Erdős-Straus Algebra with three-operator factoring and empirical results on 93,457 hard primes

The provided paper introduces the Brahim Framework Erdős–Straus Algebra, a mathematical framework designed to analyze the most difficult cases of the Erdős–Straus conjecture. The author demonstrates that the challenging "hard residues" modulo 840 are actually quadratic residues of the unit group, linking them algebraically to Heegner residues through additive negation and Frobenius exponentiation. Beyond these theoretical connections, the work presents a three-operator decomposition pipeline that successfully generated unit fraction triples for over 93,000 primes. These central algebraic theorems are rigorously verified using the Lean 4 proof assistant, ensuring their computational validity. Ultimately, the text organizes empirical data into a series of new conjectures while offering a more structured, symbolic understanding of a long-standing problem in number theory. The sources describe a three-operator decomposition pipeline (also referred to as three-operator factoring) used to construct explicit Egyptian fraction triples for "hard primes". These three sequential operators are: 1. Operator $\hat{X}$: This operator is responsible for choosing the first integer ($x$) of the triple. It employs two implementations: a modulo-8 recipe ($X̂ \text{ mod } 8$), which exploits the fact that Mordell residues are congruent to $1 \pmod 8$, and a boundary search ($X̂ \text{ boundary}$) used as a fallback for exceptional primes. 2. Operator $\hat{Y}$: Once $x$ is chosen, this operator derives the second denominator, $y$, in closed form. 3. Operator $\hat{Z}$: This operator derives the final denominator, $z$ also in closed form. Together, $\hat{Y}$ and $\hat{Z}$ are defined as the divisor decomposition operator $\widehat{YZ}$, which functions by enumerating every pair of positive integers $(y, z)$ that satisfies the equation $4/p = 1/x + 1/y + 1/z$ through the analysis of a quadratic discriminant. While these three operators form the practical decomposition pipeline, the sources also define a broader suite of five operators acting on the ring of integers modulo 840: Residue Projector ($\hat{R}$): Assigns an integer its residue class modulo 840. Additive Mirror ($\hat{W}$): Performs additive negation modulo 840. Frobenius Exponentiation ($\hat{\Phi}$): Raises an argument to the 29th power modulo 840. Radial Offset ($\hat{M}_K$): Subtracts the framework constant $K$ (107) from an integer. Divisor Decomposition $\widehat{YZ}$: The combined action of the second and third pipeline operators.