{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp06_four_proposal_generalization60_20260803","cell_id":"representation_independent_interface_contract__mathematics","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_id":"universal_completion_contract_v0","proposal_index":2,"version":0,"title":"Universal-Property Contract for Construction-Independent Metric Completions","problem":"A collaborative analysis development defines the completion of a metric space X as equivalence classes of Cauchy sequences. Subsequent theorems unfold sequence representatives, quotient relations, and the chosen lifting machinery even when their mathematical conclusions concern only the embedding of X, completeness, limits, or extension of uniformly continuous maps. A completion constructed instead from Cauchy filters or as the closure of a dense embedding therefore cannot supply those theorems without construction-specific rewrites.","actors":["Analysts proving results about metric completions","Authors maintaining the completion construction","Downstream mathematicians applying completion theorems","Implementers of alternative completion constructions","Reviewers tracking logical assumptions and theorem dependencies"],"observable_state":"Within a preselected theorem corpus, proofs refer to Cauchy-sequence representatives, quotient constructors, representative-selection lemmas, or definitional equality with the canonical completion type. An alternative pair consisting of a complete space C and a dense isometric embedding eta:X→C cannot instantiate those theorems directly, despite satisfying the intended mathematical role. It is also unclear which logical assumptions, extension laws, and uniqueness conditions a replacement must preserve.","consequence":"The chosen construction becomes an unintended premise of downstream mathematics. Results must be transported or reproved individually, and maintainers lack a representation-independent criterion for determining when another complete space is the same completion for purposes of those results.","affected_objective":"Make the selected theorems about metric completion reusable across mathematically equivalent constructions while preserving their stated assumptions and conclusions.","intervention":"Define an opaque CompletionPackage(X) by its universal behavior rather than by quotient representatives. The public contract contains a carrier C, an isometric embedding eta:X→C with dense image, a proof that C is complete, and an extension operation: for every complete metric space Y and uniformly continuous f:X→Y, extend(f):C→Y is uniformly continuous, agrees with f after eta, and is the unique map with those properties. Identity and composition coherence are explicit laws. Missing completeness or uniform-continuity premises make extension unavailable rather than producing a default map. Logical axioms, universe requirements, and any sanctioned construction-specific inspection are recorded as part of the contract. Sequence, filter, and closure constructions remain hidden implementations; each must discharge the same package obligations before downstream theorems may treat it as a completion.","structural_mapping":[{"archetype_element":"Abstract component","domain_realization":"CompletionPackage(X), characterized by its role as a completion of X rather than by the elements used to construct it."},{"archetype_element":"Observable operation surface","domain_realization":"The carrier metric space, dense isometric embedding, completeness witness, uniformly continuous extension operation, and contract-level equality of maps."},{"archetype_element":"Behavioral laws","domain_realization":"Extension agrees with the original map on X, is unique, preserves identity, and coheres with composition whenever the stated premises hold."},{"archetype_element":"State and transition semantics","domain_realization":"The package is immutable; applying extension produces a map and proof obligations rather than mutating a construction or selecting persistent representatives."},{"archetype_element":"Hidden representation","domain_realization":"Cauchy sequences, filters, quotient representatives, ambient closure points, representative-selection procedures, and internal proof terms."},{"archetype_element":"Representation invariant","domain_realization":"A concrete construction is valid only if its carrier is complete, its embedding is isometric and dense, and its extension operator satisfies agreement, uniqueness, and coherence."},{"archetype_element":"Abstraction function or semantic mapping","domain_realization":"A valid construction is mapped to its universal completion behavior; uniqueness yields the isometric equivalence commuting with the embeddings that relates any two conforming packages."},{"archetype_element":"Error semantics and side-effect limits","domain_realization":"Extension requires explicit completeness and uniform-continuity premises; unmet premises create an undischarged obligation, not an arbitrary result. All exported operations are mathematically pure."},{"archetype_element":"Conformance oracle","domain_realization":"A parameterized theorem suite verifies universal extension diagrams, uniqueness, identity, composition, density consequences, and the absence of representative access."},{"archetype_element":"Substitutability rule","domain_realization":"A completion construction may replace another for contract-parametric theorems when it discharges the package obligations with no stronger undeclared axioms; substitution is through the induced unique isometry, not definitional equality."}],"mechanism_mapping":[{"mechanism_slug":"abstract_data_type_specification","role":"Specifies completion through the abstract package and universal mapping behavior, then requires each construction to prove a validity predicate and semantic correspondence to that behavior.","counterfactual_removal":"Without an abstract specification, completeness and dense embedding would remain an informal resemblance rather than a criterion that certifies an alternative construction."},{"mechanism_slug":"opaque_type_or_module_boundary","role":"Withholds sequence, filter, quotient, and closure constructors from contract-parametric theorem proofs while exporting only the completion operations and laws.","counterfactual_removal":"Without opacity, downstream proofs could continue selecting representatives and silently restore dependence on the incumbent construction."},{"mechanism_slug":"design_by_contract_clause","role":"Assigns the extension operation explicit premises, agreement and uniqueness conclusions, and a policy that unmet premises remain proof obligations rather than yielding a fallback map.","counterfactual_removal":"Without these clauses, implementations could differ on target-space requirements, continuity strength, or extension uniqueness while claiming the same completion interface."},{"mechanism_slug":"black_box_contract_test_suite","role":"Packages a reusable collection of theorems parameterized only by CompletionPackage(X) and applies it unchanged to every proposed construction.","counterfactual_removal":"Without the common theorem suite, each construction would be validated by its own internal lemmas, providing no shared acceptance gate for downstream substitution."},{"mechanism_slug":"metamorphic_behavior_test","role":"Checks representation-independent relations between related extensions, including extension of an identity map and compatibility between extending a composite and composing extensions.","counterfactual_removal":"Without relational checks, agreement on individual embedded points could pass while global extension coherence remained wrong or unstated."},{"mechanism_slug":"representation_leakage_probe","role":"Audits theorem statements and proof dependencies for quotient constructors, representative choices, definitional equality, hidden choice principles, universe changes, or construction-specific simplification behavior.","counterfactual_removal":"Without the audit, clients could depend on hidden proof-theoretic or definitional properties even after element representations were sealed."},{"mechanism_slug":"abstraction_barrier_code_review","role":"Requires reviewers to decide whether each new completion lemma follows from the universal contract or legitimately belongs in a named construction-specific namespace.","counterfactual_removal":"Without this review boundary, convenience lemmas could gradually expose the incumbent construction and make it the unofficial interface again."},{"mechanism_slug":"reference_implementation_differential_test","role":"For one bounded test space, compares the observable embedding and extension diagrams of the incumbent sequence construction with an independently defined closure construction, treating discrepancies as questions about the contract or either construction.","counterfactual_removal":"Without one concrete cross-construction comparison, the package laws could remain internally plausible while failing to capture an observable distinction needed by the pilot theorem corpus."}],"causal_chain":["Downstream completion theorems currently use representatives and quotient machinery from one construction.","The intervention replaces construction-specific premises with a package exposing dense embedding, completeness, extension, and uniqueness.","Each construction independently proves that its internal objects realize this universal behavior and declares its logical assumptions.","The uniqueness law induces an isometric equivalence commuting with the embeddings between conforming packages.","Theorems expressed solely through the package operations transport along that equivalence without inspecting representatives.","A shared theorem suite and leakage review reject implementations or downstream proofs that rely on undeclared construction behavior.","The completion construction can then vary for the selected theorem corpus while its externally meaningful mathematics remains fixed."],"baseline":"Keep the Cauchy-sequence quotient as the canonical completion, encourage authors to avoid unfolding it when convenient, and prove ad hoc equivalence and transport lemmas whenever another construction is introduced.","nearest_rivals":["Prove a single isometric-equivalence theorem between each new completion and the canonical sequence completion, then transport downstream results individually.","Retain the canonical construction but add representative-free convenience lemmas without making the boundary opaque or defining a substitution rule.","Use a type synonym or wrapper around the sequence completion so downstream statements look abstract while definitional access to representatives remains available.","Reprove completion results separately for sequence, filter, and closure constructions.","Require only a complete carrier and dense isometric embedding, omitting an explicit extension operation and its coherence laws."],"remaining_contrastive_claim":"Compared with pairwise transport, a wrapper, or representative-free helper lemmas, the testable claim is that the universal extension and uniqueness contract is sufficient to derive a single unique isometry between two independently constructed completions and to re-establish the preselected downstream theorem statements without access to either construction's representatives or stronger undeclared axioms.","authority_safety":{"decision_authority":"Maintainers of the completion namespace or authors of the affected analysis development approve the experimental package and its assumption ledger. Owners of downstream theorems decide whether copied proofs accurately preserve their statements. Pilot conformance does not authorize replacement of the canonical completion.","authorized_first_step":"Create an isolated experimental module containing the CompletionPackage contract, adapters for the incumbent sequence construction and one independently specified closure construction, and copies of a fixed downstream theorem sample.","excluded_actions":["Changing or deleting the canonical completion construction","Rewriting theorem statements to make the pilot pass without recording the change","Adding choice, classical logic, stronger separation assumptions, or new universe constraints without exposing them in the contract","Treating isometric equivalence as definitional equality","Exposing quotient representatives through an unnamed convenience accessor","Weakening uniqueness or coherence obligations after observing an implementation failure","Migrating downstream modules outside the isolated pilot"],"halt_rollback":"Halt if either adapter requires an undeclared logical axiom, a selected theorem changes mathematical content, the purported universal extension is not unique, or proof progress requires representative access. Roll back by deleting the isolated module and copied proofs; the canonical construction and original theorem corpus remain unchanged."},"negative_tests":{"strongest_counterevidence":"Completeness plus a dense isometric embedding may already be the minimal sufficient interface: if all selected extension, uniqueness, and transport results follow uniformly from those properties, storing an extension operator and separate coherence laws in the public contract is redundant and risks over-specification.","problem_falsifier":"The dependency inventory shows that every selected theorem already depends only on completeness, density, isometry, and ordinary metric-space operations, and an alternative completion package instantiates the corpus without unfolding the canonical construction or adding transport proofs.","intervention_falsifier":"Two constructions both discharge every stated package obligation with the same declared axioms, yet a theorem phrased solely in the contract vocabulary fails to transport along the induced isometry or obtains incompatible results. This would show that the interface or its substitutability rule is insufficient.","risks":["The universal property may be stated with target objects or morphisms that are too broad, too narrow, or circularly dependent on the incumbent construction.","Packaging a chosen extension operator may over-specify data when mere existence and uniqueness would suffice.","A construction may satisfy the mathematical laws only by introducing stronger logical assumptions that are easy to hide in proof terms.","Equivalence up to unique isometry may not satisfy downstream developments that require definitional equality for reduction or automation.","The theorem suite may cover standard diagrams while omitting a coherence condition required by later arguments.","Opaque boundaries may impede legitimate construction-specific theorems and encourage informal escape hatches.","The independently specified comparison construction may share foundational definitions with the incumbent and therefore provide less independent evidence than intended."]},"next_evidence_step":"Before writing adapters, freeze 12 downstream completion lemmas spanning extension, uniqueness, density, convergence, and functoriality. Record every construction-specific dependency and every logical assumption. Define the minimal CompletionPackage contract in an isolated module, adapt the existing sequence-based completion, and specify a closure-based package for one fixed metric space with a declared dense embedding into a complete ambient space. Attempt to prove the same 12 lemma statements solely from the contract, derive the isometry between the two packages, and run the identity and composition coherence checks. Record any representative access, changed assumption, changed statement, or missing law; make no production migration.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Proposal 1 addressed formal polynomial proofs coupled to ordered-list storage and intervened with a finite-support coefficient semantics, opaque polynomial operations, and executable cross-implementation conformance tests. This proposal addresses theorem dependence on a chosen construction of metric completion. Its intervention is a universal mapping property whose uniqueness theorem induces substitution up to isometry; its causal path runs from representative-dependent completion proofs through universal characterization to theorem transport. It can be adopted solely in a metric-completion development and requires no change to polynomial types, coefficient semantics, or the proposal-1 test infrastructure.","revision_record":{"parent_version":null,"progress_targets_addressed":["Second complete proposal","Materially different mathematical problem","Distinct universal-property intervention and causal path","Explicit contrast with proposal 1","Bounded reversible evidence plan"],"conceptual_changes":["Initial version characterizes metric completion by dense embedding and universal extension rather than by its constructed elements."],"operational_changes":["Initial version limits adoption to an isolated package, two construction adapters, and 12 frozen downstream lemmas."],"evidence_changes":["Initial version adds an assumption ledger, cross-construction isometry proof, contract-parametric theorem replay, and extension-coherence checks."],"claim_changes":["The claim is restricted to the selected metric-completion corpus and does not assert novelty, prevalence, demand, or effect size."]}}