External-nonresidue shield-ratio census through `p <= 10^7`

Certificate

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Status: exact finite theorem-mining signal; replayable falsifier checked in

Date: 2026-08-15

Depends on: EXTERNAL-NR-M1-SYNCHRONIZATION.md, FAB-HARD-FIRST-FILTERS.md, external_nr_shield_ratio_probe.py

Claim boundary: this is finite evidence and a theorem falsifier. It is not a universal existence theorem and is not a proof of Erdős–Straus.

1. Rule tested

Restrict to Mordell-hard primes that survive all four exact necessary counterexample restrictions in FAB-HARD-FIRST-FILTERS.md.

For each such prime p, scan primes

\ell\equiv3\pmod4, \qquad \left(\frac\ell p\right)=-1,

in increasing order. Put

C=\frac{p+\ell}{4}.

The exact fixed-k divisor-ratio theorem gives a strong sufficient certificate if

-p^{-1}\pmod\ell

lies in the signed divisor-ratio box of C.

This census imposes the much stronger restriction that the ratio must use only the hard-shield primes:

\boxed{ 2^{z_2}3^{z_3}5^{z_5}7^{z_7} \equiv-p^{-1}\pmod\ell, }

with

-v_q(C)\le z_q\le v_q(C).

Thus no prime factor of C outside {2,3,5,7} is permitted in the winning ratio.

2. Exact finite result

On the hard-prime population through

\boxed{p\le10^7},

the census contains

20,513

Mordell-hard primes in total and

\boxed{2,173}

survivors of the four exact first filters.

Using external primes

\ell\le100,000

and allowing at most the first 300 eligible external primes for each p, the exact shield-only search found

\boxed{2,173/2,173}

successful certificates.

There were

\boxed{0}

failures in this finite census.

3. Hardest observed finite case

The largest first-hit external rank observed was

\boxed{98}.

It occurred at

\boxed{p=4,462,921}.

The successful external prime was

\boxed{\ell=2,459}.

Then

C=\frac{p+\ell}{4} =1,116,345.

The successful shield ratio had signed exponent vector

(z_2,z_3,z_5,z_7)=(0,1,1,0),

so the winning ratio was simply

\boxed{R=15}.

Equivalently,

\boxed{15p\equiv-1\pmod{2459}.}

Both 3 and 5 divide C, so this ratio lies inside the exact allowed signed divisor box.

This example is useful strategically: it demonstrates that the phenomenon is not a hidden fixed finite list of very small external primes. The adaptive external rank can grow substantially while the internal ratio remains tiny.

4. Interpretation

The finite evidence now separates the two jobs extremely cleanly:

choose an adaptive external prime ell to provide the required nonresidue modulus and force

(C/p)=(C/\ell)=-1;

use only the already-frozen hard shield {2,3,5,7} inside C to place the exact ratio

-p^{-1}\pmod\ell.

The universal theorem suggested by the data is therefore not a bounded external-prime theorem. It is the cap-free statement:

For every Mordell-hard prime surviving the exact preliminary filters, there exists some external prime <div class="math" role="math">> \ell\equiv3\pmod4,

> \quad > (\ell/p)=-1, ></div>

such that the available signed {2,3,5,7}-ratio box of <div class="math" role="math">> C=(p+\ell)/4

></div>

contains -p^{-1} mod ell.

Proving that statement would provide a strong sufficient certificate for every remaining hard prime and would close the all-prime wall after the standard reductions.

5. Exact arithmetic form of a shield-only hit

Write the signed shield ratio as

R=\frac AB,

where A,B are coprime 210-smooth integers and

AB\mid C.

Then

R\equiv-p^{-1}\pmod\ell

is exactly

\boxed{\ell\mid pA+B.}

The divisibility AB|C is exactly

\boxed{\ell\equiv-p\pmod{4AB}.}

Thus every finite hit is a pair of simultaneous exact statements:

\boxed{ \ell\mid pA+B, \qquad \ell\equiv-p\pmod{4AB}, }

with A,B supported only on {2,3,5,7}.

This form should be the starting point for the universal descent: the external prime is not arbitrary; it is simultaneously a prime factor of one tiny-shield linear form and a prescribed residue modulo the shield denominator.

6. Replay

The checked-in standalone falsifier is

external_nr_shield_ratio_probe.py.

A matching replay command is:

python3 research/erdos-straus/external_nr_shield_ratio_probe.py \
  --limit 10000000 \
  --ell-search-limit 100000 \
  --max-external 300 \
  --first-four-only

The script uses only the Python standard library and independently performs prime enumeration, the four preliminary theorem filters, Euler-criterion Legendre symbols, hard-shield valuation extraction, and exact signed-ratio dynamic programming.