2610.22388v1

Minor revision. The manuscript establishes Theorem 1.1: for every integer n ≥ 63, with m = ⌊(n−1)/2⌋, one has h(n) = L_n + 1 = 2^m + 1, and a group G with ω(G) ≤ n attains the bound if and only if G′ ≅ C_2 and G/Z(G) ≅ F_2^{2m} (in which case ω(G) = 2m + 1). The central reason for the decision is that the logical chain is sound under the stated hypotheses: finite reduction (Lemma 2.1), prime-wise isotropic-index bounds for alternating maps (Proposition 2.5 and Corollary 2.6), two-branch centralizer descent excluding non-class-two groups and nonabelian odd Sylow factors for even n ≥ 64 (Propositions 3.1, 3.6 and Lemmas 3.8–3.9), and separation of every reduced nonscalar binary map from the scalar symplectic model (Proposition 4.1 via Lemmas 4.2–4.10). The extraspecial lower bound of Lemma 2.2 matches, the odd-n case follows by monotonicity and L_{n+1} = L_n, and the rigidity statement extends to infinite groups because the finite reduction preserves G′ and G/Z(G). No mathematical gap was located in the checked steps; the only confirmed defect is bibliographic presentation of two DOI strings.

The paper is a clear advance on the classical Erdős–Neumann–Pyber circle and on the author’s own asymptotic log_2 h(n) = n/2 + O(√n (log(n+2))^3). Strengths include the clean finite reduction that carries structural invariants to infinite groups, the explicit Heisenberg model and F_{2^m}-spread that realize the extraspecial values ω = 2m + 1, a = 2^m + 1 and i = 2^m, the two-branch contraction (factors 3 and 5 with explicit index savings) that produces a concrete threshold, and the common-image fibre covers together with dual-number symplectic peeling that separate nonscalar binary maps. Appendix A documents the effective comparisons at n = 64, and the open questions correctly record the failures at n = 4 and 6 (dihedral groups of order 2q) together with the parity constraint that forces the optimal N_* to be odd.

The principal weakness is presentational rather than mathematical. In several places (Lemma 2.4 isotropic-radical gain, Proposition 2.5, Lemma 3.9) the linear product p^2 s sits adjacent to powers p^{2s} and is easy to misread; the intended and proved bound is linear (n ≥ p^2 s + 1), obtained from p^s exterior cosets each contributing at least p lifts and the comparison p^{s+1} + 1 ≥ p^2 s + 1. The same visual ambiguity appears between the linear clique size p s + 1 of Lemma 2.3 and the exponential isotropic covering number p^s + 1 of a nondegenerate scalar form. Two bibliography DOI strings ([4] Brown and [6] Erdős chapter) contain embedded spaces or line breaks that prevent resolution; the titles, venues and page ranges match known works, so the defect is typesetting, not invention. Optional reproducibility notes for the finite U_j table and the eight inequalities of Lemma A.1 would further document arithmetic already performed.

Section-by-section diagnostics confirm the same picture. Section 2 correctly encodes class-two p-groups by reduced alternating maps, proves the basic index and covering inequalities (2), supplies the extraspecial model, and obtains the prime-wise bounds of Proposition 2.5 by induction that combines coset cliques (including the exceptional binary t = 1 triple argument) with nondegenerate scalarizations. Section 3 develops elementary coset, rigidity and amplification lemmas, produces the two contraction branches of Proposition 3.6, converts them into the uniform exponential index bound of Lemma 3.7, and verifies a(G) < L_n outside the binary class-two model for even n ≥ 64. Section 4 rules out nonisotropic radicals, high-exponent targets and high-rank interactions, isolates the common-image case, and settles it by fibre covering and peeling; equality holds only for the reduced scalar form of dimension 2m. Section 5 assembles Theorem 1.1 without circularity. Appendix A closes the threshold arithmetic. The dependency graph is acyclic: no later equality statement is presupposed by the descent or the map estimates.

The explicit revision contract is limited. Required: in bibliography entries [4] and [6] (page 21), emit the DOI strings without embedded spaces or line breaks (intended forms 10.1016/0097-3165(88)90057-X for Brown and 10.1017/CBO9780511662034.004 for the Erdős chapter) and verify that a DOI resolver accepts the strings and returns the cited works. Optional but strongly recommended: typeset every occurrence of the linear product as p^2 s (thin space) or (p^2)s and, in the isotropic-radical paragraph of Lemma 2.4, record the one-line comparison p^{s+1} ≥ p^2 s; after Lemma 2.3 insert one clarifying sentence distinguishing the linear clique bound p s + 1 from the exponential scalar covering number p^s + 1; add a short reproducibility sentence in Lemma 3.7 and/or Appendix A that the U_j maxima and the eight inequalities of Lemma A.1 were obtained by exact rational evaluation at integer n ≥ 64; and, in the terminal step of Lemma 4.9, note that the isotropic-radical gain supplies 4^c + 1 before the linear weakening to 4c + 1. No change to any theorem statement, hypothesis or inequality is required.

What would alter the decision is the discovery of a genuine gap in the finite reduction, the two-branch contraction, the odd-Sylow separation, the common-image fibre covers, the symplectic peeling, or the final assembly, or the appearance of a counter-example to the linear isotropic-radical bound or to any of the Appendix A ratios under the stated premises. Cosmetic refusal to clean the two DOI strings would also block acceptance. Conversely, once the required bibliographic fix is made, the optional notation and reproducibility notes are not conditions of acceptance.

Relative to the field’s publication standard for exact extremal results in group theory and combinatorial group theory, the manuscript meets the bar for a sharp eventual formula with rigidity. The classical literature supplies the exponential sandwich h(n) = 2^Θ(n); the author’s prior asymptotic left an error term; the present work removes the error for all n ≥ 63 and classifies equality cases. The proof technique (centralizer descent plus scalarization and peeling) is new relative to the cited asymptotic work, the effective constants are tracked explicitly, and the open questions are well-posed finite targets. The result is therefore suitable for a research journal in the area after the minor revisions above.

Disclosures. This decision incorporates two independent referee reports (A and B) that examined the supplied page images; both recommended minor revision and found no mathematical error in the checked lemmas. Referee C was unavailable. A machine-assisted reference-integrity audit flagged line-broken DOI strings for [4] and [6] (fabrication_risk solely from spacing) and left [7] and [9] unresolved by automated title/author match; standing checks on 16 DOIs reported no retractions or corrections. No formal-proof artifacts or machine-checked receipts are present. The source PDF SHA256 is 3a8f786249b69e8c86949d74b35a3bd2a61e9a72268943a22122c49f009b4178. All mathematical arguments were assessed from the manuscript proofs and the named citations as used; exploratory AI assistance disclosed by the author was treated only as ordinary language/editing disclosure and did not affect the correctness judgment.

The manuscript proves Theorem 1.1: for every integer n >= 63, with m = floor((n-1)/2), the extremal function h(n) equals L_n + 1 = 2^m + 1, and equality holds if and only if G' isomorphic to C_2 and G/Z(G) isomorphic to F_2^{2m}. The lower bound is supplied by extraspecial 2-groups of order 2^{1+2m} (Lemma 2.2). The upper bound is obtained by a three-stage reduction: finite reduction of any group of finite omega (Lemma 2.1), exclusion of groups outside the class-two binary model (Proposition 3.1), and separation of nonscalar binary alternating maps from the scalar symplectic model (Proposition 4.1). The argument is self-contained once the cited classical facts (Neumann, Schur, rank-one matrix spaces, Reed-Muller distance) are granted, and the effective threshold n = 64 is documented by explicit arithmetic in Appendix A and Lemmas 3.8, 3.9 and 4.10.

Lemma 2.1 correctly reduces any group of finite omega to a finite group Q that preserves a, omega, the central quotient and the derived subgroup. Neumann supplies finiteness of G/Z(G); the finitely generated subgroup F generated by coset representatives satisfies F' = G' and Z(F) = F cap Z(G); Schur makes F' finite; killing the torsion summand of Z(F) by a suitable power N yields a torsion-free central subgroup intersecting F' trivially, so commutators (and therefore cliques and abelian covers) descend and lift bijectively. The same reduction therefore carries the rigidity statement to infinite groups without additional work.

Section 2 develops the alternating-map formalism. For a class-two p-group the commutator map beta : V x V -> W is reduced and alternating; a(P) = a(beta) and i(P) = p^{tau(beta)}. The classical extraspecial calculation (Lemma 2.2) gives the exact values omega = 2m + 1, a = 2^m + 1 and i = 2^m by Gram-matrix rank over F_2 and by a symplectic spread of F_{2^m}-lines. Scalarization (Lemma 2.4) extracts ordinary symplectic forms; the isotropic-radical gain produces the linear lower bound n >= p^2 s + 1 (typeset without a thin space, but proved via p^s exterior cosets each contributing p lifts). Proposition 2.5 then yields the prime-wise isotropic-index bounds omega(beta) >= 2 tau(beta) + 1 (p = 2), (10/3) tau(beta) (p = 3) and p tau(beta) (p >= 5) by induction on |V|, combining coset cliques for degenerate scalarizations with the nondegenerate case of Lemma 2.4. Corollary 2.6 converts these into the class-two index bound i(G) <= 2^{n/2} (and the sharper 2^{7n/24} when at least two nonabelian Sylows are present).

Section 3 excludes everything outside the binary class-two model for even n >= 64. The two-branch centralizer contraction (Proposition 3.6) produces either a threefold clique reduction with index saving 2 or a fivefold reduction with index saving 4; the uniform exponential bound i(G) <= 8 * 2^{omega(G)/2} (Lemma 3.7) is obtained by a finite check up to 31 followed by induction that the normalized contraction multipliers stay below 1 for n >= 32. Lemma 3.8 then shows a(G) < L_n whenever G' is not central, by feeding the contractions into a(G) <= n i(G) and verifying the ratios at n = 64. Lemma 3.9 removes nonabelian odd Sylow factors by the same scalarization machinery, again with explicit comparisons collected in Appendix A. The remaining groups are therefore of the form P x A with P a class-two 2-group.

Section 4 separates nonscalar binary maps. Nonisotropic scalar radicals and targets of exponent >= 4 are ruled out by Lemmas 4.2 and 4.4 (the latter using a six-point clique in each Z/2^e-plane). When the target is elementary abelian the interaction maps T_u either have rank >= 2 (Lemma 4.5, Reed-Muller) or force a common image line (Lemma 4.6). The common-image case is handled by fibre covering (Lemma 4.8) together with symplectic peeling that extracts dual-number planes while preserving the interaction kernel dimension (Lemma 4.9); the resulting clique lower bound omega(beta) >= 4s + 1 + max{s - c, 0} makes the elementary coset estimate a(beta) < L_n for n >= 64 (Lemma 4.10). Equality holds only for the reduced scalar form of dimension 2m.

The final assembly (Section 5) is immediate: for even n >= 64 the preceding propositions give a(G) <= L_n + 1 with equality precisely in the scalar binary case; the odd case follows by monotonicity of h and the identity L_{n+1} = L_n. The structural characterization follows because a normal subgroup of order two is central. No mathematical gap was found in the checked steps. The only substantive presentational issues are the ambiguous juxtaposition of products p^2 s with powers p^{2s} (easily fixed by thin spaces or parentheses) and two DOI strings that appear line-broken in the bibliography. The threshold 63 is sufficient but not claimed to be optimal; the open questions correctly record the known failures at n = 4 and 6 and the parity constraint that forces N_* odd. The result is a sharp eventual exactness theorem with a clean rigidity statement, and the proof technique (centralizer descent plus scalarization/peeling) is new relative to the asymptotic work cited as [11].

If the proof is correct, the paper replaces the earlier asymptotic log2 h(n) = n/2 + O(sqrt(n) (log(n+2))^3) by an exact eventual formula together with a complete rigidity classification of the extremal groups (including infinite ones). That closes, for all sufficiently large n, the classical Erdős question on the gap between clique and chromatic numbers of noncommuting graphs of groups. The advance is substantial within the area of covering and noncommuting-graph problems, though the threshold 63 is not claimed to be optimal and the small-n values remain open.

• Theorem 1.1 gives the exact eventual value h(n)=2^{floor((n-1)/2)}+1 for n>=63 together with a complete rigidity classification of equality cases, including infinite groups via finite reduction.
• Lemma 2.1 preserves a, omega, G' and G/Z(G), so the structural statement carries to infinite groups without extra work.
• Lemma 2.2 supplies matching extraspecial lower bounds by an explicit Heisenberg form and an F_{2^m}-spread.
• Two-branch centralizer contraction (Proposition 3.6) plus effective arithmetic at n=64 cleanly exclude non-class-two groups and odd Sylow factors.
• Common-image fibre covers and dual-number symplectic peeling separate every nonscalar binary map from the scalar model.
• Appendix A documents the concrete threshold inequalities; open questions correctly record the failures at n=4,6 and the parity constraint on N_*.