31. Erratum to Bian, Lemma 12: `C_{kappa,2}` is `-4 kappa`, not `-4`

Hunt #65 (hunts/r_2ac05f/) adjudicated two contradicting kappa = 2 tables in this repository and, in doing so, located the error that produced the disagreement. It is not in either table's code. It is in the analytic lemma both of them inherited.

Bian's Lemma 12 asserts that the second regular coefficient of the pair- correlation form factor for the zeros of xi^(kappa) is C_{kappa,2} = -4, universally in kappa. The correct value is

C_{kappa,2} = -4 kappa.

The two agree at kappa = 1, which is the whole reason this survived: every control anyone ran was a kappa = 1 control, and -4 kappa and -4 are the same number there.

This page states the correction, its three-line proof, the mechanism that produced it, and what it changes. Every number here is produced by hunts/r_2ac05f/probe.py and pinned in hunts/r_2ac05f/results.json.

Nothing here is evidence for or against RH (docs/08). The object is a form factor for the zeros of a derivative of xi, not a statement about the zeros of zeta.

1. The correction

Write P = xi'/xi, so that xi^(kappa) = Q_kappa(P) xi with Q_0 = 1 and Q_{kappa+1} = D Q_kappa + P Q_kappa. With P = L + g, L = (1/2) log(T/2pi) held constant, g = zeta'/zeta, and x = 1/L, set Qhat_kappa = Q_kappa / L^kappa. Dividing the recursion by L^{kappa+1} removes L entirely:

Qhat_0 = 1, Qhat_{kappa+1} = x D Qhat_kappa + (1 + g x) Qhat_kappa.

The three lines. The x^1 coefficient of Qhat_kappa is kappa * g, for every kappa, it accumulates once per application of the recursion, and the recursion is applied kappa times. Hence the first arithmetic coefficient of R_kappa = xi^(kappa+1)/xi^(kappa) is q_1 = kappa * (Lambda log). And since C_{kappa,i} = 2^{i-1} sum_{p+q=i-1} <q_p, q_q>,

C_{kappa,2} = 2(<q_0,q_1> + <q_1,q_0>) = -4 kappa.

The probe prints the coefficient directly, and it is -kappa on the single word (0,) at every kappa:

hunts/r_2ac05f/results.json pins kappa = 1..3 under multiplicity_witness. The kappa = 4..6 columns were re-derived for this page by calling hunts/r_2ac05f/probe.py directly (form_factor(kappa, 2) and qhat(kappa, 2)), and are recorded here as reproduced rather than as quoted from the hunt.

2. The mechanism: a dropped multiplicity

hunts/higher_xi/C2_PROVENANCE.md names the step, on thesis p. 71: the weights M(v_l) M(w_k) are present before the application of Theorem 3 and absent from the next displayed line and from eq (8.1). That multiplicity is exactly the factor of kappa. The diagnosis was recorded in this repository before this hunt ran; what was missing was an independent derivation that confirmed it rather than asserting it.

The consequence is not confined to one cell. The same missing multiplicity sits inside Qhat_kappa, therefore inside every q_j, therefore inside every C_{kappa,i}. Correcting C_{kappa,2} alone would not repair the table.

3. What it changes

The corrected kappa = 2 row is hunts/higher_xi/C2_EXACT.json, and this hunt's independent derivation reproduces it exactly at all eleven indices:

Only i = 1 survives unchanged.

4. Why it was invisible, measured rather than asserted

The published kappa = 1 row is the natural control, and it cannot see this defect. hunts/r_2ac05f/fault_check.py demonstrates that instead of claiming it, by planting the defect in the probe's own machinery:

The first row is the finding: a control that passes while the defect under audit is planted has no power over that defect. The second row is the calibration that stops the first from being explained away as a dead control.

This is the general lesson, and it is recorded as a Core candidate rather than built here: when a claim asserts a quantity is invariant in a parameter, the control must vary that parameter. Every control here was at kappa = 1, and the claim under test was a claim about all kappa.

5. Standing, and what this is not

The derivation is exact rational arithmetic throughout: no floating point, no precision parameter, no enclosures, because nothing here is a numeric estimate. It is not kernel-checked; the Lean arm carries none of it.

Its external anchor is Farmer-Gonek (arXiv:0803.0425), whose kappa = 1 row this repository owns no part of. The probe reproduces all eleven of its values exactly, including the four forced zeros, through the same code path that produces the kappa = 2 row.

The one borrowed ingredient is the basis pairing, taken from hunts/higher_xi/. It is kappa-independent by construction and calibrated by the kappa = 1 control, and the second row of the table in section 4 shows that control does go red when the pairing is corrupted, so the borrowing is anchored rather than circular. A fully independent rederivation of the pairing from the mean-value theorem was not attempted, and is the first thing to do if this correction is ever submitted anywhere.

The three implementation defects that hunts/rogue_frontier/fkappa/ found in the thesis code are not disputed by this page and are not withdrawn by it. They concern eq (10.1)'s assembly, eq (7.8)'s phantom slots and eq (6.18)'s normalization; they appear to be real findings about the code. They are simply not the whole error. An audit that reimplements a computation faithfully, finds genuine defects in it, and inherits the author's analytic lemma as an axiom cannot detect an error in that lemma, which is what happened, and fkappa/RESULTS.md §1 says so in writing: C_{kappa,2} = -4 was "carried as constants here, as Bian carried them."