Independent check of `MAJOR_ARC_EXPLICIT.md` (commit a795a3f)

2026-09-12, attempt a-0080. Scripts written fresh for this review, not reusing major_arc_explicit_probe.py: major_arc_explicit_independent_check.py (item 1) and major_arc_explicit_section5_check.py (item 5), both under this directory, results quoted below.

Summary of verdicts: (1) confirmed, (2) defective, (3) confirmed, (4) confirmed with one inherited defect from (2), (5) confirmed with one inaccurate closing sentence. The defect found in (2) does not appear to break the boxed result (S''), because the threshold c_0 is taken to be a small absolute constant well inside the corrected condition; it does break the derivation of the compatibility hypothesis (H) as written, and the document's own stated numeric bound 2/9 should read 1/6.

(1) Section 2: the explicit formula on an arc, the three integral bounds,

the zero-sum estimate, the choice of T

Verdict: confirmed.

The remainder in (1). With T = R^3, (1+N|theta|) <= 1+R/r <= 1+R and N/T = N/R^3, so the bracket (1+N|theta|)[(N/T)l^2+N^(1/4)l] is at most (1+R)(N/R^3)l^2 + (1+R)N^(1/4)l, which is << N l^2 R^{-2} for the first piece (worst case r=1) and negligible for the second, matching the document. The O(l^2) for prime powers of primes dividing r: with r <= R = e^{sigma sqrt l} < Z, the number of prime powers p^k <= N with p | r is at most omega(r) * l/log2 <= (sqrt(sigma) sqrt l) * l/log2, order l^{3/2}, each weighted by 1 in the exponential sum; O(l^2) is a valid, non-tight, upper bound for this, not an error.

Numerical check, written fresh (not major_arc_explicit_probe.py). major_arc_explicit_independent_check.py computes I_rho(theta) by composite 16-point Gauss-Legendre with panel edges spaced uniformly in the phase phi(t) (found by Newton's method, since phi is monotone for gamma,theta>=0), not uniformly in t; a first attempt with panels uniform in t badly aliased the fast oscillation near t=1 and gave ratios up to 6.5, caught by cross-checking against the closed form I_rho(0)=(N^rho-1)/ rho (exact reldiff ~1e-13 after the fix, reported in the script's own sanity-check block). On the document's own sample (60 zeros of zeta with gamma<=163.0 from mpmath.zetazero, N in {1e4,1e6}, N theta in {0,1, 10,100}, 472 integrals after discarding gamma<1):

far range (|gamma|>=4 pi N|theta|): largest |I_rho|/(N^beta/|gamma|) is 1.009918 (at N=1e4, N theta=0, gamma=101.318).
stationary range (1<=|gamma|<4 pi N|theta|): largest |I_rho|/(N^beta/ sqrt|gamma|) is 0.064235 (at N=1e4, N theta=10, gamma=65.113).

These match the document's own section 7 numbers (1.010 and 0.064) closely enough to be the same computation done independently, which confirms both the sign/normalisation of the bounds and the specific constants quoted.

The zero-sum estimate (2) and the choice of T. N(U,chi) << U log(rU) + log r is the standard zero-counting bound for L(s,chi) (Davenport, Multiplicative Number Theory, the analogue of the zeta zero-count with the log r term for small U); correctly quoted. Recomputing the two exponents: r(H_0+2) <= r*4piR/r + 2r = 4 pi R + 2r <= (4 pi+2) R, so log(r(H_0+2)) <= log R + O(1) = sigma sqrt l + O(1), giving the far-zero contribution N^{beta _chi(H_0)} <= N exp(-c l/log(r(H_0+2))) = N exp(-(c/sigma) sqrt l (1+o(1))), matching the boxed c/sigma. With T=R^3, 3rT <= 3 R * R^3 = 3R^4, so log(3rT) <= 4 log R + O(1) = 4 sigma sqrt l + O(1), giving N exp(-c l/ log(3rT)) = N exp(-(c/(4 sigma)) sqrt l (1+o(1))), matching the boxed c/(4 sigma). Both recomputations agree with the document exactly.

(2) Section 3, lemma (5): the Gauss-sum expansion, the model side H_chi,

the twisted main term, and the passage to (4)

Verdict: defective. The step from H_chi(theta) = ... + O(r E_md(1+R/ r)) to the final bound "+8R E_md" in (4) drops a factor of sqrt r (up to sqrt R at the extreme).

Where the error is. F(alpha) and H(alpha) are both represented, up to the exact prime-power term for F, as (1/phi(r)) sum_chi chi(a) tau(chi-bar) [F_chi(theta) or H_chi(theta)]; this is an exact identity for both sides (section 1's remark that both sums vanish for (b,r)>1 on the H side makes it exact there too, not just an approximation). Subtracting and bounding termwise with |tau(chi-bar)| <= sqrt r (GS) gives, correctly, |W(alpha)| <= sqrt r * max_chi |F_chi(theta) - H_chi(theta)| + O(l^2). This inequality applies sqrt r to every part of F_chi - H_chi, including whatever error is already inside H_chi. The document's own derivation of that error is H_chi(theta) = delta_chi K_N(theta) - ... + O(r E_md(1+R/r)), where the factor r comes from summing the (up to r, in fact phi(r)) residue classes b mod r that make up the character sum H_chi(theta) = sum_b chi(b) H_b(theta), each class contributing O(E_md(1+R/r)) after partial summation against e(n theta). So the error carried inside F_chi - H_chi from the model side is O(r E_md(1+R/r)), and multiplying by the sqrt r from the Gauss-sum step (as the inequality above requires) gives

sqrt(r) * r * E_md * (1+R/r) = r^{3/2} E_md + r^{1/2} R E_md,

which at the extreme r = R (both terms increasing in r on [1,R]) is O(R^{3/2} E_md), not O(R E_md). The ratio of the correct bound to the document's stated one is sqrt(r), unbounded as R -> infinity, so this is not a matter of a loose but valid inequality: R E_md is too small to be a valid upper bound for sqrt(r) * O(r E_md(1+R/r)) at r near R. By contrast the F_chi side's own error N Upsilon(r) is correctly carried with its sqrt r in (5) (the term sqrt r * N * Upsilon(r)), which shows the sqrt r was applied to one contribution and not the other, rather than omitted as a matter of principle.

The corrected statement. Equation (4) should read |W(alpha)| <= sqrt r max_chi |F_chi - delta_chi K_N + ...| + O(R^{3/2} E_md) + O(l^2), and (5)'s boxed bound should carry R^{3/2} N e^{-sqrt(l)/3} in place of R N e^{-sqrt (l)/3}.

Propagation. Squaring, this term becomes R^3 N^2 e^{-2 sqrt(l)/3} in place of R^2 N^2 e^{-2 sqrt(l)/3} in section 4's squared bound, and after the same * N l factor from int_M(|F|+|H|)^2, the corresponding exponent in (6) is 2/3 - 3 sigma in place of 2/3 - 2 sigma. At the claimed balance sigma = c_0, requiring this exponent to be at least c_0 gives c_0 <= 1/6, not the document's stated c_0 <= 2/9 (item 4 below redoes this arithmetic). Since 1/6 < 2/9, condition (H) as printed is not sufficient for the argument as printed; the corrected condition needs the extra clause c_0 <= 1/6 (or the whole minimum lowered to it, since 1/6 < 2/9). This does not by itself refute the boxed result (S''), since c_0 is described as "TT's sufficiently small c_0" and is presumably far below 1/6 in any case; it is a genuine arithmetic error in the paper's own verification of its hypothesis, not merely a stylistic slip, and the checked value 2/9 printed in both section 4 and the boxed (H) is wrong as a consequence of the missing sqrt r.

The rest of (5). The exceptional-zero matching term 1_{q~|r} sqrt r N e^{-c_0 sqrt l} is correctly built from the F_chi side alone (the model carries the same beta_e term with the same coefficient, by construction of a(n), so it cancels inside W exactly when q_e | r and chi = chi_{e,r}, per section 3's first bullet; this is unaffected by the item above). The identification of the twisted main term with I_{beta_e} (the parenthetical remark) is correct: for chi = chi_{e,r} and (n,r)=1, chi_e(n)=chi_e(b) on the class, and the untwisted-with-weight-n^{beta_e-1} sum's main term is exactly int_1^N t^{beta_e-1} e(t theta) dt = I_{beta_e}(theta) by (Md) and partial summation, as claimed.

(3) Section 3, matching the exceptional zeros with Page at (Q,T)=(R,R^3)

Verdict: confirmed. Checked the specific case named in the assignment and found it does not arise: a character mod r "induced by a real primitive character of conductor not dividing r" is a contradiction in terms (an induced character's inducing primitive character's conductor divides r by definition of induction), so this exact phrase names an empty case, not a missed one.

Looking for the substance behind the hint instead: does the three-way split (chi induced by tilde-chi=chi_e; induced by tilde-chi != chi_e; not induced by tilde-chi at all) miss a character whose TT-exceptional conductor q_e exceeds the arc cutoff R? Checked directly: if q_e > R, then q_e does not divide any arc modulus r <= R, so 1_{q_e|r}=0 on every major arc, on both sides of W. On the model side, the exceptional twist in a(n) still exists (it is defined at scale Z, not R), but section 1's own version of (Md), sum_{n<=y,n=b(r)} nu(n) chi_e(n) n^{beta_e-1} = 1_{q_e|r} chi_e(b) (y^{beta_e}-1)/(beta_e phi(r)) + O(E_md), already states the main term is zero (not just uncounted) when q_e does not divide r, with the whole sum absorbed into O(E_md); this is exactly the content of ENDPOINT_SHARP. md section 4's "twisted part, q nmid r" case (the character-orthogonality cancellation, equation (4.1)-(4.2) there), reused verbatim rather than re-derived. So the q_e>R case is already covered by (Md) as quoted, and does not need separate treatment in section 3.

The parenthetical remark ("a real zero of a real character that is not the Page-exceptional one is not exceptional for (ZF) either") is the correct invocation of Landau-Page uniqueness across the whole family of real primitive characters of conductor <=R, not just a per-character statement: (ZF) alone only asserts at most one real zero per character; the cross-character uniqueness needed to rule out a second, unrelated exceptional character is exactly what (Pg') supplies, and that is why it is invoked here rather than left to (ZF). The reduction of an imprimitive real chi mod r's real zero to its primitive inducing character's real zero is also correct: for chi induced by chi* mod q* (q*|r), L(s,chi) = L(s,chi*) * prod_{p|r,p nmid q*}(1-chi*(p)p^{-s}), and the extra Euler factors do not vanish for Re(s)>0 (each has modulus |1-chi*(p)p^{-s}|>0 there), so the real zeros of L(s,chi) in (0,1) are exactly those of L(s,chi*); likewise chi mod r is real if and only if chi* is real, since reduction (Z/r)^* -> (Z/q*)^* is surjective when q*|r, so chi real on (n,r)=1 forces chi* real on all of (Z/q*)^*.

Also checked: T in Page's box (Q,T)=(R,R^3) is irrelevant to this whole argument, since every zero under discussion in this section is real (Im=0<=T automatically); only Q=R (the conductor bound) does any work, which is consistent with T playing no role in the three bullets beyond fixing the zero-counting range for the non-exceptional zeros in section 2.

No case is missed by the three bullets.

(4) Section 4, the budget (6), the six exponents, the balance, and the

comparison with ARC_SPLIT_BUDGET.md

Verdict: confirmed, except for the exponent inherited from item (2).

Recomputing the six exponents from (6) as printed, with R=e^{sigma sqrt l}: writing each bracketed term as e^{-(rate) sqrt l},

R e^{-2 c_0 sqrt l}              -> rate 2 c_0 - sigma
R^2 e^{-(2c/sigma) sqrt l}       -> rate 2c/sigma - 2 sigma
R e^{-(c/(2 sigma)) sqrt l}      -> rate c/(2 sigma) - sigma
R^{-3}                           -> rate 3 sigma
R^2 e^{-2 sqrt(l)/3}             -> rate 2/3 - 2 sigma
R^{-1}                           -> rate sigma

matching the document's list exactly. Balancing the first and last (2 c_0 - sigma = sigma) gives sigma = c_0, at which both equal c_0, confirmed. Requiring the other four rates to be >= c_0 at sigma=c_0 reproduces the document's stated conditions exactly: 2c/c_0 - 2c_0 >= c_0 <=> c_0^2 <= 2c/3; c/(2c_0) - c_0 >= c_0 <=> c_0^2 <= c/4; 3c_0 >= c_0 automatic; and 2/3 - 2c_0 >= c_0 <=> c_0 <= 2/9, all recomputed independently and agreeing with the printed numbers. The document also correctly notes that c_0^2 <= c/4 implies c_0^2 <= 2c/3 (since c/4 < 2c/3), so only sqrt(c) /2 needs to appear in (H), not a separate bound for the second exponent.

But the fifth condition, c_0 <= 2/9, is the one item (2) shows is computed from an exponent that is wrong by a factor of R^{1/2}: with the corrected R^3 e^{-2 sqrt(l)/3} (rate 2/3 - 3 sigma), the condition at sigma=c_0 is 2/3 - 3 c_0 >= c_0 <=> c_0 <= 1/6. Section 4's own sentence "c_0<=2/9" and (H)'s boxed min(2/9, ...) should both read 1/6 in place of 2/9; this is the same defect as item (2), located here at its point of use rather than its point of origin.

The comparison with ARC_SPLIT_BUDGET.md (2 gamma/3 with gamma < c_0/ 2, giving at most c_0/3): arithmetically correct as a comparison of the two documents' own stated constants (c_0 vs 2 c_0/3, an improvement by more than 3 whenever the two c_0s are the same constant, which they are by construction here).

The repair of (1'), confirmed by direct recomputation. ENDPOINT_SHARP.md section 4's "Small y" step discards y <= N exp(-gamma sqrt l) and then falls back to the much weaker bound log y >= l/2 for everything that follows, in particular for bounding the exceptional term y^{tilde beta_y-1} <= exp(-c_0 log y/sqrt l) <= exp(-c_0 sqrt(l)/2). But the discarded range gives directly log y > log N - gamma sqrt l = l - gamma sqrt l, and using that instead: y^{tilde beta_y - 1} <= exp(-c_0 (l - gamma sqrt l)/sqrt l) = exp(c_0 gamma) * exp(-c_0 sqrt l), a fixed constant exp(c_0 gamma) times exp(-c_0 sqrt l), i.e. rate c_0 rather than c_0/2. This loses nothing elsewhere: the other steps in ENDPOINT_SHARP.md section 4 that use `log y

= l/2 (to get e^{sqrt(log y)} >= R`, etc.) only need a lower bound on

log y, and l - gamma sqrt l is a larger (better) lower bound than l/2 for fixed gamma as l -> infinity, so the sharper bound can replace l/2 everywhere in that argument without breaking anything else. This confirms both halves of the claim: ENDPOINT_SHARP.md's "Small y" paragraph does lose a factor of 2 exactly this way, and the repaired (1') (valid for any gamma < min(c_0, delta/sqrt2, 1/2)) does feed into ARC_SPLIT_BUDGET.md's own optimum (sigma=2 gamma/3) to give 2 c_0/3 in place of 2 gamma_old/ 3 with gamma_old<c_0/2, i.e. exactly a further factor 3/2 on top of the already-stated factor 3, as the document says.

(5) Section 5, the conditional lower bound and its corollary

Verdict: confirmed, with one imprecise closing sentence flagged below.

Why T needs l^4 and not l^2. Tracing the choice through: the remainder in (1) on the sub-arc is << N l^2/T; with T = q~^2 e^{kappa sqrt l} l^p, this is N e^{-kappa sqrt l} l^{2-p}/q~^2. Compared with N^{tilde beta} = N e^{-kappa sqrt l} itself (before any 1/sqrt(q~) rescaling), the ratio is l^{2-p}/q~^2. At p=2 this ratio is 1/q~^2, a fixed nonzero constant for fixed q~ (e.g. q~=3 gives 1/9), not o(1) as l -> infinity; at p=4 the ratio is l^{-2}/q~^2 -> 0 for any fixed q~. This is exactly the document's own justification once traced through (its "the ratio would be phi(q~)/q~^2, not small for q~=3" is the same point up to the precise constant in front), and l^4 is genuinely needed, not a safety margin: l^2 would leave the smallest admissible conductors (q~=3 being the smallest odd prime conductor) uncontrolled.

Page condition. Recomputed log(q~ T) = 3 log q~ + 4 log l + kappa sqrt l <= (kappa+3 eps) sqrt l (1+o(1)) using q~ <= e^{eps sqrt l}. Requiring tilde beta = 1-kappa/sqrt l > 1-b/log(q~T) reduces to kappa * log(q~T) < b sqrt l, and substituting the bound on log(q~T) gives exactly kappa (kappa+3 eps) < b for large l, matching the hypothesis.

(ZF) condition and the far-range error. Recomputed log(3 q~ T) <= (kappa+ 3 eps) sqrt l (1+o(1)), giving the far-range term N l^2 exp(-(c/(kappa+3 eps)) sqrt l). For this, multiplied by the outer sqrt(q~), to be o(N^ {tilde beta}/sqrt(q~)), the needed condition is c/(kappa+3 eps) > kappa + eps. The stated hypothesis is kappa < min(sqrt c, sqrt b) - 3 eps, i.e. kappa + 3 eps < sqrt c, i.e. (kappa+3 eps)^2 < c (the task names this condition with 2 eps; the document's own text uses 3 eps, which is a valid, slightly less tight, choice, not an error: (kappa+3eps)^2<c gives c/(kappa+3eps) > kappa+3eps > kappa+eps, i.e. more room than is needed). So the far-range error is controlled with margin to spare; no defect, and the stated 3 eps could very likely be relaxed to 2 eps but does not need to be.

Size and phase of K_N and I_beta, checked numerically (major_arc_explicit_section5_check.py, K_N by direct summation, I_beta by Simpson's rule at 4000 panels, no oscillation issue here since gamma=0 for a real zero) at N in {1e3, 2e4, 5e5}, beta in {0.9, 0.99, 0.999}, theta in {0, +-1/(16N), 1/(8N)} (the full range |theta|<=1/(8N)): |K_N(theta)|/N ranged 0.974 to 1.000 (claimed >=1/2); |I_beta (theta)|/N^beta ranged 0.975 to 1.111 (claimed >=1/4); the phase gap |arg(K_N) - pi(N+1)theta| was exactly 0 to floating precision (K_N has the closed form e^{i pi(N+1)theta} sin(pi N theta)/sin(pi theta), and the ratio is real and positive throughout this range, so the phase is exact, better than the claimed "within pi/8"), and |arg(I_beta) - pi(N+1)theta| ranged 0 to 0.021 radians (claimed <= pi/8 = 0.393). Both bounds hold with very large margin on this sample; cos(pi/4) as the resulting constant in Re(K_N-bar * I_beta) >= cos(pi/4)|K_N||I_beta| follows from the two phase gaps summing to at most pi/4 by the triangle inequality, correctly.

mu(q)=+-1 for odd squarefree q: immediate from the definition of the Mobius function, not worth a script.

The assembled lower bound vs SIEGEL_UNIFORMITY.md (23). The proven lower bound is c_3 N^{2 beta+1} q/phi(q)^3; (23)'s matching term is N^{2 beta+1} q^2/phi(q)^4. Their ratio is phi(q)/q <= 1, so the lower bound proven here is never larger than (23)'s upper bound for the same quantity, consistent with the stated reading: SIEGEL_UNIFORMITY.md shows this term would be size N^{2beta+1}q^2/phi(q)^4 if tilde chi were subtracted as exceptional; here it is not subtracted, and (7) shows the resulting error is at least of the matching order (up to the bounded factor phi(q)/q).

The corollary's arithmetic, recomputed step by step. Setting kappa = (1-tilde beta) sqrt l = (c_0+kappa'/2)/2 and solving for l gives sqrt l = (c_0+kappa'/2)/(2(1-tilde beta)), matching the stated choice exactly; kappa is the midpoint of c_0 and kappa'/2, hence strictly inside (c_0,kappa'/2) since kappa'>2c_0, confirmed. The stated equivalence q~<=e^{eps sqrt l} <=> 1-tilde beta <= eps(c_0+kappa'/2)/(2 log q~) is pure algebra from the same substitution, confirmed. The contradiction threshold: comparing (7)'s lower bound c_1 N^3 e^{-2 kappa sqrt l}/q~^2 against the assumed upper bound N^3 e^{-kappa' sqrt l} gives a contradiction exactly when q~^2 < e^{(kappa'-2 kappa) sqrt l}; substituting kappa=(c_0+kappa'/ 2)/2 gives kappa'-2 kappa = kappa'/2 - c_0, a fixed positive constant (since kappa'>2c_0), and for eps small enough that 2 eps < kappa'/2 - c_0, the assumed q~<=e^{eps sqrt l} already forces q~^2 <= e^{2 eps sqrt l} < e^{(kappa'/2-c_0) sqrt l}, supplying the contradiction as claimed.

What the corollary does and does not establish. It establishes: if E_corr^(Z)(N) << N^3 exp(-kappa' sqrt l) unconditionally for some fixed kappa'>2c_0, then no even primitive real character of odd conductor q~ has a real zero with 1-tilde beta <= c'/log q~, for q~ beyond an effective bound depending on kappa',c_0,eps. It does not establish this for odd primitive real characters (the proposition's proof uses tau(tilde chi)=sqrt(q~) for even tilde chi; the remark right after (7) explicitly says the odd case needs either an extra l^{-1} loss or a different, |W|^ 4-based, route, "neither is needed for the corollary", i.e. genuinely not done here); it does not establish anything for characters of even conductor (the argument uses q~ odd squarefree throughout, for the exact identity mu(q~)=+-1 and to avoid the separate mod-4/mod-8 cases SIEGEL_UNIFORMITY. md section 4 treats differently); and, being conditional on the assumed unconditional bound at kappa', it establishes nothing unconditionally by itself, only the stated conditional implication.

The closing sentence is not accurate as written. "That is an effective zero-free interval (1-c'/log q~,1) for real zeros of real characters, i.e. the non-existence of Siegel zeros in the classical form" drops the even/ odd-conductor qualification that the same paragraph (and the Proposition's own hypothesis) states explicitly two sentences earlier. The classical Siegel-zero statement is unqualified over all real primitive characters, both parities and every conductor with no special exclusion for the conductor's own parity; what is actually shown here, conditionally, is restricted to even primitive real characters of odd conductor. The document is internally consistent about this restriction everywhere except in this one closing sentence, which should read "for even primitive real characters of odd conductor" rather than "for real characters" generically, and should not be summarised as the classical statement without that qualifier.

What would settle the one open point

Item (2)'s defect is a bookkeeping error (a missing sqrt r), not a gap requiring new input; it is settled by the recomputation given above, and the fix is mechanical: replace 8 R E_md by 8 R^{3/2} E_md in (4)/(5), replace R^2 e^{-2 sqrt(l)/3} by R^3 e^{-2 sqrt(l)/3} in the squared bound of section 4, and replace c_0 <= 2/9 by c_0 <= 1/6 in section 4's text and in the boxed hypothesis (H). Nothing here was left unresolved.