Overview This paper establishes the minimal structural conditions under which a sequence of steps qualifies as the construction of a determinate object on a fixed domain. It is not a theory of psychological reasoning, informal inference, or a new proof calculus. It is a fixed-domain constraint theorem about admissible construction under identity-preserving conditions. The central object is the fixed-domain kernel structure[\mathcal K_D=(K,\simeq_{\mathrm{gov}},\mathcal I_D),\qquadK={\mathrm{Adm},\mathrm{St},\mathrm{Ref},\mathrm{Irr}},]where admissibility, standing, reference, and irreversibility form the governance-level closure under which identity-bearing construction has determinate mathematical status. Main Result The paper proves that any regime supporting non-degenerate, identity-preserving construction must already enforce this kernel structure. The kernel is not introduced as an optional axiom system or as one mathematical object alongside downstream representational structures. It is the formal admissibility closure under which “same target,” “same act,” “same continuation,” and “same redescription” become assessable statuses on a fixed domain. The result is eliminative and rigidity-theoretic. Any attempt to derive or generate the kernel from a governance-free prior substrate either presupposes the role it claims to generate, imports substitute governance, changes the domain of evaluation, collapses act or target identity, or reproduces the same governance work under another presentation. Structure of the Argument The paper first isolates the minimal assessability boundary for fixed-domain construction. It shows that determinate reference, standing as licensed constructional step-status, and irreversibility of act-status are jointly necessary and minimal, with admissibility arising as the governance boundary of their joint enforceability. It then establishes the core kernel results: Non-derivability from below: no member of the kernel can be produced from a weaker same-domain basis. No faithful lower generator: no governance-free prior substrate can generate the kernel while preserving the same fixed-domain target. No faithful same-domain extension: no extension can convert the kernel roles into lower theorems without illicit structure import, triviality, domain shift, or governance-equivalent collapse. Mutual closure and minimality: the kernel forms a closed necessity set at the level of governance work. Uniqueness up to governance equivalence: any adequate alternative presentation must preserve the same class of admitted constructions. Anti-weakening: no strict same-domain weakening can preserve non-degenerate construction while permitting independent admissibility-relevant grading, repair, status, or second constructional authority. Consequence Interface Once the kernel structure is fixed, the paper derives a rigid consequence interface for admissibility-assessable construction. These consequences include: bivalence of admissibility-relevant status; AMetric boundary behavior, excluding admissibility-relevant ranking, metric, selector, or parameter authority at the boundary; standing as reuse-stable admissibility; uniqueness of the admissible interior; conservation of standing; transport closure; quotient identity; domain-defined operators; no same-act repair; no independent carriers beneath standing. These results show that the kernel is not merely a local vocabulary for construction. It supplies a general constraint interface for proof-theoretic, semantic, computational, mechanized, and mathematical-physical frameworks whenever they treat outputs, models, states, proofs, or constructions as identity-bearing objects with standing on a fixed domain. Formalization An associated Lean4 formalization records the kernel structure and verifies the downstream consequence interface. The formalization does not supply a lower derivation of the kernel; the paper proves that no such derivation is available. Instead, the mechanized layer treats the fixed-domain kernel structure and its consequences as formal objects of verification once the admissibility boundary is in place. Keywords admissibility; standing; reference; irreversibility; kernel structure; fixed-domain construction; identity-preserving construction; governance equivalence; admissibility invariant; AMetric boundary; anti-weakening; non-derivability; formal construction; proof theory; proof identity; mechanized formalization; Lean4; philosophical logic