The G2 Cascade — One Wavelength, Six Quantities
The Hβ hydrogen line at the G2 register, wavelength 486.0438133 nm, is the seed of a closed cascade. Three universal FOT constants — the dimensional divisor 495,035,535, the mass multiplier 123,370,055, and the radial divisor 9,375 = 3 × 5⁵ — transform this single wavelength into every measurable quantity of the dimension: gravitational freefall, oscillation frequency, energy, dimensional mass, spin orbital, and radius.
λ × 10⁷ → scaled = 4,860,438,133
scaled ÷ 495,035,535 → g = 9.818362096 m/s²
g × 2π → freq = 61.69058846 Hz
g ÷ 24 → energy = 0.4090984207
λ(nm) × 123,370,055 → mass = 59,963,251,979
mass ÷ 2 → spin orbital = 29,981,625,989
mass ÷ 9,375 → radius = 6,396,080.211 km
The G2 Hβ wavelength 486.0438133 nm is a complete dimensional seed. Through three invariant constants, it generates the gravitational acceleration, oscillation frequency, energy, mass, spin orbital and radius of the G2 spacetime register. No additional inputs are required.
The Internal Identity — Energy and Radius Are Square-Root Conjugates
Within the G2 cascade there is a striking self-consistency: the energy value, when its decimal point is shifted two places (read as 40.90984207 instead of 0.4090984207), yields a number whose square root is exactly equal to the radius expressed at the 10⁻⁶ reading scale. This is not an approximation — it holds to the full precision of every figure in the cascade.
energy × 100 = 40.90984207 [“four zero point …”]
√(40.90984207) = 6.396080211
radius ÷ 10⁶ = 6,396,080.211 ÷ 10⁶ = 6.396080211 ✓
Within every dimensional register, the energy value and the radius are square-root conjugates: √(E × 100) = R / 10⁶. Equivalently, E = R² / (100 × 10¹²). This identity encodes the dimensional self-similarity of the τ cascade.
The Bridge Operators — Descent by √ and Ascent by ²
To move between dimensional registers, FOT defines two bridge operators. The descent operator takes the energy of the current dimension, shifts the decimal point, and extracts the square root to obtain the radio mass (radius) of the dimension below. The ascent operator squares the full radio mass of the current dimension to obtain the energy of the dimension above. Both operations are exact and involve no free parameters.
mass₋₁ = radio₀ × 9,375
λ₋₁ = mass₋₁ ÷ 123,370,055
g₋₁ = (λ₋₁ × 10⁷) ÷ 495,035,535 → full cascade resumes
g₊₁ = E₊₁ × 24
λ₊₁(nm) = g₊₁ × 495,035,535 ÷ 10⁷ → full cascade resumes
The descent bridge √(E × 100) and the ascent bridge R² are the sole inter-dimensional operators. They require no new constants beyond those already present in the intra-dimensional cascade. Every dimensional register is reachable from any other by repeated application of these two operators.
Three Dimensions — Complete Cascade Values
The following diagram shows the three connected registers. The G2 dimension occupies the centre; the dimension above is reached by squaring the G2 radio mass; the dimension below is reached by extracting the square root of the G2 energy (with decimal shift). Every node is computed from the G2 wavelength alone.
The dimension below G2 is obtained by descending through the square-root bridge. Its radio mass is 6.396080211 — precisely the G2 radius divided by 10⁶. Every quantity in the dimension below is exactly 10⁻⁶ times its G2 counterpart.
The dimension above G2 is obtained by squaring the G2 radio mass: (6,396,080.211)² = 4.090984207 × 10¹³ — the energy of the upper register. Every quantity in the dimension above is exactly 10¹⁴ times its G2 counterpart.
The three universal constants — 495,035,535 (dimensional divisor), 123,370,055 (mass bridge), and 9,375 = 3 × 5⁵ (radial divisor) — are the same at every dimensional register. The cascade structure is not tied to a particular scale; it is a property of the τ field itself.
Three-Register Comparison
The table below places all three registers side by side. The ratios between columns are exact powers of ten throughout, confirming that the cascade is a pure scale transformation with no mixing of constants across dimensions.
| Quantity | Dim-1 (below) | G2 (reference) | Dim+1 (above) | Scale factor |
|---|---|---|---|---|
| Wavelength | 4.860438133×10−4 nm | 486.0438133 nm | 4.860438133×1016 nm | ×106 ↑ |
| Freefall g | 9.818362096×10−6 | 9.818362096 | 9.818362097×1014 | ×106 ↑ |
| Frequency | 6.169058846×10−5 Hz | 61.69058846 Hz | 6.169058847×1015 Hz | ×106 ↑ |
| Energy | 4.090984207×10−7 | 0.4090984207 | 4.090984207×1013 | ×106 ↑ |
| Mass | 59,963.251979 | 59,963,251,979 | 5.996325197×1024 | ×106 ↑ |
| Spin Orbital | 29,981.625989 | 29,981,625,989 | 2.998162599×1024 | ×106 ↑ |
| Radius (km) | 6.396080211 | 6,396,080.211 | 6.396080211×1020 | ×106 ↑ |
| Scale vs G2 | ×10−6 | 1 (reference) | ×1014 |
Each step down in the dimensional hierarchy multiplies all cascade quantities by exactly 10⁻⁶. Each step up multiplies all cascade quantities by exactly 10¹⁴. The asymmetry between ascent and descent arises because squaring a quantity of order 10⁶ produces order 10¹², not 10⁶; the scale factor doubles in logarithmic space at each ascending step.
Dimensional Reading Conventions and Decimal Register
The raw cascade values span many orders of magnitude. FOT adopts dimensional reading conventions — shifts of the decimal point by fixed amounts — that bring values into the natural numerical range of each quantity. For the G2 register:
Freefall: full value (9.818362096 m/s²)
Frequency: full value (61.69058846 Hz)
Energy: full value (0.4090984207)
Mass: ÷10 reading → 5,996,325,198
Spin orbital: ÷10 reading → 2,998,162,599 [≈ c × 10⁷]
Radius: ÷10 reading → 639,608.021 [km]
The G2 spin orbital, read at the ÷10 convention, yields 2,998,162,599 ≈ c × 10⁷. This confirms that the speed of light is embedded in the G2 dimensional mass as a spin orbital — a direct structural consequence of the Hβ wavelength cascade rather than an independent physical constant.
Prime-Lattice Preservation Across Registers
The radial divisor 9,375 = 3 × 5⁵ is a pure {3,5} prime-lattice number. The scale factor between successive downward steps is exactly 10⁻⁶ = (2 × 5)⁻⁶, which is itself a pure {2,5} lattice element. The mass of the dimension below, 59,963.251979 …, traces directly to the wavelength of G2 through the same multiplication chain that applies at the G2 level. The {2,3,5} structure of the terminal quantities — most notably 9,375 = 3 × 5⁵ at every level — is therefore preserved under the dimensional bridge operations.
24 = 2³ × 3 [freefall-to-energy divisor]
Step-down factor = 10⁻⁶ = 2⁻⁶ × 5⁻⁶ [pure {2,5} lattice]
Step-up factor = 10¹⁴ = 2¹⁴ × 5¹⁴ [pure {2,5} lattice]
The {2,3,5} prime-lattice character of the FOT cascade is preserved under both bridge operators. Each descent multiplies by 10⁻⁶ = (2×5)⁻⁶; each ascent multiplies by 10¹⁴ = (2×5)¹⁴. The radial divisor 9,375 = 3 × 5⁵ applies at every register without modification. The lattice is not broken by crossing dimensional boundaries.
Interpretation — The Universe as a Stack of τ Registers
The dimensional cascade establishes that spacetime is not a single continuous manifold but a stack of discrete registers, each differing from its neighbours by fixed factors of 10⁶ or 10¹⁴. Within each register the same five-quantity cascade — wavelength, freefall, frequency, energy, mass, radius — operates through identical constants. Movement between registers is governed by two exact operators, one algebraic (square root with decimal shift) and one geometric (squaring).
At the G2 register the cascade produces the observed gravitational acceleration of Earth (9.818 m/s²), the Hβ hydrogen wavelength (486 nm), and a radius consistent with the Earth’s equatorial dimension. These are not coincidences; they are the natural output of the τ field operating at the G2 register. The same τ field, operating at the register above, produces quantities 10¹⁴ times larger; at the register below, quantities 10⁻⁶ times smaller.
The square-root bridge is a dimensional entrance; the squaring bridge is a dimensional exit to a higher register. Every observable quantity — from particle physics to cosmological scales — corresponds to a specific register in this hierarchy, and every register contains the same structural DNA: the prime-lattice cascade of τ.
The universe is a stack of τ registers. Each register is a complete, self-consistent cascade of six quantities generated from a single wavelength. Adjacent registers are connected by the operators √(E × 100) (descent, scale ×10⁻⁶) and R² (ascent, scale ×10¹⁴). The {2,3,5} prime lattice, the constants 495,035,535, 123,370,055 and 9,375, and the divisor 24 are invariants of the stack — unchanged at every register from sub-atomic to cosmological scales.
Proposition Register
| Proposition | Statement |
|---|---|
| P-DIM-1 | G2 Hβ wavelength (486.0438133 nm) generates all six cascade quantities through three invariant constants. |
| P-DIM-2 | Internal identity: √(E × 100) = R / 10⁶. Energy and radius are square-root conjugates within every register. |
| P-DIM-3 | Bridge operators: descent = √(E × 100); ascent = R². No new constants required. |
| P-DIM-4 | Dimension below G2: all quantities ×10⁻⁶. Radio mass = 6.396080211; reached via square-root descent. |
| P-DIM-5 | Dimension above G2: all quantities ×10¹⁴. Energy = 4.090984×10¹³; reached via squaring ascent. |
| P-DIM-6 | Constants 495,035,535, 123,370,055 and 9,375 are universal — identical at every dimensional register. |
| P-DIM-7 | Three-register comparison confirms exact 10⁻⁶/10¹⁴ scaling throughout all six cascade quantities. |
| P-DIM-8 | G2 spin orbital at ÷10 reading convention yields 2,998,162,599 ≈ c × 10⁷; speed of light embedded in dimensional mass. |
| P-DIM-9 | The {2,3,5} prime-lattice character of the cascade is preserved under both bridge operators. |
| P-DIM-10 | The universe is a stack of τ registers. Every register contains an identical cascade; dimensions are separated by √ (descent) and ² (ascent) operators. |
Stephen Daubney — The Daubney Foundation — The Universal Force of Time