Closed Systems from Comparison Completeness

Aligns monograph-wide status terminology and document references in the introduction, front matter, appendices, and selected chapter handoffs. The theorem stack is unchanged.

DOI assigned to v12 on June 15, 2026.

Adds local clarifying prose at selected theorem handoffs: rectangular completeness, quotient equality for supported representatives, object and morphism loci, triangle-to-loop descent, second-jet faithfulness, Einstein compatibility, local smooth realization, phase-sector scope, and numerical-calibration boundaries. The theorem stack is unchanged.

No DOI assigned to this release.

Adds declarations and source availability, a standard-results appendix, a minimal-examples and failure-modes appendix, a conventional subject index, and chapter/section running heads. The theorem stack is unchanged.

No DOI assigned to this release.

Standardizes the publication record, reader guide, principal-results list, notation conventions, standing terminology, logical-status ledger, part and chapter orientations, closing ledgers, reference indices, page style, and bibliography metadata. The theorem stack is unchanged.

No DOI assigned to this release.

Adds a reader guide, notation and conventions, standing terminology, a logical-status ledger, and back-matter reference indices for notation and recurring definitions. The Preface states the claim-status conventions; the Introduction begins the formal setup from primitive comparison data. The theorem stack is unchanged.

Adds inline citations at first load-bearing uses of external theorems, named constructions, standard frameworks, and comparison points with established literature. The bibliography and public source record are expanded accordingly. The theorem statements and proofs are otherwise preserved.

Adds Chapter 23, Observable Exhaustivity and the Automorphism Bridge. Relative to its stated data (R1)–(R3), the chapter proves that the raw limit comparison algebra exceeds every realized smooth reading, identifies the excess as the fiber-separating sector, and shows that condition (U) selects the maximal readable sector. On that sector the realized algebra is recovered as \(C(M)\), smooth derivations are identified with vector fields, and transport-stabilizing derivations are identified with the connection automorphism algebra up to gauge.

The front matter, introduction, Chapter 22 transition, conditional-completion ledger, bibliography, and source bundle were updated so the new part is integrated as a state-side reconstruction extension rather than as a premise for earlier results.

Chapter 13 was extended to decompose the second-jet faithfulness condition into torsion (T) and detectability (D), isolate the ghost failure modes, and give the canonical realization on which the second-jet comparison map is injective by construction.

Affiliation and correspondence updated to Closure Research Initiative; the PDF and source package were rebuilt from the v4 monograph source. Mathematical content unchanged from v3.

New Lemma 2.6.2 (Conservative-completion dichotomy). For any unrealized profile pair in an intrinsic, locally distinguishing comparison world, either the pair is internally excluded—its realization would violate the comparison predicate definitions—or the extension that realizes it is conservative: restriction back to U preserves the finite-coordinate comparison algebra supported in U and creates no new profile distinction among the original states. There is no third option. This splits the traditional conservativity step into a proven structural alternative.

New Corollary 2.6.3 (Closure forbids conservative omission). An intrinsic, locally distinguishing world that is closed (rectangularly complete) cannot conservatively omit any finite profile. If a profile pair could be realized without disturbing existing data, it must already be realized. This isolates the closure principle as precisely the step that rules out intrinsically undetectable omissions.

Theorem renumbered. Rectangular completeness under intrinsicality (formerly Theorem 2.6.2) is now Theorem 2.6.4, with its proof rewritten to route through Lemma 2.6.2 and Corollary 2.6.3 instead of asserting conservativity directly.

Explanatory Remark 2.6.5 added. Shows the equivalent finite-induction viewpoint (if every finite subprofile is realized, the full profile is realizable), while keeping the dichotomy as the main structural mechanism.

Scope preserved. The non-universality remark (now Remark 2.6.6) is unchanged: non-intrinsic worlds and intrinsic worlds failing local distinguishability may be open.

Cross-reference label added. The left/right profile definition (Definition 2.3.1) now carries a label so later admissibility references resolve cleanly.

Amended primitive (Definition 0.1.1). A comparison world $(U,C)$ is now required to be intrinsic: every predicate is invariant under the automorphism group of the comparison world it helps constitute, $c(gu,gv)=c(u,v)$ for all $g \in \mathrm{Aut}(U,C)$. This is a fixed-point condition—a comparison is a relation among states, not an externally imposed labeling—and is the primitive-level form of the intrinsicity clause (SP1).

Standing principles SP2 and SP3 demoted to theorems. Both labels are retained for continuity of reference but are now derived results:

• Compactness (SP2) is derived in the new Theorem 2.9.3 (Compactness from intrinsicality), supported by the finite-support lemma (2.5.11) and the new finite-orbit-separation lemma (2.5.12). Under intrinsicality, profiles that differ differ at a single coordinate, the refinement-tower limit is the direct union with no completion, and the finite-to-global boundary holds without separate hypothesis.
• Closedness (SP3) is derived in the new Theorem 2.6.2 (Rectangular completeness under intrinsicality): an intrinsic, locally distinguishing world is rectangularly complete.

New results added: Lemma 2.5.12 (finite orbit separation under intrinsicality), Remark 2.5.13 (no limit-only separation), Theorem 2.6.2, Remark 2.6.3 (scope), Theorem 2.9.3.

Updated cross-references: the SP2 and SP3 clauses (§0.2) and the summary in §0.5 now cite the derivations above. Lemmas in the former §2.9 finite-support block were relocated to §2.5 (now 2.5.11–2.5.12) so they precede their first use in §2.6; the Boolean-internalization theorem retains its place (now 2.9.5).

Scope unchanged. Rectangular completeness is not universal: non-intrinsic worlds, and intrinsic worlds failing local distinguishability, may be open. The rigid-rod, diathermal-wall, and symmetric-world examples remain genuine open systems (Remark 2.6.3).

Initial preprint; superseded archival release.