Split result
The old coefficient-tail barrier is closed. CORRECTED-F2.md gives an exact majorant for the infinite corrected series and the uniform bandwidth-one bound
\[ \sum_{i>40}|C_{2,i}||\alpha|^i<3.279\mathbin{\cdot}10^{-9}, \qquad |\alpha|\le1. \]
A different barrier remains: the available source does not justify identifying that entire coefficient-generated object with the limiting pair form factor of the zeros of xi''. BRIDGE-CLOSURE.md identifies the first failed estimate, not merely an absent uniformity argument.
First exact obstruction to the bridge
On thesis pages 32-33, the geometric truncation remainder contains
\[ \int_{T_\varepsilon}^{\infty}\frac{dv}{1+(v-t)^2}. \]
For t>=T_epsilon this is at least 1/2, but the source assigns it O((t+2)^-2). The unit interval [t,t+1] is the smallest obstruction. This invalidates the explicit-formula remainder used by the later mean square.
Correcting the absolute bound forces geometric order B=Omega(log T) on any fixed positive alpha. The source's fixed-order mean-square error can vanish only with B=O(log T/log log T). No single order selection satisfies both.
There is also a uniqueness symptom at level one:
Bian's Theorem 1 is stated for every fixed positive integer B, with only an o_(kappa,B)(1) remainder, uniformly on 0<|alpha|<1. At level one, B=1 gives
\[ \alpha-4\alpha^2+4\alpha^3, \]
while B=2 adds the nonzero term 4\alpha^5/3. Two distinct polynomials cannot both be the same limiting function on an open interval with errors that vanish as T tends to infinity. Thus the printed fixed-B statement is not a valid limit-exchange theorem for the infinite series.
The issue is separate from the missing multiplicity factors in the level-2 coefficient table. The level-1 control already exposes it.
Sharpest current statement
The exact arithmetic now supplies:
- a rational generating function for every
q_j; - exact
C[2,i]through index 40 by independent routes; - strict sign alternation for every coefficient;
- an explicit entire majorant for the full coefficient series;
- a bandwidth-one window tail bound applying to every integrable window.
For the explicit rational window in window_certificate.py, the coefficient-generated object gives the exact conditional inequality
\[ 2-D>0.923401526388517\ldots>0.9234015. \]
It is not a result about zeros of xi'' until an analytic bridge controls the terms discarded before or during the explicit-formula and mean-square passage.
Exact sufficient bridge
Let A_v be the autocorrelation of an admissible window. It would suffice to derive, directly from xi'''/xi'', a bound of the form
\[ \limsup_{T\to\infty} \left|\int A_v(\alpha) \left(F_2(\alpha,T)-\sum_{i=1}^{J}C_{2,i}|\alpha|^i\right)d\alpha\right| \le E_J(1)\|v\|_1^2. \]
At J=40, the available coefficient allowance is below 3.279e-9. The missing input is therefore not a coefficient-growth estimate. It is a uniform analytic remainder connecting the arithmetic expansion to the zero statistic.
This is Outcome C at the form-factor bridge, with the coefficient-series part of the directive completed and the first failed analytic estimate pinned.