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Constructive nonuniversal square-lift certificate results

Synthesis · hosted from the CENTL repository

Research library · Synthesis

Synthesis

Completed GitHub Actions run:

Source in the repository

Date: 2026-08-15

Status: completed green finite regression of the universal square-lift classification

Project: Free Computation Foundation / CENTL

Claim boundary: the universal classification proof is in UNIVERSAL-SQUARE-LIFT-SHADOW-CLASSIFICATION.md. The finite certificates below are regression/falsification evidence.

Workflow provenance

Completed GitHub Actions run:

workflow:     CENTL Erdős-Straus square-lift counterexamples
run id:       31852781397
head commit:  45adfb05f8f454b2564fe52338dc7255e425c05a
conclusion:   success
artifact id:  9238029825
artifact sha256:
9f221d7cf82f8a27ccd02beb35da98513852120837047fadf8ef11cff7666d87

Result

Every base

1\le j\le5000

except the three proved universal bases

\boxed{1,2,4}

received an explicit constructive certificate showing that some odd square lift escapes the base shadow.

Thus:

bases checked:                         5,000
universal bases skipped:               1,2,4
nonuniversal bases certified:          4,997
unresolved bases:                          0

For every certified base the artifact records:

  1. a Jacobi-negative residue v outside the base trap set;
  2. the positive class u=-v mod (4j-1);
  3. a prime ell == u mod (4j-1);
  4. an odd square-lift parameter c satisfying c^2 == -(4j-1)^(-1) mod ell;
  5. the lifted depth
K=((4j-1)c^2+1)/4;
  1. the check ell|K;
  2. the explicit lifted trap residue
-\ell\equiv v\pmod{4j-1},

which lies outside the base trap set.

Each tuple therefore directly witnesses failure of universal square-lift shadowing for that base.

Search scale

Across all 4,997 nonuniversal bases:

maximum progression prime used:         1,270,909
maximum progression search steps:              68
maximum constructed lift depth: 19,833,438,467,899,419

The enormous maximum lifted depth is not evidence that the theorem needs huge numbers. It reflects the deliberately direct constructive Dirichlet/reciprocity certificate mechanism.

Interpretation

The finite certification agrees perfectly with the universal theorem:

\boxed{ \text{base shadows every odd square lift} \iff j\in\{1,2,4\}. }

The key methodological point is that every nonuniversal base receives a positive counterexample certificate, rather than merely failing to pass a finite shadow scan.

This is a particularly strong regression design because any future change to the theorem machinery can be checked against thousands of explicit escape constructions.