Resolution of the 2008 higher-derivative discrepancy

Superseded in part by C2_PROVENANCE.md. This document records the first finite-row arithmetic obstruction. The later first-principles audit locates an earlier causal defect: thesis page 71 drops M(v_l)M(w_k) before equation (8.1), so the displayed higher-derivative coefficient rows are themselves incorrect from index 2 onward.

Disposition

The 0.9544 and 0.9774 proportions on page 93 of Ji Bian's 2008 thesis are chapter-11 calculation errors. They are not outputs of equation (11.5) applied to either coefficient row printed in the thesis.

This resolves the internal discrepancy. It does not determine whether the Figure 10.1 coefficients themselves are mathematically correct, and it does not supply replacement positive proportions.

Pinned source

Ji Bian, The Pair Correlation of Zeros of Derivatives of Riemann's Xi-Function, University of Rochester PhD thesis, 2008:

https://urresearch.rochester.edu/institutionalPublicationPublicView.action?institutionalItemId=5500
PDF SHA-256:
ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7

The source repository lists one public PDF version. Its original TeX file is restricted to repository administrators.

Source reconstruction

Equation (11.2) integrates the triangular Fourier kernel against

F_kappa(alpha) = spike + sum_i C[kappa,i] |alpha|^i.

For a finite coefficient row, the endpoint expression in equation (11.5) is therefore exactly

P = 1 - 2 sum_i C[kappa,i]/((i+1)(i+2)).

There is no missing factor in this passage. The same expression applied to the Figure 10.1 kappa=1 row gives

348002/405405 = 0.858405791739125...,

which matches the Farmer-Gonek first-derivative control.

First source defect: page 93 changes three signs

Figure 10.1 prints the kappa=2 entries at indices 8, 9 and 10 as

-512/45, -104/63, -416/945.

Page 93 says it is recalling Figure 10.1, but prints all three as positive. These are distinct coefficient rows, so both must be checked.

Fatal source defect: neither row gives the reported proportion

Using the Figure 10.1 signs gives

sum_i C[2,i]/((i+1)(i+2)) = 37057/73710,
P_2 = -202/36855
    = -0.005480938814272...

Using the page 93 signs gives

P_2 = -107714/184275
    = -0.584528557861891...

For kappa=3, where the two displayed rows have the same signs, exact substitution gives

P_3 = -10284002/1216215
    = -8.455743433521212...

All three values are trivial negative lower bounds. None rounds to 0.9544 or 0.9774.

An unprinted tail cannot rescue the stated calculation

Suppose an omitted weighted tail R_kappa were silently included. Equation (11.5) would require

claimed = 1 - 2 (shown weighted sum + R_kappa).

The exact tails needed to obtain the reported values are

R_2 = -8844103/18427500
    = -0.479940469407136...

R_3 = -11472730541/2432430000
    = -4.716571716760606...

Their absolute sizes are respectively

95.46485684216207% and 99.76099182302386%

of the displayed weighted sums. Page 93 explicitly obtains its percentages by assuming the coefficients after index 11 are negligible. A tail canceling almost the entire displayed sum is the opposite of that assumption.

Exact conclusion

The first local error is the three-sign mistranscription on page 93. Correcting those signs does not repair the headline. The first fatal statement is:

Applying equation (11.5) to the first eleven displayed coefficients gives 0.9544 for kappa=2 and 0.9774 for kappa=3.

It does not. The exact outputs are the negative values above. No normalization, rounding choice, or negligible omitted tail reconciles the sentence with the displayed source.

The source does not publish the arithmetic worksheet that produced the two decimals, so the particular keystroke or intermediate expression cannot be reconstructed. The source-level diagnosis is nevertheless closed: this is a chapter-11 arithmetic error, not an alternative interpretation of equations (11.2) through (11.5).

What survives

The kappa=1 control remains consistent with Farmer-Gonek.
This audit does not adjudicate the full proof of the coefficient formula or the Appendix Mathematica implementation.
The kappa=2 and kappa=3 percentage claims must be withdrawn.
The first eleven coefficients alone yield no positive simplicity bound.
Any future positive bound needs a separately justified tail, not the page 93 negligible-tail substitution.

Reproduction

.venv/bin/python hunts/higher_xi/bian_audit.py
.venv/bin/python -m pytest -q -o addopts='' hunts/higher_xi/test_higher_xi.py