Disposition
Bandwidth update. BANDWIDTH-FORENSICS.md audits the constants used below. The rho<=1/10 and |alpha|<1/100 values in this report are the first explicit promotion, not final endpoints. A finite-prime peel permits an arbitrarily small Cauchy radius on the tail-prime Euler product. RAMS2 then holds on every fixed compact rho band, and the current contour architecture reaches every compact band inside |alpha|<1/2.
URMS2-051.md subsequently removes the strict half-band RC2 loss by retaining the exact logarithmic spacings in the weighted two-range polynomial. RAMS2 at the rational far-cutoff ratio 14/5 then supports the bridge through |alpha|<=51/100.
The level-two arithmetic obstruction has a smaller exact structure than a general graph gas.
After Borel transformation in Neumann order, the denominator inverse is a rank-one monomer-dimer model. Its only non-singleton squarefree atom is the two-prime term from A*A. After taking the Hadamard square, two matchings are superposed. Every connected component is therefore one of:
- an isolated vertex;
- an alternating path;
- an even alternating cycle, with a doubled edge as the size-two cycle.
No other connected graph occurs. This classification is exact at every support size.
The resulting all-support-size squarefree majorant is summable. Repeated prime supports are handled by the same Gaussian mixture, followed by the powerful-squarefree split used at level one. On the explicit coefficient band
\[ 0<\rho=\frac{\log x}{\lambda_T}\leq \frac1{10}, \qquad \lambda_T=\frac12\log\frac{T}{2\pi}, \]
the frozen level-two square density has scale x log x. The connected term changes the coefficient, beginning at pq; it does not change the logarithmic order. Its coefficient is the existing corrected infinite F2 object, because the cluster expression is algebraically identical to the rational Q(z) before either side is expanded.
This supplies RAMS2-Cluster on a fixed positive band. Combining it with the right-line estimates already isolated in URMS2-ATTACK.md and the early-smoothed transfer of URMS1-CLOSURE.md gives the same bridge architecture for level two on every compact 0<|alpha|<1/100. The endpoint is deliberately not optimized.
The band is far too narrow for the existing simplicity window, whose support reaches about 0.68213. No higher-xi percentage follows from this closure.
1. Exact inverse without coefficient-order truncation
Use the convolution atoms
\[ A=\Lambda,\qquad B=\Lambda\log, \]
and put
\[ D(z)=(1-Az)^2+Bz^2=1-X(z), \qquad X(z)=2Az-z^2(A*A+B). \]
Every coefficient at a fixed integer receives only finitely many convolution terms. Hence the formal inverse is exact:
\[ D(z)^{-1}=\sum_{m\geq0}X(z)^{*m}. \]
Separating B from (1-Az)^2 gives the first useful closed expansion:
\[ \boxed{ D(z)^{-1} =\sum_{j,k\geq0}(-1)^j {2j+k+1\choose k}z^{2j+k}A^{*k}*B^{*j}. } \]
This is an identity of formal Dirichlet series, not a geometric cutoff. It follows from
\[ \frac1{(1-Az)^2+Bz^2} =\sum_{j\geq0}(-1)^jB^{*j}z^{2j}(1-Az)^{-2j-2}. \]
For distinct primes p_1,...,p_r, write ell_i=log p_i. Evaluating the convolution powers on squarefree support gives
\[ \boxed{ b(p_1\cdots p_r) =\Bigl(\prod_i\ell_i\Bigr) \sum_{j=0}^r(-1)^j \frac{j!(r+j+1)!}{(2j+1)!} e_j(\ell_1,\ldots,\ell_r)z^{r+j}. } \]
The direct formula, subset-convolution inversion, and the cluster formula in the next section have zero symbolic defect through five independent prime labels in rams2_cluster.py.
2. The correct Borel transform
The level-one mechanism Borel-transforms the Neumann order before taking a Hadamard square. The same operation at level two is
\[ \widehat b_t(z) =\sum_{m\geq0}\frac{t^m}{m!}X(z)^{m} =\exp_(tX(z)), \]
with the inverse recovered by
\[ D(z)^{-1}=\int_0^\infty e^{-t}\widehat b_t(z)\,dt. \]
On a squarefree prime label p, define the monomer weight
\[ m_p(t)=tz\ell_p(2-z\ell_p), \]
and on two distinct labels define the dimer weight
\[ d_{pq}(t)=-2tz^2\ell_p\ell_q. \]
Then, for any finite set S of distinct primes,
\[ \boxed{ \widehat b_t(S) =\sum_{M\text{ matching on }S} \prod_{\{p,q\}\in M}d_{pq}(t) \prod_{p\notin V(M)}m_p(t). } \]
The formula is immediate from the convolution exponential. The generator X has squarefree support of size at most two, and the size-two part is exactly -z^2 A*A.
Laplace integration gives an equivalent finite formula:
\[ \boxed{ b(S)=z^{|S|}\prod_{p\in S}\ell_p \sum_{M} (|S|-|M|)!(-2)^{|M|} \prod_{p\notin V(M)}(2-z\ell_p). } \]
This is the requested replacement for level-one multiplicativity. The disconnected background is the monomer product. The new connected object is a dimer. Larger connected supports arise only after two matching systems are superposed in the Hadamard square.
There is also a useful rank-one Gaussian form. With a standard real Gaussian G, formal Gaussian moments give
\[ \boxed{ \widehat b_t(z) =\mathbb E_G\exp_*\left( [2tz+i\sqrt{2t}\,zG]A-tz^2B \right). } \]
For each fixed G, the exponent is supported on prime powers and its Dirichlet coefficients are multiplicative. The entire failure of the level-one Borel factorization is therefore one rank-one Gaussian covariance.
3. The unit test at 6
Let ell_p=log p and ell_q=log q. In the squarefree quotient, write
\[ D=1+f_px_p+f_qx_q+f_{pq}x_px_q, \]
where
\[ f_p=-2z\ell_p+z^2\ell_p^2, \quad f_q=-2z\ell_q+z^2\ell_q^2, \quad f_{pq}=2z^2\ell_p\ell_q. \]
Route 1: direct resolvent
The x_p x_q coefficient of D^-1 is
\[ -f_{pq}+2f_pf_q. \]
Thus
\[ \boxed{ b(pq)=6z^2\ell_p\ell_q -4z^3\ell_p\ell_q(\ell_p+\ell_q) +2z^4\ell_p^2\ell_q^2. } \]
Route 2: convolution recurrence
The exact recurrence
\[ b(n)=-\sum_{\substack{d\mid n\\d>1}}(D-1)(d)b(n/d) \]
gives
\[ b(p)=-f_p,\qquad b(q)=-f_q, \]
and then the same -f_pq+2f_pf_q expression at pq.
Route 3: Borel clusters
Before Laplace integration,
\[ \widehat b_t(pq)=m_p(t)m_q(t)+d_{pq}(t). \]
The Borel connected defect is therefore exactly
\[ \boxed{ \widehat\kappa_t(p,q)=-2tz^2\log p\log q. } \]
After the common t parameter is integrated,
\[ \boxed{ \kappa(p,q)=b(pq)-b(p)b(q) =z^2\ell_p\ell_q \left[2-2z(\ell_p+\ell_q)+z^2\ell_p\ell_q\right]. } \]
At p=2,q=3, this is the existing n=6 defect. The decomposition matters:
\[ \kappa(p,q) =z^2\ell_p\ell_q(2-z\ell_p)(2-z\ell_q) -2z^2\ell_p\ell_q. \]
The first term is covariance from sharing the Laplace parameter. That effect already exists in the level-one control and is handled by retaining the Borel parameters. The second term is the new A*A connected atom. Treating their sum as one undifferentiated failure hides the actual level-two obstruction.
4. The exact Q matching formula
Let mathcal L be the coefficient derivation
\[ (\mathcal Lf)(n)=\log(n)f(n). \]
It satisfies the convolution Leibniz rule, and
\[ \mathcal L X=2zB-z^2(2A*B+C), \qquad C=\Lambda\log^2. \]
Hence the corrected arithmetic object can be written
\[ \begin{aligned} Q(z) &=-A+(\mathcal LX)D^{-1}\\ &=-A-\mathcal L\log_*D. \end{aligned} \]
This is the same rational object already stored in CORRECTED-F2.md:
\[ Q(z)=-A+ \frac{2Bz-(2A*B+C)z^2}{(1-Az)^2+Bz^2}. \]
Since
\[ \mathcal L\exp_*(tX)=t(\mathcal LX)\exp_*(tX), \]
the squarefree coefficient has the finite rooted matching formula
\[ \boxed{ Q(S)=-A(S)+\log\!\left(\prod_{p\in S}p\right) z^{|S|}\prod_{p\in S}\ell_p \sum_M(|S|-|M|-1)!(-2)^{|M|} \prod_{p\notin V(M)}(2-z\ell_p). } \]
For |S|>1, the -A(S) term vanishes. This formula shows directly why the full arithmetic coefficient has one global logarithmic mark, just as the level-one identity alpha_k(n)=log(n)Lambda_k(n)/k does.
5. Connected graphs in the Hadamard square
Use independent Borel parameters t,u. A term in
\[ \widehat b_t(S)\widehat b_u(S) \]
is a pair (M_t,M_u) of matchings. Color their edges by the parameter. Each vertex has degree at most two, with at most one incident edge of each color. Consequently every nontrivial connected component is an alternating path or an even alternating cycle.
Put
\[ h_p=2-z\ell_p, \qquad \mu(C)=z^{2|C|}\prod_{p\in C}\ell_p^2. \]
For a fixed labelled prime set C of size r, the sum of every connected two-color configuration is as follows.
For odd r>=3, only paths occur:
\[ K(C)=(-2)^{r-1}2(r-2)!(tu)^{(r+1)/2}\mu(C) \sum_{\{p,q\}\subset C}h_ph_q. \]
For even r>=2, paths contribute
\[ K_{\rm path}(C)=(-2)^{r-1}(r-2)!(tu)^{r/2}(t+u)\mu(C) \sum_{\{p,q\}\subset C}h_ph_q, \]
and cycles contribute
\[ K_{\rm cycle}(C)=2^r(r-1)!(tu)^{r/2}\mu(C). \]
The size-two cycle is the doubled edge selected in both matchings. Direct enumeration of all matching pairs agrees symbolically with these formulas for every labelled support size through six. This includes a nonzero size-three path, so discarding clusters above size two is not an admissible simplification.
The decomposition of a pair of matchings into union components is unique. It supplies the linked-cluster functional without importing a generic graph ansatz. A Mayer expansion can be applied after singleton components are factored, but it is unnecessary for the square-density bound because the matching pair can be summed directly.
6. Uniform squarefree cluster bound
Assume 0<=z log n<=rho_0. On the present positive band,
\[ |2-z\log p|\leq2. \]
For support size r, the number of matchings with a edges is
\[ N(r,a)=\frac{r!}{2^a a!(r-2a)!}. \]
The absolute one-color rooted matching mass in Q(S) is bounded by
\[ 2^rS_r, \qquad S_r=\sum_{0\leq a\leq r/2} N(r,a)2^{-a}(r-a-1)!. \]
The ratio of consecutive summands is
\[ \frac{(r-2a)(r-2a-1)}{4(a+1)(r-a-1)} \leq\frac{r}{4(a+1)}. \]
Therefore
\[ S_r\leq(r-1)!e^{r/4}. \]
The order-uniform prime-simplex majorant used in URMS1-CLOSURE.md gives, for an absolute D,
\[ \sum_{\substack{p_1\cdots p_r\leq x\\p_i\text{ distinct}}} \left(\log(p_1\cdots p_r)\right)^2 \prod_i(\log p_i)^2 \leq \frac{D^r x(\log x)^{2r+1}}{r!(2r-1)!}. \]
Combining the last three displays bounds the complete squarefree support-size r contribution by
\[ x\log x\;c_r(D\rho^2)^r, \]
where the exact finite matching coefficient is
\[ \boxed{ c_r=\frac{4^rS_r^2}{r!(2r-1)!} } \]
and the simpler bound is
\[ c_r\leq \frac{4^re^{r/2}(r-1)!}{r(2r-1)!}. \]
The successive ratio of the right side is
\[ 4e^{1/2}\frac{r}{2(r+1)(2r+1)}=O(r^{-1}). \]
Thus
\[ \sum_{r\geq1}c_r(D\rho^2)^r \]
converges for every fixed rho; in particular it is uniform on the stated band. The first exact c_r values are
\[ 4,\ 3,\ \frac{49}{45},\ \frac{289}{840},\ \frac{1681}{18900},\ \frac{11}{560},\ldots \]
and are emitted by rams2_cluster.py.
This bound sums all pairs of matchings. It does not rely on cancellation of their signs. Taking absolute values is safe after the prime-support density has been retained; taking them before that support split recreates the false x(log x)^2 dense-support bound.
7. Repeated primes
The Gaussian representation keeps prime powers local. At one prime, put ell=log p. Conditional on G, the local Euler series is
\[ \boxed{ \sum_{e\geq0}\widehat b_t(p^e)u^e =\exp\left( [2tz+i\sqrt{2t}\,zG]\ell\frac{u}{1-u} -tz^2\ell^2\frac{u}{(1-u)^2} \right). } \]
Coefficient extraction, Gaussian moments, and the Laplace integral reproduce the direct Q(z) local series at p^e for 1<=e<=4 symbolically in the control code. The identity itself holds for every e by the displayed local Euler series.
Write each integer uniquely as n=qm, where q is powerful, m is squarefree, and (q,m)=1. The conditional Gaussian coefficient factors as
\[ \widehat b_t(qm;G)=\widehat b_t(q;G) \prod_{p\mid m}\widehat b_t(p;G). \]
For the local coefficient at q, Cauchy's estimate on |u|=3/4 gives
\[ |\widehat b_t(q;G)| \leq \left(\frac43\right)^{\Omega(q)} \exp\left( 3|2tz+i\sqrt{2t}zG|\log q +12tz^2(\log q)^2 \right). \]
When z log q<=rho<=1/10, averaging the Gaussian absolute exponential costs at most 2 exp(9rho^2 t). The remaining Laplace exponent is bounded by
\[ (6\rho+21\rho^2)t\leq0.81t. \]
The residual e^{-0.19t} supplies the same factorial moment bounds as in the squarefree matching calculation, with a larger absolute constant per support vertex. For two Hadamard copies, the powerful weight is dominated by
\[ \left(\frac{16}{9}\right)^{\Omega(q)}. \]
Its harmonic mass is finite:
\[ \sum_{q\text{ powerful}} \frac{(16/9)^{\Omega(q)}}q =\prod_p\left( 1+\sum_{e\geq2}\frac{(16/9)^e}{p^e} \right)<\infty. \]
For every fixed q>1, the local coefficient tends to zero with z, because each prime in its support requires at least one factor from the exponent. Dominated convergence against the last harmonic majorant makes every mixed q>1,m>1 stratum o(x log x), uniformly on compact subbands of rho<=1/10.
The pure powerful stratum m=1 is smaller directly. Powerful integers up to x are O(sqrt(x)), and the existing exact level-two envelope on rho<1/2 gives |Q(q)|<=C_rho log q. Its square mass is
\[ O_\rho(\sqrt{x}(\log x)^2)=o(x\log x). \]
Thus repeated primes do not create a second main logarithm.
The displayed R=3/4 estimate is not a radius barrier. Fix any finite rho_0. Choose a tail radius 0<R<1 for which
\[ 2\frac{R}{1-R}\rho_0 +\left(\frac{R}{(1-R)^2}+\frac{R^2}{(1-R)^2}\right)\rho_0^2<1/2. \]
Peel off the finitely many primes p<=R^-2. For p>R^-2, the powerful Hadamard mass
\[ \prod_{p>R^{-2}}\left(1+ \sum_{e\geq2}\frac{R^{-2e}}{p^e}\right) \]
converges. Give each peeled prime its own radius R_p>p^-1/2; its exponent sum converges, and the total logarithm of the peeled set is fixed. Since z tends to zero, its extra Laplace rate is o(1) and fits in the remaining half-gap. The preceding dominated-convergence argument therefore holds on every fixed compact rho band.
8. RAMS2-Cluster
For every fixed finite
\[ 0<\rho_{\min}\leq\rho\leq\rho_0, \]
the frozen exact level-two coefficient family satisfies
\[ \boxed{ \mathcal A_{2,T}(x) =\sum_{n\leq x}|a_{2,T}(n)|^2 =x\log x\,\Phi_2(\rho)+o(x\log x), } \]
locally uniformly in rho.
For each fixed support size, the prime-simplex asymptotic gives the corresponding coefficient of Phi_2. The summable c_r majorant permits the support-size limit to pass through the series. The powerful strata vanish by the preceding dominated-convergence argument.
The cluster representation and the rational Q(z) are the same formal Dirichlet object. Therefore the resulting main series is
\[ \boxed{ \Phi_2(\rho)=\frac2\rho\,\mathcal F_2(\rho/2), } \]
with the value at zero taken by continuity. This is an identity of the exact coefficient mechanisms, not a fit to finite data. Expanding it recovers the corrected coefficient fixture through all 40 stored orders; coefficient two is the corrected value already pinned by C2_EXTENDED.json.
The connected A*A term enters this Phi_2. It is a leading-coefficient correction, not an o(x log x) term. What vanishes is the repeated-prime part after the powerful-squarefree split.
9. RC2 parameter choice and the resulting band
The right-line Wiener inverse, Hilbert multiplier, height derivative, and long-tail estimates for the full level-two resolvent are already recorded in URMS2-ATTACK.md. The early-smoothed Cauchy transfer in URMS1-CLOSURE.md depends only on those estimates and on a uniform RAMS input through the upper cutoff Y=H^beta.
One explicit non-endpoint choice is
\[ \alpha_0=\frac1{100},\qquad \delta=\frac12,\qquad \beta=\frac{11}{500}. \]
It satisfies
\[ \frac{2\alpha_0}{\delta}=0.04<0.1, \qquad \frac{2\beta}{\delta}=0.088<0.1, \qquad 2\alpha_0=0.02<\beta=0.022. \]
Choosing any sufficiently small positive contour epsilon also gives
\[ 2\alpha_0<\beta(1-2\epsilon). \]
These are exactly the three inequalities used in the level-one dyadic assembly: RAMS at the target range, RAMS at the upper cutoff, and decay of the weighted upper tail. The central segment, height-freezing commutator, archimedean derivatives, Montgomery-Vaughan off-diagonal terms, cross-range phase, and initial height interval retain the same bounds. The level-two numerator and denominator remain in their exact resummed form.
Under the hypotheses already stated for the zero-side bridge, this gives local uniformity on
\[ \boxed{0<|\alpha|<1/100.} \]
The regular limiting expression is the corrected mathcal F_2(alpha).
10. Falsification controls
The permanent controls now include:
- direct inverse, subset-convolution, and Borel-cluster agreement;
- arbitrary symbolic
pqand the specializationn=6; - isolation of the Borel pair defect
-2tz^2 log(p)log(q); - enumeration of every two-color matching pair through six labelled primes;
- the alternating-path/even-cycle closed formula;
- nonzero size-three connected support;
- repeated-prime local agreement through
p^4; - exact all-support-size majorant coefficients and ratios;
- the pre-existing rational
Q(z)identity and all 40 corrected coefficients.
The requested lesions separate as follows:
- deleting
A*Asets the Borelpqdefect to zero; - forcing multiplicativity misses the displayed
kappa(p,q); - deleting connected supports above size two misses the size-three path;
- taking a dense all-integer envelope before the support split restores the extra logarithm;
- a planted family with support coefficient
r!in place ofc_rhas nonsummable fixed-band mass.
Command:
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py -n011. Source boundary
The path/cycle classification and matching bounds are derived directly from the exact arithmetic weights. Generic polymer machinery is not needed for the main estimate.
For comparison, the hard-core polymer log can be expanded with the Kotecky-Preiss framework, and Penrose tree identities can replace connected polymer graphs by trees. Those tools apply only after the arithmetic activities above are supplied. They do not create the z, prime-simplex, or powerful-support savings. The direct matching sum is sharper here because the rank-one quadratic atom restricts the physical components to paths and cycles before any tree bound is taken.
Primary references for the generic comparison:
- R. Kotecky and D. Preiss, Cluster expansion for abstract polymer models, Communications in Mathematical Physics 103 (1986), 491-498, DOI
10.1007/BF01211762. - R. Fernandez and A. Procacci, Cluster expansion for abstract polymer models. New bounds from an old approach, Communications in Mathematical Physics 274 (2007), 123-140, arXiv
math-ph/0605041.
Neither reference supplies the arithmetic theorem. The load-bearing inputs remain the exact rank-one reduction, the prime-simplex estimate already used at level one, and the powerful harmonic majorant above.