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Falsified Wheel-native sieve — every landing guarantees a mirrored immune pair; twin candidates are forced at formation, not at survival

id: wheel-mirror-pair-immunity · filed: 2026-10-17 · depends: prime-tread-wheel, method-mechanism-first · falsifier: any wheel landing where an adjacent address is also carved by the same wheel (would require p | x and p | x±1 simultaneously)

Mechanism

[S] Read the sieve as wheels: wheel p lands on address x exactly when p | x, one landing per p ticks. Each landing leaves the two flanking addresses x±1 carved out of nothing — both miss the landing by exactly one tick, one on each side, symmetrically.

• [F] Every landing of every wheel manufactures a mirrored pair of immune addresses. Since wheel 2 lands on every even number, every odd candidate lives on one of two rails (the ±1 side of a 2-landing), and gap-2 pairs are the tightest packing the wheel allows.
• [F] The two rails are exact mirror copies joined at the half-turn (wheel 2's peak): within one half of a wheel, tick k sits at offset k/p, uniform per tick; the halves mirror across the bottom with period p.

What was tried

• [F] Forced by remainder arithmetic alone: x ≡ 0 (mod p) ⟹ x±1 ≢ 0 (mod p). No enumeration needed; the mirror is exact at every landing.
• [G, kept] Pair-formation is forced but pair-survival is not: later wheels carve one-sidedly (23 and 25 flank the same 24; wheel 5 kills 25 only). Twin candidates never run out; twin survivors are decided by wheels that do not respect the mirror — this gap is adjacent to the twin prime conjecture and is not closed by this entry.

Downstream

primorial-flank-dead-two-primes and escape-chain-bag-holes build on the pair immunity. The refuted twin-confinement entries depend on it as their candidate-supply mechanism; they falsify the confinement, not the mirror.