1{2 "title": "Sharp Zero-Forcing Bounds for Generalized Petersen Graphs: Computational Discovery and Certified Proofs for P(n,5)",3 "abstract": "The zero forcing number Z(G) of a graph is the minimum size of a set of initially black vertices from which the colour-change rule (a black vertex with exactly one white neighbour forces that neighbour black) eventually colours the whole graph; it bounds the maximum nullity of the matrices described by G. For the generalized Petersen graphs P(n,k) the sharp values were, to our knowledge, reported for k=2 (with independent confirmation only for n≤ 16), proved for k=3 and k=4 (2026), and known for k=5 only at n=11 and at n≡ 0(mod 24). We determine Z(P(n,5)) for every n≥ 11: the values are 6,8,9,8,8,8,10,10,10,10,10,11 for n=11,…,22, and Z(P(n,5))=12 for all n≥ 23, so the stabilization threshold is exactly n_0=23. The upper bound 12 is the published 2k+2 construction. The lower bound for all n≥ 26 follows the first-force boundary method of Krishnan, Ma and Zhang, adapted to k=5: we prove that the minimum boundary of an 18-vertex set in P(n,5) is at least 12 for every n≥ 26 by a column-deletion lemma that reduces the statement to the finite window 26≤ n≤ 154, which is settled by 129 unsatisfiability certificates in DRAT format, each verified by the external checker drat-trim and rebuilt from the graph definition by an independent checker. The method provably fails at n=25 for every parameter p≤ 24 (two inner pentagons give an obstruction), so n=23,24,25 and all n≤ 22 are settled by exhaustive enumeration with dihedral orbit reduction, cross-validated by a second, independently written enumeration based on first-force normalization (through n=40) and, for n≤ 23, by DRAT-certified fort-cover refutations. We also show that the k=4 parameter p=16 cannot work for k=5 at any n≥ 22, that the certified bound has zero slack, and that a pre-registered holdout of six values of n confirmed the registered conjectures. All proofs are written out; all certificates, code and a reproduction script are released.",4 "keywords": [5 "zero forcing",6 "generalized Petersen graphs",7 "minimum rank",8 "forts",9 "SAT certificates",10 "DRAT",11 "computer-assisted proof"12 ],13 "discipline": "mathematics",14 "researchMode": "theoretical",15 "secondaryModes": [16 "computational"17 ],18 "contributions": [19 "Exact determination of Z(P(n,5)) for all n >= 11 with the sharp stabilization threshold n_0 = 23 (values 6,8,9,8,8,8,10,10,10,10,10,11 for n = 11..22 and 12 for n >= 23); to our knowledge the values at n in {12,13,15,17,18,20,22,23} and the threshold statement are new, while the bound 12, n = 11, n in {14,16,19,21} (via isomorphism) and 24 | n were known.",20 "Adaptation of the Krishnan-Ma-Zhang first-force boundary method to k = 5: parameter p = 18, column-deletion reduction to the finite window 26 <= n <= 154, and 129 drat-trim-verified DRAT certificates, each independently rebuilt from the graph definition.",21 "Proof that the k = 4 parameter p = 16 fails for k = 5 at every n >= 22, and the explicit inner-pentagon obstruction showing that no p <= 24 certifies n = 25, which forces exhaustive treatment of n = 23, 24, 25.",22 "A fully certified table of exact values for 11 <= n <= 48 with independently replayed upper-bound witnesses, exhaustive or externally checked lower bounds (drat-trim-verified fort-cover refutations for n <= 23), a second independently written exhaustive enumeration through n = 40, and a pre-registered holdout confirming the registered conjectures.",23 "A falsification campaign (exhaustive validation of the SAT encodings on small instances, stress tests of the deletion lemma, negative controls on the proof checker, heuristic attacks) that found no error and showed the certified bound has zero slack."24 ],25 "methods": "Rigorous proofs of the foundational propositions (order independence of the closure, fort characterization, dihedral orbit reduction, graph facts, the isomorphism P(n,k) ~ P(n,l) for kl = 1 mod n, the 12-vertex forcing construction, the first-force boundary lemma and the column-deletion lemma for k = 5); SAT encodings (boundary formulas and fort-cover formulas) with correctness proofs; exact computation by two independently written exhaustive C enumerators (dihedral orbit reduction; first-force normalization), CaDiCaL with DRAT proof logging checked by drat-trim and re-verified by an independent checker that rebuilds every CNF from the definition; Glucose round-based encoding as an agreement test; pre-registered conjecture registry with a frozen holdout; robustness checks including negative controls on the checker and simulated-annealing attacks. Computation on one MacLustr worker node (16 cores).",26 "mainResults": "Z(P(n,5)) = 6, 8, 9, 8, 8, 8, 10, 10, 10, 10, 10, 11 for n = 11, ..., 22 and Z(P(n,5)) = 12 for all n >= 23 (sharp threshold n_0 = 23). beta_18(n) >= 12 for all n >= 26 (129 verified certificates for 26 <= n <= 154 plus the deletion lemma), beta_18(n) = 12 exactly for 26 <= n <= 40, beta_16(n) <= 11 for all n >= 22, and beta_p(25) <= 11 for every 12 <= p <= 24. All 38 tabulated values for 11 <= n <= 48 are fully certified; all six pre-registered holdout values equal 12 as predicted.",27 "limitations": "For n > 40 the lower bound rests on a single certified path (boundary DRAT certificates plus proved lemmas); for n > 154 the argument is the proved deletion lemma with no per-n certificate. No Lean formalization was attempted (toolchain unavailable), so nothing is formally verified in the proof-assistant sense. The 39 GB of DRAT proofs are not shipped; only their SHA-256 digests and a generator are released. The value Z(P(24r,5)) = 12 is cited second-hand because the primary text could not be accessed. The exhaustive oracles are cross-validated but not formally proved correct. Heuristic attacks are evidence of absence only in the heuristic sense. The parameter search covered only p <= 24 and n <= 30.",28 "dataAvailability": "No external data. All graphs are generated from the definition of P(n,5) by the released constructors; all result records (results/*.jsonl, results/*.csv, results/*.json), the conjecture registry (plan/conjectures.jsonl), the counterexample archive and the SHA-256 digests of every certificate are included in the artifact. DRAT proofs (39 GB for the boundary window; 4.0 GB for the n = 23 fort-cover formula) are regenerable with code/run_boundary_window.py and code/regen_fortcover_proof.py.",29 "codeAvailability": "All code is released under code/ (Python 3.14, python-sat 1.9.dev17 with CaDiCaL 1.9.5 and Glucose 4.1, drat-trim built from source, two C enumerators, independent checkers, figure and table generators) with pinned requirements and a one-command reproduction script code/reproduce.sh.",30 "ethics": "Pure mathematics with no human or animal subjects, no personal data and no dual-use concern. Literature identifiers were verified for all 20 cited sources; reading depth (full text, abstract, metadata) is recorded per source and second-hand attributions are labeled as such.",31 "synthetic": false,32 "language": "en"33}