Provenance correction: the completed square layer is the classical strong/Type-II route

Prior art

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Status: historical/provenance note

Date: 2026-08-15

Claim boundary: this note narrows novelty language. The square-divisor criterion and the strong/Type-II Erdős--Straus route are prior art. FCF claims only later structural synthesis as potentially novel, pending a publication-grade literature review.

1. Historical strong conjecture

Historical work discussed by Gardes distinguishes two stronger solution forms.

The strong conjecture is the assertion that one can solve Erdős--Straus with two of the three unit-fraction denominators divisible by n.

In modern standard terminology this is the prime Type-II side of the Type-I/Type-II split used elsewhere in the repository.

Therefore a theorem proving that every prime has a completed square-divisor Type-II certificate would prove a classical stronger conjecture, not merely the original Erdős--Straus conjecture.

2. Mizony's theorem is exactly the current square-completed congruence

Gardes records the following theorem of Mizony.

If there exist nonzero integers m,d such that

\boxed{d\mid m^2}

and

\boxed{4m-1\mid n+4d,}

then there is an Erdős--Straus decomposition of the strong/Type-II form.

This is exactly the current repository coordinate after renaming

\boxed{m=a,\qquad n=p.}

Thus the condition

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

must be attributed to the classical Mizony/Thépault/Rosati--Yamamoto lineage rather than presented as an FCF discovery.

3. Thépault's theorem is the divisor-complement form

The historical Thépault theorem is recorded as follows.

\boxed{b\mid a^2}

and

\boxed{4a-1\mid bn+a,}

then a strong/Type-II decomposition exists.

For prime n, Gardes proves the equivalence of the Thépault and Mizony forms by the divisor complement

\boxed{b=a^2/d.}

This is exactly the complement involution independently rediscovered in the current square-divisor analysis.

Therefore the complement equivalence itself is also prior art in this historical chain.

4. Relation to Rosati--Yamamoto

Gardes further records that for prime n, Mizony's theorem is equivalent to the first assertion of the Rosati--Yamamoto theorem governing the strong form.

Thus the square-divisor strong identity is not an isolated sufficient trick. It sits inside an established structural lineage for prime Type-II solutions.

The modern BHB-F and Bradford divisor parametrizations provide additional complete coordinates for the same overall Type-I/Type-II solution space.

5. Correct logical terminology for the repository

The repository should distinguish two lanes.

Exact Erdős--Straus lane

The exact prime ES equivalence is

\boxed{ \exists k: \{-p^{-1},-1\}\cap\mathcal R_k((p+k)/4)\ne\varnothing.}

It allows either standard Type I or standard Type II.

Strong/Type-II lane

The square-completed layer

\boxed{ S_a = \{-4d\pmod{4a-1}:d\mid a^2\}}

is the classical strong/Type-II route.

Universal coverage by the layers S_a would prove the strong conjecture for primes and hence Erdős--Straus, but it is logically stronger than the original conjecture.

This stronger route remains attractive because its exact layers possess unusually coherent multiplicative, shadow, complement, and Kneser structure.

6. What FCF should not claim

Do not claim novelty for:

the condition d|a^2;
the congruence 4a-1 | p+4d;
the equivalent Thépault condition 4a-1 | bp+a with b|a^2;
divisor complement between those two forms;
the existence of a square-divisor parametrization of strong/Type-II solutions;
the formulation of a stronger Type-II Erdős--Straus conjecture.

These belong to the established historical literature.

7. What remains potentially distinctive

The current potentially distinctive structural synthesis is instead:

identify López Type A and Type B at the same layer a as the two monotone boundary orthants of the full Mizony/Thépault square-divisor lattice;
identify the omitted strong/Type-II certificates as the mixed-sign orthants;
compute the exact mixed-parameter count

\tau(a^2)-2\tau(a)+1;

identify the full completed layer with the negative symmetric signed divisor box

S_a=-\mathcal R_{4a-1}(a);

import Kneser stabilizers into the historical strong layer;
merge those internal stabilizers with cross-layer modulus ancestry;
prove prime-index spectrum classification, prime-extension shadows, exact multiplicative-ancestry iff classification, squarefree factor lifts, and infinite fixed-quotient shadow families.

These are synthesis/structure claims only and remain potentially novel until the broader literature review is complete.

8. Preferred naming going forward

Use names such as:

Mizony/Thépault square layer;
strong/Type-II completed layer;
square-completed López boundary;
FCF orthant/Kneser/shadow synthesis.

Avoid language suggesting that FCF invented square completion itself.