R-2AC05F: the kappa = 2 table on `main` is right, and the other one inherited the defect it was auditing

Verdict: settled. An independent recomputation, written for this adjudication and importing neither disputant, reproduces hunts/higher_xi/C2_EXACT.json exactly at all eleven indices, and reproduces the externally published Farmer-Gonek kappa = 1 row exactly at all eleven indices as its control. hunts/rogue_frontier/fkappa/'s corrected kappa = 2 row is wrong from i = 2.

Grade: hardened. Two independent derivations of the same eleven rationals in exact arithmetic (higher_xi's three routes, and this probe's fourth), with an external anchor the literature fixes. Not kernel-checked; no enclosures are involved because nothing here is a numeric estimate.

Nothing here is evidence for or against RH (docs/08).

1. The table

C_{2,i}, the regular coefficients of Bian's pair-correlation form factor for the zeros of xi''. All values exact rationals.

Control row, kappa = 1, against Farmer-Gonek (arXiv:0803.0425) F_1 = |a| - 4a^2 + sum_k ((k-1)!/(2k)!)(2|a|)^{2k+1}:

Exact agreement, including the four forced zeros. Both disputants also agree on this row, so it is common ground and not itself in dispute.

2. How the probe derives it, and what it borrows

Write P = xi'/xi. Then xi^(kappa) = Q_kappa(P) xi with Q_0 = 1 and Q_{kappa+1} = D Q_kappa + P Q_kappa, so

R_kappa := xi^(kappa+1)/xi^(kappa) = P + D log Q_kappa.

With P = L + g, L = (1/2) log(T/2pi) held constant, g = zeta'/zeta, and x = 1/L, put 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,

and R_kappa = L + g + D Qhat_kappa / Qhat_kappa =: L + sum_j q_j x^j. The q_j are finite combinations of Dirichlet convolutions of A_b(s) = sum_n Lambda(n) log^b(n) n^{-s}, represented as multiset words, with g = -A_0, product = concatenation, and D (b_1..b_r) = - sum_j (b_1..b_j+1..b_r).

The form factor is then C_{kappa,i} = 2^{i-1} sum_{p+q=i-1} <q_p, q_q>.

The one borrowed ingredient is the basis pairing, taken from hunts/higher_xi/C2_EXACT.json:

<(b_1..b_r),(d_1..d_r)> = sum_{sigma in S_r} prod_j (b_j + d_sigma(j) + 1)! / (2r + sum b + sum d - 1)!, 0 on unequal lengths.

This is legitimate to borrow because it is kappa-independent by construction: it knows only about products of Lambda's and nothing about which derivative of xi produced them. It is therefore fully pinned by the kappa = 1 row, which the literature fixes independently, and §4 below shows by fault injection that the kappa = 1 control does have power against it. Everything else in probe.py was written for this adjudication.

Sanity checks along the way, all confirmed by the code: Qhat_1 = 1 + g x and Qhat_2 = 1 + 2 g x + (g' + g^2) x^2, which is exactly the object hunts/higher_xi/C2_PROVENANCE.md writes down by hand.

3. Why the other table is wrong, in one line

The x^1 coefficient of Qhat_kappa is kappa * g, for every kappa. The probe prints it:

kappa=1: -1 * (0,) kappa=2: -2 * (0,) kappa=3: -3 * (0,)

so the first arithmetic coefficient of R_kappa is q_1 = kappa * (Lambda log) and therefore

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

Derived by the probe for kappa = 1..5: -4, -8, -12, -16, -20.

Bian's Lemma 12 asserts C_{kappa,2} = -4 universally. That assertion is the dropped multiplicity, and it is exactly the defect hunts/higher_xi/C2_PROVENANCE.md names on thesis p. 71: the weights M(v_l) M(w_k) present before the application of Theorem 3 and absent from the next displayed line and from eq (8.1). higher_xi's causal diagnosis is confirmed.

And this is the specific mechanism by which fkappa went wrong. fkappa/RESULTS.md §1 states it plainly: "C_{kappa,1} = 1 and C_{kappa,2} = -4 are universal (his Lemma 12); they come from analytic main terms outside this machinery and are carried as constants here, as Bian carried them." An audit that reimplements a computation faithfully, finds three real implementation defects in it, and then inherits the author's analytic lemma as an axiom, cannot detect an error in that lemma. The three defects fkappa found look genuine on their own terms and are not disputed here (they concern eq (10.1)'s assembly, (7.8)'s phantom slots and (6.18)'s normalization). They are simply not the whole error, and correcting them while keeping -4 propagates the larger one into every corrected value from i = 3 on, because C_{kappa,2} is not an isolated cell: the same missing multiplicity sits inside Qhat_kappa and therefore inside every q_j.

So the two diagnoses are not symmetric competitors. fkappa's is a correct finding about the thesis code; higher_xi's is a correct finding about the thesis mathematics, and it dominates.

4. The control that would have caught it

fkappa/RESULTS.md §5(a) already writes the indictment of its own control set, without drawing the conclusion:

the kappa = 1 row ... is simultaneously the strongest validation of the engine and the reason none of the three defects was ever visible from the literature side.

fault_check.py measures that blindness instead of asserting it. Two plants in the probe's own machinery:

The first row is the finding. The only externally anchored control either hunt ran has zero power against the defect that decides the dispute, and the plant reproduces the published wrong value exactly. The second row shows the control is not vacuous: it does catch a corruption of the ingredient it calibrates.

The control that would have caught it, stated so it can be reused:

When a claim asserts that a quantity is invariant in a parameter, the control must vary that parameter. Compute C_{kappa,2} independently for kappa = 1, 2, 3 and assert the values are not equal.

It is three lines of the derivation above, it needs no thesis and no reimplementation, and it turns Lemma 12 from an inherited axiom into a checkable statement. fkappa ran kappa = 1..9 grids throughout and never ran this check, because the quantity it varied kappa over was always downstream of the constant it had already fixed.

This is the same failure shape the roster already carries from run 726a6b3f on finite_height_spacing_experiment: a control that "sweeps ell only, holding effective gamma at 1, so it cannot vary W at fixed x ... the control is blind to its own subject". The thread match on this brief was correct, and the shared name for it is: an invariance claim needs a control that moves the variable the invariance is asserted over.

5. What this does and does not change

• hunts/higher_xi/C2_EXACT.json and C2_EXTENDED.json, and everything on main that consumes them, stand. This adjudication is a third exact route agreeing with them.
• hunts/rogue_frontier/fkappa/'s kappa = 1 row survives in all modes: it is the Farmer-Gonek closed form and it is right.
• fkappa's reproduction of Figure 10.1 cell for cell survives as a statement about the thesis code, and is independently useful: it is how we know both hunts are reading the same published row (fkappa's skip/rows_figure kappa=2 row equals higher_xi's recorded bian_figure_10_1 column exactly, all eleven cells).
• fkappa's three implementation defects (A, B, C) are not adjudicated here. They may all be real; this hunt did not test them and takes no position. What it establishes is that correcting them is not sufficient, because the row is already wrong at i = 2, upstream of all three.
• fkappa's corrected values from i = 5 on, its i <= 20 extension, its closed generating identity C((a),(b)) = (-1)^{a+b} ab/(a+b-1), and its observed radius-1/2 coherence with Bian's picket-fence Lemma 13 are all computed on top of the wrong constant. The coherence observation is the one worth a second look: it is a real structural signal, and if it survives redoing the extension on the correct C_{kappa,2} = -4 kappa, it is evidence for the corrected diagonal rather than for fkappa's.

6. Reproduce

.venv/bin/python hunts/r_2ac05f/probe.py        # ~1 s, writes results.json
.venv/bin/python hunts/r_2ac05f/fault_check.py  # ~1 s, writes fault_check.json

Exact rational arithmetic throughout; no floating point, no mpmath, no precision parameter to get wrong.

Loose threads

1. fkappa's extension to i = 20 should be re-run on C_{kappa,2} = -4 kappa. Why it might matter: fkappa observed that its corrected diagonal has ratios converging to -2, i.e. radius of convergence exactly 1/2, matching the phase transition of Bian's own picket-fence heuristic (his Lemma 13) at |alpha| = 1/2, and that the published diagonal admits no such reading. That is a genuine internal-coherence signal, and it was computed with the wrong C_{kappa,2}. If it strengthens on the corrected constant, it is real evidence for the corrected table's structure; if it evaporates, the coherence was an artifact and the observation should be withdrawn. Either answer is worth having. First step: take fkappa's bian_engine.py --extend 20 --reading corrected, replace its hard-carried C_{kappa,2} with -4*kappa, and recompute the stable diagonal ratios r_i.

1. fkappa's three implementation defects are unadjudicated. Why it might matter: defects A, B and C are claims about Bian's code independent of the Lemma 12 error, each pinned by a finite witness (consikapa(6,2,3,2) = 1/30; |C((j),(1))| = 1 against the skip convention's 6/5, 7/6, 9/7; Lambda_2(6) log 6 = 2 log2 log3 log6 against the printed RHS 4 log2 log3 log6). They are cheap to check and nobody has. First step: check the third witness by hand against thesis (6.18), it is a single n = 6 evaluation.

1. The fkappa directory exists on no branch that has landed. Why it might matter: it was found at commit 360c545 on the hunts(rogue_frontier) checkpoint lineage, and it carries a .ext_lock file and a campaign ledger. The lab's public record does not contain it, so a reader of main cannot see either side of this dispute. First step: decide whether rogue_frontier/ lands with a correction notice pointing at this hunt, or stays unlanded; that is an operator call, not a mathematical one.

1. C_{kappa,2} = -4 kappa is a small original result about Bian's Theorem 1 and is currently recorded nowhere but here. Why it might matter: it is a one-line correction to a published lemma with a three-line proof, and the two hunts between them spent considerable compute never stating it. First step: if higher_xi's Bian findings are ever written up, state the general kappa form rather than only the kappa = 2 instance.