Unus Solus Possibilis Est: Uniqueness of the Admissible Interior

Unus Solus Possibilis Est Corpus Closure and Uniqueness of the Admissible Interior This paper presents the corpus-closure theorem for the AASC foundation. Its central claim is that, once an original qualification-and-reuse problem has been fixed independently of any proposed answer, there is exactly one complete maximal region in which every declared continuation remains qualified and warranted. That region is the greatest admissible interior. The theorem concerns the complete admissibility structure of one fixed original problem. It does not identify a unique occupant, a unique physical mechanism, or a numerical theory merely from classification. Its force is structural: no two incompatible complete admissibility answers survive within the same fixed domain and declared use contract. The Unique Admissible Interior Let the original task specify: the candidate or incidence space; immediate qualification conditions; declared continuation operations; successor relations; local and edge warrants; all required observations and identity questions. The safety operator retains candidates that are immediately qualified, satisfy every required local warrant, and have only admissible successors. The paper proves that this operator has a greatest fixed point: the complete admissible interior. This interior is: sound under every declared continuation; complete among all sound regions for the same contract; unique as the greatest and inclusion-maximal admissible region; independent of which adequate representation is used; equipped with finite unsafe-path witnesses for every excluded configuration. Every sound invariant subset lies within the greatest interior. A smaller sound region may be useful for a restricted certificate or laboratory practice, but it does not constitute the complete answer to the original qualification-and-reuse problem. The existence of a unique interior does not by itself establish that the interior is nonempty or that it contains only one occupant. Positive realization requires an independent witness, and unique occupancy requires a further identity-separation theorem. Greatestness, Generation, and Safety The paper distinguishes several notions that are often conflated: Greatest admissible interior: the full region satisfying all declared local and continuation obligations. Least generated closure: the region generated from specified seeds by allowed operations. Chosen invariant subset: a sound but potentially incomplete subregion. Minimal complete interface: the quotient retaining exactly the required original answers. Individual occupant: a particular candidate or realization within the admissible interior. These objects coincide only under additional hypotheses. A least generated closure equals the greatest admissible interior only when the relevant seed, generation, soundness, and coverage conditions have been established. A sound subset does not become complete merely because it is closed. Conversely, a greatest admissible set does not select a unique member of that set. Safety is also distinguished from liveness. Closure under every declared continuation does not imply termination, eventual success, fairness, controllability, or the attainment of a desired endpoint. Such requirements must be added to the original contract and proved separately. Exact Descent to Representations A central part of the paper establishes when admissibility descends through a quotient or description map. A surjective representation carries the entire safety operator correctly only when: immediate qualification is constant on representation fibres; every locally valid pair of representatives has the same retained successor classes; successor availability and edge warrants are preserved; the quotient can reproduce the complete continuation structure. Under these conditions, finite-depth safety approximations descend exactly, and the greatest admissible interior is transported to the quotient. When descent fails, the paper supplies an explicit witness: two representatives with the same retained description but different local validity; different successor classes; different edge warrants; or different required original answers. Forward preservation alone is insufficient. A map may send valid original transitions to valid quotient transitions while failing to reflect all quotient transitions back to valid original representatives. Exact transport requires both directions. Faithful Presentation and Corpus Closure Minimal complete interfaces are uniquely isomorphic when they preserve the same original answer profile. Faithful maps preserve and reflect admissible-interior membership when their successor and warrant conditions are matched in both directions. This gives the corpus a precise closure principle: a faithful redescription cannot create a second complete admissibility structure; a deficient representation exposes a lost distinction or failed descent condition; an omitted safe configuration defeats a completeness claim; an included unsafe configuration defeats a soundness claim; a changed target or contract is a new problem, not a same-scope alternative. The uniqueness theorem therefore applies across faithful presentations without requiring identical notation, storage, syntax, or implementation. Several Contracts on One Carrier The paper also treats several admissibility contracts defined on the same underlying carrier. It constructs: the greatest common interior satisfying all contracts; the exact condition under which that joint interior equals the intersection of the separate interiors; conservative extensions obtained by adding obligations; mixed-path witnesses showing when separately safe contracts become jointly incompatible. The intersection of individually admissible interiors is not automatically the correct joint interior. A mixed continuation may combine obligations in a way that is not visible within either contract considered separately. Joint compatibility must therefore be specified and proved at the level of the combined task. Conservation, Repair, and Fixed Targets Original-incidence conservation is kept distinct from later construction. A corrected report may improve knowledge of an old occurrence. A repaired object may satisfy a later test. A new construction may create a new admissible state. None of these changes the qualification of the original retained incidence. The paper therefore excludes retrospective repair within the same fixed target: renaming does not change an original result; a later endpoint does not erase an earlier failure; a changed contract does not provide a same-task counterexample; a new witness must be assessed under its own original conditions; a new scope requires its own result and transfer proof. Exhaustion and Corpus-Level Uniqueness Every incorrect complete classifier or proposed admissible interior has an explicit witness: a surplus point; an omitted safe point; a finite unsafe path; a failed descent condition; an information collision; an incorrect scope transfer; or an unfulfilled existence or identity obligation. These witnesses establish closure by exhaustion over the declared fixed domain. The result is not an enumeration of every conceivable formal or physical system. It is an exhaustive theorem over the independently specified admissibility-bearing variation within the fixed comparison class. Scope and Nonclaims The paper does not claim: that every admissible interior is nonempty; that the unique interior is a singleton; that one unique occupant follows from greatestness; that a mathematical quotient selects a physical realization; that safety implies termination or eventual success; that faithful description equality implies mechanism identity; that a corpus classification derives a physical law or numerical value; that all downstream uniqueness claims are established without their own domain proofs; that the complete reference edition is a new proof-assistant formalization. The theorem is fixed-domain, same-scope, and contract-relative. Its universality concerns the uniqueness of the complete admissibility structure for the same original problem. Role in the AASC Programme This paper is the corpus-closure capstone of the foundational AASC sequence: kernel necessity → AMetric boundary → complete interface → greatest admissible interior → fixed-domain exhaustion → faithful transport → corpus closure. Its central conclusion is: For one independently fixed determinate-use problem, there is one complete greatest admissible interior, unique up to faithful lawful redescription. This is the precise sense in which Unus Solus Possibilis Est: one complete admissibility structure survives the full closure requirements of the declared domain, while distinct occupants, mechanisms, representations, and physical realizations remain separate questions requiring their own proofs. Keywords Unus Solus Possibilis Est; AASC; corpus closure; unique admissible interior; fixed-domain admissibility; greatest fixed point; safe continuation; canonical quotient; faithful transport; complete interface; fixed-domain exhaustion; same-scope closure; admissibility structure; standing conservation; realization uniqueness