Consecutive binary-selector probe — Free Computation Foundation

Theorem

Write a Mordell-hard prime as

Source in the repository

Status: exact finite theorem-mining record; proposed fixed bound u<=15 falsified

Date: 2026-08-15

Depends on: BINARY-R-RESCUE.md, BINARY-R-DIVISOR-COLLISION.md

Claim boundary: finite exact evidence only. No fixed universal selector bound is claimed.

1. Consecutive parametrization

Write a Mordell-hard prime as

p=24n+1

and put

P=\frac{p-1}{4}=6n.

For every positive integer u, choose

\boxed{r_u=4u-1.}

Then r_u==3 mod 4 and the first denominator is exactly

\boxed{A_u=\frac{p+r_u}{4}=P+u.}

Thus the binary program probes the multiplicative divisor geometry of the consecutive integers

P+1,P+2,P+3,\ldots

without requiring r_u to be prime.

2. Important correction

Earlier prime-r probes skipped composite values such as

r=15,\ 39,\ 51,\ldots

but the exact signed-divisor collision theorem only requires r==3 mod4 and the relevant coprimality, not primality.

Therefore composite numerators are legitimate and must be included in theorem mining.

3. Finite signal through two million

On the exact Mordell-hard prime population below approximately 2*10^6, every tested prime had a binary rescue for some

1\le u\le15.

This initially suggested a possible fixed-selector theorem.

4. Falsification through ten million

The full hard-prime census through

p\le10^7

contains the explicit counterexample to the proposed bound

\boxed{p=8,803,369.}

For this prime, the exact binary collision test fails for every

\boxed{1\le u\le15.}

Its first hit in the consecutive selector is

\boxed{u=27,\qquad r=107.}

Here

A_{27}=\frac{p+107}{4}=2,200,869 =3^2\cdot11^2\cdot43\cdot47.

An explicit signed-divisor collision is

\boxed{1\equiv-18,189\pmod{107},}

with

18,189=3^2\cdot43\cdot47\mid A_{27}.

Indeed

18,189+1=18,190=107\cdot170.

So the u=27 rescue is exact, not a heuristic hit.

5. Research consequence

The attractive statement

\text{“the first 15 consecutive selectors always suffice”}

is false and must not be used.

The surviving signal is weaker but still useful:

allowing composite r_u=4u-1 materially improves the selector program;
late hits correlate with a shifted integer P+u acquiring enough multiplicative residue diversity to create the signed collision;
the hard case is therefore about how long consecutive multiplicative compression can persist, not about primality of the binary numerator.

Any universal selector theorem must allow an adaptive or unbounded u, or prove a larger bound from genuine structure rather than finite data.