A Fractional Interpretation of Modular Encodings via Continued Fractions

Modular encodings are traditionally interpreted through exact integer reconstruction, most commonly by the Chinese Remainder Theorem. In applications that require only positional, comparative, or structural information, however, complete reconstruction may provide more information than is immediately needed. This paper develops a fractional interpretation framework in which a residue number system (RNS) encoding is mapped to a normalized fractional image and analyzed through its continued-fraction structure. The normalized fractional images of integers in the RNS dynamic range form a finite lattice with uniform spacing equal to the reciprocal of that range. This spacing provides a natural observation scale for evaluating continued-fraction convergents. By combining classical rational-approximation properties with certified finite-precision error bounds and the intrinsic RNS resolution, the framework identifies rational approximants that are sufficient for localization or exact lifting without requiring the complete continued-fraction expansion. The selected approximants define phase-consistent rational shadows on the unwrapped real line. In the phase-stable branch, dual certification yields one-unit localization or exact nearest-integer lifting. Directed truncation may instead move a small fractional image across the lower phase boundary and make it reappear near one. An auxiliary exact-rank invariant derived from a minimally redundant modular code detects this lost integer transition. In the resulting phase-crossing branch, the directed-recovery anchor is invariant with respect to a selected continued-fraction convergent satisfying the explicit Legendre certificate (hereafter referred to as a structurally certified convergent), while the resolving power is governed by a certified enclosure of the discarded truncation tail. A fixed-point enclosure is constructed directly from local truncation remainders by combining a Cauchy–Schwarz lower bound with dyadic tabular upper majorants, without exact tail evaluation or a second CRT reconstruction. We establish continued-fraction certification conditions, localization and exact-lifting guarantees, exact phase-transition detection, phase-consistent directed recovery, fixed-point tail-enclosure bounds, and perturbation bounds for finite-precision observation and residual evaluation. Detailed numerical examples demonstrate exact-rank recovery, certified early stopping, stability under directed binary truncation, and exact recovery after a detected phase crossing. The resulting framework complements full CRT reconstruction by extracting certified intermediate information from modular encodings and by determining, for a specified output level, whether the available observation and error certificates suffice before unconditional reconstruction becomes necessary.