Nuclear binding and the Hoyle state from the pentachoron

Every transition between two spectral modes of the pentachoric networkcosts a definite energy E (nᵢ → nf) = Φ (nf²/4) − Φ (nᵢ²/4) × mₑ × (nᵢ² + nf²), where Φ (ρ) = −ln ρ + α*ρ is the Lyapunov potential of the pentachoric action, with α* = 1/ (4 ln 2) and the electron mass mₑ as sole fundamental input. The first-generation fermion modes (nₑ, nᵤ, nd) = (3, 4, 5), whose spectralindices satisfy the Pythagorean identity 3² + 4² = 5², generate three transitions. The hypotenuse e → d yields the triple-alpha threshold Q (3α) = 7. 275 MeV at +0. 56%;the long leg u → d yields the Hoyle state E* (0₂⁺, ¹²C) = 7. 654 MeV at −0. 028% (residual 2. 2 keV), identified by energy, spin-parity 0⁺ (exit multiplicity), and nucleus specificity (k = 3 cooperative threshold) ; the short leg e → u predicts0. 713 MeV (open). The channel count N = nᵢ² + nf² is derived from three constraints: extensivity, spectral disjointness, and the absence of void. The assembly formula B (k) = k B (⁴He) + (k−2) Bₛtep, with B (⁴He) = 28. 296 MeV (AME2020) as sole empirical nuclear input, reproduces all twelve α-conjugate nucleifrom ¹²C to ⁵⁶Ni with mean error +0. 11% and standard deviation 0. 60%. For comparison, the best ab initio lattice calculation of the Hoyle state (Epelbaum et al. , PRL 2011) achieves ~10% precision with substantial computationalcost. The present formula achieves 0. 028% in one line, with zero adjustable parameters. Companion script: 40 tests, all PASS.