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Status: proved lemma; see also C1-ESCAPE-CORE.md for the expanded core
Date: 2026-08-15
Claim boundary: Advances C1. Does not prove universal DSC-P, López coverage, or Erdős-Straus.
Theorem (Pullback cardinality)
\boxed{|R| \le |T_j|.}
Proof
Let t ∈ T_j. The congruence L s ≡ t − r (mod m) is solvable iff g | (t − r). When solvable, gcd(L/g, q)=1 gives a unique solution class s mod q. Distinct traps may collide, so |R| ≤ |T_j|.
Corollary (Pigeonhole escape)
|T_j| < φ(q) \implies U \setminus R \ne \emptyset.
Expanded core
Full injectivity, zero-trap, prime-modulus, and finite certificate results: C1-ESCAPE-CORE.md.
Remaining gap
When φ(q) ≤ |T_j| and 0 ∉ R, prove two-box geometry still blocks U ⊆ R.