Accuracy-Preserving Recursive Construction

• Accuracy-preserving recursive construction is an algorithmic framework that maintains key quantitative invariants—such as numerical precision, bit-complexity, and algebraic correctness—at every recursion level.
• It applies techniques like graded packing, degree projection, and controlled rounding to ensure that performance metrics remain stable even in complex applications like numerical linear algebra and coding theory.
• The approach is designed to prevent degradation or uncontrolled growth of error, making it vital for scalable distributed search, robust statistical prediction, and high-fidelity floating-point simulations.

An accuracy-preserving recursive construction is an algorithmic design guaranteeing that recursive or divide-and-conquer procedures maintain the essential quantitative invariants—such as numerical precision, bit-complexity, information rate, or algebraic correctness—across all recursion levels. Such constructions are central in numerical linear algebra, coding theory, distributed search, statistical prediction, polynomial regression, quantum dynamics, integration, and floating-point simulation. They ensure that (a) the quantitative metric of interest (accuracy, information, bit-size, recall, etc.) does not degrade (or blows up only in a provably controlled manner) as the recursion deepens, and (b) structural or optimality properties are preserved exactly or within analytically bounded error envelopes.

1. Formal Foundations of Accuracy-Preserving Recursive Construction

Defining accuracy-preserving recursive constructions requires specification of the invariants tracked by the recursion and the algebraic, analytic, or statistical mechanisms by which they are preserved.

• Graded packing and projection (GPR): In recursive algorithms for bilinear problems (e.g., matrix multiplication), the core mechanism is graded embedding: pack independent subproblems as coefficients in polynomials with distinct degrees in a formal parameter β, operate in the extended module, and extract outputs by degree projection. The three critical invariants are algebraic realizability, coefficient ownership (preventing reuse of packed temporaries), and bounded numeric magnitude at each level, ensuring a fixed β suffices for all subcalls (Uhlmann, 15 Nov 2025).
• Statistical sufficiency in recursive predictors: In time-series prediction, recursive construction defines states by partitioning pasts that induce the same prediction kernel on the future. The causal-state mapping ε is minimal sufficient, recursively updated, and accuracy-preserving if, for each new history, all informative dependence on the infinite past is retained (Shalizi et al., 2014).
• Partitioning and balanced granularity (vector search): In distributed search, a recursive bottom-up multi-level partitioning is accuracy-preserving if all search levels adopt the same “balanced” partition granularity, so the per-level recall remains fixed and the global recall composes multiplicatively, ensuring target accuracy as the number of levels grows (Xu et al., 19 Dec 2025).
• Biorthogonal polynomial recurrences: In polynomial regression, recursive upgrading and downgrading of biorthogonal polynomial bases are accuracy-preserving if the L²-least-squares error always monotonically decreases when the degree increases, and the numerical implementation avoids ill-conditioned matrix inversions (Rebollo-Neira et al., 8 Jun 2024).
• Recursive cubature and code constructions: In projective cubature and subspace code construction, recursion proceeds by lifting formulas or codes from lower dimensions (or smaller ambient spaces), ensuring each step preserves degree, distance, or embedding isometry exactly (Lyubich et al., 2013, 0806.3650).
• Floating-point recursive simulation: In numerical recursion under finite precision, rounding schemes (e.g., averaging round-down and round-up) achieve provable reductions in RMS error versus standard rounding, corresponding to one extra bit of precision per recursive level, with empirical validation (Silva et al., 2017).

2. Mechanisms for Control of Quantitative Invariants

Different mathematical domains require tailored mechanisms to maintain quantitative control as the recursion proceeds.

Domain	Control Mechanism / Invariant	Reference
Integer bilinear algebra	Bit-magnitude gap, graded β-packing	(Uhlmann, 15 Nov 2025)
Polar code construction	β-interval persistence for ordering	(He et al., 2017)
Distributed search	Balanced partition, fixed per-level recall	(Xu et al., 19 Dec 2025)
Nonlinear time-series prediction	Causal state minimal sufficient statistic	(Shalizi et al., 2014)
Biorthogonal polynomial regression	Orthogonal projections, monotonic L² error	(Rebollo-Neira et al., 8 Jun 2024)
Cubature / isometric embedding	Exact splitting, polynomial degree	(Lyubich et al., 2013)
Floating-point recursive simulation	RMS error reduction by averaged rounding	(Silva et al., 2017)

• In GPR, the bit-magnitude gap is enforced globally by choosing β ≥ 4S₀+1, so at every recursion depth ℓ, the extraction band (|L_ℓ/β|, |M_ℓ/β + L_ℓ/β²|) lies below ½: this makes mid-extraction of coefficients exact (Uhlmann, 15 Nov 2025).
• In β-expansion polar coding, at each recursion the β-interval is determined by the roots of corresponding polynomials, guaranteeing stability of channel order for all lengths greater than a threshold and thus the nested frozen set property (He et al., 2017).
• In SPIRE, distributed ANNS index accuracy stays predictable as levels are added because the multiplicative composition of per-level recalls is governed by the “balanced” partition point; this ensures both recall and per-query cost remain stable (Xu et al., 19 Dec 2025).
• In polynomial regression, recursive three-term recurrence and rank-one updates to biorthogonal duals ensure that the orthogonal projection always improves or retains L² accuracy at each stage (Rebollo-Neira et al., 8 Jun 2024).
• Averaged rounding in floating-point reduces RMS error by √2 per recursion, directly reflecting an extra bit of effective precision at each step (Silva et al., 2017).

3. Representative Algorithms and Recursion Patterns

Major frameworks and algorithmic skeletons proven to be accuracy-preserving include:

• Graded Projection Recursion (GPR): Applies to recursive bilinear algorithms, especially matrix multiplication. The recursive subroutine (GPR_PACKED) always forms new packed polynomials strictly from unscaled input blocks, projects the desired coefficient, and maintains a uniform bit-length bound. Each call's output cannot depend on packed temporaries returned by its children. The total bit complexity is quadratic up to logarithmic factors, matching the word-level cost model (Uhlmann, 15 Nov 2025).
• β-expansion for polar code ranking: The recursive formula for polarization weight is w_i(β) = ∑_k b_k β^k with the order of channels ranked stably as block length increases, independent of the communication channel as long as β lies in the same subinterval (He et al., 2017).
• CSSR for HMM-based prediction: The causal-state splitting algorithm recursively detects new states by empirical sufficiency tests on longer suffixes; each phase split preserves minimal sufficiency, and the final recursion graph enforces deterministic update of states, yielding accuracy preservation under information-theoretic criteria (Shalizi et al., 2014).
• SPIRE distributed vector index: Bottom-up construction recursively clusters data at balanced granularity, stores partition centroids at the next level, and recurses until all higher levels fit in memory. The global recall is guaranteed to be ≥ r^N for N levels with per-level recall r (Xu et al., 19 Dec 2025).
• Recursive construction of biorthogonal polynomials: The canonical three-term recurrence and analytic formulas for the biorthogonal duals allow degree increments or removals with monotonic improvement in projection accuracy and high numerical stability (Rebollo-Neira et al., 8 Jun 2024).
• Recursive projective cubature and network codes: Explicit recurrences construct higher-dimensional cubature nodes or add subspace codewords, with proofs that exactness or minimum distance are preserved at each step (Lyubich et al., 2013, 0806.3650).

4. Rigorous Guarantees and Proof Methodologies

For each domain, accuracy preservation is substantiated by mathematical theorems:

• Bit-complexity in GPR: Theorems (Lemma 2.1, Theorem 2.2, Lemma 2.3 in (Uhlmann, 15 Nov 2025)) show that with appropriate β, all intermediate and final extracted coefficients are bounded in bit-length, and no exponential blowup can occur even as recursion depth increases; extraction at every level is exact.
• Nestedness in β-expansion polar codes: For any admissible β > 1, the ranking of synthetic channels at length 2^n is exactly nested with that at 2^{n+1}; the set of polynomial inequalities defining tie-breaks further collapses to a universal interval as n grows (He et al., 2017).
• Statistical convergence in CSSR: Under finite-state, stationary, and synchronizing process assumptions, the probability that the reconstructed causal-state partition disagrees with the truth vanishes as sample size increases, and total-variation error decays as O(N^{-1/2}) (Shalizi et al., 2014).
• Rate/distance in recursive network codes: The construction's recurrence and intersection analysis prove the augmented code meets the same minimum distance as the base code, so rate increases are not accompanied by loss of correctness (0806.3650).
• Floating-point improvement: The variance of local roundoff errors for averaged rounding is half that of standard rounding, yielding one extra bit of mean precision, validated explicitly for both stable and chaotic recurrences (Silva et al., 2017).

5. Applications and Domain-Specific Instantiations

Accuracy-preserving recursive constructions are foundational in multiple fields:

• Massively scalable approximate nearest neighbor search: SPIRE achieves predictable, strictly bounded accuracy and latency for multi-billion scale indexes by recursive balanced partitioning (Xu et al., 19 Dec 2025).
• Efficient and numerically stable high-order polynomial regression: The adaptive recursive construction enables stable fits for large degrees without matrix inversion instability (Rebollo-Neira et al., 8 Jun 2024).
• Fast polar code design: β-expansion algorithms permit rapid, channel-independent construction of nested frozen sets at linear time and tight SNR performance (He et al., 2017).
• Finite-precision simulation of chaotic or sensitive recursions: Averaged rounding prevents spurious divergences and loss of stationary behavior in floating-point systems (Silva et al., 2017).
• Bilinear tensor computations in integer domains: GPR ensures claimed algebraic speedups genuinely deliver at the bit-complexity level (Uhlmann, 15 Nov 2025).
• Nonlinear prediction of discrete processes: CSSR provides statistically optimal, size-efficient models directly from data, outstripping variable-length Markov models and EM-trained HMMs (Shalizi et al., 2014).
• Projective cubature and embeddings: Recursive cubature formulae produce minimal node counts for high-dimensional integration or isometric embedding in L_p spaces (Lyubich et al., 2013).
• Network coding with random topology: Recursive code constructions yield greater rates without sacrificing essential error or erasure correction properties (0806.3650).
• Open quantum systems: Recursive perturbative expansions for time-convolutionless master equations yield generators preserving Lindblad (trace and Hermiticity) structure at every order, even when complete positivity may not be preserved locally (Colla et al., 4 Jun 2025).

6. Generalizations and Future Directions

Recent research has extended accuracy-preserving recursive techniques:

• Recursion in non-bilinear algorithms: The GPR Substitution Principle enables provably correct bit-complexity for a broad class of recurrences where non-recursive work decomposes into bit-controlled bilinear kernels (Uhlmann, 15 Nov 2025).
• Interval arithmetic and stochastic rounding: Averaged rounding and similar schemes may generalize to ODE solvers, stochastic simulations, and high-precision requirements (Silva et al., 2017).
• Polyadic and non-binary polar codes: Extensions to domains with non-binary kernels or adaptive rate-matching in polar codes (He et al., 2017) open new areas for recursive, invariant-preserving methodologies.
• Quantum information and non-Markovian dynamics: The recursive perturbative framework for master equations generalizes to arbitrary open quantum systems, allowing higher order effects and strong coupling to be systematically included while preserving key structural features (Colla et al., 4 Jun 2025).
• Adaptive model selection and downgrading: Recursive downgrading operations in polynomial regression and model compression provide dynamic control over accuracy-resource tradeoffs, with provable maintenance of approximation error bounds (Rebollo-Neira et al., 8 Jun 2024).
• Optimization of cubature and code parameters: Investigation of parameter regimes maximizing gain (e.g., choice of h_{ℓ+m} in network codes) is linked to the boundary of achievable accuracy and efficiency (0806.3650).

7. Comparison with Non-Recursive or Non-Preserving Schemes

Accuracy-preserving recursive constructions are distinguished by their explicit, often structural, invariants carried through every level of recursion. In contrast:

• Algorithms lacking coefficient ownership or global gap conditions can suffer catastrophic bit-length or numeric blowup, as the magnitude of packed intermediate data may grow exponentially, undermining their asymptotic theoretical guarantees (Uhlmann, 15 Nov 2025).
• Standard floating-point recursion accumulates roundoff errors linearly or faster with depth, and is sensitive to chaotic divergence—a single-step improvement may not suffice without recursive averaging or error control (Silva et al., 2017).
• Naive or direct cubic polynomial regression models experience severe instability and lose L² accuracy for moderate to large degrees due to Gramian ill-conditioning, in contrast with accuracy-preserving recursions (Rebollo-Neira et al., 8 Jun 2024).

In summary, an accuracy-preserving recursive construction provides a rigorous framework, in both design and proof, for achieving scalable, stable, and provably optimal or controlled accuracy in recursive algorithms across computational mathematics, coding theory, statistical learning, distributed systems, and quantum information science.