{"id":"1750a746-0e92-4627-aa28-33f1fc3af8a5","arxiv_id":"2412.20781","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every vertex of the square lattice choosing exactly two of its four neighbors to point to yields an infinite directed open path with positive probability.","lead":"A directed percolation model on the square lattice, where each vertex randomly points to exactly two of its four neighbors, is shown to have an infinite directed path with positive probability. The proof settles an open conjecture and introduces an enhancement technique for percolation with forbidden local patterns.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 6's local case checks are the load-bearing gap: Lemmas 6.7 and 6.9 construct maps Te and Se but leave essential subcases to the reader and an external website, and a hidden subcase failure would invalidate Theorem 6.1 and hence the ε=0 case of Theorem 3.1.","rationale":"The reader's weakest-assumption analysis identifies exactly the point where the proof is most vulnerable: the enhancement comparison in Section 6. I agree that the local modification maps Te and Se in Lemmas 6.7 and 6.9 are the load-bearing step. The paper's own text flags the incompleteness: Lemma 6.9 leaves verification to the reader, and Lemma 6.7 points to an external website for the full case list. Because the main theorem reduces to Theorem 3.1, which in the critical case ε=0 reduces to Theorem 6.1, a failure of these local maps would leave the central claim unproved. My reading did not uncover a different, more serious defect: the exploration algorithm, the decomposition into visited clusters via pivotal edges, the domination by constrained Bernoulli percolation, and the derivation of pc<1/2 from subexponential decay all appear internally consistent and are supported by detailed arguments. The concern is therefore not that the result is false, but that a finite, checkable case analysis is omitted at a crucial juncture. This warrants a conditional verdict, exactly as the reader concluded; no change to the verdict is needed. I would encourage the authors to supply either a fully written case analysis or a computer-assisted verification of Lemmas 6.7 and 6.9, and to make the companion website's visualizations reproducible as formal proof objects or exhaustive scripts.","tokens_in":41840,"tokens_out":5203,"duration_ms":59559,"concrete_test":"Perform an exhaustive finite-state verification of Lemmas 6.7 and 6.9. For a representative good vertical edge e, enumerate all admissible entry/exit pairs (x,y) on ∂Be consistent with deterministic rule (∗); for each pair, construct Te's p-edge pattern on the 6×5 box In(Be) exactly as prescribed and check that (a) all q-edges in In(Be) are closed, (b) the constructed open p-edge path contains exactly one forbidden pattern, associated with Q(e), and (c) for every outside configuration in PivOut_Be, setting the p-edges of In(Be) to match Te makes Q(e) pivotal for A_{p,q}. Repeat the same check for all six bad-edge categories in Lemma 6.9, including boundary-overlapping boxes and the case where o lies inside BJ(e). This is a finite case analysis (at most a few thousand entry/exit pairs per category) and can be settled by a short script or careful hand check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"To prove pc(2)<1/2, the argument must handle the critical case ε=0. This relies on Theorem 6.1, whose proof depends on Proposition 6.3, i.e., the derivative comparison ∂pΘn(p,q) ≤ C ∂qΘn(p,q). Proposition 6.3 is reduced in Lemma 6.6 to the inequalities (17) and (18), which compare pivotal probabilities for good and bad p-edges. These inequalities are proved via the local modification maps Te (Lemma 6.7) and Se (Lemma 6.9). The printed proof of Lemma 6.7 refers to the companion website for a 'concrete visualization of the construction in all possible cases' and states several subcases only heuristically; Lemma 6.9 explicitly says 'We leave the details to the reader' for the six bad-edge categories. The construction is finite but genuinely intricate: it must work for every possible entry/exit pair on ∂Be, for the cases where o lies inside Be, and for boxes Be that intersect ∂Λn. A single impossible subcase would make inequality (17) or (18) fail, so Proposition 6.3 and Theorem 6.1 would not be established, and the exponential decay of visited clusters in the original 2-neighbor model would lack its key ingredient. The rest of the proof—exploration, decomposition into visited clusters, domination by constrained bond percolation—is coherent and well supported, but this missing verification is not cosmetic: it is the exact point where the argument goes beyond standard enhancement and relies on case-specific geometry. No machine-checked proof or reproducible code is provided, so the gap is currently open.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the planar 2-neighbor directed graph percolates, i.e., pc(2) < 1/2, resolving a conjecture from [JKLT23]. The proof combines a duality and exploration algorithm for the dual forward set, a decomposition of the explored cluster into visited clusters separated by pivotal edges, a domination of each visited cluster by constrained i.i.d. bond percolation, and a block renormalization step. The critical ε = 0 case requires an enhancement argument (Section 6) showing that the forbidden-pattern constraint strictly increases the percolation threshold above 1/2. A secondary result establishes pcorn_c < 1/2 for the directed-corner model.","tokens_in":42169,"tokens_out":14018,"duration_ms":128271,"significance":"If correct, this settles a natural open problem and provides a rigorous example where a fixed-degree constraint is beneficial for percolation compared to i.i.d. directed percolation at the same expected degree. The enhancement argument for non-essential enhancements with forbidden patterns is of independent interest, as is the byproduct that bond percolation with forbidden local patterns has threshold strictly above 1/2. The high-level structure is coherent, the argument is parameter-free, and the proof invokes standard results (planar duality, the 1/2 critical threshold, and the LSS97 domination theorem) appropriately. However, the manuscript's load-bearing local modifications in Section 6 are not fully proved in the text, with a companion website and 'left to the reader' invoked for essential cases. No machine-checked proofs or reproducible code are provided, so the current version is not yet complete.","major_comments":[{"comment":"The construction of the map Te, used to prove inequality (17), is not fully specified. The printed proof gives a semi-explicit construction but explicitly refers to an external website for 'a concrete visualization of the construction in all possible cases' and treats several subcases (o inside Be, Be intersecting ∂Λn, x′ = y′, and the choice of entry/exit pairs) only heuristically. Item (iii) of the lemma is load-bearing: it must hold for every entry/exit pair and every boundary/position subcase, and a single failure would invalidate (17), hence Proposition 6.3, Theorem 6.1, and the ε = 0 case of Theorem 3.1. The companion website is not a substitute for a rigorous proof, and no machine-checked verification is supplied.","section":"§6.3.1 (Lemma 6.7)"},{"comment":"The map Se for bad edges is only sketched. For categories (a)–(e), the proof states 'We leave the details to the reader', and for category (f) it again leaves the three subcases to the reader with only a schematic figure. Since inequality (18) — and hence the bound on the total contribution of bad edges in Lemma 6.6 — depends entirely on this lemma, the proof of Proposition 6.3 is incomplete at this point. As with Lemma 6.7, this gap is not cosmetic because the local modification must work in all boundary and origin-position cases.","section":"§6.3.2 (Lemma 6.9)"},{"comment":"The exponential tail estimate for the number of pivotal edges in the directed-corner model also defers 'the verification of the intermediate cases to the reader'. While Theorem 2.8 is a byproduct rather than the main theorem, the same standard of completeness should apply, and the omitted cases are not trivial given the auto-open pivotal-edge analysis in the proof.","section":"§7 (Lemma 7.1)"}],"minor_comments":[{"comment":"The text says 'All these probabilities are smaller than qε', but in the first case the probability equals qε = 1/2 − ε/4; it should say 'at most qε'.","section":"§5.2, Step 1, inequality (10)"},{"comment":"The proof claims that In(Be) contains 71 p-edges, but the box B is defined as the 6×5 rectangle minus its four corners, which gives 63 p-edges with both endpoints in B. The bound with exponent 71 remains valid because p(1−p) ≤ 1/4, but the stated count is incorrect and should be corrected or replaced by 'at most 71'.","section":"§6.3.1, proof of inequality (17)"},{"comment":"The displayed set Ibad lists eight edges, but the entry {(−1,−1),(−1,0)} appears twice; this appears to be a typographical repetition that should be cleaned up.","section":"§6.3, definition of Ibad"},{"comment":"The display computing the conditional probability of a pivotal edge being open is garbled in the text; the final value (2/3)(1−ε/4)/(1+ε/2) is clear, but the intermediate notation should be typeset cleanly.","section":"§4.2, pivotal probability computation"},{"comment":"The notation 'z_i^* := (i+1/2, −1/2)' is used for dual start points; later in the proof the union bound is written as 3L × Ce^{-cL^{1/4}}, which is correct, but the phrase 'for some integer 0 ≤ i ≤ 3L−1' should specify that i is an index over the left side of the rectangle, since the dual path starts on that side.","section":"§3.2, Lemma 3.2 proof"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a significant open problem with a plausible and well-structured proof. The main obstacle to acceptance is the incomplete verification of the local case analysis in Lemmas 6.7 and 6.9, which are genuinely load-bearing for the ε = 0 case of Theorem 3.1. The reliance on a companion website and on 'left to the reader' for essential subcases is not appropriate for a journal proof. If the authors supply a complete written case analysis or a machine-checked verification of these lemmas, the paper would be suitable for publication. I also note that the count of 71 p-edges in In(Be) appears inconsistent with the definition of B; this is minor but should be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is real and the high-level argument holds up. The planar 2-neighbor graph percolates, and pc(2) < 1/2. This was conjectured in [JKLT23] and had been open. The proof is not a repackaging; the exploration algorithm, the pivotal-edge decomposition, and especially the non-essential enhancement argument for forbidden-pattern percolation are genuinely new. The byproduct result—that i.i.d. bond percolation with those forbidden local patterns has threshold strictly above 1/2—is also interesting.\n\nThe paper is careful about the critical case ε = 0, which is exactly where the enhancement comes in, and it handles that case in a self-contained section. The block renormalization step is standard and clean.\n\nThe soft spot is Section 6. Lemmas 6.7 and 6.9 build local modification maps Te and Se that are supposed to convert any pivotal p-edge into a pivotal q-edge by changing edges in a 6x5 box. The text gestures at the construction, refers to an external website for “all possible cases,” and in Lemma 6.9 explicitly leaves details to the reader. These maps are load-bearing: a single bad subcase would break the derivative comparison (Proposition 6.3), hence Theorem 6.1, hence the exponential decay for ε = 0. That is a genuine verification gap, not a cosmetic omission.\n\nHow much does it matter? I do not think it signals a wrong proof. The construction is explicit, the figures help, and the surrounding argument is coherent. It reads like a paper where the authors got the main theorem right and under-documented the hardest local case check. A diligent referee could either work through the cases or require the authors to expand the appendix. No code or formalization is provided, which would have helped, but that is not standard for this subfield.\n\nThe rest of the paper—domination by constrained bond percolation, the subgeometric number of pivotal edges, the duality setup—is well supported. The citation pattern is appropriate, building on [JKLT23], [AG91], [BBR14], and [Gri99] without overclaiming.\n\nVerdict: this deserves a serious referee. I would send it to peer review with the clear instruction that Section 6 must be made self-contained or the missing case checks supplied. The result is significant enough that the case-checking burden is worth it. I would cite it if I worked on directed percolation or degenerate random environments, and I would likely bring it to reading group to discuss the enhancement technique.","headline":"Genuinely new proof of the 2-neighbor percolation conjecture, with a real but non-fatal verification gap in the enhancement case analysis.","tokens_in":42761,"tokens_out":2377,"would_cite":true,"duration_ms":24672,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B43"],"pacs":[],"model":"deepseek-v4-flash","headline":"The planar 2-neighbor graph percolates: pc(2)<1/2, settling the directed percolation question.","keywords":["2-neighbor graph","directed percolation","degenerate random environment","planar duality","pivotal edges","forbidden patterns","enhancement","critical threshold"],"falsifier":"Search the finite configuration space of the 6×5 box In(Be): for each configuration outside the box, check whether the inclusion PivOut_Be ∩ RTe ⊂ {Q(e) is pivotal} holds for the map Te of Lemma 6.7. A single configuration where a good p-edge is pivotal but no deterministic local change inside the box makes Q(e) pivotal would disprove inequality (17); equivalently, compute the probability that a pivotal p-edge's associated q-edge cannot be made pivotal and show it exceeds zero for some n and e.","tokens_in":41643,"feed_emoji":"🕸️","tokens_out":5197,"duration_ms":46078,"temperature":0.7,"pith_summary":"The paper proves that the directed 2-neighbor graph on the square lattice percolates: when each vertex independently chooses exactly two of its four neighbours as outgoing edges, the origin is connected to infinity with positive probability. The precise theorem is that the critical parameter of the continuous-parameter 4p-neighbor model lies strictly below 1/2, so the model percolates already at expected out-degree two. This settles a conjecture left open in the lattice k-neighbor graph literature and distinguishes the degree-constrained model from standard i.i.d. directed bond percolation, whose critical value at the same mean degree is exactly 1/2. The proof combines planar duality, an exploration algorithm that carves the dual forward cluster into pieces separated by pivotal edges, and an enhancement comparison showing that percolation with certain forbidden local patterns is strictly harder than ordinary bond percolation.","feed_headline":"Percolation threshold drops below 1/2 for planar 2-neighbor graph","feed_subtitle":"Exactly two chosen directions per vertex still reach infinity, while independent percolation at the same mean degree does not.","key_machinery":"The load-bearing mechanism is an exploration algorithm for the dual forward set that reveals directed dual edges depth-first and counter-clockwise, subject to two rules that prevent re-visiting vertices or entering filled holes. Certain revealed edges are pivotal: the dual edge is the east side of a unit square and one of the other three sides has already been explored and found closed, which raises its open probability to roughly 2/3. The algorithm stops when a closed pivotal edge is found, so the explored cluster splits into 'visited clusters' separated by pivotal edges. Each visited cluster is stochastically dominated by the cluster of the origin in i.i.d. bond percolation at parameter 1/2−ε/4 under the constraint that paths never use a forbidden 'left-winding' pattern. An enhancement argument (Theorem 6.1) then shows that the forbidden-pattern restriction pushes the effective threshold above 1/2, giving exponential decay of visited clusters even at ε=0.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.3: pc(2)<1/2, and consequently the 2-neighbor graph percolates on $Z^{2}$. To reach it the authors work in the (2,ε)-model, where each vertex has exactly two outgoing edges plus a third with probability ε, so every directed edge is open with probability 1/2+ε/4. They prove that the dual forward set of the origin has subexponential tail decay (Theorem 3.1), and at ε=0 this requires showing that i.i.d. bond percolation with a forbidden local pattern has threshold strictly above 1/2 (Theorem 6.1). The argument also yields as corollaries that the d-neighbor graph percolates in Z^d for all d≥2 and that the directed-corner model has critical parameter below 1/2.","pith_inferences":["If the local-surgery comparison is uniform, the forbidden-pattern threshold is separated from 1/2 by a positive amount, so any planar directed model whose exploration process forbids the same left-winding pattern should percolate below 1/2; the ρ-family of isotropic degree-two models interpolating between the north-south-east-west and directed-corner models is a natural testbed.","The enhancement proof gives a finite certificate: each of the finitely many boundary and origin-position subcases in Lemmas 6.7 and 6.9 could in principle be checked by exhaustive enumeration of 6×5 boxes, which would convert the sketch into a verified inequality and possibly yield an explicit ε′.","A concrete prediction beyond the paper is that the percolation probability at p=1/2 decreases as the model interpolates from opposite-edge pairs to corner pairs, matching the conjecture that lower geometric variability favours percolation; computer simulations on the same exploration process could test this ordering before a rigorous proof is found."],"forward_implications":["The planar 2-neighbor graph percolates, so a fixed out-degree of two is enough for an infinite directed open path in Z^2.","The critical parameter satisfies 0.373 < pc(2) < 0.5, refining the earlier known lower bound with the connective constant bound of [PT00].","For every d≥2, the d-neighbor graph percolates in Z^d (Corollary 2.4).","The directed-corner model also percolates for some p<1/2 (Theorem 2.8).","At equal mean degree two, the degree-constrained model percolates while i.i.d. directed bond percolation does not: pc < piid_c = 1/2 < paon_c (Corollary 2.9)."],"supporting_citations":[{"why":"Introduces the lattice k-neighbor graph and states the conjecture that the planar 2-neighbor case percolates; the present paper settles it.","marker":"[JKLT23]"},{"why":"Introduces the continuum k-nearest-neighbor percolation model that motivates the discrete k-neighbor graph.","marker":"[HM96]"},{"why":"Provides the stochastic-domination theorem used to pass from subexponential decay of dual forward sets to percolation of the block-renormalized site model.","marker":"[LSS97]"},{"why":"Supplies the sharp-transition and exponential-decay facts for subcritical i.i.d. bond percolation used throughout Sections 5 and 6.","marker":"[Gri99]"},{"why":"The enhancement technique for strict monotonicity of critical points that the paper adapts to non-essential enhancements.","marker":"[AG91]"},{"why":"The essential-enhancements framework that the paper explicitly cannot import directly because its enhancement is not essential.","marker":"[BBR14]"},{"why":"Supplies the upper bound on the connective constant c(2) used to get the numerical lower bound pc(2)≥0.373.","marker":"[PT00]"}],"fun_headline_variants":["Two-neighbor graph percolates: threshold below 1/2","Exactly two random directions per vertex yield infinite cluster","Same mean degree, only 2-neighbor percolates on Z^2","Forbidden local patterns raise percolation threshold above 1/2","d-neighbor graph percolates for all d≥2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on the claim that inside every 6×5 box around a pivotal edge one can always locally rewrite the configuration so that a q-edge becomes pivotal while no new forbidden pattern is created; if even one boundary or origin-position subcase admits no such local rewrite, the enhancement comparison and hence the strictly-below-1/2 percolation conclusion would fail.","fun_headline_variants_meta":{"raw":{"variants":["Two-neighbor graph percolates: threshold below 1/2","Exactly two random directions per vertex yield infinite cluster","Same mean degree, only 2-neighbor percolates on Z^2","Forbidden local patterns raise percolation threshold above 1/2","d-neighbor graph percolates for all d≥2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00128,"raw_usage":{"total_tokens":5196,"prompt_tokens":870,"completion_tokens":4326,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":4235}},"tokens_in":486,"tokens_out":4326,"duration_ms":32267,"temperature":1.0,"reasoning_tokens":4235,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:12:40.470194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the finite configuration space of the 6×5 box In(Be): for each configuration outside the box, check whether the inclusion PivOut_Be ∩ RTe ⊂ {Q(e) is pivotal} holds for the map Te of Lemma 6.7. A single configuration where a good p-edge is pivotal but no deterministic local change inside the box makes Q(e) pivotal would disprove inequality (17); equivalently, compute the probability that a pivotal p-edge's associated q-edge cannot be made pivotal and show it exceeds zero for some n and e.","supporting_citations":[],"review_version":1}