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Cubic-surface form of the gcd-square rescue

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Status: proved exact reformulation and square obstruction

Date: 2026-08-15

Depends on: FAB-GCD-SQUARE-CRITERION.md

Claim boundary: this sharpens the exact rescue target. It does not prove universal existence and therefore does not prove Erdős–Straus.

1. Replace the two 3 mod 4 integers by depth coordinates

Write

\boxed{k=4u-1,\qquad d=4v-1}

with u,v>=1.

Then

kd-1 =(4u-1)(4v-1)-1 =4(4uv-u-v).

Define

\boxed{w=4uv-u-v.}

For a prime p≡1 mod4, also define

\boxed{A_u=\frac{p+4u-1}{4}.}

Then

p+k=4A_u, \qquad M=kd-1=4w.

Hence

G=\gcd(p+k,M) =4\gcd(A_u,w).

The gcd-square criterion

4M\mid G^2

therefore becomes exactly

\boxed{ w\mid\gcd(A_u,w)^2.}

2. Square-root closure

For a positive integer

w=\prod_\ell \ell^{e_\ell},

define its least square-root closure

\boxed{ \operatorname{src}(w) :=\prod_\ell \ell^{\lceil e_\ell/2\rceil}. }

This is the least positive integer h such that

w\mid h^2.

Then

w\mid\gcd(A_u,w)^2 \iff \boxed{\operatorname{src}(w)\mid A_u}.

Thus:

Surface rescue criterion

If there exist u,v>=1 such that, with

w=4uv-u-v,

one has

\boxed{ \operatorname{src}(w) \mid \frac{p+4u-1}{4}, }

then p satisfies the Erdős–Straus equation.

This is exactly the gcd-square construction in the coordinates naturally attached to the shifted cubic surface.

3. Explicit recovered parameters

Put

g=\gcd(A_u,w).

When w|g^2, the parameters in FAB-GCD-SQUARE-CRITERION.md become

\boxed{ a=\frac{A_u}{g}, \qquad b=\frac{w}{g}, \qquad c=\frac{g^2}{w}.}

Thus the leftover c is precisely the square-overlap excess.

The auxiliary quotient is

q =\frac{(4v-1)p+1}{4g}.

The resulting positive decomposition is

\frac4p = \frac1{abc} + \frac1{aqc} + \frac1{bpqc}.

4. The seductive perfect-square move is impossible

The strongest possible compression would be to make

w=s^2,

because then

\operatorname{src}(w)=s.

But this can never happen for positive u,v.

Indeed,

(4u-1)(4v-1) =4w+1.

If w=s^2, then

\boxed{(4u-1)(4v-1)=4s^2+1.}

Every odd prime divisor r of 4s^2+1 satisfies

(2s)^2\equiv-1\pmod r,

so -1 is a quadratic residue modulo r; hence

\boxed{r\equiv1\pmod4.}

Therefore every positive divisor of 4s^2+1 is 1 mod4.

But both

4u-1\equiv4v-1\equiv3\pmod4,

contradiction.

Hence

\boxed{4uv-u-v\text{ is never a perfect square}.}

This is an exact structural obstruction, not a search observation.

5. The unavoidable defect is the useful object

Write schematically

w=c s^2

when the valuation pattern permits a square part s^2 and a squarefree defect c (or use the squarefree kernel when exponents are not exactly 0/1 mod 2).

Then the square-root closure contains the defect:

\operatorname{src}(w)\supseteq c s.

So the surface criterion can succeed only when the non-square defect is absorbed into A_u.

This aligns exactly with FAB-HARD-NONRESIDUE-BRIDGE.md: on a Mordell-hard prime, a genuine coprime fab certificate cannot have its leftover squareclass supported only on the hard quadratic-residue primes 2,3,5,7; it must import an external quadratic nonresidue.

Thus the failed perfect-square idea is not wasted. It identifies the actual object that must be controlled:

\boxed{ \text{square part of }w \quad+\quad \text{one unavoidable nonresidue defect}. }

The remaining theorem target is to force that square-root closure into A_u for some adaptive surface point (u,v).

6. A useful infinite near-square subfamily

The congruence condition for u^2 | w has an exact polynomial family. Since

w=v(4u-1)-u,

requiring u^2|w gives

v\equiv u(u-1)\pmod{u^2}.

Taking the least positive representative

\boxed{v=u(u-1)}

yields

\boxed{ w=u^2(4u-5).}

More generally,

\boxed{v=u(nu-1)}

gives

\boxed{ w=u^2\bigl(4nu-n-4\bigr).}

These families isolate a large square automatically and expose a single linear defect. They are therefore natural theorem-mining families for the next stage, although no claim is made that they alone cover all hard primes.