Resummed bridge audit for the xi-double-prime form factor

Disposition

This audit closes at Outcome D, current technology barrier.

It also classifies the pinned Farmer-Gonek-Lee level-1 precedent as Classification C, a genuine gap in the printed proof with no repair located. This classification concerns the proof, not the truth of its limiting prediction. No contradictory zero statistic is known here.

The positive result of this audit is a changed representation. The corrected Q(z) object is the Taylor expansion of an exact, untruncated resolvent for xi'''/xi''. Thus geometric order B is not intrinsic to the arithmetic object. The remaining obstacle is a uniform mean-square theorem for that full resolvent.

1. Level-1 precedent

The pinned source is Farmer, Gonek, and Lee, Pair correlation of the zeros of the derivative of the Riemann xi-function, arXiv:0803.0425. The journal version is JLMS 90 (2014), 241-269.

The dependency chain in the source is:

1. Equation (7.1) writes xi''/xi' exactly in terms of L and zeta'/zeta.
2. Equations (7.5)-(7.6) replace the reciprocal by a geometric expansion of fixed order K with a right-half-plane remainder.
3. Equations (7.19)-(7.22) pass that remainder through the Cauchy kernel in the explicit formula.
4. The resulting explicit-formula error is squared and integrated in the mean-square calculation used for Theorem 1.1.

At step 3 the source obtains

\[ E_3=O\left(\frac{x^{1/2+\varepsilon}}{\varepsilon 2^K} \int_{T_\varepsilon}^{\infty} \frac{dv}{1+(v-t)^2}\right) \]

and then assigns it decay in |t|. For t>=T_epsilon, however, the integral is at least 1/2, as preserved by bridge_obstruction.py. This term is the only stated source of the later decaying geometric-tail error. It is therefore load-bearing in the printed argument.

There is a second local endpoint issue in the same passage: terms containing 1/(|t-T_epsilon|+2) are replaced by 1/(|t|+2). That replacement is also invalid near t=T_epsilon. The nondecaying E_3 term already stops the chain, so the classification does not depend on the second issue.

The fixed-order statements provide an internal control. At alpha=1/2, the printed order-1 and order-2 regular polynomials differ by exactly 1/24. They cannot both be one common pointwise limit with a remainder tending to zero at each fixed order. This shows that a nonvanishing truncation remainder is missing from the printed family of formulas. It does not determine the true limiting value.

No later erratum or independent RH-only derivation repairing this operation was located. Sodin derives critical-point limits under RH plus convergence of the full zeta-zero point process, which is a stronger statistical hypothesis, not a repair of the arithmetic theorem. Later mean-value work on fixed products of zeta'/zeta does not supply uniform control of the nonlinear reciprocal used here.

Consequently:

• the exact logarithmic-derivative identities survive;
• the fixed-order coefficient calculations survive as fixed-order arithmetic;
• the printed finite-height to infinite-form-factor passage has a gap;
• this audit does not classify the level-1 limiting prediction as false.

2. Exact representation without geometric order

Put

\[ U=\frac{\xi'}\xi=L+D, \qquad D=\frac{\zeta'}\zeta. \]

Direct differentiation gives

\[ \frac{\xi'''}{\xi''} =\frac{U^3+3UU'+U''}{U^2+U'} =U+\frac{2UU'+U''}{U^2+U'}. \]

This identity contains no coefficient-order cutoff. Split its denominator as

\[ U^2+U'=A_0+A_+, \]

where

\[ A_0=L^2+L', \qquad A_+=2LD+D^2+D', \]

and split the numerator after the leading U as

\[ 2UU'+U''=N_0+N_+, \]

with

\[ N_0=2LL'+L'', \qquad N_+=2LD'+2DL'+2DD'+D''. \]

On Re(s)>=1+epsilon, the series for D,D',D'' converge absolutely. At sufficient height, A_0 is nonzero and A_0+A_+ can be inverted in the weighted Dirichlet algebra. If a_+(n,s) denotes the coefficient sequence of A_+, its inverse b(n,s) is determined without a finite order by

\[ b(1,s)=A_0(s)^{-1}, \]

and, for n>1,

\[ b(n,s)=-A_0(s)^{-1} \sum_{\substack{d\mid n\\d>1}}a_+(d,s)b(n/d,s). \]

The full arithmetic coefficients of the resolvent are then

\[ a_2(\cdot,s) =D(\cdot)+[N_0(s)\mathbf 1+N_+(\cdot,s)]*b(\cdot,s). \]

This is an exact, height-dependent Dirichlet object on the right half-plane. It packages every q_j before the finite-height limit.

3. The corrected Q object is the frozen exact resolvent

Freeze L'=L''=0, put z=1/L, and use

\[ D=-A,\qquad D'=B,\qquad D''=-C, \]

where A=Lambda, B=Lambda log, and C=Lambda log^2. The arithmetic part of the exact resolvent becomes

\[ -A+\frac{2(L-A)B-C}{(L-A)^2+B} =-A+\frac{2Bz-(2A*B+C)z^2} {(1-Az)^2+Bz^2}. \]

The right side is exactly the corrected generating object Q(z) in CORRECTED-F2.md. resummed_bridge.py checks this symbolic identity, the two numerator/denominator splits, and the original resolvent identity.

This connection is stronger than matching forty Taylor coefficients. It identifies what must be treated in mean square: the whole convolution inverse, not a geometric polynomial whose order later has to grow.

4. What elementary absolute resummation can and cannot do

Let S_j be the sum of the absolute coefficients of the operator words in q_j. The closed formula gives exactly

\[ S_0=1,\qquad S_1=2,\qquad S_j=3\,2^{j-2}\quad(j\ge2). \]

For j>=2, the two numerator families in the closed formula contribute 2 sum_k binom(j-1,2k)=2^(j-1) and sum_k binom(j-1,2k+1)=2^(j-2), respectively. The deterministic control checks the resulting formula through order 19. Each word in q_j has word length plus total logarithmic degree equal to j+1. The elementary bound for n<=T^alpha therefore has asymptotic order ratio 4 alpha after division by L^j, where L is asymptotic to one half of log T.

Thus termwise absolute resummation is directly available only for

\[ \alpha<\frac14. \]

At alpha=1/4 this majorant loses decay. This is a boundary of the elementary absolute argument, not a singularity of the corrected coefficient series and not a zero-statistics theorem. Even below 1/4, it does not control the contour comparison, archimedean freezing, the long Dirichlet tail, or the uniform arithmetic mean square. Those are separate obligations.

5. Early smoothing and direct L2 control

The source Cauchy kernel should not be estimated pointwise after taking an absolute value. Define

\[ Z_{2,x}(s)=\sum_{\gamma^{(2)}}k(i\gamma^{(2)},s)x^{i\gamma^{(2)}} \]

with the same k as the pinned source. For any right-half-plane coefficient family a(n,s), define its two-range transform by

\ \begin{split} \mathcal D_x[a (s)=x^{-1/2}\bigg(& \sum_{n\le x}a(n,1-\bar s)(x/n)^{1-\bar s}\\ &+\sum_{n>x}a(n,s)(x/n)^s\bigg). \end{split} \]

This is the arithmetic object appearing after the source contour split, but here a=a_2 is the exact convolution-inverse sequence rather than a_K from a finite geometric expansion.

The natural replacement is to compare Z_{2,x} and mathcal D_x[a_2] directly in L2, after subtracting the explicit archimedean and diagonal terms. Parseval or Montgomery-Vaughan then has a chance to preserve cancellation that the failed pointwise bound destroyed.

No searched source provides that comparison for this height-dependent nonlinear coefficient family. Existing theorems cover fixed Dirichlet polynomials, fixed products of logarithmic derivatives, or smoothed linear statistics of the original zeta zeros. None gives a coefficient-order-uniform mean square for this convolution inverse.

6. Minimal new analytic target

The smallest replacement target found here is the following named lemma.

Uniform Resummed Mean-Square Lemma at level two, URMS2

Assume RH. Fix epsilon>0 and a compact interval K subset (0,1-epsilon). Let x=T^alpha, alpha in K, and let a_2(n,s) be the exact coefficient family above. Let A_x(s) denote the explicit archimedean and diagonal terms obtained from the poles crossed in the same Cauchy transform.

The lemma has two parts, uniform for alpha in K:

\ \frac1{T\log T}\int_T^{2T} |Z_{2,x}(\sigma+it)-A_x(\sigma+it) +\mathcal D_x[a_2 (\sigma+it)|^2dt=o(1), \]

for one, hence every, fixed 5/4<sigma<2; and

\ \frac1{T\log T}\int_T^{2T} |\mathcal D_x[a_2 (\sigma+it)|^2dt =\mathcal F_2(\alpha)+o(1), \]

after the source's exact diagonal normalization. The second equality includes all prime powers, unequal convolution lengths, off-diagonal terms, and the variation of L,L',L'' across the height interval.

The required convergence is local uniformity in alpha. A weighted variant may replace both suprema by integration against the fixed autocorrelation A_v(alpha) from window_certificate.py. That weaker form would be enough for the downstream window functional.

URMS2 implies the bridge as follows:

1. the first part replaces the invalid pointwise contour tail by an L2 comparison of the full objects;
2. the second part evaluates the resummed arithmetic side without a geometric order B;
3. Cauchy-Schwarz transfers the first comparison to the smoothed pair statistic;
4. the already bounded corrected coefficient tail identifies the resulting regular function with mathcal F_2.

The most concrete arithmetic subproblem inside URMS2 is a uniform quadratic mean for a_2(n,s). Montgomery-Vaughan handles the standard off-diagonal geometry once one has a uniform leading asymptotic and summable error for the height-dependent diagonal sums. Current fixed-word prime-number-theorem estimates do not provide that summability.

7. Literature mechanism audit

This search found useful analogies for early smoothing and mean-square organization, but no lemma with the hypotheses and uniformity above. The absence of a located repair is a literature-search result, not a claim that no repair exists.

8. Trust boundary and final status

Exact algebra and deterministic computation now cover:

• the untruncated resolvent identity;
• its convolution-inverse coefficient recurrence;
• its reduction to the corrected Q(z) object;
• the operator-word mass formula;
• the 4 alpha elementary absolute-majorant boundary;
• the counterexample to the printed contour-tail estimate.

Classical analysis is still required for:

• the full-resolvent Cauchy transform in L2;
• uniform freezing or retention of the archimedean factors;
• the resummed diagonal arithmetic asymptotic;
• prime-power and unequal-length contributions;
• the off-diagonal mean square uniformly in alpha.

No completed-CUE or finite-ell output enters this disposition. No corrected simplicity percentage or zeta rank-trace transfer is opened.

The next named target is URMS2. Reverting to a larger fixed geometric order would not address it.

The subsequent direct attack is URMS2-ATTACK.md. It constructs the natural Wiener and Hilbert-space framework but finds that both URMS1 and URMS2 still require a uniform resummed square-density estimate saving one full logarithm, plus an early-smoothed contour comparison for the exact inverse.

Pinned inputs

• Repository state at audit start: 1d850e5838b2226c1948c87ddd3c69e039901c87.
• Farmer-Gonek arXiv archive SHA-256: f6cdc7b71db06187dac655647e24312843441aa2598e41a3b53834dd1b36822f.
• Farmer-Gonek main TeX SHA-256: a3a9ac955a9c95d10d36d6b05aae05a79c6408f6e1cdcb8c14fd79f000144fb1.
• Bian thesis PDF SHA-256: ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7.

Reproduction

.venv/bin/python -m pytest -q -n0 hunts/higher_xi/test_higher_xi.py
.venv/bin/python hunts/higher_xi/resummed_bridge.py