Finite square-completion census through 50,000,000

Certificate

---

Status: exact finite computation; standalone reproducer checked in

Date: 2026-08-15

Reproducer: square_completion_probe.py

Claim boundary: this is a finite statement only. ES-SQUARE-COMPLETION-BACKBONE.md proves that completed first-hit depth is unbounded, so no finite observed ceiling is promoted to a universal theorem.

Run domain

Prime domain:

\boxed{p\le50,000,000}

restricted to Mordell-hard residue classes

\boxed{p\bmod840\in\{1,121,169,289,361,529\}.}

Number of hard primes:

\boxed{93,457.}

Completed layers searched:

\boxed{a\le1500.}

For each layer

m_a=4a-1

the exact square-completed trap is

S_a=\{-4D\pmod{m_a}:D\mid a^2\},

with the validity check

p\nmid D.

Result

Every one of the 93,457 hard primes was captured.

\boxed{\text{unresolved}=0.}

The deepest observed square-completed first hit was

\boxed{624.}

It occurred for exactly one prime:

\boxed{p=2,031,121.}

The witness is

\boxed{ a=624, \qquad D=576, \qquad m_a=2495, \qquad q=815.}

Exact checks:

576\mid624^2,

2,031,121+4\cdot576 =2,033,425 =2495\cdot815,

and, with

C=\frac{2,031,121+815}{4}=507,984,

576\mid C^2, \qquad 576+C=624\cdot815.

Thus this is simultaneously a valid square-completed congruence witness and an exact standard Type-II divisor-square witness.

The deepest witness is genuinely mixed

Factor

624=2^4\cdot3\cdot13

and

576=2^6\cdot3^2.

Relative to the midpoint exponents of 624, the square divisor 576 is

above the midpoint at 2;
above the midpoint at 3;
below the midpoint at 13.

Therefore it lies in neither López boundary orthant.

The completed depth is

\boxed{624,}

whereas a direct López Type-A/B scan gives first depth

\boxed{1403}

for the same prime.

So the finite record is a genuine mixed-sign rescue rather than an old boundary hit written in new notation.

Record frontier observed in this run

As primes are ordered increasingly, the completed-depth records are:

prime `p`	first completed depth `a`	modulus `4a-1`
1009	3	11
1201	8	31
2521	12	47
3361	25	99
33289	39	155
90841	42	167
144169	48	191
167521	65	259
225289	70	279
361321	72	287
915961	76	303
954409	84	335
1853329	96	383
2031121	624	2495

No later hard prime through 50,000,000 exceeds depth 624 in this finite domain.

Again, the unboundedness theorem proves that this plateau must eventually break.

Reproduction

From the repository root:

python3 research/erdos-straus/square_completion_probe.py \
  --prime-limit 50000000 \
  --a-max 1500 \
  --compare-ab-max 5000

Expected headline values:

hard_prime_count      93457
captured              93457
unresolved            0
max_completed_depth   624
deepest.p             2031121
deepest.a             624
deepest.D             576
deepest.m             2495
deepest.quotient      815
lopez_ab_first_depth  1403

The script emits JSON including the exact witness checks and record frontier.

Interpretation

The finite data say two things at once:

square completion is not a small cosmetic enlargement of López A/B; it can sharply reduce difficult first-hit depths;
the compression does not imply bounded latency, because the completed prime-modulus backbone proves arbitrarily large exact finite depths.

The useful proof question is therefore not

“Is completed depth bounded?”

but

“Why can no prime avoid every symmetric completed layer, even though the first successful layer can occur arbitrarily late?”