Local Estimates of Twin Primes in Blocks of Squares of Consecutive Primes

This paper investigates the local distribution of twin primes in intervals bounded by squares of consecutive primes, which we call blocks. For a pair of consecutive primes p and p+g, the associated block is Ip,g = [p², (p+g)² − 2]. We show that this choice of boundary is not arbitrary: since (p+g)² − 2 < (p+g)², every odd composite in the block has least prime factor ≤ p, so the set of primes that sieve the block is exactly {p' ≤ p}. Applying the Hardy–Littlewood asymptotic density to this structure, we obtain the heuristic estimator E_twin(p,g) = C₂ · g(p + g/2)/(ln p)², where C₂ ≈ 0.6601618 is the twin-prime constant. The estimator is validated numerically on 19,909 consecutive blocks, with p ranging from 5 to 223,577. The mean ratio of estimate to actual value is 1.000997, with standard deviation 0.0308, median 0.99995, and no block exhibiting relative error greater than 5%. Convergence in p is analysed across five ranges, with the mean bias decreasing from 25% (p < 100) to 0.013% (p > 100,000). We identify three structural regularities of the blocks: an exact combinatorial identity, a modular dominance of gaps divisible by 6 (known since Hardy–Littlewood), and a normalised decay of the density, which is an empirical observation and not a theorem. The work does not prove the twin-prime conjecture. It is an exploratory and computational note that proposes a systematic local organisation and a robust numerical validation. This is version 2 (v2) of the record. It adds the English translation of the article (Local_Estimates_Twin_Primes_Blocks.pdf), the corresponding LaTeX source (main_en.tex), and the English-language Python scripts that generated the figures. Version 1 (v1), in Portuguese, remains available in the same record.