Exact two-target divisor-square criterion — Free Computation Foundation

Theorem

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Status: proved exact reformulation

Date: 2026-08-15

Depends on: ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md, FAB-UNBOUNDED-DIVISOR-RATIO-CERTIFICATE.md

Claim boundary: this is an equivalent coordinate form of the exact prime Type-I/Type-II signed-box theorem. It does not prove universal target existence.

1. From signed exponents to divisors of C^2

Let

p\equiv1\pmod4

be prime and let

k\equiv3\pmod4, \qquad \gcd(k,p)=1.

Put

C=\frac{p+k}{4} =\prod_i r_i^{e_i}.

The signed divisor box is

\mathcal R_k(C) = \left\{ \prod_i r_i^{z_i}\pmod k: -e_i\le z_i\le e_i \right\}.

For each signed exponent vector define

u_i=e_i-z_i.

Then

0\le u_i\le2e_i

and

d=\prod_i r_i^{u_i}

runs through every positive divisor of C^2 exactly once as the exponent vector varies.

Moreover

\prod_i r_i^{z_i} =C\,d^{-1}\pmod k.

Therefore

\boxed{ \mathcal R_k(C) = \{C d^{-1}\pmod k:d\mid C^2\}.}

This is a bijective reparametrization of the signed box.

2. Type-I target

By inversion symmetry, the Type-I target may be written as

-p

instead of -p^{-1}.

Since

p\equiv4C\pmod k,

the equation

C d^{-1}\equiv-p\pmod k

is equivalent to

C d^{-1}\equiv-4C\pmod k.

Because C is a unit modulo k, this becomes

d^{-1}\equiv-4\pmod k,

\boxed{4d\equiv-1\pmod k.}

Thus:

Exact Type-I divisor-square criterion

\boxed{ \text{Type I at shift }k \iff \exists d\mid C^2: 4d\equiv-1\pmod k.}

This recovers the fixed-k divisor-square target from the earlier strong FAB lane, now as the exact standard Type-I coordinate.

3. Type-II target

The Type-II target is

-1.

Hence

C d^{-1}\equiv-1\pmod k

is equivalent to

\boxed{d\equiv-C\pmod k.}

Therefore:

Exact Type-II divisor-square criterion

\boxed{ \text{Type II at shift }k \iff \exists d\mid C^2: d\equiv-C\pmod k.}

This is the divisor-square counterpart that was missing from the one-target fixed-k formulation.

4. Exact two-target theorem

Combining the two cases with ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md gives:

Theorem — prime ES as two divisor classes inside Div(C^2)

For prime p≡1 mod4, Erdős--Straus holds for p if and only if there exists

k\equiv3\pmod4, \qquad \gcd(k,p)=1,

such that, with

C=\frac{p+k}{4},

there is a divisor d|C^2 satisfying at least one of

\boxed{4d\equiv-1\pmod k}

\boxed{d\equiv-C\pmod k.}

Equivalently,

\boxed{ \operatorname{Div}(C^2) \cap \left( \{-4^{-1}\} \cup \{-C\} \right) \ne\varnothing \pmod k.}

The first class is Type I; the second is Type II.

5. Complement involution

The divisor set has the natural involution

\boxed{d\longmapsto d^*=\frac{C^2}{d}.}

Under the signed-box map

C/d,

this is exactly inversion:

\frac C{d^*} = \left(\frac Cd\right)^{-1}.

Type I

The two orientations -p and -p^{-1} correspond to a complementary divisor pair.

d\equiv-4^{-1}\pmod k,

then

d^*=C^2d^{-1} \equiv-4C^2\pmod k,

which is the divisor coordinate of the inverse Type-I orientation.

Type II

The Type-II residue is self-inverse. If

d\equiv-C\pmod k,

then

d^* =C^2d^{-1} \equiv-C\pmod k.

Thus the Type-II target class is fixed by divisor complement.

This self-reciprocity is a useful structural distinction between the two solution types.

6. Type-II square-pair form

Suppose

d\mid C^2, \qquad d\equiv-C\pmod k.

Put

e=\frac{C^2}{d}.

Then also

e\equiv-C\pmod k.

Since de=C^2 is a square, d and e have the same squarefree kernel. Write

\boxed{d=s a^2,\qquad e=s b^2}

with s squarefree. Then

\boxed{C=sab.}

Now

d+C =s a^2+sab =s a(a+b).

Because every prime factor of sa divides C and gcd(C,k)=1, one has

\gcd(sa,k)=1.

Therefore

k\mid d+C \iff \boxed{k\mid a+b.}

So Type II may equivalently be viewed as a complementary divisor-square pair whose square roots add to a multiple of the shift.

This recovers the familiar normalized factor-pair geometry from a different direction.

7. Strategic consequence

The exact prime problem can be stated without signed exponents:

Choose an admissible shift k. The square divisor lattice of C=(p+k)/4 must hit one of two residue classes modulo k: the fixed Type-I class -4^{-1} or the moving Type-II class -C.

The two classes have different involution behavior:

Type I is a complementary pair under d -> C^2/d;
Type II is self-complementary.

This form may be better suited to divisor-distribution, reciprocity, and lattice arguments than the original signed-box notation while remaining exactly equivalent to it.