When Can an Agent Attribute Properties to Itself? A Cognitive Algebra for Identity, Integrity, and Freedom

An agent's cognitive architecture may possess every regional carrier required for a capability and still emit a wrong verdict, because the content those carriers hold is corrupted. Over the MCK observer-relative classification (Bragetti, Asaro and Italiano, Zenodo 10.5281/zenodo.21772772), version 2 formalises the boundary between what an architecture can structurally realise and what it can semantically get right. The central object is a two-layer state kappa_a(t) = (Sigma_a(t), Delta_a(t)): regional carriers versus denotational and evidence content. A capability decomposes into structural requirements (Req-Sigma) and content-correctness requirements (Req-Delta), realisability being their conjunction; this yields a three-way status (structurally untypable / typable-but-incorrect / realised) and proves that structural extension is monotone in carrier presence but not in semantic calibration. Six constructions follow: an attribute-certificate calculus for the step-relation licence slot; a self-symbol discharging the epistemogenesis conjecture and anchoring de se and reflexive certification; an other-model triad (modelled agent, modelled observation regime, predicted monitor) with the actual observation-relative monitor imported from Zenodo 10.5281/zenodo.21700205; an operator-composition typing separating structural from semantic composability; a bounded transition system making Cert / MetaCert / calibrated-world reachability decidable at finite scale; and a bounded integrity-consistency decision procedure in O(n^2 c) time. Three worked self-properties (continuity, viability, owned branching - glossed identity, integrity, freedom) receive native two-layer signatures with incomparable structural and calibration blast radii. An attention layer completes the architecture: attention as an explicit third configuration coordinate, focus shifts as a declared transition family, a same-state / different-focus theorem, an attention-gated feedback channel over temporal implication, and the activation-signature analysis: of the 32 signatures combining structural realisability, semantic adequacy, attention, reflexive availability, and behavioural expression, exactly 18 are type-consistent, and no link of the naive activation chain holds. Applied to four cognitive worlds, the two-layer state proves same-Sigma / different-Delta non-injectivity. Every finite model, witness and countermodel is exhaustively validated by the checker suite distributed with the article (v1 suite retained without regression; extended regional checker, mechanised dependency-graph extractor, and figure-consistency checker over a single-source visual specification). Version 2 supersedes v1 as an extension; v1 remains available at this concept DOI. --- SCOPE AND STATUS. This is a preprint. It has not been peer reviewed and is not under journal acceptance. --- Version note (this version): this version incorporates a citation-integrity review of the manuscript. Unsupported or inaccurate attributions -- including several DOI, author, and version-identity corrections -- were fixed, and one sentence overstating this paper's relationship to a companion paper's open conjecture was corrected to accurately state that the conjecture remains open. No change to this paper's own central definitions, theorems, or proofs is intended.