Small witnesses from the 7/8 half-plane, with explicit constants

Status: candidate, pending external review. Every theorem below is proved, given OpenAI, "The Quasi-Riemann Hypothesis", Sept 30 2026, Thm 1.1 (zero-free half-plane Re s > 7/8 for every Dirichlet L-function, zeta included; below "Sept 30 Thm 1.1"; its Corollary 1.2 is the A = 32 nonresidue bound improved here), by an ordinary written proof that no human has reviewed. The weaker companion, OpenAI, "The Quasi-Riemann Hypothesis", Oct 5 2026 (human-assisted), proves Re s > 11/12 as its Theorem 1.1 (below "Oct 5 Thm 1.1"), cites the 7/8 paper as its reference [36], and repeats the nonresidue, Miller-test and class-number consequences in its introduction; Theorem 1(d) records what it alone gives. Every numerical step in those proofs is an Arb enclosure (python-flint, ball arithmetic) or an exact integer computation. The input theorem itself is unreviewed (195 pages; kernel-checked statements only), so the composite grade is that of the weakest step: proved given an unreviewed input, with enclosure-carrying numerics. Nothing here is kernel-checked.

Reproduction (about 60 s, one core, under 1 GB):

PYTHONPATH=. .venv/bin/python hunts/qrh_nonresidue/verify.py PYTHONPATH=. .venv/bin/python -m pytest -q -n0 hunts/qrh_nonresidue/

All numbers quoted below are in verification.json.

0. Results

Notation. For a nonprincipal Dirichlet character chi mod q let n(chi) be the least positive integer n with chi(n) not in {0, 1}. It is a prime (if chi(p) is in {0,1} for every prime p <= m then chi(n) is in {0,1} for every n <= m), so n(chi) is also the least prime p not dividing q with chi(p) != 1. Write g(p) and g*(p) for the least primitive root and the least prime primitive root of a prime p, w(n) for the least Miller-Rabin (strong) witness of an odd composite n.

Theorem 1 (least character nonresidue). Given Sept 30 Thm 1.1:

• (a) For every q >= 3 and every nonprincipal chi mod q, n(chi) <= (log q)^8.
• (b) For every q >= 5, n(chi) <= (0.7 log q)^8 = 0.0576... (log q)^8. (Analytic for q >= 404; the exact table decides 5 <= q <= 403, and q = 3, 4 are the only moduli in [3, 10^4] where it fails.)
• (c) limsup over q of max_chi n(chi)/(log q)^8 is at most (96 sqrt(3)/343)^8 = 0.0030500... (kernel (s+c)^-4; the kernel (s+c)^-2 of parts (a) and (b) gives (4/7)^8 = 0.011368...).
• (d) Check against the abscissa: given only Oct 5 Thm 1.1 (Re s > 11/12), the same proof gives n(chi) <= (log q)^12 for every q >= 3, exponent 12 = 1/(1 - 11/12) in place of 8.

Consequences: (Z/qZ)^* is generated by the primes p <= (log q)^8; for p prime and k | p-1, the least k-th power nonresidue mod p is a prime at most (log p)^8; the least quadratic nonresidue satisfies n(p) <= (log p)^8 for every odd prime p.

The constant 1 in (a) cannot be lowered below 2/(log 3)^8 = 0.9425 for a statement of the form C (log q)^8 over all q >= 3: the character mod 3 has n(chi) = 2. In that sense (a) is tight at q = 3.

Theorem 2 (Miller test). Given Sept 30 Thm 1.1: every odd composite n has a strong witness w(n) <= (0.7 log n)^8 <= (log n)^8, with limsup w(n)/(log n)^8 <= (96 sqrt(3)/343)^8 = 0.00305. So Miller's deterministic test with bases a <= max(2, (0.7 log n)^8) is correct for every odd n > 1, at a cost of at most (0.7 log n)^8 modular exponentiations, O((log n)^11) bit operations with schoolbook arithmetic.

Theorem 3 (primitive roots). Given Sept 30 Thm 1.1: for every odd prime p,

g(p) <= g*(p) <= (Lambda_p log p)^8,

where Lambda_p >= 1 is the explicit sieve factor of section 4 computed from the distinct prime factors of p-1 (for example Lambda_p = 1 when p-1 is a power of 2, Lambda_p = 3 when p-1 = 2^a 3^b, and always Lambda_p <= 2^omega(p-1) - 1). Consequently g*(p) <= p^(7.68/log log p) (log p)^8 for p >= 17, an explicit subpolynomial bound. The polylogarithmic bound g*(p) << (log p)^24 with an absolute but uncomputed constant follows from Martin's shifted-sieve Proposition 3 and was stated by Guo on 8 October 2026 (section 7); this hunt did not make that constant explicit (section "The doors").

Exponent. The best exponent proved here for n(chi) and w(n) is 8 = 1/(1 - 7/8). Section 5 explains why the method cannot go below 1/(1 - theta) with a zero-free half-plane Re s > theta as its only zero information; that is a scoped statement about this information class, not an impossibility theorem.

1. The master inequality

Fix the abscissa theta in [1/2, 1) and a weight parameter c in (0, 1). In the application theta = 7/8. Put sigma0 = 2 theta + c.

1.1 Zero location (Lemma 0)

Let chi* be a primitive character of conductor q* >= 3, of parity a in {0, 1}. If L(s, chi) != 0 for Re s > theta for every Dirichlet character chi, then every nontrivial zero rho = beta + i gamma of L(s, chi*) and of zeta(s) has 1 - theta <= beta <= theta.

Proof. beta <= theta is the hypothesis. The functional equation Lambda(s, chi*) = eps Lambda(1 - s, conj chi*) with Lambda(s, chi) = (q/pi)^((s+a)/2) Gamma((s+a)/2) L(s, chi) sends rho to a zero 1 - rho of L(s, conj chi*); the hypothesis for conj chi* gives 1 - beta <= theta. For zeta use xi(s) = xi(1 - s). For theta = 7/8 the hypothesis is Sept 30 Thm 1.1 (Lean: OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, which covers every character, and OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re).

1.2 The weight (Lemma 1)

Let w(t) = t^c log(1/t) for 0 < t <= 1 and w(t) = 0 for t > 1. It is continuous, nonnegative, w(1) = 0, max w = w(e^(-1/c)) = 1/(c e) =: w_max, and its Mellin transform is W(s) = integral_0^1 w(t) t^(s-1) dt = 1/(s + c)^2 for Re s > -c. Put S(chi; x) = sum_{n <= x} Lambda(n) chi(n) w(n/x).

1.3 Explicit formula (Lemma 2)

For x > 1 and chi* primitive mod q* >= 3 with parity a, writing F(s) = -L'/L(s, chi*),

S(chi*; x) = - sum_rho x^rho W(rho) - sum_{k >= 0} x^(-a-2k) W(-a-2k)

• x^(-c) (F'(-c) + F(-c) log x),

the sum over nontrivial zeros with multiplicity, absolutely convergent. For zeta, with F_z = -zeta'/zeta,

S(1; x) = x W(1) - sum_rho x^rho W(rho) - sum_{k >= 1} x^(-2k) W(-2k)

• x^(-c) (F_z'(-c) + F_z(-c) log x).

Proof. W(s) = O(|s|^-2) on vertical lines, so Mellin inversion gives w(t) = (1/2 pi i) integral_{(2)} W(s) t^(-s) ds for every t > 0 (w is continuous and of bounded variation), and S = (1/2 pi i) integral_{(2)} F(s) W(s) x^s ds by absolute convergence. Shift the contour to Re s = -N - 1/2 along horizontal segments at heights T_j chosen with |L'/L(sigma + i T_j)| << log^2(q* T_j) for -1 <= sigma <= 2 (Davenport, Multiplicative Number Theory, chapters 17 and 19); on Re s <= -1, at distance >= 1/2 from the trivial zeros, |L'/L(s)| << log(q* |s|) (Davenport chapter 19). With |W| <= |s + c|^-2 the horizontal integrals are O(x^2 log^2(q* T_j)/T_j^2) -> 0 and the left integral is O(x^(-N-1/2) log(q* N)) -> 0. The residues are: at each zero of L, where F has residue -(multiplicity), -x^rho W(rho); at the double pole s = -c of W, d/ds [F(s) x^s] at s = -c. The trivial zeros are simple, at -a - 2k, and -c is none of them because 0 < c < 1. For zeta add the pole at s = 1 (residue x W(1)); zeta(0) = -1/2 != 0, so s = 0 is regular. The zero sum converges absolutely because sum 1/|rho + c|^2 < infinity.

1.4 Hadamard identity (Lemma 3)

For real sigma > 1 (Davenport chapter 12, with Re B(chi) = -sum Re 1/rho),

sum_rho Re 1/(sigma - rho) = Re L'/L(sigma, chi*) + (1/2) log(q*/pi)

• (1/2) psi((sigma + a)/2), sum_rho Re 1/(sigma - rho) = zeta'/zeta(sigma) + 1/(sigma - 1) + 1/sigma
• (1/2) log pi + (1/2) psi(sigma/2) (zeta),

psi the digamma function. Since |L'/L(sigma)| <= -zeta'/zeta(sigma) and psi increases on (0, infinity), the first sum is at most (1/2) log q* + A_chi with

A_chi = -zeta'/zeta(sigma0) - (1/2) log pi + (1/2) psi((sigma0 + 1)/2), A_zeta = the zeta sum at sigma0 (exact).

At theta = 7/8, c = 1/4 (sigma0 = 2): A_chi = 0.0158410371591, A_zeta = 0.0690662315300 (Arb radii below 3e-14, matching mpmath).

This identity replaces the explicit zero-counting estimate one would otherwise use (for example Bennett, Martin, O'Bryant and Rechnitzer, 2021): it prices the whole Lorentzian-weighted zero sum exactly, with coefficient 1/2 on log q and no height truncation, so no N(T, chi) error term enters.

1.5 Pointwise domination (Lemma 4)

For every zero with 1 - theta <= beta <= theta,

|x^rho W(rho)| <= x^theta K(x) Re 1/(sigma0 - rho), K(x) = sup_{1-theta <= b <= theta} x^(b - theta) (sigma0 - b)/(b + c)^2.

Proof. |x^rho| = x^theta x^(beta - theta) and, with A = sigma0 - beta and B = beta + c, |W(rho)|/Re 1/(sigma0 - rho) = (A^2 + gamma^2)/(A (B^2 + gamma^2)). Because A - B = 2(theta - beta) >= 0 this decreases in gamma^2, so its supremum is A/B^2 at gamma = 0.

If log x >= L_star := max(1/(3 theta + c - 1) + 2/(1 - theta + c), 3/(theta + c)) then K(x) = 1/(theta + c): the logarithmic derivative in b of x^(b - theta)(sigma0 - b)(b + c)^(-2) is log x - 1/(sigma0 - b) - 2/(b + c), and 1/(sigma0 - b) + 2/(b + c) is convex in b, so it is at most its larger endpoint value L_star. At theta = 7/8, c = 1/4: L_star = 88/15 = 5.8667 and K = 8/9. For smaller x, explicit.K_of_x encloses K by monotone bounds on 400 subintervals (not used in the theorems).

1.6 Residue and trivial-zero terms (Lemma 5)

From the functional equation, F(-c) = L'/L(1 + c, conj chi*) + log(q*/pi) + (1/2) psi((a - c)/2) + (1/2) psi((1 + a + c)/2) and F'(-c) = (1/4) psi'((a - c)/2) - (1/4) psi'((1 + a + c)/2) - (L'/L)'(1 + c, conj chi*). With |(L'/L)'(sigma)| <= (zeta'/zeta)'(sigma) = sum Lambda(n) log n n^-sigma,

|F(-c)| <= log q* + R1, R1 = -zeta'/zeta(1 + c) + max_a |-log pi + psi((a-c)/2)/2 + psi((1+a+c)/2)/2|, |F'(-c)| <= R2, R2 = (zeta'/zeta)'(1 + c) + max_a [psi'((a-c)/2) + psi'((1+a+c)/2)]/4.

At c = 1/4: R1 = 6.18263022331, R2 = 33.1875202227, F_z(-c) = -zeta'/zeta(-1/4) = -1.73974728516 (also obtained through the functional equation from zeta'/zeta(5/4), agreeing to 1e-14), F_z'(-c) = -0.184649743756. The trivial-zero sums are at most

T_chi(x) = max(1/c^2 + x^-2/((2-c)^2 (1 - x^-2)), x^-1/((1-c)^2 (1 - x^-2))), T_zeta(x) = x^-2/((2-c)^2 (1 - x^-2)).

1.7 Proposition 1 (master inequality)

Let q >= 3, chi nonprincipal mod q, x > 1, and let omega_x(q) count the primes p | q with p <= x. Define

Phi(x; L, omega) = W(1) x - K(x) x^theta (L/2 + A_chi + A_zeta)

• T_chi(x) - T_zeta(x)
• x^(-c) (R2 + (L + R1) log x)
• x^(-c) (|F_z'(-c)| + |F_z(-c)| log x)
• 2 omega w_max log x.

If the zero-free hypothesis holds for abscissa theta and Phi(x; log q, omega_x(q)) > 0, then n(chi) <= x.

Proof. Let chi be induced by the primitive chi* mod q* (q* | q, q* >= 3). Suppose chi(p) is in {0, 1} for all primes p <= x. For prime powers n <= x coprime to q, chi*(n) = chi(n) = 1. Hence, using Re chi* >= -1 and w >= 0,

Re S(chi*; x) >= S(1; x) - 2 sum_{n <= x, (n,q) > 1} Lambda(n) w(n/x)

= S(1; x) - 2 omega_x(q) w_max log x,

because each p | q contributes at most (log x/log p) log p w_max. By Lemmas 2 to 5, Re S(chi*; x) <= x^theta K(x)((1/2) log q* + A_chi)

• T_chi(x) + x^(-c)(R2 + (log q* + R1) log x), and

S(1; x) >= W(1) x - x^theta K(x) A_zeta - T_zeta(x) - x^(-c)(|F_z'| + |F_z| log x). Combining, with log q* <= log q, gives Phi(x; log q, omega_x(q)) <= 0.

2. Theorem 1: proof

Lemma 6 (generation). For q >= 3, max over nonprincipal chi of n(chi) equals N(q), the least prime P such that the primes p <= P with p not dividing q generate (Z/qZ)^*. Proof: if those primes generate, every nonprincipal chi is != 1 on one of them. If the primes p < N(q) generate a proper subgroup H, a nonprincipal chi trivial on H exists (duality of finite abelian groups) and has n(chi) >= N(q).

(a) Take theta = 7/8, c = 1/4, L = log q >= 5/2 and x = L^8. Then omega_x(q) <= omega(q) <= L/log 2 and log x = 8 log L >= 8 log(5/2) = 7.33 >= L_star, so K(x) = 8/9. The normalised margin is

m(L) = Phi(L^8; L, L/log 2)/L^8 = W(1) - (8/9)(1/2 + (A_chi + A_zeta)/L) - [T_chi + T_zeta]/L^8

• L^-10 (R2 + 8 (L + R1) log L) - L^-10 (|F_z'| + 8 |F_z| log L)
• (16 w_max/log 2) log L/L^7.

Every subtracted term is nonincreasing on [5/2, infinity): A_chi + A_zeta > 0; T_chi and T_zeta decrease in x; for f(L) = L^-10 (R2 + 8(L + R1) log L), L^11 f'(L) = -10 R2 - (72 L + 80 R1) log L + 8 L + 8 R1 < 0 once log L >= 1/9; the zeta residue term likewise once log L >= 1/10; and log L/L^7 decreases for L > e^(1/7). Hence m(L) >= m(5/2) for L >= 5/2, and the Arb enclosure gives

m(5/2) in [0.0923768413043 +/- 4.6e-14] > 0.

So n(chi) <= (log q)^8 whenever q >= e^(5/2), that is q >= 13. For 3 <= q <= 12 Lemma 6 and the exact table give N(q) = 2, 3, 2, 5, 3, 5, 2, 3, 2, 7 against (log q)^8 = 2.12, 13.6, 45.0, 106, 206, 350, 543, 790, 1093,

1. The binding case is q = 3.

(b) Same argument with c = 1/8 (W(1) = 64/81, K = 1, L_star = 60/7) and x = (0.7 L)^8. The normalised margin is W(1) - K/(2 kappa) - K(A_chi + A_zeta)/(kappa L) - [lower order]/(kappa L)^8 with kappa = 7/10 and A_chi + A_zeta = 0.1771 > 0; the same monotonicity checks hold (the residue derivative is negative once log(kappa L) >= 1/8), and log x = 8 log(4.2) = 11.48 >= L_star at L = 6. The enclosure gives margin 0.0260475186243 > 0 at L = 6, so n(chi) <= (0.7 log q)^8 for log q >= 6, that is q >= 404. For 3 <= q <= 403 the exact table of Lemma 6 gives N(q) <= (0.7 log q)^8 except at q = 3 and q = 4 (N = 2 and 3 against 0.12 and 0.79), which proves (b) for q >= 5.

(c) For fixed c in (0, 1) and kappa > c1(c) := (1 + c)^2/(2 (theta + c)), the normalised margin at x = (kappa L)^8 tends to W(1)(1 - c1(c)/kappa) > 0 as L grows (K(x) = 1/(theta + c) for large x and every other term is o(1)). So n(chi) <= (kappa log q)^8 for all large q. The function c1 increases on c > 0 (its derivative has the sign of 2 theta + c - 1 > 0), with infimum 1/(2 theta) = 4/7 as c -> 0. Hence the limsup is at most (4/7)^8.

The sharper constant uses the kernel W(s) = (s + c)^-4, the Mellin transform of w(t) = t^c log^3(1/t)/6. Lemma 2 holds verbatim (W = O(|s|^-4)), except that the residue at the fourth-order pole s = -c is x^(-c) sum_{j=0}^{3} binom(3, j) F^(j)(-c) (log x)^(3-j)/6. By the functional equation F(-c) = O(log q) and F^(j)(-c) = O(1) for j >= 1 (polygamma values and |(L'/L)^(j)(1+c)| <= |(zeta'/zeta)^(j)(1+c)|), so the residue is O(x^(-c) (log x)^3 log q), negligible against x^theta log q; the trivial zeros and the prime powers dividing q are O(1) and O(omega(q) log x) as before. For Lemma 4, with A = sigma0 - beta and B = beta + c, the function y -> (A^2 + y)/(A (B^2 + y)^2) of y = gamma^2 has its maximum at y = max(0, B^2 - 2A^2), with value 1/(4A(B^2 - A^2)) if B^2 > 2A^2 and A/B^4 otherwise; the two expressions agree on B^2 = 2A^2, so k(beta) = sup_gamma |W(rho)|/Re 1/(sigma0 - rho) is continuous on [1 - theta, theta]. Then K(x) = sup_beta x^(beta - theta) k(beta) tends to k(theta) as x grows (near theta by continuity, away from theta by the factor x^(beta - theta)). Choosing sigma0 = theta + (theta + c)/sqrt(3) (which is

1. maximises A (B^2 - A^2) at beta = theta and gives

k(theta) = 3 sqrt(3)/(8 (theta + c)^3). The normalised margin at x = (kappa L)^8 tends to W(1) - k(theta)/(2 kappa), positive for kappa > c1 := 3 sqrt(3)(1 + c)^4/(16 (theta + c)^3). As c -> 0 this tends to 3 sqrt(3)/(16 theta^3) = 96 sqrt(3)/343 = 0.484772237687 (Arb), whose eighth power is 0.00305001463226. A float scan over kernels (s + c)^-k (section "The doors") finds k = 4 best among integers 2 to 20.

At c = 1/4 the per-modulus bound of Proposition 1 converges visibly: x0^(1/8)/log q = 0.735, 0.705, 0.700, 0.695 at q = 10^2, 10^5, 10^10, 10^100, against c1(1/4) = 25/36 = 0.6944.

(d) Take theta = 11/12, c = 1/4 (sigma0 = 13/6, K = 6/7, L_star = 13/2, A_chi + A_zeta = -0.03250 + 0.07287 = 0.0404 > 0) and x = L^12. The normalised margin has the same form as in (a) with L^-8 replaced by L^-12; the residue term is L^-15 (R2 + 12 (L + R1) log L), decreasing once log L >= 1/14, and the other monotonicity checks carry over. At L = 2, log x = 12 log 2 = 8.32 >= L_star and the enclosure gives margin 0.1694 > 0, so n(chi) <= (log q)^12 for q >= e^2. For 3 <= q <= 7 the table of (a) gives N(q) <= (log q)^8 <= (log q)^12 because log q > 1. The measured per-modulus slope for theta = 11/12 is 11.9996 (section 6).

Corollaries. Lemma 6 turns (a) into "(Z/qZ)^* is generated by primes <= (log q)^8". For k | p - 1 a nonprincipal character of order dividing k detects k-th powers, so the least k-th power nonresidue is <= (log p)^8.

3. Theorem 2: proof

Lemma 8 (Miller liars, standard; Bach 1990). If n is odd and composite, the strong liars mod n lie in a proper subgroup H of (Z/nZ)^*. Proof. Write n - 1 = 2^s d with d odd. If n is not a prime power, let j be the largest r such that some liar b has b^(2^r d) = -1 (r = 0 works for b = -1) and m = 2^j d. Every liar a has a^m = +1 or -1: either a^d = 1, or a^(2^r d) = -1 with r <= j. So liars lie in H = {a : a^m = +1 or -1}. Write n = n1 n2 with coprime n1, n2 > 1 and choose a = b mod n1, a = 1 mod n2: then a^m = -1 mod n1 and +1 mod n2, so a is not in H. If n = p^k with k >= 2, liars satisfy a^(n-1) = 1, a subgroup of order gcd(n - 1, p - 1) < phi(n).

Proof of Theorem 2. Take a nonprincipal chi mod n trivial on H. The least a >= 2 with chi(a) != 1 is a witness: either gcd(a, n) > 1, or a is a unit outside H. That a is at most n(chi), and Theorem 1(a), (b) with q = n >= 9 apply.

4. Theorem 3: proof

Fix an odd prime p and a primitive root g0 mod p; for squarefree m | p - 1, call n m-free if p does not divide n and n is not an l-th power mod p for any prime l | m. Let rad(p - 1) = e p_1 ... p_s with the p_i the primes not in the "core" e, Omega = 2^omega(e) and delta = 1 - sum 1/p_i.

Lemma 7 (Cohen-Huczynska prime sieve, with character errors). Put S0 = S(chi0; x) and E = max over nonprincipal chi mod p of |S(chi; x)|. For squarefree m | p - 1 let N(m) be the w-weighted count of m-free n <= x. Then:

• (i) N(m) = theta(m) sum_{d | m} mu(d)/phi(d) sum_{ord chi = d} S(chi; x), theta(m) = phi(m)/m, hence |N(m) - theta(m) S0| <= theta(m)(2^omega(m) - 1) E.
• (ii) If s >= 1, the weighted count of primitive roots is at least sum_i N(e p_i) - (s - 1) N(e), which by (i) is at least theta(e)[delta S0 - E((s - 1)(3 Omega - 2) + delta (2 Omega - 1))].
• (iii) So, with Lambda = (2 Omega - 1) + (s - 1)(3 Omega - 2)/delta for s >= 1 and delta > 0, or Lambda = Omega - 1 for s = 0, a prime primitive root <= x exists whenever S0 > Lambda E.

Proof. (i) expands prod_{l | m}(1 - (1/l) sum_{chi^l = 1} chi(n)); a cyclic group of order p - 1 has phi(d) characters of exact order d. (ii) is pointwise: if n is not e-free both sides vanish; otherwise, with k primes p_i for which n is a p_i-th power, the right side is (s - k) - (s - 1) = 1 - k, which is at most the indicator of k = 0. The inequality is checked exhaustively for four primes in the test file. (iii) If a prime power l^k is a primitive root, so is l, and l <= l^k <= x. Lambda_p is the minimum of Lambda over the cores e made of the j smallest primes of p - 1 (j = 0, ..., omega(p - 1)), and Lambda_p >= 1.

Proof of Theorem 3. Every nonprincipal chi mod p is primitive with conductor p, so Lemmas 2 to 5 give E <= x^theta K(x)((1/2) log p + A_chi)

• T_chi + x^(-c)(R2 + (log p + R1) log x), and

S0 >= W(1) x - x^theta K A_zeta - T_zeta - x^(-c)(|F_z'| + |F_z| log x)

• w_max log x (the last term removes the powers of p). Put L = log p,

lambda = Lambda_p and x = (lambda L)^8. The normalised margin (S0 - lambda E)/x is

W(1) - K/2 - K A_chi/L - K A_zeta/(lambda L)

• [T_zeta + R_zeta + w_max log x]/x - lambda [T_chi + R_chi]/x,

and each subtracted term is nonincreasing in L >= 5/2 and in lambda >= 1: A_chi = 0.0158 >= 0; log x/x decreases; lambda T_chi/x <= lambda^-7 L^-8 times a decreasing function; and lambda R_chi/x equals lambda^-9 L^-10 (R2 + 8 (L + R1) log(lambda L)), whose logarithmic derivative in lambda is negative because 9 (R2 + 8 (L + R1) log L) > 8 (L + R1) for L >= 5/2 (the L-derivative is handled as in Theorem 1). At (L, lambda) = (5/2, 1) the enclosure gives 0.136300983659 > 0. Primes p < 13 are checked directly: g*(3), g*(5), g*(7), g*(11) = 2, 2, 3, 2 against bounds (Lambda_p log p)^8 = 2.12, 45.0, 1.35e6, 7.17e6 (Lambda_p = 1, 1, 3, 3).

The subpolynomial corollary uses Lambda_p <= 2^omega(p-1) - 1 (core e = rad(p-1)) and Robin's bound omega(n) <= 1.3841 log n/log log n for n >= 3, giving 2^omega(p-1) <= p^(0.9594/log log p) and the exponent 8 x 0.9594 < 7.68.

Lambda_p in practice: for p up to 10^7 the largest value is 99.9 and the record-holder p = 6525032504501281 (g = 417) has Lambda_p = 95.6.

5. Can the exponent 8 be beaten?

Not within this method, and the reason is structural. Under the hypothesis chi(p) = 1 for p <= x the prime sum S(chi*; x) equals W(1) x up to lower order, and the only information about the zero sum is that each zero has real part at most theta and that the Lorentzian weights Re 1/(sigma0 - rho) sum to (1/2) log q + O(1). A configuration with many zeros close to Re s = theta and phases aligned at the single scale x is not excluded by that information, and it balances x against x^theta log q, which is exactly x = (log q)^(1/(1 - theta)). Three variants were examined:

1. Vinogradov's smooth-number amplification needs sum_{n <= N} chi(n) = o(N) at N a power of log q. The half-plane gives that only through Lindelof-type bounds log L(sigma + it) << (log q)^((1 - sigma)/(1 - theta) + eps) (the standard Borel-Caratheodory consequence of a zero-free half-plane, not reproved here), which yield o(N) only for N >= exp((log q)^(1 - eps)). Unresolved, not refuted: it needs short character sums, a different information class.
2. Symmetry of the zeros in [1/8, 7/8] (zeros of chi and conj chi pair as rho, 1 - conj rho) is already used in Lemma 0; it removes zeros left of 1/8 but leaves the configuration above intact.
3. Weighting by x^(beta - theta) (Lemma 4) suppresses zeros left of theta, which improves constants only.

So the best exponent proved here is 8. A strictly smaller abscissa would lower it by the same formula: Guo's unreviewed claim of a zero-free region beyond 7/8, with sigma_dagger = 0.8749570194..., would give exponent 1/(1 - sigma_dagger) = 7.99725 through the identical proof (measured slope 7.9967 at log q from 10^3 to 10^4 with theta = 0.8749570195).

6. Controls

All checked by verify.py; values from verification.json.

• Constants: every Arb constant agrees with an independent mpmath evaluation to 1e-12; -zeta'/zeta(-1/4) from Arb's continuation agrees with the functional-equation route.
• Explicit formula normalisation (Lemma 2, measured): for zeta with c = 1/4, the direct prime-power sum S(1; x) and the right side built from the first 1000 zeros (cached data/zeros_1000.json), the trivial zeros and the residue at s = -1/4 agree to 6.3e-5 at x = 100 and 3.0e-4 at x = 1000, inside the truncation tail; flipping the residue's sign moves the right side by 5.2. Lemma 5's functional-equation expression for -L'/L(-1/4) matches mpmath's Hurwitz continuation to 1e-20 for the odd character mod 4 and the even quadratic character mod 5.
• Small moduli (exact subgroup generation, 3 <= q <= 10^4, 34 s): max over q of N(q)/(log q)^8 is 0.9425 at q = 3; the largest N(q) is 43, at q = 2706. Against (0.7 log q)^8 the only failures are q = 3 and q = 4.
• Exhaustive tables to 10^7 (664578 odd primes, 4335421 odd composites, 18 s): least quadratic nonresidue max 53 (p = 9257329), minimum margin (log p)^8/n(p) = 1.061 at p = 3; Bach's GRH-conditional 2 (log p)^2 also holds throughout. Least primitive root max 94 (p = 5109721), least prime primitive root max 211 (p = 4022911); Theorem 3's (Lambda_p log p)^8 has minimum margin 1.061 (p = 3), with Lambda_p <= 99.9. Least strong witness max 5 (n = 1373653; 155 composites need 3, seven need 5); Theorem 2's minimum margin (log n)^8/w(n) is 271.6 (n = 9). The exponent-1 lesion n(p) <= log p is refuted by 11002 primes in the same table.
• Literature record values, each re-derived by direct computation before use: OEIS A000229 (38 primes with record least nonresidue, up to n(70864718555231) = 163), OEIS A002229/A002230 (record least primitive roots, up to g(6525032504501281) = 417; g* is 431 there), OEIS A014233 (Jaeschke and Sorenson-Webster strong pseudoprimes to the first k prime bases, up to 3317044064679887385961981, whose least witness is 22).
• Weakened abscissa (the requested ablation): the per-modulus bound x0(log q) has local slope d log x0/d log log q equal to 7.9995, 11.9996, 15.9997 at log q in [10^3, 10^4] for theta = 7/8, 11/12, 15/16, against predictions 8, 12, 16.
• Planted fault: dropping every log q term (the zero sum) yields a q-free "bound" of 66.1; the record table refutes it at p = 48473881, where n(p) = 67. The honest bound there is 4.4e7.
• Precision: the closed-form margins are evaluated at 160 bits; their radii are below 1e-13, five orders below the margins.

7. Prior art and the search

Searched on 2026-10-08 (web search over arXiv, journal pages and Google Scholar listings; zbMATH was not queried directly; OEIS for record values):

• Exponent 1/(1 - theta) from a zero-free strip is classical in non-explicit form: Rodosskii (1954); Montgomery, Topics in Multiplicative Number Theory, chapter 9, Theorem 1, and Montgomery-Vaughan, Multiplicative Number Theory I, Theorem 13.12: n(chi) < (A delta log q)^(1/delta) with an absolute A; Pathak, arXiv:2405.13316, Theorem 2: n(chi) << (log q)^(1/delta).
• OpenAI, "The Quasi-Riemann Hypothesis", Sept 30 2026, Corollary 1.2: n(p) <= C (log p)^A with A = 32, C unstated ("no optimization is needed here"), through Bhargava-Ivanyos-Mittal-Saxena; it also notes a deterministic square-root algorithm. The Oct 5 2026 companion (11/12) remarks in its introduction that the Rodosskii method, in the form of Montgomery-Vaughan Theorem 13.12, gives a fixed power of log p, and that the half-plane yields a deterministic polynomial-time Miller test; it states no constants.
• J. Guo, "Goldbach-type representations with prescribed Liouville signs", Part II (8 October 2026, github.com/aster2024/openai-math-followups): n(p) < (A_MV/8)^8 (log p)^8 with A_MV the uncomputed Montgomery-Vaughan constant; generation of (Z/qZ)^* by primes < C (log q)^8 with C uncomputed; g*(p) << r^16 (1 + log r)^16 (log p)^8 << (log p)^24 via Martin's Proposition 3. Guo cites a 7 October memo by Sandlund deriving proper-subgroup and least prime primitive root bounds via Shoup's sieve; that memo was not read here.
• Under GRH with explicit constants: Bach (Math. Comp. 1990), 2 (log n)^2 for Miller witnesses and n(chi); Lamzouri-Li-Soundararajan (Math. Comp. 2015); Carneiro-Milinovich-Quesada-Herrera-Ramos (arXiv:2404.08380).
• Primitive roots: Wang, Shoup (Math. Comp. 1992, Proposition 1 with absolute constants c1, c2), Martin (Acta Arith. 1997, Proposition 3).

Not found in the searched set: any explicit numerical constant for the exponent-8 bound (conditional on a quasi-GRH or unconditional), the all-moduli statement n(chi) <= (log q)^8, an explicit Miller witness bound from a zero-free half-plane, or the explicit primitive-root criterion g*(p) <= (Lambda_p log p)^8. This is a report of the search, not a novelty claim. The exponents themselves are not new; the constants are this lab's original work (provenance), graded as above.

8. Grades

Standard analytic inputs used without reproof: the explicit-formula contour estimates of Davenport chapters 12, 17 and 19, the Hadamard identity, and the functional equation. The hunt does not promote its own claim: candidate, pending external review.

The doors

This hunt stops at three walls: the exponent 8 for single characters, the polylogarithmic exponent 24 for primitive roots that it did not make explicit, and the constant (96 sqrt(3)/343)^8 in Theorem 1(c).

1. Active constraints at the optimum.
2. The binding constraint for the asymptotic constant is Lemma 4 at beta = theta: the peak of |W| against the Lorentzian of width sigma0 - theta. Its shadow price is the kernel shape. A float scan over W(s) = (s + c)^-k, c -> 0, gives c1 = 0.5714, 0.5027, 0.4848, 0.4883, 0.5046, 0.5642 for k = 2, 3, 4, 5, 6, 8; k = 4 is proved in 2(c). The explicit all-moduli parts (a), (b) still use k = 2: carrying k = 4 into them needs third derivatives of F at -c and a two-regime K(x), routine but not done.
3. For the all-moduli constant the binding case is q = 3, where n(chi) = 2 and (log 3)^8 = 2.12: no constant below 0.9425 is possible.
4. For primitive roots the binding term is the sieve factor Lambda_p: the Cohen-Huczynska sieve pays 2^omega(core). Martin's shifted sieve pays only omega^2 (log omega)^2 but with uncomputed absolute constants. The door is an explicit linear (Rosser-Iwaniec or Selberg lower-bound) sieve constant at level about omega^2; with it the same Proposition 1 makes g*(p) <= C r^16 (1 + log r)^16 (log p)^8 explicit.
5. Frozen-constant inventory.
6. c = 1/4 in (a): trades K and W(1) against the residue constants R1, R2, which grow like 1/c^2 as c -> 0 (genuine trade shape: c = 1/8 is better asymptotically and worse below log q = 3).
7. sigma0 = 2 theta + c: optimal for Lemma 4 at beta = theta; a mixture of Poisson kernels at several sigma would trade the gamma-tail against the peak (unexplored).
8. Re L'/L(sigma0) <= -zeta'/zeta(sigma0): under the hypothesis chi(p) = 1 for p <= x the left side is about +zeta'/zeta(sigma0) (Rodosskii's gain), lowering A_chi by about 1.1 at sigma0 = 2; affects only the lower-order constant.
9. c in (-1/8, 0): moves the double pole of W into (0, 1/8), left of every zero, and would lower the k = 2 constant 4/7 to 49/96 = 0.5104 at the price of bounding L'/L inside the strip by its zero sum (genuine trade shape, unexplored).
10. omega(q) <= log q/log 2 in the closed form: Robin's bound would shrink that term; it is lower order.
11. Information class. Every door in 1 and 2 stays inside the data the current family reads: the zero-free half-plane, the functional equation and the Hadamard identity. Beating the exponent 8 for n(chi) requires reading more: a smaller abscissa (exponent 1/(1 - theta) moves with it; Guo's unreviewed sigma_dagger gives 7.997), zero-density information near Re s = 7/8 for a single L-function, or short character-sum bounds for Vinogradov's trick. Candidate status: (log q)^8 is the measured and proved wall for this family, not a proved ceiling.