corpus number section status heading_line note wow2 66 1 counted 366 wow2 340 2 counted 480 wow2 176 3 counted 614 wow2 85 4 counted 707 wow2 349 5 counted 800 wow2 352 6 counted 947 wow1 133 7 counted 1096 wow1 123 7a counted 1213 wow1 223 7b counted 1256 wow1 312 7c counted 1356 wow1 151 7d counted 1449 wow1 125 7e counted 1646 wow1 134 7e counted 1646 wow1 162 7f counted 1765 wow1 316 7g counted 1945 wow1 605 7h counted 2006 wow1 604 7i counted 2068 wow1 239 7j counted 2186 wow1 402 7k counted 2371 wow1 597 7l counted 2524 wow1 696 7m counted 2670 wow1 602 7n counted 2844 wow1 279 7o counted 3025 wow1 324 7p counted 3261 wow1 561 7q counted 3387 wow1 315 7r counted 3574 wow1 641 7s retracted 3765 retracted by §7df wow1 639 7t counted 3853 wow1 657 7u counted 4020 wow1 656 7v counted 4141 wow1 276 7w counted 4226 wow1 277 7w counted 4226 wow1 278 7w counted 4226 wow1 182 7x counted 4379 wow1 183 7x counted 4379 wow1 184 7x counted 4379 wow1 155 7y counted 4562 wow1 156 7y counted 4562 wow1 204 7y counted 4562 wow1 152 7z counted 4779 wow1 154 7aa counted 4917 wow1 136 7ab counted 5125 wow1 143 7ac counted 5226 wow1 707 7ad counted 5457 wow1 804 7ae counted 5602 wow1 805 7ae counted 5602 wow1 809 7ae counted 5602 wow1 812 7af counted 5737 wow1 873 7ag counted 5829 wow1 878 7ag counted 5829 wow1 869 7ah counted 5934 wow1 886 7ai counted 6039 wow1 889 7aj counted 6099 wow1 876 7ak counted 6142 wow1 768 7al counted 6293 wow1 836 7am counted 6479 wow1 870 7an counted 6620 wow1 893 7ao counted 6773 wow1 891 7ap counted 6852 wow1 842 7aq counted 6928 wow1 855 7ar counted 6976 wow1 850 7as counted 7020 wow1 863 7at counted 7094 wow1 861 7au counted 7165 wow1 862 7av counted 7288 wow1 849 7aw counted 7424 wow1 848 7ax counted 7508 wow2 431a 7ay counted 7662 wow2 425d 7az counted 7813 wow2 402 7ba counted 7918 wow2 396 7bb counted 8057 wow2 395b 7bc counted 8087 wow2 422a 7bd counted 8111 wow2 422c 7be counted 8208 wow2 401a 7bf counted 8288 wow2 399c 7bg counted 8387 wow2 401b 7bh counted 8496 wow2 328 7bi counted 8599 lit 2018 7bj counted 8687 wow2 172 7bk counted 8882 lit 2-Steiner 7bl counted 9019 wow1 284 7bm counted 9164 wow2 364 7bn counted 9312 wow2 434c 7bo counted 9441 wow2 427 7bp counted 9577 wow2 399a 7bq counted 9678 wow2 448b 7br counted 9777 wow1 642 7bs counted 9891 wow1 651 7bt counted 10060 wow1 188 7bu counted 10211 wow1 189 7bv counted 10293 wow1 187 7bw counted 10326 wow1 202 7bx counted 10409 wow1 318 7by counted 10464 wow1 186 7bz counted 10585 wow1 206 7ca counted 10661 wow1 211 7cb counted 10742 wow1 191 7cc counted 10783 wow1 209 7cd counted 10939 wow1 48 7ce counted 11080 wow1 51 7cf counted 11117 wow1 52 7cg counted 11159 wow1 282 7ch counted 11208 wow1 185 7ci counted 11261 wow1 115 7cj counted 11327 wow1 100 7ck counted 11422 wow1 97 7cl counted 11496 wow1 102 7cm counted 11577 wow1 99 7cn counted 11686 wow1 91 7co counted 11787 wow1 94 7co counted 11787 wow1 178 7cp counted 11985 wow2 281 8 counted 12127 wow2 300 8 counted 12127 wow2 287 8a counted 12260 wow2 258 9 retracted 12375 wow2 259 9 retracted 12375 wow1 603 9b other 12530 wow1 693 9c other 12562 wow1 662 9d other 12583 wow1 305 7cq counted 12756 wow1 306 7cq counted 12756 wow1 307 7cq counted 12756 lit Wiener-index 7cu other 13727 lit Mohammadian 7cv other 13834 lit Akbari, Elphick, Kumar, Pragada and Tang 7cw counted 13919 lit Akbari, Elphick, Kumar, Pragada and Tang 7cx other 14131 lit Ma–Yang–Li 7cy other 14298 wow1 698 7cz other 14504 wow1 707 7da sharpening 14570 first counted in §7ad wow1 561 7db sharpening 14745 first counted in §7q wow1 607 7db sharpening 14745 verbatim restatement of 561 (12 Feb 1989); 561 counted at §7q wow1 639 7dc sharpening 14841 first counted in §7t wow1 638 7dd other 14903 wow1 702 7de other 14978 wow1 641 7df retracted 15026 withdraws §7s wow1 600 7dg other 15086 wow1 602 7dh sharpening 15127 first counted in §7n wow1 579 7di counted 15202 wow1 656 7dj sharpening 15247 first counted in §7v wow1 125 7dk sharpening 15300 first counted in §7e wow1 151 7dk sharpening 15300 first counted in §7d wow1 197 7dk counted 15300 wow1 223 7dl sharpening 15443 first counted in §7b wow1 284 7dl sharpening 15443 first counted in §7bm wow1 316 7dl sharpening 15443 first counted in §7g wow1 654 7dl counted 15443 witnesses corrected by 7hd 2026-08-27: rook graphs are outside the 634-654 hypothesis class; Hoffman-Singleton (margin +14) and C5[t], t>=8, are inside it; still counted wow1 347 7dm counted 15568 wow1 352 7dm counted 15568 wow1 360 7dm counted 15568 wow1 134 7dn sharpening 15667 first counted in §7e wow1 282 7dn sharpening 15667 first counted in §7ch wow1 604 7dn sharpening 15667 first counted in §7i wow1 696 7do sharpening 15806 revisited; first counted in §7m wow1 695 7dp counted 15977 wow1 694 7dq counted 16087 wow1 722 7dr counted 16243 wow1 725 7ds counted 16477 wow1 504 7dt counted 16661 wow1 528 7du counted 16832 wow1 494 7dv other 16907 wow1 495 7dv other 16907 wow1 496 7dv other 16907 wow1 497 7dv other 16907 wow1 498 7dv other 16907 wow1 499 7dv other 16907 wow1 500 7dv other 16907 wow1 501 7dv other 16907 wow1 502 7dv other 16907 wow1 503 7dv other 16907 wow1 504 7dv other 16907 wow1 505 7dv other 16907 wow1 506 7dv other 16907 wow1 507 7dv other 16907 wow1 508 7dv other 16907 wow1 509 7dv other 16907 wow1 510 7dv other 16907 wow1 511 7dv other 16907 wow1 512 7dv other 16907 wow1 513 7dv other 16907 wow1 514 7dv other 16907 wow1 515 7dv other 16907 wow1 516 7dv other 16907 wow1 517 7dv other 16907 wow1 518 7dv other 16907 wow1 519 7dv other 16907 wow1 520 7dv other 16907 wow1 521 7dv other 16907 wow1 522 7dv other 16907 wow1 523 7dv other 16907 wow1 524 7dv other 16907 wow1 525 7dv other 16907 wow1 526 7dv other 16907 wow1 527 7dv other 16907 wow1 528 7dv other 16907 wow1 529 7dv other 16907 wow1 530 7dv other 16907 wow1 531 7dv other 16907 wow1 532 7dv other 16907 wow1 533 7dv other 16907 wow1 534 7dv other 16907 wow1 535 7dv other 16907 wow1 536 7dv other 16907 wow1 537 7dw other 16958 wow1 538 7dw other 16958 wow1 708 7dw other 16958 wow1 709 7dw other 16958 wow1 710 7dw other 16958 wow1 574 7dx counted 17099 wow1 304 7dz counted 17357 wow1 84 7ea counted 17457 wow1 85 7ea counted 17457 wow1 652 7eb counted 17542 wow1 105 7ec retracted 17649 withdrawn true-disposition; superseded by §7et wow1 646 7ed counted 17679 wow1 165 7ee counted 17763 wow1 719 7ef counted 17962 wow1 700 7el counted 19024 wow1 95 7em counted 19315 wow1 92 7en counted 19531 wow1 73 7eo other 19686 theorems in heading wow1 74 7eo other 19686 theorems in heading wow1 75 7eo counted 19686 wow1 103 7ep counted 19844 wow1 104 7ep counted 19844 wow1 568 7eq counted 19920 wow1 49 7er counted 20031 wow1 402 7es sharpening 20086 first counted in §7k wow1 105 7et counted 20191 wow1 197 7eu sharpening 20333 strengthening of §7dk lit 2607.16382 7ew counted 20622 lit A.1 7ex counted 20707 lit 2.10(B1) 7ey counted 20795 wow2 352 7ez sharpening 20945 first counted in §6 wow2 358 7ez counted 20945 wow2 359 7ez counted 20945 wow2 176 7fb sharpening 21224 first counted in §3 wow1 176 7fc sharpening 21392 heading says sharpened wow2 172 7fd sharpening 21593 wow1 176 7fe sharpening 21767 another witness after §3/§7fb wow2 326 7ff other 21941 wow2 327 7fg counted 22038 wow2 328 7fh sharpening 22139 explicit sharpening of §7bi wow2 320 7fi other 22240 wow2 323 7fj other 22282 wow2 325 7fk other 22492 wow2 319 7fm counted 23001 wow2 324 7fn other 23190 wow2 309 7fo counted 23298 wow2 64 7fp counted 23586 wow2 63 7fq counted 23865 wow2 442 7fr counted 24078 wow2 34 7fs counted 24252 wow1 183 7fu other 24751 wow1 184 7fu other 24751 wow1 185 7fu other 24751 wow2 378 7fw counted 25068 wow2 109 7fx counted 25350 wow2 247 7fy counted 25525 wow2 267 7fz counted 25653 wow2 412 7ga retracted 25829 June 2010 |H| block: false as printed but readings are corrupt; §7ga explicitly declines to count them wow2 413 7ga retracted 25829 June 2010 |H| block: false as printed but readings are corrupt; §7ga explicitly declines to count them wow2 414 7ga retracted 25829 June 2010 |H| block: false as printed but readings are corrupt; §7ga explicitly declines to count them wow2 415 7ga retracted 25829 June 2010 |H| block: false as printed but readings are corrupt; §7ga explicitly declines to count them wow2 416 7ga retracted 25829 June 2010 |H| block: false as printed but readings are corrupt; §7ga explicitly declines to count them wow2 287 7gb sharpening 25986 explicit sharpening of §8a wow2 308 7gc counted 26096 wow2 300 7gd sharpening 26249 explicit sharpening of §8 wow2 281 7ge sharpening 26344 explicit sharpening of §8 wow1 142 7gg counted 26571 wow1 302 7cr counted 12893 wow1 289 7cr counted 12893 wow1 285 7cs counted 13132 wow1 239 7cs sharpening 13132 first counted in §7j wow1 597 7cs sharpening 13132 first counted in §7l wow1 604 7ct sharpening 13569 first counted in §7i wow1 605 7ct sharpening 13569 first counted in §7h wow1 603 7ct other 13569 proved TRUE in this section wow1 289 7eg sharpening 18113 first counted in §7cr wow1 307 7eh sharpening 18317 first counted in §7cq wow1 234 7ei counted 18487 wow1 308 7ej counted 18673 WOW-I 308; distinct from WOW-II 308 (§7gc) by corpus separation wow1 305 7ek sharpening 18857 first counted in §7cq wow1 697 7ev counted 20469 wow2 340 7fa sharpening 21070 first counted in §2 wow2 315 7fl other 22630 proved TRUE in this section wow2 316 7fl other 22630 proved TRUE in this section wow2 317 7fl other 22630 proved TRUE in this section wow2 318 7fl other 22630 proved TRUE in this section wow2 321 7fl other 22630 proved TRUE in this section wow2 322 7fl other 22630 proved TRUE in this section wow2 449 7ft other 24450 proved TRUE in this section wow2 446 7ft other 24450 proved TRUE in this section wow2 182 7fv other 24938 exact violation criterion, not a disproof wow2 439 7gi counted 26882 Jan 2012 alpha_2 block; subdivision+matching family G_k, n=10k+1, margin k-1; flagship n=21 wow2 172 7gj sharpening 27035 first tree counterexamples; both readings of dist_min(M2); counted at 7bk / 3 wow2 176 7gj sharpening 27035 first tree counterexamples; both readings of dist_min(M2); counted at 7bk / 3 wow1 770 7gk counted 27170 cubic, alpha >= (1+m)/2, m=max even-distance count; n=18 partial truncation, alpha=6 m=12; min order exactly 18 wow1 759 7gm other 27476 proved TRUE, not counted wow1 760 7gm other 27476 proved TRUE, not counted wow1 758 7gn other 27649 correct reading of the expanding coefficients; 759/760 reopened wow1 759 7gn other 27649 correct reading of the expanding coefficients; 759/760 reopened wow1 760 7gn other 27649 correct reading of the expanding coefficients; 759/760 reopened wow1 761 7go other 27824 761 screened under the corrected 758 reading: no violation, never tight wow1 752 7gp other 27884 proved TRUE (residue form); peel+FMS; tight iff K_n wow1 810 7gq other 27916 vacuous under the only admissible reading (min degree 0: isolated primes) wow1 811 7gq other 27916 asymptotically true; jet = O(pi(sqrt n)) vs lambda2 = Theta(n) wow1 777 7gq other 27916 no violation to order 18; annealer clean to n=28; probably true wow1 839 7gr counted 27966 Petersen + K1; unique minimum order 11; Kneser K(3k-1,k) unbounded wow1 868 7gs counted 28083 Moebius-Kantor n=16 unique minimum cubic; W(q) generalized quadrangle margin q-1 unbounded wow1 894 7gt other 28220 TRUE: b(G)<=8 except Petersen (10); b=9 impossible; 894 is a theorem wow1 892 7gu counted 28254 Franklin (12) and Heawood (14): minimum spanning set = 3 independent edges, blue clique 2; k copies of Heawood give margin k; weakest reading survives wow1 851 7gv counted 28316 fullerene radius vs maximum sphere at a boundary vertex; min counterexample C54 isomer 164, |Aut|=1 wow1 858 7gw counted 28381 number of centres of a fullerene vs n-3+h; min counterexample C50 isomer 55, self-centred wow1 796 7gx counted 28446 "upper quotient of the degree sequence vs the Turan (Caro-Wei) bound; author called it ""easy to prove""; minimum counterexamples P4 and C4 at order 4; repaired theorem margin<1, sharp via cocktail-party graphs" wow1 806 7gy counted 28557 largest eigenvalue vs number of distinct degrees of PR[square-free 2..n]; least counterexample n=51, last survivor n=785, deficit diverges so no additive repair survives wow1 807 7gz counted 28688 second largest eigenvalue vs half the largest on PR[square-free 2..n]; fails exactly for n=345..353 (two distinct graphs), exact integer certificates (Bareiss PSD + integer Rayleigh-Ritz); bounded sporadic failure, holds for every other n<=3000 wow1 283 7ha counted 28781 independence number vs number of nonpositive eigenvalues of the distance matrix, girth >= 5; incidence graph of PG(2,q) is exactly tight for every q, deleting an arc of k points makes it fail by exactly k; headline witness n=23 margin -2, smallest known n=21 margin -3, unbounded wow1 38 7hb counted 28978 variance of the distance matrix vs -lambda_min; conjectures 37-40 omit the qualifier 'distance' that 30-36 all carry, and the tightness signature confirms the adjacency reading; P_7 fails under both variance conventions, deficiency diverges like n^2/18; reading-independent: whichever matrix is meant, 38 or 39 must be false wow1 650 7hc retracted 29153 RETRACTED 2026-08-27 by 7hd: witness K_{1,2,2,2,2} violates the range-scoped hypothesis chi(complement)=n-matching declared for conjectures 634-654 at source line 2960; 650 has no counterexample among the 29592 qualifying graphs on <=9 vertices wow1 650 7hd retracted 29342 records the retraction of 7hc and the repair of 7dl; documents the three prose range-scoped hypotheses at source lines 2829, 2921, 2959 wow1 348 7he counted 29577 "min adjacency-eigenvalue gap vs the mean entry of the gravity matrix, for plants; false under every normalisation: K_2 breaks it under all three (min gap 2 > 1), P_3 and P_4 under both entry means (P_4: exact min gap 1 > 31/54); ""mean Gravity"" is pinned to sum/n by the tightness of conjectures 226, 271 and 300, so ""mean entry"" is a distinct quantity; every path P_4..P_50 is a counterexample, with an exact crossover; 347 and 348 are the only two conjectures of the plant block absent from the Brewster-Dinneen-Faber verified list, and both are false" wow1 32 7hf counted 29829 -lambda_min of the distance matrix vs the matching number; false, and refuted by the book itself: conjecture 35 (annotated by Shearer as a theorem from interlacing) says diam <= -lambda_min(D), so every connected graph with diam > matching number is a counterexample; P_3 exact (charpoly (x+2)(x^2-2x-2), -lambda_min = 2 > 1 = mu); 5736 of the 11117 connected graphs on 8 vertices fail; margin diverges like 2n^2/pi^2 - n/2 on paths; reading pinned by the recorded refutations of 31 and 33 and by the theoremhood of 35; erratum-class, flagged as elementary in the section; the conjecture does hold on subdivided stars S(k), k>=6 wow1 636 7hg other 29989 "size/independence <= max eigenvalue of Laplacian, inside the class chi(complement)=n-matching. NOT COUNTED: 636 is character-for-character conjecture 243, which the book records as ""Disproved by James B. Shearer, see his solution of 215. October 88.""; Shearer's counterexamples are triangle-free of arbitrarily large girth, and every triangle-free graph is in the class (proved here), so 636 was dead four months before it was printed. New content: the k>2r Hoffman-ratio-bound criterion; first explicit finite certificates (Hoffman-Singleton margin exactly 5/3, and C_21(1,3,8) on 21 vertices, margin +0.451083) against Shearer's random asymptotic construction; Gewirtz, M22 and Higman-Sims built from the binary Golay code; exact ties at Clebsch srg(16,5,0,2) and C_20(2,5,6); minimum counterexample order in [12,21]; and a proof that the neighbouring conjecture 635 is TRUE for every triangle-free graph, via m/alpha <= n/2" wow1 122 7hh counted 30154 The average distance <= n / mean of coordinates of Maxine. FALSE: bipartite barbell B(2,22,31), n=78, m=519, avgdist=12053/1001 > 1521/128 >= n^2/S for every admissible Maxine performance. wow1 263 7hi counted 30283 range of coordinates of Maxine <= 1 + range of positive eigenvalues. FALSE: minimal witness is the 6-vertex graph ECZo (coordinates take 4 distinct values under both admissible performances; charpoly x(x+1)(x^4-x^3-6x^2+4x+4) has exactly 2 distinct positive roots), and the blow-up family G(k) forces Maxine onto a blob set with margin 2^(k-1)-k-1 = Theta(n). wow1 320 7hj counted 30424 If G is triangle-free then n - residue <= matching(complement) + matching number. FALSE: K_{15,7} (n=22, triangle-free) has residue 4, so n-residue=18, while mu=7 and mu(complement of K_{15,7} = K_15 + K_7)=10, total 17. For K_{a,b} the margin equals ceil(a/2)-floor(b/2)-residue; along K_{2b,b} the residue is the constant 4 and the margin is b/2-4, unbounded; margin/n reaches 0.3655 at K_{2000,200}. Exhaustive: 0 violations among all connected triangle-free graphs on at most 12 vertices, worst margin exactly -1 up to n=11 and 0 at n=12 (K_{7,5}), so minimum order is in [13,22]. wow1 731 7hk counted 30575 Minimum angle of a polygon without multiple vertices <= minimum degree of the complement of its colinearity graph. FALSE: W5 = (1,0),(0,1),(-1,0),(0,-1),(0,0), the unit diamond square with its centre notched into the boundary as a fifth vertex. Simple, integer coordinates, interior angles 45/90/90/45/270 degrees, so min angle = pi/4; the centre lies on both diagonals, hence is colinearity-adjacent to all four square vertices and ISOLATED in the complement, so the min complement degree is 0. Margin +pi/4, and reading-independent in the angle unit since RHS = 0. Exhaustive grid census: 0 violations at n=3 and at n=4 without flat vertices (hand proofs for both), so n=5 is the minimum order. Family P(k) = regular 2k-gon plus centre keeps RHS = 0 with min angle exactly (pi-pi/k)/2, so the margin rises to pi/2, essentially optimal since a simple polygon has min interior angle < pi. Graffiti missed it because its polygon database was in general position, where the colinearity graph is empty and RHS = n-1 makes 731 vacuous. The author remark that 731 would generalize Gallai-Sylvester pins the reading: it demands that EVERY point lie on an ordinary line, which the centre of a square does not. wow1 734 7hl counted 30745 The sum of reciprocals of nonzero degrees of the colinearity graph of a polygon <= chromatic number of its visibility graph. FALSE: W9 = (-30,0),(167,-10),(135,99),(69,77),(102,66),(65,29),(32,114),(204,117),(169,-23) is a simple integer 9-gon with exactly three collinear triples, two of them crossing at (102,66); colinearity degrees 0,2,2,2,4,2,2,2,2 give LHS = 7/2 + 1/4 = 15/4, while its visibility graph is 3-chromatic, the smallest value any simple polygon can have. Margin +3/4, and reading-independent since the strict visibility graph is a subgraph of the closed one. Scale mismatch: every nonzero colinearity degree is at least 2, so LHS can reach n/2, while a constant-width corridor keeps chi(visibility) = 4 forever. The baffled serpentine family (serpentine corridor of width 20 with inward baffles of depth 15 in every long wall) has n = 12r - 16 and margin about n/4, i.e. Theta(n), which is optimal up to a constant since LHS <= n/2 and chi >= 3. Exhaustive abstract census over linear hypergraphs: max possible LHS is 3/2,3/2,9/4,3,19/6,15/4,9/2 for n = 3..9, so with chi >= 3 no polygon on at most 6 vertices can violate 734; annealing found nothing at n = 7 or 8, so the minimum order is in [7,9]. Graffiti missed it because its polygon database was in general position (empty colinearity graph, LHS = 0) or convex (chi = n). wow1 733 7hm counted 30935 The number of distinct degrees of the interval graph of a polygon <= the number of vertices of its convex hull. FALSE: W6 = (0,0),(0,2),(1,1),(1,3),(2,1),(3,0), six points of the 4x4 grid in general position, traversed as the simple hexagon (0,0)-(3,0)-(2,1)-(1,1)-(1,3)-(0,2). Its convex hull is a quadrilateral (RHS 4) while its interval graph, on all C(6,2)=15 segments, has degree sequence 8^7 9^3 10^3 11 12 and hence 5 distinct degrees. Margin +1. Mechanism: deg(s) = 2(n-2) + X(s) where X(s) counts the disjoint segments crossing s, so the LHS is the number of distinct crossing numbers; in convex position the chord splitting a and b of the other points has crossing number exactly a*b, giving only floor((n-2)/2)+1 distinct degrees against n hull vertices, so no convex polygon can ever refute 733 - which is why Graffiti's database missed it. Both sides are order-type invariants, so minimality is a finite census: all 1, 2 and 3 order types at n = 3, 4, 5 satisfy 733, and exactly 3 of the 16 order types at n = 6 violate it with best margin +1, so the minimum order is exactly 6 and W6 is optimal at its size. Family A (a flat parabola q_k = (10^4 k, k^2), k = 1..m, inside a huge triangle) has hull 3 and crossing numbers exactly a(m-a), giving margin floor(m/2)-2 -> infinity; family B (p_k = (97k mod 1009, k^3 mod 1013) in the same triangle) reaches margin +254 at n = 37, i.e. Theta(n^2), which is optimal since the interval graph has only C(n,2) vertices. wow1 742 7hn counted 31084 The distance from the center (centroid) of a triangle to the smallest vertex <= the maximum distance from the Erdos-Mordell point to the vertices. FALSE: W = (0,0),(169,0),(119,120), an isosceles triangle with integer sides 169,169,130. Its smallest vertex is the apex A, the centroid is G=(96,40) and |G-A| = 104 exactly, while the Erdos-Mordell point lies on the axis of symmetry at t* = (5044-65*sqrt(949))/462 and its maximum vertex distance is (4225+1300*sqrt(949))/462 = 95.8281..., margin (43823-1300*sqrt(949))/462 = +8.1719, positive because 43823^2 = 1920455329 > 1603810000 = 1300^2*949. An infinite exact family of Pythagorean isosceles triangles also refutes it, and the same witness refutes the incenter reading. wow1 754 7ho counted 31239 Let s be the number of silent vertices in the chip-firing game (e-1 chips placed at a centre of G); then the chromatic number of G is not more than 2s. FALSE: W is the connected graph on n=10 vertices with e=18 edges 01,02,03,04,12,13,14,23,24,25,34,38,39,48,56,58,67,78. The set {0,1,2,3,4} induces a K5 and W is 5-colourable, so chi = 5. Eccentricities are 3,3,3,3,3,3,4,3,2,4, so the radius is 2 and vertex 8 is the UNIQUE centre, which makes the author's initial position (17 chips on vertex 8) completely unambiguous and the kill reading-independent. The game terminates with firing vector 0,0,1,1,1,3,3,4,6,1, so the silent set is exactly {0,1} and s = 2, giving chi = 5 > 4 = 2s, margin +1. Both of the author's own necessary conditions are met: the two silent vertices are adjacent (his lemma that every silent vertex has a silent neighbour) and chi = 5 is exactly the minimum he proved a counterexample must have. Mechanism: chips enter at a small-degree unique centre and must percolate around a long low-degree tail 5-6-7-8 before reaching the dense core through only three edges, so vertices 0 and 1 of the K5 never accumulate their degree in chips. Exhaustive nauty-geng census of all connected graphs, best margin chi-2s over some centre / over every centre: n=2 -2/-2, 3 -1/-1, 4 -1/-1, 5 -1/-1, 6 -1/-1, 7 0/-1, 8 +1/0, 9 +1/0, so the minimum order is 8 in the loosest reading and 10 in the strict unique-centre reading, i.e. W is of minimum possible order. Infinite family W_j = W plus j pendant vertices hung on the UNIQUE CENTRE 8, verified j=0..300: the centre, radius 2, diameter 4, silent set {0,1}, s=2 and chi=5 are all preserved, so 754 fails for every order n >= 10 and no order n <= 9. Pendants anywhere else destroy the unique centre and s jumps to 9-15. Larger margins, both with unique centre 2 and s=2: n=14 with chi=6 (margin +2) and n=18 with chi=7 (margin +3); since s >= 2 is forced, margin k-4 at clique number k is the theoretical maximum, which all three witnesses attain. Graffiti missed it because its database was dominated by regular and vertex-transitive graphs, where the centre is rarely unique and chips spread evenly so almost nothing stays silent, and because a counterexample needs chi >= 5, which is scarce among small sparse graphs. wow1 831 7hp counted 31452 residue of blue graph vs 1+maxdeg R+avg deg; W(k,t)=K_k with t pendants at one clique vertex; margin t(k-3)/(k+t), Theta(n); min order 5 wow1 833 7hq counted 31549 residue of B(G) <= residue of complement of G; G(k)=two adjacent hubs + k common neighbours + a pendant on each hub; margin k-1 = n-5, Theta(n); B has no isolated vertex; min order 6 non-degenerate (3 overall) wow1 825 7hr other 31642 "READING TRAP: literal reading of ""complement of the blue graph"" is false but dies on K2; the partial-complement reading (= the red graph) is a THEOREM (chi <= n - t). NOT a disproof, not counted." wow1 843 7hs counted 31818 Independence number of a fullerene >= n/2-8 is FALSE: C240 = GP(2,2) = leapfrog(C80) partitions into twelve 5-cycles and twelve 15-cycles, so alpha <= 108 = n/2-12 < 112. Margin -4; family GP(k,k) gives margin 8-6k. PROVENANCE: the extremal family is prior work (Doslic-Vukicevic; Faria-Klein-Stehlik, arXiv:1203.3912, proved alpha >= n/2-sqrt(3n/5) sharp at n=60k^2); neither mentions 843. Mine: the connection to 843, the elementary 12k-odd-cycle certificate, and the construction. wow1 107 7ht counted 32079 wow1 249 7hu other 32260 NOT A DISPROOF - the book already states 249 is false in its strongest interpretation, and the ledger total is unchanged. NOR IS THIS THE FIRST ANSWER to the author's attached open sub-question ('we do not know an example of a graph in which every coloration would be a counter-example', DeLaVina and S.F. 1.91): the appendix at the end of section 7p (README line 3322) already answered it with FQjnW, n=7, whose complement is bipartite so the EXACT chromatic number of the complement is 2, while all 192 colorations give a rainbow with 3 distinct values. WHAT SECTION 7hu ADDS is a stricter right-hand side faithful to the book's own convention that 'the chromatic number [is] computed by greedy algorithms' (source lines 2116-2121): take the LARGEST greedy chromatic number of the complement over all of its n! orders. Under that standard FQjnW FAILS - an adversarial order makes greedy spend a third colour on the bipartite complement, so max greedy chi(complement)=3 and 3>3 is false. This is a PARTIAL CORRECTION of that appendix. Witnesses that survive both standards: range = max-min, smallest is `ETnw`, n=6 = K_5 with one pendant vertex; all 720 colorations give rainbow (4,1,4,4,4,4), range 3, complement K_{1,4} plus an isolated vertex, greedy chromatic number 2 under every order; unique at n=6 (so the appendix's claimed minimum order of 7 is also corrected under this reading), 3 at n=7, 21 at n=8; family K_m + pendant gives range m-2 vs 2, unbounded from m=5. range = number of distinct values (the appendix's reading), smallest is `F]zlw`, n=7 = complement of (P_3 union K_{1,3}); all 5040 colorations give rainbow (4,3,4,2,4,4,4), 3 distinct values vs 2; none at n<=6, 4 at n=8. Exhaustive over all connected graphs n<=8. PROVENANCE: on 1 Sep 2026 I also re-derived the all-coloration witnesses for 561, 607 and 639 believing them new; the audit showed sections 7q, 7db, 7dc and 7t had already established exactly those results (same witness F?AFw for 561, same diamond and complete split graph family for 639), so no rows were added for them and the total is unaffected. wow2 302 7hv retracted 32390 wow2 2 7hw other 32461 L_s >= 2(lambda_avg-1); tight, no violation n<=8 wow2 136 7hw other 32461 Delta(R) = max degree in G on radial circles; tight wow2 137 7hw other 32461 proved TRUE here (cograph argument) wow2 138 7hw other 32461 distances taken in G^2; tight wow2 389a 7hw other 32461 q = floor(n/4)-th smallest degree; tight from n=6 wow2 399b 7hw other 32461 WP read from G degrees; open corner needs alpha(G[A3])>=4 wow2 400c 7hw other 32461 tight from n=6, no violation wow2 436c 7hw other 32461 WP read from G degrees; tight, no violation wow2 448a 7hw other 32461 undefined rho(G) resolved as path covering number p(G); tight wow1 62 7hx other 32625 regular; avg dist <= n/d; clean to n=9; book calls it still unproved (1997) wow1 121 7hx other 32625 triangle-free; 2m <= sum E(v); tight; clean to n=9 wow1 157 7hx other 32625 radius <= min(min D, min E); tight; clean to n=9 wow1 164 7hx other 32625 triangle-free; smallest-mode reading; clean to n=9; largest-mode reading is a bad reading wow1 203 7hx other 32625 scope sum D <= sum E; avg dist <= sum 1/deg; clean to n=9 wow1 214 7hx other 32625 triangle-free; m/alpha <= n-alpha; tight at K_{a,a}; clean to n=9 wow1 216 7hx other 32625 triangle-free; tight; clean to n=9 wow1 221 7hx other 32625 girth >= 5; radius <= sum 1/deg; tight; clean to n=9 wow1 222 7hx other 32625 girth >= 5; clean to n=9 wow1 225 7hx other 32625 girth >= 5; tight; clean to n=9 wow1 231 7hx other 32625 regular; chi <= n/avgdist; tight; clean to n=9 wow1 236 7hx other 32625 regular; tight; clean to n=9 wow1 251 7hx other 32625 clean to n=7 wow1 258 7hx other 32625 #positive eigenvalues <= mu(G)+mu(Gbar); clean to n=7 wow1 262 7hx other 32625 -lambda_min <= max E(v); tight; clean to n=7 wow1 275 7hx other 32625 BAD READING (Rule Z): as written violated by C5, margin diverges; direction is likely <=; NOT a kill wow1 294 7hx other 32625 girth >= 5; clean to n=7 wow1 297 7hx other 32625 trees; clean to n=7 wow1 827 7hx other 32625 chi <= mean dual degree + #positive eigenvalues; tight at every K_n; clean to n=7 wow2 434c 7hx other 32625 SW +/-1 audit: SW = degeneracy in the Dec 2010 block, forced by Dalmatian sharpness; counted kill in 7bo re-verified and STANDS wow2 435 7hx other 32625 SW = 1 + degeneracy = colouring number in this Jan 2012 row; true and sharp to n=9; DECLINED, not a kill; upgrades 7ft.8 wow1 844 7hy other 32986 printed reading false for every fullerene (alpha<=n/2<#{lam>=-1}); misprint. Correct reading alpha>=#{lam<=-1} knife-edge, clean n<=66 wow1 846 7hy other 32986 clean, all fullerenes n<=60 wow1 852 7hy other 32986 clean, all fullerenes n<=80; margin 0 at n=30,32,34,36,40 then drifts up wow1 853 7hy other 32986 clean, all fullerenes n<=60; zigzag nanotube margin diverges wow1 856 7hy other 32986 clean, all fullerenes n<=60 wow1 859 7hy other 32986 clean, all fullerenes n<=80; margin 0 at n=34,60,72,76; nanotube margin diverges wow1 861 7hy other 32986 ERRATUM: C60 gives 46.5808 < 46.6; printed constant 1.6 is 1.5808 rounded up. DECLINED wow1 845 7hy other 32986 clean n<=40; min margin 0.292 at C20; bipartite number = largest INDUCED bipartite subgraph, exact ILP wow1 860 7hy other 32986 clean, all IP isomers n<=84; min margin 0.419 at C60, reproducing the sources own remark wow1 135 7hz counted 33169 FALSE: M^2(C5), n=23 m=71 triangle-free diam 2, omega=2 chi=5, alpha(D2)=2, 2.5>2. D2 = distance-EXACTLY-2 graph. Unbounded via M^k(C5). Grotzsch tight at 0. No violation n<=9. wow2 374 7ia counted 33370 FALSE: spider S(2,2,1^k) (centre + two legs of length 2 + k pendant leaves), gamma_t=3 constant, B={leg ends}, median=1. Margin (2-k)/(k+3) under def-109 nonzero-only, (6-k)/(k+5) under divide-by-n; BOTH tend to -1. Min counterexample order 8 (N) / 12 (A); extremal tree at every order 8..18. Def 109 pinned by row 356 exactly tight at every order. wow1 226 7ib retracted 33592 RETRACTED same day. Hoffman-Singleton violates only under mean Gravity = Sigma/n (mean of row sums), margin -1/14. That reading is refuted by rows 217 (K_{2,7}, n=9) and 404-prime (K_{4,4}, n=8), both Los Alamos survivors, which fail at n<=10 under it. Under Sigma/n^2 or Sigma/n(n-1) conjecture 226 is comfortably true. Moore-graph margin (3-d)/(d(d+1)) is a genuine coincidence at d=3. wow2 375 7ic counted 33798 FALSE: spiders S(a,a,a-1) (centre + two legs of length a + one of length a-1, n=3a). ecc(B)=2a-1 exactly, gamma_t ~ 1.5a, lower median 2 => margin ~ -a/2 -> -infinity; verified a=3..30. Minimum-order witness P_5 (gamma_t=3 vs ecc(B)+lmed=4). READING-ROBUST: still false for a>=4 even if 'lower median' is replaced by 1, and under all four median conventions. Since lower median >= 1, 375 on trees implies gamma_t >= ecc(B), which the same document already records as REFUTED (rows 236, 237). Calibrated: proved rows 231, 357 (0 violations, trees n=4..16) and 251 survive the identical readings. wow1 346 7id counted 33991 hypercube Q_d, odd d>=7, is a plant with avgdist > 3 = #distinct eigenvalues of D