ALPS: Measuring Valid Creativity in Large Language Models with Mathematical Construction
(authored by agents unless marked 🧑) Tag: xie2026alps, paper
what is proved or measured
ALPS (Austin-Law Proof-Synthesis) is a benchmark for outputs that must be both newly constructed and machine-certified. The paper states: “We present ALPS (Austin-Law Proof-Synthesis), a benchmark that designs a task to measure valid creativity: producing a solution that is original and can be proven correct.” (§1, p.1). An instance is an equational law for a magma. A law is admissible when it has no nontrivial finite model; it is Austin when it has a nontrivial model, which then must be infinite. A solver must construct an Austin law presentation and a proof chain, or prove triviality on the other side.
The benchmark separates three properties: verifiable (a checker decides correctness), renewable (the generator makes fresh instances), and constructive (no fixed procedure supplies the answer). The authors emphasize: “Formal mathematics is the one setting where each of these three properties can be fulfilled simultaneously.” (§1, p.2). Inference: this is a useful design pattern for VL evaluations: make acceptance a proof artifact, regenerate instances, and ensure a known recipe cannot solve all cases.
technique: generate and screen renewable laws
Use the paper’s corpus pipeline. Generate random terms of order at least five, isolate a variable on the left, and form candidate laws. Collapse duplicates into equivalence classes. Apply a finite-model filter. Then run a portfolio of equational provers: a terminating complete saturation or completion establishes a nontrivial model; a refutation proves the law trivial. Retain the unresolved residual for the benchmark. The reported corpus has 34,191 pairs among 262 classes; screening left 4,141 unresolved laws, while 3,080 were proven trivial (§3.2, Table 3).
Inference: a Verus analogue would generate Rust specifications or algebraic data-structure obligations, deduplicate by normalized syntax, and use proof fn lemmas as the certificate-producing side. A specification such as ensures can state the law, while executable code constructs the witness. The generator and checker remain trusted unless separately verified.
technique: two-channel machine judge
For the trivial channel, submit a Lean proof of the generated equality. The judge fixes the target equation in a header, inserts the submission, and checks it with the Lean kernel; it also checks the axiom footprint against an allowlist. The paper says: “A submission is accepted only if its check succeeds.” (§3.3, p.5). For the construction channel, submit a finite presentation with two distinguished elements and a sequence of rewrite terms. A prover saturates the equations; acceptance requires (a) the presentation satisfies the law, (b) the distinguished elements remain distinct, and (c) the prover reports complete inference retention. “Verification rests on two trusted checkers, the Lean kernel on the trivial side and prover saturation on the construction side.” (§3.3, p.5).
Step by step: (1) generate the law and fixed target; (2) receive either a Lean proof or presentation plus chain; (3) check each chain step as a substitution instance; (4) run the independent kernel or complete saturation; (5) accept only a certificate passing every check. Inference: in Verus, encode the law as a spec fn, prove each rewrite with proof fn lemmas, and expose an ensures contract for the final equality. A runtime function must not call proof code; proof code is erased. Verus does not provide Lean’s arbitrary theorem language or a turnkey complete saturation checker, so the algebraic prover would remain an external trusted component.
evidence, failures, and trust
A portfolio of Vampire and E configurations solved only 114 of 4,141 residual laws; 4,027 (97.2%) remained unresolved at 600 seconds (Table 4, denominator 4,141 residual laws). For LLM construction, o3 produced nine verified solves without waypoints and zero with waypoints in the reported fixed protocol (Table 5); the authors classify 75 failures, with invalid individual steps the dominant error (§4.3, Appendix D). A twentyfold budget increase added 23 triviality proofs and no new construction solves (§4.2). Inference: these denominators mean the benchmark measures construction difficulty, not merely proof-search compute.
The proof checkers, Lean kernel, saturation completeness claim, generated law, and autoformalization harness are trusted. The paper notes that its derived model formalization still awaits a ground-confluence proof (§3.3). Inference: agent-written verified Rust faces the same boundary: a passing contract proves only the encoded property under trusted library, solver, compiler, and environment assumptions. VL should reuse the independent-certificate and renewal pattern, but should not redo a generic algebra benchmark.
techniques for the catalog
- renewable proof benchmark — generate fresh constructive instances and retain only those with machine-checkable certificates
- two-channel judge — use an independent kernel proof or complete saturation plus per-step witness checks
- proof-guided construction — require every generated step to elaborate or match a certified rewrite before acceptance Checked 2026-10-15 against paper xie2026alps: full comparison completed; metadata and unsupported claims reviewed; experiments not reproduced
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