URMS2-051: crossing the half-band

Disposition

The strict half-band restriction was created by a worst-frequency-spacing majorant. It is not present in the RC2 Dirichlet polynomial.

Retaining the spacing of the frequencies log n gives an off-diagonal error of order x log x, independent of the upper polynomial length. Together with RAMS2 on every fixed compact coefficient band, this gives URMS2 uniformly on compact positive bands through

\[ |\alpha|\leq\frac{51}{100}. \]

The normalized constant spectral factor at bandwidth 51/100 then gives the first corrected conditional xi-double-prime simplicity statement:

\[ \liminf_{T\to\infty} \frac{N_{2,\mathrm{simple}}(T)}{N_2(T)} \geq 0.0147728663285376\ldots, \]

under RH and the hypotheses of the rebuilt zero-side bridge. The decimal is only a rendering of the exact rational lower bound emitted by corrected_simplicity_bound().

1. The actual RC2 tail object

Fix a dyadic height block t in [U,2U], put x=H^alpha, and freeze the exact level-two coefficient family at that block. At the standard pair weight

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

the Cauchy evaluation line is sigma=3/2. The two-range arithmetic term is

\[ \mathcal D_{U,x}(3/2+it)=x^{-1/2}\left( \sum_{n\leq x}a_{2,U}(n)(x/n)^{-1/2+it} +\sum_{n>x}a_{2,U}(n)(x/n)^{3/2+it} \right). \]

After removing the harmless common phase x^(it), truncate at W and write

\[ P_{U,x,W}(t)=\sum_{n\leq W}c_n n^{-it}, \]

where

\[ c_n= \begin{cases} x^{-1}a_{2,U}(n)n^{1/2},&n\leq x,\\ x\,a_{2,U}(n)n^{-3/2},&x<n\leq W. \end{cases} \]

This is the object before Cauchy-Schwarz or a Hilbert norm is applied. Its square mean is

\[ \int_U^{2U}|P_{U,x,W}(t)|^2dt =\sum_{m,n\leq W}c_m\overline{c_n} K_U\!\left(\log\frac mn\right), \]

with exact sharp-block kernel

\[ K_U(\theta)=\int_U^{2U}e^{-it\theta}dt =e^{-3iU\theta/2}\frac{2\sin(U\theta/2)}{\theta}. \]

The diagonal is

\[ U\left[ x^{-2}\sum_{n\leq x}|a_{2,U}(n)|^2n +x^2\sum_{x<n\leq W}|a_{2,U}(n)|^2n^{-3} \right]. \]

RAMS2 evaluates this weighted Stieltjes object at the U log U scale and identifies its limit with the corrected F2 object.

The infinite remainder is

\[ R_{U,x,W}(t)=x\sum_{n>W}a_{2,U}(n)n^{-3/2-it}. \]

The right-line Hilbert bound gives

\[ \frac1U\int_U^{2U}|R_{U,x,W}(t)|^2dt \ll x^2W^{-1+2\epsilon}. \]

This is the only place where the exponent 2alpha occurs: it is the square of the upper-range factor x.

2. Exact failure of the old proof at 0.51

The old argument replaced every frequency spacing by the worst spacing at the upper endpoint W=H^gamma. It therefore demanded

\[ \frac WU=o(1),\qquad \gamma<\delta<1. \]

The infinite remainder demanded

\[ 2\alpha<\gamma(1-2\epsilon). \]

At alpha=51/100 and in the zero-epsilon limit these become

\[ \gamma>\frac{51}{50},\qquad \gamma<1. \]

The exact missing exponent is

\[ \boxed{\frac{51}{50}-1=\frac1{50}=0.02.} \]

This is the first impossible inequality in the mechanical old proof.

3. The minimal replacement lemma

For distinct real frequencies lambda_n, the spacing-sensitive Montgomery-Vaughan mean-value estimate is

\[ \int_U^{2U}\left|\sum_n c_ne^{-it\lambda_n}\right|^2dt =U\sum_n|c_n|^2 +O\left(\sum_n\frac{|c_n|^2}{\delta_n}\right), \]

where

\[ \delta_n=\min_{m\ne n}|\lambda_m-\lambda_n|. \]

Here lambda_n=log n. The elementary inequality

\[ \log(n+1)-\log n\geq\frac1{n+1} \]

gives delta_n^-1<=2n for n>=2. Hence

\[ \boxed{ \int_U^{2U}|P_{U,x,W}(t)|^2dt =U\sum_{n\leq W}|c_n|^2 +O\left(\sum_{n\leq W}n|c_n|^2\right). } \]

This is the required replacement lemma. It is no stronger than necessary and uses the actual arithmetic frequencies.

4. Mapping the coefficient sequence

The lower range contributes

\[ \sum_{n\leq x}n|c_n|^2 =x^{-2}\sum_{n\leq x}|a_{2,U}(n)|^2n^2. \]

RAMS2 and partial summation give

\[ x^{-2}\sum_{n\leq x}|a_{2,U}(n)|^2n^2 \ll x\log x. \]

The upper range contributes

\[ \sum_{x<n\leq W}n|c_n|^2 =x^2\sum_{x<n\leq W}|a_{2,U}(n)|^2n^{-2} \ll x\log x. \]

The last estimate is independent of W: partial summation is dominated by the lower endpoint x, because the weight is n^-2. Consequently

\[ \boxed{ \text{off-diagonal error}\ll x\log x. } \]

On every block U>=H^delta, its normalized size is

\[ \frac{x\log x}{U\log U}\ll H^{\alpha-\delta}. \]

Thus the true mean-value condition is alpha<delta, not gamma<delta. The upper length may cross U without creating an error at the main scale.

This gain does not rely on cancellation in the values of a_2. The essential structure is the two-range weight inherited from the early-smoothed contour transform. Replacing it by a generic unweighted polynomial restores the maximum-length obstruction.

5. Smoothing audit

Under the Fourier convention used by the contour calculation,

\[ \int_{-\infty}^{\infty}\frac4{4+u^2}e^{-i\xi u}du =2\pi e^{-2|\xi|}. \]

This one-sided exponential localization is what produces the exact weights sqrt(n)/x below x and x/n^(3/2) above x. The spacing argument keeps those weights until after the off-diagonal expansion. Removing the early smoothing removes the n^-3/2 upper weight and the W-independent estimate with it.

The exact block kernel also shows the effective near-diagonal range:

\[ \left|\log\frac mn\right|\ll U^{-1}, \qquad |m-n|\ll n/U. \]

Taking absolute values there is harmless after the spacing inequality because the coefficient weight makes the total spacing cost O(x log x). No new shifted-convolution theorem is needed.

6. Rational witness at 0.51

Choose

\[ \alpha=\frac{51}{100},\qquad \delta=\frac34,\qquad \gamma=\frac{21}{20},\qquad \epsilon=\frac1{100}. \]

The exact margins are

\[ \delta-\alpha=\frac6{25}, \]

for the spacing-sensitive mean value,

\[ \gamma(1-2\epsilon)-2\alpha=\frac9{1000}, \]

for the infinite tail,

\[ 1-\delta=\frac14, \]

for the initial height interval, and

\[ 1-\alpha(1/2+\epsilon)=\frac{7399}{10000}, \]

for the central segment and height commutator.

The largest coefficient ratio occurs at the far cutoff on the lowest block:

\[ \frac{2\gamma}{\delta}=\frac{14}{5}. \]

Finite-prime peeling supplies RAMS2 uniformly on that fixed compact ratio band. All other RC2 obligations retain positive margins.

7. URMS2-051

Assume RH and retain the hypotheses and normalization of the rebuilt RC2 identity. For every compact positive alpha-band contained in 0<|alpha|<=51/100, the exact resummed level-two pair statistic has regular term equal to the corrected F2(alpha) object, locally uniformly in alpha.

The infinite tail is o(U log U) by the 9/1000 exponent margin. The finite off-diagonal is o(U log U) by the 6/25 mean-value margin. The initial height interval, height commutator, archimedean freezing, and contour localization retain their earlier bounds.

8. First corrected xi-double-prime statement

Take

\[ v(s)=\frac{100}{51}\mathbf1_{[-51/200,51/200]}(s). \]

Its autocorrelation is the triangle

\[ A_v(\alpha)=\frac{51/100-|\alpha|}{(51/100)^2} \mathbf1_{|\alpha|\leq51/100}, \]

and its physical-space test is the normalized squared sinc used in the source simplicity argument. It is nonnegative, equals one at the origin, and has Fourier support inside the new URMS2 band.

Exact integration of the first 40 corrected coefficients, followed by the full rational tail allowance, gives

\[ D_{51/100}(v) \leq 2-0.0147728663285376\ldots. \]

The diagonal multiplicity inequality then gives, conditionally on RH,

\[ \boxed{ \liminf_{T\to\infty} \frac{N_{2,\mathrm{simple}}(T)}{N_2(T)} \geq0.0147728663285376\ldots. } \]

This is recomputed from the corrected F2, the new 0.51 bridge, and the exact window normalization. No historical decimal or finite-order contour tail enters the statement.

9. Lesions

1. Replacing delta_n^-1 by W restores gamma<delta and the 1/50 deficit.
2. Removing early smoothing removes the upper n^-3/2 weight.
3. Deleting A*A changes the corrected F2 coefficient object but is not used to manufacture the mean-value gain.
4. Undoing finite-prime peeling prevents RAMS2 from reaching the far-cutoff ratio 14/5.
5. A generic unweighted polynomial of length W has spacing cost comparable to W times its diagonal and can saturate the old bound.
6. The fixed rational witness reaches 0.499, 0.500, 0.501, and 0.510 with positive margins.

The direct frozen-recurrence experiment gives the following nonproof control at alpha=0.51:

The spacing-normalized cost decreases across this ladder while the generic lesion separates rapidly. These finite values are a falsification control, not an input to URMS2-051.

The half-band crossing is therefore caused by retaining the exact smoothed weights and logarithmic spacing. No shifted-correlation hypothesis is left.

URMS2-051-AUDIT.md independently checks the source theorem, phase algebra, weighted partial summation, height freezing at ratio 14/5, infinite tail, multiplicity normalization, and the exact rational window functional.