Binary-r rescue as a signed divisor collision — Free Computation Foundation

Theorem

Let p == 1 (mod 4) be prime and let r == 3 (mod 4) be an odd positive integer such that

Status: proved exact reformulation

Date: 2026-08-15

Depends on: BINARY-R-RESCUE.md

Claim boundary: this converts the exact binary rescue condition into a finite multiplicative collision problem. It does not prove that such a collision always occurs.

1. Setup

Let p == 1 (mod 4) be prime and let r == 3 (mod 4) be an odd positive integer such that

\gcd(r,p)=1, \qquad A_r=\frac{p+r}{4}\in\mathbb Z, \qquad \gcd(r,A_r)=1.

Put

N=N_r=pA_r.

Then

\frac4p = \frac1{A_r} + \frac rN.

By BINARY-R-RESCUE.md, the binary remainder splits if and only if there is a divisor

d\mid N^2

with

\boxed{d\equiv-N\pmod r.}

The argument below does not require r prime, only gcd(N,r)=1.

2. Every divisor of N^2 is a signed divisor ratio around N

Write

N=\prod_i q_i^{e_i}.

A divisor

d=\prod_i q_i^{f_i}, \qquad 0\le f_i\le2e_i,

can be written

\boxed{d=N\frac{x}{y}}

where x,y|N.

Indeed, for each prime put

z_i=f_i-e_i\in[-e_i,e_i].

The positive z_i exponents form x and the negative exponents form y.

Conversely, for any x,y|N,

N\frac{x}{y}

is an integer divisor of N^2: primewise its exponent lies between 0 and 2e_i.

Because gcd(N,r)=1, every divisor of N is invertible modulo r.

Therefore

d\equiv-N\pmod r

is equivalent to

Nxy^{-1}\equiv-N\pmod r,

hence to

\boxed{x\equiv-y\pmod r.}

3. Divisor-collision theorem

Let

D_r(N)=\{x\bmod r:x\mid N\} \subseteq(\mathbb Z/r\mathbb Z)^\times.

Then the following are equivalent:

the binary-r remainder r/N splits into two unit fractions;
there is d|N^2 with d == -N (mod r);
there are divisors x,y|N with

x\equiv-y\pmod r;

the divisor residue set meets its negative:

\boxed{D_r(N)\cap(-D_r(N))\ne\varnothing;}

-1 lies in the ratio set

\boxed{-1\in D_r(N)D_r(N)^{-1}.}

Thus the fixed-r ES problem is a signed divisor collision, not an arbitrary search through all divisors of N^2.

4. Signed exponent-box formulation

With

N=\prod_i q_i^{e_i},

the collision exists if and only if there are integers

-e_i\le z_i\le e_i

such that

\boxed{ \prod_i q_i^{z_i}\equiv-1\pmod r. }

This is the exact multiplicative analogue of a bounded symmetric exponent box hitting one distinguished group element.

It is structurally close to the two-box and defect-quotient machinery already developed in the Type A/B shadow program, but here the target is the actual Erdős–Straus decomposition rather than exact-depth realizability.

5. Pigeonhole escape

Since negation is a bijection of the unit group modulo r,

|D_r(N)|=|-D_r(N)|.

Therefore, if

\boxed{2|D_r(N)|>\varphi(r),}

then the two sets must intersect and the binary-r rescue is automatic.

For prime r, this is simply

\boxed{|D_r(N)|>\frac{r-1}{2}.}

Failure forces the divisor residues into at most half of the available unit classes.

This gives a clean theorem-mining target: prove divisor-residue expansion past the half-group threshold, or classify every multiplicatively compressed exception.

6. Exact reciprocity bridge for prime r

Now assume r is prime and r == 3 (mod 4).

Let q be any odd prime divisor of

A_r=\frac{p+r}{4}.

Then

p\equiv-r\pmod q.

Because p == 1 (mod 4), quadratic reciprocity gives

\left(\frac qp\right) = \left(\frac pq\right) = \left(\frac{-r}{q}\right).

Since r == 3 (mod 4), reciprocity between r and q gives

\left(\frac rq\right) = \left(\frac{-1}{q}\right) \left(\frac qr\right).

Hence

\left(\frac{-r}{q}\right) = \left(\frac{-1}{q}\right) \left(\frac rq\right) = \boxed{\left(\frac qr\right)}.

Therefore

\boxed{ q\mid A_r \Longrightarrow \left(\frac qp\right)=\left(\frac qr\right). }

So a prime factor of the shifted integer A_r is a quadratic nonresidue modulo r if and only if it is an external quadratic nonresidue modulo the hard prime p.

This is an exact bridge from FAB-HARD-NONRESIDUE-BRIDGE.md into the direct binary-r rescue.

7. Consecutive-translate interpretation

Put

A=\frac{p+3}{4}.

For

r=4t+3,

we have

\boxed{A_r=A+t.}

Thus the prime-r program probes the divisor geometry of consecutive translates of the original Eisenstein-split neighbor:

A,\ A+1,\ A+2,\ A+4,\ A+5,\ A+7,\ldots

for r=3,7,11,19,23,31,....

The finite first-hit histogram in BINARY-R-RESCUE.md is therefore evidence about a sequence of exact multiplicative collision problems on nearby translates, not a collection of unrelated congruence tricks.

8. New all-prime target

A sufficient universal theorem now has the sharply stated form:

For every Mordell-hard prime p, there exists an odd r == 3 (mod 4) with gcd(r,pA_r)=1 such that the signed divisor exponent box of <div class="math" role="math">> N_r=pA_r

></div>

contains -1 modulo r.

For prime r, failure means a multiplicatively compressed divisor set of size at most (r-1)/2; every nonresidue prime entering A_r is simultaneously an external nonresidue of p by the reciprocity bridge.

This is the current direct ES proof target.