The Alphabet of Nature: 2, 3, π, ln2 — A Closed Formal System, Its Dictionary, and Its Solution

This paper constructs the numerical content of the ΨD framework as a single formal language and presents that language's solution. The letters of the language are four symbols: 2, 3, π and ln2. Of the five axioms — finiteness, the planckon, the event counter, the binary ledger, the Main Law and period-2 identification — only 2 and ln2 descend as primary letters; 3 is the space dimension itself, π is the obligatory half-turn count of the period-2 structure, and e is not a letter but the successive-event product limit. A closed operation list (product–power, constructive sum, the Lambert W closure, the e^ (−π) gate share, the E₁ distribution, the (1−w) Jacobian writing and log₂ address reading) together with four validity checks (descent to the axioms, construction family, uniqueness and fixation before measurement) defines the language. The dictionary divides into five classes: words, syntax rules, cursors, environment quantities and unwritables. More than thirty observed quantities — including the dark-energy share, the dark-matter and baryon shares, the photon entropy, the Koide triple, the Genesis chain and the proton mass — are written in this language with zero free parameters (e. g. ΩKE 0. 68634; at 0. 22σ from Planck's 0. 6847 ± 0. 0073) ; from the universe's present temperature, with a single thermometer reading, the proton and three lepton masses are computed. Electric charges derive from pure counting with no measurement input; hydrogen neutrality is an identity. The unwritables — the prime 37, the cursor kH and the neutrino phase — establish the system's falsifiability; the Boundary Theorem shows that the language cannot derive its own cursor. The existence of the dark classes is tied to a three-channel existence equation together with four laboratory absence predictions. The structural verdict for the fine-structure constant (α⁻¹ = N/q) and the pre-announced departure predictions — −3. 3×10⁻⁷ in the photon entropy, the m_τ band, the w (z) profile and the phantom ban — take the paper beyond hypothesis and bind it to decision experiments.