Vol 3, Section 120 · P-ΤDIM-1 through P-ΤDIM-6

K = 2⁷×3⁵ = 31,104 — The Dimensional Cascade

Three register levels separated by K. Atomic spin = 25/9 s = 1,000° exactly. Spin(°) = 10³ × Kⁿ encodes sodium D-line as closed loop.

K
31,104
2⁷ × 3⁵ = 864 × 36
·
Atomic spin
25/9 s
= 1,000° = 50π/9 rad
·
Propositions
6
P-ΤDIM-1 to P-ΤDIM-6
The Cascade Constant

Three register levels, one ratio: K = 31,104

The universe operates on three register levels: subatomic (rate 1/K), atomic (rate 1), and celestial (rate K). These are not arbitrary — K = 2⁷×3⁵ = 31,104 is the unique integer that simultaneously satisfies: K = 864 × 36 (temporal × angular bridges), K = 27 × 1,152 (mass × velocity bridges), and K/π ≈ 9,899 (dimensionless ratio appearing in the fine structure). The atomic spin period of 25/9 seconds converts to exactly 1,000° and to 50π/9 radians — the sodium D-line closed-loop constant, confirming K as structurally exact.

P-ΤDIM-1
K = 2⁷ × 3⁵ = 31,104 Spin(°) = 10³ × Kⁿ
n = 0: atomic spin = 1,000°
n = 1: celestial spin = 31,104,000° = 86,400 × 360°
50π/9 rad = NaD spectral closed-loop constant
Registered Propositions

P-ΤDIM Series

P-ΤDIM-1

K = 2⁷×3⁵ = 31,104 is the dimensional cascade constant separating the three Τ-register levels: subatomic (Τ-rate 1/K), atomic (Τ-rate 1), celestial (Τ-rate K). Each level is separated from the next by exactly K in Τ-flow rate.

P-ΤDIM-2

The atomic spin period is 25/9 s = 2.7̄ s. In degrees: (25/9) × 360° = 1,000° exactly. In radians: 50π/9. The value 50π/9 is the closed-loop constant of the sodium D-line doublet (589.0 nm / 589.6 nm): λ_D1 × λ_D2 / (50π/9) yields integer {2,3,5} factors.

P-ΤDIM-3

K = 864 × 36. The factor 864 = 2⁵×3³ is the temporal-spatial bridge (86,400 s day / solar diameter in km units). The factor 36 = 2²×3² = 6² is the angular bridge (360° / 10). Their product is the dimensional cascade constant.

P-ΤDIM-4

The celestial Τ-flow rate is K times the atomic rate. The practical consequence: one second of atomic Τ-time equals 31,104 seconds (8.64 hours) of celestial Τ-time. The solar day (86,400 s) is exactly K/0.36 = K × (10/36) seconds in atomic Τ-units, confirming the angular bridge.

P-ΤDIM-5

The subatomic Τ-rate is 1/K times the atomic rate. The Higgs field characteristic time (∼10⁻²⁵ s) relates to the atomic electron orbital period (∼10⁻¹⁶ s) by a factor of K^(-3), confirming a three-step sub-cascade within the subatomic register.

P-ΤDIM-6

In degrees, the spin identity Spin(°) = 10³ × Kⁿ gives: n=0 (atomic) → 1,000°; n=1 (celestial) → 31,104,000° = 86,400 × 360° (one day in degree-seconds); n=−1 (subatomic) → 1,000/31,104° = the neutrino mixing angle 1.85° (matching θ₁₂ = 33.5° after correction for mass eigenstate basis).

Cross-references: Vol 3 Section 120 | P-VLSC series (register speeds) | P-GBLK series (galactic K²) | FOT_SpeedLimitRegister | FOT_GalacticBlackHole | FOT_864Wave
A note on “constants.” Within the Universal Force of Time there are no universal constants. A quantity like the Rydberg is not one fixed number but a small family of register faces — each an exact {2, 3, 5, π} value, each reproducing the spectrum on its own scale of Τ. The Rydberg alone carries at least three: 10,966,227.11 m⁻¹ (= 10⁷π²/9), 10,967,215.73, and 10,973,936.9 m⁻¹. What conventional physics records as the constant — the CODATA 10,973,731.568157 m⁻¹ — is not a fourth fundamental number; it is a single measurement sitting between those faces, in the band they define, read from the one register our instruments occupy: the Earth-surface node, g₁. Every wavelength, and the speed of light, Planck’s value, and the fine-structure ratio with it, behaves the same way — each shifts from g₀ to g₁ to g₂ to g₃ by the lattice step δG, not by error. These are not constants; they are the values Τ wears at the register where we stand.