Objective
Attack the open higher-derivative form-factor problem left by Ji Bian's 2008 thesis and exposed by the August 2026 rank-trace paper.
The Chapter 11 percentages are quarantined historical data. They may appear only in source-forensics checks. They are never inputs, targets, priors, normalization controls, or expected outputs.
The present obligation is to reconstruct the arithmetic coefficients of xi'''/xi'' from first principles, derive the corresponding form-factor information with a controlled tail, and use completed-CUE and direct Dirichlet recurrences only as independent falsification oracles.
C2_PROVENANCE.md closes the historical coefficient audit. It finds that thesis page 71 drops the factors M(v_l)M(w_k) when passing to equation (8.1). The corrected system first diverges at C[2,2]=-8, not the printed -4.
CORRECTED-F2.md derives the infinite corrected coefficient object. It gives the rational generating function for q_j, 40 exact coefficients, and a uniform index-40 tail below 3.279e-9 on bandwidth one. The remaining obligation is now narrower: establish a uniform analytic bridge from xi'''/xi'' to the actual zero form factor. The printed fixed-B theorem is not that bridge, as its distinct level-1 truncation polynomials already show.
BRIDGE-CLOSURE.md closes the present analytic audit at Outcome C. The first failed source estimate is the contour-tail step on thesis pages 32-33: an integral whose kernel mass stays at least 1/2 for target heights inside the integration range is assigned quadratic height decay. Correcting it exposes incompatible truncation-order requirements in the geometric tail and the fixed-order mean-square error. No fixed positive alpha band, including the target weighted window, follows from the recorded estimates.
RESUMMED-BRIDGE.md changes the representation rather than patching that argument. It derives the exact untruncated resolvent for xi'''/xi'' and shows that its frozen arithmetic part is exactly the corrected rational Q(z) object. The replacement audit closes at Outcome D: no available source gives the required uniform mean square for the full height-dependent convolution inverse. The minimal new target is named URMS2. The same source defect is load-bearing in the printed Farmer-Gonek-Lee level-1 proof, which is classified as a proof gap with no located repair, not as a contradiction of the limiting prediction.
URMS2-ATTACK.md attacks that named target and closes at Outcome D. The full inverse is bounded in a weighted Wiener algebra and on its companion Dirichlet Hilbert space. Its exact coefficient envelope converges below alpha=1/2 at level one and alpha=1/4 at level two. At that stage neither threshold was a zero-side bridge: the available resummed square sum was x(log x)^2, one logarithm above the required x log x scale. The then-smallest package was the early-smoothed contour estimate RC-kappa plus the resummed almost-prime square asymptotic RAMS-kappa.
RAMS1-ATTACK.md resolves where the elementary extra logarithm comes from. On squarefree support exactly one convolution depth survives, with an explicit factorial formula. The crude x(log x)^2 bound loses prime support density already at depth zero. That attack left mixed repeated-prime collision strata open after closing the all-depth squarefree and pure-prime-power pieces. Early smoothing also turned the false Cauchy-tail step into an exact L2 contraction. The subsequent closure below resolves both remaining level-one obligations.
The same attack finds the full second-moment generating object. The exact identity alpha_k(n)=log(n) Lambda_k(n)/k converts the ordinary resolvent into a Laplace transform of a multiplicative exponential-convolution family. Its Hadamard square has an explicit two-parameter Euler product. This reduced RAMS1 to a uniform Borel-Hadamard Tauberian problem for that product.
URMS1-CLOSURE.md closes that Tauberian problem by the unique powerful-squarefree support split. The squarefree strata give the factorial main series, while powerful corrections have finite harmonic mass and vanish by dominated convergence. The same report rebuilds RC1 on the untouched right contour line and establishes URMS1 on compact positive bands inside |alpha|<1/4. This is Outcome B: level one repaired, level two blocked. The exact new level-two target is RAMS2-Cluster, forced by the connected A*A atom at the coprime integer 6.
RAMS2-CLUSTER.md resolves that target on the explicit coefficient band 0<r<=1/10. Borel transformation turns the denominator inverse into a rank-one monomer-dimer model. In the Hadamard square its connected components are exactly alternating paths and even cycles. Their all-support-size majorant has successive ratio O(1/r); the powerful-squarefree split then removes repeated-prime strata from the main term. The connected atom changes the corrected level-two coefficient but not the x log x scale. Reusing the early-smoothed right-line architecture gives the level-two bridge on compact bands inside |alpha|<1/100. This reaches Outcome B on a deliberately narrow band. It does not reach the support of the simplicity window, so no percentage is opened.
BANDWIDTH-FORENSICS.md removes the artificial endpoint in that first promotion. The unchanged parameter inequalities reach |alpha|<1/40. Inserting the existing pointwise coefficient envelope before the far Hilbert tail reaches |alpha|<1/20. Optimizing the powerful-core Cauchy radius extends RAMS2 to rho<0.1454524603... and the full bridge to |alpha|<0.0727262301...; alpha=0.07 has an exact rational witness. Peeling off finitely many small primes then removes the fixed-rho restriction and extends the full bridge to every compact band inside |alpha|<1/2. The first failed checkpoint is 0.50, where the far-tail cutoff conflicts with the finite mean-value length. Independently, the downstream window problem has exact bounds 0.5<=beta_useful<=0.51. Cauchy-Schwarz, operator coercivity, and completion of squares exclude all admissible spectral-factor windows through 0.5; the constant window at 0.51 is useful.
URMS2-051.md crosses the remaining gap. At sigma=3/2, the two contour ranges form one polynomial with square weights n/x^2 below x and x^2/n^3 above x. Retaining the individual log n spacings in the Montgomery-Vaughan estimate makes the off-diagonal error O(x log x), independent of the far cutoff length. The old gamma<delta condition was a worst-spacing loss. The rational choice alpha=51/100, delta=3/4, gamma=21/20, epsilon=1/100 has positive margins. URMS2 now contains the closed bandwidth 0.51, and the exact constant window gives a conditional simple-zero proportion lower bound 0.0147728663285376... for xi-double-prime under RH. Bandwidth optimization stops at this first corrected theorem.
URMS2-051-AUDIT.md is the independent internal audit. It reconstructs the two-range phase, maps the exact weighted Montgomery-Vaughan theorem, adds the marked-cluster height-freezing estimate at ratio 14/5, rebuilds the window bound from C2_EXTENDED.json, and checks the multiplicity normalization. Every internal gate passes. External mathematical review remains the next promotion gate.
LEAN-FRONTIER.md records the first kernel-checked slice of this package. ZetaLean/HigherXi.lean contains the exact parameter witness, logarithmic spacing majorant, full finite Dirichlet-polynomial expansion, multiplicity normalization, and rational output positivity. It deliberately does not state URMS2-051: the weighted Montgomery--Vaughan inequality, RAMS2 asymptotic, marked-cluster freezing, contour transfer, and in-Lean coefficient/tail reconstruction remain separate formal obligations.
The completed-CUE experiment uses angular derivatives of
Z_U(theta) = constant * product_j sin((theta-theta_j)/2).The controls are forced:
- derivative level 0 reproduces the CUE form factor
min(alpha,1); - derivative level 1 reproduces the Farmer-Gonek-Lee closed form;
- derivative level 2 is compared with the independently regenerated exact finite coefficients under identical weighted observables;
- matrix size and sample count ladders separate finite-size error from signal.
The desired outputs, in order, are:
- an exact coefficient provenance table and smallest obstruction;
- an exact full-band coefficient tail and a quantified analytic bridge;
- independent numerical oracles for the corrected object;
- only after the finite-height bridge closes, a zero-statistics window result.
Scope
This hunt may write only under hunts/higher_xi/ and figures/. It may read the rest of the repository and pinned primary sources. Changes to zeta/, lean/, ontology/, harness/, or repo-level records require a separate promotion decision.
The exact pinned primary source is Ji Bian, The Pair Correlation of Zeros of Derivatives of Riemann's Xi-Function, University of Rochester PhD thesis, 2008, repository item 5500, downloaded PDF SHA-256 ec1143f4f6c83288b717cfd4cd0aa6cc620f8c68b892f510a2a8d1708f36bfb7.
Non-goals
- No claim about RH.
- No unconditional xi-double-prime constant from Bian or CUE alone. The off-line zero-side algebra and prime-side asymptotic would be separate gates.
- No coefficient fitting presented as a derivation.
- No trust in the eleven-term truncation near
alpha=1. - No historical coefficient row used outside source forensics.
- No raw characteristic-polynomial derivative substituted for the completed real trigonometric polynomial.
Vocabulary and controls
This directory is exploratory. Use exact, measured, reproduced, and consistent with. Do not use the reserved claim vocabulary banned under hunts/.
Every headline number must have an exact-rational route or an independent precision and finite-size ladder. A generated claim cannot adjudicate its own normalization.