Higher xi derivatives: corrected coefficient object

Status

The corrected level-2 coefficient series is now an understood infinite object. The present outcome is split:

1. the historical coefficient audit is closed;
2. the exact coefficient-series tail is closed;
3. the historical fixed-order bridge audit closes at Outcome C;
4. the replacement representation audit closes at Outcome D;
5. the direct URMS2 attack sharpens Outcome D to two named estimates.

The corrected arithmetic starts

1, -8, 24, -32, 64/3, -64/3, 1216/45, -256/15,
1088/63, -11776/945, 42496/4725, ...

All three independent q_j generators and both scalable mean-square routes agree exactly through index 40. The historical values 0.9544 and 0.9774 remain quarantined. They are not inputs or checks for the corrected object.

Structural result

With A=Lambda, B=Lambda log, and C=Lambda log^2, the complete formal arithmetic sequence is generated by

\[ Q(z)=-A+\frac{2Bz-(2A*B+C)z^2}{(1-Az)^2+Bz^2}. \]

This expression gives an explicit binomial formula for every q_j. It also shows that every contribution to C[2,i] has sign (-1)^(i-1), so the alternation is exact and has no internal cancellation.

C2_EXTENDED.json stores the first 40 reduced rationals and the scaled integer check C[2,i] i! / 2^(i-1).

Coefficient tail

The factorial-permanent pairing admits the rational atom weights

\[ w_0=1,\qquad w_1=5/2,\qquad w_2=11, \]

for which (a+b+1)! <= w_a w_b in all nine cases. Retaining word length through index 101 and applying a closed two-step geometric majorant afterward gives

\[ \sum_{i>40}|C_{2,i}||\alpha|^i<3.279\mathbin{\cdot}10^{-9} \quad (|\alpha|\le1). \]

The generated regular series is entire. At bandwidth one,

\[ 4.76344632220668 < \sum_{i\ge1}C_{2,i} <4.76344632876331. \]

The exact majorant, ratio formulas, and complete derivation are in CORRECTED-F2.md. The deterministic checker is corrected_form_factor.py.

Weighted window result

For any integrable window v, the omitted tail in the autocorrelation functional is at most

\[ 3.279\mathbin{\cdot}10^{-9}\|v\|_1^2. \]

The small rational Legendre window in window_certificate.py has integral and L1 norm one, with exact pointwise floor 68213/100000. Exact integration of the first 40 terms plus the full infinite tail allowance gives

\[ 2-D>0.923401526388517\ldots>0.9234015. \]

The floating optimizer gives 0.923401531890862, only 5.51e-9 higher. This is an exact conditional statement for the coefficient-generated object, not a promoted ξ'' zero statistic.

Independent numerical falsifiers

A fresh fixed-seed completed-CUE ladder used 200 samples at each of N=24,32,48. At N=48, the four raw level-2 discrepancies from the corrected pointwise values were all below one reported Monte Carlo standard error:

The same payload contains the exact level-0 expectation and the Farmer-Gonek-Lee level-1 control under identical normalization.

The direct Dirichlet recurrence remains below its limiting curves at ell=8,10,12,14. At ell=14 and alpha=0.25, its raw level-2 integral is 0.00616034 against 0.00749860 from the corrected series. Its level-1 control is 0.01226302 against the known 0.01437786. No rescaling is used.

These oracles are consistent with the corrected object at their current finite sizes. They are not proof inputs.

Remaining bridge

The first failed estimate is now exact. On Bian thesis pages 32-33, the geometric-tail contour contribution includes

\[ \int_{T_\varepsilon}^{\infty}\frac{dv}{1+(v-t)^2}. \]

For every t>=T_epsilon, its restriction to [t,t+1] is at least 1/2. The source instead bounds the full integral by O((t+2)^-2). The same step is present in the pinned Farmer-Gonek arXiv source for level one.

With the correct absolute contour bound, the geometric tail requires B=Omega(log T) on a fixed positive alpha. The recorded PNT and off-diagonal mean-square bound requires B=O(log T/log log T). These conditions are incompatible. The target autocorrelation has only a simple zero at alpha one, so weighting does not repair the available bounds.

Bian's printed fixed-B theorem cannot identify the coefficient series with the actual limiting form factor. At level one, B=1 gives alpha-4 alpha^2+4 alpha^3, while B=2 adds the nonzero term 4 alpha^5/3. Distinct polynomials cannot both equal the same limit on an open interval with only o_B(1) errors.

The remaining requirement is a new L2 treatment of the full geometric remainder, followed by a coefficient-order-uniform arithmetic mean square. The coefficient tail needed by such an argument is now explicit. At index 40, the required allowance for the rational test window is below 3.279e-9.

BRIDGE-CLOSURE.md states the exact pointwise and weighted theorem schemas, audits every error class, and records the trust boundary. No replacement ξ'' simplicity percentage is published.

Replacement representation

The coefficient-order cutoff is not intrinsic. With U=xi'/xi, direct differentiation gives the exact identity

\[ \frac{\xi'''}{\xi''} =U+\frac{2UU'+U''}{U^2+U'}. \]

On a right half-plane, the full denominator can be convolution-inverted as one height-dependent Dirichlet object. Freezing the archimedean derivatives turns its arithmetic part into exactly the corrected rational Q(z) above. resummed_bridge.py checks these identities symbolically.

This avoids the incompatible geometric-order requirements, but it exposes the actual missing estimate: a local-uniform mean-square theorem for the full resummed coefficient family. RESUMMED-BRIDGE.md names the smallest target URMS2 and records its exact implication chain. No searched source supplies that uniformity.

The level-1 Farmer-Gonek-Lee proof uses the same failed contour-tail estimate in its printed explicit formula, and the term feeds its mean-square theorem. No repair was located. This is a gap classification for the printed proof, not a claim that the level-1 prediction is false.

URMS2 attack

The full convolution inverse is well behaved on the right contour line. In the weighted Wiener algebra it satisfies an explicit resolvent bound, and it acts boundedly on the square-summable Dirichlet coefficient space. The identities

\[ \partial_t b=-b*(\partial_t A)*b \]

and

\[ \|b_t-b_u\|_1\le\|b_t\|_1\|A_t-A_u\|_1\|b_u\|_1 \]

isolate height variation without a coefficient-order cutoff. The inverse is not multiplicative; the first exact defect occurs on the coprime pair 2,3.

The reflected coefficient object resums pointwise below alpha=1/2 at level one and below alpha=1/4 at level two. The first elementary square bound was only x(log x)^2, while the two-range mean square requires x log x. The later RAMS1 closure below identifies and removes that loss at level one.

URMS2-ATTACK.md states the three URMS variants and the smallest remaining package:

1. RC-kappa, an exact early-smoothed contour comparison in L2;
2. RAMS-kappa, a uniform resummed almost-prime square asymptotic saving one full logarithm.

The follow-up RAMS1-ATTACK.md identifies the lost logarithm exactly. The pointwise majorant counts a logarithmic envelope on every integer, while the actual depth-zero and depth-one terms live on prime powers. On squarefree integers with r prime factors, alpha_k vanishes unless k=r, and then

alpha_r(n) = (r-1)! log(n) product_(p|n) log(p).

This reconstructs the fixed-depth x log x main terms and confines all non-prime cross-depth interactions to repeated-prime collision strata. The all-depth squarefree contribution has an O_r(x log x) prime-measure majorant. Pure prime powers satisfy the exact formula a_z(p^a)=log(p)((1+z log(p))^a-2) and are lower order. The remaining RAMS1 obligation at that stage was an order-uniform summable bound for mixed repeated-prime collision errors. It is not supplied by the fixed-depth source estimates.

An exact generating mechanism is available for that last obstruction: alpha_k(n)=log(n)Lambda_k(n)/k. Borel resummation turns the full ordinary inverse into a Laplace integral of a multiplicative coefficient family, and the Hadamard square of that family has an Euler product. This exposed a two-parameter Tauberian target uniform in both Laplace variables when z and s-1 are simultaneously of order 1/log x.

That target is now closed at level one. The unique decomposition n=q*m, with q powerful and m squarefree, factors the full resummed coefficient. Squarefree terms yield the infinite Farmer-Gonek-Lee series; the powerful multiplier has finite harmonic mass and tends to zero off q=1. Hence RAMS1 has the x log x asymptotic uniformly when 0<r_min<=r<=r_0<1.

Early smoothing then keeps the exact resolvent on the right contour line. The Cauchy partial fraction selects the two Dirichlet ranges by Fourier support, and the height-freezing commutator costs O(log(T)/T). Together with a separate Hilbert-space bound for the infinite upper range, this establishes URMS1 on compact positive bands inside |alpha|<1/4. The limit agrees with the infinite Farmer-Gonek-Lee regular expression.

The subsequent RAMS2-CLUSTER.md attack organizes the level-two A*A atom exactly. The Borel inverse is a rank-one monomer-dimer model, and the Hadamard square has only alternating-path and even-cycle connected components. Direct summation of all matching pairs gives a support-size majorant with ratio O(1/r). Repeated primes are lower order after the powerful-squarefree split. Thus the connected term changes the corrected coefficient but not the x log x scale on 0<r<=1/10. The early-smoothed right-line argument then gives the level-two bridge on compact bands inside |alpha|<1/100. This band does not reach the existing simplicity window, so the historical percentages remain quarantined.

Bandwidth forensics subsequently finds that 1/100 was only a convenient parameter choice. The old inequalities already reach |alpha|<1/40; an intermediate coefficient-envelope shell sharpens the infinite tail and reaches |alpha|<1/20; optimizing the repeated-prime Cauchy radius reaches |alpha|<0.0727262301.... Peeling off the finitely many primes below the tail-radius threshold removes the fixed-rho restriction and reaches every compact band inside |alpha|<1/2. The first failed checkpoint is 0.50, at the far-tail cutoff versus finite mean-value length rather than RAMS2. The downstream variational problem has the exact bounds 0.5<=beta_useful<=0.51: Cauchy-Schwarz, operator coercivity, and completion of squares exclude every admissible spectral-factor window through 0.5; the constant window at 0.51 gives a positive conditional value.

URMS2-051.md removes the half-band obstruction without a new shifted-sum input. The exact two-range weights and the individual log n spacings reduce the finite off-diagonal error to O(x log x), independent of the far cutoff. With alpha=51/100, delta=3/4, gamma=21/20, and epsilon=1/100, the mean-value margin is 6/25 and the infinite-tail margin is 9/1000. The rebuilt bridge therefore supports the bandwidth-0.51 constant window. Under RH, the resulting first corrected xi-double-prime statement has simple-zero proportion at least 0.0147728663285376... in the limiting lower sense.

URMS2-051-AUDIT.md independently reconstructs all load-bearing passages. The audit found one omitted statement, height freezing beyond the old pointwise radius, and closes it by differentiating the summable cluster majorant. It also corrects the stale cross-range phase description in URMS1-CLOSURE.md. The source theorem, spacing sum, tail, multiplicity normalization, and independent JSON-based window calculation all pass. The remaining promotion gate is external mathematical review.

A slowly growing Hilbert projection would need order only J(T)=O(log log T) on a strict sub-quarter band. The fixed-order source does not give error constants uniform at that scale. urms2_attack.py records the norm bounds, the exact nonmultiplicativity witness, the projection scale, and the independent finite resummation control.

Pinned source and historical correction

The pinned source 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

C2_PROVENANCE.md records the first causal coefficient defect: the factors M(v_l)M(w_k) disappear before equation (8.1). The first divergence is C[2,2]=-8, not -4. RESOLUTION-2008-DISCREPANCY.md preserves the separate Chapter 11 arithmetic obstruction.

Reproduction

.venv/bin/python hunts/higher_xi/bian_audit.py
.venv/bin/python hunts/higher_xi/exact_c2.py --max-index 40
.venv/bin/python hunts/higher_xi/corrected_form_factor.py
.venv/bin/python hunts/higher_xi/window_certificate.py
.venv/bin/python hunts/higher_xi/resummed_bridge.py
.venv/bin/python hunts/higher_xi/urms2_attack.py
.venv/bin/python hunts/higher_xi/cue_oracle.py --sizes 24:200,32:200,48:200
.venv/bin/python hunts/higher_xi/dirichlet_recurrence.py --ells 8,10,12,14
.venv/bin/python -m pytest -q hunts/higher_xi/test_higher_xi.py -n0