FROM THE CONSTANT TO THE ASTRONOMICAL SCALE: HIERARCHICAL ITERATION OF π BY EXPONENTIAL TRANSFORMATION

This work presents a mathematical and computational formulation based on the hierarchical transformation associated with the constant π, combining high-precision calculation, exponential transformation, iterative mapping, and a geometric interpretation associated with spiral structures. The investigation begins with the Chudnovsky formula for the high-precision calculation of π, distinguishing the number of decimal digits, the number of series terms, and the computational operations required. A finite approximation is represented by the chain N → K → C_K → π_N, where N denotes the desired decimal precision, K the number of terms, C_K the truncated Chudnovsky sum, and π_N the resulting approximation. A hierarchical reparameterization is then introduced through β = exp(π/√3) and θ = π/√3, leading to the central identity π = √3 ln(β) = √3 θ = 4270934400 / (√10005 C). From this relation, the exponential transformation F(x) = exp(x/√3) is defined and iterated according to X_0 = π and X_(n+1) = F(X_n). The computational experiment was implemented in Magma using 300 decimal digits of internal precision. The resulting sequence is X_0 = π, X_1 = exp(π/√3) = 6.1337074062362275595..., X_2 = 34.5116732582923986865..., X_3 = 450254639.3274947737116947799..., while the fourth iteration reaches the decimal scale X_4 ≈ 3.0877134664844535314... × 10^112896864, corresponding to 112896865 decimal digits. The recurrence was computationally verified for the four successive transformations, with zero numerical difference at the internal precision employed. The relation X_1 = β was likewise verified. The formulation also establishes a correspondence between the exponential parameter β, the angular parameter θ, and the spiral relation r = a exp(θ), connecting the numerical hierarchy to the geometric representation. The complete computational structure is summarized by N → K → C_K → π_N → β_N → θ_N → X_2 → X_3 → X_4. The work therefore documents a hierarchical numerical transformation in which the calculation of π constitutes the initial stage and successive applications of the exponential operator generate rapidly increasing numerical scales, without altering the constant value of π. KEYWORDS: π; CHUDNOVSKY FORMULA; EXPONENTIAL TRANSFORMATION; HIERARCHICAL ITERATION; β; θ; LATENCY; TRANSDUCTION; SPIRAL GEOMETRY; MULTIPLE PRECISION; MAGMA; NUMERICAL CALCULATION; DECIMAL SCALE