{"id":"7906ef5f-31b5-4824-9fde-cfdb76bd856d","arxiv_id":"2607.14448","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Selectable-threshold group testing achieves the counting-bound rate of 1 when thresholds are unbounded, and its fixed-threshold achievability and converse bounds meet as the defect-density exponent tends to 1.","lead":"A new group-testing model lets each pooled test choose its own threshold for how many defects make the test positive. The authors prove that with high or unbounded thresholds the information-theoretic limit of one bit per test is reachable, and they give matching algorithms and lower bounds for capped thresholds in the dense regime.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's batch-1 COMP/DD claim appears false at the stated test count, undermining the fixed-γmax achievability bound.","rationale":"The reader's weakest assumption focuses on the external SUBSET subroutine and the converse regularity assumptions. I agree those are dependencies, but the paper is explicit about them and they are plausibly acceptable. The more serious and more concrete concern is internal to the proof of Theorem 4: the claimed COMP/DD guarantee at the stated T1 is contradicted by a direct calculation of COMP's false positives under the paper's own definitions. If this calculation is correct, the fixed-γmax achievability result is not established, and the matching-with-converse claim in the dense limit loses its achievability half. Because the issue concerns a central theorem rather than a minor constant, the verdict should move from ACCEPT to CONDITIONAL, requiring the authors to either justify the [22, App. D] claim with the exact parameters used or correct the batch-1 test count and re-derive the rate. I mark agreement as 'partial' because the reader did flag reliance on [22] as a fragility, but not the specific COMP/DD inconsistency that appears most load-bearing.","tokens_in":42590,"tokens_out":34923,"duration_ms":345354,"concrete_test":"Recompute the COMP false-positive probability under the NCC parameters of App. B-C: verify whether T1 · (log 2/k) · e^{-log 2} ≥ (1+ε) ln(n/k) holds for the stated T1. Numerically, simulate the batch-1 procedure with n=10^6, k=1000, L1=10, T1≈14,388, γ=1, over 100 Monte Carlo trials. The proof requires |S_COMP|≈1000; the calculation above predicts ≈7,700. If simulation confirms the larger value, the premise of Theorem 4 is false and the proof requires substantial revision.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 4 (App. B-C) sets T1 = (1/log 2) k log2(n/k) and L1 = T1 log 2 / k = log2(n/k), then asserts via [22, App. D] that COMP produces S_COMP ⊇ S with |S_COMP| = k(1+o(1)). This is contradicted by a direct calculation using the paper's own definition of COMP (Sec. I-B). A non-defective is misclassified iff it appears in no negative test. In the NCC design, a given test contains a fixed non-defective with probability L1/T1 = log2(n/k) / (k log2(n/k)/log 2) = log 2 / k, and the test is negative (contains no defectives) with probability (1 - L1/T1)^k = (1 - log 2/k)^k → 1/2. Hence the expected number of negative tests containing that non-defective is T1 · (log 2/k) · 1/2 = 0.5 log2(n/k). The probability it is never ruled out is exp(-0.5 log2(n/k)) = (n/k)^{-1/(2 ln 2)} ≈ (n/k)^{-0.721}. Summing over the n-k non-defectives gives E|S_COMP \\ S| ≈ n (n/k)^{-0.721} = n^{0.279+0.721θ}, which is ω(k) for every θ<1. For example, n=10^6, k=1000 gives ≈7,700 false positives, while k=1000. Thus S_COMP cannot have size k(1+o(1)), and the later union bound over |S_COMP \\ S_DD| = o(k) is invalid. The paper provides no derivation for the COMP/DD claim, only a citation; as written, Theorem 4's achievability bound (Eq. 11) is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces selectable-threshold group testing (GT-ST): each binary pooled test can be assigned its own threshold γ(t) ≤ γmax, with outcome 1{ψ_t ≥ γ(t)}. The authors prove the counting bound; show rate 1 is achievable when γmax is unbounded (Theorem 2) and rate 1−O(γmax^{-1/2}) for large γmax (Theorem 3), both by a two-batch strategy built on the SUBSET result of [22]; give a finite-γmax achievability result with polynomial-time COMP/DD decoding (Theorem 4); and prove masking-based converses (Theorems 5 and 6) under near-constant weight/degree and bounded-overlap assumptions, with an optimization lemma showing that the converse is dominated by the largest threshold. The proofs are detailed and the exposition is careful.","tokens_in":43102,"tokens_out":31897,"duration_ms":308658,"significance":"If its results held, the paper would be a valuable contribution: it formulates a natural generalization of threshold group testing, gives simple rate-1 constructions in the unbounded/large-threshold regime (modulo the acknowledged external SUBSET subroutine), and provides a converse that is more general than concurrent work. The masking-based proof is sophisticated and mostly internally consistent, and the authors are explicit about assumptions and external dependencies. However, the central finite-γmax achievability theorem is unsupported as written because the batch-1 COMP/DD guarantee is quantitatively false at the stated test count; this directly undermines Eq. (11) and the claimed comparison with standard GT.","major_comments":[{"comment":"The batch-1 claim that COMP produces S_COMP of size k(1+o(1)) is not a valid consequence of [22, App. D] and is contradicted by the authors' own parameters. With T1=(1/log2)k log2(n/k) and L1=T1 log2/k=log2(n/k), the effective inclusion probability is p=L1/T1=log2/k. For a fixed non-defective, a test is negative with probability (1-p)^k→1/2, so the expected number of negative tests containing the item is T1·p/2=(1/2)log2(n/k). Hence P(item remains in S_COMP)=exp(−(1/2)log2(n/k))=(n/k)^{−1/(2 ln2)}≈(n/k)^{−0.721}, and E|S_COMP\\S|≈n(n/k)^{−0.721}=n^{0.279+0.721θ}, which is ω(k) for every θ<1. Thus S_COMP cannot have size k(1+o(1)), and the later union bound over |S_COMP\\S_DD|=o(k) is invalid.","section":"App. B-C, Theorem 4, Eq. (11)"},{"comment":"The finite-γmax claims of strict improvement over standard GT are not supported independently of the faulty batch-1 step. At the inclusion probability used in the proof, the number of batch-1 tests must be increased by a factor greater than 2ln2≈1.386 even to make the COMP false positives o(k); the DD step is also contaminated by the large false-PD set. The authors should either supply a correct batch-1 subroutine that provably meets the stated T1, or restate Theorem 4 and Figure 1 with a corrected leading term and re-evaluate the comparison.","section":"Section II-C and Fig. 1"}],"minor_comments":[{"comment":"The display γ^{-1/3} appears where the proof clearly means γ−1/3 (γ minus one third). Please fix the notation; as typeset it is ambiguous and initially reads as a tiny exponent.","section":"Lemmas 2 and 9"},{"comment":"Since [22, App. D] supplies the crucial COMP/DD guarantee for Theorem 4, the paper should state the exact lemma used, including the hypotheses and the precise sense in which S_DD and S_COMP have the claimed sizes, rather than only citing it.","section":"App. B-C, [22, App. D]"},{"comment":"The caption should specify precisely which curve from [20] is plotted and in which metric. The current description makes the claimed comparison in the dense limit difficult to verify.","section":"Fig. 1"},{"comment":"The step that discards tests whose repeated outcomes are all identical should clarify that this is valid only on the high-probability event that ψ_t lies in the interval (γ_-, γ_+); a sentence about boundary effects near γ_± would help.","section":"Theorem 3 proof"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unsupported Theorem 4. The paper's strongest contributions are the conditional rate-1 results and the converse; I would be willing to reconsider a thoroughly corrected version. The comparison to standard GT in Fig. 1 needs to be re-run after the batch-1 correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the selectable-threshold model is new, the two-batch intuition is clear, and the converse (Theorem 6 plus Lemma 1) looks like real work. But the fixed-γmax achievability, Theorem 4, has a load-bearing error as written.\n\nWhat is good: the definition of GT-ST is natural and distinct from the concurrent fixed-threshold TGT work. The unbounded-threshold result showing rate 1 conditional on [22]'s SUBSET subroutine is a clever reduction, and the paper is upfront that this subroutine is currently only proved for a computationally inefficient method. The converse generalizes the concurrent [30] converse to deterministic designs under Assumptions 1–3, and the optimization lemma showing the highest threshold dominates is clean. I found the external dependencies and the paper's own limitation statements handled honestly.\n\nWhere it breaks: in Theorem 4's proof, batch 1 sets T1=(1/log2)k log2(n/k), so L1=T1 log2/k=log2(n/k). Under the NCC design each fixed non-defective lands in a given test with probability log2/k, and a test is negative with probability (1-log2/k)^k → 1/2. Thus the expected number of negative tests containing any given non-defective is about 0.5 log2(n/k), and the probability it is never ruled out is (n/k)^{-1/(2 ln2)}, giving E|S_COMP\\S| ≈ n^{0.279+0.721θ}. That is ω(k) for every θ<1. The proof's citation to [22, App. D] does not cover the claimed simultaneous COMP/DD guarantee at this T1; the test count is simply too small for COMP on this design. Since the o(k) size of S_COMP\\S_DD is what drives batch 2, the achievability bound (11) is unsupported. This is not a minor typo—the claimed rate rests on it.\n\nThe rate-1 and large-γmax theorems inherit a different weakness that the authors themselves flag: they depend on [22]'s SUBSET, currently only proved for a computationally inefficient method. That is a condition, not a hidden flaw. If the SUBSET guarantee holds, those results are fine.\n\nBottom line: the converse and the model are worth engaging; Theorem 4 needs a corrected proof or a different first batch before it can be used.","headline":"New model and a likely-solid converse, but Theorem 4's achievability rests on a COMP/DD claim that does not hold at the stated test count.","tokens_in":43524,"tokens_out":4749,"would_cite":false,"duration_ms":49180,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that letting each pooled test choose its own positive threshold lets non-adaptive group testing reach the maximum possible rate of one bit per test when the threshold is unbounded.","keywords":["group testing","selectable thresholds","threshold group testing","non-adaptive algorithms","counting bound","achievable rates","converse bounds","pooled testing"],"falsifier":"Run the fixed-gamma_max two-batch algorithm for gamma_max=2 with n=10^6 and k=n^0.95, sweep T around the predicted value, and check whether empirical error probability drops to zero at the predicted (1/log 2) k log2(n/k) + k log k / 0.636 tests; a mismatch would falsify Theorem 4, while agreement would support the claim that selectable thresholds strictly beat standard group testing in the dense regime.","tokens_in":42496,"feed_emoji":"🧪","tokens_out":9775,"duration_ms":89124,"temperature":0.7,"pith_summary":"The paper introduces a variant of group testing in which each pooled test comes with a threshold chosen by the designer: the test is positive if and only if the number of defective items in it is at least that threshold. Its central finding is that this flexibility is powerful: with no cap on the threshold, non-adaptive strategies can recover the defective set with a number of tests equal to k log2(n/k)(1+o(1)), reaching rate 1 and matching the counting bound that no binary-outcome strategy can beat. With a fixed maximum threshold, the paper gives a two-stage polynomial-time algorithm with an explicit test-count formula, and a matching converse under regularity conditions in the dense limit k = Theta(n^theta), theta -> 1. A sympathetic reader would care because it sharpens what kinds of 1-bit test outcomes are useful, shows selectable thresholds can beat standard group testing in the dense regime, and clarifies that the largest threshold dominates information-theoretically.","feed_headline":"Selectable thresholds let pooled tests hit one bit per test","feed_subtitle":"Unbounded selectable thresholds match the counting bound; fixed thresholds beat standard group testing for dense k.","key_machinery":"The two-batch algorithm is the central construction: batch one uses standard threshold-1 group testing (COMP/DD, or a SUBSET subroutine in the unbounded case) to find a set S_DD of k(1-o(1)) defectives plus a superset containing all defectives; batch two runs near-constant-column-weight tests at threshold gamma_max, and an item is 'resolved' by a test containing exactly gamma_max-1 items from S_DD and no other uncertain item. The probability of such a test is asymptotically Poisson, nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!, which produces the explicit denominator in the rate formula. The converse rests on the masking mechanism: an item is masked when no test is informative for it (contains it","core_discovery":"The paper's central claim is that making the threshold a selectable parameter turns threshold group testing into a problem whose information-theoretic rate can reach 1 when gamma_max is unbounded, and whose fixed-threshold rate is characterized by matching achievability and converse bounds in the dense limit. The constructive idea is a two-batch scheme: first use ordinary threshold-1 group testing to identify a near-complete set of defectives, then use tests with threshold gamma_max to resolve the remaining uncertain items; the fixed-threshold analysis shows the second batch needs k log k / [nu log(1/(1 - nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!))] tests. On the converse side, the paper prove","pith_inferences":["The two-batch template suggests a general recipe: any standard group-testing method that returns a near-complete defective set can be upgraded with high-threshold resolution tests, and nothing in the analysis appears to require exactly two batches.","Because the converse assumes near-constant weights/degrees and bounded overlaps, the picture for non-regular designs such as Bernoulli-type test matrices is open; a numerical study in the dense limit would show whether the matched k log k coefficient survives without those regularity conditions.","The proven rates use only thresholds 1 and gamma_max; a natural testable extension is whether intermediate thresholds close the remaining gap for fixed theta < 1, although the paper's Lemma 1 suggests no information-theoretic gain from them.","For applications with noisy threshold devices, the idealized exact-threshold model would need calibration, and the second batch's reliance on exact counts of gamma_max-1 known defectives suggests a noise sensitivity that could be probed by simulation."],"forward_implications":["With unbounded selectable thresholds, T = k log2(n/k)(1+o(1)) tests suffice for non-adaptive recovery, matching the counting bound and making the rate exactly 1.","For fixed gamma_max, the two-batch algorithm runs in polynomial time and uses (1/log 2) k log2(n/k) + min_nu k log k / [nu log(1/(1 - nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!))] (1+o(1)) tests.","As gamma_max grows, the achievable rate approaches 1 at least as fast as 1 - O(1/sqrt(gamma_max)), and the converse shows the k log k coefficient cannot decay faster than 1/sqrt(gamma_max).","In the dense limit theta -> 1, achievability and converse for fixed gamma_max have matching k log k coefficient, so the number of tests is determined to within o(k log k).","Information-theoretically, selectable thresholds do not beat just using the largest threshold; their value is algorithmic simplicity."],"fun_headline_variants":["Selectable thresholds push group testing to one bit per test","Fixed-threshold group testing matches info bound for dense k","Unbounded thresholds let pooled tests reach maximal rate","Two-batch scheme nails one bit per test with thresholds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The rate-1 and large-gamma_max achievability claims rest on a cited SUBSET subroutine that the paper flags as only proved for a computationally inefficient method, and the matching converse additionally assumes near-constant test sizes/degrees with bounded overlaps; if either premise gives way, the corresponding headline results remain conditional.","fun_headline_variants_meta":{"raw":{"variants":["Selectable thresholds push group testing to one bit per test","Fixed-threshold group testing matches info bound for dense k","Unbounded thresholds let pooled tests reach maximal rate","Two-batch scheme nails one bit per test with thresholds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000682,"raw_usage":{"total_tokens":2920,"prompt_tokens":718,"completion_tokens":2202,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":2148}},"tokens_in":462,"tokens_out":2202,"duration_ms":15211,"temperature":1.0,"reasoning_tokens":2148,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:09:12.294754+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the fixed-gamma_max two-batch algorithm for gamma_max=2 with n=10^6 and k=n^0.95, sweep T around the predicted value, and check whether empirical error probability drops to zero at the predicted (1/log 2) k log2(n/k) + k log k / 0.636 tests; a mismatch would falsify Theorem 4, while agreement would support the claim that selectable thresholds strictly beat standard group testing in the dense regime.","supporting_citations":[],"review_version":1}