Abstract Prime gaps are traditionally studied through their global distribution, asymptotic behaviour, and conjectural models. In this work we investigate a different question: how does the prime sequence behave immediately after an unusually large normalized prime gap? Using datasets containing up to forty million consecutive primes, we examine the conditional distribution of prime gaps following events in which the normalized gap is significantly larger than average. We find that the subsequent distribution is not uniform. Instead, the probability of observing particular small gaps depends strongly on residue class modulo 210 and on the target gap being examined. The strongest effect is observed for Gap 6 following very large normalized gaps. Gap 2, Gap 4, and Gap 6 each exhibit distinct hotspot residue families and distinct collections of informative wheel offsets. We refer to these collections as local obstruction signatures. The results are empirical. No theorem is proved. The purpose of this work is to document a reproducible statistical phenomenon and identify compact local structures associated with that phenomenon. 1. Introduction Prime numbers become increasingly sparse as numbers grow larger. The average spacing between consecutive primes near a large number is approximately, yet the actual gaps fluctuate substantially. Some gaps are considerably smaller than average, while others are unexpectedly large. Most investigations focus directly on the distribution of prime gaps themselves. Questions typically concern the frequency of particular gaps, the growth of maximal gaps, bounded-gap phenomena, or the occurrence of prime constellations. The present study examines a different aspect of the prime sequence. Instead of studying gaps in isolation, we investigate what happens immediately after an unusually large normalized gap. Suppose a prime is followed by a gap substantially larger than the average spacing expected at that scale. Once the next prime finally appears, does the sequence immediately return to its ordinary statistical behaviour, or does the preceding large gap leave a detectable local signature? This question motivates the present investigation. 2. Definitions Let Δₙ = pₙ₊₁ − pₙ denote the ordinary prime gap. To compare gaps occurring at different scales, define the normalized gap Eₙ = Δₙ / log (pₙ). Values near one correspond approximately to average-sized gaps. Large values correspond to unusually sparse local regions of the prime sequence. Three thresholds were examined: Eₙ > 2 Eₙ > 4 Eₙ > 6 These define increasingly rare large-gap events. To quantify recovery after such events, define the rebound ratio Rₖ = P (next gap = k | Eₙ > A) / P (gap = k). Values greater than one indicate enhancement relative to baseline frequencies, while values below one indicate suppression. The analysis focuses on: Gap 2, Gap 4, Gap 6. Primes are additionally classified according to their residue class modulo 210 = 2 × 3 × 5 × 7. This modulus provides a natural description of the local wheel structure generated by the first four prime divisors. 3. Dataset and Computational Procedure All computations were performed on datasets containing up to forty million consecutive primes. For each prime: 1. The ordinary gap Δₙ was computed. 2. The normalized gap Eₙ was computed. 3. Events satisfying Eₙ > A were identified. 4. The subsequent gap Δₙ₊₁ was recorded. 5. The residue class modulo 210 was recorded. All rebound statistics, hotspot rankings, geometric scores, and offset searches were computed from the same dataset. The study is entirely computational and empirical. No asymptotic claims are made. 4. Conditional Gap Statistics The first threshold examined was Eₙ > 2. At this level the rebound strengths were approximately R₂ ≈ 1. 06, R₄ ≈ 1. 06, R₆ ≈ 1. 04. All three small gaps become slightly more likely following a moderately large normalized gap. The effect is measurable but weak. Recovery remains broadly distributed across residue classes and no sharply defined hotspot structure is visible. The second threshold examined was Eₙ > 4. At this level the rebound strengths become approximately R₂ ≈ 1. 08, R₄ ≈ 1. 08, R₆ ≈ 1. 06. Although the numerical increase is modest, distinct residue-dependent structures begin to emerge. Certain residue classes repeatedly exhibit stronger recovery than others. The strongest effects appear for Eₙ > 6. At this threshold the behaviour changes qualitatively. Approximate rebound strengths become R₂ ≈ 0. 33, R₄ ≈ 0. 34, R₆ ≈ 1. 27. Gap 6 becomes strongly enhanced while gaps 2 and 4 become suppressed overall. This indicates that the recovery process is selective rather than uniformly favouring small gaps. 5. Residue-Dependent Recovery To investigate the location of recovery events, primes were classified by residue class modulo 210. The analysis revealed substantial residue dependence. For Gap 6, prominent hotspot residues include 181, 23, 47, 73, 13, and 41. For Gap 2, prominent hotspots include 17, 179, 167, 197, and 191. For Gap 4, prominent hotspots include 139, 43, 67, 37, and 13. The overlap between these families is limited. This observation suggests that Gap 2, Gap 4, and Gap 6 are not responding to a single universal recovery mechanism. Instead, different target gaps appear to be associated with different residue families. One of the central observations of the study is therefore that recovery after large normalized gaps is strongly residue dependent. 6. Geometric Recovery Models The next objective was to understand why particular residues become hotspots. Inspection of the strongest Gap-6 residues suggested that many possess heavily obstructed neighbourhoods immediately preceding the first admissible recovery position. This observation motivated the introduction of two geometric quantities. The first-return score measures the extent to which nearby admissible positions are blocked before the first admissible return step. The future-richness score measures the density of admissible opportunities beyond the recovery point. The combined model may be written schematically as X₆ (r) = G₆ (r) + F₆ (r), where G₆ (r) denotes first-return structure and F₆ (r) denotes future-neighbourhood richness. This model successfully explains part of the observed residue variation. However, a substantial fraction of the variation remained unexplained. 7. Local Obstruction Signatures To identify the missing component, a systematic search over nearby wheel offsets was performed. Every nearby offset was tested to determine whether its admissibility status correlated with recovery strength. The resulting analysis revealed surprisingly compact collections of informative offsets. For Gap 6, the strongest offsets were +4, +44, −12, −20, −26. For Gap 4, the strongest offsets were +34, +8, +38, +50, −14. For Gap 2, the strongest offsets were +38, +54, −16, −4. These offsets are not recovery locations themselves. Instead, they appear to encode information about the surrounding admissibility structure generated by divisibility constraints from the primes 2, 3, 5, and 7. We refer to these collections as local obstruction signatures. The hotspot residues and the obstruction signatures represent two different descriptions of the same phenomenon. The hotspot residues identify where recovery is strongest, while the obstruction signatures identify local geometric features associated with that recovery. It is important to emphasize that these signatures were identified empirically through computational searches and are not derived from an underlying theoretical model. 8. Interpretation The observations described above suggest a coherent empirical picture. Following sufficiently large normalized gaps, the distribution of subsequent prime gaps depends not only on the size of the preceding gap but also on the surrounding local wheel geometry. One possible interpretation is that large normalized gaps create a temporary local configuration in which recovery probabilities become sensitive to nearby obstruction structure. Within this interpretation, different target gaps may respond to different aspects of the local admissibility environment. The data are consistent with such a mechanism, although no theoretical derivation is presently available. 9. Main Findings The investigation produced several notable observations. First, recovery after large normalized gaps is not random. Second, recovery depends strongly on residue class modulo 210. Third, Gap 2, Gap 4, and Gap 6 exhibit distinct hotspot residue families. Fourth, Gap 2, Gap 4, and Gap 6 exhibit distinct local obstruction signatures. Fifth, the strongest structure appears for Gap 6 at threshold Eₙ > 6. Sixth, compact collections of wheel offsets capture a substantial fraction of the observed residue variation. 10. Validation and Robustness Tests Following the initial discovery of hotspot residues and local obstruction signatures, a series of additional validation experiments was performed to assess the stability and predictive value of the observed structures. 10. 1 Out-of-Sample ValidationThe dataset of forty million consecutive primes was divided into two independent halves consisting of the first twenty million and second twenty million primes. The Gap-6 obstruction signature+4, +44, −12, −20, −26was identified and then evaluated separately on both halves of the dataset. Positive correlations between the signature score and observed recovery strength were observed in both subsets. The correlation was approximately 0. 317 in the first half and 0. 420 in the second half. These results indicate that the signature is not confined to a single subset of the data and retains predictive value across independent samples. 10. 2 Stability Across Dataset SizeTo examine finite-sample effects, the Gap-6 analysis was repeated using progressively larger
H. M. Smail (Thu,) studied this question.