says \"After those, I would stop iterative internal rewriting and move to an independent expert audit\". Noted — I'll do these edits and stop.\n\nLet me start verification. First fetch the Kontorovich blueprint zeta chapter and check Rosser theorem statement.","signature":""},{"type":"text","text":"Verifying before editing, per your instruction. Three things to check: (1) Rosser 1941 Thm 19 form and range via the Kontorovich blueprint you linked; (2) whether Voros's deformed sequence actually uses $\\Lambda_n$ (arXiv:1703.02844); (3) the E-023 certificate parameters for the MZeroProp documentation note."},{"type":"tool_use","id":"WebFetch_112","name":"WebFetch","input":{"url":"https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html","prompt":"Find any statement of Rosser's theorem about N(T), the zero-counting function: exact statement, constants (0.137, 0.443, 1.588?), the main-term formula P(T), the valid range (T >= 1467?), and whether it cites Rosser 1941 Theorem 19 or another theorem number. Also any statement relating N(T) to S(T) and a remainder term."},"extra_content":null},{"type":"tool_use","id":"Bash_113","name":"Bash","input":{"command":"cd /tmp && curl -sL -o voros2017.pdf \"https://arxiv.org/pdf/1703.02844\" && file voros2017.pdf && cat /d/Python/Quantum/research/numerics/data/E-023-assembly.json 2>/dev/null | head -50","description":"Download Voros 2017 arXiv PDF and show E-023 JSON sidecar","timeout":60000},"extra_content":null}]