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DSC-P Fragment — Coprime Active Cores

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Shadow

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Source in the repository

Status: proved theorem

Date: 2026-08-15

Claim boundary: Infinite fragment of Direct-Shadow Completeness. Not universal DSC-P. Not Erdős-Straus.

Theorem

Let a directly novel Type A/B candidate have active fixed-negative core N^{act} with pullback moduli q_j (j ∈ N^{act}). If

\gcd(q_j, q_{j'}) = 1 \quad\text{for all distinct } j,j' \in N^{\mathrm{act}},

then the candidate admits a reduced avoiding progression. Dirichlet supplies infinitely many primes of exact Type A/B depth equal to the candidate depth.

Proof chain

  1. Character-shield completeness ⇒ non-fixed-negative layers can be made Jacobi-positive (parent).
  2. Inactive fixed-negative layers (q_j = 1) exact-safe by direct novelty.
  3. Each active layer has nonempty reduced safe set S_j by C1.
  4. Pairwise-coprime CRT of the S_j by CN-coprime.
  5. Fiber reverse construction (parent).
  6. Dirichlet.

Content

This is an exact infinite theorem inside the Type A/B shadow program, not a finite census.

Limitations

  • Shared prime factors among active moduli excluded.
  • Does not force López coverage for primes outside Type A/B.
  • Does not handle composite denominators of Erdős-Straus.