Quadratic-character prior-art note

Date: 2026-08-14

Purpose: claim calibration for QUADRATIC-TRAP-SIGNATURE.md

Status: primary-source correction and novelty boundary

The quadratic-character branch must be stated carefully.

López 2024 already contains an essential part

Miguel Angel López, A Complete Congruence System for the Erdos-Straus Conjecture (arXiv:2404.01508), explicitly invokes Mordell's restriction that a residue class supporting a polynomial Erdős-Straus identity cannot be a quadratic residue modulo the identity modulus.

More specifically, López's Corollary 1 states for Type B parameters d,n that

Therefore the Type B half of the FCF/CENTL observation that trap residues have Jacobi sign -1 is prior art and must not be claimed as an FCF discovery.

The current FCF theorem packages both Type B residues -e and Type A residues -4e for every divisor e|k into one trap-set statement

The Type A half is elementary from the same reciprocity calculation and is also consonant with Mordell's general quadratic-residue restriction. It should therefore be treated as a useful unification/lemma, not advertised as a major standalone novelty claim.

Stronger candidate contribution

The potentially novel object is not merely the sign of one trap residue. It is the use of that sign inside the minimal-depth / shadow framework to construct a simultaneous character avoidance system.

For a target candidate x=r mod L, FCF introduces one Legendre-sign variable for each free odd prime coordinate and imposes, for every earlier modulus m_j, the linear condition over F_2 forcing

A solution to that finite linear system produces a reduced arithmetic progression that avoids every earlier Type A/B trap simultaneously and therefore gives infinitely many exact-depth primes by Dirichlet.

Targeted searching on 2026-08-14 did not locate this exact simultaneous character-shield construction for López Type A/B minimal witness depth, but that negative result does not establish priority.

Safe claim language

Safe:

López already records the quadratic-nonresidue property for Type B congruences. FCF is investigating a broader Type-A/B trap-set formulation and, more importantly, a simultaneous quadratic-character shield that turns exact-depth avoidance into a finite linear system over F_2 inside the C_AB/shadow framework.

Unsafe:

FCF discovered that López Type B residues are quadratic nonresidues.

Primary source

• Miguel Angel López, A Complete Congruence System for the Erdos-Straus Conjecture, arXiv:2404.01508, especially the introduction's discussion of Mordell and Corollary 1 following the Type B perfect-square theorem.

This note should be consulted before any public novelty claim involving quadratic residues, Jacobi symbols, or the character shield.