The Alpha Ladder — the Hoyle chain, and a fourth line pointing at helium-4

Carbon exists because of a single excited state. The Hoyle state of carbon-12 sits 7.654 MeV above the ground state — just 0.380 MeV above the three-alpha threshold — and that hair's-breadth resonance is the only reason three helium-4 nuclei in a star can fuse before flying apart. Hoyle predicted it in 1953 from the bare fact that carbon exists; it was found within months. This paper asks the deposit's ledger three questions about that chain, in order of cost, and reports all three answers — one of which is a hard null, stated first because it is the fence everything else stands behind. Can the ledger reach the Hoyle window? No. Null-fail, by a factor of 15 even after the P55 substitution. The same instrument limit P51 hit at and 8. Does the ledger carry the measured alpha staircase in light nuclei? Partially — and the deficit is not spread across the ladder. It is concentrated in helium-4 itself. Does this converge with the register's other findings? Yes. This is now the fourth independent line pointing at the same object: the ledger's sharpest single deficiency is a derivation of helium-4. [null-fail on the Hoyle reach; earned for the staircase measurement and the convergence; prior art credited throughout] This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.