CN Theorem — Finite Active-Core Escape

Status: pairwise-coprime case proved for all finite n; shared-factor C3/C4 certified

Date: 2026-08-15

Depends on: C1-THEOREM.md, C2-THEOREM.md

Claim boundary: Does not prove Erdős-Straus, López-all-primes, or DSC-P for unconstrained shared-prime clusters of arbitrary geometry. Closes the coprime-tower route and certifies small shared clusters.

Setup

Directly novel candidate. Active fixed-negative layers

with pullback moduli q_i > 1 and forbidden sets R_i as in C1. Let

by Theorem C1 (U_i = units mod q_i). Put Q = lcm(q_1,...,q_n).

Theorem CN-coprime (all finite n)

If gcd(q_i, q_j) = 1 for all i ≠ j, then a simultaneous reduced escape exists.

Proof

By C1, each S_i ≠ ∅. Choose a_i ∈ S_i. The system

has a unique solution mod Q = q_1···q_n by the Chinese Remainder Theorem. Moreover gcd(s,Q)=1. Hence s is reduced and avoids every R_i. QED.

Induction form. The case n=1 is C1. The case n=2 is C2-coprime. For n≥3, CRT against the product of the first n−1 moduli (inductively safe) with a_n ∈ S_n extends the escape.

Theorem CN-shared (certificates)

When the q_i share prime factors, compatibility conditions mod those primes are required.

Certificate C3 / C4 (tight, admissible)

See CN-SHARED-THEOREM.md. On the 73,814 admissible hard candidates through k ≤ 1500:

• tight q ≤ 9 triples: 3,994,891 checks, 0 failures;
• the only shared-pair failures are 21 complementary q=3 covers in the 205 family, all directly shadowed by layer 10.

Sampled-r C3/C4 scans remain valid as samples. They are not a universal shared-factor theorem. Unrestricted complementary q=3 covers exist.

Obstruction pattern

Lift-room peels every layer with φ(q)/φ(d) > |R|. The totient-ratio lemma makes this automatic whenever d < q and |R| ≤ 1. The residual tight cluster is 3-adic. Complementary covers there are either non-admissible or ancestry-absorbed (205 → 10). A fully formal proof that every q=3 layer is an absorbed child of a q=1 anchor is the remaining shared-CN path.

Corollary — Coprime active cores are DSC-P complete

Any directly novel candidate whose active fixed-negative moduli are pairwise coprime is reduced-realizable:

1. Character-shield extension (parent)
2. Inactive fixed-negative layers safe by direct novelty
3. CN-coprime simultaneous escape
4. Fiber reverse (parent)
5. Dirichlet → infinitely many exact-depth primes

Scoreboard

Next

1. Formalize shared-factor CN via simultaneous projection of thin two-box pullbacks.
2. Bound |N^{act}| in the Class-C residual (parent census: median variable negative towers ~1).
3. If |N^{act}| is uniformly bounded or generically pairwise-coprime, DSC-P collapses to closed theorems.