An explicit class-number bound from the quasi-Riemann hypothesis

Status: candidate, pending external review. Every statement below is proved, given OpenAI's Theorem 1.1 (input theorem below, unreviewed). The analytic chain is an ordinary written proof by this hunt, not independently reviewed. Every numerical constant in it is evaluated with Arb ball arithmetic (python-flint, 128 bits), so the numerical steps are enclosure-carrying; the class-number lists are exact integer computations checked against brute force, PARI and Watkins. A composite claim takes the grade of its weakest step: here the unreviewed input theorem, then the unreviewed written proof.

0. The input theorem

OpenAI, The Quasi-Riemann Hypothesis (preprint dated 30 September 2026), Theorem 1.1: every Dirichlet L-function, including zeta, has no zero with Re s > 7/8 (the pole at s = 1 of a principal character excepted). Its Lean statements (OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re, OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re) were kernel-checked by this lab on two kernels on 2026-10-08 (section 7 of the input-theorem page numbered 38, unmerged PR #271). No human has reviewed the 195-page argument. This hunt uses the theorem and does not re-check it.

The October 5, 2026 OpenAI paper (also titled The Quasi-Riemann Hypothesis) proves the weaker half-plane Re s > 11/12 as its own Theorem 1.1, cites the 7/8 paper as [36], and remarks in its introduction that Littlewood's short Euler product gives h(D) >> sqrt|D| / log log |D| "with an absolute computable implied constant", which it does not compute. Everything below is stated for a general abscissa sigma0; section 6 gives the constants for sigma0 = 11/12 as well (c = 1/16 in Theorem A, D(1000) = 10 592 194 843), so a reader who prefers the October 5 statement loses only constants, and the lists below remain complete for h <= 1000 under 11/12 alone.

Hypothesis H(sigma0). For the character in question, every zero rho = beta + i gamma of L(s, chi) with 0 < beta < 1 has beta <= sigma0. OpenAI's Theorem 1.1 gives H(7/8) for every Dirichlet character.

1. Results

Throughout, D < 0 is a fundamental discriminant, q = |D|, chi = chi_D is the Kronecker symbol (primitive, real, odd), h(D) the class number, and for D < -4 Dirichlet's formula reads h(D) = sqrt(q) L(1, chi_D) / pi.

Theorem A (clean form). Given OpenAI's Theorem 1.1, for every negative fundamental discriminant D other than -3,

L(1, chi_D) >= 1 / (10 log log |D|),

and for every negative fundamental discriminant D,

h(D) >= sqrt|D| / (10 pi log log |D|).

So c = 1/10 in L(1, chi_D) >= c / log log |D| with A = 0 and D0 = 4, and the implied constant of OpenAI's h(D) >> sqrt|D| / log log |D| is 1/(10 pi) = 0.0318. (For D = -3, -4 the class-number statement is trivial: the right side is below 1. Section 4 splits the proof into an exact range, a verified cover and an analytic tail.)

Theorem B (the explicit inequality). Assume H(sigma0) for chi_D, with eps0 = 1 - sigma0 > 0. Let 1 < y < x, a = log y, b = log x, and delta_x, delta_y > 0. Then

log L(1, chi) >= M(a, b) - ( R(x, delta_x) + R(y, delta_y) + T(x) + T(y) ) / (b - a)

where, with S(t) = sum_{p <= t} 1/p and G as in (2.3),

M(a, b) = -(1/(b-a)) int_a^b S(e^u) du

• sum_{p <= y^(1/2)} (1/(2p^2) - 1/(3p^3)) + sum_{p <= y^(1/4)} (1/(4p^4) - 1/(5p^5)),

R(t, delta) = t^(-eps0) [ (delta + eps0) G(1 + delta) int_0^delta t^(-u) (u + eps0)^(-2) du

• int_delta^inf t^(-u) G(1 + u) (u + eps0)^(-1) du ],

T(t) = (pi^2/24) t^(-2) / log t.

R is increasing in q, so a value computed at q_hi bounds every q <= q_hi.

Theorem C (the table D(h)). Given OpenAI's Theorem 1.1: if h(D) <= h then |D| <= D(h), with D(h) from dtable.json (verified for every h <= 2000). Selected values:

Corollary D (the class-number lists). Given OpenAI's Theorem 1.1, the imaginary quadratic fields of class number h are exactly those found by the search of section 5, for every h <= 1500. In particular:

• for h <= 100 the search reproduces Watkins (2004) exactly: all 100 counts (total 42 272) and all 100 largest discriminants agree with OEIS A046125 / A038552; under QRH this is an independent completion of Watkins' classification, by a two-second computation;
• for every h <= 1000 there are 4 115 897 fields in all; the largest discriminant is D = -227 932 027 (h = 996);
• for every h <= 1500 there are 9 245 562 fields; the largest discriminant is D = -562 394 347 (h = 1489); every h <= 1500 occurs;
• for the 750 odd h <= 1500 the counts (995 897 fields) agree exactly with the GRH-conditional table of Holmin and Kurlberg (section 6), computed by a different method.

The full lists are reproducible with run_search.py 1000 5433399820 OUTDIR and run_search.py 1500 12409254457 OUTDIR; their sorted texts ("|D| h" per line) have sha256 075a201c8e7cce6c0b6013423a5d96c6308452afab2fa79882388fc46d8b0c45 (h <= 1000) and c373d7c67bbecb922878481c50e6273f03ed07b93fc968a458bb0bf5389ee101 (h <= 1500), and the h <= 1000 part of the second run has the same sha256 as the first. The counts F(h) and the largest |D| for each h are in search_H1000.json and search_H1500.json.

2. Proof of Theorem B

(2.1) Explicit formula. For x > 1 and real sigma >= 1 (Perron's formula with kernel x^w / w^2, contour moved to the left; the zero sum converges absolutely; see the proof of Lemma 2.5 in Lamzouri, Li and Soundararajan, Math. Comp. 84 (2015) 2391-2412, which is unconditional up to this display, and Davenport, Multiplicative Number Theory, chapters 12, 17 and 19):

sum_{n <= x} Lambda(n) chi(n) n^(-sigma) log(x/n) = -(L'/L)(sigma) log x - (L'/L)'(sigma)

• sum_rho x^(rho - sigma) / (rho - sigma)^2
• sum_{k >= 0} x^(-2k-1-sigma) / (2k + 1 + sigma)^2,

rho over the nontrivial zeros; the last sum is over the trivial zeros -1, -3, -5, ... of an odd character.

(2.2) Two cutoffs. Subtract the same identity at y < x. The term (L'/L)'(sigma) cancels. Integrate over sigma in [1, inf). On the left, int_1^inf n^(-sigma) dsigma = 1/(n log n) and log(x/n)^+ - log(y/n)^+ = log(x/y) w(n) with

w(n) = 1 (n <= y), log(x/n)/log(x/y) (y < n <= x), 0 (n > x).

On the right, L(sigma, chi) > 0 for real sigma >= 1 and tends to 1, so int_1^inf -(L'/L)(sigma) dsigma = log L(1, chi). The interchange of the zero sum and the integral is justified by the absolute bounds of (2.4). Hence

log L(1, chi) = sum_n Lambda(n) chi(n) w(n) / (n log n) + E, |E| <= ( Z(x) + Z(y) + T(x) + T(y) ) / (b - a), Z(t) = sum_rho int_1^inf t^(beta - sigma) |sigma - rho|^(-2) dsigma.

For the trivial zeros, int_1^inf t^(-2k-1-sigma) (2k+1+sigma)^(-2) dsigma <= t^(-2k-2) / ((2k+2)^2 log t), and the sum over k is at most T(t).

(2.3) Positivity. Let xi(s, chi) = (q/pi)^((s+1)/2) Gamma((s+1)/2) L(s, chi), entire of order one. Hadamard's product and Re B(chi) = -sum_rho Re(1/rho) (Davenport, chapter 12) give, for real sigma > 1,

sum_rho (sigma - beta) / |sigma - rho|^2 = Re xi'/xi(sigma, chi) = (1/2) log(q/pi) + (1/2) psi((sigma+1)/2) + (L'/L)(sigma, chi),

every term on the left positive. Since (L'/L)(sigma, chi) = -sum_p log p * chi(p) p^(-sigma) / (1 - chi(p) p^(-sigma)) and the summand is largest at chi(p) = -1, where it equals log p / (p^sigma + 1),

sum_rho (sigma - beta) / |sigma - rho|^2 <= G(sigma) := (1/2) log(q/pi) + (1/2) psi((sigma+1)/2) - zeta'/zeta(sigma) + 2 zeta'/zeta(2 sigma).

(2.4) The zero sum. Fix t > 1 and sigma1 = 1 + delta. Assume H(sigma0).

• For sigma >= sigma1: t^(beta-sigma) |sigma-rho|^(-2) = [t^(beta-sigma) / (sigma-beta)] (sigma-beta) |sigma-rho|^(-2), and beta -> t^(beta-sigma) / (sigma - beta) is increasing on beta < sigma, so the bracket is at most t^(sigma0-sigma)/(sigma-sigma0).
• For 1 <= sigma <= sigma1: as a function of gamma^2, the ratio |sigma-rho|^(-2) / [(sigma1-beta) |sigma1-rho|^(-2)] is largest at gamma = 0, so |sigma-rho|^(-2) <= (sigma1-beta)(sigma-beta)^(-2) (sigma1-beta) |sigma1-rho|^(-2); the factor t^(beta-sigma)(sigma1-beta)/(sigma-beta)^2 is increasing in beta < sigma (logarithmic derivative log t + 2/(sigma-beta) - 1/(sigma1-beta) > 0), so it is at most its value at beta = sigma0.

Summing over rho with (2.3) at sigma and at sigma1 and integrating, with u = sigma - 1, gives Z(t) <= R(t, delta). The only property of the zeros used is beta <= sigma0, which is H(sigma0).

(2.5) The prime sum. Write the n-sum as a sum over prime powers, sum_p sum_k chi(p)^k w(p^k) / (k p^k). For each p the inner sum is smallest at chi(p) = -1 (at chi(p) = 1 it is positive; at chi(p) = 0 it is zero; the alternating series with nonincreasing terms a_k = w(p^k)/(k p^k) is negative). With chi(p) = -1 it equals -w(p)/p + sum_{k>=2} (-1)^k a_k, and the alternating tail is at least a_2 - a_3 (+ a_4 - a_5) when p^2 <= y (p^4 <= y), where a_2 = 1/(2p^2) and a_4 = 1/(4p^4) exactly and a_3 <= 1/(3p^3), a_5 <= 1/(5p^5); for p^2 > y it is nonnegative. Finally w(p) = (1/(b-a)) int_a^b 1[log p <= u] du, so sum_p w(p)/p = (1/(b-a)) int_a^b S(e^u) du. This is M(a, b). Combining with (2.2) and (2.4) proves Theorem B.

(2.6) Evaluation (lbound.py). All in Arb:

• int S(e^u) du is exact below 10^7 (S is a step function: int_a^c S(e^u) du = c S(e^c) - a S(e^a) - sum_{e^a < p <= e^c} log p / p, prefix sums over the 664 579 primes up to 10^7) and above 10^7 uses Rosser and Schoenfeld (1962), Theorem 5 (3.20): S(t) < log log t + B + 1/(2 log^2 t) for t >= 286, with the Meissel-Mertens constant B = gamma + sum_{k>=2} mu(k) log zeta(k) / k enclosed with a rigorous tail (test: it overlaps the published 50 digits).
• zeta'/zeta through Arb power series of zeta; psi through Arb's digamma.
• Both integrals in R are upper Riemann sums on a fixed rational grid (step 1/2000 on [0, 1/2], 1/500 on [1/2, 2], 1/100 on [2, 6]): on each cell the factor (u + eps0)^(-1) or ^(-2) is taken at the left end, int t^(-u) du exactly, psi((sigma+1)/2) at the right end (psi increasing) and -zeta'/zeta(sigma) + 2 zeta'/zeta(2 sigma) at the left end (it is sum_p log p/(p^sigma+1), decreasing); a cell bound is clamped at >= 0. For u >= 6, G(1+u) <= (1/2) log(q/pi) + P(7) + u/4 and the integral is done in closed form.
• delta is restricted to grid points; a, b are exact rationals chosen by a float optimiser (FloatModel, which assigns no truth; every reported number is recomputed in Arb).

3. Proof of Theorem C

dtable.json covers [10^2, 10^11] by 2041 intervals with q_hi/q_lo <= 1.01. On each, parameters (a, b, delta_x, delta_y) are chosen and Theorem B is evaluated at q_hi, where it is weakest; then h(D) >= sqrt(q_lo) L_lower / pi for every |D| in the interval. If h(D) <= h, |D| cannot lie in an interval whose verified bound exceeds h; D(h) is the right end of the last interval whose bound is <= h. Beyond 10^11, Theorem A gives h(D) >= sqrt|D| / (10 pi log log |D|) >= 3117 > 2000, increasing in |D|.

Typical parameters (verified rows): at q = 1.7 * 10^8, y = e^7.23, x = e^22.09, delta = 1/8, M = -2.724, E = 0.389, L(1, chi) >= 0.04449; at q = 9.1 * 10^9, y = e^7.95, x = e^23.1, M = -2.784, E = 0.375, L >= 0.04247. The weight is close to a pure Cesaro weight: y is small and x is a power of log q near 8.

4. Proof of Theorem A

Three ranges.

1. |D| <= 3 * 10^6, exact. h(D) is computed exactly for every one of the 911 878 fundamental discriminants (section 5) and L(1, chi_D) = 2 pi h / (w sqrt|D|). Every D with 4 <= |D| <= 3 * 10^6 satisfies L log log |D| >= 0.1 (the minimum over |D| > 4 is 0.4006); D = -3 fails (0.057), which is the stated exception (controls_exact_l.json).
2. 121 <= q <= 10^50, verified cover. 163 intervals with log q_hi / log q_lo <= 1.02; on each, Theorem B at q_hi times log log q_lo is at least 0.10030 (theorem_a.json).
3. q >= Q1 = 10^50, tail lemma. Take lambda = 1, eta = 3/20, delta_x = delta_y = 1/25 and a(q) = 8 log log q, b = (1 + eta) a. Then
4. main term: by Rosser-Schoenfeld for all t >= 286, (1/(b-a)) int_a^b S(e^u) du <= B + log a + kappa(eta) + 1/(2ab) with kappa(eta) = (1+eta) log(1+eta)/eta - 1; the P-sums are nondecreasing in a;
5. zero sums: e^(-a/8) = 1/log q, and G(sigma) <= log q (1/2 + g+(sigma)/log q) with g+ = max(0, psi((sigma+1)/2)/2 - zeta'/zeta(sigma) + 2 zeta'/zeta(2 sigma)), so R(y) <= [(delta+1/8)(1/2 + g+(1+delta)/log q) I_1(a) + int e^(-ua) (1/2 + g+/log q)/(u + 1/8) du], each factor nonincreasing in q; R(x) has the extra factor (log q)^(-eta); T and 1/(b-a) decrease.

Every piece is therefore bounded by its value at Q1, and log L(1, chi) >= K1 - log a(q) with K1 = -B - kappa - 1/(2 a1 b1) + P2(a1) - E~(Q1) = -0.189581..., where E~(Q1) = 0.036241 (Arb). So L(1, chi) >= e^K1 / (8 log log q) >= 0.10341 / log log q.

Together these give Theorem A.

Remark (the limit of the method; ordinary argument, not run in Arb). With eta -> 0, Q1 -> infinity and all prime powers kept in (2.5), the tail argument gives L(1, chi) >= (1 - eps)(1 - sigma0) zeta(2) e^(-gamma) / log log q for q >= q(eps): 0.11545 / log log q at sigma0 = 7/8, against Littlewood's GRH constant zeta(2) e^(-gamma) / 2 = 0.4618. The two differ exactly by the factor (1 - sigma0) / (1/2) = 1/4: the cutoff x must be a power 1/(1 - sigma0) = 8 of log q instead of 2. The verified values of L log log q at feasible q are larger than this limit (0.1266 at 10^6, 0.1348 at 10^12, 0.1411 at 10^100, abscissae.json), because there the optimal y is far below (log q)^8 and the weight is nearly a pure Cesaro weight.

5. The class-number computation

cn.c (exact integer arithmetic, no GRH, no floating point in any decision).

• Fundamental discriminants. n = |D| with n = 3 mod 4 squarefree, or n = 4, 8 mod 16 with n/4 squarefree; a segmented sieve removes n divisible by p^2 for odd p.
• Reduced forms. For fundamental D every form is primitive, so h(D) is the number of reduced forms (a, b, c): |b| <= a <= c, b >= 0 if |b| = a or a = c. For 4a^2 < n every b in (-a, a] with b^2 = D mod 4a gives c > a, so the number of reduced forms with first coefficient a is r(a) = #{b mod 2a : b^2 = D mod 4a}, multiplicative with r(p^k) = 1 + chi(p) for p not dividing D, r(p) = 1 and r(p^k) = 0 (k >= 2) for p | D.
• Sieve. Hence h(D) >= sum over any set of a with 4a^2 < n of r(a). For every segment the counts sum_a r(a) are streamed over all n at once (r(a) is periodic in n), over all a below sqrt(n)/2 where the primes alone cannot finish, and otherwise over the first K = 1.3 H + 40 primes; survivors continue prime by prime, and the rest of the reduced forms are counted exactly: r(a) by a multiplicative sieve for 4a^2 < n, and for n <= 4a^2 <= 4n/3 the roots of b^2 = D mod 4a by Tonelli-Shanks, Hensel lifting and the Chinese remainder theorem, each root checked.
• Checks. Every fundamental |D| <= 4000 against brute-force reduced-form counting (classno.reference_h), samples against PARI qfbclassno, the exact stage against an independent brute-force-over-b implementation inside cn.c (CN_CHECK=1), the streamed filter counts against a recomputation of every count with Jacobi symbols and no periodic patterns (CN_CHECK_STREAM=1), and the filtered runs against unfiltered full counts. The filter can only lose a field by over-counting, so the stream check is the one that guards completeness.

The runs. H = 1000 (run_search.py 1000 5433399820, one core, 9 min 59 s): 1 651 555 483 fundamental discriminants with |D| <= D(1000) = 5 433 399 820; 5 518 464 survived the streamed filter, 5 336 508 reached the exact count, 4 115 897 have h <= 1000. Time by range: below 10^7, 52 s (the list is dense there); 10^7 to 10^8, 136 s; 10^8 to 5 * 10^8, 90 s; above, 316 s (the prime stage, about 1340 periodic additions per discriminant). A separate H = 100 run to D(100) = 48 611 613 takes 1.8 s and finds the same 42 272 fields. H = 1500 (run_search.py 1500 12409254457, one core, 2327 s of chunk time over two sessions, the first stopped by a wall-clock limit and resumed from its chunk checkpoints): 3 771 960 997 fundamental discriminants, 14 817 639 filter survivors, 14 379 846 exact counts, 9 245 562 fields with h <= 1500. Its h <= 1000 and h <= 100 sub-lists have the same sha256 as the H = 1000 and H = 100 runs: no field with h <= 1000 lies in (D(1000), D(1500)], checked by computation rather than by the theorem.

The last column is the price of the analytic bound: the search must go about 24 times further than the last field it finds.

6. Controls

Bound against exact L(1, chi_D). For every fundamental D with 100 <= |D| <= 3 * 10^6 (911 847 discriminants) the exact value L(1, chi_D) = 2 pi h / (w sqrt|D|) was compared with the verified lower bound of the dtable.json interval containing |D|. No violation; the smallest ratio exact / bound is 3.888, at D = -163 (L = 0.24607 against the bound 0.06329). The class-number formula values agree with PARI's lfun(lfuncreate(D), 1) at 34 sampled D to relative 2.2e-16, an independent route to L(1, chi_D). Mosunov and Jacobson report the smallest L(1, chi_D) for |D| < 2^40 as 0.17070 at D = -107 415 709 003, where Theorem B gives about 0.041: the bound is about four times below the truth throughout the range anyone has computed. A planted inflation of the bound by the observed worst ratio times 1.01 is caught (test_a_planted_inflation_is_caught).

Weakened and strengthened abscissa (abscissae.json, Arb, verified lower bounds for L(1, chi) at a single q):

Strictly ordered at every q, as it must be. The predicted mechanism is that the cutoff x scales like (log q)^(1/(1 - sigma0)) and the constant like (1 - sigma0): at q = 10^100 the optimal log x is 37.0, 51.6 and 65.3 for 7/8, 11/12 and 15/16 (predicted ratios 1.5 and 2, observed 1.39 and 1.76), and the ratio of bounds 15/16 to 7/8 is 0.570 (limit 1/2). The 11/12 column is what the October 5 paper's own theorem gives: about 28% weaker than 7/8.

GRH calibration. At sigma0 = 1/2 the same code gives L(1, chi) >= 0.11894 at q = 10^12 and 0.08142 at q = 10^100, and L log log q -> 0.4618 (Littlewood). Lamzouri, Li and Soundararajan (Theorem 1.5, GRH, q >= 10^10) give 0.0904 at 10^12, 0.0912 at 10^20 and 0.0802 at 10^100. The two share the limit and are within 2% at 10^100; at 10^12 this method is 32% stronger, from optimising the two cutoffs rather than fixing x = (log q)^2 / 4 and from LLS's 14 log log q / log q term. Not implausibly better, which is the purpose of the control.

Class numbers. Every fundamental |D| <= 4000 against brute-force reduced-form counting; 60 sampled D near 10^6 and 1900 D from the H = 1500 output (the largest |D| for every h <= 1500 and 400 random entries, up to |D| = 562 394 347) against PARI qfbclassno, no disagreement (controls_pari_H1500.json); every streamed count in four windows (10^5 to 3 * 10^5 at H = 100; 3 * 10^8, 10^9 and 2 * 10^9 at H = 1000 or 1500, composite and prime-only modes) recomputed independently (CN_CHECK_STREAM=1); the fast exact count against the slower brute-force-over-b count inside cn.c (CN_CHECK=1) on every survivor in a window near 4 * 10^7; filtered runs against unfiltered full counts (tests: H = 7 and 60 on |D| <= 3 * 10^5; during development also H = 100 and 1000 on |D| <= 3 * 10^6). Watkins: all 100 counts and all 100 largest squarefree k for h <= 100 agree with OEIS A046125 and A038552 (from Watkins 2004), in both the H = 100, H = 1000 and H = 1500 runs. Holmin and Kurlberg: all 750 odd h <= 1500 agree with their GRH-conditional F(h) table (holmin_kurlberg_odd.json, fetched from the authors' page). After the runs, cn.c gained only the environment-gated stream check; two chunks recomputed with the new binary are byte-identical to the run files.

The October 5 abscissa, 11/12 (eleven_twelfths.json, same code with sigma0 = 11/12): the tail constant at 10^50 is 0.06959 and the cover keeps L log log q >= 1/16 from q = 100, so L(1, chi_D) >= 1/(16 log log |D|) for |D| >= 4 (the exact range is as for 7/8); D(1) = 6 390, D(100) = 95 092 214, D(1000) = 10 592 194 843. The H = 1500 search reaches 1.24 * 10^10, so the lists for h <= 1000 are complete under the 11/12 theorem as well.

7. What this improves, and the literature searched

Searched on 2026-10-08 (web search over arXiv, journal pages, zbMATH-indexed titles and author pages; the queries and hits are summarised here, not exhaustive): explicit lower bounds for L(1, chi) under a zero-free half-plane; "quasi-Riemann hypothesis" with "class number"; extensions of Watkins' h <= 100; unconditional class-group tabulations.

Found, and what this hunt adds to each:

• Littlewood (1928), GRH: L(1, chi) >= (1 + o(1)) zeta(2) / (2 e^gamma log log q). Lamzouri, Li, Soundararajan, Math. Comp. 84 (2015), Theorem 1.5: the explicit GRH form for q >= 10^10. This hunt's inequality is the same mechanism with the GRH line replaced by the half-plane, a second Cesaro cutoff that removes their (L'/L)(1, chi) term, and Hadamard positivity for every zero; its sigma0 = 1/2 specialisation is the calibration above.
• Friedlander and Iwaniec, arXiv:1701.03771, Theorem 3: Littlewood's argument under "beta > 3/4 is the only zero in Re s > 3/4", with x = (log D)^4 and unspecified constants. The half-plane version of the mechanism is therefore known; no explicit constant for it was found.
• OpenAI, October 5, 2026, introduction: h(D) >> sqrt|D| / log log |D| from its 11/12 theorem, constant "absolute computable" and not computed. Theorem A supplies the constant: 1/(10 pi) with the 7/8 input, and 1/(16 pi) with the 11/12 input (section 6).
• Watkins (2004): unconditional complete lists for h <= 100. Reproduced here exactly; under QRH this is a second, independent completion.
• Holmin, Jones, Kurlberg, McLeman, Petersen (arXiv:1510.04387): complete lists for odd h < 10^6 conditional on GRH (LLS bound, PARI under GRH, 4.5 CPU years). Odd h <= 1500 agree; this hunt replaces GRH by the 7/8 half-plane for those h and adds every even h <= 1500.
• Mosunov and Jacobson, Math. Comp. 85 (2016): unconditional class groups for all |Delta| < 2^40, organised by discriminant; no per-h lists are published. Since D(1500) < 2^40, their tabulation could in principle be re-read to the same lists (a door below).

Not found: an explicit constant for the half-plane form of Littlewood's bound, or complete lists for even h with 100 < h <= 1500 under any hypothesis. Absence in this search is a statement about the search, not a novelty claim. Within the searched set, Theorem A, the table D(h) and the QRH-conditional completion of the class-number lists for every h <= 1500 are original to this hunt (provenance), graded as stated at the top, and pending external review.

Checked and not used: a Bach-type bound from the 7/8 half-plane for the norms of primes generating the class group. The argument needs the main term x to beat the zero contribution x^(7/8) log q, so x must exceed about (log q)^8; that is below the trivial Minkowski range sqrt(q/3) only when 8 log log q < (1/2) log(q/3), that is q > about 1.7 * 10^30, even with implied constant 1. In the computed range it is useless, and the method here counts reduced forms directly, so it needs no generators at all.

8. Not done

• External review. Neither the input theorem nor this hunt's written proof has been read by a human mathematician. The chain in section 2 is the part to check: (2.1) is cited, (2.2) to (2.5) are written out here.
• Formalisation. Nothing here is in Lean; the Arb evaluation is the numerical layer, not a kernel check.
• Zero pairing. Real characters have zeros in pairs beta, 1 - beta at the same height; the positivity step charges every zero as if it sat at beta = 7/8. Using the pairing would shrink the zero term by up to about 12% at low height and more at large height; not derived.
• H beyond 1500. H = 2000 (D(2000) = 2.23 * 10^10) is estimated at about an hour of one core and was not run; the cost grows like H^3.
• Orders and group structures. Only fundamental discriminants (maximal orders) and only class numbers, not class-group structures, were tabulated.
• The full lists are not committed (49 MB and 118 MB); their sha256 and the commands that regenerate them are.

The doors

The hunt stops at H = 1500 for budget, not for a mathematical wall. What binds, and what would move it:

1. Active constraints at the optimum. (a) The zero-free abscissa itself: the constant scales with 1 - sigma0 and the cutoff with (log q)^(1/(1 - sigma0)); it is the input theorem's, frozen here. Ranked by shadow price it is first: 7/8 to 1/2 would multiply L by about 2.9 at q = 10^9 and shrink D(h) by about 8. (b) The zero term E, about 0.38 of log L at q = 10^9 (a factor e^0.38 = 1.46 in L, so about 2.1 in D(h)): the positivity bound charges every zero at beta = 7/8 and needs G(sigma) with its 1/(sigma - 1) pole, so the split point delta trades the pole against the factor (delta + 1/8). (c) The main term's prime sum, where the weight is already nearly optimal (y small, x a power of log q). (d) On the computation side, cost grows like X * H, that is H^3 (log log)^2: 10 minutes at H = 1000 became 39 minutes at H = 1500 (the survivor stage below 10^9 grew faster than estimated), and H = 2000 extrapolates to well over an hour on one core.
2. Frozen-constant inventory. sigma0 = 7/8 (input, not ours to move). The trapezoid weight (two cutoffs); a smoother weight with more cutoffs is a genuine trade (it spends main term to spread the y-term of E). The grid steps 1/2000, 1/500, 1/100 and the tail point u = 6 (cost: about 1% in R; no trade shape). The prime-power truncation at k = 5 (0.2% of L; none). T_EXACT = 10^7 and Rosser-Schoenfeld above it (Dusart's 0.2/log^3 t would gain under 0.002 in log L; small trade). The cover ratios 1.01 and the tail point Q1 = 10^50 with lambda = 1, eta = 3/20, delta = 1/25 (these set c = 1/10; c = 0.105 needs Q1 near 10^200; genuine trade between c and the size of the verified cover). The search's K = 1.3 H + 40 primes and the composite cap 4000 (speed only, never correctness).
3. Information class of each door. Zero pairing (beta and 1 - beta at one height) stays inside the data the method already reads: the functional equation of a real character. Using explicit zero counting near height 0 (Bennett, Martin, O'Bryant, Rechnitzer) reads more: its error term is comparable to the main term at small height, so it pays only with a short-interval count that is not in the literature found. Re-reading the Mosunov-Jacobson tabulation (CLGRP, unconditional to 2^40) reads more data: the single-q bound at 10^12 gives h >= 12 930 there (indicative; the verified cover stops at 10^11), so their class numbers plus Theorem B on a cover to 2^40 would complete the lists to h of order 10^4, far beyond this hunt's search, at the cost of re-running or obtaining their per-discriminant output. A smaller sigma0 is outside both: it needs a new zero-free region.

A measured optimum is not a proved ceiling: the bound's constant here is the best the float optimiser found within this family of weights and grids, at the stated q, with no proof that the family is optimal.