Linnik's constant from the 7/8 zero-free half-plane: results

Status: candidate, pending external review. Grade of every theorem below: proved (ordinary written proof, written by a model in this hunt, not yet reviewed by anyone), given OpenAI's Theorem 1.1 and the density input named in the statement. A composite claim takes the grade of its weakest step, and the weakest step here is OpenAI's 195-page argument, which no human has reviewed (its Lean statements are kernel checked on two kernels, the argument behind them is not replayed here). Nothing in this file is promoted by the hunt itself.

0. Results at a glance

p(q,a) is the least prime p = a (mod q), (a,q) = 1. "Effective" means every constant is computable in principle: no step uses Siegel's theorem or any other ineffective input. No constant is computed.

Against what. Unconditionally L = 5 (Xylouris, Bonn thesis 2011) and L < 5 (Xylouris, Chebyshevskii Sbornik 2018) are the refereed records; an AI-assisted preprint claims 3.99 (Naslund, hexagonmath.org 2610.00018v1, 2026-10-02, unreviewed). The English Wikipedia table "Linnik's theorem", read 2026-10-08, lists L = 2.4 as a "corollary of 7/8 Quasi-Riemann Hypothesis" attributed to OpenAI; that is the Ingham-Huxley exponent 12/5 (Theorem A' here). The OpenAI paper text available to this hunt states no Linnik bound (grep -i linnik on both extracted texts is empty). Theorem A lowers 12/5 to 7/3. Under GRH the exponent is 2 + eps. The improvement from 12/5 to 7/3 comes from the Chen-Gupta-Li density estimate, not from anything new about the half-plane.

The half-plane's role is qualitative. Any fixed zero-free half-plane Re s > theta with 5/7 <= theta < 1 gives the same 7/3 from current density estimates (Remark F). What the 7/8 theorem buys is that there is some such theta: it removes the Siegel zero, the log-free density estimate and the Deuring-Heilbronn repulsion from the argument entirely. OpenAI's October 5 paper's weaker 11/12 half-plane would give the same exponents.

1. Statements

Notation. chi runs over Dirichlet characters mod q; chi* is the primitive character mod q* | q inducing chi (chi* = 1, q* = 1 for the principal character, and then L(s, chi*) = zeta(s)). Z(chi*) is the multiset of nontrivial zeros rho = beta + i gamma of L(s, chi*) (for primitive chi* these are its zeros with 0 < beta < 1). For sigma >= 0, T > 0

N_q(sigma, T) = sum over chi mod q of #{rho in Z(chi*) : beta >= sigma, |gamma| <= T}.

For sigma > 0 this is CGL's sum_{chi mod q} N(sigma, T, chi): the factor prod_{p | q} (1 - chi*(p) p^-s) relating L(s, chi) to L(s, chi*) vanishes only on Re s = 0. Two hypotheses:

• Z(theta): no Dirichlet L-function vanishes in Re s > theta (pole at s = 1 of the principal character excepted). OpenAI's Theorem 1.1 is Z(7/8).
• D(A): A : (1/2, 1) -> [0, infinity), and for every sigma in (1/2, 1) and eps > 0 there is an effectively computable C(sigma, eps) with N_q(sigma, T) <= C(sigma, eps) (qT)^(A(sigma)(1 - sigma) + eps) for all q >= 1, T >= 2.

Put A*(theta) = max(2, sup over 1/2 < sigma <= theta of A(sigma)).

Theorem P (the deduction). Assume Z(theta) for some 1/2 <= theta < 1 and D(A) with A* = A*(theta) < infinity. For every eps0 > 0 there are effectively computable delta > 0 and C such that for all q >= 1, (a, q) = 1:

1. |psi_w(x; q, a) - W(1) x / phi(q)| <= C x^(1 - delta) / phi(q) for x >= q^(A* + eps0) (smoothed count, weight w of section 3);
2. there is a prime p = a (mod q) with x/2 <= p <= x for every x >= C q^(A* + eps0); in particular p(q, a) <= C q^(A* + eps0).

Theorem A. Given OpenAI's Theorem 1.1 and CGL Theorem 1.2: for every eps > 0 there is an effectively computable C(eps) with p(q, a) <= C(eps) q^(7/3 + eps) for every q >= 1 and every a with (a,q) = 1.

Proof. Z(7/8) is OpenAI's Theorem 1.1. CGL Theorem 1.2, "in all cases", reads sum_{chi mod q} N(sigma, T, chi) <= (qT)^o(1) (q^(7(1-sigma)/3) T^(2(1-sigma)) + (qT)^(30(1-sigma)/13)) for 1/2 < sigma < 1, a sum over all characters mod q (CGL section 12: the imprimitive characters are included by applying the primitive estimate to every divisor of q and summing). Since T^2 <= T^(7/3) and 30/13 < 7/3, this is D(A) with A = 7/3 constant, so A*(7/8) = 7/3 and Theorem P(2) applies. []

Theorem A'. Given OpenAI's Theorem 1.1 alone, with the classical single-modulus estimates N_q(sigma, T) << (qT)^(3(1-sigma)/(2-sigma) + eps) (Ingham type; CGL (1.4), derived there from the qT mean value theorem; the single-modulus form is classical, Montgomery's Topics, chapter 12) and N_q(sigma, T) << (qT)^(3(1-sigma)/(3 sigma - 1) + eps) (Huxley, Acta Arith. 26 (1975); CGL (1.5)): p(q, a) <= C(eps) q^(12/5 + eps). Proof: D(A) with A = min(3/(2-sigma), 3/(3sigma-1)), whose supremum on (1/2, 7/8] is 12/5 at sigma = 3/4 (section 5). []

2. Inputs, as used

• (I1) OpenAI Theorem 1.1, Lean form OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re: for every q, every chi : DirichletCharacter C q (primitive or not) and every s with 7/8 < Re s and not (chi = 1 and s = 1), L(chi, s) != 0. Used only in the form "every rho in every Z(chi*) has beta <= 7/8", and for zeta.
• (I2) Zero counting (Davenport, Multiplicative Number Theory, chapters 15 and 16; explicit forms by McCurley 1984 and Trudgian 2015): for primitive chi* mod q* and T >= 2, #{rho in Z(chi*) : |gamma| <= T} <= C3 T log(q* T), and for every real t, #{rho in Z(chi*) : |gamma - t| <= 1} <= C3 log(q*(|t| + 2)).
• (I3) Logarithmic derivative (Davenport chapters 9, 12, 16, 19): the functional equation gives, for primitive chi* with parity kappa, L'/L(s, chi*) = -L'/L(1 - s, conj chi*) - log(q*/pi) - (1/2) digamma((s+kappa)/2) - (1/2) digamma((1-s+kappa)/2), hence |L'/L(-1/2 + it, chi*)| <= C log(q*(|t| + 2)); and for every U >= 2 there is U' in [U, U+1] with |L'/L(sigma +- iU', chi*)| <= C log^2(q* U) for -1/2 <= sigma <= 2.
• (I4) Density. CGL Theorem 1.2 (statements quoted in section 4, read from the arXiv v2 PDF, 2026-07-27), and the classical (1.4), (1.5) through CGL's attribution. CGL do not discuss effectivity. Their section 12 (zero detection, the qT mean value theorem, Montgomery's fourth moment, their Theorem 1.1) uses no ineffective step that we could find; we did not audit the proof of their large-values Theorem 1.1. The best uniform single-modulus exponents A in sum_{chi mod q} N(sigma, T, chi) << (qT)^(A(1-sigma)+eps), as CGL list them: Montgomery 2.5, Forti-Viola 2.463, Jutila 2.460, Huxley 12/5, CGL 7/3. The Guth-Maynard exponent 30/13 is proved for q = 1 (Ann. of Math. 203 (2026)) and, by CGL, for T-smooth q; for general q it is open. Height range: the proof below uses CGL's bound only at heights T0 >= q^(c(eps0)) for a fixed c(eps0) > 0 (and N_q is monotone in T), so it does not lean on the small-T, large-q corner of their zero-detection argument, whose contour tails they treat as T -> infinity.
• (I5) Chebyshev. theta(y) < y log 4, hence psi(x) - theta(x) <= log 4 (sqrt x + x^(1/3) log_2 x) <= 6 sqrt x for x >= 1.

3. Proof of Theorem P

Weight. Fix w smooth on R, 0 <= w <= 1, support in [1/2, 1], w = 1 on [5/8, 7/8]. Its Mellin transform W(s) = int_0^inf w(u) u^(s-1) du is entire and W(1) >= 1/4. Write psi_w(x, chi) = sum_n Lambda(n) chi(n) w(n/x) and psi_w(x; q, a) = sum_{n = a (q)} Lambda(n) w(n/x).

Lemma 1 (decay). For each k >= 0 there is B_k (depending on w, k) with |W(sigma + it)| <= B_k (1 + |t|)^-k for -1 <= sigma <= 2; and |W(s)| <= 1 for 0 <= sigma <= 1. Proof. k integrations by parts give W(s) = (-1)^k (s(s+1)...(s+k-1))^-1 int w^(k)(u) u^(s+k-1) du. On [1/2, 1], u^(sigma+k-1) <= 4, and |s(s+1)...(s+k-1)| >= |t|^k; so |W| <= 4 ||w^(k)||_1 |t|^-k for |t| >= 1, while |W(s)| <= int w(u) u^(sigma-1) du <= 2 always and <= 1 for sigma >= 0 (as u^(sigma-1) <= 2 and int w <= 1/2). []

Lemma 2 (explicit formula). For primitive chi* mod q* >= 1 and x >= 1, psi_w(x, chi*) = [chi* = 1] W(1) x - sum_{rho in Z(chi*)} W(rho) x^rho + R, with |R| <= C_w log(q* + 2) and the zero sum absolutely convergent. Proof. Mellin inversion gives psi_w(x, chi*) = (2 pi i)^-1 int_(2) (-L'/L)(s, chi*) W(s) x^s ds. Move the line to Re s = -1/2 along the heights U' of (I3): the horizontal segments are << x^2 log^2(q*U) U^-3 -> 0 by Lemma 1. Poles crossed: s = 1 when chi* = 1 (residue W(1)x); each rho (residue -W(rho) x^rho with multiplicity); s = 0 when chi* is even and q* > 1 (trivial zero, residue -W(0), |W(0)| <= 2). zeta(0) = -1/2, the trivial zeros of zeta are at -2, -4, ..., those of odd chi* at -1, -3, ..., so nothing else lies in -1/2 < Re s < 2. On Re s = -1/2, |x^s| <= 1, (I3) and Lemma 1 with k = 2 give |int| <= C log(q* + 2). Absolute convergence: Lemma 1 with (I2). []

Lemma 3 (imprimitive characters). For chi mod q induced by chi*, |psi_w(x, chi) - psi_w(x, chi*)| <= omega(q) log x <= 2 log(q + 1) log x. Proof. The two sums differ only at n = p^j with p | q, n <= x, each weighted by at most log p, and there are at most log x / log p such j per prime. []

Lemma 4 (two zero sums). By (I2), N_q(0, T) <= C3 phi(q) T log(qT) for T >= 2, and by (I2) with Lemma 1 (k = 2), sum_{rho in Z(chi*)} |W(rho)| <= C log(q + 2).

Lemma 5 (the zero sum). Assume Z(theta) and D(A), A* = A*(theta). For X >= 2 put E_q(X) = sum_{chi mod q} sum_{rho in Z(chi*)} X^beta |W(rho)|. For every eps0 > 0 there is an effectively computable C with E_q(X) <= C X^(1 - kappa) whenever X >= q^(A* + eps0), where kappa = lambda (1 - theta)/4 and lambda = eps0/(A* + eps0).

Proof. Since q^(A* + eps0) <= X, q <= X^((1-lambda)/A*). Put eta = lambda/(4A*), T0 = X^eta, eps = lambda(1 - theta)/8, k = ceil(3/eta) + 1. If T0 < 2, then X and q are bounded in terms of eps0, and E_q(X) <= X phi(q) C log(q+2) (Lemma 4) is absorbed in C. Otherwise:

High zeros, |gamma| > T0. Lemma 1 with this k, X^beta <= X and Lemma 4 on dyadic shells: sum <= X B_k sum_{j >= 0} N_q(0, 2^(j+1) T0) (2^j T0)^-k <= C X q T0^(1-k) log(qT0) <= C X^(1 + 1/2 - 3) log X.

Low zeros, |gamma| <= T0, where |W(rho)| <= 1. Writing X^beta = X^(1/2) + int_{1/2}^{beta} X^sigma log X d sigma for beta >= 1/2, summing, and using N_q(sigma, T0) = 0 for sigma > theta (this is Z(theta)):

sum_{|gamma| <= T0} X^beta <= X^(1/2) N_q(0, T0) + log X int_{1/2}^{theta} N_q(sigma, T0) X^sigma d sigma.

First term: Lemma 4 gives <= C X^(1/2) q T0 log X <= C X^(1 - lambda/2 + eta) log X, using A* >= 2; and eta <= lambda/8.

Second term. Uniformity in sigma. D(A) is pointwise in sigma; take the grid sigma_j = 1/2 + j/J, J = ceil(A*/eps). For sigma in [sigma_j, sigma_(j+1)] with j >= 1 and sigma_j <= theta, monotonicity gives N_q(sigma, T0) <= N_q(sigma_j, T0) <= C(sigma_j, eps)(qT0)^(A* (1 - sigma) + A*/J + eps). For sigma in [1/2, sigma_1], N_q(sigma, T0) <= N_q(0, T0) <= C (qT0)^(1 + eps) and A*(1 - sigma) >= 1 - A*/J. So, with finitely many constants, N_q(sigma, T0) <= C (qT0)^(A*(1 - sigma) + 2 eps) on [1/2, theta]. Now (qT0)^A* <= X^(1 - lambda + lambda/4) and qT0 <= X^(1/2 + eta), so the integrand is at most C X^sigma (qT0)^(A*(1-sigma) + 2eps) <= C X^(1 - (3 lambda/4)(1 - sigma) + 2 eps (1/2 + eta)), largest at sigma = theta, giving <= C X^(1 - 19 lambda (1 - theta)/32) log X.

All three exponents are below 1 - kappa with room for the logarithm (the exact bookkeeping is exponents.proof_bookkeeping, pinned for six parameter sets by test_lemma5_bookkeeping_closes). []

Proof of Theorem P(1). Orthogonality, psi_w(x; q, a) = phi(q)^-1 sum_{chi mod q} conj chi(a) psi_w(x, chi), then Lemmas 3 and 2. Only chi_0 has chi* = 1, so

|psi_w(x; q, a) - W(1) x / phi(q)| <= E_q(x)/phi(q) + C log(q + 2) log(x + 2).

Lemma 5 with X = x bounds the first term. As x >= q^A* >= q^2, phi(q) <= x^(1/2) and the logarithmic term is <= C x^(3/4)/phi(q). Take delta = min(kappa, 1/4). []

Proof of Theorem P(2). With theta_w(x; q, a) = sum_{p = a (q)} log p w(p/x), 0 <= psi_w - theta_w <= psi(x) - theta(x) <= 6 sqrt x (I5). So theta_w(x; q, a) >= (x/phi(q)) (1/4 - C x^-delta - 6 phi(q) x^(-1/2)), and phi(q) x^(-1/2) <= x^(1/(A* + eps0) - 1/2) with a negative exponent since A* >= 2. Hence theta_w > 0, a prime p = a (q) lies in [x/2, x], once x >= x0(eps0); take x = x0 q^(A* + eps0). []

Checklist of the questions the mission put. The height cutoff is T0 = x^eta with eta = eps0/(4A*(A* + eps0)), so T0 = q^O(eps0); the smoothing pays for it with k = O(1/eta) derivatives of the fixed weight w. The principal character enters only through zeta in Lemma 2 (main term W(1)x, zeros of zeta bounded by Z(theta) like all others). Imprimitive characters cost Lemma 3, and their zeros in Re s > 0 are those of chi*, which is how CGL count them. No step uses Siegel's theorem, a Siegel zero, Deuring-Heilbronn or a log-free estimate: Z(theta) excludes every zero they would handle.

4. Corollaries

Corollary B (prime counts in every class). Under the hypotheses of Theorem P, for x >= q^(A* + eps0): psi(x; q, a) = x/phi(q) + O(x^(1-delta)/phi(q)) and pi(x; q, a) = li(x)/phi(q) + O(x^(1-delta)/phi(q)), delta = delta(eps0) > 0 effective. With Theorem A's inputs, pi(x; q, a) ~ li(x)/phi(q), so pi(x; q, a) is of exact order x/(phi(q) log x), uniformly once x >= q^(7/3 + eps).

Proof. Let Delta = x^-nu with nu = eta/2 (eta from Lemma 5). Take smooth w+ = 1 on [Delta, 1], supported in [Delta/2, 1 + Delta], and w- <= 1_[0,1], equal to 1 on [Delta, 1 - Delta], both with 0 <= w+- <= 1 and k-th derivatives O_k(Delta^-k). Then psi_{w-}(x; q, a) <= psi(x; q, a) <= psi_{w+}(x; q, a) + (x^(1-nu)/q + 1) log x and W+-(1) = 1 + O(Delta). Lemmas 2 and 5 used only two properties of W, both from Lemma 1: |W(rho)| <= M for 0 < beta < 1 and |W(beta + it)| <= M_k |t|^-k for |t| >= 1. For w+- integration by parts gives M = 2 nu log x + 2 and M_k = C_k x^(nu(k-1)); Lemma 2's line integral becomes O(x^(-1/2 + nu) log q) and |W+-(0)| <= nu log x + 2. With T0 = x^eta = x^(2 nu) the high zeros give X q x^(nu(k-1)) T0^(1-k) log = X q x^(-nu(k-1)) log, negligible for k large, and the low zeros carry the extra factor M = O(log x), absorbed by the power saving. So psi(x; q, a) = x/phi(q) + O(x^(1-delta)/phi(q)). Partial summation pi = theta(x;q,a)/log x + int_2^x theta(t;q,a)/(t log^2 t) dt, split at y = x^(1-nu) (trivial bound below y, where theta(t;q,a) <= (t/q + 1) log t; the asymptotic above, valid since y >= q^(A* + eps0/2) for nu small) gives li(x)/phi(q) + O(y/phi(q) + x^(1-delta)/phi(q) + log log x). []

Corollary C (almost all classes). Under the hypotheses of Theorem P, for x >= q^(A*/2 + eps0), all but C phi(q) x^-delta reduced classes a (mod q) satisfy theta_w(x; q, a) > W(1) x/(2 phi(q)), so they contain a prime in [x/2, x]. With Theorem A's inputs: x >= q^(7/6 + eps) (12/5 inputs: 6/5; smooth moduli: 15/13).

Proof. Let e0 = (psi_w(x, chi_0) - W(1)x)/phi(q). Orthogonality on (Z/qZ)^*:

sum_a |psi_w(x; q, a) - W(1)x/phi(q) - e0|^2 = phi(q)^-1 sum_{chi != chi_0} |psi_w(x, chi)|^2.

For chi != chi_0, Lemmas 2, 3: |psi_w(x, chi)| <= S_chi + C log q log x with S_chi = sum_{rho in Z(chi*)} x^beta |W(rho)|, and by Cauchy-Schwarz and Lemma 4, S_chi^2 <= (sum |W(rho)|)(sum x^(2 beta) |W(rho)|) <= C log(q+2) sum_{rho in Z(chi*)} (x^2)^beta |W(rho)|. So the right side is <= phi(q)^-1 (C log q E_q(x^2) + C phi(q) log^2 q log^2 x), and Lemma 5 with X = x^2 >= q^(A* + 2 eps0) gives E_q(x^2) <= C x^(2 - 2 kappa). By Lemma 2 for zeta and Z(theta), |e0| <= C (x^theta + log q log x)/phi(q) <= W(1)x/(8 phi(q)) for x >= x1. Now theta_w(x; q, a) <= W(1)x/(2phi(q)) forces either psi_w - theta_w >= W(1)x/(8phi(q)) at a, which by (I5) happens for at most 48 phi(q) x^(-1/2)/W(1) classes, or |psi_w(x; q, a) - W(1)x/phi(q) - e0| >= W(1)x/(4phi(q)), which by the displayed identity happens for at most C phi(q) (log q x^(-2 kappa) + phi(q) log^4 x / x^2) classes; phi(q) <= x^(1/(1+eps0)) since A*/2 >= 1. []

Corollary D (smooth moduli). For every eps0 > 0 there is eta0 > 0 such that, given OpenAI's Theorem 1.1 and CGL Theorem 1.2: if every prime factor of q is at most q^eta0, then p(q, a) <= C q^(30/13 + eps0), and Corollaries B and C hold with 30/13 and 15/13. Proof. CGL Theorem 1.2: if q is T-smooth, sum_{chi mod q} N(sigma, T, chi) <= (qT)^(30(1-sigma)/13 + o(1)). Lemma 5 uses density bounds only at T = T0 = X^eta >= q^((A* + eps0) eta); take eta0 = (30/13 + eps0) eta. The rest is Theorem P with A = 30/13. []

Corollary E (moduli with a divisor of suitable size). CGL Theorem 1.2 for 1/2 < sigma < 1 and any divisor q1 | q:

sum_{chi mod q} N(sigma,T,chi) <= (qT)^o(1) ( (q1^(1/3) qT)^(3(1-sigma)/(1+sigma)) + (qT (q1 T)^(-1/2))^(3(1-sigma)/sigma)

• (q1 T)^(-1/2) (qT)^((21-20 sigma)/6) + (qT)^(15(1-sigma)/(3+5 sigma)) ),

derived in their section 12 for 0.7 <= sigma <= 0.8 and taken from (1.4), (1.5) outside. With q1 = q^alpha and T = q^o(1) the four terms have exponents (3 + alpha)/(1+sigma), 3(1 - alpha/2)/sigma, (21 - 20 sigma - 3 alpha)/(6(1-sigma)), 15/(3 + 5 sigma) (in units of 1 - sigma; each T power is bounded by the same power of qT because alpha <= 1). The exact supremum over (1/2, 7/8] of min(3/(2-sigma), 3/(3sigma-1), max(those four, 2)) (the window [0.7, 0.8] only) is the profile Lambda(alpha):

It is minimised where 2 + alpha/3 (first term against Ingham, crossing at sigma = (3 + 2alpha)/(6 + alpha)) meets the third term's crossing value (37 + 3 alpha - sqrt(9 alpha^2 + 222 alpha - 71))/12, i.e. at alpha* = (sqrt 1081 - 31)/2 = 0.939282..., with Lambda(alpha*) = (sqrt 1081 - 19)/6 = 2.313094.... Since the first term increases and the second and third decrease in alpha, a divisor known only to lie in [q^a, q^b] is handled by taking the first at b and the others at a. Result: if q has a divisor q1 with q^0.93 <= q1 <= q^0.95, then p(q, a) <= C q^((3979 - sqrt 1432441)/1200 + eps), exponent 2.318461.... A prime modulus has only alpha in {0, 1} and gets 7/3.

Remark F (how much of the half-plane is used). The supremum defining A* is attained at sigma* = 5/7, below 7/8. For the CGL family, L(theta) = 3/(2 - theta) for 1/2 <= theta <= 5/7 (Ingham alone binds) and L(theta) = 7/3 for 5/7 <= theta < 1 (pinned at nine values of theta by test_theta_profile_seven_eighths_is_not_binding). So 7/8 is slack by 7/8 - 5/7 = 9/56; a half-plane at theta = 2/3 would give 9/4, and theta = 1/2 (GRH) gives 2, the value GRH gives directly.

Corollary G (Goldbach numbers in progressions, every modulus). Given OpenAI's Theorem 1.1 and CGL Theorem 1.2: if k (mod q) is a sum of two reduced classes, then k = p1 + p2 (mod q) with primes p1, p2 <= C q^(7/6 + eps). Proof. The number of a with a and k - a both reduced is prod_{p^e || q} p^(e-1)(p - 1 - [p does not divide k]) >= c phi(q)/log log(q + 2) when positive, and Corollary C removes at most 2 C phi(q) x^-delta of them. [] CGL Corollary 1.3 proves G(p, k) << p^(7/6 + eps) for prime moduli unconditionally (Jutila's method); the general-modulus version here is conditional on the half-plane and was not searched for separately.

5. Exact exponent algebra

exponents.py computes sup A exactly: each density exponent is a rational function of sigma, monotone on [1/2, 1] (Ingham increasing, the others decreasing), so the envelope's supremum sits at an interval end or a crossing, which sympy solves exactly; a 200001-point float grid is the independent route. Both routes agree to 2e-5 in value and 2e-3 in location (tests). Grade of this algebra: hardened (two independent routes, exact arithmetic on one). The reading of CGL's divisor form is itself checked against CGL: it reproduces their uniform 7/3, their smooth-moduli 30/13, and the three closed forms 2 + alpha/3, 3 - 3alpha/4 and B(alpha) of their display (12.6) (test_parse_reproduces_cgl_closed_forms).

The binding piece for general q is CGL's first term (q1^(1/3) qT)^(3(1-sigma)/(1+sigma)) at q1 = q, i.e. (q^(4/3) T)^(...), not their Guth-Maynard term 15/(3+5 sigma); the mission's first target (12/5 at sigma = 3/4) is superseded because CGL's term is below 12/5 there (16/7). Lesions that the instruments must and do notice: removing Huxley's piece moves L to 8/3 (Huxley carries [0.8, 7/8] in both families); a transcription slip dropping the q1^(1/3) loss moves L off 7/3; eps0 = 0 leaves Lemma 5 no saving.

6. Numerical sanity check (descriptive only)

least_primes.py computed p(q, a) for every reduced class and every 3 <= q <= 5000 (sieve to 2*10^7, 1,270,607 primes, 49 s, one process), checked against an independent brute-force walk with sympy.isprime for all classes with q <= 80 and against recomputation of selected moduli from the stored table.

• max_q max_a log p(q,a)/log q = 1.8295 at q = 5 (p(5,4) = 19); for q >= 100 the maximum is 1.7166 at q = 461 (p = 37363). No modulus reaches 2.
• The 90% quantile of log p(q,a)/log q over classes is at most 1.4919 for q >= 100, typically 1.2 to 1.4 at q near 5000.

!least primes (least_primes.png)

This is consistent with the conjecture p(q, a) < q^2 and says nothing about the constant C(eps) or about large q. The theorem's 7/3 is an upper bound from zero counting; at these sizes the actual exponents sit far below it, and setting the plotted values beside 7/3 is a picture, not evidence.

Lemma 2 calibration (measured). explicit_check.py evaluates both sides of Lemma 2 for zeta with the bump exp(1 - 1/(1 - v^2)), v = 4(u - 3/4), at x = 1000, mp.workdps(30), the zero sum truncated at the first n zeros. The residual |psi_w(1000) - (W(1)x - sum_rho W(rho) x^rho - sum_k W(-2k) x^(-2k))| is 0.0408, 0.0168, 0.00359, 0.000421 for n = 20, 40, 80, 160: it responds to the added zeros as a correct formula must. With the sign of the zero sum flipped it is 0.279 at n = 80. This pins the sign and normalisation the proof uses (one route, measured; the test reruns n = 20, 40 and the flipped sign at n = 40).

7. Prior art and what is claimed

Searched 2026-10-08: web search over arXiv and scholarly indexes (Guth-Maynard extensions to characters; Linnik's constant under a zero-free strip, half-plane, quasi-GRH or density hypothesis; almost-all least primes); zbMATH Open API (titles Linnik* constant; "least prime arithmetic progression density zero-free"; "Linnik constant conditional"; "quasi-Riemann hypothesis"); a MathOverflow-targeted query (no relevant thread found).

Found:

• Chen, Gupta, Li, arXiv:2507.08296v2 (2026-07-27), Theorem 1.2: the density input above. Their Corollary 1.4 is the conditional implication in essentially this form: if prod_chi L(s, chi) != 0 for sigma >= 1 - eta(q,T) with eta log x -> infinity, then psi(x+h; q, a) - psi(x; q, a) ~ h/phi(q) once h/phi(q) >= x^(1/2 + eps) q^(1/6) + x^(17/30 + eps); at h = x this is x >= q^(7/3 + eps'). They state it without proof ("can be deduced from Theorem 1.2 via a smoothed version of the explicit formula") and do not state a Linnik exponent. So the implication is in the literature as a conditional statement.
• Harm, arXiv:2507.15334: a conditional framework (zero-free region plus averaged density) for primes in short progressions.
• Bruna, arXiv:2603.25612: generalized Lindelof gives p(q, a) << q^(2 + eps).
• The Wikipedia table entry 2.4 for the 7/8 theorem (no derivation given there).

Contribution of this hunt, none claimed novel: the unconditional instantiation given OpenAI's Theorem 1.1 (7/3, and 12/5 on refereed density inputs alone); a complete written proof of the h = x case with the principal and imprimitive characters, the height cutoff, sigma-uniformity and effectivity handled; the identification of the binding point sigma* = 5/7 and of the fact that the half-plane is slack for theta >= 5/7; the almost-all exponent 7/6; the smooth-moduli 30/13; the divisor profile with alpha*; Goldbach numbers for every modulus. The almost-all and Goldbach statements were searched only by the queries above.

8. What is not done

• No external review of sections 3 and 4; no Lean formalisation; no constant computed.
• CGL's large-values Theorem 1.1 was not audited; Montgomery's and Huxley's papers were not read at source (their exponents enter through CGL (1.4), (1.5)).
• Effectivity of CGL's (qT)^o(1) is inferred from the shape of their section 12, not stated by them.
• No improvement of any density estimate was attempted; the doors below say where one would have to come from.
• The input theorem was used, not checked.

9. Reproduction

PYTHONPATH=. .venv/bin/python -m hunts.qrh_linnik.exponents PYTHONPATH=. .venv/bin/python -m hunts.qrh_linnik.least_primes --Q 5000 --limit 20000000 PYTHONPATH=. .venv/bin/python -m hunts.qrh_linnik.explicit_check PYTHONPATH=. .venv/bin/python -m pytest -q -n0 hunts/qrh_linnik/

The doors

The bound 7/3 is the supremum of the best published single-modulus density exponents on (1/2, 7/8]. It is a ceiling of this route with these inputs, a measured optimum of an exact computation, not a lower bound for Linnik's constant: candidate, parameter range 5/7 <= theta < 1, information class "zero counts", proof status as at the top of this file.

Active constraints at the optimum, ranked by shadow price (dL per unit of the knob, computed in exponents.py and pinned by tests):

1. CGL's gcd-twist loss, the factor q1^(1/3) in their large-values term q q1^(1/3) T N^2 V^(-4) at q1 = q. Replacing 1/3 by c gives exactly L = max(30/13, 2 + c): shadow price 1, saturating after a drop of 1/39 at c = 4/13, where the Guth-Maynard-type term 15/(3 + 5 sigma) takes over at sigma = 7/10. Moduli with a divisor near q^0.939 already escape part of it (Corollary E).
2. The Ingham-type mean value bound 3/(2 - sigma), binding at the same point. A uniform improvement Delta of its exponent near sigma = 5/7 lowers L by (3/7) Delta (ratio of the slopes 49/36 and 49/27 of the two crossing curves).
3. The half-plane: shadow price 0 on [5/7, 1); below 5/7, dL/dtheta = 3/(2-theta)^2.
4. The floor 2 from zeros near Re s = 1/2 (x^(1/2) N_q(0, T0)), slack by 1/3; it is the GRH value and no density estimate can lower it in this route.

Frozen-constant inventory (chosen, not optimised):

• eps0: trades against C(eps0) and the saving delta; no effect on L.
• eta = eps0/(4 A*(A* + eps0)), the height cutoff T0 = x^eta, and k = ceil(3/eta) + 1: trade only against the weight constants B_k; any eta > 0 works. No trade shape for L.
• The weight w (support [1/2, 1], plateau [5/8, 7/8]): fixes W(1) >= 1/4 and B_k; constants only.
• The grid J = ceil(A*/eps) and eps = lambda (1 - theta)/8: absorb losses; constants only.
• The divisor exponent alpha in CGL's bound: genuine trade shape (the first term rises with alpha, the second and third fall), optimum alpha* = 0.93928; frozen at 1 for prime moduli by arithmetic, not by choice.
• CGL's window [0.7, 0.8] for their new estimate: their choice; sigma* = 5/7 lies inside, so widening it would not move L.
• The Cauchy-Schwarz split in Corollary C: halves the exponent; a different split does not change the binding sigma*.

Information class. Doors 1 and 2 stay inside the data this route reads (counts of zeros of the family {L(s, chi) : chi mod q}): they need a better large-values estimate for character-twisted Dirichlet polynomials, and even the full density hypothesis in this class only reaches the floor 2. Door 3 also stays in the class but needs a stronger input theorem (theta < 5/7). Going below 2, or below 7/3 without new density estimates, requires reading more than zero counts: the phases conj chi(a) x^(i gamma) across characters, which the absolute values in Lemma 5 discard. Averaging over a (Corollary C) is the one place this hunt reads them, and it halves the exponent; sieve routes (Friedlander-Iwaniec) and correlation information are the other known ways in. Which door, if any, gets funded is the owner's allocation.