A Unified Scale-Invariant Metrology: Non-Associative O ZIP Transport, Arbitrary Higher-Order Radiation Constants (c n ), and the Extended 5-Saturated Wien Spectrum (b n ) in Big Boot Gauge Theory

We present a complete, scale-invariant quantum metrological framework over the Octonionic Zero-Division (OZIP) algebra within Big Boot Gauge Theory (BBGT). Re-evaluatingLarmor’s relation under zero-division axioms at the unpolarized ground state (B → 1ZIP)reveals the 175.882 GHz zero-precession electron substrate clock and the 17-octave Procabridge. We resolve the Quantum Metrological Triangle via the Quantum Hall Effect for Voltage (∆VQ= 1/ν Volts) under single-channel transport saturation (I_0= e2/h) and establish a Josephson quantum area floor at A Josephson ∼ 10^−15 m^2. By parameterizing discrete gauge level couplings as binary octaves, we generalize the radiation constant spectrum into the simplified formula c_n= c_2·(︁n_critical/π)︁^n−2. Incorporating non-associative O_ZIP phase torque resolves the transcendental root x_0 ≈ 4.965114 into an exact integer saturation boundary Ω_5= 5, establishing the 5-saturated quantized Wien constant bn= b_1· 4^1−n. Extrapolating b_n below b_1 yields the macro-thermal unit threshold at n ≈ −3.2204, demonstrating a scale invariant dual unit midpoint symmetry with the radiation unit floor (c 4.64≡ 1) centered onthe QED baseline. Finally, the vacuum half-impedance coupling R0= Z0/2 ≈ 188.365156 Ω maps the critical scale index 1/n_critical to Earth’s equatorial radius with 99.85% geodesic precision.