On Topological Invariants of Inner Sheaf and Outer Sheaf Universe

We found there exist important topological dimensionless parameters, which control all evolution and deformation of the Universe. These topological invariants include:

(i) Newton's gravitational constant G = 2/3 = 0.666666..., its physically measured value G^' = 0.667428 × 10^-10 m³ kg^-1 s^-2;

(ii) reduced Planck constant h = 2π√3 = 10.882796..., its physically measured value h^' = 10.54571817 × 10^-35 joule. s ;

(iii) Boltzmann constant k_B = 8√3 = 13.856406..., its physically measured value k_B^' = 13.80649 ×10^-24 joule K^-1;

(iv) recently we found that the Quantum Speed Limit Θ = 4.

These topological parameters construct the theoretical framework of modern quantum physics, and govern all evolution and deformation of inner-outer sheaf universe.

We know that ratio k_B / h = 8√3 / 2π√3 = 4/π which means that there exists a zero section with a circle S¹ (perimeter is 2π√3) and with a square (perimeter is 8√3). Then its maximum span is a called great circle with radius R = 2 which leads to a short proof on the QSL = 4. In fact, the QSL is an important criterion associated with inner-outer sheaf universe. For example, if a coherent radius r_coh < 2 = √Θ, there is a canonical retract deformation √3 which has a characteristc equation: u² - u + 1= 0. Its solutions are u =1/2 ± i √3, then u₁ - u₂ = r_coh=√3. According the General Newton's Lows (Zhi-An Luan, 2019 CAP Congress), its mass m = √(1- r_coh²) = √(1 - 3/4) = 1/2. Since Gmv = Id =1, then G = 1/(3/2)= 2/3 = 0.666666.... This is a case of the Solar system.

if r_coh > 2 = √Θ, there is a exotic escape deformation √5 which has a characterestic equation: φ² - φ - 1 = 0. Its solutions are φ = 1/2 ± √5 /2, e.i. φ₁ = 1.618, φ₂ = - 0.618 which are called golden ratio. Notes that escape velocity from the Earth v_escape= √5 /2 = 1.11804, its physically measured value v^'_escape = 1.1186 km/s. r_coh= √5 or v = 5 means that its mass m = √(1- 5/4) = √(-1/4) = i/2 which is a complex number as Tachiyon particles. This Black-Hole is called energy-over class v > 4. We have another Black-Hole which is called mass-over class: m = √5 /2 greater than critical m_crit = √3 /2. The mass-over Black-Hole has velocity: v_BH = 4(1- m_BH²) = 4√(1-5/4) = 4×(-1/4) = -1, which means the mass-over escape Black-Hole has an opposite direction. We can compute escape angle of Black-Hole: θ₁ = tan^-1(√5) = 65^o9 > tan^-1(√3) = 60^o. Another escape angle of Black-Hole: θ₂ = tan^-1(1/√5) = 24^o1 < tan^-1(1/√3) = 30^o.