Six Structural Dimensions and the Erdős-Straus Conjecture: Three Natural Laws Applied to a 77-Year Open Problem (v6)

The Erdős-Straus conjecture (1948) asserts that for every integer n ≥ 2, the equation 4/n = 1/x + 1/y + 1/z has a solution in positive integers. No complete proof has been given in 77 years. This paper introduces a substitution revealing a perfect square identity and a constant divergence (div V = 12), which are structural properties of the equation itself. Cases 1-5 are proved by explicit construction. Case 6 is reduced to a precisely isolated arithmetic existence lemma, supported by computational verification (0 counterexamples among 5,669 primes up to 500,000). The paper presents three natural laws: (I) Every system has content and infrastructure, each proving the other. (II) Reality is finite, algebraically: bounded infrastructure with constant coupling yields bounded content. The only objection presupposes the axiom of infinity, which itself has never been proved algebraically. (III) Exactly six structural dimensions exist; every seventh collapses. Block thinking (isolated analysis) is identified as a mathematical error: for n = 118,801 the diagonal (eight isolated paths) yields 0%, the connected equation yields 100%. Five falsification challenges are posed. The verification script is included. v6 replaces v5. Changes: finiteness equation added, block thinking framework added, challenge section expanded from 4 to 5 points.