{"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":"finite_polynomial_meaning_boundary_v0","proposal_index":1,"version":0,"title":"Finite-Polynomial Meaning Boundary for Representation-Stable Proofs","problem":"A formal mathematics library defines multivariate polynomials as normalized ordered lists of monomial-coefficient pairs. Downstream proofs unfold constructors, inspect list order, invoke normalization details, and use representation-level equality even when their mathematical claims concern only polynomial coefficients or evaluation. A proposed sparse map, dense bounded array, or expression-DAG representation therefore cannot replace the list representation without revising proofs whose conclusions are mathematically representation-independent.","actors":["Mathematicians authoring polynomial theorems","Maintainers of the formal mathematics library","Implementers of alternative polynomial representations","Reviewers responsible for admitting definitions and proofs","Downstream users importing polynomial theorems"],"observable_state":"In a bounded dependency inventory, selected downstream proofs pattern-match on the pair list, unfold normalization, refer to monomial enumeration order, or prove equality by equality of stored sequences. Re-running those proofs against a second representation fails before any mathematical disagreement has been identified. The two representations also lack a shared criterion for deciding whether edge cases such as duplicate monomials, zero coefficients, variable renaming, and multiplication denote the same abstract polynomial.","consequence":"Changing polynomial representation requires coordinated proof rewrites, while two implementations can agree on ordinary examples yet disagree on abstract meaning at edge cases. Maintainers cannot distinguish a legitimate internal re-representation from a change to polynomial semantics.","affected_objective":"Preserve the correctness and reuse of representation-independent polynomial theorems while allowing the underlying finite-polynomial representation to change.","intervention":"Introduce an opaque finite-polynomial component whose abstract value is a finitely supported coefficient function from exponent vectors in N^n to a fixed commutative ring R. Export only zero, one, monomial construction, coefficient observation, addition, negation, multiplication, variable renaming, evaluation, and extensional equality. Specify each operation by coefficient-function semantics; require finite support, dimensional validity, and preservation of the representation invariant; reject malformed imported data with a structured error and no constructed value; and declare ordering, normalization steps, node identity, stored zeros, and traversal traces non-contractual. Each concrete representation must provide a representation invariant, an abstraction function into the finite-support model, and proofs that its exported operations realize the abstract laws. A shared black-box theorem suite and executable finite-model tests gate substitution, while reviews reject downstream proofs that cross the opaque boundary.","structural_mapping":[{"archetype_element":"Abstract component","domain_realization":"The opaque type Poly(n,R), understood as a finite-support coefficient function rather than as a list, map, array, or expression graph."},{"archetype_element":"Observable operation surface","domain_realization":"Construction, coefficient lookup, ring operations, variable renaming, evaluation, and extensional equality."},{"archetype_element":"Behavioral laws","domain_realization":"Coefficient-wise addition and negation, convolution multiplication, ring identities, extensionality by coefficients, and evaluation and renaming homomorphism laws."},{"archetype_element":"Hidden representation","domain_realization":"Monomial ordering, normalization algorithm, sparse or dense storage, DAG sharing, caches, and internal proof witnesses."},{"archetype_element":"Representation invariant and abstraction function","domain_realization":"Each implementation states its valid concrete states and maps every valid state to exactly one finitely supported coefficient function."},{"archetype_element":"Error and side-effect boundary","domain_realization":"Malformed external encodings or dimension mismatches return a typed validation failure without yielding a polynomial; mathematical operations are immutable and expose no mutation trace."},{"archetype_element":"Conformance oracle","domain_realization":"One parameterized suite of algebraic theorems plus executable checks over bounded exponent vectors and finite coefficient rings, using only exported operations."},{"archetype_element":"Substitutability rule","domain_realization":"An implementation is eligible for the pilot theorem corpus only after discharging abstraction-function obligations and passing the shared public-surface suite without representation-specific exceptions."},{"archetype_element":"Leakage governance","domain_realization":"A watchlist and review rule cover ordering, printed form, error wording, traversal behavior, and definitional equality that downstream proofs might mistake for mathematical promises."}],"mechanism_mapping":[{"mechanism_slug":"abstract_data_type_specification","role":"Defines polynomials first as finite-support coefficient functions, then requires each concrete representation to supply an invariant, an abstraction function, and operation-correctness proofs.","counterfactual_removal":"Without it, the public operations would lack a common mathematical meaning, so passing examples could not certify a re-representation."},{"mechanism_slug":"opaque_type_or_module_boundary","role":"Prevents theorem authors from unfolding polynomial storage or pattern-matching on its constructors.","counterfactual_removal":"Without opacity, downstream proofs could continue making representation details into implicit premises despite the written specification."},{"mechanism_slug":"design_by_contract_clause","role":"States preconditions and postconditions for dimension-sensitive operations and fixes the behavior of malformed imports independently of any implementation.","counterfactual_removal":"Without explicit clauses, implementations could diverge on invalid variable maps, duplicate terms, or failure behavior while sharing the same signatures."},{"mechanism_slug":"black_box_contract_test_suite","role":"Applies the same exported-operation theorem and example suite to the ordered-list and alternative implementations and makes conformance a substitution gate.","counterfactual_removal":"Without a common suite, each representation would be judged by its own proofs and examples, leaving cross-implementation divergence unresolved."},{"mechanism_slug":"property_based_conformance_test","role":"Generates bounded polynomials and operation sequences to check algebraic laws and preservation of observable semantics across many paths.","counterfactual_removal":"Without generated sequences, the pilot would exercise only hand-selected cases and could miss interactions involving normalization, cancellation, or multiplication."},{"mechanism_slug":"reference_implementation_differential_test","role":"Uses a deliberately simple finite-support function model as an independent comparison oracle for bounded executable cases, comparing coefficients and evaluations rather than storage.","counterfactual_removal":"Without the model comparison, algebraic laws alone could be satisfied by two implementations sharing the same incorrect coefficient convention."},{"mechanism_slug":"representation_leakage_probe","role":"Compares all client-visible behavior with the promised contract and flags accidental observables such as term order, printer layout, and definitional equality.","counterfactual_removal":"Without the probe, clients could reconstruct and depend on hidden representation through observable but undocumented behavior."},{"mechanism_slug":"abstraction_barrier_code_review","role":"Requires reviewers to classify every new polynomial dependency as contractual mathematics or prohibited representation reach-through.","counterfactual_removal":"Without human review, technically accessible or newly introduced escape hatches could bypass the mechanical boundary and harden into proof dependencies."}],"causal_chain":["Downstream theorems currently use concrete polynomial constructors and normalization facts as proof premises.","The intervention defines polynomial meaning independently as a finitely supported coefficient function and makes concrete storage opaque.","Representation invariants and abstraction functions connect each concrete implementation to that shared meaning.","Operation clauses and algebraic laws specify what clients may infer from every exported operation sequence.","The shared conformance oracle tests and proves those obligations without inspecting representation.","Leakage checks and review prevent ordering, printing, definitional equality, or helper state from becoming substitute interfaces.","Consequently, a representation change that preserves the contract can be evaluated as a local implementation substitution rather than as an uncontrolled change to mathematical meaning."],"baseline":"Retain the normalized ordered-list definition as publicly unfoldable, document preferred lemmas informally, and repair downstream proofs individually whenever normalization or storage changes.","nearest_rivals":["Keep the ordered-list representation canonical and optimize only its internal algorithms, avoiding substitution altogether.","Add compatibility lemmas translating a new sparse representation to the existing list type while leaving downstream access to list constructors intact.","Conduct a one-time mechanical rewrite of affected proofs from list reasoning to coefficient reasoning without installing an opaque boundary or reusable conformance gate.","Represent every implementation as an explicit equivalence to the current list model and require clients to transport theorems across that equivalence."],"remaining_contrastive_claim":"Compared with a canonical representation, translation wrappers, or a one-time proof rewrite, the proposal makes one testable claim: a coefficient-function contract, enforced opacity, and a shared conformance oracle are sufficient for both an ordered-list and an independently implemented sparse-map polynomial type to support the selected representation-independent theorem corpus without those theorem proofs accessing either representation.","authority_safety":{"decision_authority":"The library maintainers who own the polynomial definition and its public theorem namespace decide whether to create the pilot boundary; owners of downstream theorem files decide whether their proofs enter the pilot. Passing the pilot does not itself authorize replacing the existing library type.","authorized_first_step":"Create an isolated experimental namespace containing the abstract contract, a wrapper around the current ordered-list implementation, an independent small sparse-map implementation, and copies of a bounded set of downstream lemmas.","excluded_actions":["Deleting or changing the existing public polynomial definition","Automatically rewriting downstream proofs outside the pilot namespace","Declaring either implementation generally substitutable from bounded tests alone","Turning current printing, ordering, normalization, or definitional-equality behavior into contractual promises without maintainer review","Weakening a failed law merely to admit an implementation"],"halt_rollback":"Stop if the pilot requires an unreviewed escape hatch, changes the statement of a selected mathematical theorem, or produces conflicting abstract results. Roll back by removing the isolated namespace and copied pilot proofs; the existing definition and downstream corpus remain untouched."},"negative_tests":{"strongest_counterevidence":"The current polynomial interface may already expose a stable coefficient-and-evaluation abstraction, with downstream list-level facts confined to implementation lemmas; if an independent representation can replace the current one by changing only an internal module binding, the proposed new boundary adds little causal leverage.","problem_falsifier":"A dependency audit of the bounded theorem corpus finds no proof that unfolds storage, depends on enumeration or normalization, or fails when the current implementation is hidden behind an existing abstract interface.","intervention_falsifier":"Both implementations satisfy the proposed laws and executable suite, yet a selected theorem expressible solely in the proposed abstract vocabulary has different truth conditions between them, or cannot be proved without representation access. Either result shows that the contract or oracle is insufficient for the claimed substitution.","risks":["The contract may over-specify coefficient ordering, printing, or equality and thereby freeze the current representation.","The contract may under-specify variable renaming, malformed inputs, coefficient-ring assumptions, or evaluation behavior, allowing divergent implementations to pass.","The simple reference model may share a semantic mistake with an implementation or be treated as authoritative when a divergence actually exposes an underspecified definition.","Generated tests over bounded rings and exponents may not exercise failures arising only in larger or different algebraic structures.","Opaque access may make representation-specific proofs awkward and encourage unofficial escape hatches.","Proof and review obligations may exceed the value of substitution for the selected theorem corpus."]},"next_evidence_step":"In an isolated namespace, inventory exactly 20 existing downstream polynomial lemmas chosen before implementation. Classify each dependency as abstract or representation-specific; write the finite-support contract; wrap the ordered-list implementation; build one independent sparse-map implementation; and attempt to re-establish the same 20 lemma statements using only the public surface. Run the shared algebraic suite and bounded differential checks over one finite coefficient ring and predeclared exponent limits. Record every failed law, required escape hatch, ambiguous contract clause, and theorem whose statement would need alteration; do not migrate production definitions.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"No prior proposal is used or compared; this is the sole proposal in the sealed output.","revision_record":{"parent_version":null,"progress_targets_addressed":["Initial complete proposal","Concrete mathematical actors and observable state","Representation-preserving causal mapping","Bounded and reversible first evidence"],"conceptual_changes":["Initial version defines polynomial meaning as a finitely supported coefficient function and separates it from storage choices."],"operational_changes":["Initial version limits work to an isolated namespace, two implementations, and 20 preselected lemmas."],"evidence_changes":["Initial version specifies a dependency audit, proof-port attempt, algebraic conformance suite, and bounded differential test."],"claim_changes":["Initial claim is limited to substitutability within the selected theorem corpus and makes no novelty, prevalence, demand, or effect-size assertion."]}}