{"id":"8b0da6d0-30d8-4a06-9666-80898560d7b3","arxiv_id":"2607.28566","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Perimeter processes of percolated triangulations equal L1 distances of spine steps of the corresponding Kreweras walk to excursion endpoints, extending to the branching case.","lead":"The paper gives an explicit formula linking the perimeter process of a percolated triangulation (along its interface or branching exploration tree) to L1 distances read off the associated Kreweras walk. This discrete identity clarifies why growth-fragmentations appear inside Brownian cone excursions in the continuum mating-of-trees picture.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the dependence on the BHS spine correspondence as the sole external pillar and correctly judges it low-risk: the correspondence is a published combinatorial fact, not an analytic extrapolation. The note’s incremental contribution—the explicit L1 formula and its branching extension—is fully checkable from the given text once that pillar is accepted. No internal inconsistency, missing case, or unstated hypothesis undermines the identity. Consequently the ACCEPT verdict stands without adjustment.","tokens_in":15154,"tokens_out":411,"duration_ms":8660,"concrete_test":"Pick any small explicit Kreweras word in K_{n,k} (e.g. the 11-letter word of Figure 7), run the iterative construction of Φ to obtain (M,σ), compute the unfilled interface peeling steps S(M,σ) and the reduced word π̂(w) side-by-side, and verify that the L1 distances ||X(T(n+1))-X(N)||_1+2 exactly reproduce the successive perimeters; any mismatch would falsify Lemma 6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identity of Theorems 1 and 10 follows once the reduced-word/spine correspondence (Lemma 8, taken from the proof of BHS23 Thm 3.6) and the unfilled-peeling equivalence (Lemma 7) are granted. Both are standard consequences of the published Bernardi–Holden–Sun bijection; the note’s own arguments (increment matching via the L1 calculation after (2), induction on monochromatic faces, and the recursive restriction for off-interface triangles) are elementary, self-contained, and free of hidden analytic or measure-theoretic assumptions. The one-white-boundary restriction is only presentational (Remark 3). No load-bearing gap appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The note establishes an explicit combinatorial identity relating the perimeter process of the percolation-interface peeling of a Dobrushin-percolated triangulation to its Kreweras walk under the Bernardi–Holden–Sun bijection. Theorem 1 states that if T(n) are the spine-step times of the walk X ending at X(N), then the interface perimeter satisfies P(n) = ||X(T(n+1)) − X(N)||_1 + 2. Theorem 10 extends the same L1 formula to every branch of the branching peeling exploration, replacing the global endpoint by suitable restricted spine times T_t and closure times D_t. The argument proceeds by identifying reduced Kreweras words with peeling steps (Lemma 6) via the spine and the unfilled peeling process, then matching increments.","tokens_in":15251,"tokens_out":1163,"duration_ms":31209,"significance":"The result supplies a clean discrete counterpart to the embedding of a self-similar growth-fragmentation inside a Brownian cone excursion obtained by Da Silva–Powell–Watson, and thereby clarifies the link between the mating-of-trees and peeling approaches to percolated triangulations. The derivation is purely combinatorial, parameter-free, and makes essential but transparent use of the published BHS bijection; once the reduced-word/spine correspondence is granted, the perimeter identity follows by elementary increment matching. The branching extension (Theorem 10) is the load-bearing discrete input needed for a prospective joint scaling limit of Kreweras walks, exploration trees, and growth-fragmentations. As a short note it is appropriately scoped and adds a concrete geometric dictionary that was previously missing.","major_comments":[{"comment":"Lemma 8 is the load-bearing bridge from the BHS bijection to the reduced-word/peeling identification, yet it is only asserted as “implied by the proof of Theorem 3.6 of [BHS23]”. For a note whose entire argument rests on this correspondence (close matchings ↔ monochromatic faces of the spine, ordered along the interface), a short self-contained extraction or a precise pointer to the relevant paragraphs/figures of BHS23 would make the claim checkable without reconstructing that proof. The subsequent induction in the proof of Lemma 6 and the recursive restriction in the proof of Theorem 10 inherit any gap here.","section":"§3.1, Lemma 8"},{"comment":"In the off-interface case of the proof of Theorem 10, after the first separating L_ℓ/R_ℓ step the argument reduces to Φ(w_{T(j_0)+1}⋯w_q) and then says “we recursively iterate the same reasoning”. It should be made explicit that each recursive call updates the ambient walk to the corresponding cone-excursion subword and that the closure times D_t(n) for later spine steps are exactly the successive close-matching c-steps of those nested excursions (so that the L1 formula continues to hold with the correct local endpoint). The current wording leaves the identification of D_t slightly implicit.","section":"§4, proof of Theorem 10"}],"minor_comments":[{"comment":"Theorem 2 is labelled “informal” and then restated as Theorem 10; consider merging into a single numbered statement (with the informal version only in the introduction) to avoid dual numbering.","section":"§1 and §4"},{"comment":"In the definition of spine steps just above Theorem 1, “cone excursion” is defined via X(m) ∈ X(n_1) + N²; the same notion reappears later with “height of the cone excursion”. A single consistent terminology (and a pointer that this matches the close-matching language of §3) would help.","section":"§1, before Theorem 1"},{"comment":"Figure 1 caption and the bottom panel are helpful; adding a short legend that the red squares are exactly the spine times T(n) would make the visual claim of the theorem immediate.","section":"Figure 1"},{"comment":"Typo: “T op left” / “T op right” in the Figure 1 caption (stray space after T).","section":"Figure 1 caption"},{"comment":"Remark 3 correctly notes that the one-white-boundary restriction is presentational; a one-sentence indication of how the L1 formula changes for general (ℓ,r) Dobrushin data (first coordinate = white boundary vertices of the unexplored region, etc.) would make the remark immediately usable.","section":"Remark 3"},{"comment":"The date on the title page reads “July 31, 2026”; confirm this is intentional.","section":"Title page"}],"recommendation":"minor_revision","confidential_remarks":"The note is a genuine and useful dictionary between two standard discrete encodings; it is not incremental padding. Fit for a combinatorics or probability journal that already publishes short notes linking bijections to peeling is good. The only editorial caution is that the dependence on a non-isolated fragment of the BHS23 proof should be tightened before final acceptance so that the note is self-contained at the level of a careful reader."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new thing here is the explicit formula: interface perimeter equals L1 distance from the spine points of the Kreweras walk to the endpoint (plus 2), and the same formula works branch-by-branch once you restrict the walk and take the right closure times. That identity is not in BHS23, BCK18 or SPW25; those papers give the bijection, the growth-fragmentation limit, and the continuum embedding. This note supplies the missing discrete geometric reading.\n\nIt does the job cleanly. Once you grant the reduced-word/spine correspondence (their Lemma 8, taken from the proof of BHS Thm 3.6) and the unfilled-peeling equivalence (Lemma 7), the rest is elementary increment matching. The L1 calculation after their display (2) is immediate, the induction on monochromatic faces is short, and the recursive argument for off-interface triangles is transparent. No free parameters, no fitting, no analytic hand-waving. Remark 3 correctly notes that the one-white-boundary restriction is only presentational. The figures make the geometry easy to check by hand on small examples.\n\nSoft spots are minor and proportional. The note treats the BHS spine correspondence as a black box rather than restating the short argument; that is appropriate for a note but means a reader who has not internalised BHS has to trust the citation. The joint-scaling remark (Remark 5) correctly flags that the missing piece is local-time convergence of spine steps, so they do not claim more than they prove. Question 4 about other models is left open, as it should be.\n\nThis is for people already working on peeling, mating-of-trees, or the SPW25 embedding. It will not convert outsiders, but anyone chasing the discrete-to-continuum dictionary will want the identity on the shelf. Math and citations look solid; the dependence on BHS is legitimate and non-circular.\n\nSend it to referees. It is a short, correct, useful note.","headline":"Clean discrete L1 identity linking Kreweras spine steps to peeling perimeters; short combinatorial note that genuinely bridges BHS and SPW25 without overclaiming.","tokens_in":15953,"tokens_out":503,"would_cite":true,"duration_ms":10268,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C10","60C05","60J80","05A19"],"pacs":[],"model":"grok-4.5","headline":"The perimeter of percolation peeling on a triangulation equals the L1 distance of spine steps on its Kreweras walk to the walk’s endpoint.","keywords":["Kreweras walks","percolated triangulations","peeling process","perimeter process","branching exploration","growth-fragmentation","Bernardi bijection","mating of trees"],"falsifier":"Take any small explicit percolated triangulation with one white boundary vertex, compute its Kreweras walk by the known bijection, mark the spine steps, evaluate the claimed L1 formula, and check whether the resulting sequence equals the perimeter process obtained by peeling the map directly along its interface.","tokens_in":15990,"feed_emoji":"📐","tokens_out":963,"duration_ms":21526,"temperature":0.7,"pith_summary":"This note proves an exact discrete identity: for a percolated planar triangulation with one white boundary vertex, the perimeter process recorded while peeling along the percolation interface is completely determined by the corresponding Kreweras walk. At each spine step of the walk (an a- or b-step not enclosed by a close matching), the perimeter equals the L1 distance from that point to the walk’s endpoint, plus two. The same geometric rule extends branch by branch to the full branching peeling exploration, using restricted spine times and the first later time the walk closes the cone containing that segment. The identity makes precise, at the discrete level, why growth-fragmentation perimeter processes appear inside correlated Brownian cone excursions in the continuum mating-of-trees picture.","feed_headline":"Peeling perimeters are L1 distances on Kreweras walks","feed_subtitle":"Spine steps of the walk give the exact discrete perimeter process of percolated triangulations","key_machinery":"Spine steps of the Kreweras walk (and the associated reduced Kreweras word). Via the Bernardi–Holden–Sun bijection they are identified with the successive bicolour triangles traversed by the percolation interface; the height of each close-matched cone excursion becomes the size of the monochromatic face filled at an Lℓ or Rℓ peeling step, so perimeter increments match the L1 increments of the reduced walk.","core_discovery":"For a percolated triangulation with exactly one white boundary vertex and its Bernardi–Holden–Sun Kreweras walk X of length N, if T(n) marks the n-th spine step of X, then the interface perimeter process satisfies P(n) = ∥X(T(n+1)) − X(N)∥₁ + 2. The same L1 formula holds for every branch of the branching peeling exploration once spine times are taken inside the walk prefix that builds the target triangle and distances are measured to the cone-closure time of that prefix.","pith_inferences":["Once spine-step local time is controlled, the three scaling limits (Kreweras walk, Brownian disk, growth-fragmentation) should converge jointly without further combinatorial input.","The first coordinate of X(T(n)) − X(N) literally counts white boundary vertices of the unexplored region, giving a direct colour-refined perimeter process.","The same geometric extraction may supply discrete precursors of the cone-free local times used in the continuum growth-fragmentation embedding."],"forward_implications":["The discrete growth-fragmentation of branching perimeters is now an explicit functional of the Kreweras walk, not merely a scaling-limit object.","Joint scaling of the Kreweras walk to a Brownian cone excursion and of the branching perimeters to the self-similar growth-fragmentation becomes a question of local-time convergence of spine steps.","The same L1 reading extends immediately to Dobrushin boundaries with several consecutive black and white vertices.","An analogous dictionary can be asked for other walk encodings of statistical-mechanics decorations on maps (e.g., hamburger-cheeseburger)."],"fun_headline_variants":["Kreweras spine steps yield exact L1 peeling perimeters","Interface perimeter equals L1 distance on Kreweras walk","Branching peeling perimeters from Kreweras L1 distances","P(n) is L1 norm to cone-closure on Kreweras spine","Percolated triangulation perimeters track Kreweras L1"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument stands or falls on the already-proved correspondence that close matchings in the Kreweras word produce exactly the monochromatic faces of the spine; if that dictionary fails for some configurations, spine steps no longer line up with peeling steps.","fun_headline_variants_meta":{"raw":{"variants":["Kreweras spine steps yield exact L1 peeling perimeters","Interface perimeter equals L1 distance on Kreweras walk","Branching peeling perimeters from Kreweras L1 distances","P(n) is L1 norm to cone-closure on Kreweras spine","Percolated triangulation perimeters track Kreweras L1"]},"model":"grok-4.5","effort":"low","cost_usd":0.003938,"raw_usage":{"total_tokens":1122,"prompt_tokens":644,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":39384000,"prompt_tokens_details":{"text_tokens":644,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":405,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":644,"tokens_out":73,"duration_ms":7813,"temperature":1.0,"reasoning_tokens":405,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T03:30:51.777885+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Take any small explicit percolated triangulation with one white boundary vertex, compute its Kreweras walk by the known bijection, mark the spine steps, evaluate the claimed L1 formula, and check whether the resulting sequence equals the perimeter process obtained by peeling the map directly along its interface.","supporting_citations":[],"review_version":1}