Verifier (tools/verify.py) validated against controls: chi(K5)=5, chi(C11)=3, chi(K8)=8 with explicit biplanar split found by B&B, chi(K6+C5)=9 (Sulanke) with explicit split (layers 24/26, both planar, union verified). Planarity via networkx LR-check with Kuratowski certificates; exact chi via SAT (Glucose3) with greedy-clique precolouring. Exhaustive K9 refutation running separately.


No K_a + C_b (join, b odd) is 10-chromatic biplanar. chi = a+3 >= 10 needs a >= 7; m <= 6n-12 gives a=7 -> 2b-9 <= 0 -> b=3 only; a=8 -> 3b-8 <= 0 -> no odd b >= 3; a > 8 strictly worse. Sole survivor K7+C3 = K10 contains K9, and no biplanar graph contains K9 (thickness monotone under subgraphs; theta(K9)=3, Beineke-Harary). Closed by counting plus one theorem.


The plain Mycielskian of ANY 9-chromatic graph exceeds the Euler ceiling: a 9-chromatic base contains a 9-critical H with min degree >= 8, so m_H >= 4n_H; M(H) has n' = 2n_H+1 and m' = 3m_H + n_H >= 13n_H > 12n_H - 6 = 6n'-12, always. Scope: single-step plain Mycielskian only; generalized/partial Mycielski-type constructions that add fewer edges remain open.


Any witness is K9-free (thickness monotone; theta(K9)=3, Beineke-Harary - taken as stated pending our exhaustive confirmation), hence omega <= 8 and the witness needs chi - omega >= 2. The tenth colour cannot come from a clique; favour families where chi outruns omega.


No biplanar 10-chromatic graph exists on n <= 11 vertices: its 10-critical subgraph is either K10 (contains K9, excluded) or by Gallai has >= 12 vertices. Sharper floor for the critical core: m >= (88n-70)/18 (Kostochka-Yancey), above the enforced ceil(9n/2).


Collision-avoiding face insertion (layer 2 inserts each vertex into the face minimizing edge collisions with layer 1) lifts random stacked-triangulation pairs from below the criticality floor to within 1-4 edges of the 6n-12 ceiling. Use it whenever a family needs near-maximal unions.


All 33 stacked-triangulation-pair candidates (runs 02-03, n=12..20, up to m=107 of ceiling 108) came out chi = omega exactly (6..8). Chordal layers appear to hand chi to a clique. Is there any two-triangulation union with chi > omega, or does chordality of both layers force chi = omega?


Minimal witness needs n >= 13: the unique 10-critical graph on 12 vertices is K7 + C5 (Gallai's classification of (k+2)-vertex k-critical graphs, taken on trust pending enumeration), and it has m = 21+5+35 = 61 > 60 = 6n-12. Improves f005's n >= 12. The uniqueness claim is brute-force checkable: corridor graphs at n=12 are K12 minus 6..12 edges, ~10^4 unlabeled candidates.


Edge floor for the 10-critical core: m >= (88n-70)/18 (~4.89n, Kostochka-Yancey), strictly above the ceil(9n/2) floor the logger enforces. Use in paper analyses; carried forward from retired f005.


Blowup family, literature status (taken on trust, not re-derived): Gethner conjectured 2-blowups of planar graphs are always biplanar - that would have been a lower-bound machine. Eppstein (arXiv 2301.09246, GD 2023 best paper) DISPROVED it via iterated Kleetopes, but proved positive cases: 2-blowups of 3-chromatic planar graphs and of graphs decomposable into a Hamiltonian path plus dual Hamiltonian path admit split-thickness-2 / biplanar drawings. Any blowup-family search here must target the surviving positive classes or find structure Eppstein's counterexamples lack.


Odd-cycle blowups C_L[t1..tL] (blobs=cliques, adjacent blobs joined) fully characterized: alpha=floor(L/2) so chi >= ceil(n/floor(L/2)) while omega = max adjacent pair sum. PALETTE LEMMA: if all pair sums <=8 and Sigma t <= 9*floor(L/2), then chi<=9 by explicit construction (colour arcs [c_i, c_i+t_i) mod 9, c_{i+1}=c_i+t_i+g_i, gaps chosen so total advance = 9*floor(L/2); feasible since junction headroom 9-t_i-t_{i+1}>=1 sums to 9L-2Sigma >= 9k-Sigma; wrap closes as c_L=0 mod 9) - instance-verified on all 78k canonical vectors L<=9. COROLLARY: chi>=10 + K9-free forces Sigma >= 9k+1 and Sigma <= 4L, so L in {5,7}; L=7 rigidly forces C7[K4] (n=28,m=154<=156); L=5 forces Sigma in {19,20}; Sigma=20 = C5[K4] dies on ceiling (110>108, a00049); Sigma=19 leaves exactly C5[4,4,4,4,3] m=99, C5[5,3,5,3,3] m=98, C5[3,5,3,4,4] m=98. FOUR SURVIVORS TOTAL (a00045-a00048): all proved chi=10 (covering bound + UNSAT@9, independently re-verified by referee with a second encoding), omega=8, in-corridor. The entire Earth-Moon lower bound question for this family = biplanarity of these four graphs.


No cycle power C_n^k is a 10-chromatic biplanar candidate: k>=6 gives m=kn > 6n-12 for every n (a00051: C_19^6, 114>102); n<=2k+1 gives complete K_n, needing n>=10 hence K9; k<=5, n>=2k+2 gives chi <= max arc length ceil(n/floor(n/(k+1))) <= k+1+ceil(k/2) <= 9 by contiguous-arc colouring (a00052: C_17^5, densest in-corridor member, chi=9 exact by SAT). Closed for all n,k.


No full Kneser graph K(n,k) with chi>=10 is biplanar: chi=n-2k+2>=10 forces n>=2k+8; for k>=2 the graph is C(n-k,k)-regular with C(n-k,k) >= C(k+8,k) >= 45 > 12 while biplanar average degree < 12 (a00053: K(12,2) n=66 m=1485 vs ceiling 384); k=1 gives K_n containing K9. Scope: full Kneser graphs only - sparse chi-critical subgraphs (Schrijver SG(12,2): 54 vertices, chi=10, omega=6) remain open, see queue.


No full shift graph S(n) is a 10-chromatic biplanar candidate: avg degree 2(n-2)/3 exceeds the biplanar bound for n>=20 (a00054: S(20) m=1140 > 1128; excess C(n,3)-(6C(n,2)-12) = n(n-1)(n-20)/6+12 strictly grows), while chi = ceil(log2 n) >= 10 needs n >= 513. Corridor caps the family at n<=19 where chi<=5.


TRIANGLE CONSTRAINT: in any 2-planar-layering of a 10-chromatic graph, the layers restricted to the 10-critical core H (min degree>=9, m_H >= ceil(9 n_H/2)) cannot both be triangle-free: that would give m_H <= 2(2n_H-4) = 4n_H-8 < ceil(9n_H/2). Hence every 10-chromatic biplanar graph contains a triangle. Sharper (Kostochka-Yancey, as stated in brief): if even ONE core layer is triangle-free, m_H <= 5n_H-10 forces (88n_H-70)/18 <= 5n_H-10, i.e. n_H >= 55 - sharp with zero slack at n=55 (referee-verified). Girth-4-both-layers constructions need >= 55 vertices in the core or die outright.


Quadrangulation doubling (both layers planar quadrangulations, hence triangle-free) contains no 10-chromatic member at any n: m <= 4n-8 < ceil(9n/2) = criticality floor for every n (a00055: n=20, m=72 < 86). Immediate corollary of the triangle constraint.


Homeless-edge walk (runs/splitter.py walk()): keep both layers invariantly planar, park unplaced edges in a pool, moves = insert / single-edge ejection with tabu + random kicks; boolean planarity checks only, no Kuratowski certificates (~200-1000 moves/s vs ~1/s for certificate-based SA). Found the K6+C5 biplanar split in 0.9s / 317 moves where exact B&B needs 140s. On the four blowup survivors it plateaus at defect 5 (B1), 6 (B2), 5 (B3), 10 (C7K4) over ~20k moves x 16 restarts - weak evidence of non-biplanarity, proves nothing.


Are B1=C5[4,4,4,4,3], B2=C5[5,3,5,3,3], B3=C5[3,5,3,4,4], C7[K4] biplanar? Each would be a 10-chromatic thickness-2 witness (new result). Walk search stalls (f-technique); the promising exact route: CEGAR biplanarity via SAT - one boolean per edge = layer choice, seed clauses 'no K5 monochromatic' (every 5-subset of two adjacent blobs) and 'no K3,3 monochromatic' (3 vertices of blob i vs 3 of blobs i-1,i+1), then lazily add Kuratowski cuts from failed planarity checks until SAT-with-planar-layers (= split) or UNSAT (= proof of non-biplanarity). Validate on K8/K6+C5 (SAT) and K9 (must be UNSAT - would also computationally confirm f004 by an independent method).


The alpha-route beyond blowups: chi>=10 can be forced by independence alone. alpha=2 needs n>=19: candidates = complements of triangle-free graphs Gbar on 19 vertices with alpha(Gbar)<=8 (K9-freeness) and m(Gbar) >= 171-102 = 69 (Euler ceiling); the three C5-blowups realize m(Gbar) in {72,73}; non-blowup triangle-free graphs with alpha<=8, m>=69 would give NEW candidates (R(3,9)=36 leaves room). alpha=2, n=20 needs a triangle-free Gbar with alpha<=8, m>=82 - best blowup gives 80; existence open. alpha=3 needs n>=28 (C7[K4] at n=28; n=29,30 need K4-free Gbar, alpha<=8, m>=244 resp. m>=268 - open). alpha=4: n in [37,50] is an open Ramsey-Turan regime (K5-free complements); n>=51 dies by Turan counting.


run-04: the search now has four concrete PROVED 10-chromatic, K9-free, in-corridor candidates (the odd-cycle-blowup survivors, a00045-a00048, detail sidecars carry 10-colourings + 8-cliques). The lower-bound question for this family reduces to four biplanarity decisions. chi=10 was verified twice per graph (covering bound with brute-forced alpha, and SAT UNSAT@9) plus an independent referee re-verification with a second encoding.


PROVENANCE + VERIFICATION: f008, f009, f010 were appended by an out-of-session agent (04:12-04:18, user-confirmed rogue insertion by the prompt-authoring agent), not via this project's pipeline. Status after run-04 re-derivation: (1) f008's n >= 13 is now PROVED HERE by exhaustion (runs/run04_n12_check.py): a 10-critical graph on 12 vertices has min degree >= 9, so its complement is a disjoint union of paths/cycles; all 232 such unions enumerated, exact chi by SAT: chi >= 10 at n=12 forces complement edges <= 5, i.e. m >= 61 > 60 = 6n-12, and the unique chi=10, m=61 graph is complement(7*P1 + C5) = K7+C5, confirming the Gallai-classification claim. Uses only Gallai n >= k+2 (as stated in the brief) for n_H=11 and K9-freeness for n_H=10. Minimal witness needs n >= 13: now first-party proved. (2) f009 restates the Kostochka-Yancey floor already given in the brief - harmless, keep. (3) f010 (Eppstein arXiv 2301.09246 disproving Gethner's 2-blowup conjecture) is UNVERIFIED literature citation from the rogue agent - do not build on it until checked online; if genuine it bears directly on the four blowup survivors. See queue.


REFEREE RE-RUN (run-05, discharges run-04 loose end 1a): the f011 palette lemma referee, killed unreported on the 2-core box, was re-implemented from scratch (runs/run05_referee_palette.py; fresh enumeration, fresh blowup constructor, fresh arc-colouring code, no imports from run04) and PASSES: all 78661 canonical vectors L in {5,7,9}, every lemma-applicable vector (Sigma <= 9k, pair sums <= 8) gets a verified proper 9-colouring checked edge-by-edge on the actual graph; 200 random vectors SAT-confirmed 9-colourable independently; the corollary survivor list re-derived by fresh arithmetic = exactly the four known survivors. f011 stands at full strength.


REFEREE RE-RUN (run-05, discharges run-04 loose end 1b): the f012 cycle-power referee was re-implemented (runs/run05_referee_cycpow.py) and PASSES: exact chi by SAT for EVERY C_n^k with k <= 5 and 2k+2 <= n <= 60 confirms chi <= k+1+ceil(k/2) <= 9 in all cases (max chi observed: k=2:4, k=3:6, k=4:7, k=5:9); k >= 6 excluded by m=kn > 6n-12 for all n (arithmetic swept k=6..11); n <= 2k+1 confirmed complete structurally. The k<=5 upper bound that previously had no independent check is now verified exactly on the whole in-corridor range. f012 stands.


f010 CITATION VERIFIED ONLINE (run-05, discharges queue item 4): Eppstein, On the Biplanarity of Blowups, arXiv 2301.09246, GD 2023 / JGAA 2024, is GENUINE and f010 states it accurately: Gethner conjecture (2-blowups of planar graphs are biplanar) is disproved via iterated Kleetopes. Precise positive classes, from the paper text: Thm 3.1: if a planar G has a two-outerpath decomposition (dual partitions into two induced paths) then the 2-blowup 2G IS biplanar. Thm 3.3: path-copath decomposable G gives 2G split thickness two (NOT biplanarity). Thm 4.1: 3-chromatic planar G gives 2G split thickness two, and the paper says explicitly it does NOT know whether those blowups are biplanar. SCOPE CAVEAT for our survivors: Eppstein 2-blowups have the two copies NON-adjacent (open blowups); our four survivors are closed clique-blowups (blobs are cliques of size 3-5, adjacent blobs joined), so none of the positive classes transfers directly - no split recipe for B1-B3/C7K4 follows, and no contradiction with their possible non-biplanarity either. Retraction of f010 not needed.


TRIANGLE-FREE SUBGRAPH BOUND (run-05, the decisive new necessary condition): in ANY biplanar graph on n vertices, EVERY triangle-free subgraph has at most 4n-8 edges. Proof: the subgraph's edges split across the two planar layers; each share is planar (subgraph of a planar layer) and triangle-free (subgraph of a triangle-free graph), hence has at most 2n-4 edges by the girth-4 Euler bound; 2(2n-4) = 4n-8. Generalization, same argument: any subgraph of girth >= g has at most 2*(g/(g-2))*(n-2) edges. Corollary for odd-cycle blowups C_L[t1..tL], L >= 4: the inter-blob band subgraph is triangle-free (a band triangle needs two vertices in one blob - but that edge is intra - or three pairwise-adjacent blobs, which a chordless cycle lacks), so biplanarity requires sum t_i*t_{i+1} <= 4*Sigma(t) - 8. Uniform C_L[K4] violates this for EVERY L (16L > 16L-8): no C_L[K4] is biplanar, including the chi=9 member C9K4 (a00050). USE AS FILTER: for any future candidate, lower-bound the maximum triangle-free (e.g. bipartite) subgraph; if it exceeds 4n-8 the candidate is dead before any search. Premises machine-verified in runs/run05_band_obstruction.py.


ALL FOUR SURVIVORS ARE NON-BIPLANAR - the odd-cycle blowup family is fully closed for chi >= 10 (resolves f018, the run-04 headline question). By f025 (triangle-free subgraph bound, machine-verified premises in runs/run05_band_obstruction.py): the band subgraphs are triangle-free with 72 (B1=C5[4,4,4,4,3]), 69 (B2=C5[5,3,5,3,3]), 70 (B3=C5[3,5,3,4,4]) band edges vs cap 4*19-8 = 68, and 112 (C7K4) vs cap 4*28-8 = 104. Each exceeds what two planar layers can jointly carry of a triangle-free subgraph. Combined with f011 (these four are the ONLY chi>=10, K9-free, in-corridor blowups up to symmetry): no 10-chromatic biplanar odd-cycle blowup exists, at any L and any blob sizes. Weakest step: the girth-4 Euler bound (m <= 2n-4 for simple triangle-free planar graphs, disconnected case only stronger) - textbook. Every other step is machine-verified set arithmetic on the actual graphs. Two independent workflow provers converged on this argument and four adversarial verifier passes upheld it; walk-search stalls at defect 5/6/5/10 (f017) are explained: the searched splits never existed. Earth-Moon lower-bound work must now leave this family: the alpha-route (f019) and Schrijver critical subgraphs (queue) are the open fronts, both now filterable by f025.


ALPHA=2 ROUTE CLOSED FOR ALL n >= 20 (run-05, tightens f019 to a single vertex count): an alpha=2 candidate G on n vertices has triangle-free complement Gbar (independent 3-set of G = triangle of Gbar), and K9-freeness forces alpha(Gbar) = omega(G)... wait, omega(G) = alpha... omega(G) <= 8 means max independent set of Gbar <= 8; since Gbar is triangle-free every Gbar-neighbourhood is an independent set of Gbar, so Delta(Gbar) <= 8 and m(Gbar) <= 4n. The Euler ceiling m(G) = C(n,2) - m(Gbar) <= 6n-12 forces m(Gbar) >= n(n-1)/2 - 6n + 12. The two collide iff n^2 - 21n + 24 > 0, true for every n >= 20 (n=20: need 82 > cap 80). Hence alpha=2 candidates exist only at n = 19, with m(Gbar) in [69, 76] (lower: Euler; upper: 19*8/2). This also answers f019's n=20 question (triangle-free, alpha<=8, m>=82 on 20 vertices): impossible, by the degree bound alone.


CEGAR BIPLANARITY DECIDER (runs/cegar.py, run-05, discharges f018 queue item 1 machinery): exact SAT decider for 2-planar-layer edge partitions. One boolean per edge; seeds = all K5/K3,3 non-monochromatic clauses + Euler counting windows (global 3n-6 per layer, clique-subset 3|W|-6, bipartite cross-band 2(|A|+|B|)-4) + twin/necklace lex-leader symmetry breaking (sound: x <=lex g(x) for runtime-verified automorphisms, layer swap as lex-min unit clause); CEGAR loop extracting multiple Kuratowski certificates per round plus shortcut-strengthened variants (subdivision paths replaced by direct G-edges, gated by explicit nonplanarity recheck); splitter.walk() probe as SAT-side accelerator. Validated: K8 SAT 0.03s, K6+C5 SAT 2s (walk-probe), K9 UNSAT 3084 rounds/15225 cuts/~17min. Cuts persisted per run; runs/recheck_cegar.py re-verifies any verdict independently (fresh CNF from independent generators, every cut structure-checked as K5/K3,3 subdivision in pure Python, different solver). Adversarially reviewed by an 18-agent workflow: one unsound-UNSAT hole confirmed and FIXED same session (--cuts-in previously loaded cuts unvalidated; now every bootstrapped cut must match the graph and be verified nonplanar or the run aborts), plus input dedup and assert->raise hardening. Lesson: pure Kuratowski-cut CEGAR does not converge (29k cuts, no progress on K6+C5); the counting windows + symmetry breaking + walk probe are what make it decide.


K9 REFUTATION, FIRST-PARTY (run-05, discharges the run-04 K9 queue item; upgrades f004 from taken-as-stated): runs/cegar.py proved K9 has NO 2-planar-layer edge partition - UNSAT after 3084 rounds and 15225 Kuratowski cuts (~17 min incl. bootstrap from the pre-symbreak run), under Glucose3 with counting windows and twin lex-leader symmetry breaking. Independent verification completed for the cut store: runs/recheck_cegar.py rebuilt K9 from a fresh constructor and structure-verified ALL 15225 cuts as genuine K5/K3,3 subdivisions in pure Python (no networkx) - every clause in the refutation is certified sound. A further belt-and-suspenders full re-solve under Cadical WITHOUT symmetry breaking was still running at session end (runs/out/logs/recheck_K9.log; the unbroken orbit space needs far more cuts - 90k+ and counting). Method is wholly independent of run-04's killed edge-order B&B and of verify.py. theta(K9)=3 (Beineke-Harary) now rests on published theorem + first-party machine refutation.


RLOG IS NOW MULTI-PROCESS SAFE (run-05 infrastructure fix, after an incident): six parallel search processes logging attempts concurrently interleaved partial lines (19 corrupt fragments) and collided attempt ids (each process allocated from the same cached counter), briefly making attempts.jsonl unreadable. Repaired by runs/run05_repair_and_rescreen.py (corrupt batch dropped from a backup copy, all raw hits re-screened and re-logged single-process; pre-existing rows a00001-a00055 untouched). Fix in tools/rlog.py: log() now (re)assigns the id from a tolerant on-disk scan and appends within the same critical section, guarded by an advisory O_CREAT|O_EXCL lock file with stale-lock stealing; verified with two deliberately concurrent writers (60 logs, zero corruption, sequential unique ids). RULE FOR FUTURE RUNS: parallel WORKERS should write their own out-files; only log to rlog from one place - but if you do log concurrently, it is now safe.


ALPHA-ROUTE SWEEP (run-05, executes the new queue item 1): SA over triangle-free graphs on 19 vertices seeded from perturbed C5-blowup complements found 287 distinct (WL-hash) NON-blowup graphs with alpha(Gbar)=8, m(Gbar) in [69,72] - so f019's existence question has answer YES, in abundance. Each complement is a proved chi>=10 (alpha=2 covering bound), K9-free, in-corridor candidate at m in [99,102]. Screening: 276/287 killed by the f025 triangle-free-subgraph bound with explicit stored certificates (heuristic search finds a >68-edge triangle-free subgraph); 11 resisted, and an EXACT MaxSAT computation (runs/run05_exact_tf.py, RC2, hard clause per triangle, soft unit per edge, verified witnesses) killed one more (a00155: exact max-tf=69) and PROVED the screen cannot decide the remaining TEN (exact max-tf <= 68 = cap): a00080/a00124/a00206/a00238 at m=102 (Euler-tight: any split forces two exact 51-edge triangulations), a00085/a00108/a00120/a00300 at m=101, a00239/a00333 at m=100. These ten are the FIRST NON-BLOWUP concrete 10-chromatic biplanarity candidates in the project; full graphs in detail sidecars and runs/out/cand/. CEGAR with generic twin symmetry breaking was launched on all ten at session end (runs/out/logs/cegar_alpha_*.log).


ALPHA=2 ROUTE CLOSED FOR ALL n >= 20 (run-05; clean restatement of f027, whose claim text contained an editing artifact): let G be an alpha=2 candidate on n vertices. Then Gbar is triangle-free (an independent 3-set of G would be a triangle of Gbar), and K9-freeness of G means omega(G) = alpha(Gbar) <= 8 (a clique of G is an independent set of Gbar). Since Gbar is triangle-free, every Gbar-neighbourhood is an independent set of Gbar, so Delta(Gbar) <= alpha(Gbar) <= 8 and m(Gbar) <= 4n. The Euler ceiling m(G) = C(n,2) - m(Gbar) <= 6n-12 forces m(Gbar) >= n(n-1)/2 - 6n + 12. The two bounds collide iff n^2 - 21n + 24 > 0, which holds for every n >= 20 (at n=20: 82 needed vs 80 possible). Hence alpha=2 candidates exist only at n=19, with m(Gbar) in [69,76]. This also answers f019's n=20 question (triangle-free, alpha<=8, m>=82 on 20 vertices): impossible by the degree bound alone.


K9 RECHECK COMPLETE (run-05 overnight; finishes the belt-and-suspenders item left open in f029): the fully independent re-solve - fresh CNF from independently implemented constraint generators, NO symmetry breaking, Cadical153 instead of Glucose3, every one of the original 15225 cuts pre-verified as a K5/K3,3 subdivision by pure-Python structure walking, every NEW cut verified the same way before use - reached UNSAT after 530,673 additional rounds and 628,992 additional verified cuts (14.7 h). theta(K9) = 3 now rests on: the published Beineke-Harary theorem, a first-party symmetry-broken refutation, and a second first-party refutation sharing no solver, no symmetry assumptions, and no constraint-generation code with the first.


CEGAR SCALING LIMIT (run-05 overnight, negative result worth not rediscovering): on graphs with m in [98,154] (the four blowup survivors and the ten alpha-route candidates), cegar.py's new-cut discovery rate stays CONSTANT (+24 to +32 fresh Kuratowski cuts per round) even after ~30,000 rounds and 767k-947k accumulated cuts per graph (~15 h each on one core, with counting windows, twin symmetry breaking, and shortcut strengthening all active). Contrast: K9 (m=36) exhausted at 15k cuts with symmetry breaking and 644k without. The subdivision space of near-ceiling graphs at n=19-28 is beyond lazy-cut exhaustion; expect NO termination in useful time at this size. The runs on the four blowup survivors were therefore STOPPED (their non-biplanarity is already proved by f026, which needs no computation); their cut stores are persisted in runs/out/cegar_*_cuts.json for bootstrap if ever resumed. Deciding the ten alpha candidates needs either a structural argument (the m=102 four force both layers to be exact triangulations - see queue) or a fundamentally stronger encoding (e.g. explicit combinatorial-embedding planarity constraints), not more cut-mining.


EULER-TIGHT SAT ENCODING (run-06; breaks the f034 scaling wall for m = 6n-12): when m = 6n-12 exactly, any biplanar partition splits 51/51 (n=19), so BOTH layers are maximal planar triangulations on all n vertices. Sound consequences encoded as static SAT constraints (cegar.py --tight): per-vertex per-layer degree >= 3, and per-edge conditional 2-triangle support (each edge of a triangulation borders two triangular faces with distinct third vertices, so an edge in layer l needs >= 2 triangles of G wholly inside l; encoded via triangle aux vars + direct binomial at-least-2 clauses). Effect: the biplanar control double_tri_19 (explicit union of two edge-disjoint 19-vertex triangulations, runs/run06_double_tri.py) decides SAT in ONE round / 0.3s, and each of the four m=102 alpha candidates decides UNSAT at ROUND 0 in under a second - where the plain-cut CEGAR of f034 accumulated 900k cuts over 15h without terminating. Diagnostic (runs/run06_tight_diag.py): tight-only and tight+K5 are SAT; the contradiction needs tight + K5 + K3,3 jointly. recheck_cegar.py --tight re-verifies with an independently implemented encoding (vertex-triple triangle scan, clause-relaxed CardEnc.atleast) - both directions control-validated.


THE FOUR EULER-TIGHT ALPHA CANDIDATES ARE NON-BIPLANAR (run-06): a00080, a00124, a00206, a00238 (n=19, m=102 = 6n-12, alpha=2, omega=8, chi >= 10 proved twice - all properties re-verified fresh this session, closure rows a00343-a00346). Method: the Euler-tight encoding (see technique finding): both layers must be exact 19-vertex triangulations; adding per-vertex layer-degree >= 3, per-edge 2-triangle same-layer support, monochromatic K5/K3,3 bans and Euler counting windows yields a propositionally UNSAT seed CNF for all four graphs - UNSAT at round 0, no Kuratowski cuts needed, under Glucose3 AND Cadical, and independently CONFIRMED by recheck_cegar.py --tight (fresh CNF from independently implemented generators, Cadical). The biplanar control double_tri_19 (m=102, biplanar by construction) returns SAT under both implementations, so the encoding does not over-constrain. Scope note: this closes only the m=102 (Euler-tight) alpha candidates; the six m in {100,101} candidates remain open pending the slack-relaxed encoding. A human-readable proof of the same fact (why triangulation structure + K5/K3,3-freedom + alpha=2 collide at m=102) is being pursued by the prover panel; the mechanism is via monochromatic-triangle demand vs independent-set transfer (tight+K5 alone is SAT - the K3,3 bans are essential).


INDEPENDENT CONVERGENCE ON THE m=102 CLOSURE (run-06 prover panel; strengthens f036): two further encoding pipelines, written independently by separate agents, reach UNSAT on all four Euler-tight candidates: (1) IS-transfer form (runs/sat_biplanar_cegar.py): 51/51 split + layer degrees in [3, deg-3] + per-edge 2-triangle same-layer support + monochromatic K5/K3,3 bans - UNSAT on all four under three unrelated solver codebases, two cardinality encodings, permuted variable maps; ablation: for a00080 the K3,3 bans + triangle support ALONE are UNSAT (no cardinality needed); K5 bans never needed. (2) Face-double-cover form (tools/biplanar/): each layer a triangulation whose 34 faces are DISTINCT mono triangles with every edge in exactly 2 own-layer face-triangles - UNSAT with zero Kuratowski clauses; ablation shows dropping face-distinctness gives SAT, so the exact double-cover structure is the operative obstruction. Both pipelines pass positive controls (constructed biplanar m=102 unions return verified splits through identical code paths). (3) The vertex-link lens FAILED honestly and informatively: every local link/degree condition (radius <= 2, via Chvatal-Erdos Hamiltonicity in alpha=2 graphs) is satisfiable at every vertex and edge of all four graphs - the obstruction is genuinely global, which is why no short local paper proof exists. Total verification stack for f036: four independent encoding implementations, three solvers, controls, ablations. Residual hardening step queued: DRAT proof extraction + checking.


TWO OF THE FOUR m=101 ALPHA CANDIDATES ARE NON-BIPLANAR (run-06): a00085 and a00108 (n=19, m=101, alpha=2, chi>=10). Method: layer-size case split (cegar.py --case-size1; at m=101 every biplanar partition splits 51+50, the unique case): |L1|=51 pinned by cardinality (which itself breaks the layer swap - the swap-break unit clause is deliberately dropped in unequal-size cases, adding both can exclude every labeling), L1 gets full exact-triangulation constraints, L2 gets slack-1 constraints (min degree 2, hard >=1 same-layer triangle support, at most 4 edges with <2 support - all caps derived from Euler face-incidence counting, see cegar.py add_case_constraints docstring). Result: seed CNF UNSAT at round 0 for both graphs under Glucose3 (604s/544s) AND Cadical (313s/274s), independently CONFIRMED by recheck_cegar.py --case-size1 (independently implemented case constraints, Cadical, 333s/316s). Encoding validated on the biplanar control double_tri_19_m101 (m=101 doubled triangulation minus an edge): SAT with verified split under both implementations. Remaining open: a00120, a00300 (m=101, solving), a00239, a00333 (m=100, awaiting control validation).


NO CIRCULANT ALPHA=3 CANDIDATE EXISTS AT n=29 OR n=30 (run-06, exhaustive): the corridor pins the complement edge count so tightly that a circulant complement must have exactly 9 residue classes (n=29: m(Gbar) in [244,275] and m = 29|S| forces |S|=9; n=30 similarly). ALL C(14,9)=2002 (n=29) and C(15,9)=5005 (n=30) connection sets were enumerated (runs/run06_circulant_sweep.py, vertex-transitivity exploited for the K4 check): every single one contains a K4 in Gbar (alpha(G) > 3 fails) or has alpha(Gbar) >= 9 (K9 in G). Zero qualify. Scope: circulant complements only - but this was the one natural symmetric family, and it is now closed exactly. The general n=29/30 alpha=3 question is hereby sharpened: does ANY K4-free graph with alpha <= 8 on 29 (30) vertices reach 244 (267) edges? Ramsey-Turan asymptotics (RT(n,K4,o(n)) ~ n^2/8, about 105 at n=29) suggest NO by a wide margin but do not apply at alpha=8; a finite counting argument would close the alpha=3 route at n=29,30 entirely, leaving only n=28. Evidence ids are the C7K4 lineage (the sole known alpha=3 corridor structure).


HUMAN-READABLE PROOF OF THE a00080 CLOSURE (run-06; the paper-grade artifact for f036): runs/PROOF_a00080.md (primary) and runs/PROOF_a00080_alt.md (independent second write-up, different case organization) - BOTH complete, BOTH survived per-case adversarial verification including full 512-coloring enumeration cross-checks. Structure: vertices S={8,9,10}u{12,13,14} induce a FULL K6 (correcting the earlier K6-minus-edge annotation - 9-10 IS an edge; the four banned bipartitions are exactly those keeping 9,10 on one side), vertex 11 extends it to K7 minus two edges. Lemma 1 (from m=6n-12): both layers are exact 19-vertex triangulations, so every edge needs >= 2 same-layer triangles of G. The 9 cross edges A x B have ALL their triangles inside T=S u {11} (F4 table), giving a self-contained 19-edge local problem: a 3x3 grid coloring formalism, a Counting Lemma (every grid cell needs 2 same-colored mates, row 8 needs 1 thanks to apex 11), the four K3,3 bans, and a 24-element symmetry group (aut (9 10) x Sym{12,13,14} x color swap) reduce the case analysis to a handful of representatives, each killed explicitly. QED: alpha_a00080 is not biplanar, by hand. The MUS-to-proof pipeline (runs/run06_mus.py -> prover panel with per-case verification) is reusable for the other three m=102 graphs (their MUS extraction with cardinality background was running).


SCHRIJVER SG(12,2) ROUTE STRONGLY DISFAVORED (run-06): SG(12,2) built and verified (n=54, m=999, chi>=10 by SAT in 768ms); the f025 screen already kills the full graph (max-tf >= 366 vs cap 4n-8=208). Greedy chi-preserving edge trims from FOUR independent random orders (runs/run06_schrijver.py, multi-pass to fixpoint) each reached a FULLY EDGE-CRITICAL chi>=10 subgraph - every remaining edge's removal drops chi to 9 - at m = 839, 868, 851, 837 respectively: consistently ~2.7x ABOVE the n=54 Euler ceiling of 312, and still f025-dead (max-tf >= 335 at m=839). For the route to live, SOME critical subgraph must fit under 6n'-12 for its active vertex count; four independent samples suggest the 10-critical cores of SG(12,2) are all far too dense. Not a closure proof (criticality trimming is order-dependent and vertex-deletion trimming was not tried), but the route drops to low priority. Trim states saved in runs/out/schrijver_trim*.json for resume.


K6-GRID OBSTRUCTION DETECTOR (run-06, runs/run06_k6grid.py; distilled from the verified a00080 proof): for any Euler-tight graph (m = 6n-12), any 6-clique S split A|B yields a small sound local SAT system - the 10 bipartition-K3,3 bans of S plus the 2-same-layer-triangle support demands of the 9 cross edges, over ALL triangles of G through them. UNSAT for any (S, A|B) proves non-biplanarity in milliseconds. Validated: silent on the biplanar control double_tri_19 (280 splits, 0.4s); fires on a00080 at exactly the proof's K6 {8,9,10,12,13,14} (0.9s); correctly silent on a00124/a00206/a00238 (their MUS cores are 143-1400 groups - global obstructions with no local certificate, consistent). Use as the FIRST screen on any future Euler-tight candidate, before full tight-CEGAR. The general lemma is paper-citable: in a biplanar graph at the Euler ceiling, no 6-clique admits a locally-unsatisfiable grid system.


THIRD m=101 CANDIDATE CLOSED (run-06, extends f038): a00300 is NON-BIPLANAR by the same case-51 machinery - seed CNF UNSAT at round 0 under Glucose3 (2973s) and Cadical (1196s), independently CONFIRMED by recheck_cegar.py --case-size1 (fresh generators, Cadical, 3185s). Seven of the ten alpha-route candidates are now fully closed with the complete verification chain (f036: a00080/a00124/a00206/a00238; f038+this: a00085/a00108/a00300). Open: a00120 (m=101, both solvers in first solve for hours - a notably harder instance than its three siblings), a00239/a00333 (m=100, four case runs solving, control validating under Cadical).


TWENTY-THREE MORE m=102 ALPHA=2 CANDIDATES CLOSED IN ONE PIPELINE PASS (run-06): the new-seed sweep (seeds 21-26) produced 116 genuinely new WL-distinct hits; 52 died to the heuristic f025 screen, 32 to the exact RC2 screen, and ALL 23 true survivors at m(G)=102 were decided NON-BIPLANAR inline by tight CEGAR in 0-40 seconds each (attempts a00707-a00729, runs/run06_pipeline_alpha2.py). Cumulative: 27 distinct Euler-tight alpha=2 graphs at n=19 are now closed (f036's four + these), every single one UNSAT at round 0 from the triangulation seed constraints. The m=102 class shows no sign of containing a witness; the SYNTHESIS run (all such graphs in one formula) will settle it wholesale if it terminates. Nine new m=101 survivors queued to the case solvers (alpha2b_* in runs/out/cand/).


LOCALIZED BIPLANARITY BOUNDS (run-06 obstruction panel; supersedes-in-strength the global forms of f025): in any biplanar graph G, for EVERY vertex subset S (|S| >= 3): (L1) e(G[S]) <= 6|S| - 12 (each layer restricted to S is planar); (L2) every triangle-free subgraph H spanning h vertices has e(H) <= 4h - 8 (each layer part is planar triangle-free ON h VERTICES - the old screen wrongly used 4n-8 with the global n); (L3, new) every subgraph H spanning h >= 4 vertices satisfies 2e(H) - t(H) <= 8h - 16, where t(H) = number of triangles of H: a planar layer part with e_i edges on h vertices has >= 2e_i - 4h + 8 triangular faces (Euler + face-length count, disconnected case only stronger), the faces are distinct triangles, and the two layers' triangle sets are edge-disjoint hence disjoint. ETERNAL COROLLARY: no biplanar graph contains C5[E_s] (independent blobs of size s, adjacent blobs completely joined) for s >= 4: h = 5s vertices, e = 5s^2 cross edges, triangle-free, and 5s^2 > 20s - 8 for all s >= 4. In particular NOTHING containing C5[K4] (or any C5 blowup with blobs >= 4) is ever biplanar - a subgraph-closed exclusion that strengthens f026's per-graph closures into a forbidden-substructure theorem. USE: screens must minimize the caps over induced/spanned subsets, not evaluate at the full n; the cheap version 4-colors Gbar and checks clique-triples.


THE ENTIRE alpha=3 n=28 SEARCH BATCH IS NON-BIPLANAR (run-06): all 535 screened candidates (runs/out/cand3/, m(G) = 140-142, K4-free complements with alpha(Gbar)=8, chi >= 10 by covering) are killed by the localized bounds f045, via TWO independent surviving arguments: (1) clique-cover lens: chi(Gbar) = 4 exactly for every batch member, so V partitions into four G-cliques (8,8,8,4); deleting the nearly-detached 8-clique leaves 20 vertices carrying 110-114 edges vs the L1 cap 6*20-12 = 108 - the global count only passed because all the Euler slack hides on one clique; (2) band lens: every member contains C5[E4] (80 triangle-free edges on 20 spanned vertices vs L2 cap 72) - the searcher had drifted onto C5[K4]-cored structures. 522/535 die by L2 alone, 13 need L3's triangle credit. Certificates per graph in runs/out/cand3_kill_certs.json; 12/12 random certificates re-verified by fresh independent code (subgraph-of-G, triangle-freeness, and the violation each checked directly). Walk farm stopped (was at defect 8-13 plateau - now explained: no splits exist). The alpha=3 n=28 route is NOT closed as a whole - only this batch; future searchers must generate AWAY from C5[K4] cores and screen with f045-localized bounds inside the loop.


ALPHA=3 n=28 SEARCH BASIN FULLY EXHAUSTED AND DEAD (run-06, extends f046 from 535 to ALL 1000 screened candidates): the localized-bound certifier (runs/run07_local_band_batch.py) killed 1000/1000 cand3 graphs - 979 by pure triangle-free certificates (L2: e(H) > 4h-8 on spanned vertices), the rest via the L3 triangle-count bound; violation margins 1-16; zero survivors across both search tranches. 10/10 fresh-sampled certificates re-verified by independent code. CONCLUSION: every graph the C7K4-seeded SA generator produces in this basin embeds a C5[E4]-type band (f045 corollary: no biplanar graph contains C5[E_s], s>=4). The alpha=3 n=28 route is not closed in general, but any future attempt needs a generator that (a) screens with the f045 localized bounds inside the search loop and (b) explicitly forbids C5-blowup band cores. Combined with f039 (n=29/30 circulants exhausted) and the Ramsey-Turan gap argument, the alpha=3 route is now the least promising small-n front; alpha=2 n=19 stragglers and the synthesis verdict remain the live questions.


a00120 AND TWO SECOND-GENERATION m=101 CANDIDATES CLOSED (run-06): (1) a00120, the last and hardest m=101 straggler from the original ten, is NON-BIPLANAR: case-51 seed CNF UNSAT under Glucose3 (19,844s - 5.5 hours, the hardest instance of the campaign) AND Cadical (5,718s); fresh-CNF recheck under Glucose3 still running at log time (Minisat22 recheck segfaulted - solver binary crash, not a soundness event; re-run under g3). Its difficulty was solver-variance on a balanced instance, not a hidden witness: structurally a near-twin of its siblings. NINE OF THE ORIGINAL TEN alpha-route candidates are now closed; only the two m=100 graphs remain. (2) alpha2b_f6621f3855e4 and alpha2b_b7a593f67a87 (new-seed m=101): UNSAT under g3 (3,015s / 8,850s) and cd15 (1,077s / 2,274s), independently CONFIRMED by fresh-CNF rechecks (1,767s / 3,436s). Seven second-generation m=101 candidates remain with their solvers.


ASSUMPTION-BASED ENCODING VALIDATION (run-06): to validate that a biplanarity constraint system does not over-constrain, do NOT re-search for a known witness split (the m=100 control burned 5+ solver-hours failing to re-find its own construction); instead ASSUME the known split as solver assumptions over the full constraint CNF - propagation-only, decides in seconds. The m=100 case-51 system was validated this way: the known 51/49 doubled-triangulation split satisfies all constraints (case cardinality + slack-derived degree/support caps + K5/K3,3 bans). Use for every future encoding: build CNF, assert known-witness literals, require SAT.


n=16 low-core reduction, clique lemmas: any H on 16 vtx with Delta<=6, s(H)<=6, m>=36 (complement of a biplanar 10-critical core candidate) is K4-free. Chain: K6/K7 excluded by hand (K6 members have <=1 external edge, so >=15 edges avoid it and any 2-matching there makes s>=7; a nu<=1 graph has <=6 edges). K5 excluded via mini-lemma V1/V1b (SAT-CEGAR, Cadical195 + Glucose42 cross-check): no graph on <=11 vtx with Delta<=6, s<=2 has >=14 edges, so K5 forces m <= 10+10+13 = 33 < 36. K4 excluded via V2w (Berge-Tutte deficiency witness encoding + savings cuts, both solvers): no graph on 12 vtx with Delta<=6, s<=3, m>=18, alpha<=8, so K4 forces m <= 6+12+17 = 35 < 36. Scripts runs/n16_minilemmas.py, runs/n16_minilemmas2.py; verdicts runs/out/candlow/n16_verdicts.txt.


NO 10-CRITICAL CORE ON 13 VERTICES (run-06 low-n program; extends f008: minimal witness now n >= 14): a 10-critical core on 13 vertices forces complement H with Delta(H) <= 3, clique-cover savings s(H) <= 3, and biplanarity forces m(H) >= 12. Exhaustive vertex-augmentation enumeration of ALL connected subcubic graphs with s <= 3 (complete: every connected graph extends from a non-cut-vertex predecessor; s is subgraph-monotone so the prune is sound) gives isomorphism-class counts 1,2,6,10,26,38,21,7,2,0,0,0 for n=2..13 - none exist beyond 10 vertices - with edge-maximal profiles s=1: m<=3 (K_{1,3}), s=2: m<=6 (K_{2,3}), s=3: m<=10. Knapsack over component multisets (<= 13 vertices, savings <= 3): MAX m(H) = 10 < 12. Cross-checks: brute-force at n<=7 reproduced identical class sets; s(G) = n - chi(complement) verified on all 112 generated graphs. K9-freeness never needed. Verified by adversarial panel (survives, no flaws).


NO 10-CRITICAL CORE ON 15 VERTICES (run-06 low-n program): via Gallai's join-decomposition theorem for critical graphs on <= 2k-2 vertices, every hypothetical 10-critical W on w <= 15 vertices decomposes as K_{t} join indecomposable critical pieces; exhausting the constraints gives exactly 11 cases over w in {10,12,13,14,15}; per-case edge accounting (cross-join edges + Dirac/Kostochka-Yancey piece minima + degree-9 demands on outside vertices) forces m(G) >= 79 > 78 = corridor ceiling in every case. The unique near-tight case (K4 join a 6-critical 11-vertex piece with m forced to 28) is killed three independent ways (Dirac 1957 m >= 29; Kostochka-Yancey m >= 29; self-contained Gallai low-vertex forest argument: 22 required forest edges vs cap 21). All lemmas corroborated computationally at small scale (90 critical graphs on <= 6 vertices; 3564 seven-vertex graphs). alpha/omega constraints never needed. Verified by adversarial panel (survives, no flaws). Full arguments in the workflow transcripts; scripts referenced in the finding trail. n=14, 16, 17, 18 agents were cut off by the session limit - resumable.


BOOKKEEPING CORRECTION + EVIDENCE-TIER CLARIFICATION (run-06, after external review; read this before trusting any earlier count or scoreboard row): (1) ARITHMETIC ERROR in f048: it said 'nine of the original ten alpha-route candidates are now closed' while also saying two remain. The correct count at that time was EIGHT of ten: four m=102 (a00080/a00124/a00206/a00238, f036) + four m=101 (a00085/a00108/a00300/a00120, f038/f043/f048), leaving the two m=100 graphs a00239/a00333 open. The closures themselves are sound; only the tally sentence was wrong. (2) FAMILY-SCOPE OVERSTATEMENT: findings f036/f038/f043/f044/f048 were filed with family='alpha2-complement', which makes rlog's scoreboard render the ENTIRE family as RULED OUT. That is false and must not be inherited: what is closed is (a) every Euler-tight m=102 candidate ever generated (45+ graphs, each individually decided), (b) six specific m=101 graphs, (c) nothing at m=100 yet, (d) NO general theorem about alpha=2 n=19 as a class - the wholesale question is exactly what the synthesis run is still deciding. The alpha=2 n=19 family remains OPEN as a family. (3) EVIDENCE TIERS: 'confidence: proved' on the SAT-based candidate closures means 'UNSAT reproduced under >= 2 independent solvers and >= 1 independently implemented encoding, with positive controls passing' - NOT DRAT-checked archival proof (DRUP certificates are archived for the m=102 four in runs/out/proofs/ but no checker has been run; queue item). The paper-grade items are f040 (human-readable proof, panel-verified), f045 (localized bounds, elementary), f025/f026, and the low-n closures f051/f052/f053/f054 (elementary + audited exhaustion). Treat the two tiers differently in any writeup.


NO 10-CRITICAL CORE ON 14 VERTICES (run-06 low-n program; with f051 this pushes the minimal witness order to n >= 15, and with f052 to n >= 16): at n=14 the complement H needs Delta(H) <= 4, s(H) <= 4 (savings), and biplanarity forces m(H) >= 19. PROVED: max m(H) under Delta <= 4 and s <= 4 on 14 vertices is exactly 17 < 19. Method: s >= nu (a matching is a clique packing) bounds the matching number, WLOG-labelling a maximum matching leaves the exposed set independent; a proved case lemma restricts exposed-side attachments per matching pair; then SAT+CEGAR per labelled space (degree cardinality + m >= k, with counterexample cuts blocking exact savings-5 clique packings extracted by an exact subset-mask DP): nu=4 UNSAT at m >= 18, nu=3 UNSAT at m >= 17, nu <= 2 analytic (m <= 14). The extremal witness at m=17 was independently re-verified end to end: its complement G has 74 edges, min degree 9, and chi(G) = 10 exactly (9-colouring UNSAT, 10-colouring SAT) - so the densest savings-4 structure misses the biplanar corridor cap 72 by exactly 2 edges. Pipeline cross-validated by reproducing the n=13 closure with identical machinery; savings DP unit-tested against hand values (K5, C5, K4,4, 2K5, K2,4, C5-blowup). alpha/K9-freeness never needed. Adversarial panel: survives, no flaws. Script: analysis/n14_candlow_decide.py (~5s rerun).


NO 10-CRITICAL CORE ON 17 VERTICES (run-06 low-n program): at n=17 the complement H needs Delta <= 7, s <= 7, alpha(H) <= 8 (K9-freeness) and m(H) in [46,59]. Closed in two stages. (1) A clique-kill chain forces H TRIANGLE-FREE: a K_q in H caps s(H-Q) <= 8-q and cross edges e(Q,O) <= q(8-q), giving per-q edge budgets; K8/K7 die by counting alone; K6/K5/K4/K3 die by SAT lemmas with exact Tutte-Berge matching certificates, clique-cover CEGAR and exact alpha clauses (L6[n=11,s<=2,m>=15], L5[n=12,s<=3,m>=22], L5'[m=21,alpha<=8], L4'[n=13,s<=4,m>=24], L3'[n=14,s<=5,m>=28,omega<=3] - the last needing 17,935 CEGAR iterations - all UNSAT). (2) The triangle-free branch dies ANALYTICALLY: triangle-freeness gives Delta <= alpha per part, Gallai-Edmonds (nu <= 7 hence deficiency >= 3) splits H into factor-critical components of size 1 or >= 5 that are pairwise non-adjacent off the A-set, alpha budgets 8 across parts, and Ramsey caps each part at R(3,alpha_i+1)-1; an exhaustive mechanized sweep over all shapes gives max m = 45 < 46. TIGHTNESS: H* = C15(1,4,6) + 2K1 satisfies every constraint (triangle-free, 6-regular, alpha=8, s=nu=7, m=45) and misses the corridor floor by exactly ONE edge - this regime is razor-thin, so the closure genuinely needed the full analysis. Machinery audited: 400 random end-to-end trials against exact subset-DP savings/matching/alpha (0 mismatches); L5/L6/L7 maxima match Chvatal-Hanson f(k,7) exactly; positive controls pass at full size. Adversarial panel: survives, no flaws. (Note: f053's parenthetical listing of low-n findings should read f051, f052 and this pair.)


n=18 IS LIVE - AND ITS FIRST THREE CANDIDATES ARE ALREADY DEAD (run-06 low-n program): unlike n=13/14/15/17, the n=18 regime ADMITS structures satisfying every necessary condition. The panel classified the triangle-free branch completely and produced explicit candidates; three were re-verified fresh here (n=18, min degree 10, omega=8, chi >= 10 by SAT UNSAT at 9): n18_001 with m=95 and n18_002/n18_003 with m=96 = 6n-12 EXACTLY. Their structure is a genuinely NEW family for this project: G = K1 join (C5-blowup with CLIQUE blobs of sizes 4,4,3,3,3) - i.e. a dominating vertex over a chi=9 blowup, which no earlier family finding covers (f002 handled K_a + C_b joins; f011/f026 handled bare odd-cycle blowups, where Sigma=17 gives only chi=9). Note the f045-L2 band check does NOT kill them (band 57 vs cap 4h-8=60 at h=17): they needed the deciders. Outcome: both Euler-tight members are NON-BIPLANAR, UNSAT at round 0 from the tight triangulation constraints in 0.6s and 6.5s under Cadical (cross-solver and independent recheck chains launched); the K6-grid detector was silent on both, so these are global obstructions. n18_001 (slack 1) is with the case-48 solver. IMPLICATION: the join-over-blowup construction is the natural chi=9-to-10 lifting mechanism at low n and it deserves systematic corridor analysis (queued) - it reaches chi=10 with far fewer vertices than the alpha route.


SCOPE MARKER for the alpha2-complement family (run-06, machine-readable companion to f053): all ruled_out findings filed against this family (f036, f038, f043, f044, f048) close SPECIFIC GRAPHS, not the family. Closed to date: every Euler-tight m=102 candidate generated (45+, each individually decided UNSAT under >= 2 solvers), six m=101 graphs, and nothing at m=100. The family-level question - does ANY alpha=2 graph on 19 vertices in the corridor admit two planar layers - is OPEN and is exactly what runs/run06_synthesis.py is deciding. rlog's scoreboard now reads 'partial closures only' for this family instead of RULED OUT.


CONSTRUCTION-FIRST SEARCH (run-07, runs/run07_alpha3_construct.py; the strategic inversion an external review correctly demanded - 699 of 742 logged attempts died on planarity, i.e. we were generating for chi and paying the NP-hard test): maintain TWO EDGE-DISJOINT PLANAR TRIANGULATIONS as a search invariant, so the union is biplanar BY CONSTRUCTION with exactly m = 6n-12 (the Euler ceiling) at every step, and biplanarity is never tested again. Then hunt the chromatic side, which is cheap. Best formulation found: at n in [28,30], alpha(union) <= 3 forces chi >= ceil(n/3) = 10 by covering alone, and omega <= 8 is automatic (K9 is not biplanar), so ANY state with no independent 4-set IS a 10-chromatic biplanar witness. Objective = number of independent 4-sets (exact, bitset, early-exit), minimised by m-preserving swaps (evict a resident, insert an edge joining two vertices of a currently-independent 4-set, planarity re-checked) under simulated annealing. Calibration and gradient: a random biplanar triangulation pair at n=19/m=102 has chi = 7 (chromatic search there is gradient-free, which is why the chi-objective version stalled); at n=30 the independent-4-set count falls 2930 -> 2123 in 180 s, a real gradient. Note this generator is DISJOINT from the basin f047 closed (those were chromatically generated, C5[K4]-cored) and cannot be killed by f045 (its localized bounds hold automatically for anything biplanar by construction). Farm running at n=28/29/30.


THE ALPHA=3 RAMSEY-TURAN COLLISION (run-07; the sharpest open question the project has, and it decides the construction-first search above): does there exist a K4-free graph on 30 vertices (resp. 28, 29) with at least 267 edges (resp. 234, 250 = C(n,2) - (6n-12)) and independence number <= 8? Equivalently: is there a biplanar-compatible alpha<=3 graph at n <= 30? The tension is explicit. Turan gives max 300 edges for K4-free on 30 vertices, attained ONLY by the balanced complete 3-partite T(30,3) - which has independence number 10, violating the alpha <= 8 requirement that K9-freeness imposes. So a witness complement must carry 89% of the Turan edge count while being structurally FAR from 3-partite; stability theory says near-extremal K4-free graphs are near-3-partite, hence have independence ~n/3 ~ 10. Either (a) an explicit construction threads this needle - and then the construction-first searcher should find it, since every state it visits is biplanar - or (b) a quantitative stability argument closes alpha=3 at n <= 30 entirely, which combined with f039 (all 7007 circulants exhausted at n=29,30) and f047 would end the alpha=3 route. This is the same shape as the queued alpha=4 question (n=37..50, K5-free, 4-partite stability) and one panel should attack both with explicit constants.


ALL THREE n=18 CANDIDATES ARE NON-BIPLANAR (run-07, completing f056): the low-n panel's n=18 structures - G = K1 join C5[cliques 4,4,3,3,3] and two variants, each verified fresh here (n=18, min degree 10, omega=8, chi >= 10 by SAT UNSAT at 9, in corridor [81,96]) - all fail. n18_002 and n18_003 sit exactly at the Euler ceiling m=96=6n-12, so both layers would have to be exact 18-vertex triangulations: UNSAT at round 0 in 0.6s and 6.5s (Cadical), cross-checked under Glucose3 and INDEPENDENTLY CONFIRMED by recheck_cegar.py --tight (fresh CNF, independent generators). n18_001 has m=95 (slack 1), needing the case-48 encoding: UNSAT at round 0 after 4,440s under Cadical (recheck launched). The K6-grid detector was silent on all three, so these are global obstructions, not local ones. SCOPE: this closes these three graphs and the specific join-over-blowup shapes they instantiate, NOT the n=18 regime, which remains LIVE - the panel's classification of the non-triangle-free branch and the other triangle-free shapes is unfinished (queued). NOTE the mechanism is worth its own family analysis: K1 join (chi=9 blowup) is the cheapest chi=9-to-10 lifting known to this project and reaches chi=10 at 18 vertices, one below the alpha=2 route's 19.


TURAN-GAP TABLE FOR THE COVERING ROUTE (run-07, exact arithmetic; frames f059 and closes one endpoint outright). For alpha(G) <= q the complement H must be K_{q+1}-free with alpha(H) <= 8 and m(H) >= C(n,2)-6n+12. Comparing that target with the exact Turan maximum e(T(n,q)) gives the deficiency d(n) a witness may use: ALPHA=3: n=28 need 222 vs max 261 (d=39); n=29 need 244 vs 280 (d=36); n=30 need 267 vs 300 (d=33). ALPHA=4: d falls monotonically 57, 54, 51, 48, 44, 40, 36, 32, 27, 22, 17, 12, 6, 0 for n=37..50. THEREFORE n=50 IS CLOSED WITH NO STRUCTURE THEORY AT ALL: the target 937 equals e(T(50,4)) exactly, and by Turan's uniqueness the only K5-free graph on 50 vertices with 937 edges is the balanced complete 4-partite T(50,4), whose independence number is 13 > 8. Contradiction. Moreover every extremal Turan graph in both regimes has alpha = ceil(n/q) in {10,...,13} > 8, so a witness must be structurally FAR from q-partite while spending at most d(n) edges - and d(n) <= 12 for n in {48,49,50}, so a stability theorem with even a weak explicit constant closes the top of the alpha=4 range. The hard end is small n (d ~ 33-57), which is exactly where Furedi-type stability (a K_{r+1}-free graph with e(T(n,r))-d edges has an r-partite subgraph missing at most d edges) must be combined with the K4-free-neighbourhood tension: internal part edges are needed to hold alpha <= 8 (a part of size s needs >= s-8 of them) but every internal edge forbids K4s, forcing missing cross edges. Quantifying that trade-off is the live panel's task.


f045 INDEPENDENTLY VALIDATED AGAINST CLASSICAL LITERATURE (run-07): the localized bound L2 (any triangle-free subgraph on h SPANNED vertices of a biplanar graph has at most 4h-8 edges) applied to complete bipartite graphs says K_{s,s} is not biplanar iff s^2 > 4(2s)-8, i.e. iff s >= 7. The classical Beineke-Harary-Moon formula theta(K_{s,s}) = ceil((s+2)/4) exceeds 2 iff s >= 7. THE THRESHOLDS COINCIDE EXACTLY: L2 allows s=4,5,6 (where theta=2, correct) and forbids s=7,8,9,10 (where theta=3, correct). So our elementary bound is tight enough to recover a published thickness theorem at precisely the right cutoff - strong external corroboration for f045 and for the closures built on it (f046/f047 killed 1000 alpha=3 candidates using exactly this inequality, 979 of them by L2 alone). Corollary forbidden-subgraph list for future screens: no biplanar graph contains K_{7,7} or K_{6,9} (or any K_{a,b} with ab > 4(a+b)-8), nor C5[E_s] for s >= 4 (f045). USE: any candidate containing a complete bipartite K_{6,9}/K_{7,7} between two vertex sets is dead on sight - this is a millisecond screen and it explains why dense alpha<=3 graphs, whose complements are near-3-partite so that G contains near-complete bipartite bands between its near-cliques, keep dying.


CONSTRUCTION-FIRST EVIDENCE THAT alpha=3 AND BIPLANARITY ARE JOINTLY IMPOSSIBLE AT n=28..30 (run-07, f058 farm results): six annealing runs (n=28,29,30 x 2 seeds, 5400s and ~5M accepted moves each) maintaining two edge-disjoint triangulations - so every state is biplanar at exactly m=6n-12 - drove the independent-4-set count down from ~2930 to a PLATEAU at 852-890 (n=28), 1134-1222 (n=29), 1062-1451 (n=30). Target is 0 (alpha <= 3). The searches froze at annealing end, so these are not hard floors, but the gap is three orders of magnitude in the wrong direction. Combined with the complementary evidence - the SAME alpha<=3 property IS abundantly satisfiable outside the biplanar class (run-06's searcher found 33k+ K4-free alpha<=8 complements at n=28), yet all 1000 screened candidates were non-biplanar (f047, 979 by L2 alone) - the picture is that each constraint alone is easy and the conjunction is impossible. CONJECTURE worth proving: every graph on 28-30 vertices with alpha <= 3 and m <= 6n-12 contains a triangle-free subgraph exceeding the 4h-8 localized cap (equivalently: contains a forbidden bipartite band such as K_{6,9}). Its complement being near-3-partite by Turan stability (f061) means G is a near-union of 3 cliques of size ~9-10, and the complete bipartite band between two such cliques is exactly a K_{9,10} - which L2 forbids outright. Making 'near' quantitative would close the entire alpha=3 route at these n.


HEREDITARY-ORDER CAPS: THE COVERING ROUTE IS A FINITE LIST OF NARROW WINDOWS (run-07 RT panel, verified; supersedes f059 which also carried two arithmetic errors - the correct targets C(n,2)-6n+12 are 222 at n=28 and 244 at n=29, NOT 234/250; f061's table had them right). THE KEY MOVE: the Euler bound is HEREDITARY (e(G[T]) <= 6|T|-12 for every T with |T| >= 3, since each planar layer restricted to T is planar). The naive Ramsey recursion is vacuous here (R(4,9) >= 73 and R(3,9) = 36 both exceed n <= 50), but adding heredity makes it bite, giving the maximum order of a biplanar-compatible graph with bounded independence: alpha <= 1 -> n <= 8 (K9 not biplanar); alpha <= 2 -> n <= 19; alpha <= 3 -> n <= 31; alpha <= 4 -> n <= 43. Since the covering bound needs n >= 10*alpha for chi >= 10, each alpha admits only a NARROW WINDOW: alpha=2 -> n = 19 exactly; alpha=3 -> n in [28,31]; alpha=4 -> n in [37,43]. This finally answers f039's open request for a finite bound on the alpha=3 route. Consequence for alpha=4: n in [44,50] is CLOSED by the order cap, and n=50 independently by Turan uniqueness (f061). CAUTION recorded honestly: a second panel agent claimed n=46 closed via a stability argument, but its own adversarial verifiers REFUTED the supporting lemma (it asserted alpha(F) >= N - nu(F), false already for K3 where alpha=1 and N-nu=2; the correct bound is nu >= ceil((N-8)/2), which halves every derived cost) - that closure was carried entirely by the error and n=46 must be treated as open on that route, though the order cap closes it anyway.


SIX NEW COVERING-ROUTE CANDIDATES FOUND AND THREE ALREADY CLOSED (run-07). The RT panel's exact search produced the first genuine witnesses to the COUNTING question at n=29 (alpha=3) and n=37/38 (alpha=4) - proving the Ramsey-Turan gate alone can never close those regimes. I re-verified all six independently (own bitset alpha/omega code + networkx + SAT): every one has chi >= 10 (UNSAT at 9), omega = 8 exactly (K9-free), min degree >= 9, and sits in the corridor. Independent hereditary-Euler peeling screen: 13 of the panel's 19 exported objects are killed outright by an induced subset violating e(S) <= 6|S|-12 (excesses up to 19 on 21-vertex subsets - the panel's own screen agreed), including the three claimed n=29 minimisers and its n=39/n=40 alpha=4 objects; six survive the screen and became real candidates. DECIDED SO FAR: rt29_t1 and rt29_t2 (n=29, m=162 = 6n-12 exactly, so both layers must be 81-edge triangulations) are NON-BIPLANAR - UNSAT at round 0 in 3.3s and 0.13s under Cadical. Notably the K6-GRID DETECTOR FIRES on rt29_t1 (K6 = {11,16,18,21,22,23} split {11,16,23}|{18,21,22}), so that graph admits a hand-checkable LOCAL proof of the same kind as f040's; rt29_t2's obstruction is global. STILL OPEN and under attack: rt29_s1/rt29_s2 (n=29, m=161, slack 1, case-81 solvers) and rt37/rt38 (n=37 m=187 and n=38 m=198 - slack 23 and 18, the ROOMIEST candidates this project has ever had, hence the best odds of actually being biplanar; 4 walk searches plus 2 CEGAR runs with walk probes are on them).


THE n=29 SLACK-1 PAIR IS ALSO NON-BIPLANAR, AND THE ROOMY alpha=4 PAIR RESISTS WALK SEARCH (run-07, extending f065): rt29_s1 (n=29, m=161, alpha=3, chi>=10, omega=8) is NON-BIPLANAR - case-81 seed CNF UNSAT at round 0 in 110s under Cadical (rt29_s2 same encoding, logged alongside). With f065's Euler-tight pair that closes all four n=29 candidates the RT panel produced, i.e. every alpha=3 candidate known at n=28 (C7K4, f026) and n=29. The alpha=3 window from f064 is n in [28,31]; n=30 and n=31 have no candidates yet and the construction-first search cannot reach alpha<=3 there (f063). STILL OPEN, and the best odds in the project's history: rt37 (n=37, m=187, ceiling 210, SLACK 23) and rt38 (n=38, m=198, ceiling 216, slack 18), both alpha=4, chi>=10, omega=8, min degree >= 9, surviving the hereditary-Euler screen. First walk searches (5400s, ~600k iterations, 860 restarts each) plateaued at defect 10 (rt37) and 15 (rt38) - so no split yet, and note the defect is LARGER than the slack-0 alpha=2 candidates ever showed (4-6), which is weak evidence against them despite the extra room. Six 12-hour walk searches and their CEGAR runs continue.


PRODUCT-COLOURING BOUND: THE FIRST THEOREM ANY LAYERED/PERIODIC CONSTRUCTION MUST SURVIVE (run-08). For ANY graphs on a common vertex set, chi(G1 u G2) <= chi(G1) * chi(G2): colour v by the pair (c1(v), c2(v)); an edge of G1 separates the first coordinate, an edge of G2 the second. CONSEQUENCE FOR THE EARTH-MOON SEARCH: since planar layers have chi <= 4 (4CT), a biplanar graph is at most 16-chromatic (weaker than Ringel's 12 from degeneracy), but more importantly chi >= 10 REQUIRES chi(L1)*chi(L2) >= 10, so the layer chromatic numbers must be {3,4} or {4,4} - IF BOTH LAYERS ARE 3-COLOURABLE THE UNION IS 9-COLOURABLE, FULL STOP. This immediately kills a whole class of otherwise attractive periodic constructions: two triangular lattices (the triangular lattice is 3-chromatic via c(i,j) = (i+j) mod 3, so ANY pair of sheared/rotated/translated triangular lattices on a shared vertex set is 9-colourable no matter how incompatible the shear); two square/hexagonal/bipartite lattices (2-chromatic, union <= 4); any pair of Eulerian planar triangulations (Heawood: a planar triangulation is 3-colourable iff every degree is even). DESIGN RULE for all layered families: at least one layer must be 4-chromatic and the (3,4) case leaves only a factor-12 ceiling, so aim for BOTH layers 4-chromatic, i.e. planar triangulations with odd-degree vertices (non-Eulerian), odd wheels, or patches carrying an odd-structure defect. Note this also explains post hoc why f007's 33 stacked-triangulation-pair unions never beat chi = omega: stacked/Apollonian triangulations are 4-chromatic but chordal, and the product bound plus chordality left no room.


CHROMATIC INEFFICIENCY OF LARGE STRUCTURED BIPLANAR GRAPHS (run-08, the decisive strategic measurement for the infinite/periodic programme). Biplanarity caps average degree below 12 HEREDITARILY (e(S) <= 6|S|-12 for every S), and in locally sparse graphs chi grows only like d/(2 ln d) - about 2.4 at d = 11.4. Measured on this project's own structured unions, all built to be biplanar by construction: two geodesic minimum-degree-5 triangulations (n=42, m=231 of ceiling 240, union minimum degree 10) give chi = 6 over 60 random permutations, and 29,749 transposition hill-climbing moves produced ZERO conflicts in any 9-colouring; two stacked/Apollonian triangulations give chi <= 8 across 11,370 tested (H,sigma) pairs at n=16..22; two random triangulations at n=19 give chi = 7. Meanwhile Sulanke's K6+C5 reaches chi = 9 on just ELEVEN vertices. CONCLUSION: chi >= 10 at average degree < 12 must come from CLIQUE RICHNESS, not from size or from long-range structural frustration - large structured constructions are chromatically inefficient, and the permuted-layer family G = H u sigma(H) (biplanar for every sigma, since sigma(H) is isomorphic to H) inherits that weakness no matter how the permutation is chosen. DESIGN CONSEQUENCE: build around K8, the largest clique any biplanar graph can contain, using it as a colour-scarcity host - with exactly 9 colours a K8 leaves exactly one free colour, so a vertex joined to 7 of its 8 vertices has exactly two admissible colours and becomes a Boolean spin. That is the clique-rich analogue of de Grey's spindling method for the 5-chromatic unit-distance graph, and it is what runs/run08_spin_gadgets.py now measures.


THE EDGE-BUDGET OBSTRUCTION, MEASURED (run-08; the quantitative core of why every route has failed, and the answer to whether colour-forcing gadgets can be assembled inside thickness 2). A gadget library was built around K8 - the largest clique any biplanar graph may contain - exploiting that with exactly 9 colours a K8 leaves exactly ONE free colour, so a vertex joined to 7 of its 8 vertices has exactly two admissible colours (a Boolean spin). 39 biplanarity certificates were produced and ALL 39 independently re-verified here (planarity of both layers, disjointness, partition). Facts established: (i) K9 MINUS AN EDGE IS BIPLANAR (two K8s sharing 7 vertices, n=9, m=35) - tight, since K9 is not; (ii) spins attach freely, up to 8 on distinct 7-subsets reaching n=16, m=84 = 6n-12 EXACTLY; (iii) BUT every pure spin assembly has chi = 8 - independent spins never interact; (iv) the chain that DOES force chi upward pays a fatal edge price. Measured on the 'positive clause' chain: k=1 gives n=11, m=50, chi=9 and sits 4 edges UNDER the ceiling - and is ISOMORPHIC TO SULANKE'S K6+C5, independently rediscovered; k=2 gives n=13, m=67, chi=9, now 1 edge OVER; k=3 gives n=15, m=86, chi=10, but 8 edges OVER the ceiling. The variant that reaches chi=9 at n=13 exactly on the ceiling (posclause_C3_k1, m=66) does so only by containing K9 (omega=9), which no biplanar graph may have. QUANTITATIVELY: each gadget stage adds 2 vertices, raising the Euler ceiling by 12, while costing 17-19 edges - a structural deficit of 5-7 edges per stage. Reaching chi=10 from the chi=9 Sulanke base needs two stages and therefore overshoots by 8 edges. This is the same accounting that killed the join family (f002), the blowups (f026) and the alpha route (f036-f066), now exhibited as an explicit budget: forcing the tenth colour costs more edges than two planar layers can carry.


LITERATURE CONTACT: OUR BAND BOUND SETTLES THE LOWER BOUND IN AN OPEN PROBLEM OF ALBERTSON-BOUTIN-GETHNER (run-08). Their paper 'The Thickness and Chromatic Number of r-Inflated Graphs' (Discrete Math 310(20):2725-2734, 2010) ends with OPEN PROBLEM 3, quoted verbatim: 'What is the thickness of C_{2k+1}[r] for any k >= 4 and any r >= 4?' Our f045-L2 bound answers the lower-bound half for the entire family, uniformly and by an elementary argument. In C_L[K_r] the inter-blob band edges number L*r^2, span all h = Lr vertices, and are TRIANGLE-FREE whenever L >= 4 (a band triangle would need three pairwise-adjacent blobs, impossible in a chordless cycle). L2 says a triangle-free subgraph of a biplanar graph has at most 4h-8 edges, so biplanarity requires L*r^2 <= 4Lr-8, i.e. L*r*(r-4) + 8 <= 0 - FALSE for every r >= 4 and every L >= 4. THEREFORE Theta(C_L[K_r]) >= 3 for all L >= 4, r >= 4. TIGHTNESS CHECK, and it is exact: at r = 3 the inequality flips (36 vs 40 at L=4), and C_n[K_3] is KNOWN to have thickness 2 for n >= 4 (ABG) - our bound fires precisely at the r=3/r=4 boundary and nowhere earlier. Verified numerically on C5[K4] (excess 8), C7[K4] (8), C9[K4] (8), C5[K5] (33). CONSEQUENCE FOR THE EARTH-MOON PROBLEM: C7[K4] - n=28, m=154, alpha=3, chi=10, Euler slack only 2, and per the run-08 literature panel the last standing published 10-chromatic thickness-2 CANDIDATE after Kirchweger-Scheucher-Szeider (SAT 2023) refuted C5[4,4,4,4,3] - is NOT biplanar. Also relevant: ABG's OPEN PROBLEM 2 asks what can be said about non-uniform inflations G[s1..sn]; for G = C5 our f011 characterised ALL chi>=10, K9-free, in-corridor members (exactly four up to symmetry) and f026 proved every one non-biplanar.


THE MINIMUM-DEGREE REFORMULATION, SETTLED (run-09; kills a tempting shortcut and answers the degeneracy question). Since every 10-critical graph has minimum degree >= 9 and subgraphs of biplanar graphs are biplanar, 'no biplanar graph has minimum degree 9' would have SETTLED Earth-Moon at chi = 9. Tempting, because no such graph was known: K8 has minimum degree 7 and Sulanke's 50-year record K6+C5 has minimum degree exactly 8. IT IS FALSE. Construction-first search (two edge-disjoint planar triangulations, so biplanarity is free, m = 6n-12 exactly, m-preserving swaps annealed on the degree deficit) produced biplanar graphs with minimum degree 9 in SECONDS at n=24,30,40; minimum degree 10 at n=24,26,30,36; and MINIMUM DEGREE 11 at n=30 and n=36 - the counting maximum, since m <= 6n-12 forces average degree < 12. All nine graphs re-verified independently here (both layers planar, edge-disjoint, degrees confirmed); files runs/out/MINDEG*.json. CONSEQUENCES: (1) the minimum-degree route to proving chi <= 9 is dead - a witness is not excluded this way, and any impossibility proof must use colouring structure, not degree counting; (2) biplanar graphs attain DEGENERACY 11, so Ringel's 1959 upper bound chi <= 12, which is exactly degeneracy+1, CANNOT be improved by the degeneracy argument - improving it requires a genuinely chromatic argument; (3) sharply confirming f068, these extremal graphs are chromatically feeble: every one has chi = 6 and clique number 5 or 6 despite minimum degree up to 11 and sitting exactly at the Euler ceiling. Maximum density and maximum minimum-degree buy nothing chromatically. The tenth colour is not bought with edges or degrees; it is bought with clique-richness, which the K9-free constraint caps at K8.


JOIN-WITH-CLIQUE FRAMEWORK: f002 GENERALISED TO ARBITRARY BASES, WITH EXPLICIT BUDGETS (run-09). Sulanke's record K6+C5 is a clique join, so the natural generalisation of the whole route is G + K_t, which is 10-chromatic exactly when chi(G) = 10-t. Two hard constraints follow. EDGE BUDGET: m(G) + t*n + C(t,2) <= 6(n+t)-12, i.e. m(G) <= (6-t)n + 6t - 12 - C(t,2). CLIQUE CAP: omega(G) <= 8-t, since omega(G+K_t) = omega(G)+t and no biplanar graph contains K9. Comparing the budget SLOPE against the Kostochka-Yancey floor for (10-t)-critical graphs settles four of the six cases outright: t=6 gives slope 0 vs floor 1.67n (this is f002, Sulanke's own shape - the base would need chi=4, omega<=2 and at most 9 edges, but the smallest triangle-free 4-chromatic graph is Grotzsch's on 11 vertices with 20 edges); t=5 slope 1 vs 2.25n; t=4 slope 2 vs 2.80n; t=3 slope 3 vs 3.33n - ALL CLOSED, the budget grows strictly slower than criticality demands. ONLY t=1 and t=2 survive: t=1 needs a 9-chromatic base with omega <= 7 and m <= 5n-6; t=2 needs an 8-chromatic base with omega <= 6 and m <= 4n-1. Both demand chi - omega >= 2 AND sparsity simultaneously, which is precisely the combination that makes such graphs hard to build (Mycielski/Kneser/Ramsey territory - and f003 already closed plain Mycielskians on Euler grounds). MEASURED TENSION at the extreme: the sparsest possible 9-chromatic graph on 11 vertices, complement(K_{2,3} + 6K1) with m = 49 = 5n-6 EXACTLY hitting the t=1 budget, has omega = 9 - it contains K9 and is dead on arrival. Minimising complement edges to keep chi high forces enormous cliques; that is the same trade-off f069 measured as an edge budget, now visible as a clique explosion. NOTE the near-miss: Sulanke's K6+C5 has m=50 at n=11 against the t=1 budget of 49, so the apex construction misses the Earth-Moon bound by EXACTLY ONE EDGE.


THE t=1/t=2 JOIN SEARCH EXECUTED: THE BLOWUP SUB-FAMILY IS EXHAUSTED, AND EVERYTHING MISSES BY ONE EDGE (run-09, executing the f072 framework). For an alpha=2 base the chromatic number is EXACTLY chi(G) = n - nu(Gbar) (savings = matching number, since a triangle-free complement has no cliques beyond edges), so the join target is not the covering bound but the sharp condition nu(Gbar) = n - (10-t), together with alpha(Gbar) <= 8-t and m(Gbar) >= C(n,2) - ((6-t)n + 6t - 12 - C(t,2)). Combined with the f064 hereditary cap (alpha <= 2 forces n <= 19) this pins the bases to n = 17,18 for t=1 and n = 15..19 for t=2. EXHAUSTIVE ENUMERATION of the natural extremal family (C5 and C7 blowups with independent blobs - the standard triangle-free graphs of small independence): exactly ONE base qualifies, C5[3,3,3,4,4] at n=17 with m(Gbar)=58 against the required 57, alpha=7. Its apex is n=18, m=96 = 6n-12, chi >= 10, omega=8 - and it is ISOMORPHIC to n18_003, already proved non-biplanar (f060). The second-best, C5[3,3,4,4,3] variants, give n18_001. So the join route at alpha=2 rediscovers exactly the graphs the low-n programme already closed: two independent derivations, same graphs, same verdict. THE RECURRING MARGIN: every remaining near-miss is short by exactly ONE edge - C5[3,3,3,3,3] at n=15 for t=2 has m(Gbar)=45 against 46; C7[1,1,1,1,1,5,6] at n=16 for t=1 has 45 against 46; and Sulanke's own K6+C5 has m=50 against the t=1 budget of 49. Three independent one-edge failures. WHAT REMAINS UNSEARCHED in this framework: non-blowup triangle-free complements with prescribed matching number (t=1/t=2 at alpha=2), and the alpha=3 and alpha=4 bases (t=1: n in [25,31] and [33,43]; t=2: n in [22,31] and [29,43]), which the hereditary caps keep finite but which no search has touched.


THE alpha=3 JOIN TERRITORY: CIRCULANTS EXHAUSTED, INDEPENDENCE IS THE BINDING CONSTRAINT (run-09, continuing f072/f073). The alpha=3 half of the join-with-clique framework requires a base G with alpha(G)=3 (so Gbar is K4-free), omega(G) <= 8-t (so alpha(Gbar) <= 8-t), and m(Gbar) >= C(n,2) - ((6-t)n + 6t - 12 - C(t,2)); the f064 hereditary caps confine it to n in [25,31] for t=1 and n in [22,31] for t=2. TWO INDEPENDENT ATTACKS, BOTH NEGATIVE. (1) Exhaustive circulant sweep: all 23,912 connection sets over n = 22..31 for both t, filtered by K4-freeness (checked at a single vertex by vertex-transitivity) and exact independence - ZERO qualify. (2) Simulated annealing over general K4-free graphs at n=25, t=1 (10.5 million moves): the edge target is EASY - it reached m(Gbar) = 207 against the 181 required - but independence stalls at alpha = 9 against the cap of 7, never closing the last two. So in this regime the edge budget is not the obstruction at all; INDEPENDENCE is. That is the same Ramsey-Turan collision as f059/f061 appearing in a new guise: dense K4-free graphs are forced toward the 3-partite Turan structure, whose independence number is about n/3 (roughly 8-10 at these sizes), well above the 6-7 that K9-freeness of the joined graph demands. NOTE R(4,8) >= 59, so K4-free graphs on 25-31 vertices with alpha <= 7 certainly EXIST - they are simply incompatible with the required density, which is exactly the collision. Unsearched remainder of the framework: alpha=4 bases (t=1 n in [33,43], t=2 n in [29,43]) and non-circulant, non-blowup structures with prescribed matching number at alpha=2.


THE alpha=4 JOIN TERRITORY SWEPT: COUNTING CLOSES A THIRD OF IT, CIRCULANTS CLOSE NOTHING ELSE (run-09, completing the f072 framework's specified regions). For alpha=4 bases the complement Gbar must be K5-free with alpha(Gbar) <= 8-t and m(Gbar) >= C(n,2) - ((6-t)n + 6t - 12 - C(t,2)); the f064 caps confine this to n in [33,43] for t=1 and n in [29,43] for t=2, i.e. 25 (t,n) cases. PURE TURAN COUNTING CLOSES 8 OF THEM with no search at all: t=1 at n=43 (need 694 vs the K5-free maximum e(T(43,4)) = 693) and t=2 at every n from 36 to 43 (deficits growing from -1 to -39). The surviving gaps shrink sharply toward the top of each range - only 4 edges at (t=1, n=42) and 3 at (t=2, n=35) - so a stability argument with even a weak explicit constant should finish those. EXHAUSTIVE CIRCULANT SWEEP over the 17 remaining cases: 119,538 connection sets, each filtered by exact independence and K5-freeness (both checked at a single vertex by vertex-transitivity) - ZERO qualify. Together with f074 (23,912 circulants, alpha=3) and f039 (7,007 circulants for the direct alpha=3 route at n=29,30), the entire circulant family is now exhausted across every specified region of both the covering route and the join framework: 150,457 connection sets, not one candidate. The binding constraint throughout is INDEPENDENCE, not edges: Turan-extremal K5-free graphs have alpha = ceil(n/4) which is 9-11 at these sizes, against the 6-7 that K9-freeness of the joined graph demands, and the annealing evidence at alpha=3 (f074: m easily 207 vs 181 needed, but alpha stuck at 9 vs cap 7) shows the same in the general non-circulant case.


THE RAMSEY-TURAN UNIFICATION: EVERY OPEN REGION OF EVERY ROUTE IS ONE QUESTION (run-10). Exploring the two corners f075 left open showed they are not two problems but one. Writing Gbar = complement(G), a covering-route witness with alpha(G) <= q needs Gbar K_{q+1}-FREE, omega(G) <= s needs alpha(Gbar) <= s, and the Euler/join budget becomes m(Gbar) >= need. So every region is an instance of RT(n, K_{q+1}, s) = the maximum edges of a K_{q+1}-free graph on n vertices with independence at most s: alpha=2 direct is (q,s,n)=(2,8,19) need 69; corner 2 is (2,7,17) need 57 and (2,6,15..19) need 46..96; alpha=3 direct (f059) is (3,8,28..31) need 222..291; alpha=3 join (f074) is (3,7,25..31) and (3,6,22..31); alpha=4 direct (f061) is (4,8,37..43) need 456..657; corner 1 (f075) is (4,7,33..42) and (4,6,29..35). A region is CLOSED OUTRIGHT, with no graph search, whenever RT(n,K_{q+1},s) < need; where RT >= need it yields candidates that still face the strictly harder biplanarity test. This also explains every failure mode observed: the binding constraint was never edges but independence, because Turan-extremal K_{q+1}-free graphs have alpha = ceil(n/q) which is 8-11 at these sizes against the 6-8 required. TOOL BUILT: runs/run10_rt.py decides RT questions exactly by SAT with CEGAR on BOTH constraint families (cliques and independent sets added lazily), since the brute-force encoding needs C(n,s+1) clauses - 118 million at n=42, s=7. Validated: it reproduces the known-achievable RT(17,K3,7) >= 58 (the C5[3,3,3,4,4] blowup) in 593 rounds. FIRST OPEN VALUE: RT(17,K3,7) is 58 or 59 - the m>=59 instance resisted both a direct 24,310-clause encoding (600s) and 5,737 CEGAR rounds with 12,917 cuts. Settling it closes or opens corner 2 by itself. Recommended next: symmetry breaking (lex-leader over vertex permutations) on the RT encoding, and consulting the Ramsey-graph literature (Radziszowski's dynamic survey) for known extremal values.


THE 'ONE EDGE' PATTERN IS SELECTION BIAS - REFUTED BY ITS OWN DISTRIBUTION (run-10, a self-correction). Five constructions had been observed to miss the Euler ceiling by exactly one edge (Sulanke's K6+C5 against the apex budget; C5[3,3,3,3,3] at n=15; C7[1,1,1,1,1,5,6] at n=16; the f069 gadget chain at its chi=9 stage; alpha=4 at n=43 against the Turan maximum), and I proposed hunting an invariant forcing (edges needed) - (6n-12) >= 1, which would have settled Earth-Moon at 9. MEASURED INSTEAD OF ASSUMED: enumerating the margin m - (6n-12) over ALL 19,875 chi>=10 constructions in the join, apex and blowup families gives a broad smooth distribution with NO spike at +1 - margin +1 accounts for 38 constructions, 0.2% of the total, comparable to neighbouring cells and dwarfed by the bulk at +10 to +30. The pattern was pure salience: near-misses are memorable and large overshoots are not. No such invariant exists, and the conjecture is withdrawn. THE SWEEP WAS STILL WORTH RUNNING: 561 apex constructions actually FIT under the ceiling, and exact re-filtering (exact clique number, exact chi >= 10 by SAT rather than the crude covering estimate) leaves EXACTLY ONE genuine chi>=10, K9-free, in-corridor candidate in the whole family - apexC5_33344 at n=18, m=95, omega=8, slack 1 - which is isomorphic to n18_001, already proved non-biplanar (f060). So the apex-over-cycle-blowup family is now exhaustively closed by enumeration rather than by sampling, and the lesson is recorded: test a suspected pattern against its own distribution before building theory on it.


THE ENTIRE alpha=2 JOIN t=2 REGION IS CLOSED BY A ONE-LINE BOUND (run-10). In a TRIANGLE-FREE graph every neighbourhood is an independent set (two adjacent neighbours would form a triangle), so Delta <= alpha, and therefore RT(n, K3, s) <= floor(n*s/2). The alpha=2 join t=2 route needs a base G with alpha(G)=2 (complement triangle-free), omega(G) <= 6 (complement independence <= 6) and m(Gbar) >= C(n,2)-(4n-1); the bound gives m(Gbar) <= 3n, and 3n < C(n,2)-4n+1 for EVERY n in the region's range 15..19 (45<46, 48<57, 51<69, 54<82, 57<96). Region closed, no search required. THE SAME BOUND SHARPENS THE TWO SURVIVING q=2 CASES rather than closing them: RT(17,K3,7) <= floor(17*7/2) = 59, and since the blowup C5[3,3,3,4,4] achieves 58 the value is EXACTLY 58 or 59 - a single bit, currently under SAT with symmetry breaking. If it is 59 the extremal graph is forced to be nearly 7-regular with degree sequence (7^16, 6), every neighbourhood a MAXIMUM independent set, and (counting from a degree-7 vertex v) the 9 vertices outside N(v)+v must carry exactly 10 internal edges and 42 edges to N(v) - a rigid structure that should be decidable. For the alpha=2 direct region RT(19,K3,8) <= 76 against need 69, and blowups already achieve 72-73, so that region genuinely contains candidates (all of which were tested and closed in runs 05-06).


rt37 AND rt38 SURVIVE EVERY LOCALIZED BOUND, AND rt38 SITS EXACTLY ON THE L2 BOUNDARY (run-10). The two roomiest candidates ever produced (rt37: n=37, m=187, ceiling 210, slack 23; rt38: n=38, m=198, ceiling 216, slack 18; both alpha=4, chi>=10, omega=8, min degree >= 9) were re-screened with EXACT rather than heuristic tests. L2 (triangle-free subgraph <= 4h-8) computed exactly by RC2 MaxSAT over the triangle constraints: rt37's maximum triangle-free subgraph is 136 against the cap 140 (4 to spare); rt38's is 144 against the cap 144 - EXACTLY ON THE BOUNDARY, the tightest non-violation the project has seen. L1 (hereditary e(S) <= 6|S|-12) tested by maximum-closure min-cut: no violating subset found for either. So both survive the entire localized-bound arsenal that killed 1000 alpha=3 candidates and every blowup, and their biplanarity remains genuinely open (plain CEGAR is at 342k/320k cuts without converging, as f034 predicts at this size). METHOD NOTE, recorded because it cost two wrong answers: the max-closure formulation of 'max over S of e(S)-6|S|' silently returns 0 for EVERY graph because the empty set attains it, and zeroing a forced vertex's cost changes the objective so the optimum collapses to a singleton. The constraint only applies for |S| >= 3, which max-closure cannot enforce; the correct exact algorithm is the (6,12) pebble game for (k,l)-sparsity. The corrected forcing version is one-sided - it detected a known-violating alpha=3 graph at |S|=20 - and both rt candidates pass it.


RT DECIDER BUILT AND THE q=2 VALUES SETTLED (run-10 infrastructure). runs/run10_rt.py and runs/run10_rt_sym.py decide RT(n,K_{q+1},s) exactly - the extremal quantity every open region reduces to (f076) - by SAT with CEGAR on BOTH constraint families (cliques and independent sets added lazily, since a static encoding needs C(n,s+1) clauses: 118 million at n=42,s=7). The symmetry-broken version adds three standard canonical-form constraints: degree monotonicity deg(0)>=...>=deg(n-1), neighbourhood normalisation of vertex 0 to an initial segment, and lex-leader predicates on adjacent transpositions. Validated: it still finds the known-achievable RT(17,K3,7)>=58 witness (55s), so the breaks do not over-constrain. PRACTICAL LIMIT MEASURED: degree monotonicity costs O(n^2) cardinality encodings and stalls the first solve at n>=22, so the larger regions run with neighbourhood normalisation only. VALUES ESTABLISHED SO FAR: RT(n,K3,s) <= floor(n*s/2) exactly (every neighbourhood of a triangle-free graph is independent), which is TIGHT at RT(19,K3,6)=57 and closes the whole alpha=2 join t=2 region (f078); RT(17,K3,7) is pinned to 58 or 59 with 58 achieved, the single undecided bit for corner 2, still running after 10,000 cuts. The recursive generalisation Delta <= R(q,s+1)-1 giving RT(n,K_{q+1},s) <= n(R(q,s+1)-1)/2 is TOO WEAK for q>=3 (it closes 1 of 45 sub-cases) because Turan already dominates it there - so the q=3 and q=4 regions genuinely need the solver or a stability argument, and four such decisions are running. INDEPENDENT CORROBORATION: the literature panel derived the same ns/2 bound and the same RT(19,K3,6)=57 value, and established the important negative that the Radziszowski survey's e(3,k,n) is the MINIMUM edge count of Ramsey graphs, not the maximum we need - so these values are not simply available from the literature.


OUR CENTRAL QUESTION HAS A NAME AND A 1964 THEOREM: ANDRASFAI'S PROBLEM (run-10 literature panel, both strands independently, adversarially verified). The quantity f076 unified everything into - RT(n,K3,s), the maximum edges of a TRIANGLE-FREE graph on n vertices with independence at most s - is exactly Andrasfai's ex(n,s), and it is NOT the Erdos-Sos-Bollobas Ramsey-Turan quantity (which requires alpha = o(n), not bounded alpha). ANDRASFAI'S THEOREM (1964): for s/n in [2/5, 1/2], ex(n,s) = n^2 - 4ns + 5s^2, with the extremal graphs being exactly the C5 BLOW-UPS - which is why every extremal object this project ever found was a pentagon blowup. Two further proven pieces (JCTB 2025, arXiv 2308.06070): ex = 3n^2-15ns+20s^2 on [3/8,2/5] and 6n^2-32ns+44s^2 on [4/11,3/8], with And(3) and And(4) blowups extremal; Andrasfai's conjecture is that the whole range n/3<s<n/2 is piecewise quadratic with breakpoints s/n = k/(3k-1) and Andrasfai-graph blowups extremal. VALUES NOW EXACT, each confirmed BOTH by the formula and by independent SAT: RT(17,K3,7) = 58 (59 is impossible - this settles corner 2's undecided bit, which our own solver had failed to crack in 10,000 cuts); RT(19,K3,8) = 73, improving on the 72 we had in hand and exhibiting a NEW extremal graph, the UNBALANCED pentagon blowup C5[4,5,4,3,3]; and RT(n,K3,6) = 3n for every n = 15..22, so the naive degree bound is tight throughout that row. Catalogue values were also read directly off McKay's Ramsey-graph collections: RT(22,K3,6)=66, RT(27,K3,7)=94, RT(35,K3,8)=140, plus the full RT(n,K4,4) row for n=10..23. IMPORTANT NEGATIVE, verified: no table or theorem anywhere in Radziszowski's survey or its references gives MAXIMUM edge counts for our (q>=3) parameters - the survey's e(3,k,n) is the MINIMUM - and McKay's catalogues cover none of the nine regions. So every q=3 and q=4 value the project needs must be computed in-house; there is no citation to wait for. Also corrected: R(4,7) is NOT 49 as an earlier brief assumed - DS1 gives 49 <= R(4,7) <= 58, still open.


THE alpha=3 JOIN REGION AT n=25 HAS A CANDIDATE, AND IT IS NON-BIPLANAR (run-11; the RT pipeline working end to end). The symmetry-broken decider returned RT(25,K4,7) >= 181 SAT after 649 rounds - so the alpha=3 join t=1 region at n=25, which had NO known construction and which the exhaustive circulant sweep (f074) had failed to populate, does contain candidates. The witness was converted automatically and verified independently here: Gbar has n=25, m=181, is K4-free with alpha EXACTLY 7 (the cap); the base G = complement has m=119 = 5n-6 EXACTLY (the t=1 budget, to the edge); and the apex G + universal vertex is n=26, m=144 = 6n-12 EXACTLY (Euler-tight), omega=8 (K9-free), chi >= 10 by SAT. Every quantity sits precisely on its bound - the region is razor-thin, which is why sampling never found it. VERDICT: NON-BIPLANAR. Because it is Euler-tight both layers would have to be exact 26-vertex triangulations, and the tight seed CNF is UNSAT at round 0 in 14s under Cadical and 7s under Glucose3, INDEPENDENTLY CONFIRMED by recheck_cegar.py --tight with freshly implemented constraint generators (26s). The K6-grid detector was silent, so the obstruction is global rather than local. METHODOLOGICAL POINT: this is the first candidate the project found by computing an extremal Ramsey-Turan value and reading a witness out of the solver, rather than by graph search - the f076 unification turning into a working pipeline. The same route should now be run at n=26..31 for alpha=3 and n=33..43 for alpha=4.


THE RT PIPELINE IS NOW AUTOMATED (run-11, runs/run11_rt_pipeline.py). The chain proven by hand in f082 - Ramsey-Turan decider returns a witness Gbar, its complement is a chi>=10 base, joining K_t gives a K9-free in-corridor candidate, and the tight/case encoding settles biplanarity in seconds because these candidates land on the Euler ceiling - now runs unattended as a daemon watching runs/out/rt/. For each new witness it (1) re-verifies K_{q+1}-freeness and alpha <= s with independent bitset code, never trusting the producing solver, (2) builds the join with t = 8-s and checks omega <= 8 and the corridor, (3) confirms chi >= 10 by SAT, (4) decides with --tight if Euler-tight else the case encoding, and (5) on SAT writes the split loudly and halts for the CLAIMING A RESULT protocol. This converts every future RT SAT into a verdict with no manual step, which matters because the RT decider is the only source of candidates in regions where graph search failed: the exhaustive 23,912-circulant sweep found nothing at n=25 while the RT route produced a candidate there whose every parameter sits exactly on its bound (f082). Six RT decisions are queued behind it at n=22,23,24,26,27 for alpha=3 and n=29,33 for alpha=4.


GALLAI-EDMONDS COLLAPSES THE n=16 SEARCH SPACE BY A THIRD (run-11). The n=16 decision needs H on 16 vertices with Delta(H) <= 6 (min degree 9 in a 10-critical core), savings s(H) <= 6 (chi = 16 - s >= 10), alpha(H) <= 8 (K9-freeness) and m(H) >= 36 (biplanarity). The naive window was m in [36,48], since Delta <= 6 caps m at 48. NEW BOUND: savings >= matching number, so s(H) <= 6 forces nu(H) <= 6; by Gallai-Edmonds nu <= 6 on 16 vertices means deficiency >= 4, i.e. some S with |S| = a leaves at least a+4 components. There are no edges between components, each S-vertex has degree at most 6, and a component of size k carries at most min(C(k,2), 3k) edges because Delta <= 6. Maximising over the component profile (one large component plus a+3 singletons) gives m <= 39 for a = 0,1,2,3, and strictly less for a >= 4 (34, 33, 36), while a >= 7 is impossible since 16-a vertices cannot host a+4 components. THEREFORE m(H) <= 39, and the window collapses from [36,48] to [36,39] - a third of the space, and the dense end where the old CEGAR was drowning is exactly what got cut. Empirically corroborated: 4,000 random graphs with n=16, Delta<=6, m>=36 gave zero violations, and the minimum matching number observed at m=44..48 was 7-8 as the bound predicts. The n=16 decider now runs with the restricted cardinality; every model it has produced still shows savings exactly 9 against the required <= 6, so UNSAT remains the expected outcome.


THE alpha=2 n=19 REGION IS BEING EXHAUSTED, NOT SAMPLED (run-11 progress record). Andrasfai's theorem pinned RT(19,K3,8)=73 (f081), which bounds the candidate window completely: every alpha=2 witness at n=19 has complement with 69 <= m(Gbar) <= 73, i.e. m(G) in [98,102]. runs/run11_enumerate_a2.py enumerates each layer up to isomorphism by SAT with blocking clauses and decides every graph as it is found. STATUS: 143 distinct Euler-tight graphs (m(G)=102) enumerated and ALL proved NON-BIPLANAR, each by the tight encoding in 0-40 seconds (both layers forced to be 51-edge triangulations, UNSAT at round 0). The enumeration is now SHARDED by deg(vertex 0) - a sound partition, since every graph has exactly one such value and triangle-free with alpha<=8 forces Delta<=8 - giving four parallel workers on the largest layer. RESOURCE ALLOCATION, chosen deliberately: the sparse layers m(G)=98..101 use WALK-FIRST rather than exact decision, because that is where a witness could actually hide (more slack) and a 90-second walk either finds a split - which IS the Earth-Moon result, self-verifying - or moves on, versus 10-50 minutes per graph for an exact UNSAT that merely confirms the expected. Note the f025 localized screen provably cannot help here (f031: exact max triangle-free subgraph <= 68 = 4n-8 for every alpha=2 n=19 graph), so the full decider is the only tool and this cost is unavoidable. If the layers exhaust, this becomes the first COMPLETE closure of the region rather than the fifth run of sampling.


THE n=16 DECISION IS NOW A SIX-CASE FINITE PROBLEM (run-11), after 720,000+ fruitless CEGAR cuts on the monolithic encoding. Two structural steps replaced the single hopeless SAT instance. (1) GALLAI-EDMONDS SPLIT: savings dominate the matching number, s(H) >= nu(H), so s(H) <= 6 forces nu(H) <= 6 and deficiency 16-2nu >= 4; Berge-Tutte then gives a set S, |S|=a, with H-S having at least a+4 odd components. Since alpha(H) >= (number of components of H-S) the profile needs a <= 4, and per-profile edge counting (all edges meeting S number <= 6a; a part of size k carries <= min(C(k,2),3k)) plus alpha(H) >= sum of the parts' alpha kills a=4,5,6 outright and leaves exactly 14 profiles - each with edge cap 36..42 against the floor 36, so the edge count is nearly forced in every one. (2) JOIN DECOMPOSITION: complementation turns the disjoint union of parts into a JOIN, and chi of a join is EXACTLY the sum, so chi(G-S) = sum_i cc(P_i) with cc the clique cover number; since chi(G) <= chi(G-S)+a, a 10-critical core needs sum_i cc(P_i) >= 10-a. Bounding each cc(P_i) arithmetically - Kostochka-Yancey on the c-critical subgraph that chi >= c forces, Brooks (a connected graph with chi = Delta >= 3 is complete, so a c-critical non-complete graph has Delta >= c), and omega(complement(P)) = alpha(P) - and running the three budgets (chi, edges, alpha) as a knapsack KILLS 8 OF THE 14 PROFILES BY COUNTING ALONE, in under a second with no solver. The 6 survivors are exactly the one-big-part-plus-singletons shapes: a=0/[13,1,1,1], a=1/[11,1x4], a=1/[10,1x5], a=2/[9,1x5], a=2/[8,1x6], a=3/[7,1x6]. Each is now a joint SAT on 63-78 edge variables instead of 120, with the edge count pinned to a window 1-4 wide. Scripts: runs/run11_n16_profiles.py (profile enumeration + joint SAT) and runs/run11_n16_ccbound.py (the arithmetic ceilings). NOT YET A CLOSURE - the 6 survivors are running. Method note worth keeping: when a CEGAR loop refutes models one at a time and the violation margin never shrinks (savings stuck at exactly 9 against the required 6 across 720,000 cuts), that is evidence the encoding is missing a structural fact, not that the instance is merely large.


THE n=16 a=0 PROFILE IS PINNED TO A SINGLE KNIFE-EDGE INSTANCE (run-11, extends f086). Literature verified online and re-checked in-house. (1) The vertex Folkman number F_v(2_6;6) = 11 (Nenov, Serdica Math. J. 35 (2009) 251-271, arXiv:0903.3812), with K1+C5+C5 the unique 11-vertex example. RE-VERIFIED HERE: n=11, m=45, omega=5, delta=8, chi=7 exactly. So 7-chromatic K6-free graphs DO exist well below 13 vertices, and NO vertex-count argument can kill the a=0 profile - the edge bound has to do all the work. (2) Kostochka-Yancey, A Brooks-type result for sparse critical graphs, Combinatorica 38 (2018) 887-934, Theorem 6: for k >= 4 a k-critical graph attains the plain KY edge bound if and only if it is k-ORE, and a non-Ore k-critical graph satisfies e >= ((k+1)(k-2)n - y_k)/(2(k-1)) with y_k = max(2k-6, k^2-5k+2). For k=7: y_7 = 16, so at n=13 the bound rises from 41 to exactly 42. (3) Every k-Ore graph contains K_{k-1}, by induction over the Ore composition (the composite contains G1-x and G2-z, each carrying a K_{k-1} by induction; this is the stronger statement that G-v contains K_{k-1} for EVERY vertex v). Hence a K6-free 7-critical graph is never 7-Ore. RE-VERIFIED HERE: Ore(K7,K7) is 7-critical on 13 vertices with exactly 41 edges, omega = 6, and no single vertex deletion destroys all its K6s. CONSEQUENCE: the a=0 profile's Q must be 7-CRITICAL with e(Q) = 42 EXACTLY, since the profile budget caps e(Q) at 42 and the theory floors it at 42. Equivalently the complement P has EXACTLY 36 edges (not the window [36,39] the monolithic encoding used, nor even the [36,37] that plain KY gives). The question is now the single instance: is there a 7-critical K6-free graph on 13 vertices with exactly 42 edges and delta >= 6? This is NOT in the literature: complete critical-graph catalogues stop at 11 vertices (Royle, via Swain-Bonnington-Farr-Morgan arXiv:2006.12741), and Gallai's join-decomposition theorem for k-critical graphs covers only n <= 2k-2 = 12, so n=13 = 2k-1 is exactly one vertex past where the structure theory stops. Sparsest known 7-chromatic K6-free graphs at n=13 carry about 57 edges, far above 42, which is suggestive of UNSAT but is not a proof. CAUTION LOGGED: a first automated attempt returned UNSAT in 7.5s and was WRONG - inverted polarity in lexicographic symmetry-breaking clauses over-constrained the instance. Any closure here must pass positive controls (re-find K1+C5+C5 at n=11/m=45, and re-find Ore(K7,K7) at n=13/m=41 when omega is relaxed to 6) before being believed.