We test whether simple low-dimensional integer relations occur among coordinatesderived from the empirical Khinchin-signature statistic Sₙ, evaluated on thecontinued-fraction partial quotients of π. Using agreement across 256-, 512-, and1024-bit runs, we isolate the maximal certified segment n=1, …, 91 and perform PSLQsearches on six fixed 3D/4D templates at 300-digit working precision with 600-digitconfirmation. Under the conservative bound maxcoeff=10³ and residual threshold10^-20, all 546 template-specific attempts are unsupported. A wide-cap diagnosticrerun with maxcoeff=10⁶ yields only high-coefficient fits, concentrated almostentirely in the 4-dimensional template, and these disappear completely when theconservative cap is restored. The outcome is therefore a reproducible, precision-controlled null result for the tested coordinate family rather thanevidence of hidden arithmetic dependence. Open directions include coordinatefamilies involving nonlinear functions of the index, algebraic combinations withadditional constants, and longer certified prefixes obtainable only through stillhigher-precision stabilization. A replication on e produces a consistent nulloutcome, supporting the generality of the methodology. Companion archival deposit for a manuscript submitted to a peer-reviewed venue; the deposited file is the anonymized form per the venue's blinding policy. For citation use the concept DOI 10. 5281/zenodo. 20173677, which always resolves to the latest version; see related-identifier entries for the SIARC-umbrella isPartOf links and the PCF-1 / PCF-2 citation graph.
papanokechi papanokechi (Thu,) studied this question.