Checking is not knowing: the blades never touch the shore — proved
Our master-formula paper found the blades: extra kill-zones floating above the shore where theories of gravity die. Three million exact checks never saw one touch the water. This paper replaces checking with knowing: a proof, built from 1,047 exact-arithmetic certificates, that no blade of any level, at any coupling, anywhere along the infinite coast, ever cuts into the land of the living.
The beautiful part
Infinitely far out, the fleet of blades sails exactly parallel to the shore: the tangency discriminant of 6ρ² − (12+4√3)ρ + (8+4√3) vanishes identically at ρ* = 1+1/√3 — zero angle between fleet and shore. The theorem survives because every actual ship is confined strictly inside the wake (windows need ρ ≤ 4/3, asymptotic tip at √(5/3) ≈ 1.291). A naive proof must fail at the tangency; the containment lemma is what makes it possible.
How it was checked (and what the check caught)
The proof is a single re-runnable script: envelope branches for small couplings plus four deep-water lemmas over ℚ(√3). An independent adversarial review then audited every implication — and caught a real bug: the prover had silently skipped 740 cells while reporting success. Fixed; the reviewer's own battery (sharing no code with the prover) re-certified every cell including the formerly uncovered ones: no counterexample. That episode ships with the paper: an exit code proves what a script checked, not what it covered.
Explain it to anyone
A house rule of this lab: every paper ships with a way to explain it to your family. Imagine a coastline where theories of gravity drown, and a fleet of blade-shaped danger zones floating above it. We had checked three million spots and the blades never touched the water. But checking is not knowing — maybe spot three million and one is different. This paper is the proof that it never happens: not anywhere, not for any blade, all the way to infinity — even though at infinity the fleet sails exactly parallel to the shore, touching it at a single unreachable point. And one more honest thing: our first version of the proof had a real bug — it skipped 740 cases while claiming victory. An independent reviewer caught it. We fixed it, re-proved everything twice, and put the story in the paper, because that is how science should work.
Honest status: the theorem covers the j=3 trajectory (the second knife); completeness for the higher knives remains a conjecture with a clean 3-million-point battery. DOI: 10.5281/zenodo.21948833.