c = 300,000 encodes particle mass and bond geometry
The speed of light, expressed in its pure FOT form c = 300,000 km/s (a pure {2,3,5} lattice number), is not simply a velocity. Normalised through the temporal base Tt = 864 and the orbital arc 360, it generates the tetrahedral bond angle, the sodium D-line wavelength, and the neutron rest mass — all to sub-ppm precision from CODATA values.
√(625/648) = 25 / (18√2) = 0.9820927516...
c → 5³√2 / 3 — the sodium wavelength
Multiplying the square root of the seed fraction by 60 gives a wavelength in exact closed form — 5³√2/3 — whose numerical value of 589,255,651 places it squarely in the sodium D-line range (589.0–589.6 nm). The factor 60 = 2²×3×5 is the FOT minutes constant.
= 1500 / (18√2)
Wavelength → 109.4653° tetrahedral bond angle
The wavelength, scaled through the multiplier 5760/π³, yields a bond angle of 109.4653° — a pure FOT tetrahedral angle with full prime-lattice ancestry. The factor 5760 = 2⁷×3²×5 is a pure {2,3,5} integer. This is not a coincidence: the spacetime boundary that generates orbital spacing generates the same bond angles at the molecular scale.
= (5³√2/3) × 5760 / π³
Tetrahedral angle comparison
| Source | Angle | Exact Form | Prime Lattice? |
|---|---|---|---|
| Conventional arccos(−1/3) | 109.4712° | arccos(−1/3) — irrational | ✗ No |
| FOT from 1080/π² | 109.4268783° | 1080/π² · 1080=2³×3³×5 | ✓ Yes |
| FOT from c-cascade | 109.4653378° | 5³√2/3 × 5760/π³ | ✓ Yes |
Neutron mass = 1200π²√2 — exact closed form
The same seed fraction, developed through a chain of pure {2,3,5} scale factors, gives the neutron rest mass to within 0.05 ppm of the CODATA 2018 value. The exact closed form — 1200π²√2 — combines a pure lattice integer (1200 = 2⁴×3×5²), the orbital geometry factor π², and the arc bridge √2.
| Step | Operation | Value | Exact Form |
|---|---|---|---|
| Seed | c / 360 / 864 | 0.9645061728 | 625/648 |
| Square root | √(625/648) | 0.9820927516 | 25/(18√2) |
| × 24 | × 2³×3 | 23.5702260 | 25×4/(3√2) |
| × 4π² | scale ×10⁷ | 9,305,152,266 | — |
| ÷ 2 | 4,652,576,133 | — | |
| × 864 | × Tₜ | 4,019,825,779 | — |
| ÷ 24 (scale ×10⁵) | 1,674,927,408 | 1200π²√2 ×10⁵ |
1200 = 2⁴×3×5² · π² from orbital geometry · √2 from arc bridge (√648 = 18√2)
M☀ through spacetime dimensions → arc value = 18×10⁹/π
As mass moves through different spacetime dimensions at planetary nodal points, the observable mass changes by the speed factor of that dimension. Starting from the solar mass base and dividing sequentially by 216 (= 2³×3³, the cubic lattice), the Mercury speed factor (125/108), and the Sun arc speed factor (25/18), the chain terminates at the arc value 5,729,577,951 — equal to 18×10⁹/π, the radian-to-degree conversion at the FOT time scale.
| Node | Divisor | Result | Exact Form |
|---|---|---|---|
| Earth (from Venus base) | — | 1,989,436,789,000 | 6.25×10¹²/π |
| ÷ 2³×3³ = 216 | 216 | 9,210,355,503 | 6.25×10¹²/(216π) |
| ÷ Mercury (125/108) | 11,574,074 | 7,957,747,154 | 25×10⁹/π |
| ÷ Sun arc (25/18) | 1,388,888 | 5,729,577,951 | 18×10⁹/π |
Arc value × 2π = 36,000,000,000 = 2²×3²×10⁹ — orbital closure identity (pure {2,3})
Orbital bond angles = molecular bond angles
At the spacetime boundary between Sun and planet, neither body moves. The dimension on the orbital boundary moves, and can create bonds at specific angles determined by the geometry of the orbit. These angles — 180°, 90°, 57.2958°, 114.5916° — correspond exactly to known molecular bond geometries. The τ mechanism is the same at all scales.
| Angle | Type | Molecular Correspondence |
|---|---|---|
| 180° | Linear (diameter crossing) | CO₂, BeCl₂, acetylene (sp) |
| 90° | Perpendicular (orbital edge) | Square planar, octahedral complexes |
| 57.2958° | Arc bond (1 radian) | Crystal lattice arc subdivisions |
| 114.5916° | Double arc (2 radians) | sp³ organic range |
Molecular bond angles are not electron orbital geometry (VSEPR theory). They are spacetime boundary arc values. The same τ mechanism that creates planetary nodal spacing creates chemical bond angles. Scale is irrelevant — the geometry is invariant across registers.
1080/π² → tetrahedral → DNA helical term 10π²/9
A second FOT tetrahedral angle arises directly from the pure lattice number 1080 = 2³×3³×5. Its cascade through inversion and scaling produces the DNA helical term 10π²/9 = 10.9662271 — which is exactly (2/25) × 1/α(FOT), linking the tetrahedral geometry of carbon bonds to the fine structure constant and DNA's helical architecture.
1/angle × 100 = π²/10.8 = 5π²/54
× 4 = 10π²/27
× 12 = 40π²/9
÷ 4 = 10π²/9 = 10.9662271 ← DNA helical term
Key Results — P-CASC-1 to P-CASC-12
c/360/864 = 625/648 = 5⁴/(2³×3⁴) — the exact irreducible seed fraction connecting the speed of light to orbital arc and temporal base.
√(625/648) × 60 = 5³√2/3 = 589,255,651 — the sodium D-line wavelength band emerges from the seed fraction via the {2,3,5} minutes constant.
wavelength × 5760/π³ = 109.4653378° — the tetrahedral bond angle in its c-cascade form. 5760 = 2⁷×3²×5 is a pure {2,3,5} multiplier.
ġ × 2π ≈ 9×10¹⁶ = FOT c² — the free-fall cascade from solar energy connects to the P-EMC-3 identity Tᴰ = Tᵐ × 9×10¹⁶.
Arc value × 2π = 36,000,000,000 = 2²×3²×10⁹ — orbital closure identity. When τ = 18×10⁹/π, the full 2π orbit exists as time, not as motion.
Neutron rest mass = 1200π²√2 = 1,674,927,407.913 (×10⁻¹⁵ kg units) — 0.05 ppm below CODATA 2018. 1200 = 2⁴×3×5²; √2 enters through the arc bridge √648 = 18√2.
All values in the solar nodal mass chain are of the form N/π where N is a pure {2,3,5} lattice integer. Spacetime dimensional transit conserves the π denominator; only the integer numerator changes.
Arc value = 18×10⁹/π = (180/π) × 10⁸ — one radian expressed at the FOT time scale × 10⁸. The radian is the ground-state bond unit of spacetime.
Planetary speed divisors (Sun 25/18, Mercury 125/108) and the cubic lattice 216 = 2³×3³ are the transition operators as mass traverses spacetime dimensions at nodal points.
Molecular bond angles (180°, 90°, 57.2958°, 114.5916°) are spacetime boundary arc values, not electron geometry (VSEPR). The same τ mechanism operates at planetary and molecular scales.
1080/π² = 109.4268783° → DNA helical term 10π²/9 = (2/25)×1/α(FOT). The tetrahedral angle, fine structure constant, and DNA helical architecture share a single prime-lattice chain from 1080 = 2³×3³×5.
Physical constants (c, mₙ, bond angles, wavelengths) are not independent quantities. They are the same cascade — c/360/864 = 625/648 — read at different scales. Time does not compute them; time IS them.