The Rydberg Constant
Derived from Free Fall

The standard physics treatment accepts R_∞ = 10,973,731.568 m⁻¹ as a measured input — a 91.1267 nm wavelength. The Universal Force of Time framework derives the Lyman series limit from first principles. Its wavelength, 91.12910749 nm, is a free-fall node on the {2, 3, 5, π} lattice: the matter limb 3⁶/8 (91.125 nm) and its δ_G sibling (91.13321497 nm), averaged — and the measured value falls exactly between the two limbs, where physical hydrogen straddles both. A second result follows: place a free-fall substitute R_g = g_UFOT × 3π × 10⁶ in the Rydberg slot, run the series formula, and the n = 3² and n = 4² slots return the speed of light and g_UFOT — both at 0.0000 ppm. The Rydberg formula is a property of the prime lattice itself, not of electromagnetism alone.

The Tau standing wave has two helical limbs: the G1 matter helix (B-DNA geometry) and the Z-DNA antimatter helix. Each carries its own pure-lattice Rydberg constant, both expressible in {2, 3} alone — no 5, no π.

The dominant prime shifts between helices: the G1 matter helix carries prime 3 (wavelength 3⁵/2 nm), while the Z-DNA antimatter helix carries prime 5 (wavelength 2³×5³/3² nm = 1000/9). This prime-shift is the spectral signature of helical handedness. The helix ratio is the exact rational Τ_RZ/Τ_R1 = 3⁷/(2×10³) = 2187/2000 — pure {2, 3, 5}, π-free.

Science's CODATA value 10,973,731.568 m⁻¹ sits below the G1 node by 18.71 ppm. Physical hydrogen straddles both helical limbs simultaneously; the measured Rydberg reflects the average register position, not either pure-lattice node. The same ~700 ppm register-crossing signature seen in crystal bond ratios appears here at reduced scale.

There is a second, deeper move. The Rydberg formula generates spectral series by varying the quantum number n. Replacing the Lyman series limit R_∞ with a free-fall Rydberg substitute R_g = g_UFOT × 3π × 10⁶ = 92,527,541.26021273, and substituting n = 3 and n = 4 in the Rydberg slots, returns — not spectral wavelengths — but the speed of light and free fall. (R_g is not the Lyman limit itself; it is a free-fall quantity built to occupy the Rydberg slot.)

The n = 4² algebraic proof is exact: R_g × 16 / (10 × 2π × 24 × 10⁵) = [75π²/8 × 10⁶ × 16] / [480π × 10⁵] = 150π × 10⁶ / 480 × 10⁵ = 25π/8 = g_UFOT. The π² in R_g and the π in the 2π denominator cancel, leaving a pure {2, 3, 5} rational closing on free fall. The structure of spectroscopy encodes free fall.

The Bohr orbital velocity v₁ = 2.180128829 × 10⁶ m/s shares an exact digit sequence with the hydrogen ground-state energy H_J = 2.180128829 × 10⁻¹⁸ J. The same nine digits appear at different powers of 10. This digit identity opens a six-step chain closing exactly on g_UFOT:

Every operator is a pure {2, 3, 5, π} lattice element: 9375 = 3×5⁵ (the universal transit operator), veil = 180/π = 2²×3²×5/π, 1200 = 2⁴×3×5². The closure 3750π⁴ / (1200π³) = 3750/1200 × π = 25/8 × π is algebraically exact — π³ cancels, leaving g_UFOT. The fine structure constant emerges as the ratio 1/α_UFOT = c_UFOT/v₁ = 137.07783890401888 (+305.319918 ppm from science).

The pure lattice number 1000/9 = 2³×5³/3² = 111.111... appears with no adjustment at three independent locations spanning geophysics, quantum mechanics, and antimatter spectroscopy. The lattice does not distinguish between scales — only unit labels differ.

Earth's meridional radius divided by the radian-degree bridge lands on the same node as the antimatter helix Lyman alpha wavelength. The hydrogen atom's n = 4 level, routed through the Shell B dimensional operator, lands there too. Three separate physical domains, three exact arrivals at 2³×5³/3². The FOT lattice is scale-invariant: 1000/9 in kilometres, in nanometres, and as a dimensionless ratio are all the same lattice node viewed at different magnifications.

Every hydrogen spectral wavelength maps to a free-fall acceleration node through a single universal constant K = 1,555,200,000/π ≈ 495,035,534.966 = (180/π) × 86,400 × 100. The free-fall law is simply g = λ/K, where λ is in metres and g in m/s².

The Lyman-beta wavelength τ_Lβ = 2¹¹ × 3⁵ = 497,664 is the free-fall anchor of the hydrogen spectrum. A triple-bridge identity connects K, τ_Lβ, and the civil calendar:

Newton's gravitational constant G is not an independent fundamental — it is the reciprocal of the Earth–Sun distance expressed in FOT lattice units (P-GRAV-3). The observed laboratory scatter of ±2,218 ppm in measurements of G is a direct consequence: G is a derived nodal quantity that varies with the Earth's orbital position, not a fixed constant. The four-band spectral free-fall ladder spans from planetary to stellar surface accelerations, each band anchored to a Lyman or Balmer series node.

The hydrogen ionisation energy closes a four-step cycle through the {2, 3} prime lattice alone — no factor of 5 or π enters — linking the quantum ground state, molar chemistry, electron-volt scale, and the solar circumference in one continuous chain.

N_A,FOT = 6.018910362 × 10²³ mol⁻¹ is the FOT Avogadro number. The J→eV operator 10368 = 2⁷ × 3⁴ is the same operator seen across the dimensional cascade. The chain closes at the solar circumference 4374 Mm = 2 × 3⁷ Mm — no factor of π appears anywhere in P-HCYC-1 through P-HCYC-3. Hydrogen's ionisation energy and the Sun's size are the same {2, 3} lattice node viewed at scales 10²⁵ apart.

Three fundamental spectral constants each have exact FOT lattice derivations that standard physics treats as measured coincidences.

The 21 cm hydrogen line — the radio-astronomy workhorse used to map the Milky Way — is not a coincidence of hyperfine splitting: it is the FOT lattice node 2⁴ × 3⁸ × π / 5⁶, verifiable to four decimal places. The water bond angle, 105.0498032° = 1036.8/π², is exactly 14400 times the fine structure constant α = 9/(125π²) — run it backwards and the atom returns its own coupling, 1/α = 14400/θ = 137.0778389. Quantum electrodynamics, the hydrogen hyperfine structure, and the water molecule share one lattice root.