# [Alchemist], 205 bytes

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 _->b+In_e+p
 p+e->p+g+f
 p+0e->q
 q+g->q+e
 q+0g->r
 r+d->r+a+h
 r+0d->s
 s+e+h->s
 s+0e+0h->Out_"true"
 s+0e+h->Out_"false"
 s+e+0h->t
 t+b->t+d+c
 t+0b->u
 u+a->u+b
 u+c->u+b
 u+0a+0c->v
 v+e->v
 v+0e->w
 w+f->w+e
 w+0f->p

[Try it online!][TIO-golfed]

## Ungolfed



 # Initialize universe b=1 and read input
 _ -> b + In_input + s0
 
 # Create copy of input
 s0 + input -> s0 + input_tmp + input_copy
 s0 + 0input -> s1
 
 s1 + input_tmp -> s1 + input
 s1 + 0input_tmp -> s2
 
 # a <- b_tmp (in the first "round" a will become 0)
 s2 + b_tmp -> s2 + a + x
 s2 + 0b_tmp -> s3
 
 # Compare input and x
 s3 + input + x -> s3
 # if x == input: it's a Fibonacci number
 s3 + 0input + 0x -> Out_"true"
 # if there are x left => we exceeded input & stop
 s3 + 0input + x -> Out_"false"
 # Otherwise continue
 s3 + input + 0x -> s4
 
 # Create copy of b
 s4 + b -> s4 + b_tmp + b_copy
 s4 + 0b -> s5
 
 # Compute b += a + b_copy
 s5 + a -> s5 + b
 s5 + b_copy -> s5 + b
 s5 + 0a + 0b_copy -> s6
 
 # restore input
 s6 + input -> s6
 s6 + 0input -> s7
 
 s7 + input_copy -> s7 + input
 s7 + 0input_copy -> s0

[Try it online!][TIO-ungolfed]

[Alchemist]: https://github.com/bforte/Alchemist
[TIO-golfed]: https://tio.run/##Nc4xDoMwEETRfo9BO0LahJ4@VY6ADCwQiUQG23B8MwRR/eeRi3VzN9n3E2LOTVm3eP0agxcPK2uPEQOp9CILRgZGKLXKip6Bw0QqHSTAMF1Qg5LvFJsirsmKa7unwc3hv13fokS0DHp0pNJJEhyDluhuqIPyscl2XnjmvG6XHQPD63Yo5XN@VlUu@8cB "Alchemist – Try It Online"
[TIO-ungolfed]: https://tio.run/##bVLbasMwDH3PV4gWdmEUnC5tYSx7GQz21E8odqIwQ@KE2Fm7/Xwmy0m8jgUCks7RkXWRdfGBjbZuHNfwbrTTstbfCIPRn9hbBJWnIE0JPcoStOkGl5xg8wIKHoh/4giZViTJGl6J5RCKtvuCtproVhAebJ/ILnsn13SL7XMCVURqmiQ2XbKZz@E5K6DiGt36l0h43oDiGMCdNuA@ECrdWwervh1MuSLKWdc1KCzaBkHcJ3brS6koRK6k/xIQEZFHbrZtOtnj1JmfEREfY69kXCb2GnRFTp4H6Am0u7Wk/aZVa2RRaDBDo7AP@WLOF5x/pN5Wrh9wFXSoE6rqK1@gxspB/gJnBLwUiCVOS4IbsK7t/ghCFKxkbVnx6PXO2vq1GafNgH@6CK@w2T8LVonNeGaBQaaalqqmhWY8OIZ389AGUqDzyXm4M3HndSTEj1M8wWP88XK49BVGBWRYz4LtfakeaQLzfhK7v77CfYj8OrYDHdshHtsidojXdojXtsBiHNMsGzdl@gM