Nuclear binding and the Hoyle state from the pentachoron

Every transition between two spectral modes of the pentachoric networkcosts a definite energy E(n_i → n_f) = [Φ(n_f²/4) − Φ(n_i²/4)] × m_e × (n_i² + n_f²),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 m_e as sole fundamental input. The first-generation fermion modes (n_e, n_u, n_d) = (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 = n_i² + n_f² is derived from three constraints:extensivity, spectral disjointness, and the absence of void. The assembly formula B(k) = k B(⁴He) + (k−2) B_step, 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.