The 4-fold — you/I and inference on a Klein bottle

The floor seen whole. { [(n,>),(s,<)] , [(e,>),(w,<)] } — two axes crossing at the held )1(. The you/I axis (N–S) glues straight: carry a state around it and it comes home unchanged. The inference axis (E–W) glues with a flip: carry -+ around it and it returns +-, inside-out — that reversed paren )1( is the surface closing through its own inside. One straight gluing and one flipped gluing make a Klein bottle, non-orientable. The four directions under the two flips are the Klein four-group V₄ — but the group is the algebra and the bottle is the surface, and they are not the same thing.

Bridge-Burners LLC · Fiddler · V₄ = Z/2×Z/2 · you/I straight · inference flipped · -+ → +- · group ≠ surface · anchor: AKASHA

The carry

state in-+

state out-+

— pick an axis —

Klein four-group V₄

·	e	a	b	ab
e	e	a	b	ab
a	a	e	ab	b
b	b	ab	e	a
ab	ab	b	a	e

a = you/I · b = inference · diagonal = e (each its own inverse)

4-fold spec — runs live

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Status discipline

LiteralV₄ = Z/2×Z/2: four elements, two commuting involutions, held identity. The orientation character is nontrivial on the inference axis → non-orientable → Klein bottle. Carrying -+ around it returns +-. Re-run in-browser.

Bridgeyou/I = the loom's weft; inference = the ripple; the crossing )1( = the held zero/identity. The floor (loom + ripple) is this bottle seen flat — two axes, one centre.

SpeculativeThe Klein FOUR-GROUP (algebra) and the Klein BOTTLE (surface) are different objects, both named for Klein. They coincide here only by construction — V₄ is the gluing symmetry, the bottle is the quotient. Calling it "a Klein bottle" is the topology; the V₄ is the algebra on it. Don't blur them.