{"id":"f1b5671d-3edb-4a42-ada6-247a5e183b8a","arxiv_id":"2607.15502","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"For each fixed 0≤k<n, the minimum lattice set containing a filled k-skeleton about every one of N centers is Θ(N^{1-(n-k)/(2n^2)}), closing the endpoint lower bound.","lead":"This paper determines the exact exponent for the size of a lattice set that must contain cube skeletons around every one of N centers: for k-dimensional skeletons in n dimensions, the answer is N^{1-(n-k)/(2n^2)} up to constants. It closes the endpoint lower bound that Thornton left open when k≥1.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the lower-bound induction, cofactor entropy step, and digit construction are internally consistent.","rationale":"The reader's verdict is ACCEPT with high confidence, and I see no reason to move it. The reader's identified weakest assumption—the lattice-only small-radius occupancy bound—is real but it is part of the theorem's hypotheses (B,S ⊂ Z^n), and the paper explicitly acknowledges it in §6. I therefore do not treat it as a load-bearing concern that threatens the central claim. I checked each of the main proof components: the uniform-cover entropy lemma, the axial midpoint bound, the cube-vertex completion via a linear change of basis, the cofactor estimate built on the completion set C, the balanced strong induction in Proposition 4.1, and the corrected Thornton digit construction in Proposition 5.2. The constants and exponents line up (β = γ + k/n), and the induction closes in both the large- and small-radius cases. I could not identify a circular step, a missing case, or an off-by-one error. The proposed concrete test would independently verify the one piece that is most amenable to a direct finite check, namely the signed-digit construction, but this is a verification step rather than a response to a suspected flaw.","tokens_in":6887,"tokens_out":37447,"duration_ms":309397,"concrete_test":"For n=2,k=1 and h=3, explicitly enumerate the digit set D_3 from Lemma 5.1 and, for every x ∈ {1,...,728}^2, compute the r_x from equation (5.1) and verify numerically that x_j − r_x and x_j + r_x both belong to D_3 for j=1,2 and that 1 ≤ r_x < 729. If any pair fails, the upper-bound construction contains a concrete bug.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof in good faith and could not find a load-bearing gap in the central claim. The main steps check out: the axial midpoint bound (Lemma 2.2) uses a correct distinct-pair count per line, the uniform-cover entropy inequality (Lemma 2.1) is valid, and the cofactor estimate (Lemma 3.1) correctly passes from cube vertices to coordinate projections via the completion set C. The strong induction in Proposition 4.1 closes: the large-radius branch yields an A_I with at least c N^γ distinct face offsets, each face contributing L^k points, and the small-radius branch uses the lattice cell bound N_m ≤ h^n ≤ ρN < N, with overlap at most 3^n, and the chosen ρ satisfies (4.4). The digit construction (Lemma 5.1) also holds; the signed-digit argument guarantees x_j ± r_x ∈ D_h with the required zero coefficient. The only delicate assumption is the lattice-only occupancy estimate in the small-radius branch (Eq. 4.8), which is an acknowledged limitation of the method for nonlattice centers, not a flaw in the stated theorem about lattice sets. No internally inconsistent or circular step was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the minimum size of a finite lattice set B that contains a filled axis-parallel discrete k-skeleton S_k(x,r_x) about each point of an N-point set S of lattice centers. The main result (Theorem 1.1) is that F_{n,k}(N) is bounded between positive constants times N^{1-(n-k)/(2n^2)} for all N. The lower bound, previously open for k≥1, is proved via a labelled Shearer entropy inequality, a midpoint estimate for cube vertices, a cofactor estimate for projections onto (n−k)-coordinate faces, and a strong induction balancing large- and small-radius cases without a logarithmic loss. The upper bound is a corrected digit construction with the same exponent.","tokens_in":7196,"tokens_out":25965,"duration_ms":202472,"significance":"The result resolves the endpoint exponent for discrete cube skeleta, closing the gap between Thornton's sub-endpoint lower bounds and his construction. The proof is self-contained and transparent: Lemma 2.1 supplies a labelled uniform-cover entropy inequality, Lemma 3.1 converts vertex data into normal cofactors, and Proposition 4.1 closes the induction with a single constant at the endpoint. The digit construction in Section 5 gives a matching upper bound and is independently verified. The paper also states its methodological limitation for nonlattice centers (§6), which is correctly scoped. If the proof is valid, this is a substantial contribution to discrete geometry.","major_comments":[],"minor_comments":[{"comment":"The sentence 'Apply Theorem 2.1 to the n coordinate-deletion sets' should refer to Lemma 2.1; there is no Theorem 2.1. The same mislabelling appears in the proof of Lemma 3.1: 'Consequently Theorem 2.1 gives' should be 'Lemma 2.1'.","section":"§2 (proof of Lemma 2.2)"},{"comment":"The sentence 'Thus the pair A, S_L satisfies the hypotheses of Theorem 3.1' should refer to Lemma 3.1, not a Theorem.","section":"§4 (proof of Proposition 4.1)"},{"comment":"In the final proof, 'Theorem 4.1 applies' and 'Theorem 5.2 supplies' are mislabelled; these are Proposition 4.1 and Proposition 5.2.","section":"§5 (proof of Theorem 1.1)"},{"comment":"The road-map paragraph refers to 'Theorems 2.1 to 2.3', 'Theorem 3.1', 'Theorem 4.1', and 'Theorems 5.1 and 5.2'. These are actually Lemmas 2.1–2.3, Lemma 3.1, Proposition 4.1, and Lemma 5.1/Proposition 5.2. Please correct the labels throughout for consistency.","section":"§1 (Introduction)"},{"comment":"The title as rendered in the full text contains a typo: 'CUBE SKELET A' should likely read 'CUBE SKELETA' or 'CUBE SKELETA' as in the abstract.","section":"Full-text title"},{"comment":"The phrase 'Thornton proved every smaller exponent' is imprecise; it would be clearer to say 'Thornton proved the lower bound for every exponent smaller than the endpoint'.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical content is sound. The main revision needed is to correct the pervasive mislabelling of Lemmas and Propositions as Theorems in the text; this is purely typographical and does not affect the validity of the results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the endpoint lower bound is real, the proof works, and the paper deserves a serious referee. I read the induction closely and couldn't find a load-bearing gap.\n\nWhat's actually new: Thornton proved all sub-endpoint exponents and constructed examples at the endpoint, but for k≥1 the matching lower bound was open. This paper provides it, with a clean labelled-Shearer step that keeps coordinate labels attached to projections, and a one-step balanced induction instead of an iterated exponent-improvement map. The construction section also corrects Thornton's digit set; the correction matters and is handled well.\n\nThe proof is self-contained: Lemma 2.1 proves the uniform-cover entropy inequality, Lemma 3.1 passes from cube vertices to cofactors without losing multiplicity, and Proposition 4.1 closes strong induction with the same constant at the endpoint. I checked the small-radius branch: the cell bound N_m ≤ h^n is exactly what makes the induction descend, and the overlap bound is the standard 3^n. The constants in (4.4) work. The digit construction in Lemma 5.1 also checks out—the cancellation argument is right.\n\nThe soft spots are minor. The paper is lattice-only; the small-radius branch genuinely needs the cell occupancy bound. That's a real limitation, acknowledged in §6, but it's not a flaw in the stated theorem. Also, in the proof of Lemma 2.2 there's a typo: it writes |C| ≥ √2 |T| ∑_L √p_L, but it should be √(2|T|) (or √2 √|T|) times the sum; the final bound is correct, so the slip doesn't affect the result. There's also a stray \"Theorem 2.1\" reference where it means Lemma 2.1. Nothing else.\n\nWho benefits: anyone working on discrete cube skeletons, square centers, or discretized maximal operators. The lattice restriction may limit direct application, but the technique—the labelled uniform-cover inequality and the balanced induction—is the kind of thing that could be reused.\n\nMy call: send it to a serious referee. If the referee goes through the induction once, they'll likely accept. I'd cite it if I worked in this area.","headline":"The paper closes the open endpoint lower bound for lattice cube skeleta with a genuinely new labelled-entropy/balanced-induction argument, and the proof is sound.","tokens_in":7644,"tokens_out":14368,"would_cite":true,"duration_ms":118199,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","52C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper determines the exact minimum size of a lattice set that carries N filled cube k-skeleta.","keywords":["cube skeleton","discrete geometry","entropy","Shearer's inequality","lattice set","square boundaries","endpoint exponent","finite cardinality"],"falsifier":"For n=2,k=1, try to build an infinite sequence of finite sets B_N⊂Z^2 with |B_N| ≤ C N^{7/8−ε} for some ε>0 that still contain a square boundary about each of N distinct lattice centers. If such a sequence existed, Corollary 1.3 would be false; the paper's construction, by contrast, matches the O(N^{7/8}) upper bound.","tokens_in":6795,"feed_emoji":"🧊","tokens_out":4472,"duration_ms":41045,"temperature":0.7,"pith_summary":"The paper establishes that the minimum size of a finite lattice set containing a filled axis-parallel discrete k-skeleton about every point of an N-point center set is, up to constants depending only on n and k, exactly N^{1-(n-k)/(2n^2)}. Earlier work produced constructions at this exponent and lower bounds at every smaller exponent, but the matching lower bound at the endpoint was open for k≥1; this paper closes that gap. The main interest is that the exponent is exact and that the proof avoids the logarithmic losses of previous approaches. In the planar case n=2, k=1 (square boundaries), the result says the carrier must have at least cN^{7/8} points, and this exponent is sharp.","feed_headline":"Cube skeleton size pinned to exact exponent","feed_subtitle":"For N centers in Z^n, the minimal carrier lies within constants of N^{1-(n-k)/(2n^2)} — and square boundaries need N^{7/8}.","key_machinery":"The central objects are the filled discrete k-skeleton S_k(x,r) (all points whose coordinates coincide with x within r on some k coordinates and equal x±r on the rest) and the cofactor estimate, which uses a labelled form of Shearer's projection inequality to convert cube-vertex entropy into a lower bound on the number of distinct (n−k)-tuples of fixed coordinates. The proof's load-bearing mechanism is a strong induction that balances two regimes: a large-radius branch where faces of length at least aN^{1/n} are disjoint, and a small-radius branch where the integer lattice confines at most h^n centers to each cell, giving a contraction that closes the induction.","core_discovery":"For fixed 0≤k<n, define F_{n,k}(N) as the least size of a set B⊂Z^n that contains a filled discrete k-skeleton S_k(x,r_x) about each of N centers x with integral radii. The paper proves two-sided bounds c N^{1-(n-k)/(2n^2)} ≤ F_{n,k}(N) ≤ C N^{1-(n-k)/(2n^2)}. The lower bound is the main work: it combines a cube-vertex midpoint estimate with a labelled uniform-cover entropy inequality to show that many labeled normal cofactors must be distinct, then a strong induction splits centers into those with radius at least aN^{1/n} and those with smaller radius. Large radii give many disjoint faces; small radii are confined to lattice cells with fewer than N centers, so induction closes with the same","pith_inferences":["The proof avoids any dyadic pigeonhole step; the same balanced local-global induction may transfer to other combinatorial incidence problems where large and small scales compete, yielding log-free bounds.","The lattice assumption is essential: for centers in R^n, local cell occupancy fails; a natural testable extension is to replace Z^n by a separated set in R^n and check whether the same exponent survives under a separated-net hypothesis.","Since the entropy argument is label-aware, it could be adapted to random-center or measure-theoretic versions of the skeleton problem where labels are preserved."],"forward_implications":["Corollary 1.3: for n=2,k=1, any lattice set containing a square boundary about each of N centers has at least cN^{7/8} points, and this exponent is attained.","For every fixed n,k, the exponent β=1−(n−k)/(2n^2) is the true growth rate, so no further improvement in the power of N is possible.","The lower bound holds for all N uniformly with constants depending only on n and k, not on the configuration.","The method eliminates the logarithmic factor that appeared in earlier near-endpoint lower bounds for the planar boundary case.","The construction in Section 5 provides an efficient carrier of size O(N^β), so the theorem is two-sided."],"fun_headline_variants":["Cube skeleton size exponent resolved","Square boundaries force N^(7/8) points","Exact exponent for cube skeleton cardinality","Cube skeleton size: exponent now known"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument requires the centers to live on the integer lattice Z^n, so that every cell of side h contains at most h^n centers and the small-radius branch can invoke induction on fewer centers; without this lattice occupancy bound, the endpoint lower bound is not established.","fun_headline_variants_meta":{"raw":{"variants":["Cube skeleton size exponent resolved","Square boundaries force N^(7/8) points","Exact exponent for cube skeleton cardinality","Cube skeleton size: exponent now known"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1344,"prompt_tokens":713,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":587}},"tokens_in":457,"tokens_out":631,"duration_ms":6910,"temperature":1.0,"reasoning_tokens":587,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:09:22.097945+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For n=2,k=1, try to build an infinite sequence of finite sets B_N⊂Z^2 with |B_N| ≤ C N^{7/8−ε} for some ε>0 that still contain a square boundary about each of N distinct lattice centers. If such a sequence existed, Corollary 1.3 would be false; the paper's construction, by contrast, matches the O(N^{7/8}) upper bound.","supporting_citations":[],"review_version":1}