Energy Balance Equation as a Physical Foundation Supporting Composite-Stroke (Quaternion) Physics

This model was not originally devised to support composite-stroke (quaternionic) physics. It was formulated as an energy (battery) charge-discharge function R = r0 exp(bθ) where the phase angle θ is introduced as a balance weight to derive a constant that remains invariant under dissipation. As a coincidental result, it is confirmed that the present energy balance equation coincides exactly with the formalism of composite-stroke physics. The unification R = r0 exp(bθ) exp(iφ) exp(jψ) → r0 exp(bθ) exp[i(φ+ψ)] under j→i replacement shows that three independently defined composite-stroke functions are projections of a single general solution. Furthermore, composite-stroke physics is consistent with quantum mechanics [Finkelstein et al. 1962; Adler 1995], and it is inferred that composite-stroke physics can be connected to macroscopic waveguide phenomena such as typhoons and to macroscopic mass bodies in the universe via geometric impedance Z(θ)=ρ(0)c0 exp(bθ). Key results:- b=√5/2=1.118 derived from regular pentagon minimum dissipation, Mach 1.5, b=cot(41.81°)- Effective resonance band ±3° (b=1.0~1.24) from Fermi Bubble 41.81°±3° and Pantheon+ 42° explains isotopic mass variance, typhoon eye stepwise convergence, absence of 1.5 hydrogen- Atomic gaps: 126 last stable (2×3²×7), 127 prime no-solution, 128=2⁷ complete resonance black hole- Mathematical closure: E_min 6.18e14 → K=62.33 → E_cum 3.85e16 → R²=1540 → E_res 9.52e17 → observed 1e18 95% without Q, τ 2~3h constant- Primordial hydrogen size recalculation: r0,H=5.29e-11m Bohr radius, L0,H=3.324e-10m, E_min,H=13.6eV auto-derived from b and R²- Geometric light speed cg=exp(bθ), cg=1 at θ=0, log origin, Big Bang origin, matches BIPM c=1 natural unit References: Hamilton 1844, Fueter 1936, Conway 1911, Silberstein 1912, Finkelstein et al. 1962/1963, Adler 1995, Sudbery 1979. Appendix A: Special atomic numbers resonance table (Fe, Cu distinct mass, gaps 43,61,85,87,127)Appendix B: Detailed derivation with primordial hydrogen atom size replacing typhoon phrasing: r0,L0 auto-derived from λ and b, H vertical mode jψ. This work continues the energy balance equation series:- 10.5281/zenodo.23051842 (previous quaternion related work)- 10.5281/zenodo.23180009 (previous energy balance related work)→ Present work 10.5281/zenodo.2335176 unifies both into supporting composite-stroke physics. Previous works derived r0,L0,H auto-derivation and E_min closure; present work replaces typhoon phrasing with primordial hydrogen atom size r0,H=5.29e-11m and derives E_min,H=13.6eV, proving b=√5/2 as unique constant from Mach angle μ=41.81°, supporting composite-stroke formalism.