I DO have a suggestion for Alan. I think Liz needs an emergency kit. A purse, sporran, or fanny pack that contains the 23rd century equivalent of a thinsulate poncho or jumper. It doesn't have to be fashionable, just something she can pull out and get into in an emergency.
It's only a problem because it's still January in the second book - in fact, the second book begins at the same scene the first book left off with. By July, this won't be a problem.
You can do anything you want with High Technology except violate the Laws of Physics.
Yes, it is possible to make a cell phone fit on a single chip and be about the size of your fingernail. However, it would have a proportionate broadcast power. The antenna on a modern cell-phone represents the second heaviest part, after the battery. The basic theory is straightforward - pass an electromagnetic signal through an antenna to broadcast the signal to the air. Pretty simple. But, the two limitations are 1) the antenna in question has to be capable of taking the wattage you're applying to it, which means it has to be of a certain mass given what kind of metal it's made out of or it will be damaged by the power you're applying to it, and 2) you have to be able to apply enough power to have your signal clearly picked up out of the general EM background "haze" of modern cities. As it stands, today's background "haze" means that a modern cell-phone has to produce several watts of power in the form of microwave energy (and you're holding the transmitter right up against your brain...).
In the future, there will be more people and more cell-phones - the background haze will get worse. The solution is already being discussed today, and is three-fold: 1) Limit the broadcast power of cell phones (both for transmission clarity and for health & safety concerns), 2) Increase the number of cell-towers so that smaller broadcast signals can be used (you don't have to broadcast so "loudly" when the nearest tower is on the same block you are), and and 3) use more excacting technology in the towers to be able to accommodate larger numbers of users and eliminate cross-talk.
The big stumbling block, however, is that a cell-phone the size of your fingernail is going to have a linear amplifier/transmitter even
smaller than that to fit inside it and accommodate something that will functino as a power source - and that means, at best, you're going to produce mere milliwatts of energy. And, it won't produce it for long, because physics limits the energy density of batteries, as well. Fuel cells could be used, yes, but physics limits their energy density, too - and limits their minimum size to even function in the first place. A fuel cell that fit into a cell phone that is the size of your fingernail would be so tiny, ordinary dust in the air could clog it's air intake and render it useless. One of the biggest reasons we havent' seen fuel cells used in modern cell-phones is simply that in test models, ordinary pocket lint screwed them up. Thus, though it is technically possible for people in Alan's day to cobble together a cell-phone on Liz' scale, it wouldn't work well because it's broadcast power would be very tiny, even given advanced technology in cell-towers. And, most importantly, it wouldn't work for long, either on duration of broadcasting, or just overall duration of the item before needing repair.
The last stumbling block is, of course, the character. 1) Even if something like this was possible, Liz herself would never use it. She values her little brain (it's her only real asset) and has no desire to hold a microwave transmitter up next to it for any real length of time. 2) From a literary standpoint, giving the character a cell-phone eliminates the idea of her own helplessness. She needs Alan far more than he needs her. However, give the little broad a cell-phone, and she could be ordering pizza and charging it to his account in-between calling the cops when she's in trouble - she is elevated from the status of "Muse" to "person in motorized wheelchair", in that while she still can't manipulate things or open doors, she is far from helpless because she can (literally) call for help.
From a literary standpoint, Liz plucks the same strings that "Lassie" does - she's intelligent, communicative, but utterly helpless in the "human" world, and situations that are easily resolved by a human border on impossible for her.
"Lassie! Timmy fell down the well! Go get help!"
"Woof!"
<scenes of Lassie running through the countryside, leaping through country fences, little Timmy's life on the line...>
And when you're watching this on the screen, it doesn't strike you as amazingly stupid - after all, you've gotten this far in the show, you've already managed to suspend your disbelief long enough to accept the notion that a collie is smarter than the little boy she hangs out with and can understand exactly what is being said when his friend tells Lassie to go get help, and what she needs to do.
Instead, it's dramatic, because of the general helplessness of Lassie to begin with. How is Lassie going to make it to help? "Well, she runs!" you say? No, sorry. Think about just how many stinkin' episodes of "Lassie" that stupid @#$%#ing dog got hung up by someone when she was on a rescue mission? The dog-catcher nabs her because she lost her collar on the fence. She gets trapped in a barn or other building looking for someone to help. She gets home and simply can't GET to someone to help, simply because she can't open doors, can't use the phone, and when Ma and Pa are off in town buying food, she not only doesn't know where they are, but doesn't know how to find them. And even when she DID find help, the big issue of her doggie-ness always came to the fore. How is Lassie going to get whoever she finds to understand the problem when her communication is limited to "Woof, woof!"
Of course, once Lassie manages to GET to help, the crisis was always resolved.
"Look, John, it's Lassie!"
"Hey, there, Lassie! You're all covered in grass, where you been? And where's Timmy?"
"Woof! Woof!"
"John, she seems to be trying to tell us something."
"Woof! Woof!" <Lassie dancing away, darting back, dancing away>
"John, I think she wants us to follow her!"
"Hmmm... Yes, and Timmy's not with her - that worries me. I'll bet that damn punk son of mine fell down the well AGAIN. Little beggar, I'll whip his ass with the belt when I find him. Come on, Lassie! Lead me to Timmy!"
It's the element of challenge that makes the situation dramatic - the character Lassie is challenged by a situation that even a child would not be challenged by. For example:
"Billy! Timmy fell down the well! Go get dad!"
"Okay!"
<scenes of little Billy running back to the farm, then shouting "Dad! Dad! Timmy fell down the well!">
See what I mean? It's the relative challenge that makes the scenario dramatic. Little Billy doesn't have a whole lot of drama when all he has to do is run home and yell for Dad, so it's always either A.) he has to run home and get Dad before little Timmy
drowns, or B.) the Villain captures little Billy and detains him, not believing (or not caring) when he explains the situation.
"But Mister Parkins, Timmy needs help!"
"Yeah, right, kid. Look - I'm the Truant Officer, and I'll lose my job if I don't take you in. You and Timmy and Johnny have been skipping school, it's time for you to pay the piper."
"But Timmy might DROWN!"
"Sure, sure. And I'll bet I find him and Johnny hanging around the apple tree again, like last time."
Again - see what I mean?
Well, by extension, it's not a big deal for a human to be stuck outside in January with no warm clothes. Hell, in extremis, you can just knock on the neighbor's door and ask to call a cab. Liz can't even do that - she is utterly dependent on Alan for the basics of survival. It's her helplessness that makes these simple scenes dramatic - and giving her a cell-phone eliminates that helplessness, and renders it extremely anticlimactic.
"Look, Mister Parkins - here's my phone. Call Timmy - he's down the well. He'll tell you!"
"Well, alright." <beep-boop-beep-beep-beep-boop-beep-beep!> <ring!> <ring!>
"Hello? Yeah, Timmy? This is Mister Parkins. Where the hell are you, you little punk? Oh... I see, you're down the well on the Forsythe farm? Alright - I'll come get you. But boy, are you kids gonna get it for skipping school again!" <hands phone back to Billy> "Okay, kid, let's go save your friend!"