Single-active Class-A local escape

Status: proved universal local lemma

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Coordinator: primary research coordinator

Partner input: Operator-02 active fixed-negative split and valuation criterion

Claim boundary: this lemma resolves the unique active fixed-negative row locally under a Class-A hypothesis. It does not by itself prove the full Class-C system, universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

Read with:

• CLASS-C-C1-SINGLE-ACTIVE.md
• OPERATOR-COORDINATION.md
• operator-02/DIAMOND-FIXED-NEGATIVE-PULLBACK-SPLIT.md
• operator-02/DIAMOND-VALUATION-CRITERION.md

1. Setup

Fix a directly novel admissible target candidate

Let j<k be an earlier layer and put

Its exact pulled-back Type A/B forbidden set is

Assume j lies in the active fixed-negative core and that its valuation excess is Class A only:

Equivalently, no prime absent from L survives into q_j; activity comes only from higher powers of primes already fixed by the target progression.

2. Local escape theorem

Theorem

If the target candidate is not directly shadowed by layer j and

then there exists

such that

Moreover every such choice is automatically reduced at every prime dividing q_j.

Proof

Direct shadowing by j is exactly the statement that every parameter residue modulo q_j is forbidden:

The candidate is directly novel, so in particular it is not directly shadowed by j. Therefore

Choose

By definition of the exact pullback, this gives

Now let p|q_j. By the Class-A hypothesis, p|L. Hence

Since the target progression is admissible,

so p does not divide r. Thus

This holds for every p|q_j, so every local escape residue is automatically reduced on the active layer's prime support. QED.

3. CRT independence corollary

Corollary

Under the theorem's hypotheses, the local escape choice

can be combined by CRT with arbitrary parameter congruence conditions whose moduli are coprime to q_j.

Proof

This is the ordinary Chinese remainder theorem. The point here is structural: Class-A activity lives entirely on prime-power lifts of coordinates already contained in L, while genuinely new parameter coordinates away from those primes are coprime to q_j. QED.

4. What the lemma removes

For a candidate with

let j0 be the unique active fixed-negative layer.

If j0 is Class A only, then:

1. the unique active fixed-negative row itself always has an exact local escape;
2. reducedness cannot eliminate that escape;
3. any remaining obstruction must come from compatibility with other exact rows sharing the same small prime-power coordinates, or from nonfixed rows surviving the finer reductions.

So the unique active row is not, by itself, a local covering obstruction.

This is weaker than full C1 because the complete residual system can contain nonfixed exact constraints.

5. Finite k <= 1500 signal

The coordinated primary replay of the frozen k<=1500 bundle found, before independent workflow verification is promoted:

exactly one active fixed-negative layer: 2,770 candidates
Class-A-only source:                     2,770
Class-B or mixed source:                     0
observed q_j0:                           3, 5, 9
minimum reduced-safe residues mod q_j0:     2

Thus every single-active candidate in that finite range lies inside the theorem above.

These counts remain finite evidence until the dedicated Class-C census workflow completes and freezes its independent verifier/artifact.

6. A sharper finite observation to test

The same replay indicates that after exact fiber peeling the unique active fixed-negative row survives into the final nonempty residual kernel only very rarely, while most residual edges in this single-active population come from nonfixed earlier layers.

If independently verified, that means the next C1 proof target should not be phrased as merely "escape the unique active row." The theorem above already does that locally. The real task is:

prove that the active-row escape can always be coordinated with the surviving nonfixed exact rows.

That is a much sharper statement.

7. Falsifier and boundary

The theorem would fail only if one of its explicit hypotheses failed:

• the candidate were directly shadowed, making R_j full;
• a prime dividing q_j were absent from L (Operator-02 Class B), so reducedness and free-coordinate compatibility would require additional work;
• or the target progression were not reduced/admissible.

No claim is made for Class-B activity here.

The lemma is elementary once the active-core/valuation language is isolated. Its value is as a rigorous reduction inside the coordinated DSC-P proof architecture, not as a stand-alone resolution of the Erdős-Straus problem.