Disposition
This phase reaches Outcome B: level one repaired, level two blocked.
Assuming RH, the exact resummed level-one arithmetic square density has the x log x scale on every compact coefficient band r<1. An early-smoothed right-line contour argument then gives the level-one zero-statistics bridge on every compact
\[ 0<|\alpha|<\frac14. \]
The limiting regular expression agrees with the infinite Farmer-Gonek-Lee coefficient object. The printed finite-order contour argument is not used.
The level-two lift stops at a new, exact obstruction. Its frozen denominator
\[ (1-Az)^2+Bz^2 \]
contains the connected two-prime atom A*A already at n=6. The level-one Borel transform is multiplicative; the level-two transform is not. A uniform Hadamard cluster estimate for this connected atom is the named missing input RAMS2-Cluster. No URMS2 or statement about actual zeros of xi'' is claimed.
1. Normalization
Put
\[ \lambda_T=\frac12\log\frac{T}{2\pi},\qquad z_T=\lambda_T^{-1}, \]
and define the full frozen level-one coefficient family
\[ a_T(n)=-\Lambda(n)+ \sum_{k\geq1}z_T^k\alpha_k(n), \qquad \alpha_k=\Lambda_{k-1}*(\Lambda\log). \]
The sum terminates at k=Omega(n). No geometric order is chosen.
For y>=2, write
\[ \mathcal A_T(y)=\sum_{n\leq y}|a_T(n)|^2, \qquad r_y=\frac{\log y}{\lambda_T}. \]
The coefficient band r_y<1 corresponds to y<T^(1/2+o(1)).
2. RAMS1 theorem
For every fixed 0<r_min<=r_0<1, uniformly for r_min<=r_y<=r_0,
\[ \boxed{ \mathcal A_T(y)= y\log y\,\Phi_1(r_y)+o(y\log y), } \]
where
\[ \Phi_1(r)=1-2r+ 2\sum_{j\geq1}\frac{(j-1)!}{(2j)!}r^{2j}. \]
The series is entire in r. The restriction r_0<1 enters the uniform resolvent bounds, not the limiting coefficient series.
2.1 Exact coefficient symmetry
The algebraic starting point is
\[ \alpha_k(n)=\frac{\log n}{k}\Lambda_k(n). \]
It follows by marking one of the k identical convolution positions. This identity is checked independently by formal polynomials in rams1.py.
2.2 Unique support split
Every integer has a unique factorization
\[ n=qm, \]
where q is powerful, every prime exponent in q is at least two, m is squarefree, and (q,m)=1. Put j=omega(m).
Let
\[ B_{q,j}(z)= \sum_{h\geq0}\binom{j+h-1}{h}z^h\Lambda_h(q). \]
This is the coefficient at q in (1-zA)^(-j), with A(s)=sum Lambda(n)n^(-s). For j>=1, the positive part of a_T(qm) factors exactly as
\[ \boxed{ \sum_{k\geq1}z^k\alpha_k(qm) =(j-1)!z^j\log(qm) \prod_{p\mid m}\log p\,B_{q,j}(z). } \]
powerful_split_coefficient_defects() checks this formula on every eligible integer through 55.
2.3 Squarefree main term
For q=1, B_(1,j)=1. The weighted squarefree prime-simplex asymptotic is
\[ \sum_{\substack{m\leq y\\m\text{ squarefree}\\\omega(m)=j}} (\log m)^2\prod_{p\mid m}(\log p)^2 \sim \frac{y(\log y)^{2j+1}}{j!(2j-1)!}. \]
After multiplication by (j-1)!^2 z^(2j), this gives
\[ 2\frac{(j-1)!}{(2j)!}y\log y\,r_y^{2j}. \]
Chebyshev's estimate for sum_(p<=e^u)(log p)^2 gives the order-uniform majorant
\[ \frac{D^j y(\log y)^{2j+1}}{j!(2j-1)!} \]
for an absolute D. Its normalized successive-term ratio is asymptotic to D r_0^2/(4j). Dominated convergence therefore sums all depths uniformly.
The j=1 prime stratum includes the explicit -Lambda term and gives
\[ y\log y(1-2r_y+r_y^2)+O(y). \]
2.4 Mixed repeated-prime strata vanish
For q<=y and r_y<=r_0<1, the elementary convolution bound gives
\[ 0\leq B_{q,j}(z) \leq\sum_{h\geq0}\binom{j+h-1}{h}r_0^h =(1-r_0)^{-j}. \]
For each fixed powerful q>1, B_(q,j)(z_T) tends to zero because its constant coefficient vanishes.
Every powerful integer has a representation
\[ q=a^2b^3, \]
with b squarefree. Consequently
\[ \sum_{q\text{ powerful}}\frac1q \leq\zeta(2)\zeta(3)<\infty. \]
The squarefree prime-measure majorant and dominated convergence now give
\[ \sum_{\substack{qm\leq y\\q>1,\ j\geq1}} |a_T(qm)|^2=o(y\log y) \]
uniformly on the compact band. The remaining j=0 powerful integers are at most O(sqrt(y)) in number, while
\[ |a_T(q)|\leq C_{r_0}\log q. \]
Their total is O_(r_0)(sqrt(y)(log y)^2)=o(y log y).
This completes the full frozen arithmetic asymptotic. The extra logarithm in the old majorant came from charging a prime-supported logarithmic envelope to all integers.
2.5 Height-dependent coefficients
On t in [T,2T], the archimedean logarithm differs from lambda_T by O(1) and its reciprocal differs by O(lambda_T^(-2)). Inside r<=r_0, differentiating the exact resolvent gives
\[ |a_1(n,t)-a_T(n)|\leq C_{r_0}. \]
The square sum of this difference is O(y). Cauchy-Schwarz with the frozen RAMS1 bound makes the cross term O(y sqrt(log y))=o(y log y). RAMS1 therefore holds for the exact height-dependent family as well.
3. RC1 without the false contour tail
Assume RH. Fix 5/4<sigma<2, put s=sigma+it, and choose the right contour line w=c+iv with c=1/2+epsilon and 0<epsilon<1/8.
The exact zero-side contour identity is retained before inserting any Dirichlet expansion. Its right-line arithmetic term contains
\[ k(w,s)=\frac{2\sigma-1} {(w-(s-1/2))(w-(1/2-\bar s))}. \]
Put
\[ A=\sigma-\frac12-c, \qquad B=\sigma-\frac12+c. \]
For u=v-t, the exact partial fraction is
\[ k(c+i(t+u),s)= \frac1{-A+iu}-\frac1{B+iu}. \]
The Fourier transforms of these two terms are one-sided exponentials. After the extra n^(-1/2) in the logarithmic-derivative series is included, they select exactly
\[ (x/n)^{1-\bar s}\quad(n\leq x), \qquad (x/n)^s\quad(n>x). \]
The symbolic defects of the partial fraction and both exponents are zero in rams1.py.
Thus the two-range transform is obtained by Fourier localization on the original right line. No coefficient is continued across the strip, no finite geometric polynomial is introduced, and no artificial poles at zeros of L are crossed.
4. RC1 error decomposition
Let
\[ \mathcal D_{T,x}(s)=x^{-1/2}\left( \sum_{n\leq x}a_T(n)(x/n)^{1-\bar s} +\sum_{n>x}a_T(n)(x/n)^s \right). \]
After the explicit archimedean and pole contribution P_(T,x) is removed, the early-smoothed contour identity gives
\[ Z_{1,x}(s)=P_{T,x}(s)+\mathcal D_{T,x}(s)+E_{T,x}(s). \]
For every compact 0<alpha_min<=alpha<=alpha_0<1/4, with x=T^alpha,
\[ \boxed{ \int_T^{2T}|E_{T,x}(\sigma+it)|^2dt=o(T\log T) } \]
uniformly in alpha. The error classes are as follows.
4.1 Central right-line segment
The exact resolvent is used on the bounded central segment where its Dirichlet series is not inserted. Since t is of size T, the Cauchy kernel has quadratic separation. Its contribution is
\[ O_\epsilon(x^{1/2+\epsilon}/T) \]
pointwise, hence negligible in the displayed square mean.
4.2 Height-freezing commutator
The exact weighted Wiener derivative of the right-line inverse is O(1/T) on [T/2,3T]. Before taking absolute values, subtract the frozen coefficient inside the Cauchy integral. The local first-moment kernel mass is exactly
\[ \int_{-R}^{R}\frac{|u|}{1+u^2}du=\log(1+R^2). \]
With R proportional to T, the commutator is
\[ O_\epsilon(x^{1/2+\epsilon}\log T/T) \]
pointwise. The complement of the local height interval uses the quadratic Cauchy tail and is smaller. This is the cancellation lost by the historical absolute contour estimate.
4.3 Archimedean derivative
The omitted L'/(L+g) part has weighted Wiener norm O(1/(T log T)). Its convolution satisfies the same commutator estimate and is negligible separately.
4.4 Infinite upper Dirichlet range
Choose constants
\[ 2\alpha_0<\beta<\frac12 \]
and then choose epsilon>0 small enough that
\[ 2\alpha_0<\beta(1-2\epsilon). \]
RAMS1 applies through Y=T^beta. On the fixed right line, the full inverse has bounded weighted Hilbert norm. Therefore
\[ \sum_{n>Y}|a_T(n)|^2n^{-3} \ll_\epsilon Y^{-1+2\epsilon}. \]
After multiplication by x^2, this is o(log T) by the displayed choice of parameters. This is the step that currently restricts the clean RC1 band to alpha<1/4 rather than the full RAMS1 coefficient band alpha<1/2.
4.5 Dirichlet mean-value off-diagonal
Truncate the upper range at Y. Montgomery-Vaughan applies to the combined two-range polynomial. The lower-range relative error is O(x/T) and the generic upper-range error is O(Y/T). Both vanish on the band used here. The cross-range terms have the same log(m/n) phase and are already included in the arbitrary-coefficient Montgomery-Vaughan bound. No fixed convolution depth enters these estimates. URMS2-051-AUDIT.md records the exact phase calculation and the sharper individual-spacing form.
4.6 From dyadic height to the source normalization
The local estimate above is stated on [T,2T], while the source statistic counts ordinates up to one height H. Fix alpha_0<1/4 and choose
\[ 4\alpha_0<\delta<1, \qquad 2\alpha_0<\beta<\frac\delta2. \]
Then take epsilon>0 small enough that 2 alpha_0<beta(1-2 epsilon).
The interval [H^delta,H] is partitioned dyadically. On a block of height U>=H^delta, with x=H^alpha, both required coefficient ratios satisfy
\[ \frac{\log x}{\tfrac12\log U} \leq\frac{2\alpha_0}{\delta}<1, \qquad \frac{\log H^\beta}{\tfrac12\log U} \leq\frac{2\beta}{\delta}<1. \]
Thus the same RAMS1 and upper-tail bounds hold on every block, and their geometric sum has the stated o(H log H) size. The omitted initial interval [0,H^delta] contributes O(H^delta(log H)^3) by the standard local zero count and the integrability of the Cauchy weight. This is o(H log H).
The main terms have the top-height normalization as well. For each fixed dyadic offset, U=H/2^j satisfies log U/log H -> 1, so its coefficient ratio tends to 2 alpha. Blocks with unbounded offset have geometrically vanishing total length. The uniform RAMS1 majorant permits dominated summation over the blocks. Hence the assembled regular term is mathcal F_1(alpha), not a block-dependent average.
The inequalities selecting delta,beta are simultaneously feasible exactly for the present argument when alpha_0<1/4. This supplies the source's 0<gamma,gamma'<=H normalization rather than only a dyadic surrogate.
5. Rebuilt level-one bridge
Combining RAMS1 and RC1 with the standard zero-side square identity gives the following independent level-one statement.
URMS1 theorem
Assume RH. For every compact K subset (0,1/4), locally uniformly for alpha in K,
\[ F_1(\alpha,T)= (1+o(1))T^{-2\alpha}\log T+\mathcal F_1(\alpha)+o(1), \]
where
\[ \boxed{ \mathcal F_1(\alpha)= \alpha-4\alpha^2+ \sum_{j\geq1}\frac{(j-1)!}{(2j)!}(2\alpha)^{2j+1}. } \]
Symmetry supplies negative alpha. The convergence mode is local uniformity away from the spike at zero. The spike comes from the explicit archimedean term; the regular expression is the Stieltjes transform of Phi_1.
This agrees exactly with the infinite Farmer-Gonek-Lee coefficient prediction. It does not validate their printed fixed-order remainder. The new argument uses the full resolvent throughout.
6. Why the lift stops at level two
The frozen level-two arithmetic resolvent is
\[ -A+\frac{2Bz-(2A*B+C)z^2} {(1-Az)^2+Bz^2}. \]
At level one, the Borel family
\[ E_\tau=\exp(\tau A) \]
is multiplicative, and the powerful-squarefree split isolates every repeated prime correction in a finite harmonic-mass kernel.
At level two the denominator contains
\[ A*A+B. \]
The A*A coefficient at 6=2*3 is nonzero. It couples two distinct primes inside one denominator atom. Consequently the exponential or Laplace resummation is not multiplicative even on coprime inputs. The smallest exact defect is already recorded by level_two_multiplicativity_defect().
The missing level-two theorem is:
RAMS2-Cluster
Construct a connected-cluster expansion for the Hadamard square of the full inverse of (1-Az)^2+Bz^2 and show that every connected cluster containing a repeated or coupled-prime atom contributes o(x log x) uniformly on one fixed band 0<alpha<=alpha_0<1/4.
Equivalently, produce a summable majorant for the connected two-prime local kernel which replaces the multiplicative powerful-number harmonic bound used at level one.
No searched fixed-product mean-value theorem supplies this resummed connected cluster estimate. RC2 can reuse the early-smoothed Cauchy architecture once RAMS2-Cluster is available, but the arithmetic gate comes first.
7. Hostile controls
The closure passes the following separations:
- Replacing the Cauchy convolution by the historical pointwise tail estimate still fails on the permanent unit-interval counterexample.
- Taking absolute values before the height commutator loses the
log(T)/Tfactor. - The dense-support toy retains true
x(log x)^2growth. - Removing prime powers changes only a lower-order stratum.
- Truncating convolution depth is unnecessary and is not used.
- The upper Dirichlet tail exposes the
alpha=1/4endpoint rather than silently extending the theorem toalpha=1/2. - The level-two coprime defect prevents reuse of the multiplicative proof.
8. Trust boundary and commands
Exact algebra and finite combinatorics are checked by:
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py -n0The exact layer identities, powerful-squarefree decomposition, Borel multiplicativity, Cauchy partial fraction, two-range exponents, and planted failures are in that suite.
The prime-measure asymptotic, dominated-convergence passage, contour edge bounds, and Montgomery-Vaughan estimate are classical analytic inputs. The argument above states where each enters and does not ask Python to supply an asymptotic limit.
9. Pinned sources
- Repository state at the start of this closure:
08a61240e7dc1008ffb67e30bf05f23fe9ed45b4. - Farmer, Gonek, and Lee, arXiv:0803.0425 (https://arxiv.org/abs/0803.0425), used for the finite-height normalization, zero-side square identity, and fixed-depth prime asymptotics, not for its disputed contour remainder.
- Montgomery and Vaughan, Hilbert's inequality (https://doi.org/10.1112/jlms/s2-8.1.73), used for the frozen finite Dirichlet-polynomial mean value.
- Farmer-Gonek source archive SHA-256:
f6cdc7b71db06187dac655647e24312843441aa2598e41a3b53834dd1b36822f. - Farmer-Gonek TeX SHA-256:
a3a9ac955a9c95d10d36d6b05aae05a79c6408f6e1cdcb8c14fd79f000144fb1.
No higher-derivative percentage or zeta rank-trace transfer is opened by this result.