The Geometric Proof of Legendre's Conjecture \

Abstract This paper proves Legendre’s Conjecture by demonstrating that the growth of the spa tial field |Δ𝑛| = 2𝑛 + 1 invariably exceeds the occupancy capacity of the composite set 𝑆𝑐𝑜𝑚𝑝 derived from the High-Index Power Products of primes ≤ 𝑛. Unlike asymptotic approximations, this ”Universal Correction” method utilizes the deterministic structure of square intervals to show that the residual count 𝐶𝑛 ≥ 1 for all 𝑛. 摘要本文透過論證空間場域|Δ𝑛|=2𝑛+1的增長始終超過由質數≤𝑛之「高指數 冪次乘積」所產生的合數集合𝑆𝑐𝑜𝑚𝑝的佔用容量，從而證明了勒讓德猜想。不同 於漸近估計法，此「一律通用」的修正法利用平方區間的確定性結構，證明了對於 所有𝑛，殘餘數量𝐶𝑛≥1恆成立。 Keywords: Legendre＇s Conjecture, Prime Distribution, Square Interval, 2𝑛 + 1, High Index Powers, Precision Correction Method. 關鍵字：勒讓德猜想、質數分布、平方數區間、2𝑛+1、高指數冪次、精密補差法。