Result
The displayed kappa=2 row in Bian Figure 10.1 does not survive independent reconstruction. The first divergence is already
\[ C_{2,2}=-8, \]
not -4. Of the first eleven entries, only C_{2,1}=1 matches the printed row.
This is Outcome B1. The first causal defect is visible on thesis page 71. The exact expansion of A(x) contains the weights
\[ M(v_l)M(w_k). \]
They are present in the line immediately before the application of Theorem 3, then absent from the next displayed line and from equation (8.1). The constant C(v_l,w_k) defined in equation (7.8) does not contain these weights. Equations (9.6), (10.1), Lemma 12, Figure 10.1, and the Chapter 11 percentages inherit that omission.
Pinned inputs
- Audited repository state:
922884d1649999e1960cdc46c31a4ffaa25f4c6a. - Ji Bian thesis PDF SHA-256:
ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7. - Farmer-Gonek arXiv source archive SHA-256:
f6cdc7b71db06187dac655647e24312843441aa2598e41a3b53834dd1b36822f. - Farmer-Gonek main TeX SHA-256:
a3a9ac955a9c95d10d36d6b05aae05a79c6408f6e1cdcb8c14fd79f000144fb1.
The thesis archive exposes the PDF publicly. Its original TeX attachment is access-restricted, so the source-level audit is bounded by the PDF.
The one-line obstruction
Put g=zeta'/zeta, Dg=g', and x=1/L. Ignoring only terms lower by a power of the height, the exact logarithmic-derivative identity gives
\[ R_2=\frac{\xi'''}{\xi''} =L+g+D\log\left(1+2gx+(g'+g^2)x^2\right). \]
Therefore
\[ R_2=L-\Lambda+2(\Lambda\log)L^{-1}+O(L^{-2}) \]
as a formal Dirichlet series. Write these first two arithmetic coefficients as q_0=-Lambda and q_1=2 Lambda log. The leading prime-number-theorem pairing is
\[ \langle\Lambda,\Lambda\log\rangle=1. \]
With the thesis normalization log(x)/L=2|alpha|,
\[ C_{2,2} =2\bigl(\langle q_0,q_1\rangle+\langle q_1,q_0\rangle\bigr) =2(-2-2)=-8. \]
Bian's equation (2.21) independently exposes the same factor: for kappa=2, its first term from f'/(1+f) is 2g'/L. The printed -4 cannot follow from that equation.
Independent exact routes
exact_c2.py implements three coefficient generators.
Route A starts from
\[ R_0=L-\Lambda, \qquad R_{j+1}=R_j+D(R_j)/R_j, \]
and performs formal power-series inversion under Dirichlet convolution.
Route B starts from
\[ 1+2gx+(g'+g^2)x^2 \]
and expands its logarithmic derivative by separate operator-word code. It does not call Route A's expression or series helpers.
Route C uses the independently derived rational generating object
\[ Q(z)=-\Lambda+ \frac{2(\Lambda\log)z- [2\Lambda*(\Lambda\log)+\Lambda\log^2]z^2} {(1-\Lambda z)^2+(\Lambda\log)z^2} \]
and expands its quadratic denominator by a closed binomial formula. It shares neither series-inversion implementation.
The mean-square constant retains the original two literal checks through index 11: relative permutations and subset-permanent dynamic programming. Two additional implementations scale to index 40: a column-type count recurrence and an independent enumeration of 3-by-3 contingency tables. For basis words beta=(b_1,...,b_r) and delta=(d_1,...,d_r), both calculate
\[ K(\beta,\delta)= \frac{\displaystyle\sum_{\pi\in S_r} \prod_{j=1}^r(b_j+d_{\pi(j)}+1)!} {(2r+|\beta|+|\delta|-1)!}. \]
Words of unequal length pair to zero. Finally,
\[ C_{2,i}=2^{i-1}\sum_{p+q=i-1}\langle q_p,q_q\rangle. \]
All coefficient routes agree exactly through index 40. Both scalable mean-square routes agree there, and both literal routes agree through index 11. C2_EXTENDED.json is the authoritative 40-entry fixture. The infinite structure and tail are in CORRECTED-F2.md.
Level-1 normalization control
Running Route A for one derivative reproduces the Farmer-Gonek series exactly:
\[ 1,-4,4,0,\frac43,0,\frac{16}{45},0, \frac8{105},0,\frac{64}{4725}. \]
This is the same eleven-entry kappa=1 row printed by Bian. Thus the level-2 disagreement is not caused by the definition of L, the factor 2|alpha|, or the form-factor normalization.
Exact table
| i | Route A | Route B | Bian Figure 10.1 | Status |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | MATCH |
| 2 | -8 | -8 | -4 | MISMATCH |
| 3 | 24 | 24 | 4 | MISMATCH |
| 4 | -32 | -32 | -16 | MISMATCH |
| 5 | 64/3 | 64/3 | 28 | MISMATCH |
| 6 | -64/3 | -64/3 | 16 | MISMATCH |
| 7 | 1216/45 | 1216/45 | 544/45 | MISMATCH |
| 8 | -256/15 | -256/15 | -512/45 | MISMATCH |
| 9 | 1088/63 | 1088/63 | -104/63 | MISMATCH |
| 10 | -11776/945 | -11776/945 | -416/945 | MISMATCH |
| 11 | 42496/4725 | 42496/4725 | 6688/1575 | MISMATCH |
C2_EXACT.json is the historical comparison fixture through index 11. C2_EXTENDED.json is the corrected machine-readable fixture through index 40. Rational strings, not decimal renderings, are the stored values.
Consequence for the historical calculation
The Chapter 11 problem is upstream of its endpoint arithmetic. The printed higher-derivative coefficient rows were generated from equation (10.1) after the multiplicity weights had disappeared. They are not coefficients of the independently reconstructed xi'''/xi'' arithmetic object.
The historical 0.9544 and 0.9774 remain quarantined. They are not inputs, targets, normalization controls, or expected oracle outputs.
Reproduction
.venv/bin/python hunts/higher_xi/exact_c2.py --max-index 40
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py