CRPT ω-Tower

A self-contained formal development of CRPT (Compositional Recursive Projection Theory): a projection-axiom framework over a two-aspect substrate (convergent ↓_M and persistent ∞_M), with canonical-form theory, horizon classification, an observer architecture, instance algebra, a categorical self-structure, the Reification/Tower construction, and a self-application rank (Rω) in which the theory is applied to itself.

Status

v1.0.0-rc8 — release candidate. rc8 ports the presentation to the theory's own structure and completes the provenance account:

• The documents are the eight genuine ranks (R0–R6, Rω), not ten presentation levels — level → rank throughout, with the merged ranks co-defined.
• The convergent/persistent partition is named the aspect duality and identified with the WF–NWF (projection) duality (WF-NWF-Dual, Fold-Unfold).
• model → instance and lift → reification are carried through in full — prose, tags, and notations — keeping the model-theory locus (R4.3) where instances are proved to be models; Reify-Earned discharges the reification name.
• Min-Interp states what CRPT is, minimally — the recursive projection of a structure onto its discernible content — crediting discrete dynamical systems among its imports.
• The sameness schema (Sameness: bisimilar / isomorphic / faithful-embedding) classifies fibers, instances, and the five provenance statuses uniformly; Fiber-Unif proves these are one construction — a fiber of recursive projection — and every construct's header carries its sameness tier.
• A final correctness audit verifies 0 broken links, 0 duplicate labels/tags, every status obligation discharged, and every theorem proved or derived.

Earlier: rc7 resolved the external-review findings (tower-growth trichotomy, completed axiom matrix, repaired instance algebra, scoped observer naturality, the five-status provenance system, the PA-namespace discipline, the rank stratification). rc6 audited the projection-axiom basis, the WF/NWF duality, and the categorical self-structure for internal consistency, and introduced the Self-Sectioning theory (Rω.8) that reads the spec as a pure-WF CRPT instance whose sections are fibers of the Source-anchor projection. rc4 derived the observation labelling from PA-Prod and resolved the two-mode question; rc3–rc2 established the labelled bisimilarity, aspect-aware equivalences, and the instance-neutral, legacy-free, engine-free presentation.

Reading order

Start with the front matter, then the ranks in order:

1. Preface — informal orientation
2. Reading Guide — suggested paths by background
3. Index · Global Map · Common Pitfalls
4. Ranks: R0 · R1 · R2 · R3 · R4 · R5 · R6 · Rω

Reference

• Theorem & Definition Index — every labelled item by rank, with tags
• Definitions Index
• Release Notes

Conventions

Every formal item carries a stable label R⟨rank⟩.⟨section⟩.⟨kind⟩⟨n⟩ (e.g. R1.2.Ax1, R2.3.T9) and a short tag (e.g. `PA-WN`, `WF-NWF-Dual`). Cross-references cite both, as `Tag` (Label), and every cross-reference is a hyperlink to the cited construct's subsection (projection axioms are cited without backticks, PA-X (Label), likewise hyperlinked). Labels are placed by the fiber discipline of Self-Sect (Rω.8.T1): a construct lives in the section of its primary Source anchor, and the section map (CRPT_OMEGA_TOWER_SECTION_MAP.md) records the resulting census. Each construct's header table carries its Type · Label · Tag · Notation · Status · Sameness, where the Status is one of five provenance values (Status (Rω.8.D3)) — Imported (exact transport), Specialized (import restricted to a CRPT domain), Reframed (maximal import + minimal stated delta, with a Reframing Note), Derived (composition of priors, with its chain), Novel (genuinely new, with a Novelty Note crediting its basis) — each with a mandatory body obligation, auditable by census, and the Sameness is the construct's tier under the sameness schema relative to its Source (Sameness (Rω.8.D4)): Imported ≅, Specialized ↪, Reframed ≈, Derived ≅∘, Novel ∅. The PA-* tag namespace is reserved for the nine projection axioms (Ax-NS (R1.2.R2)).