V182 Revised — Rank‑One Even Endpoint Completion in a Finite Toeplitz‑White Explicit‑Formula Model

Description This revised paper refines the finite Toeplitz‑white matrix program. It withdraws the earlier emphasis on a uniform spectral gap and instead highlights the robust finite phenomenon: a rank‑one even endpoint completion. The work proves exact finite‑dimensional reductions, records reproducible numerical audits, and clarifies why the correct target is sign control rather than a uniform gap. It does not prove the Riemann Hypothesis.

🔹 Main Contributions

• Finite Toeplitz‑white model defined from a Paley‑Wiener bump, truncated prime‑power weight, raw Toeplitz form, and Gram matrix.

• Whitening via Cholesky factor to obtain a symmetric Toeplitz‑white matrix.

• Endpoint functionals corresponding to evaluations at ±i/2, leading to endpoint‑free compression.

• Schur complement inertia accounting showing exactly one negative eigenvalue, localized at the even endpoint.

• Rank‑one even endpoint completion: adding the square (ea+eb)(ea+eb)T removes the negative sector.

• Raw/whitened equivalence between endpoint penalties and generalized inequalities.

🔹 Revision Highlights

• Stress test at k = 8192 showed the trace‑normalized gap drops by an order of magnitude.

• The uniform‑gap narrative is withdrawn.

• The robust finite mechanism is sign control via rank‑one even endpoint completion.

🔹 Numerical Evidence

• One negative eigenvalue persists through k = 8192.

• Endpoint‑free compression remains positive.

• Negative mode is aligned with the even endpoint channel.

• Raw generalized penalty coefficient Beab ≈ 1 across tested scales.

• Coupling ratio between endpoint and interior blocks decays steadily with k.

🔹 Interpretation

• The correct asymptotic target is sign control, not a positive uniform gap.

• The negative sector is rank‑one and even‑endpoint dominated.

• The unit coefficient in the completion term suggests an underlying analytic identity.

🔹 Open Problems

• Prove the unit even endpoint completion for all finite sections.

• Explain analytically why the coefficient is exactly one.

• Develop a positive spectral‑integral representation of the completed form.

• Control finite‑section edge effects and bridge numerical audits to symbolic identities.

• Establish continuum transfer if possible.

🔹 Conclusion The revised V182 paper narrows its claim: no uniform gap is asserted. The robust finite phenomenon is the rank‑one even endpoint completion, which controls the entire negative sector. Proving this completion inequality analytically is the next meaningful step. Even if proved, it remains a finite or asymptotic sign mechanism, not a proof of RH.

👉 Key message: V182 Revised demonstrates that the finite Toeplitz‑white model stabilizes through a rank‑one even endpoint completion. The emphasis shifts from spectral gap to sign control, marking a clearer and more precise theorem target.

📩 Verification Note: For reproducibility, audit files should include raw and whitened matrices, endpoint vectors, and compression data. Independent verification can be requested via 24ping@naver.com.