A classification of the objecthood-status of declared mathematical entity-commitment tokens, not a verdict on what mathematical objects in fact exist. The corpus fixes what a record is of (Paper 13), when a declared structure is objective (Paper 15), when an existence-claim is true (Paper 20), when a belief is knowledge (Paper 21), when a postulate may be adopted (Paper 63), what a term refers to (Paper 75), and the necessity-status of a commitment (Paper 76) — but nowhere types the history-internal, system-relative objecthood-status of a declared mathematical entity-token. While no such layer exists, a Global Platonist — an external authority who fixes, from outside a history, which mathematical objects exist and what is genuine abstract reality — works by default. Paper 76 reserved this exact seat, at classification grade, by name. This paper types that layer. Four type facts stand in print and bind the paper: "Typed is not adopted; classified is not answered. " / "An objecthood-status is a type, not a world-side existence verdict. " / "An eliminability-status is a checked certificate, not a decided fact. " / "Objecthood is relative to a declared system, never simpliciter. " The bearer is not an object. An entity-commitment token is a recorded commitment, in a declared system, to use a symbol or construction as object-involving at a specified grain — never a presupposed abstract object. The construction is an objecthood-status grounding object O (e) = (Sdecl, Intro, Elim, Indiv, Use, gamma) — a token's declared recursively-presented systems, its recorded introduction-provenance, an elimination-certificate store with its checker rules, a declared individuation grain, its use record, and a token-individuation grain — with a normal form and a barred-fields box. On it, a three-axis ObjecthoodProfile (e; S) = (I, E, N) is typed by a certificate-based decidable test: the E-axis (eliminability-status) is read by checking a declared certificate — an elimination certificate, or a grade-bounded indispensability certificate consisting of an essential-occurrence witness plus a bounded no-elimination exhaustion — never by deciding real eliminability; its positive cell typed-indispensable-at-grade-g-in-S always means indispensability certified at the declared grade g, not unbounded or real indispensability. The I-axis (introduction-mode) is set-valued over the recorded introductions, with a named typed-no-introduction-record cell when the record is declared-empty. The N-axis (individuation-status) is laddered over a fixed identity-question domain split into representation-identity and structural-role-identity tiers: typed-structure-relative fixes structural role by a declared isomorphism criterion while representation-identity stays open, and is thereby distinct from typed-underdetermined, where neither tier is fixed. Certificate conflicts route to a diagnostic outcome, not a status. The contribution beyond "eliminability-in-S restated" (the K0 guard) is that the I- and N-axes are computed from history-internal introduction-provenance and the declared individuation grain, not from the eliminability fact: two tokens both typed-indispensable-at-grade-g-in-S can carry different profiles, and the paper exhibits explicit non-collapse witness pairs over a fully declared theory fragment. The multi-axis profile locates the indispensabilist, nominalist, abstractionist, structuralist, and underdetermination positions as distinct typed profile points, so the output is an objecthood-status profile, not only an eliminability bookkeeping. The ceiling is declared and patrolled: typing is not adopting; classifying is not answering. The substantive what-in-fact-exists question is declined, not refuted; Platonism and nominalism are declined either way; truth, objectivity, knowledge, reference, necessity, and adoption stay at their owners' seats; strict recovery of Paper 13's per-record semantics is exact (Lemma R-es). The predicted landing is ObjecthoodProfile-layer = scoped, with a null via K0 an honest branch and a Platonist-dependence theorem (K3) declared and patrolled. Nothing is adopted; no truth, satisfaction, or world-side existence verdict is claimed. The paper engages the ontology-of-mathematics literature (Carnap 1950; Putnam 1971; Field 1980; Benacerraf 1965; Benacerraf 1973; Shapiro 1997; Wright 1983; Quine 1951) as pressure-test and boundary only, operationalizing Carnap's internal/external relativity with a decidable test, granting Putnam's indispensability pressure a checked status while withholding the realist conclusion, admitting Field's nominalist paraphrase as one elimination certificate, typing Benacerraf's identity problem as underdetermination rather than resolving it, locating Shapiro's structuralism and Wright's abstractionism as profile cells, keeping the introduction tags honest against Quine's analyticity critique, and leaving Benacerraf's access problem at the knowledge seat.
Tomoyuki Uchida (Thu,) studied this question.