Hunt #121: what moves under QRH(theta), with every exponent explicit

2026-10-08. Everything on this page is conditional on QRH(theta), the unreplayed claim defined in MISSION.md. If that claim is withdrawn or refuted by an independent replay, every statement below is void and this page says so now. Nothing here is a result, nothing here is evidence about RH, and the strip is not RH: 7/8 is not 1/2, and the preprints say so themselves.

Update, later on 2026-10-08: the claim has since been replayed here, Lean's kernel and the independent NanoDa kernel accepting its statements (docs/38 section 7). The argument has not been reviewed by any person, so every statement below keeps the hypothesis exactly as written.

Grades used: derived (an ordinary argument on the cited inputs, written out below, reviewed by nobody), measured (one mpmath route, dps 30, pinned by test_qrh_conditional.py), cited (somebody else's statement, with its location). A composite statement takes the grade of its weakest step; the weakest step everywhere on this page is the hypothesis.

0. The one step every consequence uses, and its dependencies

Under QRH(theta): uniformly in q >= 1, x >= 2 and (a, q) = 1,

psi(x; q, a) = x / phi(q) + O(x^theta (log qx)^2), (R)

with an absolute implied constant. This is the standard explicit-formula argument (Davenport, Multiplicative Number Theory, chapters 19 and 20), and it is stated here with its dependencies rather than assumed:

1. Zero counting (Davenport ch. 16): N(T, chi) << T log(qT), and the number of zeros of L(s, chi) with ordinate in [t, t + 1] is << log(q(|t| + 2)).
2. The truncated explicit formula for primitive chi (Davenport ch. 19, and ch. 17 for zeta): for x >= 2, T >= 2, psi(x, chi) = -sum_{|gamma| < T} x^rho / rho + (remainders), where the remainders Davenport displays are all O(x T^{-1} log^2(qx) + x^{1/4} log x + log(qx)).
3. Where the hypothesis enters. The functional equation pairs each zero rho of L(s, chi) with 1 - conj(rho), also a zero of L(s, chi), so under QRH(theta) every nontrivial zero has 1 - theta <= Re rho <= theta. Hence |x^rho / rho| <= x^theta / |rho| with |rho| >= 1 - theta, and by (1), sum_{|gamma| < T} 1/|rho| << log^2(qT). Taking T = x, psi(x, chi) - delta_chi x << x^theta log^2(qx), where delta_chi is 1 for the principal character and 0 otherwise.
4. Imprimitive characters and orthogonality (Davenport ch. 19 end and ch. 20): psi(x, chi) differs from psi(x, chi*) by O(log q log x), and psi(x; q, a) = (1/phi(q)) sum_chi conj(chi(a)) psi(x, chi). The phi(q) terms each carry the bound of step 3, the 1/phi(q) cancels the count, and (R) follows.
5. From psi to pi. Partial summation and psi - theta << x^{1/2} give pi(x; q, a) = Li(x)/phi(q) + O(x^theta log x), one log fewer. At theta = 11/12 this is the shape of the preprint's Corollary 1.2 (x^{11/12} log x, absolute effective constant). The preprint's "effective" is its word; the chain above yields an absolute constant that nobody here has computed.

Grade: derived, on cited chapters. Limitations: step 2's remainder list is quoted as a shape, not transcribed; nothing above was checked against the preprint's own proof of its corollary.

What (R) is, measured against the hunt's existing inputs.

• Level. (R) is nontrivial only where x^theta (log qx)^2 < x / phi(q), that is q <= x^{1 - theta} / log^2 x: pointwise level 1/8 at theta = 7/8, 1/12 at 11/12, with a power saving.
• Siegel-Walfisz (SW) as UPPER_BOUND.md states it: q <= L^B, error O(N L^{-H}), ineffective. Under QRH(theta), (R) is stronger in the error (a power of N against a power of log N), in the range (a power of N against a power of log N) and in effectivity. Everywhere (SW) is used, (R) supersedes it.
• Bombieri-Vinogradov gives level 1/2 on average over q with a logarithmic saving; (R) gives level 1/8 pointwise with a power saving. Neither implies the other. Bombieri-Vinogradov is not an input anywhere in UPPER_BOUND.md, RESULTS.md, RANK3_SCOPE.md, SW_EFFECTIVE.md, the two SW_*_SPLICE.md files or SIEGEL_UNIFORMITY.md (grep, 2026-10-08).
• Barban-Davenport-Halberstam, the input of the RANK3_BDH_* documents (not read here beyond their names), bounds sum_{q <= Q} sum_a |psi(x; q, a) - x/phi(q)|^2 << xQ log x for x L^{-A} <= Q <= x. Squaring (R) and summing over the ~Q^2 pairs gives Q^2 x^{2 theta} L^4, which beats xQ only for Q < x^{1 - 2 theta}: never, for theta > 1/2. In BDH's own range the hunt's existing input is strictly stronger than what QRH(theta) provides. Below that range, (R) beats the unconditional x^2 L^{-A} variance only for Q <= x^{1 - theta} L^{-2 - A/2}, level 1/8 again.

1. Under QRH(7/8): rank 1 moves a quarter power and stops three quarters short

UPPER_BOUND.md section 7 identifies the q = 1 mixed moment through the exact identity (29),

T_N = sum_{t <= N} Delta(t)^2 + sum_{t < N} (Delta(N) - Delta(t))^2, Delta(t) = psi(t) - t,

and bounds it by (SW) at q = 1 as T_N <<_H N^3 L^{-2H}. SW_EFFECTIVE.md sharpens the input to the classical N exp(-c_1 sqrt(log N)) and records the component as "a full power of N short of (31)", the unproved target T_N <<_eps N^{2 + eps}.

Under QRH(7/8): (R) at q = 1 gives max_{t <= N} |Delta(t)| << N^{7/8} L^2. Each of the 2N - 1 terms of (29) is at most (2 max |Delta|)^2, so

T_N << N * (N^{7/8} L^2)^2 = N^{1 + 7/4} L^4 = N^{11/4} (log N)^4.

With every exponent explicit: the power of N is 1 + 2 theta = 11/4, the saving over the recorded N^3 is 2 - 2 theta = 1/4 of a power, and the distance to (31) is 1 + 2 theta - 2 = 2 theta - 1 = 3/4 of a power. The gap narrows and does not close. Under QRH(11/12): N^{17/6} L^4, saving 1/6, gap 5/6. Under theta = 1/2, RH, the gap is 0 and the bound is the N^2 L^4 that section 7 already records under RH.

Two facts about the shape of this gap, both derived here:

• The exponent is the exact content of the information class. If the supremum Theta of the real parts of zeta's zeros equals theta, then by the classical oscillation theorem (Ingham, ch. V: psi(x) - x = O(x^a) forces Theta <= a) Delta(N) is not O(N^{theta - eps}), and (29)'s first completed square gives T_N >= ((N + 1)/2) Delta(N)^2 >= N^{1 + 2 theta - eps} infinitely often. So nothing that reads only "Theta <= theta" can beat N^{1 + 2 theta - eps} at rank 1; the quarter power above is all a 7/8 strip can be worth there.
• No fixed strip closes rank 1. Section 7 states that (31) implies Delta(N) = O(N^{1/2 + eps}), which by the same oscillation theorem forces Theta = 1/2, i.e. RH. A hypothesis consistent with Theta = theta > 1/2 therefore cannot prove (31). The gap 2 theta - 1 vanishes only at theta = 1/2, and at theta = 1/2 the strip is RH.

Grade: derived (exact exponent arithmetic on a displayed identity, with the mean-square route checked to give the same power: sum_{t <= N} t^{2 theta} L^4 is again N^{1 + 2 theta} L^4). Pinned: test_exponents_under_the_seven_eighths_strip.

2. Under QRH(7/8): the completed bound (1) moves from a logarithmic saving to N^{35/12}

UPPER_BOUND.md (1) is E(N) <<_C N^3 (log N)^{-C} for every fixed C, proved in its section 5 with polylogarithmic arcs Q = floor(L^B). Its "The doors" entry records that the term binding (1) is the minor-arc fourth moment (20), I_Q << (N^3 / Q + N^{13/5}) L^6, which takes no prime-counting input at all: it is Vaughan's bound (V) times the second moment. So at polylog Q the strip changes nothing in (1): the major-arc term (19) gets smaller and the minor arcs still cost N^3 L^6 / Q. What the strip changes is how large Q may be.

Under QRH(theta), with Q = N^a: every exponent is explicit.

(17), strip version. On the arc |beta| <= Q/(qN) around a/q, split F_N by residue class, insert (R) through the Gauss sums (Davenport ch. 26 does this for the major arcs of the three-primes problem; the Gauss sum of any character mod q to a reduced residue has modulus at most q^{1/2}), and partially sum against e(t beta) at the cost of a factor (1 + 2 pi N|beta|):

R_{q,a}(beta) := F_N(a/q + beta) - mu(q) K_N(beta)/phi(q) << q^{1/2} (1 + N|beta|) N^theta L^2,

absolute constant. On the arc N|beta| <= Q/q, so sup over all arcs of |R| is << Q N^theta L^2 (the q = 1 arc).

The cross term of (18). With |K_N(beta)| <= min(N, 1/(2|beta|)), int_{|beta| <= Q/(qN)} |K_N|^2 (1 + N|beta|)^2 dbeta << N + N Q/q << NQ/q. Per arc, 8 int |P|^2 |R|^2 << (mu^2(q)/phi(q)^2) q N^{2 theta} L^4 * NQ/q; summed over the phi(q) residues and over q <= Q, with sum mu^2(q)/phi(q) << log 2Q,

sum_{q,a} 8 int_{I_{q,a}} |P|^2 |R|^2 << Q N^{1 + 2 theta} L^5.

The fourth power of (18), two routes that agree at the balance point. Route 1, pointwise: int_{|beta| <= Q/(qN)} (1 + N|beta|)^4 dbeta << Q^5 / (q^5 N), so per arc 2 int |R|^4 << q^2 N^{4 theta} L^8 * Q^5 q^{-5} N^{-1}; over residues and q, sum q^{-2} converges, giving Q^5 N^{4 theta - 1} L^8. Route 2, through the second moment: sum_{q,a} int |R|^4 <= (sup |R|)^2 sum_{q,a} int (2|F|^2 + 2|P|^2) << Q^2 N^{2 theta} L^4 * (d_N + N log 2Q) << Q^2 N^{1 + 2 theta} L^5.

The rest of (12). The minor arcs are (20) unchanged, << (N^3/Q + N^{13/5}) L^6, valid for any Q with 2Q^2 < N since Dirichlet approximation then puts (V) in its own range. The leakage (13) is << N^3 Q^{-2} L^3, the tail (6) is << N^3 / Q^2, both needing only 2Q^2 < N and Q <= sqrt N.

Balance. With Q = N^a the constraints that each major-arc term stay below the minor-arc N^{3 - a} are: cross term, a + 1 + 2 theta <= 3 - a, i.e. a <= 1 - theta; fourth power, route 1, 5a + 4 theta - 1 <= 3 - a, and route 2, 2a + 1 + 2 theta <= 3 - a, both i.e. a <= (2 - 2 theta)/3. The fourth-power constraint binds for every theta < 1 (pinned for theta in [0.50, 0.99]). So

a = (2 - 2 theta)/3, E(N) << N^{3 - a} (log N)^6 = N^{(7 + 2 theta)/3} (log N)^6,

the logarithm not optimized. Under QRH(7/8): a = 1/12 and E(N) << N^{35/12} (log N)^6, a saving of 1/12 of a power over the unconditional (1), 11/12 of a power short of the N^{2 + eps} that would force RH, and 5/12 of a power behind the N^{5/2} L^c that CHHL report under GRH for Dirichlet L-functions (UPPER_BOUND.md section 1, item 1), on a weaker hypothesis. Under QRH(11/12): a = 1/18, N^{53/18} L^6. At theta = 1/2 the same route gives N^{8/3}, which is weaker than CHHL's conditional 5/2: the route is lossy, and the strip buys exactly its own a and nothing more.

Bootstrap, so nobody tries it. UPPER_BOUND.md section 1 records that CHHL's Theorem 2, E(N) = Omega(N^{1 + 2 Theta - delta}), turns any bound E(N) << N^c into Theta <= (c - 1)/2. Feeding the conditional bound back: Theta <= ((7 + 2 theta)/3 - 1)/2 = (2 + theta)/3, which exceeds theta by 2(1 - theta)/3 > 0 for every theta < 1. Under QRH(7/8) the bootstrap says Theta <= 23/24, weaker than the hypothesis. The map drifts toward 1 and never toward 1/2; pinned.

Grade: derived, by this hunt, on UPPER_BOUND.md's displayed inequalities (12), (13), (6), (18), (20) and the Gauss-sum major-arc approximation; reviewed by nobody. Limitations: UPPER_BOUND.md's own grade is a written argument with one model review and no kernel check, and this inherits it; the (1 + N|beta|) factor from partial summation is what makes the fourth-power term bind, and a mean-value treatment of int |R|^4 on major arcs (not attempted) is the one place a sharper conditional exponent might come from; no numerical check of a conditional asymptotic exists or could.

3. Under QRH(7/8): rank 2 does not move

Rank 2 of the doors table is the minor fourth moment I_Q and the fourth residual moment Z_{q > R_0}, both O(N^{13/5} L^6) by (22) and (28), both resting on Vaughan's bound (V). (V) is an unconditional exponential-sum estimate through Vaughan's identity and reads no zero-free region; the N^{4/5} term the text traces to its Type I/II balance at U = V = N^{2/5} has no theta in it. Under QRH(7/8) nothing derived here changes (V), I_Q or Z_{q > R_0}. This is a statement about the inputs of this hunt, not a theorem that a strip cannot reach a minor arc by some other route.

Grade: derived (a reading of which inputs carry theta). Pinned only in the sense that strip_exponents carries no rank-2 field.

4. Under QRH(7/8): rank 3 does not move in the square-root-arc configuration

RANK3_SCOPE.md prices the moments at 2 <= q <= R_0 = Q/L, Q = floor(sqrt N / 3), and names Route A, "(SW)-type uniformity to q up to a positive power of N", as the missing input. (R) is exactly such an input, but only to level 1 - theta = 1/8, while R_0 is of order N^{1/2}/L: for N^{1/8} < q <= R_0 the remainder N^{7/8} L^2 exceeds the main term N/phi(q) and (R) says nothing. And for q <= N^{1/8} the configuration itself defeats it: section 6's arcs have width Q/(qN) with Q of order sqrt N, so the partial-summation factor (1 + N|beta|) <= Q/q costs sqrt N, and the cross-term computation of section 2 above gives sum_{2 <= q <= Q'} U_{(q)} << Q N^{1 + 2 theta} L^5 = N^{13/4} L^5 for any Q' <= Q: worse than the trivial N^3 log Q. Under QRH(7/8), in the arc configuration of UPPER_BOUND.md section 6, rank 3 does not move. In the polylog configuration of section 5 the range "2 <= q <= R_0" does not exist as a separate bucket: the strip widens the major arcs to N^{1/12} and the price of that is section 2.

Grade: derived. Limitation: the RANK3 route documents were not read, so whether any of them already carries a partial-summation-free treatment that (R) would feed is not known here.

5. Under QRH(7/8): Theorem A becomes N^{7/4} L^4 and (T) does not follow; Theorem B is capped, not improved

RESULTS.md section 17, Theorem A: W(N) << N^{Theta + Theta_chi} (log N)^4 unconditionally, with Theta, Theta_chi the suprema of the real parts of the zeros of zeta(s) and L(s, chi_3); the text calls the unconditional form empty when Theta + Theta_chi = 2.

Under QRH(7/8), for zeta and for L(s, chi_3): Theta <= 7/8 and Theta_chi <= 7/8, so

W(N) << N^{7/4} (log N)^4, T(N) = I(N) + O(N^{7/4} log^4 N),

the first instance in which Theorem A's exponent is below 2 without assuming a Riemann hypothesis. The statement (T) needs O(N^{1 + eps}); the distance is 2 theta - 1 = 3/4 of a power, the same number as rank 1 and for the same reason: the double sum over pairs (rho, rho') carries N^{beta + beta'}, and nothing in "Theta <= theta" keeps beta + beta' below 2 theta. Under QRH(11/12): N^{11/6} L^4, gap 5/6.

RESULTS.md section 18, Theorem B: E(N) = Omega(N^{1 + 2 Theta_chi - eps}) along an unexhibited sequence. Under QRH(7/8): Theta_chi lies in [1/2, 7/8] instead of [1/2, 1], so the lower bound Theorem B can deliver is capped at N^{11/4 - eps}. A cap on a lower bound makes it weaker, not stronger; of the three gaps tb_bind.py records (no constant, no exhibited sequence, Theta_chi unknown), the strip narrows the third and touches neither of the others. CHHL's own Omega(N^{1 + 2 Theta - delta}) is capped the same way.

FRONTIER_2026_09_12.md's RH-bearing scalar: B_N = N R(N) + R(N)^2/2 + O(N log^2 N), M_N = 2|B_N|^2/N, and RH iff M_N <<_eps N^{2 + eps}. Under QRH(7/8): R(N) << N^{7/8} L^2 gives M_N << N^{11/4} L^4, 3/4 of a power short: it is the rank-1 quantity N R(N)^2 under another name and moves by the same quarter power. The same page's D_N target, |D_N| <<_eps N^{1/2 + eps}, gets only the envelope bound: with |R(u)| << u^theta L^2 and K = floor(sqrt N), |D_N| << N^{(1 + theta)/2} L^2 = N^{15/16} L^2 under QRH(7/8), 7/16 short, and that page records that the envelope is not what binds on that route (the multiplier zeta(rho) is), which a strip does not touch.

Grade: derived (substitution into displayed theorems). Pinned: theorem_A_exponent, theorem_B_cap, frontier_M_N_exponent, frontier_D_N_exponent.

6. Under QRH(theta): the "ineffective constants" caveat goes, the exponent does not

UPPER_BOUND.md (1) is stated "with ineffective constants", and SW_EFFECTIVE.md shows the q = 1 line never needed Siegel-Walfisz. The remaining ineffectivity sits in (SW) at 2 <= q <= L^B in section 5, where Siegel's theorem is the only unconditional tool. Under QRH(theta): the hypothesis excludes every zero with Re s > theta, hence every Landau-Siegel zero (document 38 section 3 notes that the 7/8 statement implies the uniform Siegel bound with c = (log 3)/8), and (R) replaces (SW) with an absolute constant. So (1), and the conditional N^{35/12} L^6 of section 2, carry constants that are effective in principle under the hypothesis. Nobody has computed one. SW_EFFECTIVE.md's own sentence stands: effectiveness and sufficiency are different questions, and only the first moved.

SIEGEL_UNIFORMITY.md's displayed bound (1) carries two terms "present only if the exceptional character/zero (q, chi, beta)" with beta > 1 - c_0/log Z exists, Z = exp(L^{1/10}). Under QRH(theta): beta <= theta, so once 1 - c_0/log Z > theta, that is for all sufficiently large N, no such zero exists and the bound reduces to its no-exception form N^3 exp(-c_kappa (log N)^kappa), which that document itself says "is not a fixed power saving". Section 2's N^{35/12} is stronger than it under the same hypothesis.

Grade: derived from headers and displayed statements; SIEGEL_UNIFORMITY.md was read only at its header.

7. The two crossover heights

Where does x^theta (log x)^2 become the smaller bound against an explicit, published de la Vallee Poussin remainder? SW_EFFECTIVE.md names no explicit constant ("cannot quote a citation-checked numeric digit"), so one is imported: Johnston and Yang, arXiv 2204.01980, Theorem 1.1, for all x >= 2,

|psi(x) - x| <= 9.39 x (log x)^{1.515} exp(-0.8274 sqrt(log x)),

read from the arXiv HTML rendering on 2026-10-08 and hedged to that reading (journal version not checked). The strip side has an absolute constant that nobody has computed; it is set to 1 by labelled convention, the same convention SW_EFFECTIVE.md uses for its K, and the table shows what the convention costs. In u = log x the comparison is the sign of

g(u) = log C + (theta - 1) u + (2 - 1.515) log u - log 9.39 + 0.8274 sqrt(u),

negative where the strip-shaped bound is smaller. Roots by grid scan (pitch 1/4) and bisection, dps 30; measured.

(a) At theta = 7/8 the crossover is log x = 35.11, x about 1.8 x 10^15. (b) At theta = 11/12 it is log x = 98.22, x about 4.6 x 10^42. Both with C = 1. The lower root at C = 1 (u = 6.8, x about 900) is where both bounds exceed x itself and say nothing; between the two roots the explicit unconditional bound is the smaller one, and that interval contains the whole of SW_EFFECTIVE.md's measured ladder 10^5 <= N <= 10^7 (pinned): on every height the hunt has measured, the strip-shaped bound is not the binding one. Each factor of e in C moves the 7/8 crossover by about 24 in u near C = 1; the number is a convention-dependent order of magnitude, not a constant of nature. Against the bare shape x exp(-0.8274 sqrt(log x)) with K = 1 (a second convention, also measured) the crossovers are u = 167.7 and 313.9.

Grade: measured, pinned to 1e-9 in u. Limitation: one published remainder, one convention for the strip side; a sharper published remainder moves the heights up and the exponents of sections 1 to 6 not at all.

8. Li's criterion under a strip: nothing usable at finite n

zeta/li.py carries lambda_n = sum_rho [1 - (1 - 1/rho)^n] over the nontrivial zeros, paired symmetrically, and Bombieri and Lagarias (1999): RH iff lambda_n >= 0 for every n. For rho = beta + i gamma,

|1 - 1/rho|^2 = ((1 - beta)^2 + gamma^2) / (beta^2 + gamma^2),

below 1 iff beta > 1/2. The functional equation pairs a zero at beta + i gamma, beta > 1/2, with one at rho' = (1 - beta) + i gamma, and for the partner

r^2 := |1 - 1/rho'|^2 = 1 + (2 beta - 1) / ((1 - beta)^2 + gamma^2) > 1,

the algebra checked numerically at three points to 1e-35. The pair's contribution to lambda_n is 2 - 2 r^n cos(n phi') with phi' = arg(1 - 1/rho'), bounded below by 2 - 2 r^n, which is as negative as r^n is large.

Under QRH(theta): beta <= theta, so every zero at height gamma has

r <= r_cap(theta, gamma) = sqrt((theta^2 + gamma^2) / ((1 - theta)^2 + gamma^2)),

and that is the whole of what a strip says about lambda_n. Three consequences, all derived:

• A strip bounds neither the number nor the heights of hypothetical off-line zeros. The sum over zeros of 1 diverges, so "the growth rate of lambda_n" from per-zero caps is not even a well-posed quantity without a zero-density input, which a strip is not.
• At small height the cap is useless: r_cap(theta, 0) = theta/(1 - theta) = 7 at theta = 7/8 (11 at 11/12), so a hypothetical zero near the real axis would double its factor at n below 1. What excludes that is the verified height, not the strip: RH is verified to T_0 = 3 x 10^12 (Platt and Trudgian 2021, cited from docs/05), and above T_0 the cap is 1 + 4.2e-26 under QRH(7/8) against 1 + 5.6e-26 with no strip at all.
• For gamma >> 1, log r_cap(theta, gamma) tends to (2 theta - 1)/(2 gamma^2), against 1/(2 gamma^2) with no strip. The strip multiplies the per-zero growth exponent of a hypothetical off-line zero by 2 theta - 1, 3/4 at theta = 7/8 and 5/6 at 11/12, and does nothing else. Any finite-n positivity statement for lambda_n rests on the verified height and on a count of zeros above it; the strip improves the hypothetical negative part of such a statement by that constant factor and supplies no lower bound on lambda_n at any n on its own.

Numeric illustration (measured, dps 30, a what-if at the first ordinate):

So the honest answer the task anticipated is the one the algebra gives: nothing usable at finite n, because |1 - 1/rho'| exceeds 1 by an amount the strip caps only through the factor 2 theta - 1, with the height and count of such zeros outside its reach. The equivalence RH iff Re(xi'/xi) > 0 on Re s > 1/2 and Li's criterion are statements at 1/2 and are untouched (document 38 section 6).

9. Does not move, cited and not recomputed

• The de Bruijn-Newman record. Under a zero-free half-plane Re s > 7/8, Lambda <= 9/32 = 0.28125 (document 38 section 6, from de Bruijn 1950 Theorem 13 in the normalisation of docs/05); the record is 0.2 (Platt and Trudgian 2021, docs/05). 9/32 > 0.2, so the record does not move; only the comparison is done here.
• The registered and candidate simple-zero proportions (README.md, hunts/four_point_pressure/) take no zero-free strip as input; nothing moves (document 38 section 6).
• Lambda_DH (hunts/lambda_dh_bounds/, hunts/lambda_dh_exact/, docs/29): Davenport-Heilbronn is a linear combination of two L-functions without an Euler product; the claim says nothing about it in either direction (document 38 section 6).
• Hunt #118's out-of-band wall (HANDOFF.md, walls section): the two inputs it names as reopening it, an unconditional in-band evaluation of the ordinate pair correlation and an unconditional out-of-band upper bound on the form factor, are not consequences of a strip by anything derived here. Not a claim that they cannot be.
• The equivalence RH iff Re(xi'/xi) > 0 on Re s > 1/2 (stated in hunts/epp_herglotz/RESULTS.md) is a statement at 1/2 and is untouched (document 38 section 6). Until 2026-10-09 this entry and the end of section 8 cited it as PR #268's (hunts/rh_resolve/); that PR records the equivalence as classical, not as its own, and its own audit of 2026-10-04 withdrew its finite Li enclosure.

10. The table

The 11/12 de Bruijn-Newman cell is document 38's formula with the half-width 2 (11/12 - 1/2) = 5/6 in docs/05's normalisation, Lambda <= (5/6)^2 / 2 = 25/72; the arithmetic is done here, not cited, and nothing else on this page depends on it.

11. Kill conditions, evaluated

None fired. The hypothesis stands exactly where document 38 left it: an ordinary mathematical argument of about 195 pages that no person is known to have reviewed, plus a Lean development whose compile is asserted by its producer. If an independent replay withdraws or refutes it, every "Under QRH" sentence on this page is void, the table's "moves" column is empty, and the only surviving content is section 0's dependency list, section 7's two root-findings as arithmetic, and section 8's algebra.

The doors

This hunt measures no ceiling of its own; it re-prices other hunts' ceilings under a hypothesis. The analysis below is therefore about the conditional exponents as functions of theta, which is the one parameter the hypothesis has.

1. Active constraints at the optimum

Ranked by the slope of the conditional exponent in theta, the only shadow price available.

2. The frozen-constant inventory

3. The information class

• Rank 1's door needs Theta = 1/2 read into the argument: RH itself, outside every strip. The door is the hypothesis, not a technique.
• The circle-total door stays inside the data a strip supplies ("every Delta(x; q, a) is pointwise small") but its binding term reads only sup |R|; reading int |R|^4 as a mean value over the arcs would read more than pointwise size, of the Bombieri-Vinogradov kind, which a strip does not carry and which the hunt's existing inputs carry only on average at level 1/2 with a logarithmic saving.
• Rank 2's door reads (V), outside any zero-free input.
• The Li door reads the count of zeros above the verified height, a zero-density datum; a strip carries none.
• The non-movers of section 9 read no zero-free region at all, which is why they do not move and why a replay of the claim, either way, leaves them where they are.