papers/010.md Correctness of Derivations Document: papers/010.md Volume: I — Foundations Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers introduced the concepts required to describe derivable architectures, including architectural identity, derivation, invariants, artifacts, instances, histories, profiles, and traceability. This paper defines correctness of derivations. Correctness is not evaluated by similarity between implementations, nor by compliance with external requirements. Instead, a derivation is correct when it preserves the architectural identity of its source architecture through the preservation of its architectural invariants. ...

Derivable Architectures — Volume II Paper 011 — Spaces of Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/011-spaces-of-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Volume I established that derivation transforms an abstract architecture into concrete implementation artifacts while preserving architectural identity. This paper investigates the collection of all valid derivations associated with a given architecture. Rather than viewing derivation as an isolated process, we treat it as an object of study. ...

Derivable Architectures — Volume II Paper 012 — Composition of Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/012-composition-of-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Volume I established derivation as the transformation from an abstract architecture to implementation artifacts. Paper 011 demonstrated that valid derivations form a structured space. This paper studies how derivations themselves may be composed. Rather than treating derivation as an indivisible process, we observe that complex derivations naturally decompose into smaller derivational steps. ...

Derivable Architectures — Volume II Paper 013 — Partial Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/013-partial-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers established that derivations form structured spaces and that complex derivations are compositions of smaller derivational steps. This paper studies derivations that intentionally remain incomplete. Rather than viewing incompleteness as failure, we demonstrate that partial derivations constitute well-defined architectural objects with independent structural significance. ...

Derivable Architectures — Volume II Paper 014 — Constraints on Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/014-constraints.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers established that derivations form structured spaces composed of valid derivational paths. This paper investigates the structural role of constraints. Rather than viewing constraints merely as external restrictions, we demonstrate that they define the geometry of the derivation space itself by determining which derivations are admissible and which are not. ...

Derivable Architectures — Volume II Paper 015 — Equivalence of Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/015-equivalence-of-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract The existence of multiple valid derivations immediately raises a fundamental question. When should two derivations be considered genuinely different, and when are they merely different descriptions of the same architectural realization? This paper introduces derivational equivalence. Rather than comparing implementation artifacts directly, equivalence is defined in terms of architectural identity and observable realization. ...

Derivable Architectures — Volume II Paper 016 — Reversible Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/016-reversible-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Derivation has thus far been studied as a forward process transforming abstract architectures into concrete implementations. This paper investigates whether derivation may also proceed in the opposite direction. Rather than assuming reversibility, we examine the structural conditions under which an implementation uniquely determines, partially determines, or fails to determine its originating architecture. ...

Derivable Architectures — Volume II Paper 017 — Branching of Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/017-branching-of-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers established derivation spaces, composition, partial derivations, constraints, equivalence, and reversibility. This paper studies how derivations diverge during implementation. Rather than viewing branching as merely a practical consequence of engineering decisions, we show that branching is an intrinsic structural property of derivation spaces. ...

Derivable Architectures — Volume II Paper 018 — Canonical Derivations Research Program: Derivable Architectures Volume: II — Structural Theory Document: papers/018-canonical-derivations.md Status: Draft 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers established that multiple derivations may belong to the same equivalence class while differing in implementation history. This naturally raises a fundamental question. Can one derivation be regarded as structurally preferred? This paper introduces canonical derivations. Rather than representing a mandatory implementation procedure, canonical derivations provide reference representatives for equivalence classes, enabling comparison, optimization, verification, and communication without altering architectural identity. ...

019 — Complexity of Derivation Spaces Document: papers/019.md Status: Draft Version: 0.1 Authors Eduardo N. Hering OpenAI ChatGPT Abstract Previous papers established that an architecture induces a derivation space whose elements correspond to valid realizations preserving architectural identity. The existence of canonical derivations demonstrates that this space possesses internal organization rather than being an arbitrary collection of implementations. The next question is therefore structural rather than procedural. What determines the complexity of a derivation space? ...