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Prior-art matrix for WS-CAND-003

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Prior art

This file records the targeted primary-literature pass used to calibrate claims around CAB, exact trap cardinality, congruence-layer shadowing, the depth-spectrum / survival-process framework, and the later character,…

Source in the repository

Search date: 2026-08-14

Status: working research record. A negative search result is not proof of novelty or priority.

This file records the targeted primary-literature pass used to calibrate claims around C_AB, exact trap cardinality, congruence-layer shadowing, the depth-spectrum / survival-process framework, and the later character, multiplicative-quotient, and infinite-shadow-family results.

Primary sources reviewed

Miguel Angel López, 2024

A Complete Congruence System for the Erdos-Straus Conjecture https://arxiv.org/abs/2404.01508

Relevant prior art:

  • Type A and Type B solution classes;
  • congruence characterizations involving 4dn-1;
  • conjecture that every prime has a Type A or Type B solution;
  • relation to Type II solutions and classical Mordell restrictions;
  • experimental Type A/B coverage reported for the first 10,000 primes;
  • quadratic-nonresidue observations in the Type A/B setting;
  • mutual-inverse behavior of the two divisor residue families.

Not claimed by FCF: Type A, Type B, their basic congruence formulas, the general use of congruence systems, the classical quadratic-residue restrictions, or the inverse relationship itself.

Miguel Angel López, 2022

Structure and form of the solutions of the Erdos-Straus conjecture https://arxiv.org/abs/2206.10319

Relevant prior art:

  • structural classification of solution forms;
  • solutions of the form (du,dv,duv);
  • structural treatment of difficult primes including 1009.

Serge E. Salez, 2014

The Erdős-Straus conjecture: New modular equations and checking up to N=10^17 https://arxiv.org/abs/1406.6307

Relevant prior art:

  • modular equations and optimized computational sieving;
  • verification of the original Erdős-Straus conjecture through 10^17.

Consequence for WS-CAND-003: the 10^7 finite run is not a novelty claim about the verification range of Erdős-Straus itself.

Christian Elsholtz and Terence Tao, 2011

Counting the number of solutions to the Erdos-Straus equation on unit fractions https://arxiv.org/abs/1107.1010

Relevant prior art:

  • analytic study of the number and distribution of Erdős-Straus solutions over primes;
  • solution-counting framework and asymptotic bounds.

Christian Elsholtz and Stefan Planitzer, 2018

The number of solutions of the Erdős-Straus Equation and sums of k unit fractions https://arxiv.org/abs/1805.02945

Relevant prior art:

  • bounds and algorithms for enumerating unit-fraction decompositions;
  • distributional results in reduced residue classes.

R. C. Vaughan, 1970

On a problem of Erdős, Straus and Schinzel, Mathematika 17 (1970), 193-198.

Relevant prior art:

  • classical sparse-exception / almost-all results for Erdős-Straus;
  • large-sieve methods showing that the unresolved population is very thin.

Consequence for the current program: a density-one solvability statement is not, by itself, a novelty claim.

Christian Elsholtz, 2001

Parametric-solution work strengthening sparse-exception bounds for generalized unit-fraction problems.

Relevant prior art:

  • parametric families can yield strong upper bounds for exceptional sets;
  • asymptotic sparsity of failures is classical territory.

Bernd R. Schuh, 2025

The Erdös-Straus Conjecture and Pythagorean Primes https://arxiv.org/abs/2503.11672

Relevant prior art:

  • additional structured parametrizations and congruence restrictions for a prime subclass;
  • algorithmic determination of parameters in those forms.

Xiaoping Xu, 2026

Congruence Classes of Supporting the Erdös-Straus Conjecture I: Tame Solutions https://arxiv.org/abs/2605.23601

Relevant prior art:

  • current congruence-class research;
  • tame/wild solution classification for primes of the form 24m+1;
  • parametrized congruence families supporting solutions.

M. Bello-Hernández, M. Benito, E. Fernández, 2026

A Divisor Parametrization for the Erdős--Straus Conjecture https://arxiv.org/abs/2606.10922

Relevant prior art:

  • divisor-based parametrization of unit-fraction decompositions;
  • comparison with established Type I/II descriptions.

Targeted novelty questions

The search specifically looked for prior definitions or equivalent constructions corresponding to:

  1. the first-hit invariant

C_AB(p) = min { k >= 1 : p mod (4k-1) in {-d,-4d : d | k} };

  1. a record sequence for the minimum product dn among López Type A/B witnesses;
  1. the exact trap-set cardinality

|T_k| = 2*tau(k) - 1 - 1_{4|k}*tau(k/4);

  1. an explicit partial order, graph, or inclusion theory in which one Type A/B congruence layer is redundant because its complete hard-class-compatible CRT fibre is already contained in an earlier Type A/B trap layer;
  1. a distinction between direct shadowing and collective union-shadowing of Type A/B congruence classes;
  1. an irredundant-core reduction of the López Type A/B congruence system based on those shadow relations;
  1. the exact-depth spectrum

D = { k : infinitely many primes satisfy C_AB(p)=k };

  1. an exact finite-depth survivor process for López Type A/B minimal depth with

delta_K, mu_k = delta_(k-1)-delta_k, and h_k = mu_k/delta_(k-1);

  1. a prime-modulus exact-depth backbone in which 4k-1 prime makes the target layer an independent CRT coordinate and gives a closed conditional hazard;
  1. a structural-gap versus finite-latency distinction for minimal Type A/B depth, including finite modular certificates proving infinite exact-depth realization;
  1. a prime-modulus survivor core whose remaining finite Type A/B witnesses must occur at composite target moduli, organized as a composite-rescue problem;
  1. threshold pre-shadow load measuring how conditioning on early survival deletes later admissible Type A/B candidate classes;
  1. a vector-valued Legendre-signature formulation in which the complete Type A/B trap signature is one affine subspace eta_k+V_k;
  1. character-level or signature-level direct-shadow completeness, in which collective character obstruction collapses to direct immutable layers;
  1. a multiplicative subgroup envelope T_k subset -H_k generated by the prime divisors of k, together with a quotient factorization separating quadratic-signature and higher-order multiplicative information;
  1. a higher-codimension proper-Jacobi-ancestor theorem for Type A/B signature layers;
  1. an infinite exact power-of-two shadow lattice based on T_{2^a}=-<2> modulo 2^(a+2)-1 and the divisibility relation a+2 | b+2;
  1. the resulting infinite family, and density-one subsequence statement, of structurally impossible power-of-two minimal depths;
  1. the binary-versus-odd-prime-power trap-coset dichotomy.

The targeted searches did not surface these exact objects as an integrated Type-A/B-specific framework. A second search specifically using combinations of “Erdős-Straus”, powers of two, 4k-1, Mersenne moduli, Type A/B, shadow/redundancy, and congruence containment likewise did not surface the exact Mersenne shadow-lattice theorem recorded here.

That is encouraging but insufficient for a priority claim. Equivalent ideas could exist under different notation, in non-arXiv literature, theses, books, conference material, computational notes, or unpublished work.

Important novelty exclusions

The following must not be advertised as FCF discoveries by themselves:

  • density-one solvability of Erdős-Straus;
  • the fact that simple divisor/congruence criteria solve almost all integers or primes;
  • general sieve survival probabilities;
  • general congruence redundancy or covering-system ideas;
  • CRT independence for a new coprime modulus;
  • Dirichlet realization of a reduced arithmetic progression;
  • elementary group-index factorization or character duality in isolation;
  • the classical identity 2^u-1 | 2^v-1 iff u|v in isolation;
  • quadratic reciprocity or Jacobi-symbol facts in isolation.

The candidate novelty is the way these ingredients arise as a coherent obstruction theory inside the López Type A/B minimal-depth system.

Current strongest candidate

The strongest responsible mathematical novelty statement is now broader than the original shadow-only wording:

FCF has identified a potentially novel Type-A/B-specific minimal-depth framework for the Erdős-Straus problem, combining C_AB, congruence-layer shadowing, exact-depth spectra and arrival, exact survivor mass/hazard, a prime-modulus backbone, character and multiplicative quotient structure, and explicit infinite structural-gap families.

This should still be described as potentially novel until a publication-grade literature review and external mathematical review are complete.

Integration record

The current synthesis is DIAMOND.md. The moving edge is CURRENT-FRONTIER.md.

New theorem records include:

Next prior-art work

A publication-grade review should expand beyond keyword search and trace citations backward and forward from López 2022/2024, Salez, Mordell-related congruence results, Monks/Velingker, Vaughan, Elsholtz/Tao, Elsholtz/Planitzer, and modern divisor parametrizations.

The new group/character results also warrant searching broader literature under:

  • multiplicative characters of divisor-generated subgroups;
  • quadratic-signature affine subspaces;
  • subgroup coset coverings;
  • Mersenne divisibility lattices in congruence sieves;
  • redundant congruence systems with 2^n-1 moduli;
  • covering systems and subgroup-generated residue sets;
  • stopping depths / first-hit times in arithmetic sieves.