{"id":"06c76ae9-ed28-4c75-8979-1d5657e01b08","arxiv_id":"2607.16997","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"A formal proof's route novelty can be scored as the surprisal of its dependency footprint under a prior built from an earlier Mathlib snapshot; the score matches expert majority choice in 53/76 pairs and shows an endpoint-calibration tendency with the absolute score gap.","lead":"PriorProof is a new method that scores how nonstandard a formal proof's route is, by measuring how surprising its use of existing library facts looks under a prior built only from earlier theorems. In a blinded test against expert mathematicians it agrees on 53/76 pairs (69.7%), with the most reliable agreement (84.2%) at the largest score gaps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Detour normalization is the load-bearing gap: the nested redundancy backstop is dormant on live proofs, so Eq. (4) can reward eliminable detours and may not measure proof-route nonstandardness.","rationale":"The reader's weakest assumption identifies §3.7's directness premise; I agree and sharpen it. The single most load-bearing condition is that the proof term as written is a faithful record of the route. The paper's own Table 1 shows the only mechanism that could enforce this—nested redundancy detection—is dormant (0 subterm records), and §7 concedes the family hierarchy is organizational. Together these mean Eq. (4) can be inflated by a proof that re-derives an already-available theorem through rare namespaces; such a proof would receive a high surprisal score despite following no genuinely nonstandard route to the result. This matters most for the paper's intended downstream use (§6): machine-generated proofs are precisely where eliminable detours are common. The endpoint-calibration result and the decomposability claim inherit this fragility. My proposed check—turning on subterm extraction and rerunning the redundancy backstop—would settle whether any live detours exist and whether removing them changes the agreement tables. If the tables move, the headline should be weakened to 'dependency-footprint surprisal under a namespace prior' unless normalized. Since the paper already discloses these limitations, the reader's CONDITIONAL verdict is the right one; no further adjustment is needed.","tokens_in":15888,"tokens_out":11005,"duration_ms":110616,"concrete_test":"Implement proof-subterm extraction for the 248 scored targets (e.g., traverse elaborated terms using Lean's pp.all or the kernel's Expr recursion), run the existing nested redundancy check from §3.7 against L<bt, and flag any sub-derivation whose conclusion is definitionally equal to a pre-bin theorem. Recompute S_t(D) after masking or removing flagged detours, then rerun the Table 9 quartile agreement and Table 5 overall agreement. If the largest-gap bin's 16/19 or the 53/76 majority agreement changes materially, the endpoint-calibration claim is confounded by detour inflation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.7 defines the directness premise: the score is meaningful only when the proof is reasonably direct. The mechanism intended to enforce this—the nested redundancy backstop—is inactive in the reported corpus: Table 1 records 0 proof-subterm rows, so no live proof can be checked for sub-derivations that re-derive an available pre-bin theorem through unusual machinery. Consequently an eliminable detour contributes its rare dependency families to Eq. (4) with positive weights and high surprisal, inflating S_t(D) without any change in the mathematical route. The 'zero live redundancy hits' is vacuous: there are zero opportunities to hit. This is not a peripheral caveat; the paper's motivating downstream use is scoring machine-generated proofs (§6), exactly the regime in which such detours are expected. The family buckets themselves are inherited Mathlib namespaces, which §7 concedes are 'organizational rather than a canonical ontology of mathematical technique'; the score therefore conflates namespace rarity and proof presentation with route nonstandardness. Until the nested backstop is live or the score is shown invariant to eliminable detours, the headline claim that S_t(D) is a measure of proof-route nonstandardness—rather than dependency-footprint surprisal under a namespace prior—is not established for the intended population.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces PriorProof, a mechanical, point-in-time measure of proof-route nonstandardness for Lean/Mathlib. For a theorem, it extracts a weighted dependency-family footprint from the elaborated proof term, builds a statement-conditioned smoothed prior over families from an earlier quarterly snapshot (with pre-bin retrieval and per-bin encoder training), and scores the weighted surprisal. In a blinded topology study, agreement with the three retained raters' majority is 53/76 distinct pairs (69.7%), with 11/12 on canonical contrasts and 42/64 on stratified pairs; score-gap quartiles show an endpoint tendency (largest bin 84.2% vs. smallest 63.2%), while the best language-model baseline reaches 60/76 (78.9%) with McNemar p=0.210. The paper carefully presents the measure as a decomposable reliability signal, not a universal ranking metric.","tokens_in":16154,"tokens_out":6222,"duration_ms":61211,"significance":"If the construct is accepted, PriorProof offers a genuinely novel, leakage-controlled operationalization of a concept (time-relative proof-route nonstandardness) that is otherwise difficult to mechanize. The strengths are real: the temporal leakage discipline is meticulous (pre-bin slicing, per-bin encoders, reuse counts excluding target snapshots); the method requires no hand-built ontology or human labels for scoring; the paper releases code under MIT; and the evaluation is unusually honest, reporting nonmonotone bin results, a nonsignificant LM comparison, and a claim-to-evidence map. The endpoint-calibration tendency for score-gap quartiles, if replicated, is a practically useful reliability signal for triaging formal proofs. The manuscript is not a claim to beat LLM judges but a proposal for a mechanically decomposable, time-anchored signal.","major_comments":[{"comment":"The directness assumption is load-bearing for the central claim. The paper asserts that S_t(D) in Eq. (4) measures proof-route nonstandardness, but the only mechanism designed to exclude eliminable detours—the nested redundancy backstop—has zero live coverage (Table 1: 0 records with extracted proof subterms). Consequently, a proof that re-derives an available theorem through exotic machinery would have its score inflated by rare dependency families, with no change in mathematical route. This is especially relevant to the motivating downstream use on machine-generated proofs (§6), where detours are expected. The limitation is acknowledged in §7 but the abstract and title still claim the construct. I ask the authors to either (a) activate subterm extraction on a sample and report the redundancy backstop's live behavior, (b) demonstrate score invariance under detour-adding transformations","section":"Section 3.7 / Table 1"},{"comment":"The human ground truth rests on post hoc exclusion of one rater. The paper states that outcome statistics were first computed with all four raters, and the exclusion rule (consistency statistic) was constructed after inspection. While the timing evidence and the extreme consistency scores for the screened rater make the exclusion plausible, the headline agreement figures (53/76, 42/64, etc.) are based on a panel selected in light of the outcomes. To make the central validation more robust, the authors should report all-four-rater aggregates (at least in an appendix) and provide a sensitivity analysis under alternative inclusion criteria. Without this, the risk of selection bias in the principal empirical result remains unquantified.","section":"Section 5.2"},{"comment":"Construct validity of the family buckets is an open concern. The measure's interpretability claim—that the score decomposes into terms w_i[-log q_t(f_i|D)] and reflects mathematical technique—depends on Mathlib namespaces being a reasonable proxy for technique. The paper itself concedes in §7 that the hierarchy is 'organizational rather than a canonical ontology.' This creates a risk that the metric conflates namespace rarity (a presentation artifact) with route nonstandardness. The human agreement is encouraging, but it does not isolate whether the raters are responding to the same signal. I request a diagnostic that separates these components, e.g., computing the correlation between S_t(D) and the average namespace-prefix rarity of the footprint, or a stratified analysis on pairs matched for namespace rarity but differing in dependency composition.","section":"Section 3.8 / Section 7"}],"minor_comments":[{"comment":"The language-model baseline lacks exact prompt strings and date-pinned model identifiers, and the two identical-setting runs differ on 17/100 presentations. The paper is transparent about this, but as the baseline is central to the empirical comparison, consider including the exact prompt templates in an appendix (the human prompt is given, but the LM prompt variants are only described).","section":"Section 4.4 / Appendix F"},{"comment":"The no-namespace ablation shows a positive mean Δsurprisal but only 40% of declarations move upward. This nuance is stated but could be interpreted explicitly in the text as evidence that namespace smoothing serves mainly as a small floor in the topology slice.","section":"Table 3"},{"comment":"The 'cross-bin overlap' metric and the prespecified 0.75 threshold are not defined. Please specify how overlap is computed (e.g., neighbor sets, embedding cosine similarities) so that the threshold decision is interpretable.","section":"Section 3.3"},{"comment":"Equation (1) uses 'L< b(t)' with italic L; the notation is slightly confusing because 'L' is both a library and the less-than relation. A clear calligraphic or bold symbol would improve readability.","section":"Section 3.1"},{"comment":"The phrase 'after repeat collapse' could be clarified in the main text to indicate that the quartile bins are on the 76 distinct pairs, not on the 100 presentations. The figure and table are clear, but the narrative should reinforce this.","section":"Section 5.5"}],"recommendation":"major_revision","confidential_remarks":"This is a careful, transparently written paper with a genuinely novel operationalization and strong leakage discipline. The major concerns are the dormant redundancy backstop and the post hoc rater exclusion; both are acknowledged but not resolved. The central claim can be defended by either activating the backstop on a sample or narrowing the construct's label. The paper is not fatally flawed, but it needs additional evidence or a more precise claim before it can be accepted. Recommend major_revision rather than reject because the methodological framework is sound and the empirical work, while noisy, is honestly reported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"PriorProof is worth your time: it is a genuinely new operationalization — quarterly point-in-time weighted surprisal of a proof's dependency footprint against a statement-conditioned prior built only from earlier Mathlib snapshots — and the paper respects its own boundaries. The leakage control is unusually careful: per-bin encoders, pre-bin reuse counts, and a strict pre-bin slicing discipline. The blinded human study is small but transparently reported, with the post hoc rater screening disclosed rather than hidden.\n\nWhat impressed me: the score is mechanically decomposable, and the paper makes no overreach — it frames the score-gap result as an endpoint tendency, not a calibration curve. The canonical contrast check (11/12) and the stratified pair agreement (65.6%) are what they are: moderate but real evidence that the signal tracks something humans recognize.\n\nSoft spots, in proportion. The most serious is the directness/normalization premise, which the stress-test correctly identifies. Section 3.7 says the score is meaningful only when the proof is 'reasonably direct,' but the nested redundancy backstop is dormant — Table 1 shows zero proof-subterm records — so eliminable detours can inflate surprisal without changing the mathematical route. The paper acknowledges this in Limitations, but it remains a load-bearing gap for the stated downstream use of scoring machine-generated proofs, exactly the regime where detours are likely. That said, it does not sink the paper; it defines the boundary of the current claim.\n\nTwo smaller issues: the rater screening rule was constructed post hoc, and the released pipeline cannot regenerate the reported human/model aggregates because responses are private. Both are disclosed, and the paper gives enough detail to mount a fresh study. The LM baseline having a higher nominal agreement (78.9% vs 69.7%) with McNemar p=0.210 is not a flaw — the authors say it plainly — but it does temper the 'reliability indicator' framing.\n\nOverall: this is a serious, honest paper. The construct is narrow and the evidence is modest, but the method is reusable and the reporting sets a good standard. I'd send it to a thoughtful referee, and I'd read a follow-up that actually switches on subterm-aware normalization and re-runs the study in a second domain.","headline":"A careful, honestly bounded operationalization of proof-route novelty with a real directness gap that limits its use on machine-generated proofs.","tokens_in":16729,"tokens_out":2051,"would_cite":true,"duration_ms":20489,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A mechanical score can flag when a formal proof takes a nonstandard route relative to its own time.","keywords":["proof-route nonstandardness","surprisal","Lean theorem prover","Mathlib","formal proof dependencies","point-in-time evaluation","novelty measurement","retrieval-conditioned prior"],"falsifier":"Take a theorem whose standard proof is well known, prepend a deliberately contorted detour that re-derives an available lemma using rare imported machinery, and check whether PriorProof's surprisal score rises while human raters still judge the route standard; if the score rises sharply, the directness assumption is violated in a way that breaks the metric's reliability on exactly the machine-generated proofs it is meant to monitor.","tokens_in":15652,"feed_emoji":"📐","tokens_out":3943,"duration_ms":33419,"temperature":0.7,"pith_summary":"PriorProof tries to establish that 'how nonstandard is this proof's route' can be operationalized as a number. For a theorem in the Lean proof assistant, it reads the proof term, extracts the families of prior machinery the proof actually uses, builds a probability distribution over those families from an earlier quarterly snapshot of the library and similar theorem statements, and scores the weighted surprisal. The paper claims this yields a leakage-controlled, decomposable signal that agrees with expert judgments on just under 70% of 76 distinct proof comparisons, with agreement rising to 84% when the score gap between the two proofs is large. The intended use is not a universal novelty ranking but a reliability indicator that tells reviewers which comparisons are worth inspecting.","feed_headline":"A surprisal score flags nonstandard proof routes","feed_subtitle":"PriorProof scores Lean proofs against earlier library snapshots, and the score gap tells you when to trust it.","key_machinery":"The central object is the weighted surprisal S_t(D) = sum_i w_i [-log q_t(f_i | D)] (Eq. 4): the observed proof's dependency families f_i, weighted by w_i, are scored against a mixture prior q_t(f | D) (Eq. 3) combining retrieved-neighbor, namespace, module, and global family distributions, all fit from a quarterly snapshot strictly before the proof's bin. The footprint is produced by extracting constants actually used in the proof term, filtering plumbing, unfolding recent helpers to an established-machinery frontier, and backing off to families with at least five prior occurrences. The absolute gap between scores of two proofs (Eq. 5) is the confidence signal.","core_discovery":"The paper's load-bearing claim is that time-relative proof-route nonstandardness reduces to weighted surprisal of a proof-term dependency footprint under a statement-conditioned, hierarchically smoothed prior built only from earlier library data (Eq. 4). The prior is induced by retrieving similar prior theorem statements from a pre-bin snapshot; the footprint is read mechanically from the elaborated proof term. In a blinded topology study with three retained raters, the metric agrees with the rater majority on 53/76 distinct pairs (69.7%), including 11/12 canonical contrasts and 42/64 stratified pairs; score-gap quartiles show an endpoint tendency, with 16/19 agreement in the largest-gap bin","pith_inferences":["Applied to a second formal library or another Mathlib domain, the same pipeline would test whether the endpoint-calibration tendency generalizes or is an artifact of the topology slice.","The score could be used as a diversity-seeking reward in proof search to nudge provers toward routes that earlier libraries would not predict, though the paper's own redundancy caveats would need to be enforced to prevent gaming.","Scoring historical proofs against their own earlier snapshots could trace when a once-surprising technique became standard, making the measure a tool for the history of mathematical method.","If nested proof-subterm extraction were enabled (currently dormant), the redundancy backstop could catch padded proofs, which may sharpen the gap-reliability signal in machine-generated proofs."],"forward_implications":["Proof-route novelty in formal mathematics can be measured without hand-built ontologies, human labels, or access to the proof that is being scored.","The absolute score gap between two proofs indicates which pairwise comparisons are reliable: in this sample, large-gap comparisons agree with the expert majority more often, while small-gap comparisons are near chance.","Every score decomposes into named dependency families with their surprisals, so a high score can be audited to see which unexpected machinery is driving it.","Because scoring happens after proof checking, the method can compare human, machine-generated, and search-variant proofs of the same theorem under the same temporal counterfactual.","The metric's per-bin training discipline prevents future proof structure from leaking into the prior through retrieval or encoder fine-tuning."],"fun_headline_variants":["Surprisal score pegs nonstandard Lean proof routes","Time-anchored proof novelty: surprisal against old Mathlib","PriorProof: scores proof routes vs. prior snapshots, no labels","69.7% expert agreement on nonstandard proof-route calls","No labels needed: surprisal scores nonstandard Lean proofs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The score is meaningful only when the proof term as written is a reasonably direct record of the route taken, but the live pipeline records no proof subterms, so a proof that pads its route with an eliminable detour through exotic machinery could be scored as novel rather than redundant.","fun_headline_variants_meta":{"raw":{"variants":["Surprisal score pegs nonstandard Lean proof routes","Time-anchored proof novelty: surprisal against old Mathlib","PriorProof: scores proof routes vs. prior snapshots, no labels","69.7% expert agreement on nonstandard proof-route calls","No labels needed: surprisal scores nonstandard Lean proofs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001812,"raw_usage":{"total_tokens":7061,"prompt_tokens":931,"completion_tokens":6130,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":6049}},"tokens_in":675,"tokens_out":6130,"duration_ms":39913,"temperature":1.0,"reasoning_tokens":6049,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:19:17.698669+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a theorem whose standard proof is well known, prepend a deliberately contorted detour that re-derives an available lemma using rare imported machinery, and check whether PriorProof's surprisal score rises while human raters still judge the route standard; if the score rises sharply, the directness assumption is violated in a way that breaks the metric's reliability on exactly the machine-generated proofs it is meant to monitor.","supporting_citations":[],"review_version":1}