C1 Escape Core Theorems

Status: superseded in part by C1-THEOREM.md (keep as lemma sheet)

Date: 2026-08-15

See C1-THEOREM.md for the assembled escape theorems including thinness, program-range closure for j ≤ 1500, and small-φ certificates.

Universal lemmas retained

• 0 ∉ T_j
• |R| ≤ |T_j| ≤ 2τ(j)
• ψ injective; image progression ≡ r (mod g)
• pigeonhole when |T_j| < φ(q)
• thinness: |R| ≤ δ_g(j;a)+δ_g(j;b) (two divisor classes only)
• T_j ⊆ -D_j
• q = 3 difference obstruction

Finite certificates

• Zero unit covers for j ≤ 1500, all odd q with φ(q) ≤ 64
• Zero unit covers for φ(q) ≤ 48, j ≤ 3500 along ancestry APs

Residual open strip

j > 1500 and φ(q) ≤ |T_j| — only remaining local obstruction for universal C1.