The corrected level-2 coefficient object and its remaining bridge

Result

The corrected coefficients define an absolutely convergent power series on |alpha|<=1. In fact, the majorant below has infinite radius. The first 40 coefficients are exact, and the regular series after index 40 satisfies

\[ \sum_{i>40}|C_{2,i}|\,|\alpha|^i \le E_{40}(\alpha) \le E_{40}(1) <3.279\mathbin{\cdot}10^{-9}. \]

The exact value of E_40(1) is emitted by corrected_form_factor.py and stored as an exact rational in the window check. Thus the coefficient-generated object

\[ \mathcal F_2(\alpha)=\sum_{i\ge1}C_{2,i}|\alpha|^i \]

is rigorously controlled on the full bandwidth. Its endpoint lies in

\[ 4.76344632220668 < \mathcal F_2(1) < 4.76344632876331. \]

This closes the coefficient-series tail problem. It does not by itself identify mathcal F_2 with the limiting pair form factor of the zeros of xi''. The uniform analytic passage missing from the source remains a separate bridge.

Exact generating mechanism

Use the commutative Dirichlet-convolution atoms

\[ A=\Lambda,\qquad B=\Lambda\log,\qquad C=\Lambda\log^2. \]

If q_j denotes the coefficient of L^(-j) in the arithmetic part of xi'''/xi'', then the formal object

\[ Q(z)=\sum_{j\ge0}q_jz^j \]

has the rational expression

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

Equivalently, for n>=1,

\[ \begin{split} q_n={}&2\sum_{j=0}^{\lfloor(n-1)/2\rfloor} (-1)^j {n-1\choose2j}A^{*(n-1-2j)}B^{(j+1)}\\ &-\sum_{j=0}^{\lfloor(n-2)/2\rfloor} (-1)^j {n-1\choose2j+1}A^{*(n-2-2j)}*B^{*j}*C. \end{split} \]

exact_c2.py regenerates this sequence in three independent ways: the successive logarithmic-derivative recurrence, the direct operator-word expansion, and this closed binomial expression. They agree exactly through index 40. Two separate compressed mean-square calculations also agree exactly.

Every word beta in q_n has entries in {0,1,2} and satisfies

\[ \operatorname{len}(\beta)+|\beta|=n+1, \qquad \operatorname{sgn}[q_n(\beta)]=(-1)^{n-\operatorname{len}(\beta)}. \]

In a nonzero pairing contributing to C[2,i], the two words have equal length and their powers sum to i-1. Hence every summand has the same sign,

\[ \operatorname{sgn} C_{2,i}=(-1)^{i-1}. \]

This is an exact structural statement, not an observed sign pattern.

Exact coefficient fixture

The formula is

\[ C_{2,i}=2^{i-1}\sum_{p+q=i-1}\langle q_p,q_q\rangle. \]

For paired words beta,delta of equal length r, the weighted-degree identity reduces the denominator in the mean-square constant to exactly i!:

\[ 2r+|\beta|+|\delta|-1=i. \]

C2_EXTENDED.json stores indices 1 through 40 as reduced rationals. It also stores the integers

\[ C_{2,i}\,i!/2^{i-1}, \]

which give a compact independent serialization check. Indicative generation cost on the recorded machine was 0.08 s, 2.13 MB through index 20 and 1.40 s, 34.15 MB through index 40. These timings are not part of the mathematical claim.

Tail majorant

For exponents a,b in {0,1,2}, set

\[ w_0=1,\qquad w_1=5/2,\qquad w_2=11. \]

All nine inequalities

\[ (a+b+1)!\le w_aw_b \]

hold by direct rational comparison. Therefore the factorial permanent for two length-r words is at most

\[ r!\,w(\beta)w(\delta). \]

The checker retains the exact word length in this bound through index 101. For the remainder, put n=i-1,

\[ R=\lfloor n/2\rfloor,\quad L=\lceil n/4\rceil,\quad M=R-L+1,\quad a=5/2,\quad K=47/4. \]

The binomial formula, binom(n,j)<=2^n, and monotonicity of r!a^(-2r) give, for i>=26,

\[ |C_{2,i}|\le B_i= \frac{(n-1)M4^nK^2a^{n-2-2R}R!}{(n+1)!}. \]

The two-step ratio B_(i+2)/B_i splits into four exact rational expressions according to n mod 4. Their successive differences have numerators

\[ \begin{array}{c|l} 0&-32\\ 1&-2(16m^4+64m^3+88m^2+46m+7)\\ 2&-8(16m^3+100m^2+176m+87)\\ 3&-32(8m^3+36m^2+52m+25), \end{array} \]

over positive denominators. Each residue-class ratio therefore decreases. The largest starting value for n>=101 is

\[ \rho=\frac{5202}{64375}<0.081. \]

The two parity tails are consequently geometric. The exact bound used by the checker is

\[ E_{40}(\alpha)= \sum_{i=41}^{101}U_i\alpha^i+ \frac{B_{102}\alpha^{102}+B_{103}\alpha^{103}} {1-\rho\alpha^2}, \]

where U_i is the exact length-retaining permanent majorant. The same ratio tends to zero, so the regular coefficient series is entire as a complex power series.

Direct weighted consequence

For an integrable window v with autocorrelation A_v, the bandwidth-one tail contributes at most

\[ E_{40}(1)\int_{-1}^{1}|A_v(u)|\,du \le E_{40}(1)\|v\|_1^2. \]

window_certificate.py uses the exact rational window

\[ v(s)=P_0(2s)-\frac{185616}{10^6}P_2(2s) -\frac{111471}{10^6}P_4(2s) -\frac{20783}{10^6}P_6(2s). \]

Since |P_j(x)|<=1, this window has the exact positive floor 68213/100000, integral one, and L1 norm one. Exact polynomial integration of the first 40 terms gives

\[ D_{40}=1.0765984703331668\ldots. \]

Adding the full tail allowance gives

\[ D<1.076598473611483\ldots, \qquad 2-D>0.923401526388517\ldots>0.9234015. \]

The floating window optimizer finds 0.923401531890862; its gap above the exact rational lower calculation is 5.51e-9, split between window rounding and the uniform tail allowance.

This is an exact conditional calculation for the coefficient-generated object. It is not yet a ξ'' simplicity result because the bridge in the next section remains open.

The remaining analytic bridge

Bian's printed fixed-B statement cannot be used as a uniform tail theorem. For level one, B=1 already gives the polynomial

\[ \alpha-4\alpha^2+4\alpha^3, \]

while B=2 adds the nonzero term 4\alpha^5/3. Both cannot equal the same limiting form factor with only an o_B(1) error on a fixed open alpha interval. This inconsistency exists before the level-2 coefficient correction.

What is still required is one of the following:

1. a corrected explicit-formula theorem with a remainder uniform in B;
2. a direct mean-square derivation whose discarded arithmetic terms are bounded by the coefficient majorant above;
3. a weighted version of that derivation controlling the full admissible window functional.

A sufficient theorem schema is

\[ \limsup_{T\to\infty} \left|\int r(\alpha) \left(F_2(\alpha,T)-\sum_{i=1}^{J}C_{2,i}|\alpha|^i\right)d\alpha\right| \le E_J(1)\|v\|_1^2, \]

for the autocorrelation kernel generated by every admissible v. At J=40, the right side for the explicit rational window is below 3.279e-9.

Until such a bridge is derived, the strict 0.9234015 number remains a conditional consequence of identifying the actual form factor with mathcal F_2, not a promoted statistic for zeros of xi''.

Independent numerical falsifiers

The corrected object was compared without rescaling to both existing oracles. For completed CUE at N=48, 200 fresh fixed-seed samples gave:

All four raw discrepancies are below one reported Monte Carlo standard error. The same run includes level-0 and Farmer-Gonek-Lee level-1 controls.

For the direct Dirichlet recurrence at ell=14, the raw cumulative values are still below their limiting controls:

No ad hoc rescaling is applied. The finite-ell deficit remains visible in both levels. These comparisons do not supply the missing analytic bridge.

Reproduction

.venv/bin/python hunts/higher_xi/exact_c2.py --max-index 40
.venv/bin/python hunts/higher_xi/corrected_form_factor.py
.venv/bin/python hunts/higher_xi/window_certificate.py
.venv/bin/python hunts/higher_xi/cue_oracle.py --sizes 24:200,32:200,48:200
.venv/bin/python hunts/higher_xi/dirichlet_recurrence.py --ells 8,10,12,14
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py -n0