External-nonresidue × hard-shield fixed-k probe

Theorem

---

Status: exact finite exploratory signal; standalone falsifier checked in

Date: 2026-08-15

Depends on: FAB-UNBOUNDED-DIVISOR-RATIO-CERTIFICATE.md, FAB-HARD-NONRESIDUE-BRIDGE.md, FAB-MIRROR-CHARACTER-OBSTRUCTION.md

Claim boundary: this is not a universal theorem and not a proof of Erdős-Straus. The zero-failure observation below is finite and must remain labelled computational until promoted by a theorem.

1. Motivation

The mirror-character theorem proves that reflecting a shifted factor whose prime support is already quadratic-residue-safe cannot supply the fixed-k target. Therefore the fixed-k construction must deliberately import external nonresidue support.

For a Mordell-hard prime p, call a prime

\ell\ge11

external when

\boxed{\left(\frac\ell p\right)=-1.}

The hard shield gives

\left(\frac2p\right) = \left(\frac3p\right) = \left(\frac5p\right) = \left(\frac7p\right)=+1.

This suggests combining one genuinely new nonresidue prime with the existing small shield.

2. Exact rule tested

For each hard prime p:

list the first eight primes ell>=11 with (ell/p)=-1;
for each such ell, try

\boxed{k=m\ell,\qquad m\in\{1,3,5,7\},}

retaining only k==3 mod4;

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

apply the proved fixed-k theorem exactly:

\boxed{ \exists u\mid C^2,\qquad4u\equiv-1\pmod k. }

A hit therefore gives a genuine sufficient Egyptian-fraction certificate. The only non-theorem part is the assertion that this menu always contains a hit.

3. Exploratory finite result through 10^7

On the six Mordell-hard classes modulo 840, the exploratory exact run covered

\boxed{20,513/20,513}

hard primes p<=10^7.

Observed first-hit multiplier counts were:

m = 1 : 11,935
m = 3 :  6,000
m = 5 :    867
m = 7 :  1,711

The successful external-nonresidue rank was heavily front-loaded:

rank 1 : 18,774
rank 2 :  1,458
rank 3 :    216
rank 4 :     46
rank 5 :     12
rank 6 :      6
rank 7 :      1

No first hit in this run required the eighth nonresidue, although the checked-in falsifier deliberately keeps an eight-prime menu for margin.

For the successful shifted factor C=(p+k)/4, the gcd with the hard shield 105=3*5*7 was distributed as:

gcd(C,105)= 1 : 2,747
             3 : 6,415
             5 : 4,231
             7 :   527
            15 : 6,187
            21 :    35
            35 :   371

So most hits acquire a 3 or 5 coordinate, but the mechanism is not merely divisibility by 105: 2,747 successful cases had gcd(C,105)=1.

4. Why this signal is stronger than another bounded box

The menu is not a generic search over arbitrary (a,b).

It has the specific structural form

\boxed{ \text{external nonresidue} \times \text{hard-shield multiplier }\{1,3,5,7\}. }

The external factor supplies the quadratic sign that the mirror theorem proves is necessary; the multiplier comes entirely from the already-frozen small-prime hard shield.

This makes the observed rule a plausible theorem target rather than a naked numerical cutoff.

5. Standalone replay

The exact finite falsifier is checked in as

external_nr_fixed_k_probe.py.

It uses only the Python standard library and independently performs:

prime enumeration;
Mordell-hard residue filtering;
Euler-criterion Legendre symbols;
exact factorization of C;
exact residue dynamic programming over all divisors of C^2.

Recommended replay:

python3 research/erdos-straus/external_nr_fixed_k_probe.py \
  --limit 10000000 \
  --nr-count 8 \
  --json

A hosted/independent replay should be frozen before treating the finite counts as a formal certificate artifact.

6. The theorem we actually need

The finite observation points to the following sharply stated target.

External-nonresidue shield-rescue target

For every Mordell-hard prime p, there exists an external prime ell>=11 with

\left(\frac\ell p\right)=-1

and a multiplier

m\in\{1,3,5,7\}

such that

k=m\ell\equiv3\pmod4

and, with

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

the square-divisor box hits the fixed target:

\boxed{ \exists u\mid C^2: 4u\equiv-1\pmod k. }

Proving this statement would give a pointwise sufficient fab certificate for every Mordell-hard prime. Combined with the classical reductions for the non-hard classes and scaling from prime divisors, it would close Erdős-Straus.

The number 8 from the finite probe is not part of the theorem target. It is evidence that the external prime may be chosen very early, not a claimed universal bound.

7. Immediate proof question

For k=m ell, the CRT decomposition of the target separates into:

an external ell coordinate where (ell/p)=-1 forces a nontrivial character condition on

C=(p+m\ell)/4;

a tiny m coordinate supported entirely on 3,5,7.

The next proof step is therefore to determine whether the divisor-square residue set of C can fail simultaneously on both coordinates for every external nonresidue ell.

That is the present one-shot target.