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Falsified Wheel-native sieve — a killed primorial flank always certifies ≥2 primes in the danger zone; the +1 escape is untouchable past a finished base

id: primorial-flank-dead-two-primes · filed: 2026-10-17 · depends: wheel-mirror-pair-immunity · falsifier: survived — all 13 primorial flanks to 41# fully factored (14 dead flanks, every factor > l, Ω ≥ 2, no prime-square flank observed)

Mechanism

[S] Let P be the product of all wheels ≤ l (a completed base). The addresses P−1 and P+1 are invisible to every wheel in the base: each lands dead-center on P, its next landing is a full wheel away, so none can touch the ±1. Every completed base manufactures a twin-candidate pair immune to the entire base (Euclid's escape restated on the wheel).

• [F] If a flank P±1 is composite, its smallest factor q cannot divide P (q ∤ P and q | P±1 is impossible), so q > l. The cofactor ≥ q, hence a dead flank certifies Ω ≥ 2 primes (with multiplicity), all in the danger zone (l, √(P±1)]. Distinct-p version (ω ≥ 2) fails only for a prime-square flank q² — possible in principle, never observed.
• [F, unconditional] Double +1 escape: P+1 forces a prime q₁ > l; (P·q₁)+1 forces a second one q₂ ∉ {primes ≤ l} ∪ {q₁}. At least two primes beyond every layer exist whether or not any flank ever dies.
• [G, kept] Whether dead flanks keep occurring forever is unproven — it is unknown whether infinitely many primorial primes exist. Observed flank survival 42% (10/24 non-trivial flanks prime).

What was tried

Enumeration the same morning the claim was stated: trial-division factorization of every primorial flank 2# … 41# (both flanks each). 14 dead flanks, all factors > l, Ω values 2,2,3,2,3,2,3,2,2,4,3,3,3 (mean ≈ 2.6), every smallest killer inside (l, √P]. Prediction check: bookkeeping expects 14.77 surviving flanks of 26, observed 12 (~1σ). Residual checks T1 (survival rate matches slice-bookkeeping) and T2 (executioners are ordinary mid-size wheels, 13/14 within 20·l) both pass.

Downstream

escape-chain-bag-holes uses the immunity's conditionality (immunity holds only for wheels in the base). If a primorial-prime counterexample ever appeared above 41#, only the [G] note changes; the Ω ≥ 2 proof is untouched.