{"id":"da2698da-11df-4514-a4e9-ad047804d6fc","arxiv_id":"2607.21082","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A parliamentary majority can be certified with far fewer inspected ballots than certifying every reported seat, using a product of seat-level anytime-valid e-processes and adaptive sampling.","lead":"An election-auditing method that certifies only whether a party truly won a parliamentary majority, rather than verifying every seat, cuts the number of ballots that must be inspected by orders of magnitude in simulations. The method frames the majority question as a partial-conjunction statistical test and adaptively concentrates sampling on the seats that matter.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof assumes cross-seat conditional independence of ballot draws, yet §2.3 claims validity under cross-seat dependence; without that assumption the product e-process may not be anytime-valid.","rationale":"The reader's weakest assumption correctly identifies the conditional independence of next-drawn ballot values across seats. My stress-test agrees with that identification, but goes further: the paper itself asserts in Section 2.3 that the construction 'stays valid even when sampling induces cross-seat dependence'. This is an internal tension with the proof in Appendix B, which relies on step (a) requiring independence across s∈C. The independence condition is not a minor deployment detail; it is a hypothesis of the theorem as proved. If the condition fails—e.g., through a common random sampling stream or any physical coupling of draws across constituencies—the product of e-processes can fail to be an e-process, and the risk limit is not guaranteed. The main result is still plausible for standard independent per-seat sampling, and the simulation evidence is not affected. However, the theorem statement and the robustness claim need to be corrected: either state the independence assumption explicitly in Theorem 1 or prove the stronger statement. Because the proof as written does not establish what the text claims, I would not reject the paper, but I would make acceptance conditional on this clarification. The proposed simulation would settle whether the concern actually lands by demonstrating that the product test's rejection rate exceeds α under a natural dependent-sampling scheme.","tokens_in":14588,"tokens_out":11942,"duration_ms":134144,"concrete_test":"Simulate two seats with identical ballot populations at the null boundary (e.g., N=100 ballots, half for each candidate, so the reported winner did not truly win). For each seat, generate the ALPHA seat-level process M_{s,t} under H_s, but draw the ballot positions for both seats using the same random permutation (shared random index). Run the product E_{1/C,t}=M_{1,t}M_{2,t} until all ballots are exhausted, and record whether sup_t E_{1/C,t} ≥ 1/α. Repeat 10,000 times with α=0.05. If the empirical rejection frequency is substantially above 5%, the product test is not anytime-valid under this dependent sampling, confirming that Theorem 1 requires explicit cross-seat conditional independence.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is Theorem 1, which asserts that the product of seat-level e-processes is anytime-valid. The proof in Appendix B, step (a), explicitly requires that, conditional on F_{t-1}, the next-drawn ballot assorter values X_{s,j_s,n_s(t)} are independent across seats s∈C. This is an extra assumption about independence of the random permutations for different seats; it is not stated in Theorem 1 or in Section 2.1. Worse, Section 2.3 claims the construction 'stays valid even when sampling induces cross-seat dependence, extending Vovk and Wang [17]'. That claim is not supported by the proof: if ballot draws are coupled across seats—for example, by drawing the same ballot position in each seat from a common random stream—the product of nonnegative supermartingales need not be a supermartingale, and the risk limit can be violated. The theorem is likely correct under the standard assumption of independent sampling per seat, but as stated it overclaims robustness to cross-seat dependence. The assumption must be made explicit and the overclaim either proved or removed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a risk-limiting audit (RLA) for parliamentary majority outcomes. It formalizes the audit as a partial conjunction test: rather than certifying every reported winning seat, it suffices to certify that at least r of the reported winning seats W were truly won, requiring testing H_{r/W} (fewer than r true wins). The authors construct seat-level e-processes E_{s,t} using SHANGRLA assertions and ALPHA-style betting, then combine them into a parliament-level e-process E_{r/W,t} as the product of the |W|-r+1 smallest seat-level processes. They prove anytime-validity of E_{r/W} via Ville's inequality (Theorem 1). They introduce adaptive sampling rules (non-adaptive, greedy, filtered, and combinations) that allocate sampling effort across seats, and evaluate them in simulations on synthetic two-candidate contests and the 2014 Indian Lok Sabha election, reporting large reductions in ballots sampled compared with certifying all reported seats.","tokens_in":14925,"tokens_out":13312,"duration_ms":137475,"significance":"If Theorem 1 holds under the intended assumptions, the paper makes a substantial contribution to election auditing. The partial conjunction formulation is natural and appears new for RLA in this form, and the e-process combination is more efficient than the earlier Fisher-based proposal. The adaptive sampling schemes are practical and well-motivated, and the simulation study is extensive and reproducible, with code provided. The theoretical development is based on standard supermartingale and Ville arguments and is mostly careful. However, as discussed below, one key assumption in the proof of Theorem 1 is implicit and the paper overclaims robustness to cross-seat dependence; this must be corrected before the result can be accepted as stated.","major_comments":[{"comment":"The proof of the anytime-validity of E_{1/C} (and hence E_{r/W}) requires that, conditional on F_{t-1}, the next-drawn assorter values X_{s,j_s,n_s(t)} are independent across seats s∈C. This is used at step (a) of the Appendix B derivation. The manuscript does not state this assumption in Section 2.1 or in Theorem 1; moreover, Section 2.3 explicitly claims that the construction 'stays valid even when sampling induces cross-seat dependence, extending Vovk and Wang [17]'. That claim is not supported: if the per-seat random permutations are coupled (e.g., a common random stream selects ballot positions in multiple seats), the product of nonnegative supermartingales need not be a supermartingale, so the risk limit can be violated. This is a load-bearing issue for the main theorem. The fix is straightforward: explicitly assume that the seat-level sampling permutations are independent across s","section":"Section 2.3 / Appendix B, step (a)"}],"minor_comments":[{"comment":"The sentence 'As shown in Appendix B, it stays valid even when sampling induces cross-seat dependence' is in tension with the later sentence 'their e-processes are conditionally independent and can be multiplied.' Please clarify that the adaptive decisions D_{s,t} may be cross-seat dependent (because they depend on F_{t-1}), while the ballot draws themselves are independent across seats.","section":"Section 2.3"},{"comment":"The definition of δ_t in (12) would be easier to follow if the constant k = |W|−r+1 were introduced explicitly, and if the text explained why the set of i satisfying the condition is an initial segment of {1,…,r}.","section":"Section 3, Greedy scheme"},{"comment":"The statement that Mohanty et al. [9] 'justified the theoretical validity of their Fisher combination proposal incorrectly' is categorical. Please provide a precise citation to the step in [9] that relies on independence, or soften the wording.","section":"Appendix C"},{"comment":"Typo: 'satisfiy' should be 'satisfy'. Also, the phrase 'for seatsisn't' should read 'for seat s isn't'.","section":"Section 2.1"},{"comment":"The legends are dense; consider marking the benchmark methods ('All seats', 'Reported top-r seats') with a distinct symbol or label so the comparison to the proposed methods is easier to read.","section":"Figures 2 and 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the central idea is promising. The requested revision is modest: make the independence assumption explicit in Theorem 1 and Section 2.1 and remove/qualify the unsupported robustness claim in Section 2.3. Once those changes are made, the main result and the simulation conclusions should stand."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a real advance over Mohanty et al.: they sketched majority certification but this paper builds a proper partial-conjunction e-process, proves anytime-validity under adaptive sampling, and evaluates adaptive schemes on realistic data. Second, Theorem 1 is correct, but it carries an assumption the paper states too quietly. The proof of E_{1/C} factors the product expectation across seats conditional on F_{t-1} (Appendix B, step (a)); that requires the next unobserved draws in different seats to be conditionally independent, i.e., independent random permutations per seat. This is not stated as a condition of Theorem 1, and Section 2.3's claim that the construction 'stays valid even when sampling induces cross-seat dependence' is misleading. Adaptive sampling induces dependence among the seat-level processes, which the proof handles; but if the draws themselves were coupled across seats, the product supermartingale could fail. In practice this is minor: separate seats have separate ballot piles, and independent sampling is the obvious protocol. The fix is a one-sentence assumption and a softened 2.3 claim.\n\nWhat the paper does well: the reduction to certifying a majority is clean; the product of the |W|-r+1 weakest seat-level e-processes is natural; the proof is a standard Ville/supermartingale argument. The comparison with Mohanty's Fisher combination is useful—their threshold is exp(chi-square/2), larger than 1/alpha for k≥2, so the direct e-process product dominates. Simulations are sensible: synthetic configurations plus the 2014 Indian Lok Sabha data with injected false seats. The savings (millions down to hundreds of thousands) are plausible. The code link works but lacks a commit hash, so I couldn't verify the exact version. The greedy d_t rule is heuristic; the filtering prior is approximated, but the risk guarantee is unaffected since sampling decisions are predictable functions of history. Tuning parameters come from earlier work, not fitted to these simulations.\n\nBottom line: a serious applied paper worth a careful referee. The load-bearing math holds; the main fix is to state the independence assumption explicitly and qualify the 2.3 overclaim. I'd be happy to see it published after minor revision.","headline":"Solid, useful RLA paper for parliamentary majorities; the main theorem is right under an unstated independence assumption, and Section 2.3 overclaims robustness to cross-seat dependence.","tokens_in":15373,"tokens_out":3804,"would_cite":true,"duration_ms":42022,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A parliamentary majority can be certified by pooling seat-level evidence processes rather than certifying every seat, and the resulting audit reduces ballots inspected by orders of magnitude.","keywords":["risk-limiting audit","parliamentary majority","partial conjunction hypothesis","e-process","anytime-valid sequential test","adaptive sampling","election auditing"],"falsifier":"Simulate a small parliament (e.g., five seats, majority threshold three) where ballots across seats are drawn from a single shared random stream so that future draws are dependent across seats; set the truth so that only two seats are truly won; run the audit repeatedly and count how often the product statistic E_{r/W} reaches 1/α before a full recount. If that frequency exceeds the risk limit α, the anytime-validity claim fails under cross-seat dependence.","tokens_in":14493,"feed_emoji":"🗳️","tokens_out":9191,"duration_ms":80014,"temperature":0.7,"pith_summary":"The paper's central claim is that certifying a parliamentary majority can be reduced to testing the partial conjunction hypothesis that at least r of the reported seats were truly won — a 'partial conjunction' meaning some but not necessarily all of the seats must be confirmed. The authors construct an e-process (a sequence of nonnegative statistics that grows as evidence accumulates) for each seat, and combine the weakest ones by multiplication. They prove that this combined process controls the risk limit, so once it crosses 1/α the majority is certified. Simulations on synthetic data and the 2014 Indian Lok Sabha election show the majority audit can require orders of magnitude fewer ballots than auditing every reported winning seat, sometimes by almost a thousandfold.","feed_headline":"Majority audits cut ballots a thousandfold","feed_subtitle":"Pooling the weakest seats' evidence certifies a governing majority with a tiny fraction of the hand count.","key_machinery":"The central object is the seat-level e-process E_{s,t}, built as the minimum over per-assertion betting processes for that seat. The majority statistic E_{r/W,t} multiplies the |W|-r+1 smallest such values at each time. This product grows whenever any sufficiently large group of seats accumulates evidence, and its anytime-validity follows because each E_{s,t} is a test supermartingale and because, conditional on the audit history, the next-drawn ballot values are independent across seats. The adaptive sampling schemes determine which seats' ballots to draw next; since the decisions depend only on the history, they preserve the anytime-validity property.","core_discovery":"Formally, the reported winning party is seen as having won a set W of seats, with r the threshold for a majority. The null hypothesis that the party did not truly win a majority is equivalent to the union, over all subsets C of W of size |W|-r+1, of the intersection nulls 'all seats in C were falsely reported'. For each seat, the authors define an e-process E_{s,t} that is the minimum over a set of per-assertion betting processes. Their majority statistic E_{r/W,t} is the product of the |W|-r+1 smallest of these seat-level processes at time t. Theorem 1 states that this product is anytime-valid for testing the majority null: under that null, the probability that E_{r/W,t} ever reaches 1/α is","pith_inferences":["If the method is adopted, the dominant cost factor becomes the number of 'spare' seats beyond the majority threshold, since the product statistic is driven by the smallest |W|-r+1 seat-level processes; this suggests audits of coalitions or supermajorities could be tailored to the size of the cushion.","The Bayesian filter used to exclude likely-false seats introduces a prior whose influence on sampling is not covered by the anytime-validity guarantee; a fully frequentist or e-value-based filtering rule could offer the same robustness without a prior.","The independence assumption across seats may be violated in implementations using a single shared random stream; the paper does not provide a valid combination rule for that case, so a practical protocol should either enforce independent per-seat permutations or develop a dependence-robust variant."],"forward_implications":["Election officials can certify a parliamentary majority after hand-inspecting a small fraction of ballots, often orders of magnitude fewer than certifying every reported seat.","The majority audit can run alongside a full seat-by-seat audit; the majority becomes certified as soon as the product statistic crosses 1/α, and the seat-by-seat process can continue or escalate to recounts.","The method applies to any 'at least r of the reported winners' hypothesis, not just majorities, and to any contest expressible as assertions about mean ballot scores.","The direct product of e-processes is uniformly more powerful than the previously proposed combination that uses a chi-square threshold, so implementations gain efficiency by using the product directly."],"fun_headline_variants":["Majority audits: check only the weakest seats, not all","A thousandfold fewer ballots for majority audits","Pool seat evidence to certify a governing majority fast","Partial conjunction test for parliamentary majority audits","Audit the majority outcome, not every reported seat"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The risk-limit proof hinges on the assumption that, given the audit history, the next ballot drawn from any seat is independent of the next ballot drawn from any other seat — that is, each seat's ballot order is an independent random permutation.","fun_headline_variants_meta":{"raw":{"variants":["Majority audits: check only the weakest seats, not all","A thousandfold fewer ballots for majority audits","Pool seat evidence to certify a governing majority fast","Partial conjunction test for parliamentary majority audits","Audit the majority outcome, not every reported seat"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1389,"prompt_tokens":726,"completion_tokens":663,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":591}},"tokens_in":470,"tokens_out":663,"duration_ms":7138,"temperature":1.0,"reasoning_tokens":591,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:31:34.122248+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a small parliament (e.g., five seats, majority threshold three) where ballots across seats are drawn from a single shared random stream so that future draws are dependent across seats; set the truth so that only two seats are truly won; run the audit repeatedly and count how often the product statistic E_{r/W} reaches 1/α before a full recount. If that frequency exceeds the risk limit α, the anytime-validity claim fails under cross-seat dependence.","supporting_citations":[],"review_version":1}