Nothing here is a result about the Riemann Hypothesis. This is the report of one attack on one mechanism, and the mechanism did not survive it. No progress on RH was made. The value of the run, if it has any, is that the death is located precisely rather than described.
The mechanism was written first, in MISSION.md, in a commit containing nothing else. What follows is what happened to it.
Verdict, in one paragraph
The mechanism claimed that Euler-product positivity (Lambda(n) >= 0, called EPP here) plus the functional equation compose into Re (xi'/xi) >= 0 on Re s > 1/2, which is equivalent to RH. It is false. The composition step is refuted by an explicit function that carries every hypothesis the mechanism uses, carries each of them in larger quantity than zeta does, and has zeros off its own critical line unconditionally:
W_a(s) = zeta(s+a) zeta(s-a), Xi_a(s) = xi(s+a) xi(s-a), a = 1/4.
Xi_a(1-s) = Xi_a(s) exactly; Xi_a(1/2+it) is real; W_a has a scalar Euler product; its log-derivative coefficients are Lambda(n)(n^a + n^{-a}) >= 0, which is EPP with room to spare; its archimedean term is exactly twice zeta's and positive; and its zeros are the points rho +- a, so Hardy's theorem puts infinitely many of them on Re s = 1/4 and on Re s = 3/4. The mechanism's conclusion is therefore false for a function satisfying its hypotheses, and no repair internal to those hypotheses can exist.
What separates zeta from W_a is not a positivity at all. It is a normalisation, in four faces of one shift: the size of the coefficients, the abscissa of the Dirichlet series, the location of the poles, and Re mu_j >= 0 in the gamma factor. Once the mechanism is repaired by importing it, the mechanism has nothing left to run on, for a reason that stage F measures: the inequality it must prove is an identity on the boundary of the region where it must hold, and its entire content is the first-order term inside, which is the zero-counting measure. That is the exact step, and it is unavoidable in the sense given in section 6.
1. The target, measured (stage A)
Re (xi'/xi) is the quantity whose sign the mechanism was trying to force. It behaves exactly as the mechanism's first sentence says, which is a check on the arithmetic and evidence for nothing.
| measurement | value |
|---|---|
xi'/xi by explicit split vs by contour differentiation of xi | agree to 7.6e-42 |
zeta.core.xi vs the probe's own xi | 1.7e-32 |
max abs Re (xi'/xi) on Re s = 1/2, five heights | 2.3e-41 |
min Re (xi'/xi) over 49 points with 1/2 < sigma <= 3 | +4.6e-06, at sigma = 0.5001 |
max Re (xi'/xi) over 9 points with sigma < 1/2 | -4.7e-06 |
The vanishing on the line is F(1-s) = -F(s), and it is why the mechanism believed the functional equation was giving it exact boundary data. It is, and section 6 is about what that data is worth.
2. The rival carries every hypothesis (stage B, a = 1/4)
| hypothesis the mechanism uses | measurement for W_a |
|---|---|
exact s -> 1-s functional equation for the completion | max relative defect 1.2e-16 over four points |
real on the critical line (a Hardy-style Z) | max relative imaginary part 0.0 |
| real, non-negative Dirichlet coefficients | min over n <= 60 is 1.0 |
| a scalar Euler product | truncation defect 3.1e-06 at 400 primes, s = 2.5+1.5i |
| EPP: non-negative log-derivative coefficients | min over n <= 200 is -2.0e-34, i.e. zero to working precision |
Re (Xi_a'/Xi_a) = 0 on Re s = 1/2, the boundary data the mechanism leans on | max abs value 1.3e-51 over four heights |
| the same, by a second route | recursion from the coefficients vs the closed form Lambda(n)(n^a+n^{-a}): max difference 1.3e-15 |
The EPP measurement was made twice on purpose. The closed form is a derivation; the recursion b_n log n = sum_{d|n} Lambda_F(d) b_{n/d} reads only the Dirichlet coefficients and never sees an Euler product, so it is the route the battery uses on the rivals.
3. The kill (stage C)
| measurement | value |
|---|---|
abs Xi_a(3/4 + i gamma_1) | 8.1e-22 |
abs Xi_a(1/2 + i gamma_1), same height, on the line | 1.2e-07 |
zeros of Xi_a in the box [0.1, 0.9] x [10, 25], argument principle | 4 |
sign changes of the real Z on Re s = 1/2 over 10 < t < 25, 900 samples | 0 |
Re (Xi_a'/Xi_a) at sigma = 3/4 - 0.001, t = gamma_1 | -998.0 |
the same at sigma = 3/4 - 0.01 / 0.05 / 0.12 | -97.9 / -17.8 / -5.7 |
Four zeros in the box and none on the line. For comparison the standing Davenport-Heilbronn window in zeta/epstein.py has five in the box and three on the line. The mechanism's conclusion, Re (xi'/xi) >= 0 to the right of the critical line, is violated by three orders of magnitude at a point where the mechanism's hypotheses all hold.
Note that the off-line zeros of Xi_a do not depend on any assumption about zeta. Xi_a(s) = 0 exactly when s = rho +- a for a zero rho of xi, and Hardy's theorem gives infinitely many rho with Re rho = 1/2. So infinitely many zeros of Xi_a sit at distance a from its own critical line, whatever is true of the rest of zeta's zeros.
4. The mechanism's two positivities are shared, and larger (stage D)
| quantity | zeta | W_a |
|---|---|---|
min eigenvalue of the 7x7 Toeplitz matrix [A(sigma + i(t_j - t_k))], sigma = 2.5 | +0.0053 | +0.0166 |
Re G(0.6 + 10i) | 0.234 | 0.468 |
Re G(0.6 + 50i) | 1.037 | 2.074 |
Re G(0.6 + 1000i) | 2.535 | 5.070 |
Both Bochner matrices are positive semidefinite, as EPP forces. Both archimedean terms are positive and grow. The rival has more of each, and its conclusion is false. A mechanism whose inputs are monotone in these two quantities cannot be repaired by making either of them bigger.
5. Gate #3, and the truncation that flips it (stage E)
zeta.epstein.battery was run on EPP, read from the Dirichlet coefficients alone. The verdict depends on where the coefficient scan stops, and the dependence is not cosmetic.
| truncation | verdict | shared with |
|---|---|---|
n <= 40 | does not distinguish | epstein_1_1_6 |
n <= 200 | distinguishes | none |
The detail, at n <= 200:
| function | EPP | first negative n | min Lambda_F |
|---|---|---|---|
| zeta | holds | none | -9.2e-41 |
| Davenport-Heilbronn | fails | 3 | -6.78 |
Epstein (2,1,3), non-principal | undefined: a_1 = 0 | n/a | n/a |
Epstein (1,1,6), principal | fails | 48 | -19.88 |
W_a | holds | none | -2.0e-34 |
Three things are worth recording exactly as they are.
The 40-coefficient reading is a false pass for the rival, and it was the first reading this run took. It said EPP is shared with a function that violates RH, which would have killed the mechanism at gate #3 for the wrong reason. The principal Epstein form's first negative log-derivative coefficient is at n = 48. Unique factorisation in the principal-class ideal monoid of discriminant -23 first fails at n = 36, where (2)(3) = (p2 p3bar)(p2bar p3), and the measured Lambda_F(36) there is 0 to 4.6e-41; Lambda_F is also 0 at 24, 32 and 40. So the coefficients sit on the boundary of the claim for a while before crossing it, which is precisely why a short scan reads as a pass. A coefficient claim run through the battery therefore carries its truncation as part of the claim.
The non-principal Epstein form has no log-derivative Dirichlet series in this normalisation at all. a_1 = 0, so the recursion is undefined rather than negative. Reporting that as "the claim fails" is correct for the battery's boolean and wrong as a description, so both are recorded.
EPP passes gate #3 and the mechanism still died. That is the methodological content of the run: the standing rivals are linear combinations of legitimate Euler products, so they cannot test a hypothesis that a genuine scalar Euler product also satisfies. W_a is such a rival and the battery does not contain one. This is an observation about the battery, not a proposal; the hunt does not promote it and did not pursue it.
6. Why the step is unavoidable (stages F and G)
W_a differs from zeta in more than one way, so the repair has to be named carefully. It has degree 2 rather than 1; its poles sit at 1 +- a rather than at 1; its coefficients grow like n^a, so its Dirichlet series converges only for Re s > 1 + a; and its gamma factor has Re mu_j = -a/2 < 0. Those are four faces of the same shift, and none of them is a positivity. Every positivity-shaped hypothesis the mechanism named is shared, and shared with more of it (section 4).
Suppose then that the mechanism is repaired by adding the shift-excluding hypothesis in any of its faces: the coefficients do not grow, equivalently the Dirichlet series converges for Re s > 1, equivalently Re mu_j >= 0. No rival can then be exhibited, since exhibiting one would refute RH, and a degree-1 element of the Selberg class is zeta or a shifted Dirichlet L-function by Kaczorowski-Perelli. What is left of the mechanism is measured here, and it is nothing.
The inequality is an identity on the boundary. The functional equation gives Re A(1/2+it) = Re G(1/2+it) exactly, for every t. That is the same fact as Re F = 0 on the line, measured in stage A at 2.3e-41. The mechanism must prove Re A <= Re G on Re s > 1/2, and its target holds with zero margin along the entire boundary of that region. An inequality proved by bounding one side loses something; here there is nothing to lose.
Just inside, the margin is the zero-counting data itself. Re F(1/2+eps+it) is linear in eps with slope sum_rho abs(s - rho)^{-2}:
t | Re F / eps at eps = 1e-3 | at 1e-5 | at 1e-7 |
|---|---|---|---|
| 7 | 0.0621641 | 0.0621641 | 0.0621641 |
| 20 | 1.0768414 | 1.0768423 | 1.0768423 |
| 50 | 20.048188 | 20.048571 | 20.048571 |
and the same slope, assembled from the ordinates by a disjoint route (mpmath.zetazero for the first 300 zeros plus the Riemann-von Mangoldt density tail), is 0.0621709 against the F-route's 0.0621641 at t = 7. The margin the mechanism has to produce is the zeros, to five digits. This is the circularity, as a number.
EPP's bound on the prime side is off by a growing factor. All EPP gives is abs A <= sum Lambda(n) n^{-sigma}, which diverges at sigma = 1/2. Cut at the approximate-functional-equation length N ~ sqrt(t/2pi):
t | N | EPP triangle bound | actual partial sum | archimedean target | bound / target |
|---|---|---|---|---|---|
| 1e3 | 12 | 4.26 | 1.43 | 2.53 | 1.68 |
| 1e4 | 39 | 9.65 | 4.60 | 3.69 | 2.62 |
| 1e6 | 398 | 37.30 | 3.70 | 5.99 | 6.23 |
| 1e8 | 3989 | 123.67 | 8.22 | 8.29 | 14.92 |
The actual prime sum tracks the target, as it must, since the two are equal on the line. The bound EPP supplies exceeds the target by a factor growing like sqrt(N)/log t, without bound. So the mechanism needs square-root cancellation in the prime sum, and square-root cancellation in the prime sum is RH.
That is the unavoidability, stated plainly: EPP is a statement about signs and the target is a statement about cancellation. The functional equation converts neither into the other, because what it supplies at the critical line is an equality, not slack.
7. What this run does not say
- It does not say anything about whether RH is true.
- It does not say the Euler product is irrelevant to RH. It says that
Lambda(n) >= 0, together with the functional equation and the Euler product itself, is not enough, with a witness. W_ais not offered as new. The one-sided shiftzeta(s - delta)is already in this tree atdocs/18section 6 anddocs/24section 6, which cite Conrey-Ghosh 1992 remark 2 for the observation itself; the symmetric product is the standard reason the Selberg class carries a Ramanujan axiom. What the run did with it is the attack, not the object.- No claim here has been through the funnel or been promoted. Per
hunts/README.md, a hunt cannot say yes about itself, and this one is saying no about itself anyway.
8. Reproduce
.venv/bin/python hunts/epp_herglotz/probe.py # writes results.json, ~15 min9. Afterword, added after the run closed
The run stopped where its terms said to stop, with the observation in section 5 recorded and not acted on. The operator then asked for the instrument to be fixed, so it was, outside this hunt and in the core:
zeta.epsteincarriesshifted_interface, andzeta.epstein.batteryruns it as a third rival by default.claim_multiplicativityconsequently no longer distinguishes, and both readings are pinned intests/test_epstein.py.docs/09section 5.1 carries the construction and the scope caveat.claim_euler_product_positivityandlog_derivative_coefficientsare public, so the claim this hunt died on is now a standing gate-#3 instrument with its truncation as an explicit argument.batteryrecords a claim that raises rather than dying on it, and refuses to reportdistinguisheswhen a rival could not be evaluated.
None of that is a result about RH either. It is the tool that killed this hunt's mechanism, made available to kill the next one faster.