The bidi bottle — every edge io, lifted to the torus
Make each edge bidirectional: (n(io(s(io(e(io(w(io()))))o — every direction carries both in and out. A one-way pass on the inference axis still flips orientation, so the surface is still the bottle. But io is out-and-back — two flips, which cancel — so to keep orientation consistent you carry it as state: two sheets, the orientation double cover. Crossing inference once moves you between sheets; cross twice and you're home. And the orientation double cover of the Klein bottle is the torus. So bidi on every edge lifts the bottle to an orientable torus. One-way is the bottle (v1, half-duplex); bidi is the torus (v2, full-duplex). Same surface, walked honestly.
one-way inference → flip (bottle)
io = out + back → flip² = keep
carry orientation = 2 sheets
double cover of Klein = torus
χ = 2·0 = 0 · orientable
bidi spec — runs live
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Status discipline
Literalio round-trips preserve orientation (flip²=id); one-way inference still flips. Carrying orientation = the 2-sheet orientation double cover, which for the Klein bottle is the torus (χ=0, orientable). Re-run in-browser.
Bridgeio = v2 full-duplex (both ways at once); one-way = v1 half-duplex. The held centre )1( is the identity on both sheets — it survives the lift.
Speculative"Bidi heals the bottle into a torus" is the reading. The mechanism is standard: the orientation double cover of K is T². The cover is orientable; the base still is not. Don't claim the bottle became orientable — you lifted to a cover that is.