With the plug disconnected, your AB, AC and BC resistance readings should be 0.1 to 1.0 Ohm. I'm pretty sure a 1 on the far left of the screen of a multimeter indicates the resistance is off the scale. It does not mean 1 Ohm, most meters start at a scale of 0-200 on the smallest range. You are measuring where the wires come through the vee of the engine, not where they go into the main harness. The connectors on yours look pretty shoddily done, and clearly one of them has massively overheated because it's gone brown and toasty, as has the reg/rec plug. Many things can cause this, from a dodgy reg/rec to the effect of corrosion on the connectors causing an increase the the resistance and therefore excessive heat buildup.
You say the battery is weeping. I'd dispose of it purely on those grounds, and I'd be willing to bet that this is one of the reasons it is not holding charge. However, this is probably a result of the real problem, not the cause. You won't find the issue until you have sorted the wiring. There are replica Honda reg/rec connectors on eBay from a Spanish seller. I've used one and while it's not absolutely stunning, for a mere three quid it is a bargain and they arrive in a few days putting many UK based sellers to shame. http://cgi.ebay.co.uk/ws/eBayISAPI.dll? ... OC:GB:3160 Also, my preference would be to get a three pin plug for the alternator, although there is no reason you can't use bullets or individual spades if you prefer. I got one of these http://www.ebay.co.uk/itm/3-Way-Electri ... 1e78ab8721 along with some other bits from the same seller, and again it's not as high quality as the genuine article, however it's two quid. Honda actually do a repair kit for the alternator side of the loom and it's something daft like £30-40!
So, as with the previous advice, cut back and crappy wires until you find good, uncorroded strands inside. On my alternator wiring, this meant several inches! Use good quality automotive cable to replace what you cut out. Don't use cheap stuff, don't use stuff that's not specifically automotive, and use stuff with at least the same core area as what comes out. If you have to replace any of the red or green on the reg/rec plug this means 30A cable. Once you have the wiring sorted out, you can test the circuit properly, with a new battery. Your previous test readings seem to imply that the leaking one is not holding charge. Even if it was, it is leaking, so IMHO it is only fit for scrap.
The first test is to see if the reg/rec is overcharging the system. Overcharging can cause battery damage, and it can potentially burn the connectors as well. With the lights on, at 5000rpm you should read 13.5 to 15.5V DC at the battery terminals. Any higher, and the reg/rec is not reducing the output of the charging system enough and it must be replaced. I am pretty sure that if for some reason the alternator was producing excessive output, it would still be dissipated (or the unit would fry?), but I stand to be corrected. You could always check using the method below. Over time, this causes the battery to become damaged internally, and as a result it becomes less able to hold charge for any period of time and eventually won't hold anything at all. Under 13.5 and the system is not as efficient as it should be, and when you start getting under 13V you might as well think about walking. Usually, low voltage indicates poor wiring. This can usually be sorted by making sure all the connectors are nice and fresh, which you will hopefully have done
However, it is not out of the question that there is a wiring issue somewhere in the middle of the loom, or the alternator may not be putting out what it should be.Disconnect the alternator, and check the resistances as above. Then set the meter to AC on the 0-200 scale. At 5000rpm, each pair should produce 40-50 volts. If the voltage and resistance are in spec, I'd also check at the reg/rec plug for the same readings. They should be as near identical as makes no difference. If they are significantly different, it would suggest the wiring between the plugs needs investigating for damage or corrosion. In my case, there was 0.1 Ohm difference on one of the pairs, and I seem to recall the voltage was identical to a tenth or two.
If you find no obvious problem there, plug everything back in and measure the DC output at the reg/rec plug. Then measure it at the battery. A lot of people find that wiring directly from the reg/rec to the battery and bypassing the main wiring loom can make good gains. In my case, I went from barely being in spec to being right in the middle. To do this, you need some 30A automotive wire (I used both red and black to avoid confusion), an inline fuseholder rated at 30A, and some M6 battery terminals. At least, that's what I used. I ran both the red and black back to the battery. I guess you could use green instead of black as that's what Honda use for the negative side on the connector. Anyway, the black goes directly back to the battery. The red needs to have the fuseholder somewhere in it, I chose to put it at the battery end, and tuck it under the battery strap. Do not be tempted not to use one. If you were to get a dead short it would let this new wire draw as much current as it possibly could until it either melted, caught fire, or did something similar to another component on the bike. That is why there is a 30A fuse next to the solenoid, and you are bypassing it with this circuit. Also, make VERY sure you cap off the old wires from the reg/rec to the loom. The earth won't do any damage itself, but if the positive finds the earth, or finds earth by touching a metal part of the bike, you'll have a dead short and blow the main fuse.
Well, that's another chapter of War and Peace... lol
