Results: one form per discriminant, not forty-one

2026-09-10. Grade: measured. The load-bearing quantity is a(1), an exact integer count of representations of 1, and everything else here is downstream of it. Nothing bears on RH (docs/08) and nothing here is a new theorem: the facts used are classical, and what the hunt adds is a measurement of which rows of a published table those facts entitle.

This page was rewritten after an independent adversarial audit. The audit is in AUDIT.md with its attacks and its verdict, survives with corrections. It found the claim standing after eight attacks and the write-up wrong in eleven places, including two counts and one sentence that presented an algebraic identity as a coincidence. Every correction is applied below and the three carrying numbers were recomputed here before being accepted.

Landed on main 2026-10-10 as Hunt #127 (it was #124 on its branch, and #119 in its issue #93 comment of 2026-09-10); the front door is docs/40. test_euler_defect_axis.py in this directory now recomputes every number on this page from the forms, checks the artifacts against that recomputation, and checks that the page still states the numbers it pins. One stated count did not reproduce and is marked in section 2; the permutation p-value in section 5 is a Monte Carlo estimate and is marked as one. The repair in PROPOSAL.md was applied by #296 before this landing (section 6).

1. The verdict

forms in the published table 41 forms whose a(1) equals 1 (the principal ones) 14, one per discriminant forms whose a(1) equals 0 27 non-principal forms repairable by rescaling 0 of 27 published axis 0 .. 36.0644 axis over the entitled rows, cutoff n < 61 0, then 2.9608 .. 5.0847 the same axis at cutoff n < 401 0, then 5.0196 .. 10.5748

The composite-defect discriminator is defined for a Dirichlet series with a(1) = 1. A binary quadratic form represents 1 exactly when it is the principal form of its class group, so the recursion is entitled to run on one form per discriminant and on the class-group sum, and on nothing else.

The qualitative reading of docs/34 section 6 survives on the rows entitled to it: class number one gives an Euler product and a composite 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 does not survive is the sentence "every individual form is loud", the numerical axis, and the experimental design issue #93 builds on top of that axis.

2. What a(1) = 0 does to the recursion

The recursion solves

a(n) log n = sum_{d | n} c(d) a(n/d)

by isolating the d = n term as c(n) a(1). With a(1) = 0 that term vanishes and c(n) is not determined.

The residual of the identity at the returned c is not an independent check on this, and the first version of this page said it was. Written out,

R = max_{2 <= n < 61} | a(n) log n - sum_{d | n} c(d) a(n/d) | = |1 - a(1)| * max_{2 <= n < 61} |c(n)|,

identically, because the recursion's own sum omits exactly the d = n term. Measured on all 41 forms, the difference between R and that product is 0.0 exactly on 40 rows and 1.6e-30 on the remaining one. So R is a rescaling of max |c| by |1 - a(1)| and it adds nothing to knowing a(1).

Not reproduced, 2026-10-10. The split "40 rows and 1.6e-30 on the remaining one" does not come back, and the command that produced it is not recorded (RUNS.md, the audit manifest, names a one-off -c). Recomputed from probe.py at its own precision: comparing the float R the probe reports with the float product gives 0.0 on all 27 non-principal rows and on 6 of the 14 principal ones; on the other 8 principal rows the product is exactly zero, so the difference is R itself, between 7.9e-31 and 3.2e-30. Compared at working precision, the identity holds to below 3e-29 on all 41 rows. The conclusion of the paragraph stands, since R still carries a(1) and nothing more; the count does not, and it is graded as not reproduced rather than edited to match. Pinned in test_the_residual_is_a_rescaling_of_max_c.

a(1) is the whole of it, it is an integer, and it is 1 on exactly the fourteen principal forms:

For the record, and because the first version leaned on it: the largest published defect (36.0644, d = -15, form (2,1,2)) is also the largest R (189.6888). That is not two measurements agreeing. They are two norms of the same coefficient vector, correlated at 0.99 across the 27 rows with a shared argmax by construction, and the sentence claiming otherwise is withdrawn.

3. What the published numbers actually are

They are not noise, and calling those rows empty was wrong.

Because the recursion never uses a(1) on its right-hand side (every divisor d < n gives n/d >= 2), running it on a with a(1) = 0 returns exactly the coefficients of the series with a(1) replaced by 1. That series is

ftilde(s) = 1 + Z_Q(s),

and the published number for each non-principal form is precisely its composite defect. Measured on all 27 rows against a Dirichlet-logarithm route: worst absolute disagreement 0.0.

So the correct statement is not that the rows carry no information. It is that they describe 1 + Z_Q(s), a series with no functional equation, no Euler product on either reading, and no connection to the zeros of Z_Q, rather than the Epstein zeta function the table names.

No rescaling reaches Z_Q itself. Dividing through by the least represented value m would restore a(1) = 1 only if every represented value were a multiple of m. Enumerated over n < 61 for all 27 non-principal forms: none is. (2,1,2) of d = -15 represents 2 and 3; (2,-1,3) of d = -23 represents 2 and 3. The audit went further and asked whether any integer-indexed c solves the identity for a non-principal form: least squares over all c(1..60) with c(1) free leaves a residual of 2.18 to 5.79, so the system is inconsistent and no such c exists.

4. The control that says the fault is not in the instrument

For class number one, zeta_Q = w * zeta * L(chi_d) exactly, which forces

c(n) = Lambda(n) (1 + chi_d(n)).

That prediction uses no recursion. Computed with the Kronecker symbol from its definition and von Mangoldt from trial division, it agrees with the recursion's c on all five class-number-one discriminants to at worst 2.8e-30. The audit attacked the Kronecker implementation directly, against an independent one, and did not break it.

5. The corrected axis, and what it costs issue #93

Issue #93 proposes the family as "a defect axis from 0 to 36 with class number as the knob". Three things go wrong with that, and they are not the same thing.

The range is not 0 to 36, and it is not any particular range. The defect is an unnormalised L2 norm over composites below a cutoff, so it grows with the cutoff: the band is 2.96 .. 5.08 at n < 61, 3.78 .. 6.17 at 121, 4.18 .. 7.15 at 201 and 5.02 .. 10.57 at 401. Its width relative to its mean moves too, 0.58, 0.52, 0.54, 0.78. The axis has no scale, which is a worse problem than having the wrong endpoints, and it is the first thing to fix if the family is to be used as an axis at all.

The ordering is nearly stable, and the "nearly" is the part worth stating. The first version of this section said the nine discriminants "rank in exactly the same order at n < 61 and at n < 401". They do not, and that sentence was false when it was written. Six of the nine positions hold at every cutoff: -15 at the top, and -31, -39, -47, -71, -95 in that order at the bottom. All of the movement is inside the -20, -23, -24 block, where -23 climbs as the cutoff grows. Against n < 61 the rank correlation is 0.9500 at 121 and 0.9833 at 201 and at 401, and the ordering then settles: n < 201 and n < 401 agree exactly, rho = 1.0000. So the quantity does order these subjects, and it needs a couple of hundred composites before it does. ordering.py recomputes all of it from artifacts/cutoff.json, which is the same artifact the false sentence was read off.

Class number is not separable from the discriminant on nine points. Over the nine loud rows corr(defect, h) = -0.69 and corr(defect, log|d|) = -0.82, both negative, so the direction is opposite to the one the issue's framing implies. But corr(h, log|d|) = +0.97 on these nine points, tie-corrected Spearman puts the two at -0.979 and -0.983 against the defect, and the audit's permutation test for the gap between them gives p = 0.113 (a 200,000-draw Monte Carlo estimate; the exact value over all 9! orderings of the defect, computed 2026-10-10, is 0.1124, inside three standard errors of it). So the claim that the discriminant explains it better than the class number does is withdrawn: nine points cannot separate two predictors that correlate at 0.97. What is supported is the sign and the ordering, not the attribution.

The sparsity explanation is withdrawn too. The first version said the principal form of a large discriminant represents fewer integers below the cutoff and so has a sparser coefficient sequence. Tested against that proxy: corr(defect, count of represented n < 61) = +0.47, R^2 = 0.22, t = 1.39, and the coefficient reverses sign once log|d| enters the model. The mechanism may be true; this data does not show it.

Disposition. The compute job issue #93 prices, off-line zeros for every discriminant against the defect, would be run against an axis whose scale is set by an arbitrary cutoff and whose two candidate knobs cannot be told apart on nine points. That is worth knowing before buying the compute rather than after. Normalising the defect and building a family with a real knob comes first; the door in section 7 says how.

One caveat this hunt inherits and did not state. docs/34's gloss on F4, that c(n)/sqrt(n) is the weight of frequency log n in the zero distribution, reads the explicit formula on the critical line. The rows this hunt keeps are exactly the h > 1 principal forms, whose Epstein zeta functions classically have zeros off it. The gloss therefore needs its own statement for these subjects, and neither docs/34 nor this page supplies one. (2026-10-10: docs/34 now does. #296 added to F4 that the 1/sqrt(n) weight is the normalisation for zeros on the critical line, and that for these subjects the defect is a statement about the coefficients alone.)

6. What is pinned, and what a repair would move

tests/test_zeta_temperament.py contains

assert d15["max_form_defect"] > 30

which pins 36.0644, a number belonging to 1 + Z_Q(s) for the form (2,1,2). The same test's structural assertions all still hold on the entitled rows: class number one below 1e-25, loud above 1.0 (corrected rows run 2.96 to 5.08), and every class-group sum below 1e-25. So the guard that would change is the one naming 30, and the guards carrying the section's actual claim are unaffected.

This hunt does not make that change. tests/, zeta/ and docs/ are outside what a hunt may write. The proposed repair is in PROPOSAL.md.

Applied 2026-10-10 by #296 (commit da94b5a), before this hunt was landed. The > 30 assertion is gone; tests/test_zeta_temperament.py pins the principal-form column of docs/34 table E7, the superseded numbers as composite defects of 1 + Z_Q(s), and the 14 / 27 split, and probe_euler_discriminator.py refuses a series with a(1) != 1. What #296 did differently from the proposal is in PROPOSAL.md's closing note.

7. The doors

1. Active constraint. The normalisation a(1) = 1. It binds on 27 of 41 rows, and its shadow price is not that those rows are empty but that they describe a different function: 1 + Z_Q instead of Z_Q. The distinction matters because the first is checkable and wrong, which is a cheaper defect to find than noise.
2. Frozen-constant inventory.
3. The cutoff NMAX = 61. The one frozen constant with real trade shape, and the measurement above prices it: the band roughly doubles from cutoff 61 to 401 while the ordering moves by one adjacent transposition and is fixed from cutoff 201 upward. Anything using this quantity as an axis needs it normalised, by the number of composites in range or by the same norm of the prime-power part, so that two subjects can be compared without agreeing on a cutoff first. That is one line of code and it is the highest-value change available here.
4. The unit count w(d), which is what makes a(1) = 1 for the principal form. Not free: any other choice breaks the normalisation.
5. The principal form as the family's representative. The alternative inside the hypothesis is the class-group characters: L(s, chi) = sum_Q chi(Q) zeta_Q(s) has a(1) = 1 for every chi and an Euler product for every chi, so it gives defect zero everywhere and no axis at all. That is why the family has no knob, and the audit checked it rather than taking it on the hunt's word. (Scope, 2026-10-10: the audit checked it, and test_every_class_group_character_has_an_euler_product now checks it, on class numbers 2, 3 and 4 only, at d = -15, -20, -24, -23, -31, -39, with real and complex characters; the test adds a planted table that is not a character, which is loud. The characters at h = 5, 7, 8 are checked by neither.)
6. Information class. A defect axis with a real knob needs a family whose members all satisfy a(1) = 1, share a functional equation, and vary continuously in multiplicativity. The Epstein family supplies the first two on one form per discriminant and does not supply the third. The route that does is a Davenport-Heilbronn-type construction at several moduli, where the mixing of two characters is the knob by construction and the functional equation is imposed by solving for the mixing constant, exactly as zeta/epstein.py already does at modulus 5. That reads no new information; it builds a family instead of borrowing one.