Cunningham Chain Immunization Dashboard

Small-prime divisibility analysis for first-kind CC roots


produced by cc_immunization_analysis — or auto-loaded from immunization_all.json if present

1. Immunization Heatmap — immune% by CC Length x Prime

0% 100% Immune = (p+1) ≡ 0 mod q — chain can never be killed by q

2. Dominant Immune Combinations

Each root has an immune fingerprint: the set of primes q where (p+1) ≡ 0 mod q. These are the most common fingerprints.

3. Why These Rates? — Multiplicative Order of 2

qord(2,q)|⟨2⟩| Safe residuesImmune%Safe% Expected imm%
(among safe)		 Mechanism

4. Survival Mechanism Breakdown

Three ways a root survives prime q:

IMMUNE (p+1) ≡ 0 mod q — q never divides any member

COSET (p+1) mod q ∉ ⟨2⟩ — 2j·(p+1) never ≡ 1

BEYOND — kill position j ≥ chain length

5. Residue Distribution — (p+1) mod q

Distribution of (p+1) mod q across all roots. Residue 0 = immune. Residues in ⟨2⟩ mod q = would-be killers (but filtered out by chain existence).

6. Kill Position Map

Bright = kill possible at this position. Dashed line = CC10 boundary. Positions left of boundary are forbidden for valid chains.

7. Summary Table

qord(2,q)ImmuneImmune%SafeSafe%Random 1/q

CC18 Campaign — Cunningham Chain Constructor — Immunization Analysis