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Prior-art calibration for the square-completed Type-II route

Prior art · hosted from the CENTL repository

Research library · Prior art

Prior art

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Source in the repository

Status: source/provenance note

Date: 2026-08-15

Claim boundary: this note deliberately narrows FCF novelty language. The divisor-of-a-square Type-II mechanism is prior art. Any FCF novelty claim must concern later structural synthesis only after a publication-grade review.

1. The square-divisor mechanism is old prior art

The current FCF route uses positive integers a,d with

d\mid a^2

and a congruence equivalent to

4a-1\mid ap+d.

This mechanism must not be advertised as an FCF discovery.

A source trail reaches at least to Léon Thépault's 1979 work, recorded in later historical analysis of the Erdős--Straus problem.

The theorem is stated in the form:

for a prime n≡1 mod4, if there exist positive a,b such that b|a^2 and 4a-1 divides bn+a, then an Erdős--Straus decomposition exists.

The explicit decomposition recorded with that theorem is

\boxed{ \frac4n = \frac1{an} + \frac{4a-1}{n(a+bn)} + \frac{b(4a-1)}{a(a+bn)}.}

Thus divisor-square Type-II constructions were already present decades before the current work.

2. Exact equivalence with the current congruence coordinate

The FCF square-completed congruence is written as

\boxed{ 4a-1\mid p+4d, \qquad d\mid a^2.}

Since

4a\equiv1\pmod{4a-1},

this is equivalent to

\boxed{4a-1\mid ap+d.}

Let

d^*=\frac{a^2}{d}.

Because d is a unit modulo 4a-1, multiplying

ap+d\equiv0

by a/d gives

\boxed{d^*p+a\equiv0\pmod{4a-1}.}

This is exactly the Thépault shape with

\boxed{b=d^*.}

Therefore the present square-completed coordinate and the historical Thépault condition are related by the divisor-complement involution.

The underlying existence mechanism is prior art.

3. 2017/2023 public discussion

A 2017 Mathematics Stack Exchange question studied the integrality condition

\boxed{ \frac{d+an}{4a-1}\in\mathbb Z}

with square-divisor conditions.

A 2023 answer explicitly relates the resulting condition D|A^2 to the classical Type-II form and derives the square-divisor parameter from Mordell-style Type-II variables.

This is non-peer-reviewed discussion, but it is additional evidence that the square-divisor coordinate itself was already recognized before the current project.

4. Bradford 2024

Kyle Bradford's 2024 preprint

Elemental Patterns from the Erdős Straus Conjecture, arXiv:2403.16047,

proves necessary and sufficient divisor-of-a-square modular descriptions in terms of the smallest denominator x.

For prime p, Bradford uses divisors

d\mid x^2

and gives separate modular conditions corresponding to standard Type I and Type II, with a one-to-one correspondence to Erdős--Straus solutions.

This is a different coordinate system from the 4a-1 square-completed López layer, but it confirms that complete square-divisor descriptions of prime ES solutions are established prior art.

5. Bello-Hernández--Benito--Fernández 2026

The 2026 divisor-parametrization paper

A Divisor Parametrization for the Erdős--Straus Conjecture, arXiv:2606.10922,

provides another complete divisor-based coordinate system and compares it with standard Type I/II descriptions.

The FCF two-target signed-box theorem and divisor-square forms should be presented as reformulations/syntheses against this complete modern background, not as the first divisor parametrization of ES.

6. What remains potentially distinctive in the current work

The targeted prior-art pass above changes the responsible novelty boundary.

Do not claim novelty for:

  • the condition d|a^2;
  • a Type-II solution generated from a divisor of a square;
  • divisor complement by itself;
  • a complete divisor-of-a-square description of Type I/II solutions;
  • the general use of modular divisor criteria for Erdős--Straus.

The current potentially distinctive synthesis is narrower:

  1. place Thépault's complete square divisor lattice on the same layer a as López's 2024 Type-A/B congruence system;
  2. prove that López Type A and Type B are exactly the two monotone boundary orthants of that centered divisor box;
  3. identify the omitted Type-II certificates as mixed-sign/cross-orthant parameters;
  4. derive the exact mixed-parameter count
\tau(a^2)-2\tau(a)+1;
  1. identify López A/B mutual inversion as the restriction of the global divisor-complement involution;
  2. prove the exact finite-group identity
S_a=-\mathcal R_{4a-1}(a),

merging the completed López layer with the repository's Kneser signed-box machinery;

  1. combine that internal Kneser geometry with the pre-existing cross-layer shadow/ancestry framework.

These items are potentially novel structural synthesis, not established priority claims.

7. Terminology correction

Future research notes should prefer language such as:

  • Thépault square-completed layer;
  • square-completed López layer;
  • FCF orthant/Kneser synthesis;

rather than wording that implies FCF originated the square-divisor criterion.

The theorem files already carry claim-boundary disclaimers; this note supplies the explicit provenance correction.

8. Publication-grade follow-up

A proper priority review should trace:

  1. the original Thépault publication in Pour la Science / Gardner's reporting and any surviving primary text;
  2. Gardes' historical thesis transcription and analysis;
  3. Mordell's Type-II parametrization;
  4. Elsholtz--Tao's solution parametrizations;
  5. the 2017 public square-divisor discussion;
  6. Bradford 2024;
  7. López 2022/2024;
  8. Bello-Hernández--Benito--Fernández 2026;
  9. Schuh 2025 and related Pythagorean-prime square-divisor parametrizations.

Until that review is complete, the orthant/signed-box merger should be described as potentially novel rather than first or unique.