Level-two bandwidth forensics

Half-band update. URMS2-051.md retains the individual log n spacings and the exact two-range coefficient weights in the finite mean square. Its off-diagonal error is O(x log x), independent of the far cutoff length. This removes gamma<delta, reaches bandwidth 0.51, and opens the first corrected conditional xi-double-prime simplicity statement.

Disposition

The promoted endpoint 1/100 was not a theorem boundary. It was the result of one convenient dyadic parameter choice.

Four successive ranges are now separated.

1. The proof as written in RAMS2-CLUSTER.md, with its tail split unchanged, works on every compact band

\[ 0<|\alpha|<\frac1{40}. \]

1. Inserting the already available pointwise coefficient envelope between the RAMS2 range and the far Hilbert tail extends the bridge to

\[ 0<|\alpha|<\frac1{20}. \]

1. The repeated-prime estimate in RAMS2-CLUSTER.md fixed the local Cauchy radius at 3/4. Optimizing that radius, without changing the cluster representation, extends RAMS2 to every

\[ 0<\rho<\rho_*, \qquad \rho_*=0.1454524603084045\ldots, \]

and the bridge to every compact

\[ \boxed{0<|\alpha|<\rho_*/2 =0.0727262301542022\ldots.} \]

The round milestone |alpha|<0.07 has an entirely rational parameter witness.

1. The apparent condition R>1/sqrt(2) comes only from applying the same Cauchy radius to the prime 2 and the entire Euler product. Peel off the finitely many primes p<=R^-2, then use radius R on the remaining primes. Since E_R(rho) tends to zero with R, RAMS2 holds on every fixed compact rho band. The current contour architecture consequently reaches every compact band

\[ \boxed{0<|\alpha|<\frac12.} \]

The first requested checkpoint not reached is now 0.50, as a strict endpoint. The binding conditions are 2alpha<gamma for the far tail and gamma<delta<1 for the finite mean value. The squarefree path/cycle series and the powerful support no longer bind at any fixed rho.

No corrected xi-double-prime simplicity theorem follows on the enlarged band. Cauchy-Schwarz, an exact operator-norm bound, and completion of squares exclude every admissible spectral-factor window through bandwidth 1/2. A constant window works at 0.51. Current exact downstream bounds are

\[ \boxed{\frac12\leq\beta_{\rm useful} \leq\frac{51}{100}.} \]

The upper bound uses the normalized constant window at bandwidth 0.51 and gives the exact positive conditional lower value

\[ 0.0147728663285376\ldots. \]

Thus the next useful target is not the historical 0.68213. The current bridge ends exactly where a useful window first becomes possible. The first exact candidate is at 0.51. Reaching it requires a stronger far-tail norm or a long-polynomial mean-value replacement, as stated in section 11.

1. The map from alpha to rho

Let the source height be H, the Fourier scale be

\[ x=H^\alpha, \]

and discard the initial interval below H^delta, where 0<delta<1. On a dyadic block of height U>=H^delta, the frozen archimedean parameter is

\[ \lambda_U=\frac12\log\frac{U}{2\pi}. \]

Therefore the coefficient ratio entering RAMS2 is

\[ \rho_x(U)=\frac{\log x}{\lambda_U} =\frac{2\alpha\log H}{\log U+O(1)}. \]

At top height U comparable with H, this tends to 2 alpha. Uniformly on all retained dyadic blocks,

\[ \boxed{\rho_x(U)\leq\frac{2\alpha}{\delta}+o(1).} \]

There is no factor ten in this identification. The arithmetic band rho<0.1 naturally permits alpha<0.05 before any tail estimate is imposed.

If an upper arithmetic cutoff is Y=H^beta, its coefficient ratio is

\[ \rho_Y(U)\leq\frac{2\beta}{\delta}+o(1). \]

If a second envelope cutoff is W=H^gamma, its ratio is similarly at most 2 gamma/delta.

2. Dependency graph

bandwidth_forensics.py emits the same graph as machine-readable records.

3. Where 0.1 became 0.01

The previous report chose

\[ \alpha_0=0.01,\qquad \delta=0.5,\qquad \beta=0.022. \]

The inequalities checked there were

\[ \frac{2\alpha_0}{\delta}=0.04<0.1, \qquad \frac{2\beta}{\delta}=0.088<0.1, \qquad 2\alpha_0=0.02<\beta=0.022. \]

No lemma states alpha<0.01. The first appearance of that number is this parameter assignment.

Keeping the same old tail split, the existence conditions are

\[ 2\alpha<\beta(1-2\epsilon), \qquad \beta<\frac{\rho_R\delta}{2}, \qquad \delta<1. \]

Letting epsilon tend to zero shows that parameters exist exactly when

\[ 4\alpha<\rho_R\delta<\rho_R. \]

For rho_R=0.1, the natural endpoint of the unchanged architecture is

\[ \boxed{\alpha<0.025.} \]

As a minimal witness that 0.01 was not a boundary, alpha=0.011 admits

\[ \delta=\frac{18}{25},\qquad \beta=\frac{29}{1000},\qquad \epsilon=\frac7{116}. \]

The exact positive margins are

\[ \frac5{72},\qquad \frac7{360},\qquad \frac7{2000} \]

for the target RAMS ratio, cutoff RAMS ratio, and far-tail exponent.

Classification of 1/100: D, a safety and convenience choice. The 1/40 endpoint that replaces it in the unchanged proof is produced by B/C, a sufficient non-sharp tail inequality obtained by applying the Hilbert norm immediately after the RAMS cutoff.

4. Continuation scan

The table distinguishes the report as previously promoted, the old lemmas with parameters reselected, the intermediate-shell improvement, the uniform Cauchy-radius optimization, and the finite-prime split.

The first failed requested checkpoint after every current optimization is 0.50.

5. The binding RC2 estimate

At the standard smoothing weight

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

the Cauchy evaluation point is sigma=3/2. The upper Dirichlet range has square weight x^2n^-3. The old proof placed the Hilbert bound immediately after Y:

\[ \sum_{n>Y}|a(n)|^2n^{-3} \ll Y^{-1+2\epsilon}. \]

Multiplication by x^2 forces Y beyond x^2, which is the condition beta>2alpha.

Changing sigma would change the zero-pair smoothing weight. It is therefore not a free optimization for the stated F2; the standard weight fixes this part of the calculation.

6. The intermediate-shell repair

Insert a second cutoff

\[ x<Y=H^\beta<W=H^\gamma. \]

RAMS2 is used only through Y. Between Y and W, use the already proved pointwise level-two envelope. On every compact coefficient band below rho=1/2, it gives

\[ \mathcal A_{2,U}(u)\leq C u(\log u)^2. \]

Partial summation gives

\[ \begin{aligned} \int_{Y-}^{W}u^{-3}\,d\mathcal A_{2,U}(u) &\leq C Y^{-2}\left[ \frac32(\log Y)^2+rac32\log Y+\frac34 \right]. \end{aligned} \]

After multiplication by x^2, this shell is negligible when

\[ \alpha<\beta, \]

not only when 2alpha<beta.

Beyond W, retain the Hilbert estimate:

\[ x^2\sum_{n>W}|a(n)|^2n^{-3} \ll x^2W^{-1+2\epsilon}. \]

It is negligible when

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

The pointwise envelope permits 2gamma/delta<1/2. This far condition allows alpha<delta/8, so it is nonbinding while RAMS2 has rho_R=0.1.

The complete parameter conditions reduce to

\[ \alpha<\beta<\frac{\rho_R\delta}{2}, \qquad 2\alpha<\gamma(1-2\epsilon), \qquad \frac{2\gamma}{\delta}<\frac12, \qquad \delta<1. \]

For rho_R=0.1, they are feasible exactly on compact bands

\[ \boxed{\alpha<0.05.} \]

At alpha=0.049, one exact witness is

\[ \delta=\frac{99}{100},\quad \beta=\frac{197}{4000},\quad \gamma=\frac{691}{4000},\quad \epsilon=\frac1{16}. \]

Every bandwidth-bearing margin is positive; the smallest is 1/4000 in beta-alpha.

This is a level-two bridge improvement. No cluster estimate was changed to obtain it.

7. The apparent next binding estimate: powerful Borel/Laplace mass

After the shell repair, the target RAMS2 condition 2alpha/delta<rho_R binds. The squarefree cluster majorant does not. Its support-size ratio tends to zero for every fixed rho, and the exact low components remain:

• size two: the pq connected atom and doubled edge;
• size three: alternating paths;
• size four: alternating paths and cycles.

The restriction comes from the repeated-prime Cauchy estimate. On a local circle |u|=R, define

\[ a(R)=\frac{R}{1-R}, \qquad b(R)=\frac{R}{(1-R)^2}. \]

The powerful local factor has Laplace exponent bounded by

\[ \boxed{ \mathcal E_R(\rho) =2a(R)\rho+[b(R)+a(R)^2]\rho^2. } \]

If one radius is imposed at every prime, the Hadamard powerful harmonic mass requires R^-2<2, hence

\[ R>\frac1{\sqrt2}. \]

The previous report chose R=3/4, for which

\[ \mathcal E_{3/4}(\rho)=6\rho+21\rho^2. \]

It then selected the convenient subband rho<=0.1, where the exponent is at most 0.81. Even at fixed R=3/4, the actual strict endpoint is

\[ \rho<\frac{\sqrt{30}-3}{21} =0.1179631226215077\ldots. \]

Optimizing R down to 1/sqrt(2) gives the limiting rate

\[ (2+2\sqrt2)\rho+(7+5\sqrt2)\rho^2. \]

Let rho_* be its positive root. Then every rho<rho_* admits some R>1/sqrt(2) with a positive Laplace gap. Explicitly,

\[ \rho_* =\frac{\sqrt{(2+2\sqrt2)^2+4(7+5\sqrt2)}-(2+2\sqrt2)} {2(7+5\sqrt2)} =0.1454524603084045\ldots. \]

This extends RAMS2 and, through section 6, RC2 and URMS2 to

\[ |\alpha|<0.0727262301542022\ldots. \]

For the round milestone alpha=0.07, choose

\[ R=\frac{71}{100},\qquad \rho_R=\frac{143}{1000}. \]

The exact Laplace margin is

\[ \frac{3854691}{841000000}>0. \]

Together with

\[ \delta=\frac{283}{286},\quad \beta=\frac{563}{8000},\quad \gamma=\frac{11079}{57200},\quad \epsilon=\frac1{16}, \]

all RAMS, shell, envelope, far-tail, and dyadic margins are strictly positive.

Finite-prime peeling removes the fixed-rho endpoint

The last restriction still uses more uniformity than the Euler product needs. Fix any finite rho_0 and choose 0<R<1 so small that

\[ \mathcal E_R(\rho_0)<1/2. \]

Put P=R^-2 and split the powerful support into primes at most P and primes larger than P. On the tail-prime product,

\[ \prod_{p>P}\left(1+\sum_{e\geq2}\frac{R^{-2e}}{p^e}\right) \]

converges, because each local geometric series converges and its first term is O_R(p^-2). The Laplace rate is at most E_R(rho_0)<1/2.

There are only finitely many primes at most P. Give each such prime its own radius R_p>p^-1/2. Its exponent sum converges. The total logarithm of this finite prime set is constant, so its contribution to the Laplace rate is o(1) as z tends to zero. It is absorbed by the remaining half of the Laplace gap. Dominated convergence on the powerful strata then runs exactly as before.

This establishes RAMS2 on every compact fixed rho band. For example, split_powerful_witness(21/10) chooses a rational tail radius, a finite prime cutoff, and a Laplace rate below 1/2 using exact arithmetic.

With RAMS2 available at the target, the intermediate cutoff, and the far cutoff, choose for any alpha<1/2

\[ \delta=\frac{2\alpha+1}{2},\qquad \gamma=\frac{2\alpha+\delta}{2},\qquad \beta=\frac{\alpha+\gamma}{2}. \]

Then

\[ \alpha<\beta<\gamma<\delta<1, \qquad 2\alpha<\gamma. \]

A sufficiently small positive epsilon gives 2alpha<gamma(1-2epsilon). These are the shell decay, far-tail decay, finite mean-value, and initial-interval conditions. Therefore RC2 and URMS2 hold on every compact band 0<|alpha|<1/2. The exact witness at alpha=0.499 is emitted by prime_split_parameter_witness.

8. Path and cycle audit

The directive's requested low-order separation is already exact in rams2_cluster.py.

For a labelled support C, the path and cycle weights are separate formulas. Direct enumeration through six prime labels has zero symbolic defect against those formulas. The full all-matching support majorant begins

\[ 4,\quad3,\quad\frac{49}{45},\quad\frac{289}{840},\quad \frac{1681}{18900},\quad\frac{11}{560},\ldots \]

and has ratio O(1/r). Evaluating support sizes two, three, and four exactly therefore cannot enlarge the present endpoint: those pieces are already kept without the high-order majorant. After finite-prime peeling, neither the powerful core nor the path/cycle enumeration binds on a fixed rho band.

Deleting A*A still removes the Borel pq defect, forcing multiplicativity still misses kappa(p,q), and removing clusters above size two still misses the size-three path. None of these lesions is hidden by the bandwidth optimization.

9. Downstream window obstruction

Let an admissible real spectral factor v have support of length beta and

\[ \int v=1. \]

The physical-space test |v_hat(u)|^2 is nonnegative and equals one at the origin. Its autocorrelation A_v is supported on [-beta,beta], but it may have either sign. The downstream denominator is

\[ D_\beta(v)=\int v(s)^2\,ds +2\int_0^\beta\mathcal F_2(\alpha)A_v(\alpha)\,d\alpha. \]

The exact coefficient and tail majorants show

\[ \mathcal F_2(\alpha)>0 \qquad(0<\alpha\leq1/2). \]

The deterministic check forms the rational lower function from the first 40 coefficients, the exact majorants through index 101, and the two-residue geometric tail. After multiplying by the positive denominator and dividing by alpha, the resulting degree-102 polynomial has every Bernstein coefficient positive on [0,1/2]. The smallest coefficient is

\[ 0.0049495354041559\ldots>0. \]

Cauchy-Schwarz gives

\[ \int v^2\geq\frac1\beta, \qquad |A_v(\alpha)|\leq\int v^2. \]

Put

\[ I_\beta^+ =\sum_{i=1}^{40}\frac{C_i\beta^{i+1}}{i+1} +\beta\,\mathcal R_{40}(\beta), \]

where R_40 is the exact absolute tail majorant. Positivity on the half-band and the two inequalities above give, for every such v,

\[ D_\beta(v) \geq (1-2I_\beta^+)\int v^2 \geq\frac{1-2I_\beta^+}{\beta}. \]

At beta=1227/2500, exact rational evaluation gives

\[ \frac{1-2I_{1227/2500}^+}{1227/2500} =2.0002085758065586\ldots>2. \]

Thus no admissible spectral-factor window of bandwidth at most 0.4908 can give a positive simplicity expression. For the exact integral, half-band positivity makes 1-2beta-2 integral_0^beta F2 decrease with beta, so the endpoint calculation covers every smaller bandwidth.

The remaining interval closes by retaining the positive interaction of the constant part. Write v=1/beta+w, with integral w=0, and let T_beta be the integral operator with kernel F2(|s-t|). Schur's bound gives

\[ \|T_\beta\|\leq2I_{1/2}^+<1. \]

Completing the square in w bounds the possible improvement over the constant factor by

\[ \frac{4(I_{1/2}^+)^2} {(1227/2500)(1-2I_{1/2}^+)}. \]

Meanwhile the exact constant interaction at beta=1227/2500, with its full tail allowance, remains a lower bound throughout [1227/2500,1/2]. After division by the largest possible beta^2, the exact difference between this interaction and the completion correction is

\[ 0.0233227357071382\ldots>0. \]

Since 1/beta>=2, every admissible real spectral factor through beta=1/2 has D_beta(v)>2. Therefore

\[ \boxed{\frac12\leq\beta_{\rm useful}.} \]

This is a mathematical lower barrier for the downstream variational problem, not a weakness of the chosen Legendre basis.

The unrestricted exploratory quadratic solve uses 400 midpoint cells. At beta=0.499 it returns denominator 2.0287201304..., hence simplicity value -0.0287201304...; its minimizing spectral factor is still positive. At beta=0.51 it returns 0.0147819191..., consistent with the exact constant window below. This scan finds no candidate inside the newly reached band, but it is not used as a proof of nonexistence.

10. A useful window at 0.51

Take the normalized constant window

\[ v_\beta(s)=\frac1\beta \mathbf1_{[-\beta/2,\beta/2]}(s). \]

Then

\[ A_{v_\beta}(\alpha)=\frac{\beta-\alpha}{\beta^2} \quad(0\leq\alpha\leq\beta). \]

This is exactly the triangular transform in Bian's Chapter 11 test. The thesis chooses the squared-sinc physical-space kernel immediately before equation (11.2) and computes its triangular Fourier transform in the paragraph that follows. Equation (11.2) is the resulting pair-sum asymptotic. The constant v_beta above has that triangle as its autocorrelation, so this candidate does not enlarge the historical admissible class.

For beta=51/100, exact integration of the first 40 corrected coefficients, plus twice the full tail allowance, gives

\[ 2-D_{51/100}(v_{51/100})

0.0147728663285376.

\]

The exact rational value is emitted by constant_window_bound(51/100). Consequently

\[ \boxed{1/2\leq\beta_{\rm useful}\leq51/100.} \]

This is a coefficient-side statement. It becomes an xi-double-prime theorem only after the zero-statistics bridge reaches that bandwidth.

The primary source used for this comparison is Ji Bian, The Pair Correlation of Zeros of Derivatives of Riemann's Xi-Function, University of Rochester PhD thesis, 2008. The audited PDF SHA-256 is

ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7

11. The replacement lemma now required

The current rigorous bridge stops at alpha=1/2, exactly where the universal window obstruction ends. The current exact candidate is at 0.51.

RAMS2 is no longer the obstacle at 0.51: finite-prime peeling supplies every fixed ratio needed at the target and intermediate cutoffs. The conflicting RC2 conditions are instead

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

At alpha=0.51, the first demands gamma>1.02 in the zero-epsilon limit, while the finite mean-value estimate demands gamma<1. This is the first binding line.

Strong far-tail alternative

Establish, for some eta>1.02,

\[ \sum_{n>Y}|a_{2,T}(n)|^2n^{-3} \ll Y^{-\eta+o(1)} \]

outside the RAMS band. The cutoff condition becomes 2alpha<eta*beta, reducing the amount of arithmetic continuation required.

Long-polynomial mean-value alternative

Replace the current O(W/U) relative error by an early-smoothed estimate that remains o(1) for W=H^gamma at some gamma>1.02. This permits the existing quadratic tail decay to start beyond x^2 without requiring W<U.

The present obstruction is therefore a proof-technology barrier: the far-tail norm or the length range of the finite mean-value theorem. No divergent squarefree cluster, powerful-core divergence, or resolvent singularity has appeared.

12. Commands and controls

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

The controls cover:

• the complete inequality graph;
• an exact witness at alpha=0.011;
• strict failure of the old endpoint at 1/40;
• an exact shell witness at alpha=0.049;
• an exact optimized-Cauchy witness at alpha=0.07;
• a finite-prime split witness at rho=2.1;
• an exact contour witness at alpha=0.499;
• every requested continuation checkpoint;
• positivity of the corrected regular object on [0,1/2];
• the universal lower bound beta_useful>=1/2;
• the exact constant-window upper bound beta_useful<=0.51;
• all prior coefficient, cluster, contour-lesion, and level-one controls.