Guo's Hypothesis 11.1: which mean value it is, what implies it, and 328/375

Hunt #136, lane b78-amplification, 2026-10-21. Lane output, unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how the hunt ended. Every fraction and decimal on this page is recomputed by price.py and pinned by test_price.py; integers are line numbers or the integer coefficients of a quoted identity, which the tests check against the source.

Grades as in MISSION.md: cited (a line of a source file, quoted, with the file's SHA-256 checked by price.verify_citations), derived (exact arithmetic here, pinned by a test), inferred (a reading or a standard step that no computation here checks), measured (float). Nothing on this page bears on real part 1/2, and no abscissa below is claimed as established: every one is conditional on an unproved hypothesis or on inputs this lane does not check.

Sources (read as data, outside the tree):

1. The inequality, exactly

Guo, section 11, lines 33 to 46 (cited). Fix a finite-order ray character nu with its zero-extended presentation, a finite set S of primes containing its modulus support and the primes over 6, a compact interval I = [a, b] in (0, infinity), and a nonnegative radial Schwartz majorant Phi with Phi >= 1 on [0, 1] and compactly supported Fourier transform. Put

A_u(D) = sum over (n, S) = 1 of mu(n) nu(n) chi_n(u) W(q_n / D), M(D, H) = (1/D) sum over u in O of Phi(q_u / H) |A_u(D)|^2.

Hypothesis 11.1. For every such datum and every eps > 0 there are J and C with M(D, H) <= C H D^eps ||W||^2_{C^J(I)} for all D >= 2, all D^(99/100) <= H <= D^(101/100) and all W in C_c^infinity(I) (line 43 and line 45, cited).

In the paper's notation, M_u(D; W) = D^(-1/2) sum mu(n) nu(n) chi_n(u) W(q_n/D) (line 5225, cited), so the hypothesis is the bound sum_u Phi(q_u/H) |M_u(D; W)|^2 << H D^eps. It is the paper's equation raw-moment (lines 12353 to 12357) with its range H >= D^(1+c) extended down to H >= D^(99/100) (derived from the two cited displays): a Moebius-weighted mean square of sextic-character polynomials of length D, over every row u of norm at most about H, at diagonal size, with slightly fewer rows than columns. The paper obtains raw-moment from Lemma 17.1 at Z = H, m = 1, no prime slots (lines 12345 to 12350, cited); Lemma 17.1's first width condition r + 2z <= m - c_1 (line 9254, cited) is then r = log_H D <= 1 - c_1, and the hypothesis asks for r <= 100/99. Guo states that the 11/12 companion paper's Proposition 3.1 proves the case H = D^(1 + theta), 0 < theta <= 1/10 (line 33, cited; the companion was not read in this lane). Two further differences are cosmetic: the smooth majorant Phi in place of the sharp ball, and the explicit C^J norm of W.

Its use (line 61 to 63, cited) is Lemma 17.6's proof with H' = max(2U, D^(99/100)) in place of H = max(2U, D^(1+c)): then H'/P = U^(1/6) H'^(5/6) and the amplified exponent becomes e_99(r) = max{1, 1/6 + 33r/40} (derived: (5/6)(99/100) = 33/40). In price.py this is the law Law.threshold(99/100), slope 33/40 and intercept 1/6.

2. Does a published theorem imply it?

No, in the searched set, and for two reasons that are checkable here.

(a) No mean value for arbitrary coefficients can give it (derived, one line). For any row family containing u = 1, where chi_n(1) = 1 for every column n, an inequality sum_u |sum_n c_n chi_n(u)|^2 <= Delta sum_n |c_n|^2 valid for all c forces Delta >= N, the number of columns: take c_n = 1 and keep only the row u = 1, which contributes N^2. Applied to the Moebius coefficients, such an inequality therefore bounds M(D, H) by a constant times D at best, never by H D^eps when H = D^(99/100). The two named results are of this kind:

• Heath-Brown's cubic large sieve (Israel J. Math. 120 (2000), Theorem 2; quoted by Baier and Young at line 195 of their source, and in its sextic primary-convention form as the paper's own Lemma sextic-large-sieve, line 4717, both cited; Heath-Brown's paper itself was not read): arbitrary coefficients on squarefree elements, constant M + N + (MN)^(2/3). At H rows and D columns with H = D^(99/100) this is at least (HD)^(2/3), which exceeds H.
• Baier and Young, "Mean values with cubic characters" (arXiv:0804.2233v3, whose source was read; published in J. Number Theory 130 (2010) per a web search), Theorem 1.3 (line 171) and its sextic extension Theorem 1.4 (line 205), cited: primitive cubic, and sextic, Dirichlet characters to moduli q in (Q, 2Q], rational squarefree m in (M, 2M], arbitrary coefficients, constant min{Q^(5/3) + M, Q^(4/3) + Q^(1/2) M, Q^(11/9) + Q^(2/3) M, Q + Q^(1/3) M^(5/3) + M^(12/5)}. Every branch is at least M, as (a) says it must be; the family is also a different one (rational columns, Dirichlet characters), so it does not even address the rows u in Z[w].

(b) The hypothesis alone already implies a zero-free half-plane (derived from its own statement; the passage from smoothed partial sums to the Dirichlet series is the standard Mellin step, inferred). Keep only the row u = 1 in M(D, H) and take H = D^(99/100): |A_1(D)|^2 <= C D^(199/100 + eps), so sum mu(n) nu(n) W(q_n/D) << D^(199/200 + eps) for every W, and 1/L^S(s, nu) is holomorphic in Re s > 199/200. Using the rows u = a^6 as well, which is Lemma 17.6's argument at U = 1 with the hypothesis in place of raw-moment, gives |M_1(D)|^2 << H^(5/6) and the abscissa 1/2 + (5/12)(99/100) = 73/80. Every finite-order Hecke character of Q(sqrt(-3)) is such a nu, so Hypothesis 11.1 by itself gives every finite-order Hecke L-function of that field, and through the paper's transfer (lines 310 to 318) every Dirichlet L-function, no zeros in Re s > 73/80. A zero-free half-plane of any width was open before the 2026 preprints (the 7/8 preprint presents its theorem as resolving that question, lines 113 to 116, cited), so no earlier theorem can imply the hypothesis without having implied that. Of the 2026 results read by this hunt (the 7/8 preprint, Liu, Xu and Sang, Guo, the tracker records; FOLLOWUPS.md), none states a mean value of this shape with fewer rows than columns; the 7/8 theorem itself only bounds each row pointwise, giving H D^(3/4) (inferred from the reading; this is not a search of the whole literature).

The same test applied to the paper's own input is a consistency check, not a new claim: Lemma 17.6 at U = 1 reads |M_1(D)|^2 << D^(5(1+c)/6), which is a zero-free half-plane Re s > 11/12 for every L(s, nu), the abscissa of the paper's Part I. A raw moment valid for H >= D^theta gives 1/2 + 5 theta/12 this way (price.trivial_route_abscissa): 11/12 at theta = 1, 73/80 at theta = 99/100, 7/8 at theta = 9/10, 7/9 at theta = 2/3 (derived). Conversely GRH for the family implies the hypothesis (shift to Re s = 1/2 + eps; inferred). So it sits between GRH and a zero-free half-plane, and is at least as deep as the latter.

3. The claimed consequence 328/375, recomputed

Guo's proposition prop:conditional (lines 50 to 99, cited), checked step by step in exact arithmetic:

• Geometry (lines 81 to 83): ell = 21/125, b = 11/100 give l_x = 361/1000, l_y = 471/1000, h = 807/1000 and sigma = 11/12 - ell/4 = 328/375 (derived).
• Counts (lines 73 to 75): with e_99, the crossing of the short count with the long branch 1/6 + (33/40 - delta) t is t_99 = 1 + (D_x/120 + delta P_x/2)/J_99 and the excess over 1 - delta is T_99 = (97/120 - delta) delta P_x / (2 J_99), J_99 = (33/40 - delta) D_x + delta P_x (derived: price.excess with Law.threshold(99/100) equals T_99, and the minimisation over t done directly lands on t_99).
• Endpoint (Guo 06_geometry.tex line 15): E_99 = -1/4 + b/6 + 5 ell/4 - b delta/2 - ell (1/2 - x) delta + h T_99, which is hunt #134's E_* on the face M + ell = 1.
• The polynomial identity of lines 86 to 89, `20000(-108 J_99 E_99) = 526080 delta^2 - 406288 delta
• 79365 + y(1292640 delta^2 - 603308 delta + 72930) + y^2 (920640 delta^2 - 19144 delta) + 322560 y^3 delta^2 with y = 1/2 - x, holds exactly: both sides have degree at most 2 in delta and 3 in y`, and they agree on a 13 by 5 rational grid (derived).
• The three completed-square remainders of lines 92 to 94 are exactly 45529856, 13108397936 and 1763264 (derived).
• The bound E_99 <= -7/42768 on 0 <= delta <= 3/4, 0 <= x <= 1/2 (line 98) is decided true by this lane's own exact rectangle procedure, which does not use Guo's squares (derived).

So, given Hypothesis 11.1 and the rest of Guo's envelope (whose other inputs are the paper's lemmas, not checked here), the endpoint inequality at 328/375 holds. 328/375 is a rational point, not the envelope's infimum. Under e_99 the exact optimum of the same endpoint system (hunt #134's frame with delta <= 3/4, which is Guo's criterion of 06_geometry.tex line 22) is

2305/2529 - 8 sqrt(22906)/32877 = 0.8745999251,

at ell = 0.1682669662, b = 0.1093307117, inside Guo's geometry box 1/6 <= ell <= 17/100, 1/10 <= b <= 13/100 (derived; the sandwich of LOSS_MAP.md section 4 agrees). Inside Liu's refined comparison system the same hypothesis gives the fixed point 0.8744476409, at kappa = 0.7488952817, which lies below the kappa range 7497/10000 <= kappa <= 3/4 in which Liu's tangent family is stated (derived; Liu's admissibility would have to be re-checked there). Both are conditional numbers about comparison systems, not claims.

For scale: the tracker's record is 8748087/10^7 = 0.8748087 (FOLLOWUPS.md, cited there). The raw-moment threshold that would reach it is theta = 0.9958275690 in hunt #134's system and 0.9982147398 in Liu's (derived, LOSS_MAP.md section 4): Guo's 99/100 is well past both.

4. The rival gate

What the hypothesis and its use rely on, explicitly: the sixth-power identity psi_{u a^6}(m) = psi_u(m) 1_{(m,a)=1} (paper line 12420), which is the multiplicativity of the residue symbol in its upper argument together with chi^6 = 1; the Moebius function's multiplicativity and squarefree support (the exact column identity, line 12428); and |chi| <= 1 (the coefficient mass d(a), line 12430).

• W_a = zeta(s + a) zeta(s - a), a > 3/8. Over F the rival is zeta_F(s + a) zeta_F(s - a); its inverse coefficients mu_a, of 1/(zeta_F(s + a) zeta_F(s - a)), are multiplicative but not bounded: mu_a(p) = -(q_p^a + q_p^(-a)). For a > 33/80 the hypothesis fails for this rival by the argument of section 2(b): the column identity still holds for a multiplicative sequence supported on cube-free ideals, with coefficient mass << P^eps when a < 1/2, so the hypothesis would make zeta_F(s - a) zero-free in Re s > 73/80, that is zeta_F = zeta L(s, chi_-3) zero-free in Re w > 73/80 - a < 1/2, against Hardy's theorem for zeta (the argument of section 2(b), its Mellin step inferred; Hardy cited from the literature). For 3/8 < a <= 33/80 failure is not shown here. Its diagonal alone, (1/D) sum_{n <= D} mu_a(n)^2, grows like D^(2a) against the D^eps per row that the hypothesis allows (measured over the rational integers: the ratio to D^(2a) is 0.87 at a = 0.4 and 0.73 at a = 0.45, D = 20000), but in the range D <= H <= D^(101/100) the off-diagonal is not controlled by anything checked here.
• Epstein zeta of discriminant -15. The coefficients r_Q(n)/2 of Q = x^2 + xy + 4y^2 are not multiplicative: a(6) = 2 while a(2) a(3) = 0 (derived, zeta.epstein), so the column identity has no analogue. Applied to the inverse coefficients of Z_Q, the row u = 1 of the hypothesis would make 1/Z_Q(s) holomorphic in Re s > 199/200, against the zeros of Z_Q in Re s > 1 (Davenport and Heilbronn 1936, cited from the literature as zeta/epstein.py does, not recomputed here).

So the hypothesis is not a generic inequality: it fails for the Epstein rival and for W_a at least when a > 33/80. That is the gate it must pass to be worth proving; it says nothing about whether it is true.

5. Limitations

• Guo's other inputs (Lemma 17.1, Lemma 18.1 at kappa = 3/4, the detector, the uniform side conditions on his geometry box) are taken as stated; nothing here checks them.
• Heath-Brown 2000 was read only through two quotations; the companion paper's Proposition 3.1 was not read.
• Section 2(b) uses the standard step from bounds on smoothed partial sums to holomorphy of the Dirichlet series; it is inferred, not formalized.
• "No published theorem implies it" is a statement about the results named here and the structural reasons (a) and (b), not a search of the literature.