Reflection as a Resource: Stratified Representability, Fixed Points Under Restricted Internalization, and a Selector-Strength Hierarchy Paper 28 of the NEMS Suite
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2026
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| author | Spivack, Nova |
| author_facet | Spivack, Nova |
| contents | Paper 26 (General Self-Reference Calculus) showed that any system with full representability (repr-spec) satisfies the Master Fixed-Point Theorem (\MFP-1). Paper 27 (Closure Audits) formalized when determinacy is genuine vs. silently outsourced. The present paper fills the graded middle ground: how much internalization is enough for which fixed points, and how that maps to selector strength (NEMS IIa/IIb) in an abstract, non-physics way. We parameterize representability by a class \RepClass \subseteq (\SRICod \to \SRIObj): a system may only internalize transformers in \RepClass. The key notion is diagonal closure: \RepClass is closed under the diagonalization template F \mapsto (c \mapsto F(\ulcorner \run(c,c)\urcorner)). We prove the Diagonal Closure Theorem: if \RepClass is diagonally closed, then every F \in \RepClass has a mixed fixed point p \simeq F(\ulcorner p\urcorner). When \RepClass is not diagonally closed, we prove a formal method-level separation: identity-only on \mathbbN with F \in \RepClass but G_F \notin \RepClass, so the diagonal construction cannot produce a fixed point via \repr(G_F). This yields a resource theory of reflection: levels of internalization correspond to achievable fixed-point guarantees. Full \SRIz is the top level; we deliver a Lean-proved strict separation (identity-only) and roadmap further hierarchies. We bridge to Closure's internality predicate and to SelfReference's MFP-1, showing that Reflection extends both conservatively. The development is mechanized in Lean 4 (without custom axioms beyond Lean/mathlib; classical choice used only where explicitly stated), as the Reflection library in nems-lean. Trust boundary. Stratified representability and diagonal-closure hypotheses are explicit; the strict separations are about the formal \RepClass constraints, not every physical "reflection" metaphor. Mechanization is nems-lean . See . |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19429771 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Reflection as a Resource: Stratified Representability, Fixed Points Under Restricted Internalization, and a Selector-Strength Hierarchy Paper 28 of the NEMS Suite Spivack, Nova reflection diagonal closure stratified representability NEMS reflexive reality self-containment preprint Paper 26 (General Self-Reference Calculus) showed that any system with full representability (repr-spec) satisfies the Master Fixed-Point Theorem (\MFP-1). Paper 27 (Closure Audits) formalized when determinacy is genuine vs. silently outsourced. The present paper fills the graded middle ground: how much internalization is enough for which fixed points, and how that maps to selector strength (NEMS IIa/IIb) in an abstract, non-physics way. We parameterize representability by a class \RepClass \subseteq (\SRICod \to \SRIObj): a system may only internalize transformers in \RepClass. The key notion is diagonal closure: \RepClass is closed under the diagonalization template F \mapsto (c \mapsto F(\ulcorner \run(c,c)\urcorner)). We prove the Diagonal Closure Theorem: if \RepClass is diagonally closed, then every F \in \RepClass has a mixed fixed point p \simeq F(\ulcorner p\urcorner). When \RepClass is not diagonally closed, we prove a formal method-level separation: identity-only on \mathbbN with F \in \RepClass but G_F \notin \RepClass, so the diagonal construction cannot produce a fixed point via \repr(G_F). This yields a resource theory of reflection: levels of internalization correspond to achievable fixed-point guarantees. Full \SRIz is the top level; we deliver a Lean-proved strict separation (identity-only) and roadmap further hierarchies. We bridge to Closure's internality predicate and to SelfReference's MFP-1, showing that Reflection extends both conservatively. The development is mechanized in Lean 4 (without custom axioms beyond Lean/mathlib; classical choice used only where explicitly stated), as the Reflection library in nems-lean. Trust boundary. Stratified representability and diagonal-closure hypotheses are explicit; the strict separations are about the formal \RepClass constraints, not every physical "reflection" metaphor. Mechanization is nems-lean . See . |
| title | Reflection as a Resource: Stratified Representability, Fixed Points Under Restricted Internalization, and a Selector-Strength Hierarchy Paper 28 of the NEMS Suite |
| topic | reflection diagonal closure stratified representability NEMS reflexive reality self-containment preprint |
| url | https://doi.org/10.5281/zenodo.19429771 |