Let's start from the beginning.
First, we'll need the following information:
1. Who did you buy this system from?
2. What "code" sequence is the controller indicating at this point?
3. What "code" sequence did the controller indicate prior to the hall sensor replacement?
4. Please further explain what you mean by "I have mutilated the bread board". If you're talking about the controller circuit board then, that's probably all or part of where you problem lies at this point.
I'd further suggest that you check the following:
1. Make sure that the system battery is fully charged by carefully checking the voltage output (at the battery itself).
2. Test the system with the appropriate system battery.
3. Double check any and all system fuses.
4. Make sure that the brake cutoff/cutout switches (if applicable) are operating correctly.
5. Double check every wire for both continuity, short circuits, and shorts to ground.
6. Double check every wire connection.
7. Double check that the system is configured for the correct battery voltage (as per "itselectric's" advice).
I would suspect a bad throttle and/or controller if system passes the above tests. The "ebikesSF" website has several technical articles that might be of help:
http://ebikessf.com/ttm/menu-technicalarticles
http://ebikessf.com/bmc-throttle-pinout
http://ebikessf.com/BMC-50A-controller-outline
Note: pay close attention to the throttle pinout voltage and ground values as per the throttle pinout link above. You can test the throttle by performing the general hall effect throttle testing steps below:
1. The throttle ground should have continuity to ground (i.e. the controller/bike frame/hub motor axle).
2. The "+5" pin (from the controller throttle connector) should show 5 volts. If not, the controller or controller wiring and/or wire connections are probably bad.
3. The "speed signal/hall sensor output" pin (from the throttle connector) should sweep from 0 volts to ~5 volts as you open and close the throttle. Suspect a bad throttle or bad throttle wires and/or connections if this test shows no voltage or a constant voltage while turning the throttle.
Here's a link with some general DC brushless motor design, construction, and testing information:
http://docs.google.com/viewer?a=v&q=cache:J75OcM5UaeMJ:www.gobrushless.com/GBL_single_v2.pdf+troubleshooting+a+brushless+DC+motor&hl=en&gl=us&pid=bl&srcid=ADGEESg8LiR1kgBmWk7ociOnCBRpeglzKO5xbAkOMKPXWny0NVBnEpH-j3fKEoJdN-7A1ein9lm84tvYjUxHMhzti52AQ8nt-HX8o7PmRw_6Xaku8BFT05vqt3k0Tn76EI54RokbPmYH&sig=AHIEtbRdN_rlxPPCMdxvEi8P-VAfTxfnZw
Meanwhile, maybe some of us can find out what the controller "codes" are for your BMC controller...
BTW, electrical/electronic troubleshooting can be frustrating at times. Hang in there, you'll figure this thing out.