Conjecture 1.3 says that any fractional single-source flow can be re-routed so each commodity travels a single path, with no arc's load changing by more than the largest single demand. Over roughly 25 hours across 24–25 July 2026 the goal was to refute it, following two routes: an implication-gadget counterexample, and driving a cost-preserving violation constant β* above 2 to trigger the Traub–Vargas Koch–Zenklusen contrapositive.
Neither worked. No counterexample was found, and not one close call — every certified instance stayed comfortably inside the bound. What the campaign produced instead is a reduction, one small unconditional theorem, and a ledger of everything that failed with the argument that killed it.
| Result | Status | Where |
|---|---|---|
| 2-path instances normalise to out-trees (splice closure) | Proved | SUMMARY.md §2, Lemma 4 |
| Restricted to those, Conjecture 1.3 is a linear-discrepancy question on network matrices with demand-scaled columns | Proved | SUMMARY.md §3, Lemma 5 |
| Equal demands are always safe (Hoffman–Kruskal integrality) | Proved | SUMMARY.md §4, Lemma 6 |
| k = 3: worst case is exactly R = 3/4, census complete through m = 10, exact branch-and-bound on both maximal classes | Proved, unconditional | K3_PROOF.md |
| k = 4: R = 4/5 attained by an equal-demand instance | Lower bound proved | SUMMARY.md §5 |
| R_max(k) = k/(k+1) for all k, hence sup = 1 | Open (Conj. 12.2) | SUMMARY.md §8 |
| Neither literature cites the other: no mention of Doerr, "linear discrepancy", or "totally unimodular" in the three relevant flow papers | Observed, full-text searched | SUMMARY.md §4, References |
The full table with killing arguments is in NOGOS.md (20 entries, NG-1 through NG-20, plus Lemmas 1–11). Four claims were proposed confidently and then refuted; all four were ours.
| # | Claim | Verdict |
|---|---|---|
| NG-1 | Ceilings-only parallel-chain pigeonhole, any m ≥ 2 chains | Impossible: summation gives (m−1)d₁ > m·d_max. Corollary: floors are essential |
| NG-3 | Pure coverage-pigeonhole gadgets | Unrealisable: antichain mass bound Σx(a) ≤ Σdᵢ |
| NG-4 | β* above 2 via odd-hole / clique conflict ladders | Self-limiting at β* = 7/6; never approaches 2 |
| NG-6 | Minimal k = 3 single-floor implication gadget | Impossible for every demand vector (blocking-cost summation) |
| NG-7 | A reported counterexample, t = 1/8 > 0, two independent verifiers agreeing | False positive, caught. Both verifiers scored a declared path menu; the conjecture quantifies over all graph paths. 256 real routings, 23 satisfying, t = −3/8 |
| NG-9 | Out-trees whose tree+chords graph is K4-minor-free | Already proved by Majthoub Almoghrabi–Skutella–Warode. 85.3% of enumerated structures |
| NG-14 | k = 4 by the same census method | Compute no-go: 1.24×10¹⁰ configurations, ~2,070 h, plus a 2ᵏ-deep search |
| NG-15 | R ≤ 3/4 for all k (ours) | Refuted at k = 4: an equal-demand instance attains 4/5 exactly |
| NG-19 | The k+1 sorted "staircase" roundings always suffice (ours) | Refuted at k = 8: staircase-best reaches 1.64·d_max |
| NG-20 | The tightness construction, claimed as new (ours) | Prior art: Morell–Skutella 2022 Fig. 3, cited by TVZ §6. Rediscovered independently |
Everything decides in exact rational arithmetic (fractions.Fraction, and sympy's exact simplex for LPs); floats only ever screened candidates. Needs scipy and sympy, no credentials, no network.
python3 tests/test_calibration.py # 5 gates python3 test_rybin.py # 13 anchor assertions python3 k3_proof.py # k=3 census + exact branch-and-bound python3 fast_census.py # census closure through m=10 python3 ladder.py # beta* mechanism ladder