RESULTS: `rh_moonshot`, six RH-direct mechanisms (hunt #150, first wave)

Synthesis, 2026-10-26. Written by the synthesis session, which built none of the six constructions and ran none of the challenges. Inputs: the six NOTES.md on hunt/rh-moonshot; the rival-gate challenges (branch hunt/rh-moonshot-challenge-rival, head 7c3523396, files */challenge/CHALLENGE_RIVAL.md and CHALLENGE_RIVAL_STATUS.md); the circularity challenges (branch hunt/rh-moonshot-challenge-circularity, head 264a74658, files */challenge/CHALLENGE_circularity.md). Both branches were merged into hunt/rh-moonshot without conflict. Nothing here is reviewed by any person.

Grades are those of MISSION.md: measured, hardened, derived (with its review status), kernel-checked; cited (read in a primary source at the stated place) and inferred (a reading no computation checks) as the mechanisms use them. Nothing in this hunt is kernel-checked. A composite claim takes the grade of its weakest step. Dispositions are ALIGNMENT.md section 5's four (refuted, restricted class obstructed, unresolved, paused by allocation) plus alive: a stated next step and no located failure.

The answer, in the scope of this hunt

No route to RH was established in the searched set under the stated inputs. This is the bounded fact ALIGNMENT.md section 5.1 asks for, about six specific constructions at their stated parameters, and it is not a statement that no route exists. Sorted by section 5.1's four headings:

• (a) Refuted, at stated parameters. Two absolute-value bounds on Li's arithmetic part (li_arithmetic); the implication "scalar Euler product, nonnegative log-derivative coefficients and the functional equation give lambda_n >= 0", by W_{1/4} at n = 8260 (li_arithmetic, hardened); the same package as a source of the Hermite-Biehler property, by W_{1/4} at h = 1/5 (de_branges, measured); a function-field implication, by the golden fake (positivity_geometry, derived, self-reviewed); the tower premise that sigma_0(k) falls to 1/2, inside the order-k template (probe_tower); and self-adjointness as a discriminator, by DH (spectral_operator).
• (b) Restricted classes with proved obstructions. The order-k power-residue template of the 7/8 assembly (probe_tower; its transfer to actual order-k theta systems is inferred); the CCM operator's self-adjointness step, by Connes and van Suijlekom's theorem for every even-simple real distribution (spectral_operator).
• (c) Unresolved or alive. ccm_arithmetic (alive), li_arithmetic, positivity_geometry, de_branges (unresolved; the last recorded as a reformulation with nothing left in its family), and spectral_operator's link (3), which is ccm_arithmetic's open step in operator language.
• (d) Not searched here. Every construction outside these six, among them Suzuki's de Branges space of the Weil form, Bombieri-Stepanov's surface-free route, Berry-Keating and Sierra operators, and probes whose principal frequencies have density above 1/2.

1. ccm_arithmetic: CCM step 2, the ground-state transform converging to Xi

The construction (ccm_arithmetic/NOTES.md). The even-sector truncated Weil form Q_c on [0, log c] in a band of N cosines, assembled in Arb balls from the archimedean place, the pole and Lambda(n), n <= c. Its ground state v_1 is compared with the radical vector of Xi (a diagnostic that reads Xi values, never zeros, and never enters the form). The chain: (1) the ground state is simple and even for every c; (2) capture of v_c by Xi times even polynomials, with F_v -> Xi locally uniformly on a neighbourhood of |Im z| < 1/2; (3) Prop. 4.1 of Connes and van Suijlekom plus Hurwitz give RH. Link (2) is the distinguishing one.

Builder: unresolved. On c <= 60, N <= 128, zeta's ground state is captured (dist^2 to R_4 at most 8.3e-11), its transform's real zeros match Xi's below 100 to 7.8e-16, and the approach rate kappa is near 0.8. DH, Epstein (1,1,6) and W_{1/4} each lose capture at a located window (31 to 33; 29 to 59/2; 2 to 3) where the form goes negative; the Dedekind zeta of Q(sqrt -23) keeps it to c = 60. The perturbative capture bound is vacuous at m = 2 and m = 4.

Rival gate: unresolved. An independent 70-digit Galerkin assembly reproduced the transitions: DH at c = 31 has lowest value -3.81e-30 and misfit 1.1e-4 (still captured), at c = 33 misfit 0.9999; Epstein changes sign between 29 and 29.5; W_{1/4} at c = 2 has misfit 0.19, at c = 3 misfit 0.999. zeta(s - 0.1) fails the fixed-half even sector (evenness defect 0.0099 at the first node); recentred at 0.6 its matrix is zeta's to 1e-50, so it meets the conclusion and is not a counterexample.

Circularity: unresolved, with one grade lowered (C1). No zero or RH input in the form, checked by hand at ccm_common.py lines 171-229. But "lambda_1 > 0" is not ball-carried positivity: a positive Rayleigh ball bounds the lowest eigenvalue from above only, and the band-N matrix is a principal submatrix of every larger band, so lambda_1(N) is an upper bound on the window infimum. It falls monotonically in N on every window of zeta, Dedekind and DH, by 112 orders from N = 32 to 128 at c = 60. So zeta's positivity and simplicity are band-level and measured; the rivals' negative values stand. C2: capture co-locates with positivity on this grid, so a proof of link (2) may have to deliver all-window Weil positivity (kill condition 3), undecided. Prior art: CCM's own abstract says a proof of this convergence would establish RH; Zhu (arXiv:2608.24827) finds the window infimum positive but falling doubly exponentially, which agrees with C1; no published rival test of step 2 was found.

Synthesis: alive. The challengers' "unresolved" and this "alive" do not conflict: both say the chain is unproved for any infinite family of c, and MISSION.md reserves alive for a construction with a stated next step and no located failure, which is this one's state. No step fails for zeta on the grid; each rival tested fails at link (2) where it lacks a named hypothesis (DH and Epstein the Euler product, W_{1/4} unitary local roots); and the next steps (section 8) are concrete. C1 is accepted: wherever this page or the NOTES say zeta's form is positive, read "the band-N lowest eigenvalue is positive, measured". One datum bears on C2 and is shared by builder and rival challenger: DH at c = 31, N = 64 is still captured while its form already takes a negative value, so on that cell capture is not the same event as positivity. One cell does not decide C2; the fine grid of section 8 does.

Grades: link (1) measured, band-level; link (2) measured on c <= 60, N <= 128, with the relative error on Im z = 0.45 at best 7.3e-2 and no established decay; link (3) cited (Prop. 4.1, Commun. Math. Phys. 406 (2025) 312) plus classical Hurwitz; rival negative values ball-carried for stated vectors, W_{1/4}'s sign change hardened. Composite: measured.

2. de_branges: the Lagarias structure functions E_h

The construction (de_branges/NOTES.md). E_h(z) = xi(1/2 + h - iz), h > 0, with the Hermite-Biehler defect D_h measured by an Arb route and an mpmath route. (a) E_h is Hermite-Biehler if and only if xi has no zero with Re rho > 1/2 + h (derived, self-reviewed); (b) from the 7/8 strip, E_h is Hermite-Biehler for every h > 3/8 and A_h, B_h have only simple interlacing zeros on the line (derived, conditional on the strip, which is kernel-checked here and pending human review); (c) no single Euler factor is Hermite-Biehler. The form differs from what Conrey and Li refuted (positivity conditions on the space H(xi(1 - iz)), not the property of E_h at h < 1/2).

Builder: a reformulation under kill condition 3, unresolved as a route. By (a) the property is exactly a zero-free half-plane, so the family delivers only the strips it is given and reads only the zero locus.

Rival gate: unresolved. Independent defects at a failure point and a control point per rival: DH flips at h between 0.25 and 0.375 at its zero, Epstein between 0.375 and 0.5, W_{1/4} between 0.2 and 0.3 (so the scalar Euler product with nonnegative coefficients and the functional equation does not give the property), zeta(s - 0.1) lacks the fixed functional equation. The challenger declined to upgrade the reformulation to a restricted-class impossibility.

Circularity: the builder's disposition stands. B1: statement (a) is close to Sondow-Dumitrescu (|xi| increasing along horizontal half-lines in a zero-free right half-plane, Period. Math. Hungar. 60 (2010)) and to Lagarias's 2006 chapter (a de Branges space for L exists if and only if RH holds for L); the exact form at given h was not found in print. B2: by Saias and Weingartner (Acta Arith. 140 (2009), abstract re-read), DH has zeros with real part in (1, 1 + eta), so by (a) E_h^DH fails for every h <= 1/2; the grid's passes at h = 3/8 and 1/2 are a finite stand-in that the theorem overrides. B5: a Chowla-Selberg census of the Epstein zeta in [0.55, 1.6] x [0.5, 60.5] finds five off-line zeros, not the NOTES' two; the one at 33.650 lies inside the builder's grid (x <= 40), whose step of 1/2 missed it (defect +0.457 at x = 33.6501, negative at the grid points 33.5 and 34.0). A run-time trap around the builder's defect records no zero read.

Synthesis: unresolved, recorded as a reformulation; the family is exhausted. All three sessions agree, and the evidence that decides it is the exact equivalence (a), whose converse the circularity challenger rechecked: nothing in this family is weaker than quasi-RH at the matching abscissa. Not alive, because no next step is left in the family; not restricted class obstructed, because (a) is an equivalence, not a theorem that rules a class out. B2 and B5 are accepted. The NOTES' Epstein failure list is corrected by a dated note at the end of de_branges/NOTES.md.

Grades: (a) derived, reviewed by a second session (the circularity challenger rechecked the converse); (b) derived, self-reviewed, conditional on the 7/8 strip; zeta's 2828 grid points hardened at the points, a sampled statement; B2 cited from an abstract. Composite: derived, conditional on the 7/8 strip. (Corrected 2026-10-31: 2827 of the 2828 points are hardened; at the removable point s = 1 (h = 3/10, y = 1/5, x = 0) the Arb ball is undecided and the negative sign is the mpmath route's alone, so measured, as de_branges/NOTES.md says. Section 10.)

3. li_arithmetic: a uniform bound on Li's arithmetic part

The construction (li_arithmetic/NOTES.md). lambda_n = A(n) + T(n) (archimedean and arithmetic parts, identity defect 8.1e-28), and the prime-side form lambda_n >= 0 if and only if P_n <= M(n), with P_n = int_2^oo (psi(x) - x) L_{n-1}^{(2)}(log x) dx / x^2. Measured for n <= 5000: |T(n)| <= 13.6 (zeta), 19.2 (DH), against A(n) up to 1.6e4.

Builder: unresolved, two sub-routes refuted. The pointwise bound by |E| exceeds M(n) at every n <= 5000 where M(n) > 0 (least ratio 11.5), even with Schoenfeld's RH envelope; the mean-square bound fails outside n in [32, 1258]. W_{1/4} has lambda_8260 = -122.32 with the failure in its arithmetic part.

Rival gate: unresolved. An independent recurrence finds DH's first negative log-derivative coefficient at 3 and off-support at 6, Epstein's at 48 and 6, none for zeta, W_{1/4} or zeta(s - 0.1) through 60; an independent zero-side sum gives lambda_8260(W_{1/4}) = -122.315821669551207. W_{1/4} refutes the coarse package; a theorem that adds unitary local roots is unformulated.

Circularity: unresolved, the target a reformulation. L1: the RH-conditional inputs (Schoenfeld, Cramer's constant) sit only in the refuted routes; the positive half of the mean-square bullet ("at most M(n) on [32, 1258]") is conditional and is no evidence. L2: the artifact's P_n is M(n) - lambda_n with lambda_n from the xi side, so "P_n <= M(n) for n <= 5000" is lambda_n >= 0 restated; an independent prime-side integral to 10^8 matches it within 0.04 for n <= 1000 (measured, the tail beyond 10^8 unbounded). L3: W_{1/4}'s zero-side cross-check assumes on-line zeros above height about 1419 (verified to 3e12). L4: that DH has some negative lambda_n is a theorem (Bombieri-Lagarias multiset theorem); only the index 328997 is extrapolated. Computed new: zeta(s - delta), delta = 1/4, has lambda_n > 0 for every n (derived, self-reviewed by that session; measured for n <= 30) and violates RH. Prior art: Voros's sharpening (arXiv math/0506326) is the published form of the builder's onset heuristic; Johansson has lambda_n with interval values to n = 100000.

Synthesis: unresolved; the target is recorded as a reformulation until an estimate is offered; the two absolute-value sub-routes and the coarse-package implication are refuted at their stated parameters. The challengers agree with each other and with the builder; L2 lowers the builder's "prime integral satisfies P_n <= M(n)" to the xi-side statement above n = 1000. The zeta(s - delta) result sharpens the target: Li positivity implies RH only through the functional equation, which in this split enters through A(n), so a proof of P_n <= M(n) must use the functional equation as well as psi.

Grades: split derived, self-reviewed, identity measured; Laguerre identity derived, measured by a second route for n <= 1000; W_{1/4}'s sign hardened (two Cauchy radii plus the zero route, reproduced by the rival challenger); onset heuristic inferred, published in Voros's form. Composite: measured.

4. positivity_geometry: Weil's proof for curves, carried over

The construction (positivity_geometry/NOTES.md). The finite-field shadow (Gram matrix of {F1, F2, Frobenius graphs} on C x C from brute-force counts, exact inertia, 8 curves of genus 1 and 2: one positive eigenvalue, every fake two); the number-field target H_Q, equivalent to Weil positivity; the window-2g lemma (in the shadow, positivity on the first 2g Frobenius powers is RH for C, sharp in genus 2 and 3); the golden fake Z(T) = (1 - 5T + 5T^2)/((1 - T)(1 - 5T)) over F_5, with rationality, the curve functional equation and poles, nonnegative closed-point counts in every degree and a quadratic-character Euler product of conductor 4, whose Frobenius moduli are sqrt 5 phi^(+-1). Euler-product step: G1 (nonnegative prime-power intersections), plus G2 (one pole plane); the engine G3 (Hodge index on an ambient surface) uses no Euler product and is missing for zeta.

Builder: unresolved, with a restricted-class obstruction for arguments confined to G1, G2, the functional equation and finiteness of rationality type that "transfer verbatim" to function fields.

Rival gate: unresolved, the obstruction narrower. Independent exact recomputation of the golden fake (counts, closed points to degree 80, moduli to 30 digits, margin -5) and of the G1 table; W_{1/4} and zeta(s - 0.1) pass G1, so G1 is a precondition. The restricted-class obstruction stays at the builder's review grade.

Circularity: unresolved; the restricted-class label withdrawn to a refutation (D2). D1: the claim that no degree-1 Q-analogue of the golden fake exists is conditional on GRH for Dirichlet L-functions. D2: "transfers verbatim" defines no class; what the golden fake establishes is a counterexample to one function-field implication, {rationality, curve functional equation, curve pole structure, nonnegative integer closed-point counts in every degree, a +-1 local factor per place of F_5(t) with conductor 4} does not give RH. D3: Ihara zeta functions of non-Ramanujan regular graphs (and Weng's non-abelian zeta functions) are published neighbours, not the exact form. H_Q as a reformulation is in print (Connes 1999; Connes, Consani and Marcolli, arXiv math/0703392). The window-2g lemma is adjacent to Caratheodory-Fejer, not shown to follow from it. Recomputed golden fake to degree 300; window lemma checked on 120 arbitrary closed multisets.

Synthesis: unresolved for the mechanism; the golden fake recorded as a refutation of the function-field implication D2 lists. The builder and the rival challenger disagree with the circularity challenger only about the label. ALIGNMENT.md section 5, item 2, decides it: a restricted-class obstruction needs an explicitly defined class and a theorem whose hypotheses are shown to apply, and "arguments that transfer verbatim" is not an explicit class. The golden fake does fully support disposition 1 for the stated implication. D1 is accepted: the scope boundary over Q is conditional on GRH. Over Q, zeta(s - delta) passes G1 and G2 and violates RH while lacking the centred functional equation, which supports the builder's "preconditions, not engine" without the function-field detour.

Grades: shadow on 8 curves hardened (exact integers, two inertia routes); window-2g lemma derived, reviewed by a second session, with measured checks; golden fake in every degree derived, self-reviewed, its finite part recomputed exactly by both challengers; H_Q equivalent to RH cited. Composite of the refutation: derived, self-reviewed.

5. spectral_operator: the CCM zeta spectral triple

The construction (spectral_operator/NOTES.md). CCM's operator D'' on E_N / C xi, self-adjoint for W = T - eps_N when T is even-simple (CCM Lemma "key", Theorem "finmain"), built from the same prime, pole and archimedean data. Even-simplicity hardened by Arb LDL^T inertia on 7 cells; spectrum by two routes agreeing to 2e-26. For zeta at (60, 100) all 46 zeros below 138 are matched to 4.6e-40, far beyond the Gram-point control (median distance 0.76). For DH, at (60, 100) its form has negative directions, yet the operator is self-adjoint with real eigenvalues inside the gap of its off-line pair.

Builder: a located obstruction, recorded as a reformulation for the self-adjointness step; the RH content is link (3), the convergence, which is ccm_arithmetic's step 2.

Rival gate: unresolved; self-adjointness refuted as a discriminator. Independent assembly and operator for DH (13, 40), Epstein (13, 20), W_{1/4} (2, 8): commutator defects 6e-60 to 2e-55, real spectra, nearest eigenvalues at distance 0.46, 0.67 and 0.34 from the off-line zeros. Recentred zeta(s - 0.1) is zeta. One small cell each, measured.

Circularity: restricted class obstructed for the self-adjointness step; unresolved for the mechanism. P1: provenance verified at run time, which is stronger than the builder's text scan: traps on every zero routine of mpmath, Arb, zeta.zeros and the xi and theta functions, every zero-like module constant on the definition path poisoned (including DH's off-line ordinates, which wp_common carries), every file open logged. Nothing fired, nothing was touched, no file was opened, for zeta and for DH; both planted controls were caught. The 4e-41 match is therefore an output, not imported zeros. (Corrected 2026-10-31: the Arb traps were not set in the run the challenger recorded; the committed trap code could not set them, nor complete that run. Re-run on 2026-10-31 with every trap set: the same null result. Section 10.) P2: self-adjointness is Connes and van Suijlekom's theorem with no arithmetic input. P3: a second eigensolver agrees.

Synthesis: restricted class obstructed for the self-adjointness step; link (3) unresolved and merged with ccm_arithmetic. The two challengers differ only in the label for the same fact, and the circularity challenger's label is the one the evidence supports: there is a primary-source theorem (Connes and van Suijlekom, Prop. 4.1, with CCM's Lemma "key") for an explicit class (every real distribution whose truncated form is even-simple), and its hypotheses are shown to apply to DH on three hardened cells. The rival challenger's "refuted as a discriminator" is the instance of that theorem on DH. The mechanism is not ranked separately below, because its only open step is ccm_arithmetic's link (2) read through the operator.

Grades: even-simplicity hardened, band-level (the C1 caveat applies); self-adjointness cited, checked numerically to 1e-60; spectrum against zeros measured (float-grade eigenvector, 42-digit targets); provenance clean by three methods. Composite: measured. (Corrected 2026-10-31: of the three methods, the run-time traps first ran with their Arb part set on 2026-10-31; section 10.)

6. probe_tower: probes of growing degree in the 7/8 assembly

The construction (probe_tower/NOTES.md). Reading each 6 of OpenAI's 7/8 paper as the character order k gives sigma_0(k) = 1 - 1/k + (M - 2 ell)/(2k), strictly increasing in k and tending to 1; LP walls 1 - 2/k (dual length) and 1 - 4/(5k) (floor rows), so the best order is k = 2 at 3/5, for which no system exists; the formal k = 1 is Littlewood's RH-equivalent Moebius bound. The Euler-product step (E_k) is k-independent; DH and Epstein fail it at their zeros, DH for every finite deletion set by Saias-Weingartner; W_a passes it and is held off by unitarity, a shift of exactly 1/4.

Builder, rival gate and circularity: restricted class obstructed. The rival challenger rebuilt the formulas and the dual identity in exact rationals (1,080 identity instances) and the rival rows; the circularity challenger rebuilt the walls by exact vertex enumeration and DH's correction at its zero by a Hurwitz route. P1: every wall rests on the inferred k-reading, and the agreement with hunt #136's probe lane checks algebra, not the reading. P2: some cited sources were uncommitted when cited; hunt #136's checkpoint 57c7024c1 was local only when checked. New: zeta(s - delta) passes (E_k) exactly and every wall rises by delta, the cleanest unitarity rival. Prior art: Liu (arXiv 2610.12234) proves optimality of about 0.874957 within an explicitly specified exponent system; no published order-k tower was found.

Synthesis: restricted class obstructed, with the class as probe_tower/NOTES.md section 9 defines it, the tower premise refuted inside it, and the transfer to actual order-k theta systems inferred. All three sessions agree. Its hand-off is a half-plane step below 13/15 that belongs to hunt #136, not an RH step.

Grades: walls derived, symbolic in k, rechecked by two independent routes; k-reading inferred; rival rows measured; Saias-Weingartner cited from its abstract. Composite: inferred.

7. Ranking of the survivors by distance to a proof of RH

The survivors are the constructions not obstructed in their RH form: ccm_arithmetic (with spectral_operator's link (3)), li_arithmetic, positivity_geometry and de_branges. All four are far from a proof. The ranking is by what is missing and whether the missing step is known to be as hard as RH.

1. ccm_arithmetic. The missing step is one convergence statement. It is not known to be equivalent to RH (C2 is open), it is measured to hold for zeta on every cell to c = 60, and it separates at located windows all three rivals the hunt names while a second Euler product (Dedekind) passes. It is also the only mechanism whose discriminating step the hunt located by computation on every rival rather than by an equivalence. Against it: C1 (positivity is band-level and falling in N), the vacuous perturbative bounds, and the risk that capture is Weil positivity in disguise. (Corrected 2026-10-31: "measured to hold for zeta on every cell to c = 60" overstates the grid. Capture and the on-line zero match are measured there; the convergence, in the form Hurwitz needs, is not. Section 10.)
2. li_arithmetic. The target is RH-equivalent by construction, and every absolute-value route fails by factors from 11.5 to 1.6e4 on the measured range. What is missing is a signed cancellation estimate that uses unitarity and the functional equation (zeta(s - delta) and W_{1/4} show both are needed), of the depth of RH itself. It ranks above the next two because the missing inequality is explicit and each candidate can be tested cheaply against W_{1/4}'s failure at n = 8260.
3. positivity_geometry. The target is RH-equivalent and the ingredient that proves it in the shadow, an ambient surface with Riemann-Roch, does not exist for Spec Z. Its first next step is cheap and decisive for one candidate (the pairing of two scaling correspondences against Lambda(n) n^(-1/2)), but a match would only move the problem to Riemann-Roch on the square.
4. de_branges. In the Lagarias family the property is quasi-RH at each abscissa, so the family adds nothing to RH; a different structure function (Suzuki's space, which assumes RH, or his screw-function form) would be a new construction, not this one.

8. The next construction step for the top two, and what decides it

ccm_arithmetic: the fine DH grid on c in (30.6, 33) at N = 64 and N = 128 (NOTES section 9, step 2). Measure, on one grid, the sign change of the band-N form, the window where capture is lost (misfit against DH's own radical), and the gap scale (lambda_2 - lambda_1)/||dA/dy||. What decides it: if capture is lost at a c that stays above the sign change by a margin that does not shrink as N grows (c = 31 already shows capture with a negative value at N = 64), capture is strictly weaker than window positivity and the target is not Weil's criterion restated; if the loss of capture converges onto the sign change as N grows, C2 holds on this evidence and link (2) is recorded as a reformulation (kill condition 3). Either outcome is a disposition change. Under C1 the grid must also report lambda_1 against N, so that a sign is read only where it is stable in N. The step after it, whichever way this goes: test the semilocal unitary decomposition (U-S) on v_c for zeta and Dedekind against W_a at its threshold c_a, the arithmetic input the unitarity dial points to.

li_arithmetic: a mean-value bound on the kernel-weighted prime sum (NOTES section 9, step 1, its arithmetic alternative): bound sum_m Lambda(m) m^(-1/2) e^(i t log m) weighted by the Laguerre kernel's phase 2 sqrt(n u), by a large-sieve or mean-value inequality, and compare with M(n). What decides it: the bound must beat M(n) on 1259 <= n <= 5000, where the mean-square route fails, and its W_{1/4} analogue must fail at or before n = 8260; a bound whose W_{1/4} analogue still holds at 8260 uses only the coarse package that W_{1/4} refutes, and is refuted with it before any zeta computation.

9. What the record keeps regardless of RH

These are outputs of the hunt that stand whatever happens to the mechanisms. None is kernel-checked, and none is claimed novel in the literature unless a search is stated.

1. Five off-line zeros of the Epstein zeta of x^2 + xy + 6y^2 (discriminant -23) in [0.55, 1.6] x [0.5, 60.5], counted by the circularity challenger's argument-principle census (windows of height 10 from 0.5: 0, 1, 1, 1, 2, 0; de_branges/challenge/):

The four below the first are new to this tree: absent from hunt #120's census (which covered Re s >= 0.8751 only), from plocal_rivals, zeta.epstein and docs/23. None is enclosed: the evidence is Newton roots with small values and winding counts, and no in-tree route evaluates this function on segment balls (hunt #120's ball route inflates an input radius of 1e-4 to an output radius of 1e67 at height 15). No literature novelty is claimed: one search was run for the zero at 29.55 (Hejhal's ICM 1986 survey and five papers on other forms, none listing it) and none for the other three.

1. W_{1/4} = zeta(s + 1/4) zeta(s - 1/4) has its first negative Li coefficient at n = 8260, lambda_8260 = -122.32 (215 negative values to n = 9000, least -87409.5 at n = 8977). Hardened: Cauchy routes at two radii agree to 2.6e-48 relative, and a zero-side route (which assumes on-line zeros above height about 1419) gives -122.3158; the rival challenger reproduced -122.315821669551207 without mechanism code. The failure is in the arithmetic part, abruptly (T_W(8260) = -55949 against A_W(8260) = 55827).
2. zeta(s - delta) keeps every lambda_n > 0 while violating RH. Its completion's zeros all lie on Re s = 1/2 + delta, so |1 - 1/rho| < 1 for each and every lambda_n = sum_rho [1 - (1 - 1/rho)^n] is positive (derived, self-reviewed by the circularity challenger; measured for n <= 30 at delta = 1/4). It lacks the centred functional equation, so it is not a counterexample to Li's criterion; it shows the criterion carries RH only through that functional equation. The same function passes G1 and G2 (positivity_geometry) and the Euler step (E_k) (probe_tower), which makes it the cleanest one-factor rival for the unitarity hypothesis.
3. The golden fake over F_5: rational, with the curve functional equation and poles, nonnegative closed-point counts in every degree and a +-1 local factor per place of F_5(t) of conductor 4, and Frobenius moduli sqrt 5 phi^(+-1). Derived for all degrees, self-reviewed; recomputed exactly to degree 300 by a second session. Original to this hunt; published neighbours (Ihara, Weng) are not the exact form.
4. The window-2g lemma for curves: positivity of the projected Toeplitz form on the first 2g Frobenius powers is equivalent to RH for C, and 2g is sharp in genus 2 and 3. Derived, reviewed by a second session; adjacent to Caratheodory-Fejer.
5. The W_a unitarity dial (ccm_arithmetic): W_a's truncated Weil form first goes negative at c_a in (2.3887, 2.3906) for a = 1/4, moving by about +0.15 per halving of a, with W_0 = 2 Q_zeta to 3.9e-180. Hardened (Arb LDL^T).
6. DH meets no Hermite-Biehler structure of the Lagarias family for any h <= 1/2, by Saias-Weingartner and statement (a) (cited plus derived).
7. The CCM operator's provenance is clean by three methods, including run-time traps with poisoned constants, so its 4e-41 agreement with zeta's first zero (and 4.6e-40 for 46 zeros below 138 at (60, 100)) is an output of primes up to c, the pole and the archimedean place. Measured; a reproduction of CCM, no novelty claimed. (Corrected 2026-10-31: in the run first recorded, the run-time method lacked its Arb traps; the re-run with every trap set gives the same result. Section 10.)
8. A citation in this tree is wrong (de_branges/NOTES.md section 1, confirmed by the circularity challenger): hunts/rogue_frontier/FRONTIER_MAP.md, territory F, gives Lagarias's 2006 chapter a wrong title and an equivalence it does not state. That file is outside this hunt's scope; the operator holds the correction.

10. Corrections (2026-10-31)

Added by the session landing this hunt (land-361), which built, challenged and synthesized none of it, after checking the synthesis's claims against their artifacts. The text above is left as written, with a dated pointer at each place corrected.

1. The Arb traps of spectral_operator's P1 were not active in the recorded run. circ_spo.py as committed set them by setattr on python-flint's acb and arb, which are immutable extension types: the call raises TypeError in every release checked (0.5.0, 0.6.0, 0.7.1, 0.8.0, 0.9.0), each of which has all three routines. It sat outside the try/finally, so no run of that code reached the definition path with the Arb traps set, and the planted control could not run either. The trap now replaces the type wherever a pure-Python module names it, sets everything inside the try/finally, and is checked by a planted Arb fault (seven calls, all stopped) and by a source check for the one route it cannot see. Re-run on 2026-10-31 with every trap set (28 routines, the three Arb ones among them): no trap fired, no poisoned constant was touched, no file was opened, for zeta and for DH; both planted controls caught. So P1, section 5's "provenance clean by three methods" and item 8 of section 9 stand, now on a run in which every trap they name was set; grade unchanged, measured. Details, the evidence and a run manifest: spectral_operator/challenge/CHALLENGE_circularity.md, "Correction (2026-10-31)". The same page corrects P3: the first eigenvalue at (13, 16) differs from gamma_1 by -1.41e-17, not -8.5e-16 (the reference was held at 53 bits).
2. ccm_arithmetic's step 2 is not "measured to hold for zeta on every cell to c = 60" (section 7, item 1, and the case-log entry). A convergence statement cannot be measured to hold on finitely many cells, and the grid does not show its distinguishing part. From ccm_arithmetic/data/measure_zeta.json (30 cells, c <= 60, N <= 128): capture holds on every cell (dist^2 to R_4 at most 8.3e-11); every zero of Xi below the band cutoff is matched by a real zero of F_v on the 24 cells with c >= 13 (to 7.8e-16 where N >= 64), and on the six cells at c = 6 and 9 only 6 to 18 of them are; and on Im z = 0.45, inside the neighbourhood of |Im z| < 1/2 on which Hurwitz needs locally uniform convergence, the relative error is 7.3e-2 at best over the 30 cells, and at N = 128 it rises again from 7.3e-2 (c = 50) to 0.27 (c = 60). Read section 7's item 1 as: "capture and the on-line zero match are measured on the grid; the convergence is not, and its off-axis part is not seen to decay". Section 1's grades already say this; the ranking is left as the synthesis wrote it, and whether it survives this reading is the owner's call. Pinned in ccm_arithmetic/test_ccm_arithmetic_data.py.
3. The run record's test counts did not reproduce. "Every challenge directory (80 tests) ... pass": on 2026-10-31 (python-flint 0.9.0), before the fix in item 1, 73 of the 80 passed and 7 in spectral_operator/challenge failed; the mechanisms' own 166 gave 165 passed and 1 skipped (probe_tower's citation check, which needs a pinned copy of OpenAI's paper that is not in the tree). After the fix the challenge directories hold 83 tests and all pass.
4. One of de_branges's 2828 zeta grid points is measured, not hardened (section 2, grades). de_branges/data/grid_zeta.json records 0 positive and 1 undecided Arb sign, at s = 1 (h = 3/10, y = 1/5, x = 0), where the mpmath route gives -0.0055; the NOTES disclose it. Pinned by test_de_branges.py (at most one undecided per h).

Checked and left as written: the W_{1/4} sign at n = 8260 (two Cauchy radii agree to 2.6e-48 relative, the zero-side route gives -122.3158, li_arithmetic/artifacts/wa_quarter.json); the spectral cells' numbers (46 = 46 eigenvalues against transform zeros within 2.0e-26 at (60, 100), spectral_operator/data/); the W_{1/4} dial bracket (2.3887, 2.3906), which is band-level at N = 32 (ccm_arithmetic/data/wa_dial.json); and the bounded verdict itself, "no route to RH was established in the searched set", which claims no more than its scope.

The doors

This hunt measured where six constructions stop, so it owes the door analysis. Candidate status: every stopping point below is a measured or derived wall of a specific construction at its stated parameters, not a proved ceiling on RH; information class is given per door; proof status as graded in sections 1 to 6. The latest door gets no automatic priority: allocation is the owner's.

Active constraints at the optimum

Ranked by how directly each binds a survivor's open step:

1. The uniform step (all windows, all n, all h, all k). Every survivor stops at a finite range: c <= 60 and N <= 128 (ccm_arithmetic), n <= 5000 and x <= 10^8 (li_arithmetic), |x| <= 100 (de_branges), seven (c, N) cells (spectral_operator). It binds everywhere and no measurement relaxes it.
2. Capture co-located with positivity (ccm_arithmetic). The perturbative bound eta / (mu - lambda_1) is at least 2.8e4 because mu, the bottom of the form off R_m, is itself exponentially small (lambda_2's eigenvector is also radical times a polynomial); the leakage eta, about 1e-35, is not what binds. Rigidity shows any proof of capture must read the prime powers exactly, at a scale shrinking exponentially in c.
3. Band-level positivity (C1). lambda_1(N) is an upper bound on the window infimum and still falling at N = 128, so no window sign for zeta is read from this grid.
4. Absolute value against oscillation (li_arithmetic). The true P_n is small by cancellation of the Laguerre kernel against psi(x) - x; pointwise bounds exceed M(n) by at least 11.5, the mean-square tail (3.99e4 against 7.7e3 at n = 5000) binds from n = 1259.
5. The zero-free abscissa (de_branges). The family reaches h > 3/8 only because the 7/8 strip exists; its slope is one for one, h = sigma - 1/2.
6. The missing ambient surface (positivity_geometry, G3) and the comparison pole 1/k (probe_tower, shadow price -3/4 at the paper's geometry), both outside the survivors' data.

The frozen-constant inventory

The information class

• Inside the data the families already read. The fine DH grid and the saturation in N (primes up to c, the pole, the archimedean place); the U-S test on v_c (local factors, the same data); a mean-value bound on the kernel-weighted prime sum (prime powers only); the operator run on Epstein (1,1,6), W_{1/4} and zeta(s - delta) below and above their capture-loss windows, of which the rival challenger ran one small cell each (DH (13, 40), Epstein (13, 20), W_{1/4} (2, 8), recentred zeta(s - 0.1) (13, 16)) but not the comparison across the windows; the census of the Epstein zeta above height 60.5.
• Reading more. Enclosures of the Epstein zeros, which need a route that evaluates the D = -23 function on segment balls, which this tree does not have; a signed bound on P_n zero by zero, which reads the zeros and is a different information class from the prime side; any move of de_branges's h-range, which needs a new zero-free strip; positivity_geometry's ambient surface, an object no current data supplies; and the half-plane step below 13/15, which belongs to hunt #136.
• Not done, named by the challengers. Enclosures of the five Epstein zeros, a census above height 60.5, and the Epstein, W_{1/4} and zeta(s - delta) rivals through the spectral operator across their capture-loss windows. Each is in the first class except the enclosures.

Run record

id: rh_moonshot-synthesis-2026-10-26
hunt: rh_moonshot
started: 2026-10-26T16:14+00:00
finished: 2026-10-26T16:24+00:00
ran:
  - git merge --no-edit origin/hunt/rh-moonshot-challenge-rival (no conflict)
  - git merge --no-edit origin/hunt/rh-moonshot-challenge-circularity (no conflict)
  - nice -n 10 .venv/bin/python -m pytest -q -n0 on each of the six */challenge directories and each mechanism's own test_*.py files, one process per directory (12 processes in sequence)
  - .venv/bin/python -m pytest -q -n0 tests/test_hunt_doors.py tests/test_hunt_probe_discipline.py tests/test_huntspec.py tests/test_docs_numbering.py
  - .venv/bin/python scripts/make_context.py --check
outcome: both challenges merged; every challenge directory (80 tests) and every mechanism's own tests (166) pass with -n0, plus the repo checks test_hunt_doors, test_hunt_probe_discipline, test_huntspec, test_docs_numbering and make_context --check; six dispositions recorded (ccm_arithmetic alive; de_branges, li_arithmetic, positivity_geometry unresolved; spectral_operator self-adjointness restricted class obstructed with link (3) unresolved; probe_tower restricted class obstructed); no route established in the searched set
artifacts:
  - hunts/rh_moonshot/RESULTS.md
  - hunts/rh_moonshot/de_branges/NOTES.md (dated correction appended)

Corrected 2026-10-31: the outcome's test counts did not reproduce; 73 of the 80 challenge tests passed and the mechanisms' 166 gave 165 passed and 1 skipped. Section 10, item 3.