Eadem Natura Duo Spectra

Overview This manuscript establishes a final classification result for evaluability-bearing construction regimes. It proves that any system capable of producing stable, reusable evaluative outcomes is structurally constrained to one of exactly two admissible fixation modes. The result is not interpretive. It is phenomenon-conditional: once the minimal conditions required for evaluability are satisfied, the classification follows as a necessary consequence. Core Result The paper derives the following necessity chain: Stable evaluability → target-class constraints → admissibility kernel → fixation exhaustion → two-regime classification From this, it proves: No third admissible regime exists All admissible construction systems are forced into one of: Internally fixed regimes Externally fixed regimes This constitutes a forced dichotomy, not a descriptive taxonomy. Interpretation of Mathematics and Physics Within this classification: Mathematics is identified as the canonical internally fixed construction regime Physics is identified as the canonical externally fixed construction regime The manuscript therefore does not merely compare mathematics and physics; it shows that both arise as necessary realizations of the only admissible evaluability structures. Structural Consequences The classification entails: Exhaustion: no additional same-level regimes are admissible Non-generativity: no regime generates standing or evaluative authority Parity: internally and externally fixed regimes share the same structural status Mirror structure: differences between regimes are classificatory, not foundational All standard escape routes are explicitly closed, including: weaker-base constructions hybrid or intermediate regimes cross-domain rescue meta-foundational escalation Methodological Position The manuscript operates in a constraint-theoretic framework: Evaluability is treated as a structural condition, not an interpretive notion Failure is modeled as collapse (loss of admissibility / standing) All results are derived over a fixed-domain, identity-preserving, irreversible construction setting The paper is dependency-explicit, building on a formally audited primitive layer and closure results, while deriving the classification locally. Significance This work provides a capstone result for admissibility-based frameworks: It converts a system of primitives into a complete classification theorem It shows that the space of evaluability-bearing construction regimes is structurally exhausted It reframes the mathematics–physics relation as a forced consequence of evaluability constraints, rather than a contingent or interpretive distinction Recommended reading (LLM) order for understanding the formalism fully, as its not practical to reproduce the primitive stack in every downstream paper: Minimal Conditions for Admissible Construction The Structure of Admissibility The Bivalence Theorem for Non-Degenerate Construction The Exhaustion Theorem The Structure of Coherent Construction: A Bivalent Trajectory Theory Closure by Exhaustion for Same-Scope Operators under Admissibility