{"id":"ac5628b7-9df7-48f8-99cb-5c23480860ae","arxiv_id":"2608.11500","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A new verifiable proportionality axiom, FJR+, dominates both FJR and EJR+, is checked by linear programming, is achieved by a greedy rule, and admits a priceable completion.","lead":"This paper introduces FJR+, a proportional-representation rule for approval-based committee elections that is stronger than the well-known FJR and EJR+ rules yet can still be checked and computed in polynomial time. It provides a greedy algorithm that always meets FJR+, along with a way to fill the remaining committee seats so voters' equal budgets pay for the outcome, and it extends the same idea to participatory budgeting with project costs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Priceability does not imply the sub-core under the paper's definitions; a concrete 6-voter election makes Theorem 4.5's sub-core claim false.","rationale":"The reader identified Proposition 4.4 as an unproved external lemma and flagged the sub-core conclusion as the weakest assumption, but stopped short of finding that the lemma is actually false under the paper's Definitions 2.6 and 2.7. The counterexample above shows that priceability does not imply the sub-core with a weak Hare inequality: a committee can be priceable and yet admit a group S and comparison set T with |S|=q_H|T| and A_i∩W strictly contained in A_i∩T for every i in S. The same example arises as the output of RBG on a simple profile, so the flaw is not confined to an exotic priceable committee; it directly refutes Theorem 4.5's claim that the proposed rule always returns a committee satisfying the sub-core. The FJR+ verification and the RBG FJR+ theorem appear intact, and the priceability completion may also be correct, but the abstract and Theorem 4.5 claim FJR+, the sub-core, and priceability together. Since the sub-core half of that central claim is false, the paper cannot be accepted as is. A revision could potentially remove the sub-core guarantee or replace Proposition 4.4 with a correct implication under a stricter sub-core definition (e.g., a strict inequality |S|>q_H|T|), but as written the central result is materially incorrect.","tokens_in":24347,"tokens_out":16157,"duration_ms":139592,"concrete_test":"Run the proposed rule (RBG(q_H) followed by any size-k completion, including Algorithm 2) on the 6-voter, k=3 election with A1=A2=A3=A4={a,b,d} and A5=A6={c}. For the returned committee W, check Definition 2.7 with S={1,2,3,4}, T={a,b,d}; if W contains exactly {a,b,d}, instead check S={5,6}, T={c}. The violation confirms that Theorem 4.5's sub-core guarantee fails. Independently, verify Proposition 4.4 by testing the priceable committee W={a,b,c} from the same profile against the same S,T.","verdict_should_be":"REJECT","load_bearing_attack":"Proposition 4.4 ('every priceable committee lies in the sub-core') is false as stated. Consider n=6, k=3, candidates {a,b,c,d}, with approvals A1=A2=A3=A4={a,b,d} and A5=A6={c}. Let W={a,b,c}. Then q_H=2. W is priceable at price 2: voters 1 and 2 pay 1 each to a, voters 3 and 4 pay 1 each to b, voters 5 and 6 pay 1 each to c; every voter spends exactly 1, and the only unelected candidate d has approvers with total remaining budget 0 < 2. Yet S={1,2,3,4} and T={a,b,d} satisfy |S|=4=q_H|T|, and for each i in S, A_i∩W={a,b} is a strict subset of A_i∩T={a,b,d}. Thus W violates Definition 2.7. This is not a contrived completion: on this election RBG(q_H) cannot buy anything at level 3; at level 2 each of a,b,d has score 2, so RBG selects one of them, and every possible level-1 completion is priceable and fails the sub-core (either via S={1,2,3,4},T={a,b,d} or via S={5,6},T={c}). Hence the proof of Theorem 4.5 is not merely missing a justification for Proposition 4.4; the proposition itself is invalid, and the sub-core part of the central theorem is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces full justified representation+ (FJR+), a fractional strengthening of FJR and EJR+ for approval-based committee elections. A level-ℓ fractional violation is defined by voter weights z_i, candidate weights y_c, and assignment variables x_ic subject to (C1)-(C5) and Z ≥ q_H Y; the Droop version uses Z > q_D Y. The authors show that FJR+ implies FJR and EJR+ and is strictly stronger than their conjunction (Proposition 3.8), characterize FJR+ on party-list profiles by lower quota (Proposition 3.9), and give polynomial-time verification via one LP per level (Theorem 3.4). The main computational result is Theorem 3.5: Residual-Budget Greedy with price q_H returns a partial committee P such that every size-k completion satisfies FJR+, and with price q_D gives Droop-FJR+. The paper then proves that the payments make P affordable (Proposition 4.2), completes P by sequential Phragmén to a priceable committee (Theorem 4.3), and, using the known implication from priceability to the sub-core (Proposition 4.4), obtains a polynomial-time committee rule satisfying FJR+ and the sub-core, priceable whenever at least k candidates are approved (Theorem 4.5). Section 5 gives separation and monotonicity results, and Section 6 extends the framework to approval-based participatory budgeting with arbitrary project costs: PB-FJR+, PB-RBG, and a priceable PB-sub-core completion (Theorems 6.3, 6.4, 6.7).","tokens_in":24613,"tokens_out":28382,"duration_ms":243756,"significance":"If the results hold, which I believe they do, the paper makes a substantial contribution: it identifies a natural proportionality axiom that is strictly stronger than FJR and EJR+ yet simultaneously admits polynomial-time verification and polynomial-time construction, and it carries the same guarantee over to participatory budgeting with arbitrary project costs. The proof strategy is convincing: the RBG potential F(P,b)-ρY(P) with identity (9), the nondecreasing-threshold argument for Phragmén completion (Lemma C.1), and the budget-accounting proofs in the PB section are all coherent. I specifically examined the alleged counterexample to Proposition 4.4 raised in review; it is arithmetically invalid, since with n=6, k=3, q_H=2 and |T|=3 the required coalition size is q_H|T|=6, while the proposed S has size 4; the alternative S={5,6}, T={c} does not give a strict inclusion. Hence the sub-core conclusion of Theorem 4.5 is not undermined. The paper ships complete appendices, explicit constructions for strictness, and polynomial-time algorithms with stated running times.","major_comments":[],"minor_comments":[{"comment":"The sub-core implication in Proposition 4.4 is imported without proof from Dong and Peters [2026]; because it is load-bearing for Theorem 4.5, a one-paragraph proof or a precise theorem reference would improve self-containedness.","section":"Section 4.3, Proposition 4.4"},{"comment":"In the proof of Proposition 3.9, the displayed inequality appears garbled as 'jknj n k'; it should read 'k n_j/n < ℓ', and the surrounding derivation should be typeset more clearly.","section":"Appendix B.4, Proposition 3.9"}],"recommendation":"accept","confidential_remarks":"The review stress-test's main attack on Proposition 4.4 fails by a simple arithmetic error; I see no residual correctness issue. The only verifiability caveat is the reliance on the very recent Dong and Peters [2026] citation for Proposition 4.4; it might be worth asking the authors to add a proof or a precise theorem pointer before publication, though the statement is correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result is real and worth engaging with. The paper introduces FJR+, a fractional-weight strengthening of FJR that also implies EJR+, gives a polynomial-size LP verification, and proves that every execution of the broader RBG rule returns a partial committee whose every size-k completion satisfies FJR+. That completion freedom is the key move, and using it to run sequential Phragmén to get priceability is a nice piece of work. The PB extension with project-specific costs is coherent and not bolted on. I spot-checked the main examples and the potential argument in Theorem 3.5; the proofs in the appendix are detailed and the identity in (9) does the work it claims.\n\nThe soft spots are real but not load-bearing. The sub-core conclusion depends entirely on Proposition 4.4, an imported lemma attributed to Dong and Peters without proof. For a paper that otherwise proves its central claims carefully, that is a genuine presentation gap; I would want the proof in an appendix or a precise pointer to a theorem statement. I also checked the stress-test counterexample to Proposition 4.4. It does not work: in that example q_H = 2, so for T = {a,b,d}, q_H|T| = 6, not 4, and the group S = {1,2,3,4} does not satisfy the sub-core size condition. The alternative S = {5,6}, T = {c} has equality rather than strict inclusion. So the stress-test concern does not land, and priceability implying the sub-core remains plausible and consistent with the literature. The other caveat is minor: the Table 1 citation for coNP-completeness of sub-core verification looks like it points to a paper about auditing core stability rather than the sub-core as defined here; worth fixing but not threatening.\n\nThis paper is for researchers in computational social choice who care about verifiable proportionality axioms. It delivers a new definition that strictly dominates the previous pair, with polynomial-time verification and computation, and it is honest about what is new at the FJR level versus what it credits to Ai and to Frank-Peters. I would send it to a serious referee. It deserves a careful reading, and the only required revision is to nail down the priceability-to-sub-core implication rather than importing it silently.\n\nRecommendation: accept into the review process, with a request to prove or precisely cite Proposition 4.4 and to re-check the Table 1 attribution.","headline":"FJR+ is a genuinely new, verifiable strengthening of FJR and EJR+ with a real RBG completion theorem; the sub-core part rests on an unproved imported lemma, but the stress-test counterexample to that lemma does not survive contact with the paper's definitions.","tokens_in":25183,"tokens_out":5513,"would_cite":true,"duration_ms":55507,"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 fractional strengthening of full justified representation is both checkable and satisfiable in polynomial time.","keywords":["approval-based committee elections","full justified representation","FJR+","EJR+","residual-budget greedy","priceability","sub-core","participatory budgeting"],"falsifier":"Enumerate all small profiles (say $n \\le 5$, $k \\le 3$) and check whether every priceable committee lies in the sub-core; a single priceable committee outside the sub-core refutes the external lemma on which Theorem 4.5 rests. Alternatively, run the linear program $\\mathrm{LP}_\\ell(W)$ on every size-$k$ completion of the set returned by $\\mathrm{RBG}(q_H)$ on the profile of Proposition 3.8 and look for an optimum at least $q_H$.","tokens_in":24078,"feed_emoji":"🗳️","tokens_out":9233,"duration_ms":71427,"temperature":0.7,"pith_summary":"Approval-based committee elections ask how to convert voter approvals into a size-$k$ committee that is proportionally representative. The strongest satisfiable justification axiom, full justified representation (FJR), can be met in polynomial time, but checking a proposed committee is coNP-complete. This paper defines FJR+, a strict strengthening of FJR and EJR+ stated with fractional voter and candidate weights, and shows it can be verified by linear programming and satisfied by a greedy rule. The main theorem proves that Residual-Budget Greedy (RBG) returns a partial committee $P$ such that every size-$k$ completion satisfies FJR+, which leaves the remaining seats free to be chosen for priceability; the completed committee lies in the sub-core, and a cost-based version works for participatory budgeting. If true, this removes the verification-versus-computation gap while making the proportionality requirement stronger, not weaker.","feed_headline":"Stronger fair representation, verified and built in polynomial time","feed_subtitle":"Residual-budget greedy fixes a partial committee whose every completion meets the new axiom; the rest is priceable.","key_machinery":"The load-bearing object is Residual-Budget Greedy (RBG) with candidate price $\\rho$. It walks levels $h=k,k-1,\\dots,1$; a candidate $c$ not yet chosen is affordable when $\\mathrm{score}_h(c;P,b)=\\sum_{i \\in N_c: u_i(P)<h} b_i/(h-u_i(P)) \\geq \\rho$, and when chosen its payments satisfy the same individual caps and sum to $\\rho$. The proof tracks $F(P,b)-\\rho Y(P)$, where $F$ sums $b_i t_i/a_i$ over weighted voters with gap $a_i=\\ell-u_i(P)$ and remaining assignment $t_i$; the identity showing that this quantity never decreases is the engine that rules out every fractional violation for every completion. Verification of FJR+ rests on normalizing total candidate weight to $1$ and maximizing total voter weight by linear programming for each level.","core_discovery":"FJR+ is defined by fractional violations. At a representation level $\\ell$, a committee $W$ has a violation when there are nonnegative voter weights $z_i$, candidate weights $y_c$, and assignments $x_{ic}$ such that only voters with $u_i(W)<\\ell$ receive weight, each such voter is assigned exactly $\\ell z_i$ units from approved candidates, assignments to candidate $c$ never exceed $y_c$, assignments to an elected candidate $w$ never exceed $z_i$, and total voter weight $Z$ is at least $q_H = n/k$ times total candidate weight $Y$. Binary variables recover exactly FJR violations; a single positive candidate recovers EJR+ violations; and an example shows FJR+ is strictly stronger than FJR plus EJR+ together. The paper proves that RBG run with price $q_H$ returns a set $P$ such that every size-$k$ committee $W \\supseteq P$ satisfies FJR+, and that the Droop version with strict inequality satisfies Droop-FJR+. Sequential Phragm\\'en then completes $P$ to a priceable committee, which, by the priceability-to-sub-core implication, lies in the sub-core; this is stated as Theorem 4.5, with a participatory-budgeting analogue in Theorem 6.7.","pith_inferences":["The LP-based verification defines a quantitative FJR+ gap, the maximum ratio $Z/Y$ across levels; that number could rank nearly proportional committees or guide search when exact FJR+ is too demanding.","FJR+ is not monotone: adding approvals for already elected candidates can create a fractional violation, so dynamic or perpetual settings would need rerunning the rule or a monotone variant rather than incremental updates.","The single-voter knapsack hardness for additive utilities suggests that exact polynomial FJR+ computation is limited to approval utilities, making approximation guarantees or restricted utility classes the natural next target in participatory budgeting.","The completion freedom invites a testable extension: among the priceable completions of a fixed RBG partial committee, one can optimise utilitarian welfare and measure whether priceability and FJR+ are preserved in practice."],"forward_implications":["Any rule that returns an FJR+ committee automatically satisfies FJR and EJR+, so the stronger axiom comes with both classical guarantees at no extra computational cost.","Because every size-$k$ completion of the RBG partial committee is FJR+, the completion step can be optimised for priceability, utilitarian welfare, or other objectives without sacrificing FJR+.","When at least $k$ candidates receive an approval, the completed committee is priceable, and priceability implies the sub-core, so the same outcome satisfies FJR+ and the sub-core simultaneously.","At the Droop quota $n/(k+1)$ with a strict group-size inequality, the same algorithm satisfies Droop-FJR+, and the equality case cannot be ruled out in general.","In approval-based participatory budgeting with arbitrary project costs, PB-RBG returns a feasible partial outcome whose every feasible superset is PB-FJR+, and the natural continuation is priceable and lies in the PB sub-core."],"supporting_citations":[{"why":"It introduces EJR+ and the Greedy Justified Candidate Rule, the property and rule that FJR+ strengthens.","marker":"[Brill and Peters, 2023]"},{"why":"It introduces FJR in participatory budgeting and the Greedy Cohesive Rule, supplying the FJR definition and the cost-based priceability used in Section 6.","marker":"[Peters et al., 2021]"},{"why":"It describes the concurrent descending-budget rule from which RBG is derived by relaxing the eligibility of approvers.","marker":"[Ai, 2026]"},{"why":"It provides a concurrent polynomial-time FJR rule, establishing the baseline FJR computation result that FJR+ is compared with.","marker":"[Frank and Peters, 2026]"},{"why":"It proves that verifying FJR is coNP-complete, the computational gap that polynomial-time FJR+ verification closes.","marker":"[Kalayci et al., 2025]"},{"why":"It defines priceability and the Method of Equal Shares, the budget interpretation used for the completion guarantee.","marker":"[Peters and Skowron, 2020]"},{"why":"It defines sequential Phragm\\'en's load-balancing rule, which completes the partial committee to a priceable one.","marker":"[Brill et al., 2017a]"},{"why":"It is the source of the stated but unproved lemma that every priceable committee lies in the sub-core, the load-bearing external step for Theorem 4.5.","marker":"[Dong and Peters, 2026]"},{"why":"It introduces the sub-core, the group-deviation guarantee that Theorem 4.5 combines with FJR+.","marker":"[Munagala et al., 2022]"},{"why":"It supplies the Droop-quota conventions and the strict inequality used in the Droop-FJR+ result.","marker":"[Casey and Elkind, 2025]"}],"fun_headline_variants":["New FJR+ axiom: verifiable and satisfiable in polynomial time","RBG partial committee: every completion satisfies FJR+ and is priceable","FJR+ strengthens FJR and EJR+ yet stays polynomial-time checkable","Residual-budget greedy yields a partial committee all of whose completions meet FJR+"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sub-core guarantee is imported from an unproved external lemma that every priceable committee lies in the sub-core; if that implication is false or needs extra conditions, the sub-core parts of Theorems 4.5 and 6.7 collapse.","fun_headline_variants_meta":{"raw":{"variants":["New FJR+ axiom: verifiable and satisfiable in polynomial time","RBG partial committee: every completion satisfies FJR+ and is priceable","FJR+ strengthens FJR and EJR+ yet stays polynomial-time checkable","Residual-budget greedy yields a partial committee all of whose completions meet FJR+"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000308,"raw_usage":{"total_tokens":1794,"prompt_tokens":1014,"completion_tokens":780,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":630,"completion_tokens_details":{"reasoning_tokens":694}},"tokens_in":630,"tokens_out":780,"duration_ms":23996,"temperature":1.0,"reasoning_tokens":694,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:08:50.508629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all small profiles (say $n \\le 5$, $k \\le 3$) and check whether every priceable committee lies in the sub-core; a single priceable committee outside the sub-core refutes the external lemma on which Theorem 4.5 rests. Alternatively, run the linear program $\\mathrm{LP}_\\ell(W)$ on every size-$k$ completion of the set returned by $\\mathrm{RBG}(q_H)$ on the profile of Proposition 3.8 and look for an optimum at least $q_H$.","supporting_citations":[{"cited_title":"A Polynomial-Time Rule Satisfying Full Justified Representation","cited_arxiv_id":"2608.05397","evidence_quote":"It provides a concurrent polynomial-time FJR rule, establishing the baseline FJR computation result that FJR+ is compared with."},{"cited_title":"Full proportional justified representation","cited_arxiv_id":null,"evidence_quote":"It proves that verifying FJR is coNP-complete, the computational gap that polynomial-time FJR+ verification closes."},{"cited_title":"An axiomatic analysis of proportionality notions in approval-based multiwinner voting","cited_arxiv_id":null,"evidence_quote":"It is the source of the stated but unproved lemma that every priceable committee lies in the sub-core, the load-bearing external step for Theorem 4.5."},{"cited_title":"Auditing for core stability in participatory budgeting","cited_arxiv_id":null,"evidence_quote":"It introduces the sub-core, the group-deviation guarantee that Theorem 4.5 combines with FJR+."}],"review_version":1}