This study aims to structurally examine why loss of access to Bitcoin is permanent and to assess claims regarding the possibility of recovery, master keys, or backdoors in the Bitcoin system. This study departs from the general trend that positions Bitcoin solely as a monetary innovation, while its most radical aspect lies in its system design. Bitcoin is analyzed as a non-linear formal system that defines ownership through cryptographic facts, rather than through identity, intent, or social context. The methodology used is a conceptual-design analysis based on cryptographic studies, protocol architecture, empirical precedents of human error, and the implications of network consensus. The main results demonstrate that irreversibility and loss of access are not system failures, but rather direct consequences of Bitcoin's design axioms, which tie ownership to high-entropy private keys and reject all forms of discretionary authority. The analysis also shows that claims of backdoors, whether mathematical, implemental, hardware, or temporal, are structurally incoherent without undermining the fundamental assumptions of cryptography and decentralized consensus. The study's conclusions affirm that Bitcoin's resilience stems not from adaptive flexibility but from deliberate rule rigidity. This research's contribution lies in mapping Bitcoin as a system design artifact that resists recovery, while also expanding the Bitcoin discourse from the monetary realm to the study of system design, applied cryptography, and the philosophy of technology.