Recurrence-Equivalence Exponent

• Recurrence-equivalence exponent is a principled quantifier that converts qualitative or asymptotic recurrences into explicit finite scaling laws, enabling precise computations in diverse fields.
• In analytic number theory, it defines the minimal exponent for converting prime recurrences into effective algorithms, directly impacting predictions of prime gaps and sequence generation.
• In deep learning and dynamical systems, the exponent quantifies resource scaling and mixing rates respectively, offering actionable insights for model design and system analysis.

The recurrence-equivalence exponent is a principled quantifier that translates qualitative or asymptotic recurrence relations—arising in diverse mathematical, computational, and statistical contexts—into effective, finite scaling laws or computational procedures. Across analytic number theory, linear recurrences, dynamical systems, and modern machine learning, the recurrence-equivalence exponent isolates the minimal parameter (typically an exponent or power) at which recurrence-based constructions achieve an explicit, verifiable equivalence to an otherwise limiting or iterative process. This concept thus fundamentally connects the effectiveness of recurrence strategies with the quantitative growth, prediction, or computational resource requirements of the underlying system.

1. Analytic Number Theory: Golomb–Keller Recurrence and Prime Generation

In prime number theory, the recurrence-equivalence exponent $s_n$ emerges as the minimal exponent rendering analytic recurrences for prime sequences explicit and exact. The original (asymptotic) Golomb–Keller recurrence generates the $(n+1)$th prime by

$$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$

where $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$. To make this formula effective (i.e., computable for finite $s$), one defines $s_n$ as the minimal $s > 1$ such that $h_n(s) = \left(D_n(s) - 1\right)^{-1/s}$ falls into $(p_{n+1}-1, p_{n+1}]$. The explicit finite recurrence then becomes

$$p_{n+1} = \left\lceil \left(-1 + \zeta(2p_n)\prod_{j=1}^n\left(1 - p_j^{-2p_n}\right)\right)^{-1/(2p_n)} \right\rceil$$

for all $(n+1)$0, where $(n+1)$1 is always sufficient by Bertrand’s postulate. This explicit value of $(n+1)$2 is the recurrence-equivalence exponent for this setting; it replaces the formal limit with a provable finite threshold at which the recurrence, in a single step, yields the next prime in sequence (Cloitre, 23 Jul 2025).

Empirical evidence suggests that the effective bound is not tight, with $(n+1)$3 often significantly less than one. Should $(n+1)$4 hold universally, the method would be both computationally superior and more directly reflective of local prime gaps.

2. Deep Learning: Scaling Laws and Model Sharing

In modern LLM design, the recurrence-equivalence exponent $(n+1)$5 quantifies how architectural recurrence (i.e., parameter sharing via looped or depth-recurrent blocks) maps to effective model capacity. The empirical Chinchilla scaling law

$(n+1)$6

relates the validation loss $(n+1)$7 to model architecture and training budget, with $(n+1)$8 and $(n+1)$9 representing the parameters in unique and recurrently shared blocks, respectively, and $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$0 the recurrence count. The exponent $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$1 formalizes the fractional equivalence: $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$2 yields full equivalence to an unshared model; $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$3 implies no benefit from recurrence. The measured value $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$4 establishes that each recurrence is worth $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$5 unique blocks in terms of empirical performance, quantifying the cost of parameter sharing at fixed compute (Schwethelm et al., 22 Apr 2026).

The recurrence-equivalence exponent thus allows principled model comparison, design optimization, and interpretable resource scaling for looped architectures. It also predicts downstream task behavior, with observed validation loss angles tracking $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$6 most strongly on parametric knowledge tasks.

3. Linear Recurrences: Quantitative Growth and Rate Bounds

The recurrence-equivalence exponent also quantifies the maximal (effective) exponential rate of growth in non-degenerate linear recurrence sequences. For a sequence

$$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$7

with dominant root modulus $$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$8, define

$$p_{n+1} = \lim_{s \to \infty} \left(D_n(s) - 1\right)^{-1/s},$$9

as the recurrence-equivalence exponent (here denoted $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$0) of $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$1. The effective theorem of Noubissie establishes that, for all such sequences, $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$2: for every $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$3, only finitely many $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$4 satisfy $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$5, with explicit upper bounds on such $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$6 (Noubissie, 13 Apr 2025).

This result confirms the expectation that “generic” non-degenerate linear recurrences grow at the full exponential rate $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$7, and that no smaller exponent is possible in effective big-O bounds.

4. Dynamical Systems: Quantitative Recurrence and Hitting-Time Exponents

In ergodic theory and smooth dynamics, the recurrence-equivalence exponent (also called the quantitative recurrence or hitting-time exponent) is defined for metric dynamical systems as

$$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$8

where $$D_n(s) = \zeta(s) \prod_{j=1}^n \left(1 - p_j^{-s}\right)$$9 denotes the first hitting time of $s$0 within $s$1 of $s$2. These exponents quantitatively calibrate the local recurrence rate and typically coincide with the local dimension of the invariant measure for mixing systems, but are heavily modulated by Diophantine properties in toral extensions (Galatolo et al., 2011).

The recurrence exponent appears as an upper bound in decay-of-correlation estimates, for instance, as

$s$3

when the relevant Diophantine type is $s$4. Thus, the recurrence-equivalence exponent connects local metric, probabilistic, and arithmetic properties of the system with observable mixing rates.

5. Algorithmic Implementation: Power Series and the Differential Transformation Method

In algorithmic contexts such as the Differential Transformation Method (DTM) for nonlinear ODEs, the recurrence-equivalence exponent underpins the conversion of nonlinear terms (power and exponential) into efficient, explicit recurrences for the coefficients. Miller’s recurrence for the $s$5th power,

$s$6

and the analogous recurrence for the exponential demonstrate that exponentiation operations become linear-time, one-pass recurrences—a major efficiency gain over naive iterative expansion (Finkel, 2010). This “recurrence-equivalence” effectively quantifies when the operation $s$7 or $s$8 becomes computationally equivalent to a recurrence, rather than repeated symbolic multiplication.

6. Comparative Summary of Definitions and Interpretations

Context	Exponent Notation	Quantifies
Analytic prime recurrence	$s$9	Minimal $s_n$0 giving exact next prime
Looped neural networks	$s_n$1	Equivalence of $s_n$2 recurrences to capacity
Linear recurrences	$s_n$3	Minimal rate exponent for exponential growth
Dynamical systems (hitting-time)	$s_n$4	Local recurrence/hitting-time scaling law
Series exponentiation (DTM)	—	O(N) recurrence equivalence for exponentiation

Across these settings, the recurrence-equivalence exponent provides a unifying conceptual and quantitative tool for making asymptotic, iterative, or shared-structure processes explicit, efficient, and effectively analyzable.

7. Open Problems and Research Directions

Major open problems concerning recurrence-equivalence exponents include sharpening provable bounds (e.g., establishing $s_n$5 in the prime recurrence context (Cloitre, 23 Jul 2025)), elucidating the precise dependence of $s_n$6 on architecture and optimization in machine learning (Schwethelm et al., 22 Apr 2026), and characterizing the recurrence exponents for exceptional or degenerate dynamical systems (Galatolo et al., 2011). The effective, explicit calibration of these exponents continues to play a critical role in analytic number theory, recurrence sequence growth, learning theory, deterministic chaos, and algorithmic symbolic computation.