This hunt may not write zeta/, tests/ or docs/. What follows is the change it would make if it could, stated precisely enough to be argued with.
1. hunts/zeta_temperament/probe_euler_discriminator.py
Add the residual check to the probe itself and refuse a form it is not entitled to, rather than returning a number for it:
def spectrum(a, nmax=NMAX):
if abs(a[1] - 1) > mp.mpf(10) ** -25:
raise ValueError(
"the recursion isolates c(n) a(1); a(1) = %s does not determine c"
% mp.nstr(a[1], 6))
...and report, per discriminant, principal_form_defect in place of max_form_defect. The per-form loop then runs over the forms that represent 1, which is one form per discriminant.
A guard that cannot fire is worth nothing, so it ships with the planted fault that fires it: the form (2,1,2) of discriminant -15.
2. tests/test_zeta_temperament.py
The line
assert d15["max_form_defect"] > 30pins 36.0644, the value with residual 189.6888. It has no replacement at the same strength, because nothing in the corrected axis reaches 30. The assertions that carry the section's claim need no change: class number one below 1e-25, loud above 1.0 (the corrected rows run 2.96 to 5.08), and every class-group sum below 1e-25.
The replacement worth having is not a looser threshold on the same quantity. It is the check that would have caught this:
def test_every_reported_defect_solves_the_identity_it_claims():
"""A composite defect is a statement about c; c is defined by
a(n) log n = sum_{d|n} c(d) a(n/d). A row whose residual is nonzero is
not a defect at all."""3. docs/34 section 6, table E7
The per-form column becomes the principal-form column, with a correction notice in the section's own style recording what the earlier table reported and why. The section's argument is unaffected: the discriminator separates zeta from a rival sharing its functional equation, class number one supplies an Euler product, the class-group sum restores one. Only the numbers attached to non-principal forms go.
4. Issue #93
The open question stands, but its proposed design does not. The paragraph "The Epstein family supplies a defect axis from 0 to 36 with class number as the knob" wants replacing with the corrected axis and three narrower observations that replace it: the band is 2.96 .. 5.08 at cutoff 61 and 5.02 .. 10.57 at cutoff 401, so the axis has no scale; the ordering of the nine discriminants holds in six of nine positions at every cutoff and is fixed from 201 upward, so it does order them once there is enough of it; and the two candidate knobs correlate with each other at +0.97 over those nine points, so nine points cannot say which one is doing the work.
5. Disposition of the compute job
The off-line-zero measurement issue #93 prices is not worth buying against this axis, and normalising the defect comes before either. The band runs 2.96 .. 5.08 at cutoff 61 and 5.02 .. 10.57 at cutoff 401 while the ordering of the nine discriminants moves by a single adjacent transposition, so the quantity orders subjects and does not scale them. Normalise it first, by the number of composites in range or against the same norm of the prime-power part.
Then, if the question is worth pursuing, the family to build is a Davenport-Heilbronn construction at several moduli, where the mixing constant is a genuine knob and the functional equation is imposed by a linear solve, as zeta/epstein.py already does at modulus 5. That is a construction, not a borrowing, and it is the door section 7 of RESULTS.md names.
Applied, 2026-10-10
Added when the hunt was landed on main. The sections above are left as they were written on 2026-09-10, when this was "the change it would make if it could" and the hunt's issue #93 comment of that date called the repair "deliberately not applied". Both were true then.
The repair was applied by #296 (commit da94b5a) on 2026-10-10, before this directory reached main. It changed docs/34, hunts/zeta_temperament/probe_euler_discriminator.py, hunts/zeta_temperament/results_euler.json and tests/test_zeta_temperament.py. Sections 1 and 2 went in as written: the same a(1) guard in spectrum, principal_form_defect in place of max_form_defect, the per-form loop over the forms that represent 1, the planted fault (2,1,2) of d = -15, the > 30 line removed, and test_every_reported_defect_solves_the_identity_it_claims added under that name. Where it went beyond or against this page:
- It followed
AUDIT.mdwhere section 3 and the audit conflict. Section 3 says the argument ofdocs/34section 6 is unaffected and only the non-principal numbers go. The audit's last correction says that is too broad, because the sentence "every individual form is loud" goes too. #296 withdrew that sentence with a dated note, rewrote "the composite lines of the individual forms cancel" (only the principal form has composite lines of its own), and added the hedge toF4for zeros off the critical line that the audit asked for andRESULTS.mdsection 5 says was missing. - It kept the superseded numbers inline. Table E7 prints each corrected row as, for example,
5.0847 (was 36.0644), beside a dated correction notice, rather than replacing the column and recording the old values only in the notice. - It added checks this page did not ask for: the probe raises unless exactly one reduced form per discriminant represents 1, the planted fault is run on every probe run and recorded in
results_euler.json, and the tests pin the corrected column, the six superseded numbers as composite defects of1 + Z_Q(s), and the 14 / 27 split. - Section 4 was not carried out as an edit. Issue #93's text was left as it is;
docs/34's Scope paragraph points at the issue comment of 2026-09-10 instead. - Section 5 is untouched. The defect is not normalised and the Davenport-Heilbronn family at several moduli has not been built.