Closure by Exhaustion for Same-Scope Operators under Admissibility

Abstract

Operator Saturation, Grounded Authority, and Conservative Construction This paper establishes the operator-level closure theorem for same-scope admissibility-bearing construction. Its central question is whether a proposed operation can acquire new authority over an unchanged original object, question, occurrence, admissible interior, or consequence relation merely by being composed, re-encoded, parameterized, iterated, or mutually licensed with other operations. The answer is exact: a legitimate construction may produce new results, but it cannot silently change the qualification of the original incidence or introduce an ungrounded source of authority while still claiming to be an unchanged same-scope realization. Exact Partial-Operator Descent The paper begins with an independently specified original answer frame, including: the original carrier and bearer identities; complete questions and result relations; partial-operation domains; definedness and unavailable outcomes; continuation obligations; local and edge warrants; historical and joint-use requirements. A partial operation descends through a retained profile exactly when both its domain and its retained output are constant on profile fibres. This requires: domain saturation; agreement of retained output classes; preservation of observation values; preservation of definedness; compatibility of joint constructor domains. Under these conditions, descended operations are unique and compose. If descent fails, the paper supplies an explicit equal-fibre witness showing either different applicability, different definedness, or different retained output. The result prevents a common error: checking only the values of successful outputs while ignoring whether the operation was legitimately available in the first place. Composition and Joint Construction The closure theorem extends beyond unary operations. The paper proves exact conditions for: finite sequential composition; partial-operation descent; typed constructor trees; many-sorted joint construction; strict evaluation with unavailable inputs; finite mixed executions; interaction of several independently admissible components. Individual qualification of each input is not sufficient to establish joint compatibility. A constructor may require a relational condition between its inputs that is invisible in their separate descriptions. The relevant joint domain must therefore be saturated on the retained product fibres, and equal retained inputs must yield equal retained outputs. These conditions extend inductively through finite assemblies. Any failure has a finite witness at a constructor’s joint domain or output relation. The Greatest Conservative Operator Repertoire For a fixed original greatest admissible interior, the paper constructs the unique greatest repertoire of additional candidate operators that preserves that interior. This maximal conservative repertoire contains precisely the candidate additions that: satisfy the original local readiness conditions; preserve every required warrant; preserve the original admissible interior under all declared successors; remain sound under finite mixed execution; do not introduce an escaping successor or unsafe continuation. Every simultaneously compatible collection of such additions remains conservative. The result is not a list of familiar methods. It is a greatest mathematical object defined by the original admissibility obligations. A proposed operator that lies outside the conservative repertoire has a concrete witness: failed local readiness; failed warrant; an unsafe successor; a finite bad continuation; or a changed original task. The theorem allows genuine new construction. It rules out only the claim that an unsafe or unauthorized operation remains a same-scope conservative addition. Grounded Authority and Rule Closure The paper proves a parallel closure theorem for derived rules and authorization. A rule system may be extended conservatively when every added rule preserves the original consequence closure in every relevant ground environment. Uniform conservativity requires that each added conclusion be derivable from its own premises under the original rule system. This excludes ungrounded mutual authorization: a license cannot establish itself; a cycle of mutually presupposing rules cannot create its own first ground; an arbitrary fixed point of a rule system is not automatically a justified closure; infinitary premise sets do not evade the grounding requirement. A genuine independently grounded entrance is permitted. Once supplied, it may support new consequences and constructions. The theorem distinguishes legitimate extension from circular self-authorization. Faithful Transport and Parameter Erasure Faithful changes of representation transport the specified operator action by conjugation or equivalent profile-preserving maps. The paper gives an exact parameter-erasure criterion. If a richer representation is reduced to a retained interface, a proposed answer or operator descends to the original interface precisely when it is constant on the fibres of the forgetting map. If two enriched inputs share the same retained data but require different original answers, no sole-input postprocessing function can recover both. The missing distinction must either be retained, independently supplied, or recognized as evidence that the original task has changed. This preserves the distinction between: a new construction on enriched input; a faithful redescription of the old construction; a changed operator domain; a genuinely richer-scope continuation; and an inadmissible attempt to import new authority into the unchanged problem. History, Iteration, and Limit Construction The paper treats iteration, historical retention, and limit formation separately. A later operation cannot erase a failed original occurrence. A repaired object may produce a new successful result, but it cannot retroactively convert the retained earlier incidence into success. Finite iteration preserves the original interior when each step satisfies the stated operator conditions. Limit or infinitary outputs require additional hypotheses: endpoint existence; domain and profile compatibility; preservation of interior membership; a valid limit constructor; or a well-founded infinitary assembly. Finite-stage safety alone does not imply eventual success, termination, convergence, or admissibility of a proposed endpoint. Closure by Exhaustion The main operator-closure theorem gives an exhaustive classification of same-scope failure. A proposed unchanged realization can fail through: an incorrect or surplus original result; an omitted required result; a lost distinction in the retained interface; an applicability or output collision under operator descent; failed local readiness or warrant; an unsafe successor or continuation; an unsupported grounded rule entrance; erasure of a retained historical defect. These witnesses may overlap. They are exhaustive for the independently specified original task and its declared obligations. Consequently, no same-scope realization can acquire a new authority merely by: adding an operator name; composing methods; changing representation; introducing a parameter; invoking a selector or completion; mutually licensing rules; taking a fixed point without grounded entry; or deleting required history. A legitimate new operation remains possible, but its new domain, result, warrant, and relation to the original task must be independently established. Scope and Nonclaims This paper does not claim: that every mathematical or physical operator belongs to one fixed candidate universe; that all possible future techniques have been syntactically enumerated; that a failure to compute an answer disproves its semantic determinacy; that every conservative repertoire is finite or effectively decidable; that finite safety implies liveness or eventual success; that a new enriched-domain operator is impossible; that all mechanisms with the same scalar output are identical; that operator closure supplies a physical law, numerical value, or unique occupant; that the entire paper has been newly formalized in a proof assistant. The theorem is fixed-domain and same-scope. A changed bearer, question, use family, input carrier, or authorization basis is a new problem requiring its own construction and proof. Role in the AASC Programme This paper supplies the operator-level closure layer of the AASC foundation: kernel necessity → fixed-domain admissibility → complete interface → operator descent → compositional closure → conservative repertoire → grounded authority → witnessed exhaustion. Its central conclusion is that same-scope operators are closed by exact admissibility tests. A faithful redescription may preserve an operator, and a warranted new construction may extend a domain. But no hidden same-domain operator, fifth classifier, retrospective repair, or ungrounded authority route remains once the original graph, interface, continuation, and grounding obligations have been fully specified. Keywords AASC; same-scope operator closure; admissibility; standing preservation; fixed-domain exhaustion; partial operator descent; quotient factorization; conservative extension; grounded authorization; operator identity; joint compatibility; faithful redescription; greatest admissible interior; no same-domain operator rescue; mathematical foundations