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.
This criterion unifies the conceptual framework developed throughout Volume I.
1. Motivation
The ICA engineering process repeatedly required implementation decisions whose correctness could not be determined by examining implementation artifacts alone.
Instead, correctness was established by returning to constitutional definitions and verifying that architectural intent had been preserved.
This observation suggests that correctness is fundamentally an architectural property rather than an implementation property.
2. Observations
Observation 1
Implementation reviews consistently evaluated conformity to architectural concepts rather than textual similarity.
Observation 2
Implementation artifacts frequently differed while remaining architecturally acceptable.
Observation 3
Implementation corrections generally restored architectural consistency without altering architectural identity.
Observation 4
Successful derivations preserved the constitutional relationships established by the architecture.
Observation 5
Architectural traceability made correctness verifiable after derivation had completed.
3. Definitions
Definition 1 — Correct Derivation
A correct derivation is a derivation that preserves the architectural identity of its source architecture.
Definition 2 — Incorrect Derivation
An incorrect derivation is a derivation that fails to preserve architectural identity.
Definition 3 — Architectural Validation
Architectural validation is the evaluation of a derivation with respect to architectural correctness.
Definition 4 — Architectural Conformance
An artifact or instance is architecturally conformant when it participates exclusively in correct derivations.
4. Discussion
Correctness should not be confused with functional success.
A system may operate successfully while violating its architecture.
Conversely, a correctly derived realization may fail because of defects unrelated to architecture.
Similarly, legal compliance, software quality, organizational efficiency, and implementation elegance are distinct evaluation criteria.
Architectural correctness concerns only one question:
Has the architectural identity of the source architecture been preserved?
This definition deliberately separates architecture from all domain-specific notions of quality.
It therefore applies independently of technology, legislation, engineering discipline, or organizational context.
5. Propositions
Proposition 1
Correctness is determined by preservation of architectural identity.
Proposition 2
Preservation of architectural invariants is necessary for correctness.
Proposition 3
Traceability provides observable evidence supporting correctness.
Proposition 4
Distinct profiles may all represent correct derivations of the same architecture.
Proposition 5
Correctness is independent of implementation similarity.
6. Corollaries
Corollary 1
Two implementations may differ substantially while both being correct.
Corollary 2
Textual equivalence does not imply architectural correctness.
Corollary 3
Implementation diversity is compatible with architectural uniformity.
Corollary 4
Architectural correctness is preserved under every evolution that preserves architectural identity.
7. Conjectures
Conjecture 1
Correct derivations constitute an equivalence class induced by architectural identity.
Conjecture 2
Correctness may eventually admit complete formal verification.
Conjecture 3
Architectural correctness may become the primary criterion for comparing independent realizations of a common architecture.
8. Open Questions
- Can correctness always be mechanically verified?
- Which architectural invariants are sufficient to establish correctness?
- Can correctness be quantified?
- Does every derivable architecture admit a complete correctness criterion?
9. Status
This paper completes the conceptual foundation of Volume I.
Correctness is defined entirely in terms of concepts introduced in previous papers.
No additional primitive concepts are required.
The resulting framework explains the engineering observations obtained during the construction of the Institutional Continuity Architecture while remaining independent of the institutional domain.
Volume II will investigate the structural, algebraic, and formal properties of Derivable Architectures.