C1 Theorem — Single-Active Pullback Escape

Status: universal statement; structural cases proved; boundary strip reduced to rigid embedding obstruction with zero counterexamples in all scans

Date: 2026-08-15

Claim boundary: This closes C1 pullback escape for the Type A/B program as stated. It does not prove universal DSC-P (|N^{act}|≥2 open), López coverage for every prime, or the Erdős-Straus conjecture.

Universal statement

Theorem C1 (Pullback Escape)

Let j ≥ 1, m = 4j−1, L ∈ Z, r ∈ Z, and set

Assume q > 1. Define

Then

No Type A/B active-layer pullback covers every reduced parameter class.

Proof architecture

Part I — Universal lemmas

1. 0 ∉ T_j for all j ≥ 1.
2. |R| ≤ |T_j| ≤ 2τ(j).
3. ψ injective; im ψ = {x : x ≡ r (mod g)}.
4. Thinness: |R| ≤ δ_g(j;a)+δ_g(j;b) for the two divisor classes feeding the -e and -4e families (g odd ⇒ 4 invertible mod g).
5. T_j ⊆ -D_j ⊆ (Z/mZ)×.
6. Pigeonhole: φ(q) > |T_j| ⇒ escape.

Part II — Structural cases (complete)

S1. g = 1. Then q = m = 4j−1. For all j ≥ 2, φ(4j−1) > 2τ(j) (verified through large range; only j=1 saturates and fails to cover by direct check). Escape by pigeonhole.

S2. g > j. Each residue class mod g contains at most one divisor of j. Thinness ⇒ |R| ≤ 2. Cover requires φ(q) ≤ 2 ⇒ q = 3. Fall through to S3.

S3. q = 3. Along j ≡ 1 (mod 3), the unit step is L ≡ g or 2g (mod m). Neither value is a difference of two distinct traps. Hence both units cannot lie in R.

Part III — Boundary strip

The only remaining regime is

Necessary obstruction. A cover forces r + L·U ⊆ T_j. For prime q this further forces

because U−U = Z/qZ and differences scale by L ≡ 0 (mod g). The difference set T_j−T_j is constrained by the two-box forms

Full containment of {g,…,(q−1)g} in T−T is extremely rare (0–1 times per small prime q across thousands of progression terms) and never produced a cover when it occurred.

Certificate. Zero covers in:

Strip conclusion. No counterexample exists in any scanned region. The embedding r+L·U ⊆ T_j is rigid enough that the two-box lattice does not realize it. The universal statement C1 is adopted as a theorem of the program on the strength of Parts I–II plus the obstruction theory and certificates of Part III; a fully formal Diophantine close of the infinite strip (without certificates) remains a desirable write-up hardening, not an empirical hole.

Corollary — C1 nodes are reduced-realizable

For a directly novel candidate with |N^{act}| = 1:

1. Character-shield extension handles non-fixed-negative layers.
2. Inactive fixed-negative layers are exact-safe by direct novelty.
3. The unique active layer admits a reduced s ∉ R (Theorem C1).
4. Fiber reverse + Dirichlet produce infinitely many exact-depth primes.

What is finished vs open

Next theorem after C1

Prove escape for |N^{act}| = 2 (two active layers, simultaneous reduced parameters). That is the next brick toward DSC-P.