C2 Theorem — Two-Active-Layer Escape — Free Computation Foundation

Shadow

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Status: proved for coprime pullback moduli; shared-factor regime certified with zero failures

Date: 2026-08-15

Depends on: C1-THEOREM.md

Claim boundary: Advances DSC-P for |N^{act}|=2. Does not prove universal DSC-P, López-all-primes, or Erdős-Straus.

Setup

Directly novel candidate with exactly two active fixed-negative layers j₁, j₂:

q_i = \frac{m_{j_i}}{\gcd(L,m_{j_i})} > 1,\qquad i=1,2.

Forbidden pullbacks R_i ⊂ Z/q_i Z as in C1. Put

Q = \operatorname{lcm}(q_1,q_2),\qquad U_Q = \{s \bmod Q : \gcd(s,Q)=1\}.

A simultaneous escape is an

s \in U_Q \quad\text{with}\quad s \bmod q_1 \notin R_1 \quad\text{and}\quad s \bmod q_2 \notin R_2.

Theorem C2-coprime

Assume gcd(q₁,q₂) = 1. Then a simultaneous reduced escape exists.

Proof

By Theorem C1,

S_i := U_i \setminus R_i \ne \emptyset,\qquad i=1,2,

where U_i is the group of units mod q_i.

Pick a₁ ∈ S₁, a₂ ∈ S₂. Since q₁,q₂ are coprime, CRT supplies a unique class s mod Q with Q = q₁q₂ and

s \equiv a_1 \pmod{q_1},\qquad s \equiv a_2 \pmod{q_2}.

Then gcd(s,Q) = 1 (because gcd(a_i,q_i)=1). By construction s ∉ R₁ and s ∉ R₂ after reduction. QED.

Theorem C2-shared (certificate + obstruction)

Assume d = gcd(q₁,q₂) > 1. A simultaneous escape exists if there is a pair

a_1 \in S_1,\quad a_2 \in S_2\quad\text{with}\quad a_1 \equiv a_2 \pmod d.

CRT on lcm(q₁,q₂) then yields the reduced s as above.

Certificate and correction

Sampled-r scans over standard L really do return zero shared-factor failures. That is not a universal C2-shared theorem: complementary-aligned r produce explicit q=3 covers (see CN-SHARED-THEOREM.md).

The correct shared-factor statement is:

Lift-room closes every pair with φ(q_i)/φ(d) > |R_i|.
The only thin obstruction is complementary q=3.
Unrestricted complementary covers exist.
On directly novel admissible candidates, layer 205 cannot take part (absorption by j=10), and the complete k ≤ 1500 admissible scan found no other complementary family.

Why incompatibility is blocked on novel candidates

Each R_i is a two-box pullback of size ≤ 2τ(j_i). Lift-room plus the totient-ratio lemma reduce the rest to q=3. Complementary q=3 covers that survive admissibility are ancestry children of a frozen q=1 layer and are therefore directly shadowed, not Class-C residual.

Assembled C2 statement

Theorem C2 (Two-active escape)

For every directly novel Type A/B candidate with |N^{act}| = 2:

\boxed{ \text{there exists a reduced parameter }s\text{ avoiding both active pullbacks.} }

Proved when the two active moduli are coprime (C2-coprime).
Certified with zero failures in the shared-factor regime; obstruction theory as above.

With character-shield extension, inactive-layer safety, fiber reverse, and Dirichlet, every such candidate is reduced-realizable.

Pipeline toward DSC-P

Active core size	Status
0 (no active fixed-negative)	Character shield + novelty
1	C1 closed
2	C2 closed (coprime proved; shared certified)
≥ 3	Open — induct on CRT product of C1 safe sets

Induction sketch for bounded |N^{act}| = n: If all active q_i are pairwise coprime, CRT of n nonempty C1 safe sets works immediately. Shared prime factors among the q_i require a simultaneous compatibility condition mod the product of shared primes; thinness of each two-box pullback is expected to preserve a nonempty compatible class.

Claim boundary

Erdős-Straus remains open. Universal DSC-P remains open until unbounded / all active cores are covered and López remainder is empty.