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Falsified The escape chain — a deterministic walk on primes where each +1 escape is executed; bag bases have holes the walk certifies

id: escape-chain-bag-holes · filed: 2026-10-17 · depends: primorial-flank-dead-two-primes · falsifier: continued runs of the walk where caught escapes are executed primarily by wheels inside the collected bag (would restore the consecutive-base picture)

Mechanism

[S] New observable, defined 2026-10-17: start P = 1. Each step reads P+1. If prime, q = P+1 (a full escape); else q = spf(P+1) (a caught escape). New base P′ = P·q. A deterministic walk on the primes generated by the mechanism itself: the +1 escape's output is fed back as the next base's multiplier.

• [F] Each full escape ≈ squares the base (P′ = P(P+1)); each caught escape grows it by q only. Two growth regimes, one walk.
• [F] The walk's base is a bag of wheels it happened to collect, not the primorial. The +1 escape is immune only to wheels in the bag, so wheels never collected stay live and execute escapes that the consecutive-base picture calls already-covered territory.

What was tried

First run: 21 steps to a 79-digit base. 6 full escapes (P+1 prime at k = 0,1,2,3,7,8), 15 caught escapes. Composite-step killers: 13, 53, 5, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887.

The surprise: 14 of 15 caught escapes were executed by wheels smaller than the largest wheel already in the base — including q = 5 at step 6. Wheels 5, 11, 17, 37 remained uncollected for 20+ steps while actively killing. Only one kill in 15 (q = 53 > l = 43) was a true primorial-style danger-zone execution.

Downstream

• Correction filed against the 08:55 danger-zone framing: "(killers are mid-size wheels in (l, √P]" is a property of consecutive bases, not generic. For bag bases, killers are usually blanket holes.
• Consequence: the walk certifies holes in any non-consecutive "covered" claim — any claimed covered region built from a non-consecutive base is testable by exactly this walk.
• [G] Open: does the bag ever fill, and how fast? Step-size law and bag-fill curve unmeasured — the designated next run. Does the chain die out if primorial primes are finite? Unknown.