{"id":"e3ef4f15-1719-4314-b6ab-0a291941443e","arxiv_id":"2606.30954","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In complete b-metric spaces, perimeter-contracting maps without 2-cycles have iterates that become graphic contractions when s q^n <1 (yielding 1–2 fixed points); with a 2-cycle, even iterates do likewise under s q^2 <1.","lead":"Maps that shrink triangle perimeters in complete b-metric spaces have high iterates that act as path-wise contractions, forcing either one or two fixed points or exactly one isolated 2-cycle. The result extends fixed-point tools to spaces (such as certain L^p and sequence spaces) where ordinary metrics fail.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript cleanly extends Cvetković's metric-space results to complete b-metric spaces. Continuity of MCPTs is established first (Prop. 3.1) by a careful subsequence argument that never assumes joint continuity of d; the graphic-contraction estimates for high iterates (Thm 3.2) and for even iterates when a 2-cycle is present (Thm 3.14) follow by direct perimeter iteration plus the elementary inequalities of Lemma 2.9; the period dichotomy (no periods >2, mutual exclusion of fixed points and 2-cycles) is elementary and independent of s. The external Petruşel–Petruşel theorem is invoked only after its hypotheses (closed graph + sλ<1) have been secured inside the paper, so the dependence is not load-bearing in the sense of an unverified assumption. Examples confirm both that MCPTs properly contain graphic contractions and that the thresholds s q^n<1 and s q^2<1 are sharp. Presentational defects (duplicated abstract, self-referential sentence, acronym drift) do not affect the mathematics. Consequently the reader's ACCEPT / high-confidence verdict stands.","tokens_in":19729,"tokens_out":619,"duration_ms":5820,"concrete_test":"Independently re-derive the graphic estimate (3.4) from the n-fold perimeter inequality (3.3) using only Definition 2.1, Lemma 2.9 and the b-triangle inequality, without invoking any external fixed-point theorem; if the rearranged bound d(f^n x,f^{2n}x)≤(s q^n)/(1-s q^n)d(x,f^n x) fails to hold under s q^n<1, the central claim collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption flag (dependence on Petruşel–Petruşel for closed-graph graphic contractions under sλ<1) is correctly noted but does not create an internal soft spot. Proposition 3.1 proves continuity of any MCPT by sequential tracking (Lemma 2.2) without assuming simultaneous continuity of d; closedness of G_f (and of G_{f^n}) then follows from the Hausdorff property of b-metric spaces. The coefficient bound s q^n <1 is obtained by elementary limit, and the graphic estimate itself is derived from the perimeter inequality plus the elementary bounds of Lemma 2.9. The external theorem is therefore applied only after its hypotheses have been verified inside the paper. No hidden gap appears in the derivation of the graphic inequality, the period dichotomy, or the cardinality statements.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies mappings that contract perimeters of triangles (MCPTs) on complete b-metric spaces with coefficient s ≥ 1. Under the exclusion of prime period-two orbits, it proves that iterates f^n become continuous graphic contractions whenever s q^n < 1, so f is weakly Picard and 1 中 |Fix(f)| ≤ 2. When a 2-cycle is present and s q^{2} < 1, the even iterates f^{2n} are continuous graphic contractions and the map has exactly two periodic points forming that unique 2-cycle. Continuity of any MCPT is established by sequential tracking (Lemma 2.2), periods greater than 2 are ruled out, fixed points and 2-cycles are shown to be mutually exclusive, and examples (including a shift map) demonstrate that MCPTs properly contain graphic contractions and that the parameter thresholds are sharp.","tokens_in":19920,"tokens_out":801,"duration_ms":7447,"significance":"The work cleanly extends Cvetković’s metric-space results on perimetric contractions and their iterates to the b-metric setting, correctly handling the lack of joint continuity of d via sequential inequalities. The continuity proposition, the period dichotomy, the graphic estimates derived from Lemma 2.9, and the cardinality statements are self-contained once the external graphic-contraction theorem of Petruşel–Petruşel is invoked under verified hypotheses. Concrete examples establish that the class of MCPTs is strictly larger than graphic contractions and that the thresholds s q^n < 1 and s q^{2} < 1 are optimal. The contribution is a solid, technically careful generalization that completes the structural picture for this multi-point contraction class in b-metric spaces.","major_comments":[],"minor_comments":[{"comment":"Abstract and title use both “CPTM” and “MCPT”; the body consistently uses MCPT. Standardize the acronym throughout.","section":null},{"comment":"Theorem 3.2, Step 2: the parenthetical “exactly as in the proof of Theorem 3.2” is a self-reference; rephrase to “as in the estimate leading to (3.4)” or similar.","section":null},{"comment":"Proposition 3.1, Step 4: the passage from (3.1) to the limsup inequality is correct via Lemma 2.2, but a one-sentence reminder that the left-hand side is independent of l would improve readability.","section":null},{"comment":"Example 3.3: the verification that d_b is complete is clear, yet the claim that the triangle inequality for the original d holds for all triples with k < 4 is asserted without listing the six triples; a short table or explicit check would remove any residual doubt.","section":null},{"comment":"Several bibliographic entries (e.g., [7], [12], [24]) appear with future or non-standard volume data; verify final publication details before typesetting.","section":null},{"comment":"Notation for the perimeter P(x,y,z) is introduced twice (once in the preliminaries and again before Definition 2.8); a single definition suffices.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a careful and essentially complete extension of known metric-space results. The dependence on the Petruşel–Petruşel graphic-contraction theorem is legitimate once continuity and the coefficient bound are established inside the paper; no hidden gap remains. Fit for a solid specialized journal in fixed-point theory or generalized metric spaces is good. Minor presentational polishing is all that is required."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a competent, self-contained extension of Cvetković’s metric-space results on mappings that contract triangle perimeters (MCPTs) to complete b-metric spaces. The new content is real: under the sole exclusion of prime period-2 orbits, every iterate f^n with s q^n < 1 is a continuous graphic contraction, so the map is weakly Picard and 1 ≤ |Fix(f)| ≤ 2; when a 2-cycle is present and s q^{2} < 1, the even iterates become graphic contractions and there are exactly two periodic points forming that single cycle. The continuity proof (Prop. 3.1) correctly uses sequential limsup tracking (Lemma 2.2) instead of joint continuity of d, the period >2 prohibition is elementary and independent of s, and the graphic estimates follow from the perimeter inequality plus the elementary (s+1)/s bounds. Examples (shift map, discrete 3-point space) show both that MCPTs properly contain graphic contractions and that the parameter thresholds are sharp.\n\nThe dependence on the Petruşel–Petruşel graphic-contraction theorem is not a hidden gap: the paper first proves continuity of any MCPT (hence closed graph of the iterates) and obtains the coefficient bound by elementary limit, so the external hypotheses are verified inside the manuscript. Presentational defects exist—duplicated abstract, acronym flip between CPTM/MCPT, one self-referential proof sentence—but they are cosmetic and do not touch the mathematics.\n\nThis is honest incremental work for people already working with b-metrics and multi-point contractions. It does not reorganize nonlinear analysis, but a specialist can trust the statements and build on them. I would send it to referees; the core arguments hold.","headline":"Solid, usable extension of Cvetković’s perimeter-contraction iterates to b-metrics; the sequential-tracking arguments work and the 2-cycle dichotomy is clean.","tokens_in":20516,"tokens_out":463,"would_cite":true,"duration_ms":4434,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H05","47J25"],"pacs":[],"model":"grok-4.5","headline":"In complete b-metric spaces, mappings that shrink triangle perimeters become graphic contractions on high enough iterates, so they are weakly Picard with at most two fixed points (or exactly one 2-cycle).","keywords":["b-metric space","mapping contracting perimeters of triangles","graphic contraction","weakly Picard operator","periodic point","fixed-point set","iterate"],"falsifier":"Exhibit a complete b-metric space and a perimeter-contracting map with no 2-cycle for which some iterate with s q^{n} < 1 fails to be a graphic contraction, or for which the fixed-point set is empty or has more than two points.","tokens_in":20630,"feed_emoji":"△","tokens_out":662,"duration_ms":4941,"temperature":0.7,"pith_summary":"The paper studies mappings that contract the perimeter of every triangle of three distinct points in a complete b-metric space. These multi-point conditions are independent of ordinary pairwise contractions. When the map has no period-2 orbits, every sufficiently high iterate becomes a continuous graphic contraction (provided the product of the b-coefficient s and the nth power of the contraction constant is less than 1). Consequently the map is weakly Picard and its fixed-point set has cardinality between 1 and 2. When a genuine 2-cycle is present and s q^{2} < 1, the even iterates become graphic contractions and the map has exactly those two periodic points. The proofs carefully avoid simultaneous continuity of the b-metric by using sequential tracking inequalities, and explicit examples show that the class of perimeter contractions properly contains the graphic contractions.","feed_headline":"Perimeter-shrinking maps become graphic contractions on high iterates","feed_subtitle":"In complete b-metric spaces they are weakly Picard with 1–2 fixed points, or exactly one 2-cycle","key_machinery":"The perimeter-contraction inequality P(f(x),f(y),f(z)) ≤ q P(x,y,z) for distinct triples, combined with sequential tracking bounds that replace simultaneous continuity of the b-metric; these force high iterates (or even iterates) to satisfy the graphic-contraction estimate d(f^{n}(x),f^{2n}(x)) ≤ λ d(x,f^{n}(x)) with λ < 1.","core_discovery":"Under the sole exclusion of prime period-two orbits, every iterate f^{n} of a mapping that contracts perimeters of triangles becomes a continuous graphic contraction once s q^{n} < 1; the map is therefore weakly Picard and satisfies 1 ≤ |Fix(f)| ≤ 2. When a 2-cycle exists and s q^{2} < 1, the even iterates f^{2n} are continuous graphic contractions and the map possesses exactly two periodic points forming that single 2-cycle.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["CPTM iterates become graphic contractions once sq^n <1","Perimeter maps are weakly Picard with 1–2 fixed points","No period-2: high CPTM iterates act as graphic contractions","With a 2-cycle, even CPTM iterates turn graphic under sq^2<1","b-metric perimeter contractions yield exactly one 2-cycle or 1–2 fixed points"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The whole existence argument rests on an external theorem that a continuous graphic contraction with closed graph and sλ < 1 already possesses a fixed point in a complete b-metric space.","fun_headline_variants_meta":{"raw":{"variants":["CPTM iterates become graphic contractions once sq^n <1","Perimeter maps are weakly Picard with 1–2 fixed points","No period-2: high CPTM iterates act as graphic contractions","With a 2-cycle, even CPTM iterates turn graphic under sq^2<1","b-metric perimeter contractions yield exactly one 2-cycle or 1–2 fixed points"]},"model":"grok-4.5","effort":"low","cost_usd":0.004354,"raw_usage":{"total_tokens":1385,"prompt_tokens":893,"num_sources_used":0,"completion_tokens":103,"cost_in_usd_ticks":43540000,"prompt_tokens_details":{"text_tokens":893,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":389,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":893,"tokens_out":103,"duration_ms":4605,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T10:26:06.567866+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a complete b-metric space and a perimeter-contracting map with no 2-cycle for which some iterate with s q^{n} < 1 fails to be a graphic contraction, or for which the fixed-point set is empty or has more than two points.","supporting_citations":[],"review_version":2}