E Pluribus Unum: Hadron Masses from Discrete Computation on a Spiral N9 Graph with Clockwork Mechanism

Face A: The Discrete Computational Core (The Clockwork Hadron Masses)

This foundational paper develops a non-variational framework for hadron physics, demonstrating that particle masses emerge as invariant arithmetic residues of a discrete recursive computational process on a 9-node spiral nonagon graph (N_9). Operating with an exact phase quantum step (\Delta = 1/7) and a clockwork mechanism, the model avoids empirical fitting parameters by completely deriving all relative values from the rigid structural triad (3,7,11). The proton mass is structurally fixed by the Z_3-invariant node 3. The neutron emerges at node 6 with a doubling of the separation parameter (\varepsilon_n = 2\varepsilon_p = 2/693), and the Delta resonance stabilizes at node 9 via the internal interaction of the reconciling sub-bond structure (b_3 = 77). This formalization reinterprets quantum uncertainty (\varepsilon \neq 0) not as an absolute informational limitation, but as the mandatory ontological condition preventing systemic vacuum collapse.  

Face B: The Cosmological Scale & Companion Theory Integration

Conversely, the text maps the structural closure of the N_9 spiral topology onto high-dimensional differential geometry and macro-astrophysical persistence frameworks. The algebraic sub-bond (b_3 = 77 = 7 \times 11) translates directly into the topological register of the compact 7-manifold X_7 with holonomy G_2, exactly matching its split third Betti cohomology H^3(X_7) = 33 \oplus 44 = 77. At cosmological scales, the structural scaling relation C = 1/(3V^2) = 1/(27 \cdot 77^2) \approx 6.25 \times 10^{-6} dictates the extreme stability parameters governing millisecond pulsar timing modulations inside the companion IUVLAV model at Level k=22. The paper provides testable predictions challenging standard \Lambda\text{CDM} paradigms, reframing gravity as computational space curvature, and predicting precise higher-order mass thresholds (Level k=4) near 9.9\text{ GeV}.