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Library · hunts/higher_xi/BRIDGE-CLOSURE.md

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

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Outcome C

The corrected coefficient series is not presently identified with the pair form factor of the zeros of xi'' by the available analytic argument.

This document audits the historical fixed-order architecture. The subsequent replacement audit is RESUMMED-BRIDGE.md. It derives an exact untruncated resolvent and closes at Outcome D because the corresponding uniform resummed mean square is not currently available.

The first failed estimate occurs before the coefficient-order limit. On Bian thesis pages 32-33, the geometric remainder in the approximate Dirichlet series contributes the contour kernel

\[ I_V(t)=\int_V^\infty\frac{dv}{1+(v-t)^2}, \qquad V=T_\varepsilon. \]

The source asserts I_V(t)=O((t+2)^-2). In fact,

\[ I_V(t)=\frac\pi2+\arctan(t-V). \]

For every t>=V, the interval [t,t+1] lies inside the integration range and

\[ I_V(t)\ge \int_t^{t+1}\frac{dv}{1+(v-t)^2}\ge\frac12. \]

Thus no constant independent of t can make I_V(t)<=C/(t+2)^2. The same contour-tail estimate occurs in the pinned Farmer-Gonek-Lee arXiv source for the level-1 calculation and is load-bearing in its printed explicit-formula to mean-square chain. No later repair was located in the replacement audit. This classifies the printed proof as having a gap; it does not contradict the limiting prediction. bridge_obstruction.py preserves the rational unit-interval counterexample.

This failure invalidates the stated explicit formula remainder and every later pair-form-factor passage that uses it. It does not show that the corrected infinite series is the wrong limiting object. A replacement argument could still establish that identification. None is supplied by the pinned sources.

1. The bridge theorem that is actually required

Assume RH. Write the zeros of xi'' as

\[ \frac12+i\gamma^{(2)}, \]

with ordinates counted according to multiplicity. Let

\[ w(u)=\frac4{4+u^2} \]

and, for T>=2, define

\[ F_2(\alpha,T)= \frac{2\pi}{T\log T} \sum_{0<\gamma^{(2)},\gamma'^{(2)}\le T} T^{i\alpha(\gamma^{(2)}-\gamma'^{(2)})} w(\gamma^{(2)}-\gamma'^{(2)}). \]

The normalization, smoothing, multiplicity convention, and height interval are exactly those in Bian equations (1.4), (5.5), and (9.6).

Put

\[ D_T(\alpha)=T^{-2|\alpha|}\log T, \qquad \mathcal F_2(\alpha)=\sum_{i\ge1}C_{2,i}|\alpha|^i. \]

The coefficient series and its uniform tail are established independently in CORRECTED-F2.md.

Pointwise bridge theorem

The strongest useful pointwise statement would be:

For every compact K contained in (0,1), \[ \lim_{T\to\infty} \sup_{\alpha\in K} |F_2(\alpha,T)-D_T(\alpha)-\mathcal F_2(\alpha)|=0. \]

Symmetry then gives the negative band. This is local uniform convergence away from the spike and away from the endpoint. A claim uniform on the entire open interval is stronger and is not supported by an estimate stated only for x<=T^(1-epsilon).

Weighted bridge theorem

For the August-2026 application, the smaller sufficient statement is:

Let v be the explicit rational window in window_certificate.py and \[ A_v(\alpha)=\int_{\mathbb R}v(s)v(s-\alpha)\,ds. \] Then \[ \lim_{T\to\infty}\int_{-1}^{1}A_v(\alpha)F_2(\alpha,T)\,d\alpha =A_v(0)+ \int_{-1}^{1}A_v(\alpha)\mathcal F_2(\alpha)\,d\alpha. \]

The first term on the right is exact because

\[ \lim_{T\to\infty}\int_{-1}^{1}A_v(\alpha)D_T(\alpha)\,d\alpha=A_v(0). \]

This weighted theorem alone would justify the downstream rational window calculation. It is not currently established.

2. Required finite-height decomposition

A valid proof must derive, before taking any limit,

\[ F_2(\alpha,T)= D_T(\alpha)+P_B(\alpha) +E_{\rm diag}+E_{\rm trunc}+E_{\rm contour} \quad +E_{\rm PNT}+E_{\rm pp}+E_{\rm off}+E_{\rm edge}. \]

The classes are:

The source establishes only fixed-order bounds for most of these classes. It does not establish a valid bound for E_trunc after the contour replacement.

3. Exact finite-height arithmetic origin

For derivative level two, write

\[ g=\frac{\zeta'}\zeta, \qquad L(s)=\frac{h'}h(s), \qquad x=L(s)^{-1}. \]

Since xi=h zeta, the exact logarithmic-derivative identity is

\[ \frac{\xi'''}{\xi''}(s) =L(s)+g(s)+D\log\left([L(s)+g(s)]^2+L'(s)+g'(s)\right). \]

After factoring out L(s)^2, this becomes

\[ L+g+2L'/L+ D\log\left(1+2g/L+[g^2+g'+L']/L^2\right). \]

The arithmetic q_j sequence comes from the terms containing g and its derivatives. Terms containing L' are archimedean remainders. On every fixed vertical strip used by the contour, the source has L'=O(1/t) and L of logarithmic size, which yields the recorded O(1/(t log t)) class before the contour convolution. Expanding the arithmetic part generates q_0,q_1,q_2,...; its exact rational mechanism is the Q(z) identity in CORRECTED-F2.md.

On Re(s)>=1+epsilon, the geometric expansion has ratio below 1/2 once the height exceeds the explicit source threshold. At geometric order B, its pointwise remainder is bounded by

\[ O\left(\frac{1}{\varepsilon^2 2^B}\right) +O\left(\frac{1}{\tau\log\tau}\right). \]

This estimate is on the right half-plane. It is not yet an estimate for the zero-side pair correlation. Moving it through the contour kernel is the first load-bearing operation.

4. First failed operation

On thesis page 32 the first remainder contributes

\[ E_3\ll \frac{2^{-B}}{\varepsilon^2}x^{1/2+\varepsilon} \int_{T_\varepsilon}^{\infty} \frac{dv}{1+(v-t)^2}. \]

Page 33 replaces this by

\[ E_3\ll_{\varepsilon,B}\frac{x^{1/2+\varepsilon}}{\tau^2}. \]

The kernel calculation above shows that this implication is false for the later range t>=T_epsilon. The strongest bound obtained by absolute values is

\[ E_3\ll_{\varepsilon} 2^{-B}x^{1/2+\varepsilon}. \]

The adjacent archimedean remainder also requires a convolution estimate in the contour height rather than pulling the target-height factor outside the integral. The geometric remainder is already sufficient to stop the proof.

The published finite polynomials expose the same failure without inspecting the contour. At derivative level one and alpha=1/2, the order-1 and order-2 polynomials differ by exactly

\[ \frac1{24}. \]

They cannot both be the limit of the same F_1(alpha,T) with remainders that vanish for each fixed order. The missing nonvanishing truncation remainder is therefore forced by uniqueness of limits.

5. Incompatible order requirements

Let H=log T, put x=T^alpha, and fix alpha>0. The corrected absolute contour bound gives the mean-square scale

\[ U_{\rm trunc}(T,B)= 4^{-B}T^{(1+2\varepsilon)\alpha}/H. \]

Making this displayed upper bound vanish requires geometric order linear in H.

The recorded mean-square and prime-number-theorem error for kappa=2 has normalized scale

\[ U_{\rm PNT}(T,B)= T^{\alpha-1}H^{4B+2}, \]

before accounting for its additional unknown B-dependent constant. Making this bound vanish forces

\[ B=O(H/\log H)=o(H). \]

No order selection B=B(T) makes both displayed bounds tend to zero on any fixed positive alpha. Indeed, the second condition makes

\[ \log U_{\rm trunc} =(1+2\varepsilon)\alpha H-o(H)-\log H\to+\infty. \]

The conflict is stronger after restoring the untracked constants. It is not resolved by the coefficient-series tail, which controls the limiting leading constants rather than these finite-height remainders.

The exponent bookkeeping only leaves room for a band shrinking on the scale alpha=O(1/log log T), and even there the untracked constants prevent a theorem. No fixed positive bandwidth follows from the recorded estimates.

6. Why the target weighting does not repair the source bound

The rational target window satisfies

\[ v(1/2)=v(-1/2)=\frac{68213}{100000}. \]

Consequently, as u tends to zero from above,

\[ A_v(1-u)=left(\frac{68213}{100000}\right)^2u+O(u^2). \]

Thus the autocorrelation has only a simple zero at the band edge. The source derivation itself stops at x<=T^(1-epsilon), so it supplies no endpoint- uniform estimate. Even if its displayed PNT error scale is extended to the edge, absolute integration gives

\[ H^{4B+2}\int_0^1 A_v(\alpha)T^{\alpha-1}\,d\alpha \asymp H^{4B}. \]

It does not vanish even at fixed positive B. A smooth window vanishing to arbitrarily high order at the edge could improve this particular endpoint calculation, but it would not resolve the geometric-tail versus order-growth conflict on an interior positive alpha interval.

Therefore Outcome B is not available from the current estimates, including for the specific rational window.

7. Prime powers and off-diagonal terms

The arithmetic classes omitted from the leading coefficient object are visible in the source proof:

  1. Square-free equal-length pairings generate the constants C[2,i].
  2. Nonsquare-free integers, including prime powers, lose one logarithm for each fixed word pair and enter the error term in the source's Lemma 11.
  3. Unequal convolution lengths have zero leading constant but remain in the same lower-log error class.
  4. The m!=n terms in the Dirichlet-polynomial mean square are bounded by a Hilbert-type inequality, producing the displayed O_B(x(log x)^(4B+3)) scale.
  5. Mixed ranges n<=x<m contribute a similar fixed-order bound.

Every statement is fixed-order. Their implicit constants include composition counts, multinomial weights, and constants from prime-sum estimates. The proof contains no sequence K_i satisfying

\[ |C_i(T)-C_i|\le K_i/H, \qquad \sum_iK_i|\alpha|^i<\infty. \]

This is the precise coefficient-order dominated-convergence estimate that is missing after the contour error is repaired.

8. Smallest sufficient replacement estimates

A pointwise bridge would follow from both of the following, uniformly on every compact K inside (0,1):

  1. an L2 tail estimate for the exact, untruncated logarithmic-derivative remainder, \[ \lim_{B\to\infty}\limsup_{T\to\infty} \sup_{\alpha\in K} \frac1{T\log T} \int_0^T|\mathcal E_B(T^\alpha,t)|^2dt=0; \]
  2. a summable finite-height arithmetic majorant, \[ |C_i(T)-C_i|\le E_i(T), \qquad \lim_{T\to\infty}\sup_{\alpha\in K} \sum_{i\ge1}E_i(T)|\alpha|^i=0. \]

For the weighted bridge it would suffice to replace the two suprema by the single estimate

\[ \lim_{B\to\infty}\limsup_{T\to\infty} \left|\int_{-1}^{1}A_v(\alpha) [F_2(\alpha,T)-D_T(\alpha)-P_B(\alpha)]d\alpha\right|=0. \]

The already established coefficient tail then permits P_B to be replaced by mathcal F_2, with an explicit error below 3.279e-9 after index 40.

These are theorem schemas, not consequences of the current source estimates.

9. Oracle consequence

The completed-CUE and finite-ell Dirichlet ladders remain consistent with the corrected coefficient object at their recorded sizes. The failed contour bound changes no oracle parameter and suggests no rescaling. Because neither bridge theorem was obtained, the oracles have no promotion role in this outcome.

10. Trust boundary

The closure audit separates into four regimes:

The exact obstruction is small enough for Lean, but the active hunt scope does not authorize changes under lean/, and formalizing it would not repair the missing analytic estimate. The paper proof should be repaired before expanding the formal boundary.

11. Consequence

The coefficient-generated object must continue to be labeled separately from the actual xi'' zero form factor. The conditional 0.9234015 window number does not advance to a zero-statistics result.

The next mathematical target is no longer coefficient generation. It is an L2 estimate for the full geometric remainder after the contour transform, with order-uniform control strong enough to coexist with the off-diagonal prime-sum bounds.

Pinned sources

Reproduction

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