Independent audit of URMS2-051

Verdict

URMS2-051 survives the independent internal audit.

The six load-bearing gates are:

This is an internal mathematical audit. Independent external review remains separate work.

1. Source theorem and exact hypothesis map

The primary analytic input is H. L. Montgomery and R. C. Vaughan, Hilbert's Inequality (https://doi.org/10.1112/jlms/s2-8.1.73), Journal of the London Mathematical Society (2) 8 (1974), 73-82.

Its weighted form gives, for distinct real frequencies lambda_n,

\[ \int_A^{A+U}\left|\sum_n c_ne^{-it\lambda_n}\right|^2dt =U\sum_n|c_n|^2 +O\left(\sum_n\delta_n^{-1}|c_n|^2\right), \]

where delta_n is the nearest-neighbor frequency spacing. Translation of the interval from [0,U] to [A,A+U] changes each coefficient by a unit complex phase and does not change the estimate.

For lambda_n=log n, the frequencies are distinct. Since

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

one has delta_n^-1<=2n for n>=2. The n=1 term has fixed spacing log 2 and is absorbed in the same absolute constant. Therefore

\[ \int_U^{2U}\left|\sum_{n\leq W}c_nn^{-it}\right|^2dt =U\sum_{n\leq W}|c_n|^2 +O\left(\sum_{n\leq W}n|c_n|^2\right). \]

No condition W<U occurs in this theorem.

2. Independent phase reconstruction

At sigma=3/2, the lower contour exponent is -1/2+it and the upper exponent is 3/2+it. Including the outside factor x^-1/2 gives

\[ x^{-1/2}(x/n)^{-1/2+it} =\frac{\sqrt n}{x}(x/n)^{it}, \]

and

\[ x^{-1/2}(x/n)^{3/2+it} =\frac{x}{n^{3/2}}(x/n)^{it}. \]

After the common unit phase x^it is removed, both ranges have frequency log n with the same sign. exact_two_range_phase_defects() returns two symbolic zeros.

The reflected level-two logarithmic derivative causes no magnitude mismatch. Differentiating xi(1-s)=xi(s) three times gives

\[ \frac{\xi'''(1-s)}{\xi''(1-s)} =-\frac{\xi'''(s)}{\xi''(s)}. \]

The reflection changes a sign and conjugates at the paired point. The weighted Hilbert inequality permits arbitrary complex coefficients, so both operations preserve every square estimate.

3. Diagonal and off-diagonal weights

With the frozen level-two coefficients, put

\[ |c_n|^2= \begin{cases} x^{-2}|a_2(n)|^2n,&n\leq x,\\ x^2|a_2(n)|^2n^{-3},&x<n\leq W. \end{cases} \]

The diagonal is U sum |c_n|^2. It is the weighted RAMS2 Stieltjes object whose limit is the corrected F2.

The weighted off-diagonal cost below x is

\[ x^{-2}\sum_{n\leq x}|a_2(n)|^2n^2. \]

If A(y)=sum_(n<=y)|a_2(n)|^2<=C y log y, partial summation gives

\[ x^{-2}\int_{1-}^{x}u^2dA(u) \leq Cx(3\log x+2). \]

Above x, the cost is

\[ x^2\sum_{x<n\leq W}|a_2(n)|^2n^{-2}. \]

The same calculation gives

\[ x^2\int_x^W u^{-2}dA(u) \leq Cx(3\log x+2). \]

The bound is independent of W; the decreasing weight makes the lower endpoint dominant. Thus the full off-diagonal error is O(x log x). On U>=H^delta, division by the main scale U log U leaves O(H^(alpha-delta)).

This also resolves the lower/upper cross terms. They are entries of the same arbitrary-coefficient polynomial and are already included in the weighted Hilbert inequality. No separate shifted-correlation hypothesis is used.

4. Height-freezing gate

The earlier reports documented height freezing only inside the old elementary pointwise radius. The 0.51 witness uses the larger fixed ratio 14/5, so this gate needed a new argument.

Differentiate the Borel monomer-dimer representation with respect to the frozen parameter z. A derivative marks one monomer or dimer factor. In the Hadamard square, a marked connected component gains at most a polynomial factor in its support size. The unmarked support majorant has successive ratio O(1/r), so every polynomially marked series is still summable on a fixed compact rho band.

For powerful supports, choose the tail-prime Cauchy radius for a slightly larger compact band than 14/5. Cauchy's derivative estimate changes only the fixed constant. The finitely peeled primes again contribute a finite factor. The differentiated powerful-squarefree majorant is therefore summable.

The resulting marked-cluster estimate is

\[ \sum_{n\leq y}|\partial_z a_{2,z}(n)|^2 \ll_{\rho_0}y(\log y)^3. \]

Across one dyadic height block,

\[ |z(t)-z_U|\ll(\log U)^{-2}. \]

For log y bounded by a fixed multiple of log U, the mean-value theorem in z gives

\[ \sum_{n\leq y}|a_2(n,t)-a_{2,U}(n)|^2 \ll_{\rho_0}\frac{y}{\log U}. \]

Applying the two contour weights gives frozen-difference diagonal O(1/log U) and spacing cost O(x/log U). Cross terms with the main coefficient family are smaller by Cauchy-Schwarz. The L' and L'' pieces carry additional factors U^-1 and U^-2 and are smaller.

Thus height freezing remains negligible at the far-cutoff ratio 14/5.

5. Infinite remainder

For Re(s)=1+epsilon, the exact inverse has a uniformly bounded weighted Wiener norm. Therefore

\[ \sum_{n>W}|a_2(n,s)|n^{-3/2} \leq W^{-1/2+\epsilon} \sum_{n>W}|a_2(n,s)|n^{-1-\epsilon}. \]

This is a pointwise bound before the height integral. After multiplication by x and squaring,

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

The rational witness

\[ \alpha=51/100,\quad \gamma=21/20,\quad\epsilon=1/100 \]

has exponent margin

\[ \gamma(1-2\epsilon)-2\alpha=9/1000. \]

The cross term between the finite polynomial and this remainder is negligible by Cauchy-Schwarz.

6. Independent window computation

independent_window_bound() does not call constant_window_bound. It:

1. reads the 40 rational coefficients from C2_EXTENDED.json;
2. integrates the triangular autocorrelation term by term;
3. rebuilds indices 41 through 101 from the Fock majorants;
4. rebuilds the two residue-class tails from the coarse ratio;
5. doubles the full absolute tail allowance.

It returns the same exact rational denominator upper bound as the theorem code and the positive lower value

\[ 0.0147728663285376\ldots. \]

This agreement uses a stored coefficient artifact rather than calling the coefficient derivation used by the first route.

7. Multiplicity and normalization

Let the distinct zeros have multiplicities m_j. The zero count with multiplicity is N_2=sum m_j, and the diagonal of the pair sum is sum m_j^2. For every positive integer m,

\[ \mathbf1_{m=1}\geq2m-m^2. \]

Summing gives

\[ N_{2,\mathrm{simple}} \geq2N_2-\sum_jm_j^2. \]

The squared-sinc physical kernel is nonnegative, bounded by one, and equals one on the diagonal. Hence the full pair sum bounds sum m_j^2 from above. After division by N_2, the lower proportion is exactly 2-D_beta(v).

The spike at alpha=0 supplies A_v(0). URMS2 is used for the regular part away from zero, with dominated passage supplied by the existing uniform RAMS2 majorant. The triangular autocorrelation vanishes at |alpha|=0.51, so no endpoint atom is omitted.

8. Remaining trust boundary

The audit found and closed the missing marked-cluster freezing argument. It also corrected a stale sentence in URMS1-CLOSURE.md that described the cross-range phase as log(mn); after the exact two-range transform, all finite terms belong to one log(m/n) polynomial.

The theorem package is now internally self-consistent and source-grounded. It has not received independent external mathematical review. That is the next promotion gate, not further bandwidth optimization.

Reproduction

.venv/bin/python hunts/higher_xi/urms2_051_audit.py
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py -n0