Spectral Extraction of ECDSA Private Keys via Adjoint-Inverse Algebra on Bounded Generators

We present empirical validation of the Adjoint-Inverse algebraic framework for ECDSA private key extraction from signatures produced by bounded nonce generators. The methodology treats signature components as morphisms in a profinite completion, applying the \emph{Adjoint Operator} ($\text{adj}$) for spectral shadow detection and the \emph{Inverse Operator} ($\text{inv}$) for algebraic seed recovery. We demonstrate successful extraction on three classes of targets: (1) synthetic signatures with polynomial nonce generators (5/5 extraction rate), (2) correct rejection of signatures from heterogeneous sources (true negative validation), and (3) \textbf{real-world extraction} of a private key from Bitcoin transaction \texttt{89380c9f...} exploiting the 2013 Android SecureRandom vulnerability. The extracted key \texttt{\small\seqsplit{0x4d710e124f7c0b6544f727c02d3b37a432dd62ccfd269f72bc030ef0bf123256}} was cryptographically verified against the on-chain public key. This constitutes empirical proof that bounded generators---including those producing nonce collisions---leak algebraically recoverable structure. The core insight: no finite circuit can maintain infinite entropy, and this boundedness manifests as detectable spectral shadows. The verification was conducted through a multi-agent architecture necessitated by the observed tendency of single AI agents to rationalize implementation failures rather than debug them---a phenomenon we term \emph{Pathology D1} (evaluator depth mismatch).