The Ladder of Truncation L3: Zero-Reference Spectral Zeta Construction and the Weil Bridge—O-L3 Debt Decomposition and First-Layer Closure

This paper registers the debt re-registration and spectral-landing closure of thethird-layer debt O-L3 in the number-theoretic truncation-coordinate registry. Theoriginal debt required constructing a compact operator that does not cite zero dataand whose spectrum is exactly the set of reciprocals of nontrivial zeros. The coreconclusion of this paper is that the millennium-level open weight of this debt lies onthe positivity side of the Weil functional; the spectral landing can be independentlyclosed as a parallel asset, but the original debt as a whole cannot be registered asthe sum of the spectral landing and the positivity criterion. Specifically, constructthe unbounded self-adjoint operator H on ℓ2(N) by Hen = (logn)en. For ℜs > 1,its exponential semigroup e−sH is trace class and Tr(e−sH) = ζ(s); the input ofthis construction does not invoke zero coordinates, and its zero-reference qualification is borne by the functional-equation route, with the Hadamard productpresented only as an output of the continued function. Object-type re-registrationshows that the compact-operator object of the original debt has not been delivered: H is an unbounded self-adjoint operator whose spectrum is {logn} ratherthan {1/ρ}; between the two spectral diagrams there is currently no registeredzero-free-input conversion. The precise content of the debt re-registration theorem is that the open weight of O-L3 is equivalent to the Weil positivity criterion,not that O-L3 as a whole is strictly logically equivalent to the positivity criterion.The obstruction map registers four coordinates—spectral rigidity, the distinctionbetween statistics and coordinates, circularity risk, and the spectral-diagram conversion gap—which jointly characterize the current unreachability of the directconstruction route. The machine-check protocol rechecks trace truncation and zerocalibration readings within a finite window, but does not bear proof. The status ofthe closing series is: spectral landing is closed in parallel, Weil positivity remains amillennium-level open node; re-registration is an asset, the bridge is a legacy.