Hunt #127, hunts/euler_defect_axis/. The measurements and the doors are in hunts/euler_defect_axis/RESULTS.md; the audit that rewrote it is in AUDIT.md; the repair it proposes to files a hunt may not touch is in PROPOSAL.md. This page is the front door.
Landed on main 2026-10-10. The hunt was written on the branch
claude/repo-hunt-content-2smbid, where it was Hunt #124 and this page was numbered 38; its issue #93 comment of 2026-09-10 called it hunt #119. On main those are three other things (#119 islambda_dh_exact, #124 isqrh_class_number, and number 38 is the quasi-Riemann claim), so the hunt is #127 and this page is 40. The repair inPROPOSAL.mdreached main first, in #296 on 2026-10-10; its closing note says what #296 did differently. The measured numbers on this page are recomputed byhunts/euler_defect_axis/test_euler_defect_axis.py(the Davenport-Heilbronn linen = 51is pinned bytests/test_zeta_temperament.py). One count did not reproduce as stated and one sentence was false when written; both are marked where they appear.
Grade: measured. Nothing on this page bears on RH (docs/08), and no new theorem is claimed: the mathematics used is classical, and what is new is a measurement of which rows of a published table that mathematics entitles.
1. The discriminator, and its one hypothesis
docs/34 section 6 records a genuinely good idea. For a Dirichlet series f = sum a(n) n^-s, write -f'/f = sum c(n) n^-s. Then f has an Euler product exactly when c is supported on prime powers, and Weil's explicit formula makes c(n)/sqrt(n) the weight of frequency log n in the distribution of f's zeros. So a nonzero c at a composite is a failure of multiplicativity, read off the zeros. For zeta the coefficients are von Mangoldt's and the composite part vanishes; for the Davenport-Heilbronn function, which shares zeta's functional equation and violates RH, it does not, and its loudest single line is n = 51 = 3 x 17.
The idea needs one thing, and says so: a(1) = 1. The coefficients c are recovered by a recursion that isolates the d = n term of
a(n) log n = sum_{d | n} c(d) a(n/d)
as c(n) a(1), and divides.
docs/34 then reports the quantity for 41 reduced binary quadratic forms across fourteen discriminants, with a headline of 36.0644 at d = -15, and issue #93 reads the family as "a defect axis from 0 to 36 with class number as the knob", proposes an experiment on top of it, and prices the compute.
2. A binary quadratic form represents 1 exactly when it is principal
That is classical, and it settles the table. The principal form of each discriminant represents 1; the others do not. So a(1) = 1 on fourteen rows and a(1) = 0 on twenty-seven, the d = n term vanishes on those twenty-seven, and the recursion divides by zero in the only sense available to it: it never forms the quotient at all, and returns a number anyway.
Measured, not read off the source: a(1) is exactly 1 on the fourteen principal forms and exactly 0 on the other twenty-seven, and no rescaling repairs any of them, because a rescaled series needs every represented value to be a multiple of the least one and none of the twenty-seven has that property. The form (2,1,2) of d = -15, whose row is the table's loudest, represents 2 and 3.
The qualitative reading of docs/34 survives on the rows entitled to it: class number one gives an Euler product and a defect at machine zero, class number above one gives a principal form that is loud, and the class-group sum returns to machine zero every time. What goes is the sentence "every individual form is loud", the numerical axis, and the experiment issue #93 builds on it.
3. Then an adversary read the write-up
The first version of the hunt's result page was given to a separate agent with no part in producing it, told to break the claim rather than review it. It ran eight attacks. The claim survived all eight. The write-up did not.
It found the counts wrong. The page said 44 forms and 30 non-principal. The class numbers sum to 41, of which 27 are non-principal. Two numbers, in the verdict box, wrong.
It found the load-bearing measurement was not one. The page reported the residual of the identity as an independent check that the recursion had no right to run. Written out, R = |1 - a(1)| * max_n |c(n)| identically, because the recursion's own sum omits exactly the term the residual is testing for. Measured on all 41 forms: the difference between the two is 0.0 exactly on 40 rows. (Not reproduced, 2026-10-10: recomputed from the probe's output, the difference is 0.0 on all 27 non-principal rows and on 6 of the 14 principal ones; on the other 8 the product is exactly zero and the difference is R itself, between 7.9e-31 and 3.2e-30. The identity holds to working precision on every row; the count 40 does not reproduce.) So the residual is a rescaling of a norm the recursion computed, it is downstream of a(1), and it adds nothing to knowing a(1). The page had presented an algebraic identity as two measurements agreeing.
It found the strongest sentence was the emptiest. "The largest published defect is the largest residual" reads as a coincidence detected by a second route. It is two norms of the same vector, correlated at 0.99 with a shared argmax by construction.
It found the withdrawn numbers were not empty. The page said those rows carry no information. They carry precise information about the wrong function. Because the recursion never touches a(1) on its right-hand side, running it on a series with a(1) = 0 returns exactly the coefficients of the same series with a(1) replaced by 1, which is 1 + Z_Q(s). The published number for each non-principal form is that series' composite defect, exactly: measured on all 27 rows, worst disagreement 0.0. A series with no functional equation and no relation to the zeros of Z_Q.
It found two explanations unsupported. The page said the discriminant explains the defect better than the class number does. On nine points the two predictors correlate at +0.97, their rank correlations against the defect are -0.983 and -0.979, and a permutation test for the gap gives p = 0.113 (a 200,000-draw estimate; the exact value over all 9! orderings is 0.1124). Nine points cannot separate them. The page also offered a sparsity mechanism; against its own natural proxy that gives R^2 = 0.22 and the coefficient reverses sign once the discriminant is in the model.
It found the axis has no scale. The defect is an unnormalised sum over composites below a cutoff, and the cutoff was frozen at 61. Move it to 401 and the band runs 5.02 to 10.57 instead of 2.96 to 5.08. The ordering of the nine discriminants nearly holds: six of the nine positions are the same at every cutoff measured (61, 121, 201, 401), all of the movement is inside the -20, -23, -24 block, and the ordering is fixed from cutoff 201 upward. So the family orders subjects and does not scale them, which is a sharper statement than "the range is 0 to 5" and a more useful one, since the page was using the range to cancel a compute job. *(Corrected 2026-10-10, when this page was landed: it read "The ordering of the nine discriminants is identical at both cutoffs." That was false when written: -23 and -24 trade places between 61 and 401. The hunt's ordering.py caught it on 2026-09-10 and the sentence was fixed in RESULTS.md, PROPOSAL.md and the case log, but not on this page.)*
Every correction is now in the result page, and the three carrying numbers were recomputed here before being accepted rather than taken on the audit's word.
4. What is left standing
The claim: the composite-line discriminator is entitled to one form per discriminant. Eight attacks, including an independent reimplementation of the Kronecker symbol and a least-squares search for any integer-indexed c that solves the identity for a non-principal form (residual 2.18 to 5.79, so none exists). It did not move.
The disposition: the compute job issue #93 prices should not be bought against this axis, and normalising the defect comes before either. A family with a real knob is a Davenport-Heilbronn construction at several moduli, where the mixing constant is the knob by construction, which zeta/epstein.py already does at modulus 5.
The pinned number: tests/test_zeta_temperament.py asserts d15["max_form_defect"] > 30, which pins 36.0644, which belongs to 1 + Z_Q(s). The test's structural assertions all still hold on the corrected rows. That is one line, and changing it is the tree's decision, not the hunt's. (The tree decided on 2026-10-10: #296 removed that assertion, and tests/test_zeta_temperament.py now pins the principal-form column of table E7 and pins the superseded numbers as composite defects of 1 + Z_Q(s).)
5. Why this page exists
Because the interesting part is not that a table had bad rows. It is that the check which would have caught them is one line long, is implied by the method's own stated hypothesis, and nobody ran it. And then, that a write-up of that finding contained eleven overclaims of its own, including one that dressed an identity up as a corroboration, until something with no stake in it went looking.
Both halves are the same lesson and the second is the harder one to arrange.