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

Description This paper serves as the final status report for the Reburn/V182 finite Toeplitz‑white project. It defines a reproducible finite Toeplitz‑white matrix model inspired by the Weil explicit formula, records confirmed numerical facts, and carefully separates valid finite observations from refuted proof mechanisms. The emphasis is conservative: no claim of proving the Riemann Hypothesis, no uniform spectral gap, and no zero‑side reconstruction theorem.

🔹 Confirmed Finite Facts

• One negative eigenvalue: For all tested scales (k = 512, 1024, 2048, 4096, 8192), both raw and whitened matrices have exactly one negative eigenvalue.

• Endpoint‑free compression positivity: Compression remains positive in tested cases.

• Even endpoint localization: The unique negative mode is strongly aligned with the even endpoint direction.

🔹 Refuted Proof Mechanisms

• Uniform trace‑normalized gap: rejected; trace normalization is unstable and conceptually too strong.

• Off‑line perturbation sensitivity: refuted; null heights respond similarly, showing generic perturbation sensitivity rather than zero‑specific fidelity.

• Zero‑side matrix reconstruction: identified as a wrong target; the explicit formula is scalar, not entrywise matrix.

• Rank‑one sign‑count split: refuted; negative Toeplitz part is high‑rank, cancellation not domination.

🔹 Surviving Core

• The finite Toeplitz‑white construction is reproducible.

• Exactly one negative eigenvalue persists across all tested scales.

• Endpoint‑free compression is positive.

• Even endpoint direction captures the negative mode.

• Unit completion remains conjectural, not proved.

🔹 Remaining Mathematical Problem

• Indefinite Toeplitz inertia problem: Explain analytically why a highly sign‑changing symbol, with both positive and negative Toeplitz parts of high rank, nevertheless produces exactly one negative square in the tested first‑zero‑free window.

• This requires tools from indefinite Toeplitz theory, Krein space methods, and negative‑square counting, beyond classical Grenander–Szegő asymptotics.

🔹 Interpretation Principles

• Finite confirmation is not asymptotic proof.

• A failed mechanism can coexist with a true theorem.

• Endpoint facts are not automatic consequences of inertia.

• Closure is appropriate: the project is complete as a finite audit, not as a proof of RH.

🔹 Conclusion The V182_2 report closes the finite Toeplitz‑white project. It confirms reproducible inertia and endpoint phenomena, rejects unstable or invalid proof routes, and isolates the surviving open problem: the indefinite Toeplitz inertia puzzle. The project’s value lies in clarifying what is true, what is false, and what remains to be explained.

👉 Key message: V182_2 demonstrates that the finite Toeplitz‑white model consistently shows one negative eigenvalue and endpoint localization, but the mechanism is not rank‑one domination. The unresolved challenge is to explain this high‑rank cancellation analytically.

📩 Verification Note: For reproducibility, audit files and numerical safeguards are documented. Independent verification can be requested via 24ping@naver.com.