q=3 Strong Absorption

Status: theorem

Date: 2026-08-15

Depends on: CN-SHARED-THEOREM.md (205→10 special case)

Claim boundary: Kills a large family of complementary q=3 threats on directly novel candidates. Does not classify base q=3 layers. Does not prove DSC-P or Erdős-Straus.

Setup

Layer j with m_j = 4j−1. Write q_j = m_j / gcd(L, m_j). Interest is the case q_j = 3, i.e.

so 3 ∦ necessarily, but the pullback modulus is exactly 3 and L is divisible by every prime-power factor of m_j except a residual factor 3.

Theorem (Strong q=3 absorption)

Let j > i ≥ 1 satisfy:

1. m_i | (m_j / 3),
2. T_j \bmod m_i \subseteq T_i (trap reduction / divisor-child inclusion).

If a candidate progression has q_j = 3 and R_j ≠ ∅, then the progression is directly shadowed by layer i (in particular it is not directly novel).

Proof

q_j = 3 means gcd(L, m_j) = m_j/3. Hypothesis (1) gives m_i | (m_j/3), hence m_i | L. Therefore

the progression is frozen modulo m_i:

Nonempty R_j supplies some parameter with x ∈ T_j. Reducing modulo m_i and applying (2) yields x ∈ T_i. But every point of the progression has the same residue modulo m_i, so the whole progression lies in T_i. QED.

Corollary

On any directly novel candidate, a layer j with a strong absorption ancestor cannot participate in a complementary q=3 cover.

Special case recovery

For j = 205, m_{205} = 819, m_{205}/3 = 273, and m_{10} = 39 | 273 with trap reduction T_{205} \bmod 39 \subseteq T_{10}. The theorem recovers the parent 205 → 10 absorption.

Census through j ≤ 1500

Among layers with j ≡ 1 (mod 3) (necessary for 3 | m_j):

Every strong-absorb layer is novel-impossible as a q=3 complementary participant.

Remaining q=3 threat

Complementary covers can only involve:

• base layers (no reducing ancestor), and/or
• weak-only layers (ancestor exists but may not freeze when q_j=3).

Admissible scan through k ≤ 1500 found the only complementary failures inside the 205 family (strong-absorbed). Extending that scan past 1500 and classifying base/weak pairs is the next finite certificate.

What this is not

• Not a proof that every q=3 layer is absorbed.
• Not a universal shared-factor CN theorem.
• Not Erdős-Straus.