Inverse Structural Reconstruction of Ramanujan's 1/π Series

Ramanujan's seventeen series for 1/π have a mature direct theory: their hypergeometric forms, modular transformations, complex-multiplication evaluations, and broad Ramanujan--Sato generalizations are classical or now systematically classified. We study a different, inverse question. If one is given exact coefficient data from a corpus of such series, can the upstream differential structure be recovered, and can an entire omitted subfamily be reconstructed from only a short prefix of each withheld series? We study the one-parameter Ramanujan term family tₙ=B(xn+1)cₙ(λ)zⁿ, cₙ₊₁(λ)cₙ(λ) =(n+12)[(n+12)²-λ]/(n+1)³, which is equivalent to the standard ₃F₂ representation after λ=δ² and $x=A/B$. We prove an exact finite-prefix identifiability theorem: under explicit nonvanishing assumptions, the first four terms, assumed nonzero, determine at most three admissible parameter tuples (x,λ,z,A,B) through one explicit cubic equation; if exactly one candidate satisfies the stated admissibility conditions, then the entire infinite term sequence is uniquely determined. In the holdout experiment the same one-parameter cubic family is independently recovered from the training coefficient data; the withheld parameter value is not supplied to that reconstruction. We then perform a whole-fiber holdout experiment on Ramanujan's 1914 corpus. Equations (31) and (32), comprising the complete λ=1/36 fiber, are removed before the upstream family is reconstructed. The remaining fifteen formulas yield three distinct observation-free cubic kernels whose coefficient vectors lie on a single affine line Pᵤ(n)=n³+32n²+un+ u2-14. For each withheld formula, four terms suffice to recover a unique real root and hence the exact hidden values λ=1/36, $(A,B,z)=(15,2,2/27)$ and $(33,4,4/125)$, respectively. The recovered recurrence determines every subsequent coefficient exactly. The modular and CM evaluations on the right-hand sides are kept logically separate. Existing modular theory supplies independent evaluation certificates after the series-side reconstruction; we make no novelty claim for those evaluations or for the hypergeometric classification itself. To our knowledge, the corpus-level whole-fiber holdout problem and the explicit four-term identifiability result below have not previously been formulated for Ramanujan's seventeen 1/π series.