A Fractional Interpretation of Modular Encodings via Continued Fractions

Abstract

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.

1. Introduction

Residue number systems (RNSs) represent an integer by a vector of residues with respect to pairwise coprime moduli. Their carry-free and channel-parallel arithmetic makes them attractive in coding theory, cryptography, computer algebra, and large-scale exact computation. The classical foundations of RNS arithmetic and its architectural development are well established [1,2,3,4,5,6,7,8,9,10]. However, comparison, ordering, scaling, division, sign determination, and reverse conversion require global information that is not directly available from the individual residue channels. Recent work on rank evaluation, reverse conversion, scaling, and related nonmodular operations has continued to address this difficulty [11,12,13,14,15,16].

The standard solution is exact reconstruction, most commonly through the Chinese Remainder Theorem (CRT). Yet many computational tasks do not initially require the complete integer. A comparison, range decision, approximate scaling step, or early algorithmic branch may need only reliable positional or structural information within the dynamic range. This observation has motivated approximate and floating-output CRT methods, as well as robust CRT constructions for imperfect residue data [17,18,19,20,21,22,23,24,25,26]. Such approaches show that a modular encoding may be mapped to a numerical image and exploited before full reconstruction. The central question of the present paper is whether that image can be treated not merely as a terminal approximation, but as a structured mathematical object admitting certified rational interpretation.

A normalized CRT identity provides the natural starting point. After absorbing the modular correction factors into normalized residues, division by the dynamic range M yields an aggregate whose fractional component is

$$\beta \left(X\right)={\displaystyle \frac{X}{M}}.$$

The admissible fractional images form the finite lattice

$$\left\{0,{\displaystyle \frac{1}{M}},\dots ,{\displaystyle \frac{M-1}{M}}\right\},$$

with intrinsic spacing $1/M$. Thus, the RNS itself supplies a natural observation scale at which numerical and rational interpretations should be assessed.

We use continued fractions to reveal the rational structure of this fractional image. Their approximation and recognition properties are classical [27,28], and continued-fraction techniques also underlie classical rational reconstruction from modular images [29,30,31]. The novelty here is not a new continued-fraction theorem. It is the integration of classical rational approximation with the intrinsic RNS spacing and certified finite-precision error budgets. This resolution-aware viewpoint identifies rational approximants that are already sufficient for localization or exact lifting, without requiring the deepest convergent or exact reproduction of the observed fraction.

For a selected rational fraction $p/q$, the framework constructs a phase-consistent rational shadow on the unwrapped real line and, when permitted by the available certificate, a corresponding lifted rational representative in the integer domain. These objects provide complementary levels of interpretation: the shadow preserves the rational and directional structure of the fractional observation, whereas the representative supplies an integer-domain localization or exact value. The resulting output may therefore range from structured rational information to one-unit localization or exact recovery.

Finite-precision evaluation introduces an additional boundary difficulty. Under directed truncation, a sufficiently small fractional image may lose one integer transition and reappear near the upper boundary of $[0,1)$. The principal fractional observation alone then cannot distinguish a genuinely large value from a small value that has crossed the lower phase boundary. To preserve the validity of the fractional interpretation in this critical region, the framework uses an auxiliary exact-rank invariant obtained by extending the primary RNS with the redundant modulus 2. This invariant detects the lost transition and determines whether the selected rational approximation belongs to the phase-stable branch or must be interpreted through phase-corrected directed recovery.

The exact-rank mechanism is therefore not the primary object of the paper, but a supporting device that makes the fractional interpretation phase consistent under directed truncation. In the phase-stable branch, continued-fraction certification yields one-unit localization or exact nearest-integer lifting. After a detected crossing, the sum of a phase-consistent rational shadow and its signed residual reduces to the invariant anchor $\tilde{\beta }\left(X\right)-c\left(X\right)$; consequently, the resolving power of the directed recovery interval is governed by the certified enclosure of the discarded truncation tail rather than by deeper continued-fraction expansion. The framework may use a supplied two-sided tail enclosure or construct a fixed-point enclosure directly from local truncation remainders. A singleton admissible set gives exact recovery, whereas a wider certified interval returns the complete finite set of admissible integers.

The main contributions are as follows.

• We formulate a fractional interpretation of modular encodings based on the exact relation $\beta \left(X\right)=X/M$ and identify the intrinsic lattice spacing $1/M$ as the natural certification scale.
• We integrate continued-fraction recognition with certified observation-error budgets and the intrinsic RNS resolution, yielding rational approximants whose sufficiency is certified for a specified output level without requiring exact reconstruction of the observed fraction.
• We introduce phase-consistent rational shadows on the unwrapped real line and distinguish them from lifted rational representatives in the integer domain. For phase-stable observations, we establish sufficient conditions for one-unit localization and exact nearest-integer lifting, together with an explicit perturbation bound.
• We incorporate an auxiliary exact-rank invariant, obtained by extending the primary RNS with the redundant modulus 2, to detect the loss of one integer transition under directed truncation and thereby preserve phase consistency of the fractional interpretation.
• For detected phase crossings, we prove that the directed-recovery anchor is invariant with respect to the selected structurally certified convergent. The continued-fraction stage may therefore stop at the first such convergent, while the resolving power is transferred to the certified truncation-tail enclosure.
• We construct a fixed-point tail enclosure directly from local truncation remainders. A Cauchy–Schwarz lower bound is combined with an upper bound obtained from a dominant-loss anchor and dyadic tabular majorants of common precision $\lceil {\log }_2{m}_{\max }\rceil$.
• We provide detailed numerical examples covering independent exact-rank recovery, exact and finite-precision fractional interpretation in the phase-stable branch, certified early stopping, and exact recovery after a detected phase crossing without exact tail evaluation or a second CRT reconstruction.

The remainder of the paper is organized as follows. Section 2 presents the computational motivation and complexity considerations. Section 3 formulates the problem setting. Section 4 develops the normalized proper-fraction representation and its phase-consistency and directed-recovery mechanisms. Section 5 establishes continued-fraction certification, phase-stable lifting, and perturbation bounds. Section 6 gives numerical examples for both phase branches. Section 7 discusses related methods, recovery boundaries, implementation implications, and limitations. Section 8 concludes the paper.

2. Computational Motivation and Complexity Considerations

Exact Chinese Remainder Theorem (CRT) reconstruction operates at the scale of the full dynamic range and therefore involves large-integer multiplication, modular reduction, and carry-propagating arithmetic whose cost grows with the total bit length of the represented integer [32,33]. This cost is unavoidable when the complete reconstructed value is required. Many intermediate operations, however, need only partial global information, such as comparison, ordering, range classification, approximate scaling, verification, or an early algorithmic decision based on the position of the encoded integer within the RNS range.

Approximate and floating-output CRT methods address this distinction by mapping a residue vector to a bounded numerical image that can support comparisons, range checks, or early termination without immediate full reconstruction [17,18,19,20,21,24]. The present framework follows the same general computational motivation but assigns the fractional image a further interpretive role. Rather than treating it as a terminal approximation, we analyze its continued-fraction structure relative to the intrinsic RNS spacing $1/M$ and certified finite-precision error bounds.

This transition can replace unconditional full-range reconstruction at selected decision point by with adaptive sequence of bounded proper-fraction accumulation and rational certification. Continued-fraction convergents are generated successively and need be evaluated only until the structural recognition condition and the task-specific resolution condition are met [27,28]. In the phase-stable branch, the computation stops when a convergent supplies the requested rational shadow, one-unit localization, or exact lifted representative. In the phase-crossing branch, it stops at the first structurally certified convergent because the sum of its phase-consistent shadow and signed residual is the invariant anchor $\tilde{\beta }\left(X\right)-c\left(X\right)$; further resolving power is then supplied by the certified truncation-tail enclosure.

Finite-precision evaluation requires additional care near the boundary of the principal fractional interval. Under directed truncation, a sufficiently small fractional image may lose one integer transition and reappear near one. In this critical case, the fractional observation alone does not determine the correct interpretation branch. The framework therefore supplements the fractional computation with an auxiliary exact-rank invariant derived from modular overflow information and one redundant parity residue [11,12]. This rank branch can be evaluated in parallel with the proper-fraction accumulation and is used only to preserve phase consistency: it determines whether ordinary phase-stable lifting remains valid or whether the observation must enter the directed recovery procedure.

After a detected crossing, the discarded tail may be enclosed directly from the local truncation remainders. The proposed fixed-point construction uses short accumulations, one Cauchy–Schwarz lower bound, a single anchor channel associated with the largest relative local loss, and dyadic tabular majorants whose common precision is $\lceil {\log }_2{m}_{\max }\rceil$. For a fixed anchor, the construction requires a linear number of local operations on bounded-width quantities and does not form the full positional result. Thus the auxiliary rank and tail-enclosure mechanisms protect and complete the fractional interpretation in the exceptional boundary regime rather than replacing it as the primary computational pathway.

Any computational benefit is therefore task dependent. The framework does not claim to dominate optimized CRT reconstruction in every setting, nor does it eliminate full reconstruction when the complete integer is unconditionally required. Binary accumulation of directedly truncated proper fractions is performed by additions and shifts on prescribed fixed-point operands, while direct CRT reconstruction generally requires modular products, reductions, and arithmetic at the scale of the full dynamic range. This distinction can reduce the cost of forming and interpreting the global quantity in the stated arithmetic model, but it does not reduce the input precision required for certified recovery. Full CRT reconstruction may instead be deferred or avoided when resolution-aware rational interpretation, certified early recovery, or candidate reduction already supplies the required information.

For fixed moduli, the normalized residue contributions, auxiliary rank and parity data, truncation remainders, and dyadic majorants are drawn from finite sets and are compatible with lookup-table and fixed-point implementations. Continued-fraction evaluation is adaptive, and both phase branches admit explicit stopping conditions. These properties are suitable for parallel and hardware-oriented realizations, although the practical advantage depends on the modulus set, target precision, available parallelism, table organization, and the proportion of operations that can exploit partial or early-certified information.

3. Problem Setting

3.1. Modular Data and Fractional Interpretation Model

Let

$$X\in \{0,\dots ,M-1\}$$

(1)

be represented by its residues with respect to the pairwise coprime odd primary moduli

$$m_1,\dots ,m_k,M:={\displaystyle \prod _{i=1}^k}{m}_i,$$

(2)

where M is the primary RNS dynamic range [1,4]. Throughout the paper, the primary moduli are indexed in strictly increasing order and satisfy

$$m_1<m_2<\cdots <m_k,m_{\ell }>\ell -1,\ell =1,\dots ,k.$$

In particular, the final modulus is the largest primary modulus and obeys $m_k\ge k$. This ordering is used by the exact-rank recovery construction in Section 4.3.

The primary residue vector is

$$r_i:=X\mathrm{mod}{m}_i,i=1,\dots ,k.$$

For each primary modulus, let

$$M_i:={\displaystyle \frac{M}{m_i}},{\mu }_i:=M_i^{-1}\mathrm{mod}{m}_i,$$

and define the normalized residue

$$x_i:={⟨r_i{\mu }_i⟩}_{m_i},0\le x_i<m_i,$$

(3)

where ${⟨·⟩}_{m_i}$ denotes the least nonnegative residue modulo $m_i$. These bounded local quantities form the proper-fraction aggregate

$$\alpha \left(X\right):={\displaystyle \sum _{i=1}^k}{\displaystyle \frac{x_i}{m_i}}.$$

(4)

Section 4 establishes the decomposition

$$\alpha \left(X\right)=\rho \left(X\right)+\beta \left(X\right),\beta \left(X\right)={\displaystyle \frac{X}{M}},$$

(5)

where $\beta \left(X\right)\in [0,1)$ is the exact fractional image and $\rho \left(X\right)=\lfloor \alpha \left(X\right)\rfloor$ is its associated exact normalized rank. The principal object to be interpreted is the fractional image, whose admissible values lie on the lattice with spacing $1/M$.

3.2. Finite-Precision Observation and Auxiliary Phase Control

In a finite-precision implementation, each proper fraction $x_i/m_i$ is evaluated by directed truncation. The resulting aggregate is modeled as

$$\tilde{\alpha }\left(X\right)=\alpha \left(X\right)-\tau \left(X\right),\tau \left(X\right)\ge 0,$$

(6)

with observed components

$$\tilde{\rho }\left(X\right):=⌊\tilde{\alpha }\left(X\right)⌋,\tilde{\beta }\left(X\right):=\left\{\tilde{\alpha }\left(X\right)\right\}.$$

(7)

The implementation is assumed to provide, or to permit the construction of, a certified bound

$$0\le \tau \left(X\right)\le {\tau }_{\max }<1.$$

(8)

When directed recovery is required, a two-sided enclosure

$${\tau }_{\min }\le \tau \left(X\right)\le {\tau }_{\max }$$

may be supplied directly or constructed from the local truncation remainders by the fixed-point procedure of Section 4.6. The condition $\tau \left(X\right)<1$ ensures that truncation can remove at most one integer transition from the aggregate.

The principal observation $\tilde{\beta }\left(X\right)$ does not reveal whether such a transition has been lost. In particular, a small exact fractional image may cross the lower phase boundary and reappear near the upper boundary of $[0,1)$. To preserve the phase consistency of the fractional interpretation in this critical case, we assume that the input code is the minimally redundant RNS obtained by augmenting the primary residue vector with the additional residue channel

$$r_0:=X\mathrm{mod}2.$$

(9)

Thus, $r_0$ is supplied as part of the encoded input and is not inferred from the primary residue vector. Because the primary moduli are odd, the redundant modulus 2 is coprime to every $m_i$. It does not enlarge the source range (1); it supplies only the auxiliary one-bit invariant used to recover the exact normalized rank [11,12].

Three rank-related quantities must be distinguished. The exact normalized rank is

$$\rho \left(X\right):=⌊\alpha \left(X\right)⌋.$$

(10)

The modular rank algorithm of Section 4.3 first produces the inexact rank ${\hat{\rho }}_k\left(X\right)$. Since the set of k primary moduli is fixed throughout the paper, we subsequently write

$$\hat{\rho }\left(X\right):={\hat{\rho }}_k\left(X\right).$$

Its binary correction is

$$\delta \left(X\right):=\rho \left(X\right)-\hat{\rho }\left(X\right)\in \{0,1\}.$$

By contrast, $\tilde{\rho }\left(X\right)$ in (7) is the integer part of the truncated proper-fraction aggregate, not an output of the modular rank algorithm. Accordingly,

$$\delta \left(X\right)=\rho \left(X\right)-\hat{\rho }\left(X\right),c\left(X\right):=\rho \left(X\right)-\tilde{\rho }\left(X\right),$$

(11)

compare the exact rank with two different quantities and need not coincide. The bit $\delta \left(X\right)$ corrects the inexact modular rank, whereas $c\left(X\right)\in \{0,1\}$ records whether directed truncation has removed one integer transition. Thus, exact-rank recovery is used here only as an auxiliary phase-control mechanism for the fractional interpretation.

Primary residue arithmetic, modular inverses, and the parity residue are assumed exact. Finite precision enters through the directed evaluation of the proper-fraction contributions and, when used, the dyadic tabular majorants for the truncation-tail enclosure. If the signed rational residual used later is rounded, quantized, or evaluated independently, its residual-evaluation error is assumed to have a certified enclosure and is incorporated into the recovery interval of Section 4.6.

3.3. Required Certified Outputs

Given the modular encoding, its finite-precision fractional observation, and the applicable uncertainty bounds, the objective is to extract the strongest certified rational interpretation supported by the available information without unconditional full multiprecision CRT reconstruction. Specifically, the framework is required to:

• identify a rational fraction whose continued-fraction structure and relevance at the intrinsic scale $1/M$ are certified;
• construct the corresponding phase-consistent rational shadow;
• use the auxiliary exact-rank invariant to determine whether the observation is phase-stable or has crossed the lower phase boundary; and
• return, according to the phase and the available error certificates, one of the following:

  (a)

  a lifted rational representative with a certified one-unit localization bound;

  (b)

  the exact integer X obtained by phase-stable lifting or directed recovery; or

  (c)

  the complete finite set of admissible integer candidates when uniqueness is not certified.

Section 4 and Section 5 establish these outputs in two stages. Section 4 derives the normalized proper-fraction representation, exact phase-transition test, directed-recovery intervals, and fixed-point tail enclosure. Section 5 supplies continued-fraction certification, branch-specific stopping conditions, phase-stable lifting, and perturbation bounds.

4. Structural Transition from Modular Encoding to Fractional Representation

Section 3 introduced the normalized residues, proper-fraction aggregate, and finite-precision observation model. We now derive the fractional representation and the auxiliary phase information required for its correct interpretation. Continued-fraction selection and certification are deferred to Section 5.

4.1. Normalization of the Modular Encoding

With the notation of (2)–(3), the CRT representation can be written as

$$X={⟨{\displaystyle \sum _{i=1}^k}{x}_i{M}_i⟩}_M.$$

Hence, for a unique integer $\rho \left(X\right)$,

$${\displaystyle \sum _{i=1}^k}{x}_i{M}_i=X+\rho \left(X\right)M.$$

Since $M_i/M=1/m_i$, division by M gives

$${\displaystyle \sum _{i=1}^k}{\displaystyle \frac{x_i}{m_i}}=\rho \left(X\right)+{\displaystyle \frac{X}{M}}.$$

Thus the normalized CRT identity is expressed entirely through bounded channel-wise contributions. Its fractional component can therefore be evaluated without first forming the full positional CRT reconstruction.

4.2. Proper-Fraction Aggregation and the Fractional Image

Using the aggregate $\alpha \left(X\right)$ defined in (4), the preceding identity yields the decomposition already stated in (5):

$$\alpha \left(X\right)=\rho \left(X\right)+\beta \left(X\right),\beta \left(X\right)={\displaystyle \frac{X}{M}}.$$

Because $0\le X<M$, the principal fractional image satisfies $0\le \beta \left(X\right)<1$, and 

$$\rho \left(X\right)=\lfloor \alpha \left(X\right)\rfloor ,\beta \left(X\right)=\left\{\alpha \right(X\left)\right\}.$$

Its admissible values form the lattice

$$\left\{0,{\displaystyle \frac{1}{M}},\dots ,{\displaystyle \frac{M-1}{M}}\right\},$$

whose spacing $1/M$ is the intrinsic RNS resolution. The fractional image is the principal object interpreted below; the integer component $\rho \left(X\right)$, although intrinsic to the composition, is subsequently used as an auxiliary phase invariant.

4.3. Auxiliary Exact-Rank Recovery for Phase Control

The finite-precision integer part $\tilde{\rho }\left(X\right)$ in (7) need not equal the exact rank $\rho \left(X\right)$ in (10). To make the latter independently available, we extend the primary RNS by the redundant modulus 2, as in (9), and use the rank-recovery method of [11,12]. The binary correction $\delta \left(X\right)$ remains distinct from the phase indicator $c\left(X\right)$, as specified in (11).

Theorem 1 

(Exact rank recovery in a minimally redundant RNS). Assume the ordered primary moduli of Section 3.1, so that $m_k\ge k$, and let

$$M_{k-1}:={\displaystyle \prod _{i=1}^{k-1}}{m}_i.$$

For the largest primary modulus $m_k$, define the lookup-table values

$$R_{i,k}\left(r_i\right)=⌊{\displaystyle \frac{m_k{x}_i}{m_i}}⌋,i=1,2,\dots ,k,$$

where, in particular,

$$R_{k,k}\left(r_k\right)=x_k=\left(M_{k-1}^{-1}{r}_k\right)\mathrm{mod}{m}_k.$$

Define the inexact rank and the associated parity bits by

$$\hat{\rho }\left(X\right):=⌊{\displaystyle \frac{1}{m_k}}{\displaystyle \sum _{i=1}^k}{R}_{i,k}\left(r_i\right)⌋,{\hat{\rho }}^{\left(0\right)}\left(X\right):=\hat{\rho }\left(X\right)\mathrm{mod}2,$$

$$x_i^{\left(0\right)}:=x_i\mathrm{mod}2,i=1,\dots ,k.$$

The binary correction

$$\delta \left(X\right)=\left(r_0+{\displaystyle \sum _{i=1}^k}{x}_i^{\left(0\right)}+{\hat{\rho }}^{\left(0\right)}\left(X\right)\right)\mathrm{mod}2\in \{0,1\}$$

recovers the exact normalized rank:

$$\rho \left(X\right)=\hat{\rho }\left(X\right)+\delta \left(X\right)=⌊\alpha \left(X\right)⌋.$$

Proof. 

Set

$${\eta }_i:=\left\{{\displaystyle \frac{m_k{x}_i}{m_i}}\right\},0\le {\eta }_i<1.$$

Since

$$R_{i,k}\left(r_i\right)={\displaystyle \frac{m_k{x}_i}{m_i}}-{\eta }_i,$$

we obtain

$$\hat{\rho }\left(X\right)=⌊\alpha \left(X\right)-{\displaystyle \frac{1}{m_k}}{\displaystyle \sum _{i=1}^k}{\eta }_i⌋=\rho \left(X\right)+⌊\beta \left(X\right)-{\displaystyle \frac{1}{m_k}}{\displaystyle \sum _{i=1}^k}{\eta }_i⌋.$$

The ordering assumption gives

$$0\le {\displaystyle \sum _{i=1}^k}{\eta }_i<k\le m_k.$$

Together with $0\le \beta \left(X\right)<1$, this implies

$$-1<\beta \left(X\right)-{\displaystyle \frac{1}{m_k}}{\displaystyle \sum _{i=1}^k}{\eta }_i<1.$$

Hence the last floor is either $-1$ or 0, and therefore

$$\rho \left(X\right)=\hat{\rho }\left(X\right)+\delta \left(X\right),\delta \left(X\right)\in \{0,1\}.$$

This is the one-bit rank correction used in [11,12].

The exact CRT identity underlying the normalized aggregate is

$$X={\displaystyle \sum _{i=1}^k}{M}_i{x}_i-\rho \left(X\right)M,M_i={\displaystyle \frac{M}{m_i}}.$$

Since all primary moduli are odd, both M and every $M_i$ are odd. Reduction modulo 2 therefore gives

$$r_0\equiv {\displaystyle \sum _{i=1}^k}{x}_i^{\left(0\right)}+\rho \left(X\right)\left(\mathrm{mod}2\right),$$

where subtraction and addition are identical modulo 2. Substituting $\rho \left(X\right)=\hat{\rho }\left(X\right)+\delta \left(X\right)$ yields

$$\delta \left(X\right)=\left(r_0+{\displaystyle \sum _{i=1}^k}{x}_i^{\left(0\right)}+{\hat{\rho }}^{\left(0\right)}\left(X\right)\right)\mathrm{mod}2.$$

Thus the parity relation uniquely determines the missing binary correction. Finally, $\alpha \left(X\right)=\rho \left(X\right)+X/M$ with $0\le X/M<1$, and therefore $\rho \left(X\right)=\lfloor \alpha \left(X\right)\rfloor$.    □

Remark 1 

(Parallel evaluation). For each $r_i$, the table value $R_{i,k}\left(r_i\right)$ and parity bit $x_i^{\left(0\right)}$ can be generated simultaneously. The accumulations for $\hat{\rho }\left(X\right)$ and $\delta \left(X\right)$ may therefore proceed in parallel with the truncated proper-fraction accumulation.

The recovered value $\rho \left(X\right)$ will be compared with $\tilde{\rho }\left(X\right)$ to detect whether directed truncation has removed one integer transition.

4.4. Truncated Aggregation and Phase-Transition Detection

The observation model (6)–(8) assumes directed truncation and a certified total truncation tail $0\le \tau \left(X\right)<1$. For binary channel precisions $L_i$, it follows from

$${\tilde{t}}_i:=2^{-L_i}⌊2^{L_i}{\displaystyle \frac{x_i}{m_i}}⌋,0\le {\displaystyle \frac{x_i}{m_i}}-{\tilde{t}}_i<2^{-L_i},$$

so that

$$\tilde{\alpha }\left(X\right)={\displaystyle \sum _{i=1}^k}{\tilde{t}}_i,\tau \left(X\right)={\displaystyle \sum _{i=1}^k}\left({\displaystyle \frac{x_i}{m_i}}-{\tilde{t}}_i\right),$$

and therefore

$$0\le \tau \left(X\right)<{\displaystyle \sum _{i=1}^k}{2}^{-L_i}.$$

For a common precision L, the sufficient condition $L\ge \lceil {\log }_{2}k\rceil$ guarantees $\tau \left(X\right)<1$. With $\tilde{\rho }\left(X\right)$ and $\tilde{\beta }\left(X\right)$ defined in (7), set

$$c\left(X\right):=\rho \left(X\right)-\tilde{\rho }\left(X\right).$$

Theorem 2 

(Exact detection of a lost integer transition). Let

$$\alpha \left(X\right)=\rho \left(X\right)+\beta \left(X\right),\beta \left(X\right)={\displaystyle \frac{X}{M}}\in [0,1),$$

and let

$$\tilde{\alpha }\left(X\right)=\alpha \left(X\right)-\tau \left(X\right),0\le \tau \left(X\right)<1.$$

Then

$$c\left(X\right)=\rho \left(X\right)-⌊\tilde{\alpha }\left(X\right)⌋\in \{0,1\},$$

and, more precisely,

$$c\left(X\right)=\left\{\begin{array}{cc}0,\hfill & \tau \left(X\right)\le \beta \left(X\right),\hfill \\ 1,\hfill & \tau \left(X\right)>\beta \left(X\right).\hfill \end{array}\right.$$

Moreover, the observed fractional part satisfies the exact identity

$$\tilde{\beta }\left(X\right)=\beta \left(X\right)-\tau \left(X\right)+c\left(X\right).$$

Equivalently,

$$\tilde{\beta }\left(X\right)-c\left(X\right)=\beta \left(X\right)-\tau \left(X\right).$$

Thus, $c\left(X\right)=1$ detects exactly one integer transition lost during the truncated accumulation.

Proof. 

Using $\alpha \left(X\right)=\rho \left(X\right)+\beta \left(X\right)$, we obtain $\tilde{\alpha }\left(X\right)=\rho \left(X\right)+\beta \left(X\right)-\tau \left(X\right).$ Because $0\le \beta \left(X\right)<1$ and $0\le \tau \left(X\right)<1$, we have $-1<\beta \left(X\right)-\tau \left(X\right)<1.$ Hence,

$$⌊\beta \left(X\right)-\tau \left(X\right)⌋\in \{-1,0\},$$

and therefore $\tilde{\rho }\left(X\right)=\rho \left(X\right)+⌊\beta \left(X\right)-\tau \left(X\right)⌋$. It follows that

$$c\left(X\right)=-⌊\beta \left(X\right)-\tau \left(X\right)⌋\in \{0,1\}.$$

If $\tau \left(X\right)<\beta \left(X\right)$, then $\beta \left(X\right)-\tau \left(X\right)\in (0,1),$ so its floor is zero, $c\left(X\right)=0$, and

$$\tilde{\beta }\left(X\right)=\beta \left(X\right)-\tau \left(X\right)\in (0,1).$$

At the boundary $\tau \left(X\right)=\beta \left(X\right)$, one has

$$\tilde{\alpha }\left(X\right)=\rho \left(X\right),\tilde{\rho }\left(X\right)=\rho \left(X\right),\tilde{\beta }\left(X\right)=0,c\left(X\right)=0.$$

Thus equality with the boundary does not represent a lost integer transition. If $\tau \left(X\right)>\beta \left(X\right)$, then

$$\beta \left(X\right)-\tau \left(X\right)\in (-1,0),$$

so its floor is $-1$, $c\left(X\right)=1$, and 

$$\tilde{\beta }\left(X\right)=1+\beta \left(X\right)-\tau \left(X\right)\in (0,1).$$

Therefore,

$$c\left(X\right)=1⟺\tau \left(X\right)>\beta \left(X\right),$$

whereas the boundary case $\tau \left(X\right)=\beta \left(X\right)$ belongs to the phase-stable branch.

Finally,

$$\begin{array}{cc}\hfill \tilde{\beta }\left(X\right)& =\tilde{\alpha }\left(X\right)-\tilde{\rho }\left(X\right)\hfill \\ \hfill & =\rho \left(X\right)+\beta \left(X\right)-\tau \left(X\right)-\left(\rho \left(X\right)-c\left(X\right)\right)\hfill \\ \hfill & =\beta \left(X\right)-\tau \left(X\right)+c\left(X\right),\hfill \end{array}$$

which proves both stated identities.    □

Corollary 1 

(Zero-rank phase safety). If $\rho \left(X\right)=0$, then $c\left(X\right)=0$. Hence, directed truncation cannot produce a phase crossing when the exact normalized aggregate has zero rank.

Proof. 

If $\rho \left(X\right)=0$, then

$$\alpha \left(X\right)=\beta \left(X\right).$$

Since every truncated term is nonnegative,

$$0\le \tau \left(X\right)\le \alpha \left(X\right)=\beta \left(X\right).$$

Theorem 2 therefore gives $c\left(X\right)=0$. Equivalently,

$$\tilde{\alpha }\left(X\right)=\beta \left(X\right)-\tau \left(X\right)\ge 0,$$

so the truncated aggregate cannot cross the lower integer boundary.    □

Remark 2 

(Characterization of the phase-critical region). Theorem 2 and Corollary 1 show that a lost integer transition is possible only when both $\rho \left(X\right)\ge 1$ and

$$\beta \left(X\right)={\displaystyle \frac{X}{M}}<\tau \left(X\right)$$

hold. If only the upper bound $\tau \left(X\right)\le {\tau }_{\max }$ is available, then every potentially phase-critical integer must satisfy

$$1\le X<M{\tau }_{\max },$$

or, equivalently,

$$1\le X\le ⌈M{\tau }_{\max }⌉-1.$$

When $M{\tau }_{\max }\le 1$, this integer set is empty. Otherwise, the lower-range bound is necessary but not sufficient: integers in this range with $\rho \left(X\right)=0$ remain phase-safe. Thus, phase criticality is controlled jointly by the small fractional image and a positive exact rank, not by the magnitude of X alone.

Remark 3 

(Constant residue tails for small integers). Assume that the primary moduli are arranged in increasing order and that $X<m_k$. If j is the smallest index such that $m_j>X$, then

$$r_j=r_{j+1}=\cdots =r_k=X.$$

Thus, every sufficiently small integer produces a constant tail of residues, whose length equals the number of primary moduli exceeding that integer. In particular, if $X<m_1$, then

$$r_1=r_2=\cdots =r_k=X.$$

For large ordered moduli sets, integers in the lower region identified in Remark 2 may therefore exhibit long constant residue tails. Such a tail provides a low-cost structural indicator of a small encoded integer, but it is not by itself an exact certificate. If its common value is r, the candidate $X=r$ must be verified against the complete residue vector by checking

$$r_i=r\mathrm{mod}{m}_i,i=1,2,\dots ,k.$$

If all equalities hold, uniqueness of the residue representation over $[0,M)$ implies $X=r$. This structural indicator complements, but does not replace, exact-rank recovery.

The two cases of Theorem 2 are summarized by $\tilde{\beta }\left(X\right)-c\left(X\right)=\beta \left(X\right)-\tau \left(X\right)$, which supplies the phase-corrected fractional anchor used in the next subsection.

4.5. Phase-Consistent Rational Shadow

Let $p/q$ be a rational approximation to $\tilde{\beta }\left(X\right)$, where

$$q>0,0\le {\displaystyle \frac{p}{q}}\le 1,$$

and define the signed approximation residual

$$d:=\tilde{\beta }\left(X\right)-{\displaystyle \frac{p}{q}}.$$

Its sign is retained because truncation is one-sided.

Definition 1 

(Phase-consistent rational shadow). With the phase indicator $c\left(X\right)$ from Theorem 2, define the phase-consistent rational shadow

$$s\left(X;{\displaystyle \frac{p}{q}}\right):={\displaystyle \frac{p}{q}}-c\left(X\right).$$

Its scaled form is

$$S\left(X;{\displaystyle \frac{p}{q}}\right):=Ms\left(X;{\displaystyle \frac{p}{q}}\right).$$

For a continued-fraction convergent $p_j/q_j$, we use the abbreviations

$$s_j\left(X\right):=s\left(X;{\displaystyle \frac{p_j}{q_j}}\right),S_j\left(X\right):=S\left(X;{\displaystyle \frac{p_j}{q_j}}\right).$$

Remark 4 

(Terminological hierarchy). The fractional image is the exact or observed fractional quantity induced by the modular encoding. A rational shadow $s\left(X;p/q\right)$ is a phase-consistent rational interpretation of that image on the unwrapped real line; it is not yet an integer reconstruction. Its scaled form $S\left(X;p/q\right)=Ms\left(X;p/q\right)$ provides an integer-scale anchor. In the phase-stable branch, nearest-integer lifting may convert this anchor into a lifted rational representative. After a detected phase crossing, the same shadow instead anchors a directed recovery interval, which yields either an exactly recovered value under the corresponding uniqueness certificate or the complete finite set of admissible integer candidates. Thus, rational shadow, scaled shadow, lifted representative, recovered value, and candidate set denote distinct levels of certified interpretation and are not interchangeable.

The phase correction preserves the signed rational-approximation residual:

$$\left(\tilde{\beta }\left(X\right)-c\left(X\right)\right)-s\left(X;{\displaystyle \frac{p}{q}}\right)=\tilde{\beta }\left(X\right)-{\displaystyle \frac{p}{q}}=d.$$

(12)

Combining (12) with the phase-corrected identity following Theorem 2 gives

$$X=M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+d+\tau \left(X\right)\right).$$

(13)

Thus, rational-approximation error and directed truncation tail remain algebraically separated.

The definitions yield the following cancellation:

$$s\left(X;{\displaystyle \frac{p}{q}}\right)+d=\left({\displaystyle \frac{p}{q}}-c\left(X\right)\right)+\left(\tilde{\beta }\left(X\right)-{\displaystyle \frac{p}{q}}\right)=\tilde{\beta }\left(X\right)-c\left(X\right).$$

(14)

Therefore, the choice of $p/q$ controls the rational representation of the phase-consistent shadow, but it does not alter the exact phase-corrected anchor when the signed residual d is retained exactly. The role of a convergent therefore depends on the phase branch. In the phase-stable branch, the magnitude of d remains decisive for localization and exact nearest-integer lifting. After a phase crossing has been detected, however, the shadow and residual provide only a decomposition of the invariant quantity $\tilde{\beta }\left(X\right)-c\left(X\right)$; the width and resolution of the directed recovery interval are then governed by the certified enclosure of the truncation tail. Thus, the phase-crossing branch may stop at the first structurally certified convergent and redirect the remaining certification effort to the tail enclosure.

If $c\left(X\right)=1$ and $p/q<1$, then

$$s\left(X;{\displaystyle \frac{p}{q}}\right)={\displaystyle \frac{p}{q}}-1<0.$$

This negative value does not represent a negative reconstruction of the source integer. It is an oriented representative on the unwrapped real line, recording that directed truncation has displaced the observed fractional phase through the lower boundary of the principal interval. It must not be reduced modulo 1 or modulo M, because such a reduction would erase the detected transition and recreate the false large representative.

Moreover, when a crossing is detected,

$$s\left(X;{\displaystyle \frac{p}{q}}\right)+d=\tilde{\beta }\left(X\right)-1=\beta \left(X\right)-\tau \left(X\right)<0.$$

By Remark 2, $c\left(X\right)=1$ implies $\rho \left(X\right)\ge 1$ and hence $X>0$; in particular, the zero codeword is never a phase-crossing case.

Remark 5 

(Exact and implemented residuals). Equation (14) assumes that d is formed from the same $\tilde{\beta }\left(X\right)$ and $p/q$ as the shadow. An independently quantized residual is treated in Lemma 1.

4.6. Directed Recovery After a Detected Phase Crossing

The phase-consistent shadow from Section 4.5 supplies an oriented anchor on the unwrapped real line. We now determine when a certified enclosure of the accumulated truncation tail converts this anchor into exact integer recovery. Throughout this subsection,

$$c\left(X\right)=1,$$

so a loss of one dynamic-range revolution has been detected by Theorem 2.

Writing

$$y:=\tilde{\beta }\left(X\right),$$

Theorem 2 gives, in the crossing branch,

$$y=1+\beta \left(X\right)-\tau \left(X\right),\beta \left(X\right)=y-1+\tau \left(X\right).$$

Consequently, a certified tail enclosure

$${\tau }_{\min }\le \tau \left(X\right)\le {\tau }_{\max }$$

induces the phase-corrected enclosure

$$\beta \left(X\right)\in [y-1+{\tau }_{\min },y-1+{\tau }_{\max }]\cap [0,1).$$

Multiplication by M gives exactly the directed integer-scale interval introduced below. The intersection with $[0,1)$, equivalently with $[0,M)$ after scaling, is essential near both endpoints of the dynamic range.

Assume first that the accumulated truncation tail satisfies the two-sided enclosure

$$0\le {\tau }_{\min }\le \tau \left(X\right)\le {\tau }_{\max }<1.$$

(15)

Using (13), define

$$I_{\mathrm{pc}}:=[A_{\min },B_{\max }]\cap [0,M),$$

(16)

where

$$\begin{array}{cc}\hfill A_{\min }& :=M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+d+{\tau }_{\min }\right)=M\left(\tilde{\beta }\left(X\right)-c\left(X\right)+{\tau }_{\min }\right),\hfill \\ \hfill B_{\max }& :=M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+d+{\tau }_{\max }\right)=M\left(\tilde{\beta }\left(X\right)-c\left(X\right)+{\tau }_{\max }\right).\hfill \end{array}$$

By (14), both endpoints depend on the selected rational fraction only through the invariant combination $s\left(X;p/q\right)+d=\tilde{\beta }\left(X\right)-c\left(X\right)$; hence they are unchanged when one structurally certified convergent is replaced by another and the corresponding signed residual is retained.

Then

$$A_{\min }\le X\le B_{\max },$$

(17)

with exact endpoint distances

$$X-A_{\min }=M\left(\tau \left(X\right)-{\tau }_{\min }\right),B_{\max }-X=M\left({\tau }_{\max }-\tau \left(X\right)\right).$$

(18)

Thus the excess of the right endpoint over X is precisely the unused part of the upper truncation allowance.

Theorem 3 

(Exact right-endpoint recovery after a detected phase crossing). Let $X\in \{0,\dots ,M-1\}$ and let the exact-rank test give $c\left(X\right)=1$. If (15) holds and

$${\tau }_{\max }-{\tau }_{\min }<{\displaystyle \frac{1}{M}},$$

(19)

then $I_{\mathrm{pc}}$ contains exactly one admissible integer, namely X, and 

$$X=\lfloor B_{\max }\rfloor =⌊M\left(\tilde{\beta }\left(X\right)-c\left(X\right)+{\tau }_{\max }\right)⌋.$$

(20)

Proof. 

The unreduced interval has width

$$B_{\max }-A_{\min }=M\left({\tau }_{\max }-{\tau }_{\min }\right)<1.$$

By (17), it contains the integer X and therefore cannot contain another integer. Intersecting with $[0,M)$ preserves X and cannot introduce an additional integer. Furthermore, (18) gives

$$0\le B_{\max }-X\le M\left({\tau }_{\max }-{\tau }_{\min }\right)<1.$$

Hence $X\le B_{\max }<X+1$, proving (20).    □

Remark 6 

(Minimal pointwise condition). For a fixed source value, the floor formula requires only

$$0\le {\tau }_{\max }-\tau \left(X\right)<{\displaystyle \frac{1}{M}}.$$

(21)

Condition (19) is stronger but operational: it certifies (21) without knowing the exact value of $\tau \left(X\right)$.

Proposition 1 

(Fixed-point enclosure of the truncation tail). Assume a common binary truncation precision L, and define the local truncation remainders

$$e_i:=2^L{x}_i\mathrm{mod}{m}_i,0\le e_i<m_i,i=1,\dots ,k.$$

Then the accumulated truncation tail is

$$\tau \left(X\right)=2^{-L}T,T:={\displaystyle \sum _{i=1}^k}{\displaystyle \frac{e_i}{m_i}}.$$

Let

$$S:={\displaystyle \sum _{i=1}^k}{e}_i,W:={\displaystyle \sum _{i=1}^k}{e}_i{m}_i.$$

In the phase-crossing branch, $S>0$ automatically: if $S=0$, then all $e_i=0$, so $\tau \left(X\right)=0$ and Theorem 2 gives $c\left(X\right)=0$. Hence

$$T\ge T_{\min }:={\displaystyle \frac{S^2}{W}}.$$

Choose an anchor channel $\ell \in \{1,\dots ,k\}$. For each i, perform the local Euclidean division

$$m_{\ell }{e}_i=q_i^{(\ell )}{m}_i+s_i^{(\ell )},0\le s_i^{(\ell )}<m_i,$$

and set

$$Q_{\ell }:={\displaystyle \sum _{i=1}^k}{q}_i^{(\ell )}.$$

Fix the common tabular precision

$$h:=⌈{\log }_2{m}_{\max }⌉,m_{\max }:=\underset{i}{\max }{m}_i,$$

and define the directed dyadic majorants

$$b_i^{(\ell )}:=⌈{\displaystyle \frac{2^h{s}_i^{(\ell )}}{m_i}}⌉.$$

Then

$$T\le T_{\max }^{(\ell )}:={\displaystyle \frac{1}{m_{\ell }}}\left(Q_{\ell }+2^{-h}{\displaystyle \sum _{i=1}^k}{b}_i^{(\ell )}\right).$$

Consequently,

$$2^{-L}{T}_{\min }\le \tau \left(X\right)\le 2^{-L}{T}_{\max }^{(\ell )}.$$

In the detected phase-crossing branch $c\left(X\right)=1$, with $y=\tilde{\beta }\left(X\right)$, the source integer therefore belongs to the certified interval

$$I_{\ell }^{\mathrm{tail}}:=M\left[y-1+2^{-L}{T}_{\min },y-1+2^{-L}{T}_{\max }^{(\ell )}\right]\cap [0,M).$$

and the complete admissible candidate set is

$${\mathcal{C}}_{\ell }^{\mathrm{tail}}:=I_{\ell }^{\mathrm{tail}}\cap \mathbb{Z}.$$

In particular, if 

$$\left|{\mathcal{C}}_{\ell }^{\mathrm{tail}}\right|=1,$$

then its unique element is X; otherwise, ${\mathcal{C}}_{\ell }^{\mathrm{tail}}$ is the complete certified set of integer candidates supplied by this enclosure.

Proof. 

By the definition of directed binary truncation,

$${\displaystyle \frac{x_i}{m_i}}-2^{-L}⌊2^L{\displaystyle \frac{x_i}{m_i}}⌋={\displaystyle \frac{e_i}{2^L{m}_i}}.$$

Summation over all channels gives $\tau \left(X\right)=2^{-L}T$.

Because $S>0$ in the phase-crossing branch, the Cauchy–Schwarz inequality yields

$${\left({\displaystyle \sum _{i=1}^k}{e}_i\right)}^2={\left({\displaystyle \sum _{i=1}^k}\sqrt{e_i{m}_i}\sqrt{{\displaystyle \frac{e_i}{m_i}}}\right)}^2\le \left({\displaystyle \sum _{i=1}^k}{e}_i{m}_i\right)\left({\displaystyle \sum _{i=1}^k}{\displaystyle \frac{e_i}{m_i}}\right).$$

Hence $T\ge S^2/W=T_{\min }$.

The local Euclidean divisions give

$${\displaystyle \frac{e_i}{m_i}}={\displaystyle \frac{1}{m_{\ell }}}\left(q_i^{(\ell )}+{\displaystyle \frac{s_i^{(\ell )}}{m_i}}\right).$$

Moreover, by the definition of $b_i^{(\ell )}$,

$${\displaystyle \frac{s_i^{(\ell )}}{m_i}}\le {\displaystyle \frac{b_i^{(\ell )}}{2^h}}.$$

Summing these inequalities proves

$$T\le T_{\max }^{(\ell )}.$$

Finally, when $c\left(X\right)=1$, Theorem 2 gives

$${\displaystyle \frac{X}{M}}=\beta \left(X\right)=y-1+\tau \left(X\right).$$

Substitution of the certified tail enclosure and intersection with the dynamic range $[0,M)$ give the stated interval and candidate set.    □

Remark 7 

(Anchor selection and tabular cost). The enclosure is valid for every anchor channel ℓ. A deterministic low-cost choice is

$${\ell }_{\ast }\in \underset{1\le i\le k}{\mathrm{arg\; max}}{\displaystyle \frac{e_i}{m_i}},$$

that is, a channel with the largest relative local truncation loss. The comparison can be performed without division by cross-multiplication. The choice

$$h=⌈{\log }_2{m}_{\max }⌉$$

provides a single dyadic denominator for all directed majorants. The values $b_i^{(\ell )}$ can be read from local tables, so the enclosure for a fixed anchor requires only a linear number of bounded-width local operations and short accumulations. The proposition does not assert that ${\ell }_{\ast }$ minimizes the interval width for every residue vector; it provides a reproducible anchor tied to the channel that contributes the largest relative truncation loss.

Lemma 1 

(Recovery under an inexact signed residual). Let the implemented signed residual satisfy

$$\hat{d}=d+{\epsilon }_d,{\epsilon }_{-}\le {\epsilon }_d\le {\epsilon }_{+}.$$

(22)

Under (15), define

$$I_{\mathrm{pc}}^{\left(d\right)}:=[A_d,B_d]\cap [0,M),$$

where

$$\begin{array}{cc}\hfill A_d& :=M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+\hat{d}-{\epsilon }_{+}+{\tau }_{\min }\right),\hfill \\ \hfill B_d& :=M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+\hat{d}-{\epsilon }_{-}+{\tau }_{\max }\right).\hfill \end{array}$$

If

$$({\tau }_{\max }-{\tau }_{\min })+({\epsilon }_{+}-{\epsilon }_{-})<{\displaystyle \frac{1}{M}},$$

then $I_{\mathrm{pc}}^{\left(d\right)}$ contains exactly one admissible integer, namely X, and 

$$X=\lfloor B_d\rfloor .$$

Proof. 

Equation (22) implies

$$\hat{d}-{\epsilon }_{+}\le d\le \hat{d}-{\epsilon }_{-}.$$

Combining this with (15) and (13) gives $A_d\le X\le B_d$. The interval width is

$$B_d-A_d=M\left[({\tau }_{\max }-{\tau }_{\min })+({\epsilon }_{+}-{\epsilon }_{-})\right]<1,$$

so X is its unique integer. In addition,

$$\begin{array}{cc}\hfill B_d-X& =M\left[{\epsilon }_d-{\epsilon }_{-}+{\tau }_{\max }-\tau \left(X\right)\right]\hfill \\ \hfill & \le M\left[({\epsilon }_{+}-{\epsilon }_{-})+({\tau }_{\max }-{\tau }_{\min })\right]<1.\hfill \end{array}$$

Since $X\le B_d$, it follows that $X\le B_d<X+1$ and therefore $X=\lfloor B_d\rfloor$.    □

Corollary 2 

(Symmetric residual-error bound). If

$$|\hat{d}-d|\le {\delta }_d,$$

then exact right-endpoint recovery is guaranteed by

$${\tau }_{\max }-{\tau }_{\min }+2{\delta }_d<{\displaystyle \frac{1}{M}},$$

and is given by

$$X=⌊M\left(s\left(X;{\displaystyle \frac{p}{q}}\right)+\hat{d}+{\delta }_d+{\tau }_{\max }\right)⌋.$$

Remark 8 

(Scope of the residual-error lemma). If d is evaluated exactly from the same quantities used to form the shadow, then (14) applies and no separate residual-error allowance is required. Lemma 1 is needed only when d is quantized, rounded, transmitted, or otherwise evaluated with an independent certified error.

For an arbitrary directed interval (16), define the complete admissible candidate set

$${\mathcal{C}}_{\mathrm{adm}}:=I_{\mathrm{pc}}\cap \mathbb{Z}.$$

The uniqueness certificate is

$$|{\mathcal{C}}_{\mathrm{adm}}|=1.$$

For explicit endpoint evaluation, define

$$N_{\min }:=⌈\max \{0,A_{\min }\}⌉,N_{\max }:=\min \{M-1,\lfloor B_{\max }\rfloor \}.$$

Then

$${\mathcal{C}}_{\mathrm{adm}}=\left\{\begin{array}{cc}\{N_{\min },N_{\min }+1,\dots ,N_{\max }\},\hfill & N_{\min }\le N_{\max },\hfill \\ ⌀,\hfill & N_{\min }>N_{\max }.\hfill \end{array}\right.$$

Accordingly, the interval uniquely determines X if and only if

$$N_{\min }=N_{\max },$$

in which case

$$X=N_{\min }=N_{\max },{\mathcal{C}}_{\mathrm{adm}}=\left\{X\right\}.$$

Consequently, the subunit-width condition is sufficient but not necessary. Writing

$$W_{\tau }:=M({\tau }_{\max }-{\tau }_{\min }),$$

we distinguish three regimes: $W_{\tau }<1$ gives the universal certificate of Theorem 3; $W_{\tau }=1$ may place two consecutive integers at closed endpoints; and $W_{\tau }>1$ is inconclusive from width alone, because endpoint placement may leave either one or several integer candidates. Section 6 illustrates the singleton case after fixed-point tail enclosure; when the resulting interval contains several integers, the same construction returns their complete certified set.

The directed-recovery results above assume that a rational fraction has passed the structural continued-fraction test. Section 5 supplies that certification and the branch-specific stopping rules: phase-stable lifting depends on the residual accuracy of the selected convergent, whereas phase-crossing recovery may stop at the first structurally certified convergent and transfer the remaining certification task to the truncation-tail enclosure.

5. Continued-Fraction Certification of Rational Shadows

5.1. Scope and Input from the Phase-Consistent Framework

Section 4 establishes the fractional representation used throughout the remainder of the paper. Its principal object is the exact fractional image

$$\beta \left(X\right)={\displaystyle \frac{X}{M}},X\in \{0,\dots ,M-1\},$$

(23)

whereas directed truncation produces the observed fractional part $\tilde{\beta }\left(X\right)$ and the phase-transition indicator $c\left(X\right)\in \{0,1\}$. These quantities satisfy

$$\tilde{\beta }\left(X\right)-c\left(X\right)=\beta \left(X\right)-\tau \left(X\right),$$

where $\tau \left(X\right)\ge 0$ is the accumulated truncation tail. Thus, the fractional observation remains the object to be interpreted, while the auxiliary phase indicator from Section 4 preserves its correct unwrapped interpretation if directed truncation removes one integer transition.

The purpose of the present section is to develop the continued-fraction extraction and certification stage of the framework. The number-theoretic ingredients are standard facts from continued fractions and Diophantine approximation [27,28]; the contribution here is their integration with the intrinsic RNS resolution, certified observation-error budgets, and the auxiliary phase information supplied by Section 4. The role of the selected convergent is branch dependent. In the phase-stable branch, its residual accuracy determines whether localization or exact lifting is certified. In the phase-crossing branch, continued-fraction generation stops at the first structurally certified convergent, because the phase-corrected anchor is invariant and the remaining resolving power is supplied by the truncation-tail enclosure.

Continued fractions are applied to $\tilde{\beta }\left(X\right)\in [0,1)$. If $p/q$ is a rational fraction selected from this analysis, its phase-consistent rational shadow is, by Section 4.5,

$$s\left(X;{\displaystyle \frac{p}{q}}\right)={\displaystyle \frac{p}{q}}-c\left(X\right).$$

We reserve the term shadow for this real-valued, phase-consistent object. An integer obtained by lifting a certified rational fraction will instead be called a lifted rational representative. This terminological separation is essential: a rational shadow records the unwrapped phase, whereas a lifted representative is an integer-domain output produced under an explicit recovery or localization certificate. For the terminological summary below, write $y:=\tilde{\beta }\left(X\right)$ and let $p_j/q_j$ denote a selected continued-fraction convergent of y.

Remark 9 

(Hierarchy of uncertainty terms). The uncertainty quantities used below refer to different stages of the interpretation pipeline and are not interchangeable. The accumulated truncation tail $\tau \left(X\right)$ measures the one-sided loss incurred when the proper-fraction contributions are evaluated by directed truncation. In the phase-stable branch, the observation budget u bounds the total discrepancy between the available observation y and the exact fractional image $\beta \left(X\right)$; it may therefore already include the accumulated truncation tail together with any additional measurement or arithmetic uncertainty. The signed rational-approximation residual

$$d_j:=y-{\displaystyle \frac{p_j}{q_j}}$$

measures the oriented discrepancy between the available observation and the selected convergent. If this residual is itself evaluated inexactly, the resulting residual-evaluation error is enclosed separately. Accordingly, the truncation tail must not be added separately to the phase-stable budget

$$E_j:=u+\left|d_j\right|.$$

After a detected phase crossing, this phase-stable budget is no longer used: the accumulated truncation tail enters explicitly through its certified directed enclosure, while any residual-evaluation error remains a separate quantity.

5.2. Intrinsic RNS Resolution

By (23), the exact fractional image belongs to the finite lattice

$${\mathcal{B}}_M:=\left\{0,{\displaystyle \frac{1}{M}},{\displaystyle \frac{2}{M}},\dots ,{\displaystyle \frac{M-1}{M}}\right\}.$$

The spacing between adjacent admissible images is therefore

$${\Delta }_M:={\displaystyle \frac{1}{M}}.$$

This spacing is the intrinsic resolution of the RNS fractional domain. It is set by the dynamic range rather than by the arithmetic format used to evaluate the normalized aggregate.

The distinction between intrinsic and computational resolution is important. Approximations at scales substantially below ${\Delta }_M$ cannot distinguish additional source integers within the fixed RNS range, although they may still improve the numerical representation of the observed fraction. Conversely, for sufficiently large M, finite-precision arithmetic may be coarser than ${\Delta }_M$ and therefore become the effective observation limit. Accordingly, all localization and recovery statements below combine an approximation error with a certified observation-error budget instead of treating the continued-fraction residual in isolation.

5.3. Continued-Fraction Approximation and Recognition

Let $y\in [0,1)$ and let $p/q$ be a reduced rational fraction with $q>0$. Continued fractions provide two logically distinct kinds of statements. The first describes an approximation property possessed by every convergent; the second recognizes a reduced rational fraction as a convergent from a sufficiently strong approximation inequality.

Lemma 2 

(Approximation property of convergents)). If $p/q$ is a convergent of the continued-fraction expansion of y, then

$$\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{q^2}}.$$

(24)

The converse of Lemma 2 does not hold in general, so (24) is not an algorithmic recognition criterion.

Lemma 3 

(Legendre recognition criterion). Let $p/q$ be reduced and $q>0$. If 

$$\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{2q^2}},$$

(25)

then $p/q$ is a convergent of the continued-fraction expansion of y.

Lemma 3 supplies a strict sufficient Legendre certificate for the selected rational approximation. If a fraction is generated directly by the continued-fraction recurrence, its status as a convergent is already known and no separate recognition test is logically required. The additional strict test (25) is retained here deliberately: it attaches a uniform explicit certificate and the quantitative residual bound used in the subsequent RNS-scale analysis. Consistently with the terminology introduced in the Abstract, a structurally certified convergent is therefore a generated continued-fraction convergent that additionally satisfies (25). Failure of this sufficient test does not remove the continued-fraction status of a generated convergent; it only means that the explicit Legendre certificate used in the subsequent analysis is unavailable for that convergent.

5.4. Dual Certification at the RNS Resolution

Let

$$y:=\tilde{\beta }\left(X\right)$$

be the observed principal fractional part. Throughout this subsection, assume that the observation is phase-stable,

$$c\left(X\right)=0,$$

and that an application-dependent bound

$$\left|\beta \left(X\right)-y\right|\le u$$

(26)

is available. For directed truncation with $c\left(X\right)=0$, one may take $u={\tau }_{\max }$ whenever $0\le \tau \left(X\right)\le {\tau }_{\max }$. The quantity u may also incorporate additional measurement or arithmetic uncertainty.

Definition 2 

(Dual-certified rational fraction). A reduced rational fraction $p/q$, with $q>0$, is called dual-certified for localization relative to $(y,u,M)$ if

1.

it satisfies the structural certificate

$$\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{2q^2}};$$

(27)

2.

it satisfies the localization certificate

$$u+\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{M}}.$$

(28)

It is called dual-certified for exact phase-stable lifting if the stronger condition

$$u+\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{2M}}$$

(29)

holds in place of (28).

The two certification components serve different purposes. Condition (27) identifies genuine continued-fraction structure, while (28) relates the selected fraction to the discrete RNS scale. Neither condition subsumes the other. Their crossover occurs near $q=\sqrt{M/2}$ when the observation budget is negligible: below this denominator scale the resolution threshold is tighter, whereas above it the Legendre threshold is tighter.

Remark 10 

(Hierarchy of certification levels). The certificates used in this section answer different questions and should not be conflated. The structural certificate confirms that the selected rational approximation has the required continued-fraction structure. In the phase-stable branch, the resolution certificates compare the phase-stable budget $E_j=u+\left|d_j\right|$ from Remark 9 with the intrinsic RNS spacing: the condition

$$E_j<{\displaystyle \frac{1}{M}}$$

provides a localization certificate, whereas the stronger condition

$$E_j<{\displaystyle \frac{1}{2M}}$$

provides an exact-lifting certificate. After a detected phase crossing, the structural certificate retains its role, but the phase-stable resolution certificates are no longer applied. Instead, exact recovery requires a uniqueness certificate for the directed recovery interval: its intersection with the admissible integer range $[0,M)$ must contain exactly one integer. A subunit interval width alone guarantees at most one integer and is therefore not, by itself, a complete uniqueness certificate.

Proposition 2 

(Localization supplied by dual certification). Assume $c\left(X\right)=0$ and (26). If $p/q$ is dual-certified for localization relative to $(y,u,M)$, then

$$\left|\beta \left(X\right)-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{M}}$$

(30)

and therefore

$$\left|X-M{\displaystyle \frac{p}{q}}\right|<1.$$

(31)

Proof. 

The triangle inequality and (26) give

$$\left|\beta \left(X\right)-{\displaystyle \frac{p}{q}}\right|\le \left|\beta \left(X\right)-y\right|+\left|y-{\displaystyle \frac{p}{q}}\right|\le u+\left|y-{\displaystyle \frac{p}{q}}\right|.$$

Condition (28) proves (30); multiplying by M and using $X=M\beta \left(X\right)$ proves (31).    □

5.5. Phase-Stable Integer Lifting

Ordinary nearest-integer lifting is valid only when the observed fraction and the selected rational fraction belong to the correct principal phase. We therefore restrict this subsection to the phase-stable case

$$c\left(X\right)=0.$$

(32)

For a rational fraction $p/q$, define its lifted rational representative by

$$R_{p/q}:=⌊M{\displaystyle \frac{p}{q}}⌉,$$

(33)

where $\lfloor x\rceil$ denotes any integer nearest to x. In the localization regime a half-integer tie may be resolved either way, and either choice satisfies the stated one-unit error bound. Under the strict half-spacing condition used for exact recovery, the nearest integer is unique, so no tie-breaking convention is required.

Corollary 3 

(Localization and exact phase-stable lifting). Assume (32) and (26).

If $p/q$ is dual-certified for localization, then

$$|X-R_{p/q}|\le 1.$$

(34)

If $p/q$ is dual-certified for exact phase-stable lifting, then

$$R_{p/q}=X.$$

(35)

Proof. 

By Proposition 2, $|X-Mp/q|<1$. Nearest-integer rounding can therefore produce only X or an adjacent integer, proving (34). Under (29), the same triangle-inequality argument gives $|X-Mp/q|<1/2$, which proves (35).    □

Remark 11 

(Detected phase crossings). If $c\left(X\right)=1$, nearest-integer lifting of $Mp/q$ or of the scaled shadow $S\left(X;p/q\right)$ is not the appropriate recovery mechanism. The selected fraction and its signed residual determine the invariant phase-corrected anchor

$$s\left(X;{\displaystyle \frac{p}{q}}\right)+d=\tilde{\beta }\left(X\right)-1,$$

while the certified truncation-tail enclosure determines the directed recovery interval of Section 4.6. Exact recovery then follows either from the subunit-width right-endpoint certificate or, more generally, from a singleton admissible candidate set. If several integers remain, the complete certified candidate set is returned instead.

5.6. Stability of Certified Rational Representatives

We now quantify the effect of bounded observation error in the phase-stable regime. Let

$$y=\beta \left(X\right)+\epsilon ,\left|\epsilon \right|\le {\epsilon }_{\max },c\left(X\right)=0.$$

(36)

This model describes perturbations on the ordinary real line without wrapping through a phase boundary.

Theorem 4 

(Error bound for a phase-stable lifted representative). Let $p/q$ be a reduced rational fraction satisfying the Legendre certificate with respect to y,

$$\left|y-{\displaystyle \frac{p}{q}}\right|<{\displaystyle \frac{1}{2q^2}},$$

(37)

and let $R_{p/q}$ be defined by (33). Then

$$|X-R_{p/q}|<M{\epsilon }_{\max }+{\displaystyle \frac{M}{2q^2}}+{\displaystyle \frac{1}{2}}.$$

(38)

Proof. 

From (36) and (37),

$$\left|\beta \left(X\right)-{\displaystyle \frac{p}{q}}\right|\le |\beta \left(X\right)-y|+\left|y-{\displaystyle \frac{p}{q}}\right|<{\epsilon }_{\max }+{\displaystyle \frac{1}{2q^2}}.$$

Multiplication by M gives

$$\left|X-M{\displaystyle \frac{p}{q}}\right|<M{\epsilon }_{\max }+{\displaystyle \frac{M}{2q^2}}.$$

By nearest-integer rounding,

$$\left|R_{p/q}-M{\displaystyle \frac{p}{q}}\right|\le {\displaystyle \frac{1}{2}}.$$

The triangle inequality proves (38).    □

The three terms in (38) have distinct origins: observation error amplified by M, certified rational-approximation error decaying quadratically with q, and the half-unit discretization cost of nearest-integer lifting. Increasing the denominator reduces only the second component. For detected phase crossings, bounded truncation and residual errors are instead handled by the directed interval and the inexact-residual lemma of Section 4.6.

5.7. Certified Interpretation Procedure

Algorithm 1 summarizes the phase-aware interpretation procedure. It takes the fractional observation and auxiliary phase indicator supplied by Section 4. In either branch, the first Legendre-certified convergent supplies a certified phase-consistent rational shadow if that is the requested output level. In the phase-stable branch, continued-fraction generation otherwise proceeds until the requested localization or exact-lifting certificate is met. In the phase-crossing branch, generation stops at the first Legendre-certified convergent, after which the procedure constructs a certified tail enclosure and returns either the unique admissible integer or the complete finite candidate set supported by that enclosure.

Throughout Algorithm 1, the structural Legendre test and the phase-stable budget use the exact rational residual

$$d=\tilde{\beta }\left(X\right)-{\displaystyle \frac{p}{q}},$$

computed from the finite-precision observation and the generated convergent. The residual-error enclosure is required only if a separately quantized, rounded, transmitted, or otherwise inexact copy of this residual is used to construct the phase-crossing recovery interval, as in Lemma 1. Likewise, tail-enclosure data are required only when the phase-crossing branch proceeds beyond rational-shadow output.

Algorithm 1 Phase-aware continued-fraction certification and recovery

Require:  M, $\tilde{\beta }\left(X\right)$, $c\left(X\right)$, and the requested output level; if phase-stable localization or exact lifting is requested, a certified phase-stable observation-error bound u; if phase-crossing recovery beyond rational-shadow output is requested, either a certified two-sided tail enclosure or the local truncation remainders required by Proposition 1; if an inexact residual is used in that recovery interval, a certified residual-error enclosure Ensure:  A certified rational shadow, an exact recovered integer, a one-unit localization, a complete finite admissible candidate set, or an explicit insufficient- or inconsistent-certificate declaration   1: Generate the reduced convergents $p/q$ of $\tilde{\beta }\left(X\right)$ successively   2: for each successive convergent $p/q$ do   3:     Compute the residual $d=\tilde{\beta }\left(X\right)-p/q$   4:     if $\left|d\right|\ge 1/\left(2q^2\right)$ then   5:         Continue with the next convergent   6:     end if   7:     Form the phase-consistent rational shadow $s\left(X;{\displaystyle \frac{p}{q}}\right)=p/q-c\left(X\right)$   8:     if a certified rational shadow is the requested output level then   9:         return $s\left(X;{\displaystyle \frac{p}{q}}\right)=p/q-c\left(X\right)$, together with $d=\tilde{\beta }\left(X\right)-p/q$ and the active phase indicator $c\left(X\right)$ 10:     end if 11:     if $c\left(X\right)=0$ then                                                                        ▹ phase-stable branch 12:         Set $E=u+\left|d\right|$ 13:         if exact lifting is requested and $E<1/\left(2M\right)$ then 14:            return $X=⌊Mp/q⌉$ 15:         else if one-unit localization is requested and $E<1/M$ then 16:            return $R_{p/q}=⌊Mp/q⌉$ with $|X-R_{p/q}|\le 1$ 17:         end if 18:     else                                                                   ▹ detected phase-crossing branch 19:         Stop the continued-fraction generation at this first structurally certified convergent 20:         if a certified two-sided tail enclosure is not supplied then 21:            Construct $[{\tau }_{\min },{\tau }_{\max }]$ from the local truncation remainders using Proposition 1 22:         end if 23:         if the exact residual d is used in the recovery interval then 24:            Construct $I_{\mathrm{rec}}:=I_{\mathrm{pc}}$ from (16) 25:         else 26:            Obtain $\hat{d}$ and a certified enclosure ${\epsilon }_{-}\le \hat{d}-d\le {\epsilon }_{+}$ 27:            Construct $I_{\mathrm{rec}}:=I_{\mathrm{pc}}^{\left(d\right)}$ using Lemma 1 28:         end if 29:         Form ${\mathcal{C}}_{\mathrm{adm}}:=I_{\mathrm{rec}}\cap \mathbb{Z}$ 30:         if $|{\mathcal{C}}_{\mathrm{adm}}|=1$ then 31:            return the unique element of ${\mathcal{C}}_{\mathrm{adm}}$ 32:         else if $|{\mathcal{C}}_{\mathrm{adm}}|>1$ then 33:            if a finite admissible candidate set satisfies the requested output level then 34:                return the complete admissible candidate set ${\mathcal{C}}_{\mathrm{adm}}$ 35:            else 36:                return the declaration that exact recovery requires a tighter certified tail or residual-error enclosure 37:            end if 38:         else 39:            return the declaration that the supplied certificates are inconsistent 40:         end if 41:     end if 42: end for 43: return the declaration that the requested certificate has not been met by any processed convergent

The procedure makes the branch-dependent hierarchy explicit. Continued fractions provide the rational interpretation, the RNS spacing $1/M$ determines the relevance of a phase-stable approximation, and the exact-rank invariant selects the mathematically valid recovery branch. In the phase-stable branch, the residual accuracy of the selected convergent controls localization or lifting. In the phase-crossing branch, the convergent supplies an invariant anchor, while the certified tail and residual-error enclosures determine the admissible integer set. Exact-rank recovery therefore preserves phase consistency without replacing the continued-fraction interpretation as the principal structural operation.

5.8. Computational Requirements and Stopping Depth

For a platform-independent requirement analysis, let

$$B:=⌈{\log }_{2}M⌉$$

denote the bit length of the RNS dynamic range, and let J denote the number of continued-fraction convergents processed before the applicable stopping certificate is reached. For a common directed-truncation precision L, the accumulated tail satisfies

$$0\le \tau \left(X\right)<k{2}^{-L}.$$

Consequently, a sufficient design condition for the universal a priori truncation budget to lie below the phase-stable exact-lifting threshold is

$$k{2}^{-L}<{\displaystyle \frac{1}{2M}},$$

which is equivalent to

$$L>{\log }_2\left(2kM\right).$$

The smallest integer precision satisfying this strict inequality is

$$L_{\min }=⌊{\log }_2\left(2kM\right)⌋+1.$$

Thus the common contribution precision scales as

$$L=B+O\left(\log k\right).$$

This condition concerns only the a priori truncation budget; the actual phase-stable exact-lifting certificate remains $u+\left|d\right|<1/\left(2M\right)$, and may therefore depend on the residual of the selected rational approximant.

The channelwise proper-fraction accumulation and exact-rank evaluation each require a linear number of local operations in k, and a parallel reduction tree may reduce the accumulation depth to logarithmic order in k. For a fixed anchor, the phase-crossing tail enclosure likewise requires $O\left(k\right)$ bounded-width local operations and short accumulations. If J convergents are processed, the continued-fraction stage performs $O\left(J\right)$ recurrence and certificate steps on integers whose bit lengths grow with the partial denominators. The common tabular-majorant precision is

$$h:=⌈{\log }_2{m}_{\max }⌉.$$

For fixed moduli, the modular inverses, local rank values, parity data, and dyadic majorants are precomputable; the resulting memory–arithmetic tradeoff remains dependent on the modulus set, table organization, operand representation, and target architecture. These estimates count arithmetic operations and recurrence steps, not full bit complexity. The cost of each operation depends on operand width, which in turn depends on the moduli and the required precision. The stated parallel depth counts reduction stages rather than hardware-independent bit-operation depth. No measured execution-time or universal performance advantage over optimized CRT reconstruction is claimed.

6. Numerical Validation

This section summarizes the numerical validation of the proposed framework. The complete modular, Euclidean, continued-fraction, truncation, and tail-enclosure calculations are given in Appendix A. The main text retains only the quantities required to verify the logical progression from exact phase control to branch-specific certified recovery.

6.1. Common Modular Setting and Validation Scenarios

All validation scenarios use the primary RNS

$$(m_1,m_2,m_3,m_4)=(11,13,15,17),M=36465$$

(39)

together with the redundant modulus 2 required for independent exact-rank recovery. Two source integers are considered.

The first source value

$$X=17293$$

is used to validate the phase-stable branch in three successive stages: independent exact-rank recovery, continued-fraction interpretation of the exact fractional image, and certified lifting after directed binary truncation. These stages correspond to the detailed computations in Appendix A.1, Appendix A.2 and Appendix A.3.

The second source value

$$X=19$$

is used to validate the phase-crossing branch. Directed truncation at a deliberately low precision removes one integer transition from the normalized aggregate. The independently recovered exact rank detects the crossing, after which a fixed-point enclosure of the discarded tail yields a unique admissible integer. The complete calculation is given in Appendix A.4.

Thus the numerical sequence exercises the two successful exact-recovery branches of Algorithm 1: exact lifting in the phase-stable branch and singleton recovery after a detected phase crossing. The general theory and Algorithm 1 additionally define rational-shadow output, one-unit localization, finite multiple-candidate output, and explicit declarations for insufficient or inconsistent certificates; those outcomes are not instantiated numerically in the present article. The two demonstrated recovery paths follow the common logical chain

$$\begin{array}{cc}\hfill \mathrm{exact}\mathrm{rank}⟶\mathrm{phase}\mathrm{detection}& ⟶\mathrm{continued}\text{-}\mathrm{fraction}\mathrm{stopping}⟶\hfill \\ \hfill & ⟶\mathrm{branch}\text{-}\mathrm{specific}\mathrm{certified}\mathrm{recovery}.\hfill \end{array}$$

6.2. Phase-Stable Validation

6.2.1. Independent Exact-Rank Recovery

For $X=17293$, the primary residue vector and redundant parity residue are

$$(r_1,r_2,r_3,r_4)=(1,3,13,4),r_0=1.$$

The normalized residues are

$$(x_1,x_2,x_3,x_4)=(3,12,13,7).$$

With reference modulus $m_4=17$, the local rank contributions are

$$(R_{1,4},R_{2,4},R_{3,4},R_{4,4})=(4,15,14,7),$$

which give

$$\hat{\rho }\left(X\right)=⌊{\displaystyle \frac{4+15+14+7}{17}}⌋=2.$$

The redundant parity correction is $\delta \left(X\right)=0$, and therefore

$$\rho \left(X\right)=2.$$

(40)

This calculation uses only bounded local table values, one short accumulation, and the redundant parity residue; no full CRT reconstruction is required. The detailed local computation is given in Appendix A.1.

6.2.2. Exact Fractional Image and Early Certified Stopping

Using (40), the exact normalized aggregate gives

$$\beta \left(X\right)={\displaystyle \frac{17293}{36465}}.$$

Its continued-fraction expansion begins

$$\beta \left(X\right)=[0;2,9,4,1,\dots ],$$

and the first convergent satisfying both the structural and exact-lifting certificates is

$${\displaystyle \frac{p}{q}}={\displaystyle \frac{46}{97}}.$$

Indeed, with $u=0$,

$$d:=\beta \left(X\right)-{\displaystyle \frac{46}{97}}={\displaystyle \frac{31}{3537105}},$$

and

$$\left|d\right|<{\displaystyle \frac{1}{2·{97}^2}},\left|d\right|<{\displaystyle \frac{1}{2M}}.$$

Consequently, Corollary 3 gives

$$R_{46/97}=⌊M{\displaystyle \frac{46}{97}}⌉=17293=X.$$

The continued-fraction process therefore stops at $46/97$; the terminal exact fraction is unnecessary for recovery. The complete Euclidean decomposition and convergent table are given in Appendix A.2.

6.2.3. Directed Binary Truncation

We next replace the exact local fractions by directed binary truncations at the smallest common precision for which the a priori tail budget lies below the exact-lifting threshold:

$$L=19,u:={\displaystyle \frac{4}{2^{19}}}={\displaystyle \frac{1}{131072}}<{\displaystyle \frac{1}{2M}}.$$

The resulting observed fractional part is

$$y:=\tilde{\beta }\left(X\right)={\displaystyle \frac{124317}{262144}}.$$

The exact and truncated observations share the continued-fraction prefix

$$[0;2,9,4,1],$$

so they produce the same fifth convergent $46/97$. For the truncated observation,

$$d:=y-{\displaystyle \frac{46}{97}}={\displaystyle \frac{125}{25427968}},$$

and the complete phase-stable budget is

$$u+\left|d\right|={\displaystyle \frac{319}{25427968}}<{\displaystyle \frac{1}{2M}}.$$

The same lifted representative is therefore still exact:

$$R_{46/97}=17293=X.$$

Thus directed truncation changes the terminal continued fraction but preserves the first convergent meeting the prescribed exact-lifting certificate. The local truncations, exact verification tail, Euclidean decomposition, and complete convergent table are given in Appendix A.3.

Taken together, the three phase-stable computations show that the exact-rank mechanism, continued-fraction early stopping, and finite-precision lifting remain logically separate but mutually compatible. The rank is recovered independently, the rational structure is certified at the intrinsic RNS scale, and the same certified convergent survives the prescribed directed truncation.

6.3. Phase-Crossing Validation

We now retain the modulus system (39) but choose the nontrivial small source integer

$$X=19.$$

Its residue and normalized-residue vectors are

$$(r_1,r_2,r_3,r_4)=(8,6,4,2),(x_1,x_2,x_3,x_4)=(2,11,4,12).$$

The independent modular rank computation gives

$$\hat{\rho }\left(X\right)=1,\delta \left(X\right)=1,\rho \left(X\right)=2.$$

Choose the deliberately low common binary precision

$$L=11.$$

The local truncation remainders are

$$(e_1,e_2,e_3,e_4)=(4,12,2,11),$$

and the truncated accumulation is

$$\tilde{\alpha }\left(X\right)={\displaystyle \frac{4095}{2048}}=1+{\displaystyle \frac{2047}{2048}}.$$

Hence

$$\tilde{\rho }\left(X\right)=1,y:=\tilde{\beta }\left(X\right)={\displaystyle \frac{2047}{2048}},$$

and the exact-rank comparison detects the lost integer transition:

$$c\left(X\right)=\rho \left(X\right)-\tilde{\rho }\left(X\right)=1.$$

The observed fraction has the continued fraction

$$y=[0;1,2047].$$

The initial convergent $0/1$ fails the structural certificate, whereas

$${\displaystyle \frac{p}{q}}={\displaystyle \frac{1}{1}}$$

is the first structurally certified convergent. Its signed residual and phase-consistent shadow are

$$d=-{\displaystyle \frac{1}{2048}},s\left(X;{\displaystyle \frac{1}{1}}\right)=0,$$

so

$$s\left(X;{\displaystyle \frac{1}{1}}\right)+d=-{\displaystyle \frac{1}{2048}}=y-c\left(X\right).$$

By (14), this anchor is independent of the particular structurally certified convergent. The continued-fraction stage therefore stops at $1/1$, and the remaining certification task is transferred to the truncation-tail enclosure.

For the Cauchy–Schwarz lower bound,

$$S:={\displaystyle \sum _{i=1}^4}{e}_i=29,W:={\displaystyle \sum _{i=1}^4}{e}_i{m}_i=417,$$

which gives

$$T\ge T_{\min }:={\displaystyle \frac{S^2}{W}}={\displaystyle \frac{841}{417}}.$$

The dominant-loss rule selects the second channel,

$${\ell }_{\ast }=2,m_{{\ell }_{\ast }}=13.$$

Using the common majorant precision

$$h=⌈{\log }_{2}17⌉=5,$$

the local Euclidean divisions and dyadic tabular majorants yield

$$T\le T_{\max }^{\left(2\right)}:={\displaystyle \frac{431}{208}}.$$

Therefore,

$${\displaystyle \frac{841}{417·2048}}\le \tau \left(X\right)\le {\displaystyle \frac{431}{208·2048}}.$$

Substitution into the phase-crossing recovery formula gives

$$\begin{array}{cc}\hfill I_2^{\mathrm{tail}}& =36465\left[-{\displaystyle \frac{1}{2048}}+{\displaystyle \frac{841}{417·2048}},-{\displaystyle \frac{1}{2048}}+{\displaystyle \frac{431}{208·2048}}\right]\hfill \\ \\ \hfill & =\left[{\displaystyle \frac{644215}{35584}},{\displaystyle \frac{625515}{32768}}\right]=[18.104063\dots ,19.089203\dots ].\hfill \end{array}$$

Its admissible integer set is

$${\mathcal{C}}_2^{\mathrm{tail}}:=I_2^{\mathrm{tail}}\cap \mathbb{Z}=\left\{19\right\}.$$

Consequently, $X=19$.

This certificate uses neither exact evaluation of the discarded tail nor a second CRT reconstruction. Exact recovery is supplied by the combination of exact phase detection and the bounded-width fixed-point tail enclosure. The continued-fraction stage provides the phase-consistent rational shadow and may stop at the first structurally certified convergent; because the resulting directed-recovery anchor is convergent-invariant, deeper continued-fraction expansion does not tighten the final recovery interval. If the same construction produced several admissible integers, Algorithm 1 would return their complete certified set. The complete rank computation, local truncations, anchor divisions, dyadic majorants, exact-tail verification, and adjacent-precision boundary check are given in Appendix A.4.

7. Discussion and Related Work

The framework developed in Section 4, Section 5 and Section 6 changes the role of fractional RNS information. Rather than treating a finite-precision fractional output as an approximate substitute for the encoded integer, it interprets that output at the intrinsic lattice scale $1/M$ and asks what can be certified from the available precision. This viewpoint supports several distinct outcomes: structural rational information, one-unit localization, exact phase-stable lifting, exact directed recovery, or a complete finite set of admissible integer candidates. Full CRT reconstruction remains the appropriate operation when the exact integer is unconditionally required; the present framework instead targets settings in which a weaker certified output, or an exact value obtained by early recovery, already satisfies the computational task.

A second conceptual point is the separation of rational structure, global phase, and directed truncation loss. Continued fractions organize the rational information visible in the fractional observation. The exact-rank invariant, obtained by extending the primary RNS with the redundant modulus 2, detects whether directed truncation has removed one integer transition. Once a crossing is detected, the selected structurally certified convergent and its signed residual determine an invariant phase-corrected anchor, whereas the resolving power is supplied by a certified enclosure of the discarded tail. The proposed fixed-point enclosure constructs this information from local truncation remainders by combining a Cauchy–Schwarz lower bound with dyadic tabular upper majorants. Thus, the rank mechanism preserves the correct branch, the continued fraction supplies the rational interpretation, and the tail enclosure determines the admissible integer set without exact tail evaluation or a second CRT reconstruction.

7.1. Relation to Floating-Output and Approximate CRT Methods

RNS comparison, scaling, division, and other nonmodular operations commonly require magnitude information that is unavailable within independent residue channels. Floating-output and approximate CRT methods address this need by evaluating a normalized numerical image suitable for comparison, range decisions, or approximate conversion. The present framework shares that fractional starting point but differs in the status assigned to the result.

Here, the observed fraction is interpreted relative to the discrete spacing $1/M$ and accompanied by explicit uncertainty certificates. Continued fractions provide a hierarchy of rational shadows in the phase-stable branch, where generation may stop as soon as the requested localization or exact-lifting condition is met. Near the lower phase boundary, the independently recovered exact rank detects whether directed truncation has removed one integer transition. In the resulting phase-crossing branch, the rational anchor is invariant under the selected structurally certified convergent, while a certified tail enclosure determines the directed recovery interval. Consequently, an observation numerically close to one is not automatically converted into a large integer representative when it may originate from a small value displaced across the boundary.

This branch separation also clarifies the relation to direct scaling. If a direct method is supplied with the same exact phase indicator and the same certified observation or tail enclosure, it may certify the same integer-domain localization or exact value. Continued fractions are therefore not claimed to be uniquely necessary for integer recovery. Their distinct role is to organize the fractional observation into a denominator-ordered hierarchy of structurally certified rational shadows and to provide adaptive stopping at the requested output level. In the phase-crossing branch, this structural output is retained, but the uniqueness of the recovered integer is determined by the phase-corrected observation and the certified tail enclosure rather than by the depth of the continued-fraction expansion. The significance of this hierarchy is not that it replaces all alternative recovery procedures. Rather, it introduces an explicit certified-information layer between the residue encoding and unconditional reconstruction. At this layer, the available modular information can support different certified outputs, including rational shadows, localization, exact lifting, directed recovery, or finite candidate sets, depending on the available uncertainty certificates. Continued fractions provide the organizing structure that links these output levels to a hierarchy of rational approximants of increasing complexity. From this viewpoint, exact integer recovery is only one possible endpoint of the interpretation process rather than the sole objective of the representation.

This distinction leads naturally to graded outputs. Depending on the available certificates, the method may return a rational shadow, a localized or exactly lifted integer representative, an exactly recovered value from a directed interval, or the complete finite set of admissible integer candidates. The framework therefore makes explicit which conclusions are justified by the available resolution instead of collapsing all finite-precision outputs into a single approximate estimate.

The associated perturbation bounds extend this certification logic to finite-precision observation and residual evaluation. In the phase-stable branch, uncertainty enters the localization or lifting budget; after a detected crossing, tail and residual-evaluation enclosures determine the certified recovery interval.

7.2. Relation to Classical Rational Reconstruction and Continued Fractions

Classical rational reconstruction addresses a different inverse problem. Given modular information and suitable numerator–denominator bounds, it seeks an exact rational number compatible with that modular image. Euclidean and continued-fraction methods are central to this task, beginning with the reconstruction procedures of Wang and subsequent developments in modular computer algebra [29,30,31].

The present framework does not infer an unknown exact rational target directly from a residue modulo a large integer. It begins with a fractional observation derived from an RNS encoding and asks which visible rational structure is both certified and operationally sufficient at the scale of the encoded integer. The rational approximant is therefore an intermediate interpretive object rather than necessarily the final reconstructed quantity.

Nor does the novelty lie in the generation of continued-fraction convergents, whose approximation and recognition properties are classical [27,28]. The contribution is their integration with the intrinsic RNS spacing, certified finite-precision uncertainty, and exact phase control under a detected boundary crossing. Continued fractions supply the rational hierarchy; the RNS geometry and the requested task determine where that hierarchy may be stopped and what integer-domain conclusion may be drawn.

Table 1 summarizes these distinctions. The entries describe typical roles rather than asserting identical behavior for every implementation within a broad method class.

Table 1. Comparison of exact reconstruction, approximate RNS interpretation, classical rational reconstruction, and the proposed phase-consistent framework.

Method	Primary Input or Object	Phase Information	Typical Output	Certification Scale	Exact-Recovery Role
Exact CRT	residue vector	implicit in full reconstruction	exact integer in $[0,M)$	modulus product M	primary objective
Floating-output or approximate CRT	residue vector or normalized numerical image	generally not explicit	magnitude estimate, comparison, range decision, or approximate value	implementation precision or application bound	implementation dependent
Classical rational reconstruction	modular image with size bounds	not an RNS phase variable	exact rational number, when uniquely reconstructible	numerator–denominator bounds relative to the modulus	primary objective
Proposed fractional-interpretation framework	fractional observation	auxiliary $c\left(X\right)$ when needed	shadow, lifted representative, exact value, or candidate set	$1/M$ plus certified error budgets	branch-specific certification; exact recovery when the corresponding uniqueness certificate holds

7.3. Operational Scope and Implementation Implications

The framework is most useful when an application requires a certified decision rather than the unconditional recovery of every bit of the encoded integer. Examples include range decisions, order or sign-related tests, bounded localization, exact early recovery, and candidate reduction before residue-domain verification. In such settings, a rational shadow can serve as a certified intermediate information object when the requested task has been specified together with a decision rule showing that the available shadow-level certificate is sufficient for that task. The present paper establishes the representation and certification layer; task-specific sufficiency for a concrete application must be verified separately. If every invocation must return the exact integer regardless of the available observation precision or certificate width, exact CRT reconstruction remains the natural baseline.

The framework may therefore be viewed as a certified interpretation layer over distributed modular data. Its role is not to replace exact reconstruction, but to determine whether the available finite-precision observation and phase information are sufficient for a requested output level. Depending on the satisfied certificate, the layer may return a phase-consistent rational shadow, a one-unit localization, an exact lifted integer, or a complete finite set of admissible candidates; otherwise, it explicitly reports that the requested conclusion has not been certified. In this sense, the output is determined by certified information sufficiency rather than by an unconditional requirement to reconstruct the complete positional value.

The two numerical scenarios in Section 6 illustrate this interpretation without constituting separate end-to-end applications. In the phase-stable scenario, the observed data already support exact early lifting through the convergent $46/97$, which satisfies the prescribed structural and exact-lifting certificates. In the boundary-sensitive scenario, the principal fractional observation lies near one, but exact phase control and the directed tail enclosure identify the singleton admissible set $\left\{19\right\}$. These examples show how the same modular observation pipeline can support different certified conclusions according to the available precision and the detected phase branch. Concrete decision procedures for range, ordering, classification, early acceptance or rejection, and candidate reduction remain application-dependent extensions of the present certification framework.

For fixed moduli, the normalized residue contributions, local rank contributions, parity information, local truncation remainders, and dyadic majorants are drawn from finite sets and are therefore compatible with lookup-table realization. Fractional accumulation and exact-rank accumulation may proceed in parallel. In the phase-stable branch, continued-fraction generation stops when the requested resolution certificate is met. In the phase-crossing branch, it stops at the first structurally certified convergent, after which a fixed-point tail enclosure can be formed by bounded-width local divisions, table lookups, and short accumulations. The common majorant precision

$$h=⌈{\log }_2{m}_{\max }⌉$$

provides a uniform dyadic format tied to the largest primary modulus. This organization suggests low-latency implementations based on parallel modular channels, fixed-point accumulation, local tables, and branch-specific early termination.

These structural advantages do not by themselves establish a universal speedup. Performance depends on the modulus set, precision format, parallel resources, memory organization, stopping frequency, table cost, and the cost of any final candidate verification. Increasing the tabular precision may tighten a tail enclosure, but it also widens the stored constants and local accumulators; the proposed common precision therefore prioritizes a uniform bounded-width implementation rather than unconditional minimization of every recovery interval. The present results provide the correctness conditions needed for such realizations, while comparative hardware and software evaluation remains an empirical question.

The practical trade-off relative to conventional CRT reconstruction is application dependent. Exact CRT directly produces the complete positional integer and remains the natural solution when that output is always required. The proposed framework introduces additional interpretation overhead through normalized fixed-point accumulation, exact-rank phase control, continued-fraction processing, and, after a detected crossing, construction of the truncation-tail enclosure and admissible candidate set. In return, it may provide certified intermediate outputs, branch-specific stopping, and, in some cases, exact recovery before unconditional positional reconstruction is performed. Whether these capabilities compensate for the associated overhead depends on the requested output level, modulus architecture, precision requirements, available parallelism, and the frequency with which the relevant certificates are satisfied. For fixed modulus sets, part of the additional cost may be amortized through precomputation and channel-level parallelism. The potential benefit is therefore adaptive certification and possible deferral of full reconstruction rather than a universal reduction in arithmetic cost. Meaningful quantitative comparison consequently requires common modulus sets, arithmetic formats, requested output levels, precomputation policies, and implementation resources.

7.4. Limitations and Future Work

The guarantees require certified uncertainty bounds. Without a valid bound on the discrepancy between the exact and observed fractional images, a continued-fraction pattern may still be computed, but its integer-domain interpretation is not certified. In the phase-crossing branch, the same requirement applies to the directed tail enclosure and to any additional residual-evaluation error. An incorrect enclosure may exclude the true integer, whereas a valid but coarse enclosure may leave several admissible candidates.

Precision scaling is a second limitation. As M grows, the lattice spacing $1/M$ may fall below the resolution of a fixed arithmetic format. No deeper continued-fraction expansion can recover information already lost in the initial observation. Very large dynamic ranges may therefore require adaptive observation precision, interval arithmetic, or exact fixed-point accumulation. In the phase-crossing branch, increasing the table precision h can tighten the tail enclosure, but it also increases table-entry and accumulator widths. The common majorant precision used in the fixed-point tail enclosure provides a uniform dyadic format tied to the largest modulus; it is not claimed to minimize every resulting interval.

A third limitation occurs when the directed recovery interval contains several admissible integers. The method then returns the complete certified candidate set but cannot select one value without additional information. This is an intentional output of the theory rather than a failure of the algorithm. Possible extensions include residue-domain verification of the remaining candidates, additional external constraints, and tighter analytic or tabular tail enclosures. Adaptive refinement of the truncation or table precision may also be studied, but it is not part of the fixed-precision procedure analyzed here.

A fourth limitation concerns anchor selection. Choosing a channel with the largest relative local truncation loss gives a deterministic and inexpensive rule, but the present results do not prove that this channel minimizes the recovery-interval width for every residue vector. Characterizing an interval-optimal anchor, or a low-cost approximation to it, remains an open implementation problem.

An additional direction concerns systematic validation across different modulus architectures and dynamic-range scales. The present work uses a single primary modulus configuration to isolate and validate the proposed certification mechanisms. A broader study should examine families of pairwise coprime modulus sets with different numbers and relative sizes of moduli and increasing dynamic-range bit lengths $B=\lceil {\log }_{2}M\rceil$. Random inputs should be complemented by boundary-biased samples whose exact fractional images lie near an integer transition. Such an evaluation should quantify phase-stable and phase-crossing frequencies, certification and recovery rates, candidate-set cardinalities, continued-fraction stopping depths, and the precision required for each certified output level. Establishing these empirical trends across modulus families remains future work.

Further work should examine hardware-oriented realizations, quantitative comparison with optimized CRT reconstruction, cryptographic-scale modulus sets, and the use of minimal redundancy to protect other nonmodular RNS operations. In particular, scaling, division, and Montgomery-type transformations may admit analogous exact global invariants that preserve phase or range information while retaining the parallel structure of the residue representation.

A complementary direction is to develop the information-processing interpretation of rational shadows beyond the representation and certification layer established here. This includes task-specific decision procedures, quantitative notions of certified information sufficiency, and the use of partial modular information in range, ordering, classification, early acceptance or rejection, and candidate-reduction problems. Such developments should determine which global decisions can be justified from a finite-precision modular observation before complete positional reconstruction becomes necessary.

8. Conclusions

The main outcome of this work is a change in how modular encodings may be interpreted. Instead of treating the normalized fractional image only as an approximate numerical output derived from the CRT identity, the proposed framework treats it as a structured numerical object whose continued-fraction content can be assessed at the intrinsic resolution of the residue system. This creates an intermediate interpretation layer between independent residue channels and unconditional integer reconstruction.

Within this layer, the strength of the conclusion adapts to the information actually available. In the phase-stable branch, a structurally certified convergent may provide a phase-consistent rational shadow, one-unit localization, or exact nearest-integer lifting. Directed truncation may instead remove one integer transition and move a small fractional image across the lower phase boundary. The auxiliary exact-rank invariant detects this event and selects the phase-crossing branch. There, the sum of the selected rational shadow and its signed residual is invariant with respect to the structurally certified convergent, so continued-fraction generation may stop at the first such convergent.

After a detected crossing, the resolving power is supplied by a certified enclosure of the discarded truncation tail. The proposed fixed-point construction derives this enclosure directly from local truncation remainders by combining a Cauchy–Schwarz lower bound with dyadic tabular upper majorants. It uses a common bounded-width format tied to the largest primary modulus and does not require exact tail evaluation or a second CRT reconstruction. The resulting directed interval yields either exact recovery through a singleton admissible set or the complete finite set of certified integer candidates.

This viewpoint clarifies both the value and the limits of partial modular information. Continued fractions do not recover detail absent from the observation, and a rational approximation is operationally meaningful only relative to the RNS spacing and a certified uncertainty budget. Likewise, a coarse but valid tail enclosure may leave several candidates, while an invalid enclosure may destroy the certificate. When the available bounds are sufficient, full reconstruction can be deferred or avoided; when they are not, the framework states the remaining ambiguity explicitly rather than concealing it behind an unsupported point estimate.

The resulting connection between modular arithmetic, rational approximation, exact phase control, and fixed-point tail enclosure suggests a broader program in which nonmodular RNS operations are organized around task-dependent certified information rather than automatic complete reconstruction. Future work will investigate hardware-oriented realization, quantitative comparison with optimized CRT methods, residue-domain verification of candidate sets, interval-optimal anchor selection, cryptographic-scale modulus sets, and the use of minimal redundancy to protect other nonmodular RNS operations.

Appendix A.1. Detailed Exact-Rank Computation

We first isolate the auxiliary rank computation that will be used in the subsequent fractional examples. Consider the RNS with pairwise coprime odd moduli

$$(m_1,m_2,m_3,m_4)=(11,13,15,17),M={\displaystyle \prod _{i=1}^4}{m}_i=36465,$$

and extend the primary RNS with the redundant modulus 2 required by Theorem 1. Let

$$X=17293.$$

Its primary residue vector and redundant parity residue are

$$(r_1,r_2,r_3,r_4)=(1,3,13,4),r_0=X\mathrm{mod}2=1.$$

For the primary moduli, the CRT cofactors and their inverses are

$$(M_1,M_2,M_3,M_4)=(3315,2805,2431,2145),$$

$$({\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4)=(3,4,1,6).$$

Hence the normalized residues defined by (3) are

$$(x_1,x_2,x_3,x_4)=\left({⟨1·3⟩}_{11},{⟨3·4⟩}_{13},{⟨13·1⟩}_{15},{⟨4·6⟩}_{17}\right)=(3,12,13,7).$$

We now compute the exact rank by the modular rank algorithm of Theorem 1, without reconstructing X by the CRT. Taking the final primary modulus $m_4=17$, the lookup-table contributions are

$$R_{i,4}\left(r_i\right)=⌊{\displaystyle \frac{17x_i}{m_i}}⌋,i=1,2,3,$$

and

$$R_{4,4}\left(r_4\right)=x_4.$$

Their values are displayed in Table A1.

Table A1. Local contributions to the exact-rank computation for $X=17293$.

i	${\mathit{m}}_{\mathit{i}}$	${\mathit{r}}_{\mathit{i}}$	${\mathit{x}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i},4}\left({\mathit{r}}_{\mathit{i}}\right)$
1	11	1	3	$\lfloor 17·3/11\rfloor =4$
2	13	3	12	$\lfloor 17·12/13\rfloor =15$
3	15	13	13	$\lfloor 17·13/15\rfloor =14$
4	17	4	7	$x_4=7$

Table A1. Local contributions to the exact-rank computation for $X=17293$.

i	${\mathit{m}}_{\mathit{i}}$	${\mathit{r}}_{\mathit{i}}$	${\mathit{x}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i},4}\left({\mathit{r}}_{\mathit{i}}\right)$
1	11	1	3	$\lfloor 17·3/11\rfloor =4$
2	13	3	12	$\lfloor 17·12/13\rfloor =15$
3	15	13	13	$\lfloor 17·13/15\rfloor =14$
4	17	4	7	$x_4=7$

Therefore, the inexact rank is

$$\hat{\rho }\left(X\right)=⌊{\displaystyle \frac{1}{17}}{\displaystyle \sum _{i=1}^4}{R}_{i,4}\left(r_i\right)⌋=⌊{\displaystyle \frac{4+15+14+7}{17}}⌋=⌊{\displaystyle \frac{40}{17}}⌋=2.$$

To determine whether the binary correction is required, we use

$$(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)},x_4^{\left(0\right)})=(1,0,1,1),{\hat{\rho }}^{\left(0\right)}\left(X\right)=0.$$

The correction bit is consequently

$$\begin{array}{cc}\hfill \delta \left(X\right)& =\left(r_0+{\displaystyle \sum _{i=1}^4}{x}_i^{\left(0\right)}+{\hat{\rho }}^{\left(0\right)}\left(X\right)\right)\mathrm{mod}2\hfill \\ \hfill & =(1+1+0+1+1+0)\mathrm{mod}2=0.\hfill \end{array}$$

Thus the exact normalized rank is

$$\rho \left(X\right)=\hat{\rho }\left(X\right)+\delta \left(X\right)=2.$$

This computation uses only bounded local table values, one short accumulation, and the redundant parity residue. In particular, the rank is obtained independently of full CRT reconstruction. The same modular data and the recovered exact rank are retained in Appendix A.2 and Appendix A.3, where the fractional image is interpreted first without truncation and then under directed binary truncation.

Appendix A.2. Exact Fractional Image and Early Certified Stopping

We now retain the modular data and the exact rank recovered in Appendix A.1, but examine the fractional component without truncating the normalized proper fractions. The purpose of this example is twofold. First, it shows the complete Euclidean and continued-fraction computation for the exact fractional image. Second, it identifies the first convergent that already contains all information required for exact lifting, thereby making the early-stopping mechanism explicit.

For

$$(m_1,m_2,m_3,m_4)=(11,13,15,17),(x_1,x_2,x_3,x_4)=(3,12,13,7),$$

the exact normalized aggregate is

$$\begin{array}{cc}\hfill \alpha \left(X\right)& ={\displaystyle \frac{3}{11}}+{\displaystyle \frac{12}{13}}+{\displaystyle \frac{13}{15}}+{\displaystyle \frac{7}{17}}\hfill \\ \\ \hfill & ={\displaystyle \frac{90223}{36465}}=2+{\displaystyle \frac{17293}{36465}}.\hfill \end{array}$$

The exact rank computed independently in Appendix A.1 is $\rho \left(X\right)=2$. Hence the exact fractional image is

$$y:=\beta \left(X\right)=\alpha \left(X\right)-\rho \left(X\right)={\displaystyle \frac{17293}{36465}}.$$

Since no truncation or other observation error is present,

$$\tilde{\beta }\left(X\right)=\beta \left(X\right),u=0,c\left(X\right)=0.$$

Thus this example isolates the continued-fraction stage from all finite-precision effects.

Appendix A.2.1. Euclidean Decomposition

Because y is rational, its continued fraction is obtained exactly by the Euclidean algorithm. The successive divisions are

$$\begin{array}{cc}\hfill 36465& =2·17293+1879,\hfill \\ \hfill 17293& =9·1879+382,\hfill \\ \hfill 1879& =4·382+351,\hfill \\ \hfill 382& =1·351+31,\hfill \\ \hfill 351& =11·31+10,\hfill \\ \hfill 31& =3·10+1,\hfill \\ \hfill 10& =10·1+0.\hfill \end{array}$$

The same computation is displayed in Table A2, where each quotient becomes the next partial quotient of the continued fraction.

Table A2. Exact Euclidean decomposition of $\beta \left(X\right)=17293/36465$.

Step	Dividend/Divisor	Quotient	Remainder
0	$36465/17293$	2	1879
1	$17293/1879$	9	382
2	$1879/382$	4	351
3	$382/351$	1	31
4	$351/31$	11	10
5	$31/10$	3	1
6	$10/1$	10	0

Table A2. Exact Euclidean decomposition of $\beta \left(X\right)=17293/36465$.

Step	Dividend/Divisor	Quotient	Remainder
0	$36465/17293$	2	1879
1	$17293/1879$	9	382
2	$1879/382$	4	351
3	$382/351$	1	31
4	$351/31$	11	10
5	$31/10$	3	1
6	$10/1$	10	0

Therefore,

$$y={\displaystyle \frac{17293}{36465}}=[0;2,9,4,1,11,3,10].$$

Appendix A.2.2. Generation of All Convergents

The convergents are generated by the standard integer recurrences for continued fractions [27,28]:

$$p_j=a_j{p}_{j-1}+p_{j-2},q_j=a_j{q}_{j-1}+q_{j-2},$$

with

$$p_{-2}=0,p_{-1}=1,q_{-2}=1,q_{-1}=0.$$

For the partial quotients

$$(a_0,a_1,\dots ,a_7)=(0,2,9,4,1,11,3,10),$$

the recurrences give

$$\begin{array}{cccccccc}\hfill {\displaystyle \frac{p_0}{q_0}}& ={\displaystyle \frac{0}{1}},\hfill & \hfill {\displaystyle \frac{p_1}{q_1}}& ={\displaystyle \frac{1}{2}},\hfill & \hfill {\displaystyle \frac{p_2}{q_2}}& ={\displaystyle \frac{9}{19}},\hfill & \hfill {\displaystyle \frac{p_3}{q_3}}& ={\displaystyle \frac{37}{78}},\hfill \\ \\ \hfill {\displaystyle \frac{p_4}{q_4}}& ={\displaystyle \frac{46}{97}},\hfill & \hfill {\displaystyle \frac{p_5}{q_5}}& ={\displaystyle \frac{543}{1145}},\hfill & \hfill {\displaystyle \frac{p_6}{q_6}}& ={\displaystyle \frac{1675}{3532}},\hfill & \hfill {\displaystyle \frac{p_7}{q_7}}& ={\displaystyle \frac{17293}{36465}}.\hfill \end{array}$$

Table A3 records the complete integer recurrence together with the signed residual

$$d_j:=y-{\displaystyle \frac{p_j}{q_j}}$$

and the two certification tests. Since $u=0$, exact phase-stable lifting requires

$$|d_j|<{\displaystyle \frac{1}{2M}}={\displaystyle \frac{1}{72930}}.$$

Table A3. Convergents of $17293/36465$ and their certification status.

j	${\mathit{a}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	${\mathit{d}}_{\mathit{j}}=\mathit{y}-{\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	(27)	(29)
0	0	$0/1$	$17293/36465$	yes	no
1	2	$1/2$	$-1879/72930$	yes	no
2	9	$9/19$	$382/692835$	yes	no
3	4	$37/78$	$-3/24310$	no	no
$\mathbf{4}$	$\mathbf{1}$	$\mathbf{46}/\mathbf{97}$	$\mathbf{31}/\mathbf{3}\mathbf{537}\mathbf{105}$	$\mathbf{yes}$	$\mathbf{yes}$
5	11	$543/1145$	$-2/8350485$	yes	yes
6	3	$1675/3532$	$1/128794380$	yes	yes
7	10	$17293/36465$	0	yes	yes

Table A3. Convergents of $17293/36465$ and their certification status.

j	${\mathit{a}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	${\mathit{d}}_{\mathit{j}}=\mathit{y}-{\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	(27)	(29)
0	0	$0/1$	$17293/36465$	yes	no
1	2	$1/2$	$-1879/72930$	yes	no
2	9	$9/19$	$382/692835$	yes	no
3	4	$37/78$	$-3/24310$	no	no
$\mathbf{4}$	$\mathbf{1}$	$\mathbf{46}/\mathbf{97}$	$\mathbf{31}/\mathbf{3}\mathbf{537}\mathbf{105}$	$\mathbf{yes}$	$\mathbf{yes}$
5	11	$543/1145$	$-2/8350485$	yes	yes
6	3	$1675/3532$	$1/128794380$	yes	yes
7	10	$17293/36465$	0	yes	yes

The first four convergents do not satisfy both conditions simultaneously. In particular, $0/1$, $1/2$, and $9/19$ satisfy the structural certificate but remain too coarse at the intrinsic RNS scale, whereas $37/78$ satisfies neither of the two strict inequalities. The next convergent is

$${\displaystyle \frac{p_4}{q_4}}={\displaystyle \frac{46}{97}}.$$

Its signed residual is

$$d_4={\displaystyle \frac{17293}{36465}}-{\displaystyle \frac{46}{97}}={\displaystyle \frac{31}{3537105}}.$$

The structural certificate holds because

$$|d_4|={\displaystyle \frac{31}{3537105}}<{\displaystyle \frac{1}{2·{97}^2}}.$$

Moreover, the exact-lifting certificate is already satisfied:

$$u+|d_4|={\displaystyle \frac{31}{3537105}}<{\displaystyle \frac{1}{72930}}={\displaystyle \frac{1}{2M}}.$$

Thus $46/97$ is the first convergent that is simultaneously certified as continued-fraction structure and sufficiently accurate for exact phase-stable lifting.

Since $c\left(X\right)=0$, the phase-consistent rational shadow is simply

$$s\left(X;{\displaystyle \frac{46}{97}}\right)={\displaystyle \frac{46}{97}}.$$

On the integer scale,

$$M{\displaystyle \frac{46}{97}}=36465{\displaystyle \frac{46}{97}}=17293-{\displaystyle \frac{31}{97}},$$

so Corollary 3 gives

$$R_{46/97}=⌊36465{\displaystyle \frac{46}{97}}⌉=17293=X.$$

Appendix A.2.3. Why the Computation Stops at 46/97

The remaining convergents in Table A3 are included only to display the complete exact continued fraction and to verify the position of the selected fraction within it. They are not needed by the algorithm. Once $46/97$ has passed both strict certification tests, exact lifting has already been proved: no later convergent can strengthen the conclusion $R_{46/97}=X$. Continuing through $543/1145$, $1675/3532$, and finally the terminal fraction $17293/36465$ would merely refine a rational approximation after the source integer has already been recovered exactly. Therefore, in an actual execution of Algorithm 1, the continued-fraction generation terminates at

$${\displaystyle \frac{46}{97}},$$

and the final three rows of Table A3 are never computed.

Appendix A.3. Directed Binary Truncation in the Phase-Stable Branch

We now repeat the fractional computation of Appendix A.2 for the same residue system and the same source integer, but replace each exact normalized proper fraction by its directed binary truncation. This allows the effect of finite precision to be seen directly, while keeping the modular data, the exact rank, and the target integer unchanged.

Retain

$$(m_1,m_2,m_3,m_4)=(11,13,15,17),(x_1,x_2,x_3,x_4)=(3,12,13,7),$$

$$M=36465,X=17293,\rho \left(X\right)=2.$$

The precision is chosen from the intrinsic lifting scale before the continued-fraction computation is started. Since four directedly truncated fractions are accumulated, the a priori tail budget is

$$u={\displaystyle \frac{4}{2^L}}.$$

For the strict lifting condition

$$u+\left|d\right|<{\displaystyle \frac{1}{2M}}$$

to be feasible for any candidate, it is necessary that

$$u<{\displaystyle \frac{1}{2M}}.$$

Thus

$${\displaystyle \frac{4}{2^L}}<{\displaystyle \frac{1}{2·36465}},\mathrm{or}\mathrm{equivalently}{2}^L>8M=291720.$$

Now

$$2^{18}=262144<291720<524288=2^{19}.$$

Therefore, $L=19$ is the smallest common binary precision for which the a priori truncation budget lies strictly below the intrinsic lifting threshold $1/\left(2M\right)$. We accordingly choose

$$L=19,2^L=524288,$$

and define the directed truncations

$${\tilde{t}}_i:=2^{-19}⌊2^{19}{\displaystyle \frac{x_i}{m_i}}⌋,i=1,2,3,4.$$

This preliminary choice only makes certified lifting possible; the full condition, including the residual $\left|d\right|$ of the selected convergent, is verified below.

Appendix A.3.1. Directed Truncation of the Local Fractions

The four integer numerators are computed independently:

$$\begin{array}{cccc}\hfill ⌊2^{19}{\displaystyle \frac{3}{11}}⌋& =142987,\hfill & \hfill 2^{19}·3& =11·142987+7,\hfill \\ \hfill ⌊2^{19}{\displaystyle \frac{12}{13}}⌋& =483958,\hfill & \hfill 2^{19}·12& =13·483958+2,\hfill \\ \hfill ⌊2^{19}{\displaystyle \frac{13}{15}}⌋& =454382,\hfill & \hfill 2^{19}·13& =15·454382+14,\hfill \\ \hfill ⌊2^{19}{\displaystyle \frac{7}{17}}⌋& =215883,\hfill & \hfill 2^{19}·7& =17·215883+5.\hfill \end{array}$$

Consequently,

$${\tilde{t}}_1={\displaystyle \frac{142987}{524288}},{\tilde{t}}_2={\displaystyle \frac{483958}{524288}},{\tilde{t}}_3={\displaystyle \frac{454382}{524288}},{\tilde{t}}_4={\displaystyle \frac{215883}{524288}}.$$

Table A4 displays the local truncations together with the discarded binary remainders. The latter are included to make the direction and magnitude of every local error explicit.

Table A4. Directed binary truncation of the normalized proper fractions at precision $L=19$.

i	${\mathit{x}}_{\mathit{i}}/{\mathit{m}}_{\mathit{i}}$	$\lfloor 2^{19}{\mathit{x}}_{\mathit{i}}/{\mathit{m}}_{\mathit{i}}\rfloor$	$2^{19}{\mathit{x}}_{\mathit{i}}\mathrm{mod}{\mathit{m}}_{\mathit{i}}$	${\tilde{\mathit{t}}}_{\mathit{i}}$
1	$3/11$	142 987	7	$142987/524288$
2	$12/13$	483 958	2	$483958/524288$
3	$13/15$	454 382	14	$454382/524288$
4	$7/17$	215 883	5	$215883/524288$

Table A4. Directed binary truncation of the normalized proper fractions at precision $L=19$.

i	${\mathit{x}}_{\mathit{i}}/{\mathit{m}}_{\mathit{i}}$	$\lfloor 2^{19}{\mathit{x}}_{\mathit{i}}/{\mathit{m}}_{\mathit{i}}\rfloor$	$2^{19}{\mathit{x}}_{\mathit{i}}\mathrm{mod}{\mathit{m}}_{\mathit{i}}$	${\tilde{\mathit{t}}}_{\mathit{i}}$
1	$3/11$	142 987	7	$142987/524288$
2	$12/13$	483 958	2	$483958/524288$
3	$13/15$	454 382	14	$454382/524288$
4	$7/17$	215 883	5	$215883/524288$

Appendix A.3.2. Binary Accumulation and Phase

Because all four terms have the common denominator $2^{19}$, their accumulation is purely integral:

$$\begin{array}{cc}\hfill \tilde{\alpha }\left(X\right)& ={\displaystyle \sum _{i=1}^4˜}{t}_i\hfill \\ \hfill & ={\displaystyle \frac{142987+483958+454382+215883}{524288}}\hfill \\ \hfill & ={\displaystyle \frac{1297210}{524288}}=2+{\displaystyle \frac{248634}{524288}}\hfill \\ \hfill & =2+{\displaystyle \frac{124317}{262144}}.\hfill \end{array}$$

Thus the rank extracted from the truncated accumulation is

$$\tilde{\rho }\left(X\right)=2=\rho \left(X\right).$$

No turn is lost, and therefore

$$c\left(X\right)=\rho \left(X\right)-\tilde{\rho }\left(X\right)=0.$$

The observed fractional part is

$$y:=\tilde{\beta }\left(X\right)={\displaystyle \frac{124317}{262144}}.$$

For each directed truncation,

$$0\le {\displaystyle \frac{x_i}{m_i}}-{\tilde{t}}_i<2^{-19}.$$

Hence the total discarded tail satisfies the a priori bound

$$0\le \tau \left(X\right)<u,u:={\displaystyle \frac{4}{2^{19}}}={\displaystyle \frac{1}{131072}}.$$

For verification only, the local integer remainders in Table A4 give the exact tail

$$\begin{array}{cc}\hfill \tau \left(X\right)& ={\displaystyle \frac{7}{11·2^{19}}}+{\displaystyle \frac{2}{13·2^{19}}}+{\displaystyle \frac{14}{15·2^{19}}}+{\displaystyle \frac{5}{17·2^{19}}}\hfill \\ \hfill & ={\displaystyle \frac{36787}{9559080960}}<{\displaystyle \frac{1}{131072}}=u.\hfill \end{array}$$

As required by the phase-stable relation,

$$\beta \left(X\right)=y+\tau \left(X\right).$$

The algorithm, however, uses only the computable bound u, not the exact value of $\tau \left(X\right)$.

Appendix A.3.3. Euclidean Decomposition of the Truncated Observation

The continued fraction of y is again obtained entirely by integer arithmetic. The Euclidean divisions are

$$\begin{array}{cc}\hfill 124317& =0·262144+124317,\hfill \\ \hfill 262144& =2·124317+13510,\hfill \\ \hfill 124317& =9·13510+2727,\hfill \\ \hfill 13510& =4·2727+2602,\hfill \\ \hfill 2727& =1·2602+125,\hfill \\ \hfill 2602& =20·125+102,\hfill \\ \hfill 125& =1·102+23,\hfill \\ \hfill 102& =4·23+10,\hfill \\ \hfill 23& =2·10+3,\hfill \\ \hfill 10& =3·3+1,\hfill \\ \hfill 3& =3·1+0.\hfill \end{array}$$

They are summarized in Table A5.

Table A5. Exact Euclidean decomposition of the truncated observation $y=124317/262144$.

Step	Dividend/Divisor	Quotient	Remainder
0	$124317/262144$	0	$124317$
1	$262144/124317$	2	$13510$
2	$124317/13510$	9	2727
3	$13510/2727$	4	2602
4	$2727/2602$	1	125
5	$2602/125$	20	102
6	$125/102$	1	23
7	$102/23$	4	10
8	$23/10$	2	3
9	$10/3$	3	1
10	$3/1$	3	0

Table A5. Exact Euclidean decomposition of the truncated observation $y=124317/262144$.

Step	Dividend/Divisor	Quotient	Remainder
0	$124317/262144$	0	$124317$
1	$262144/124317$	2	$13510$
2	$124317/13510$	9	2727
3	$13510/2727$	4	2602
4	$2727/2602$	1	125
5	$2602/125$	20	102
6	$125/102$	1	23
7	$102/23$	4	10
8	$23/10$	2	3
9	$10/3$	3	1
10	$3/1$	3	0

Therefore,

$$y=[0;2,9,4,1,20,1,4,2,3,3].$$

The first five partial quotients coincide with those of the exact fraction in Appendix A.2:

$${\displaystyle \frac{17293}{36465}}=[0;2,9,4,1,11,3,10],$$

$${\displaystyle \frac{124317}{262144}}=[0;2,9,4,1,20,1,4,2,3,3].$$

The two expansions first differ only after the partial quotient $a_4=1$. Consequently, they generate the same convergents through $p_4/q_4=46/97$.

Appendix A.3.4. Generation of All Convergents

The convergents are generated by the standard integer recurrences for continued fractions [27,28]:

$$p_j=a_j{p}_{j-1}+p_{j-2},q_j=a_j{q}_{j-1}+q_{j-2},$$

with

$$p_{-2}=0,p_{-1}=1,q_{-2}=1,q_{-1}=0.$$

For the partial quotients

$$(a_0,a_1,\dots ,a_{10})=(0,2,9,4,1,20,1,4,2,3,3),$$

the complete sequence is

$$\begin{array}{cccccccc}\hfill {\displaystyle \frac{p_0}{q_0}}& ={\displaystyle \frac{0}{1}},\hfill & \hfill {\displaystyle \frac{p_1}{q_1}}& ={\displaystyle \frac{1}{2}},\hfill & \hfill {\displaystyle \frac{p_2}{q_2}}& ={\displaystyle \frac{9}{19}},\hfill & \hfill {\displaystyle \frac{p_3}{q_3}}& ={\displaystyle \frac{37}{78}},\hfill \\ \\ \hfill {\displaystyle \frac{p_4}{q_4}}& ={\displaystyle \frac{46}{97}},\hfill & \hfill {\displaystyle \frac{p_5}{q_5}}& ={\displaystyle \frac{957}{2018}},\hfill & \hfill {\displaystyle \frac{p_6}{q_6}}& ={\displaystyle \frac{1003}{2115}},\hfill & \hfill {\displaystyle \frac{p_7}{q_7}}& ={\displaystyle \frac{4969}{10478}},\hfill \\ \\ \hfill {\displaystyle \frac{p_8}{q_8}}& ={\displaystyle \frac{10941}{23071}},\hfill & \hfill {\displaystyle \frac{p_9}{q_9}}& ={\displaystyle \frac{37792}{79691}},\hfill & \hfill {\displaystyle \frac{p_{10}}{q_{10}}}& ={\displaystyle \frac{124317}{262144}}.\hfill \end{array}$$

Table A6 shows the signed residual

$$d_j:=y-{\displaystyle \frac{p_j}{q_j}}$$

for every convergent and applies both certification conditions with the nonzero tail budget $u=1/131072$.

Table A6. Convergents of the truncated observation and their certification status.

j	${\mathit{a}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	${\mathit{d}}_{\mathit{j}}=\mathit{y}-{\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	(27)	(29)
0	0	$0/1$	$124317/262144$	yes	no
1	2	$1/2$	$-6755/262144$	yes	no
2	9	$9/19$	$2727/4980736$	yes	no
3	4	$37/78$	$-1301/10223616$	no	no
$\mathbf{4}$	$\mathbf{1}$	$\mathbf{46}/\mathbf{97}$	$\mathbf{125}/\mathbf{25}\mathbf{427}\mathbf{968}$	$\mathbf{yes}$	$\mathbf{yes}$
5	20	$957/2018$	$-51/264503296$	no	yes
6	1	$1003/2115$	$23/554434560$	yes	yes
7	4	$4969/10478$	$-5/1373372416$	yes	yes
8	2	$10941/23071$	$3/6047924224$	yes	yes
9	3	$37792/79691$	$-1/20890517504$	yes	yes
10	3	$124317/262144$	0	yes	yes

Table A6. Convergents of the truncated observation and their certification status.

j	${\mathit{a}}_{\mathit{j}}$	${\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	${\mathit{d}}_{\mathit{j}}=\mathit{y}-{\mathit{p}}_{\mathit{j}}/{\mathit{q}}_{\mathit{j}}$	(27)	(29)
0	0	$0/1$	$124317/262144$	yes	no
1	2	$1/2$	$-6755/262144$	yes	no
2	9	$9/19$	$2727/4980736$	yes	no
3	4	$37/78$	$-1301/10223616$	no	no
$\mathbf{4}$	$\mathbf{1}$	$\mathbf{46}/\mathbf{97}$	$\mathbf{125}/\mathbf{25}\mathbf{427}\mathbf{968}$	$\mathbf{yes}$	$\mathbf{yes}$
5	20	$957/2018$	$-51/264503296$	no	yes
6	1	$1003/2115$	$23/554434560$	yes	yes
7	4	$4969/10478$	$-5/1373372416$	yes	yes
8	2	$10941/23071$	$3/6047924224$	yes	yes
9	3	$37792/79691$	$-1/20890517504$	yes	yes
10	3	$124317/262144$	0	yes	yes

As in the exact computation, none of the first four convergents satisfies both conditions. The next one is again

$${\displaystyle \frac{p_4}{q_4}}={\displaystyle \frac{46}{97}}.$$

Its signed residual with respect to the truncated observation is

$$d_4={\displaystyle \frac{124317}{262144}}-{\displaystyle \frac{46}{97}}={\displaystyle \frac{125}{25427968}}.$$

The structural certificate holds because

$$|d_4|={\displaystyle \frac{125}{25427968}}<{\displaystyle \frac{1}{2·{97}^2}}.$$

More importantly, the discarded tail is fully covered by the lifting certificate:

$$u+|d_4|={\displaystyle \frac{1}{131072}}+{\displaystyle \frac{125}{25427968}}={\displaystyle \frac{319}{25427968}}<{\displaystyle \frac{1}{72930}}={\displaystyle \frac{1}{2M}}.$$

Thus every exact fractional value compatible with the directed-truncation budget lies within one half of an RNS unit from the rational shadow $46/97$.

Since $c\left(X\right)=0$, the phase-consistent rational shadow remains

$$s\left(X;{\displaystyle \frac{46}{97}}\right)={\displaystyle \frac{46}{97}}.$$

Therefore,

$$R_{46/97}=⌊36465{\displaystyle \frac{46}{97}}⌉=17293=X.$$

Appendix A.3.5. Interpretation and Early Stopping

The exact and truncated observations have different terminal continued fractions, but they share the complete prefix $[0;2,9,4,1]$. This common prefix produces the same fifth convergent $46/97$. In the exact computation of Appendix A.2, that convergent already satisfies the intrinsic lifting scale $1/\left(2M\right)$. After directed binary truncation, the stronger quantity $u+|d_4|$ still remains below the same scale. Hence the finite-precision observation preserves not only the candidate itself, but also its exact lifting certificate.

The later convergents are displayed in Table A6 only to show the complete continued fraction of the binary observation and to make the divergence from Appendix A.2 visible. They are not generated in an actual certified execution. Once $46/97$ has passed both tests, exact recovery of X is already guaranteed, so continuing to $957/2018$, $1003/2115$, or the terminal dyadic fraction $124317/262144$ has no algorithmic purpose. The certified continued-fraction stage therefore terminates at

$${\displaystyle \frac{46}{97}}.$$

Appendix A.4. Phase Crossing and Fixed-Point Tail Enclosure

The preceding examples remain in the phase-stable branch, where the rank extracted from the directedly truncated accumulation agrees with the exact rank. We now retain the same modulus system but choose a small nontrivial source integer for which directed truncation crosses the preceding integer boundary. The purpose of this example is to demonstrate that the phase transition can be detected exactly and that the discarded tail can be enclosed tightly by bounded-width local computations, without evaluating the tail exactly and without performing a second CRT reconstruction.

Consider again

$$(m_1,m_2,m_3,m_4)=(11,13,15,17),M=36465,$$

but now let

$$X=19.$$

Its residue representation is

$$(r_1,r_2,r_3,r_4)=(8,6,4,2).$$

Unlike a source integer smaller than every primary modulus, this value is not directly visible in any individual residue channel. Using the CRT inverses from Appendix A.1,

$$({\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4)=(3,4,1,6),$$

we obtain the normalized residues

$$\begin{array}{cc}\hfill (x_1,x_2,x_3,x_4)& =\left({⟨8·3⟩}_{11},{⟨6·4⟩}_{13},{⟨4·1⟩}_{15},{⟨2·6⟩}_{17}\right)\hfill \\ \hfill & =(2,11,4,12).\hfill \end{array}$$

Appendix A.4.1. Independent Exact-Rank Computation

We first recover the exact rank by the modular procedure of Theorem 1. With the reference modulus $m_4=17$, the local rank contributions are

$$\begin{array}{cc}\hfill R_{1,4}\left(r_1\right)& =⌊{\displaystyle \frac{17·2}{11}}⌋=3,\hfill \\ \hfill R_{2,4}\left(r_2\right)& =⌊{\displaystyle \frac{17·11}{13}}⌋=14,\hfill \\ \hfill R_{3,4}\left(r_3\right)& =⌊{\displaystyle \frac{17·4}{15}}⌋=4,\hfill \\ \hfill R_{4,4}\left(r_4\right)& =x_4=12.\hfill \end{array}$$

Hence

$$\hat{\rho }\left(X\right)=⌊{\displaystyle \frac{3+14+4+12}{17}}⌋=⌊{\displaystyle \frac{33}{17}}⌋=1.$$

The redundant parity residue and the local parity bits are

$$r_0=X\mathrm{mod}2=1,(x_1^{\left(0\right)},x_2^{\left(0\right)},x_3^{\left(0\right)},x_4^{\left(0\right)})=(0,1,0,0),$$

and

$${\hat{\rho }}^{\left(0\right)}\left(X\right)=1.$$

Therefore,

$$\begin{array}{cc}\hfill \delta \left(X\right)& =\left(r_0+{\displaystyle \sum _{i=1}^4}{x}_i^{\left(0\right)}+{\hat{\rho }}^{\left(0\right)}\left(X\right)\right)\mathrm{mod}2\hfill \\ \hfill & =(1+0+1+0+0+1)\mathrm{mod}2=1,\hfill \end{array}$$

and the exact rank is

$$\rho \left(X\right)=\hat{\rho }\left(X\right)+\delta \left(X\right)=2.$$

For verification, the exact normalized aggregate is

$$\begin{array}{cc}\hfill \alpha \left(X\right)& ={\displaystyle \frac{2}{11}}+{\displaystyle \frac{11}{13}}+{\displaystyle \frac{4}{15}}+{\displaystyle \frac{12}{17}}\hfill \\ \hfill & =2+{\displaystyle \frac{19}{36465}},\hfill \end{array}$$

so

$$\beta \left(X\right)={\displaystyle \frac{19}{36465}}.$$

Thus the exact fractional image lies close to the lower boundary of the normalized dynamic range, where a directed truncation may produce a phase crossing.

Appendix A.4.2. Directed Binary Truncation and Phase Detection

Choose the common binary precision

$$L=11,2^L=2048.$$

The four local truncations and their discarded integer remainders are

$$\begin{array}{cccc}\hfill ⌊2048{\displaystyle \frac{2}{11}}⌋& =372,\hfill & \hfill 2048·2& =11·372+4,\hfill \\ \hfill ⌊2048{\displaystyle \frac{11}{13}}⌋& =1732,\hfill & \hfill 2048·11& =13·1732+12,\hfill \\ \hfill ⌊2048{\displaystyle \frac{4}{15}}⌋& =546,\hfill & \hfill 2048·4& =15·546+2,\hfill \\ \hfill ⌊2048{\displaystyle \frac{12}{17}}⌋& =1445,\hfill & \hfill 2048·12& =17·1445+11.\hfill \end{array}$$

Accordingly,

$$(e_1,e_2,e_3,e_4)=(4,12,2,11),$$

where

$$e_i:=2^{11}{x}_i\mathrm{mod}{m}_i.$$

The binary accumulation gives

$$\begin{array}{cc}\hfill \tilde{\alpha }\left(X\right)& ={\displaystyle \frac{372+1732+546+1445}{2048}}\hfill \\ \hfill & ={\displaystyle \frac{4095}{2048}}=1+{\displaystyle \frac{2047}{2048}}.\hfill \end{array}$$

Hence

$$\tilde{\rho }\left(X\right)=1,y:=\tilde{\beta }\left(X\right)={\displaystyle \frac{2047}{2048}}.$$

Comparison with the independently recovered exact rank yields

$$c\left(X\right)=\rho \left(X\right)-\tilde{\rho }\left(X\right)=2-1=1.$$

Thus the observed fractional part near 1 is not interpreted as a source integer near the upper end of the dynamic range. The exact-rank invariant identifies the phase-crossing branch and redirects the observation to the lower end of the range through

$$\beta \left(X\right)=y-1+\tau \left(X\right).$$

Appendix A.4.3. Continued-Fraction Shadow and Early Stopping

The observed fractional part has the exact continued fraction

$$y={\displaystyle \frac{2047}{2048}}=[0;1,2047].$$

Its convergents begin with

$${\displaystyle \frac{0}{1}},{\displaystyle \frac{1}{1}},{\displaystyle \frac{2047}{2048}}.$$

The initial convergent $0/1$ does not satisfy the structural certificate, since

$$\left|y-{\displaystyle \frac{0}{1}}\right|={\displaystyle \frac{2047}{2048}}\nless {\displaystyle \frac{1}{2}}.$$

The next convergent,

$${\displaystyle \frac{p}{q}}={\displaystyle \frac{1}{1}},$$

is the first structurally certified one because

$$\left|y-{\displaystyle \frac{1}{1}}\right|={\displaystyle \frac{1}{2048}}<{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2·1^2}}.$$

The signed observation residual is

$$d=y-{\displaystyle \frac{p}{q}}={\displaystyle \frac{2047}{2048}}-1=-{\displaystyle \frac{1}{2048}}.$$

Since $c\left(X\right)=1$, the phase-consistent rational shadow is

$$s\left(X;{\displaystyle \frac{1}{1}}\right)=1-1=0.$$

Therefore,

$$s\left(X;{\displaystyle \frac{1}{1}}\right)+d=0-{\displaystyle \frac{1}{2048}}=-{\displaystyle \frac{1}{2048}}=y-c\left(X\right).$$

By (14), this phase-corrected anchor is independent of the particular structurally certified convergent used to represent the observation. Hence the terminal convergent $2047/2048$ need not be generated: the continued-fraction stage stops at the first structurally certified convergent $1/1$, and the remaining resolving power is supplied by the certified enclosure of the positive truncation tail.

Appendix A.4.4. Lower Tail Bound from the Cauchy–Schwarz Estimate

Following Proposition 1, define

$$T:={\displaystyle \sum _{i=1}^4}{\displaystyle \frac{e_i}{m_i}},\tau \left(X\right)=2^{-11}T.$$

The short accumulations required for the lower estimate are

$$S:={\displaystyle \sum _{i=1}^4}{e}_i=4+12+2+11=29$$

and

$$\begin{array}{cc}\hfill W:={\displaystyle \sum _{i=1}^4}{e}_i{m}_i& =4·11+12·13+2·15+11·17=417.\hfill \end{array}$$

Therefore,

$$T\ge T_{\min }:={\displaystyle \frac{S^2}{W}}={\displaystyle \frac{{29}^2}{417}}={\displaystyle \frac{841}{417}}.$$

Appendix A.4.5. Upper Tail Bound from the Dominant-Loss Channel

The relative local truncation losses are proportional to

$${\displaystyle \frac{4}{11}},{\displaystyle \frac{12}{13}},{\displaystyle \frac{2}{15}},{\displaystyle \frac{11}{17}}.$$

The largest is $12/13$, so the deterministic anchor rule of Remark 7 selects

$${\ell }_{\ast }=2,m_{{\ell }_{\ast }}=13.$$

The required local Euclidean divisions are

$$\begin{array}{cc}\hfill 13·4& =4·11+8,\hfill \\ \hfill 13·12& =12·13+0,\hfill \\ \hfill 13·2& =1·15+11,\hfill \\ \hfill 13·11& =8·17+7.\hfill \end{array}$$

Thus

$$(q_1^{\left(2\right)},q_2^{\left(2\right)},q_3^{\left(2\right)},q_4^{\left(2\right)})=(4,12,1,8),$$

$$(s_1^{\left(2\right)},s_2^{\left(2\right)},s_3^{\left(2\right)},s_4^{\left(2\right)})=(8,0,11,7),$$

and

$$Q_2:={\displaystyle \sum _{i=1}^4}{q}_i^{\left(2\right)}=25.$$

The common tabular precision is fixed by the largest modulus:

$$h:=⌈{\log }_2{m}_{\max }⌉=⌈{\log }_{2}17⌉=5.$$

Hence the directed dyadic majorants use the common denominator $2^h=32$. Their integer numerators are

$$\begin{array}{cc}\hfill b_1^{\left(2\right)}& =⌈{\displaystyle \frac{32·8}{11}}⌉=24,\hfill \\ \hfill b_2^{\left(2\right)}& =0,\hfill \\ \hfill b_3^{\left(2\right)}& =⌈{\displaystyle \frac{32·11}{15}}⌉=24,\hfill \\ \hfill b_4^{\left(2\right)}& =⌈{\displaystyle \frac{32·7}{17}}⌉=14.\hfill \end{array}$$

Therefore,

$$T\le T_{\max }^{\left(2\right)}:={\displaystyle \frac{1}{13}}\left(25+{\displaystyle \frac{24+0+24+14}{32}}\right)={\displaystyle \frac{1}{13}}\left(25+{\displaystyle \frac{31}{16}}\right)={\displaystyle \frac{431}{208}}.$$

Combining the two independent estimates gives the certified tail enclosure

$${\displaystyle \frac{841}{417·2048}}\le \tau \left(X\right)\le {\displaystyle \frac{431}{208·2048}}.$$

Appendix A.4.6. Certified Integer Recovery

Since $c\left(X\right)=1$, $y=2047/2048$, and

$$s\left(X;{\displaystyle \frac{1}{1}}\right)+d=-{\displaystyle \frac{1}{2048}},$$

Proposition 1 gives

$$\begin{array}{cc}\hfill I_2^{\mathrm{tail}}& =36465\left[-{\displaystyle \frac{1}{2048}}+{\displaystyle \frac{841}{417·2048}},-{\displaystyle \frac{1}{2048}}+{\displaystyle \frac{431}{208·2048}}\right]=\left[{\displaystyle \frac{644215}{35584}},{\displaystyle \frac{625515}{32768}}\right].\hfill \end{array}$$

Numerically,

$$I_2^{\mathrm{tail}}=[18.104063\dots ,19.089203\dots ].$$

This interval lies at the very beginning of the full dynamic range

$$[0,M)=[0,36465)$$

and contains exactly one integer. Consequently,

$${\mathcal{C}}_2^{\mathrm{tail}}=I_2^{\mathrm{tail}}\cap \mathbb{Z}=\left\{19\right\},$$

and the source integer is recovered exactly: $X=19$.

No exact evaluation of T, no exact reconstruction of the discarded tail, and no second CRT reconstruction are used in this certificate. The lower endpoint follows from two short accumulations and the Cauchy–Schwarz estimate; the upper endpoint uses one dominant-loss anchor, local Euclidean divisions, and dyadic tabular majorants with the prescribed common precision

$$h=⌈{\log }_2{m}_{\max }⌉=5.$$

This is the smallest common word length satisfying $2^h\ge m_i$ for every primary channel and therefore provides one uniform dyadic table format across the complete modulus set. Smaller values of h still define valid but coarser directed majorants and do not satisfy this full-range common-format criterion. If the same construction produced an interval containing several integers, the framework would return their complete certified set, as prescribed by Algorithm 1.

Appendix A.4.7. Verification of the Phase Boundary

The exact tail is not required by the recovery procedure. It may nevertheless be evaluated after the fact to verify the numerical example:

$$\tau \left(X\right)={\displaystyle \frac{1}{2048}}\left({\displaystyle \frac{4}{11}}+{\displaystyle \frac{12}{13}}+{\displaystyle \frac{2}{15}}+{\displaystyle \frac{11}{17}}\right)={\displaystyle \frac{75377}{74680320}}={\displaystyle \frac{1}{2048}}+{\displaystyle \frac{19}{36465}}.$$

Thus

$$\tau \left(X\right)>\beta \left(X\right)={\displaystyle \frac{19}{36465}},$$

which confirms the detected phase crossing. At the next binary precision $L=12$, the directed accumulation becomes exactly $\tilde{\alpha }\left(X\right)=2$, so

$$\tilde{\beta }\left(X\right)=0,\tau \left(X\right)=\beta \left(X\right),c\left(X\right)=0.$$

This adjacent-precision comparison illustrates the boundary statement of Theorem 2: strict inequality $\tau \left(X\right)>\beta \left(X\right)$ produces the crossing branch, whereas equality belongs to the phase-stable branch.

This calculation completes the detailed numerical record supporting Section 6. The four appendix subsections respectively verify independent exact-rank recovery, certified early stopping for the exact fractional image, preservation under directed truncation of the same rational representative selected at the prescribed output level, and exact recovery after a detected phase crossing by fixed-point enclosure of the discarded tail.