I really like reading questions that start with “It may sound really stupid”; it reminds me of myself when I firstly got into Bitcoin, a year ago.
but let's say I have a cold wallet with 1 bitcoin on it.
You shouldn't be picturing bitcoins as actual coins inside wallets. They're not hidden into your computer. On the contrary, they're publicly announced and everyone can observe your transactions. The wallet from your computer generates private keys and then it derives from them their addresses, which are the invoices. When you give to someone your address, you're the only person who can prove the ownership of that address. Is it understandable 'till now? Alright.
The Bitcoin network is consisted of computers running a bitcoin client, the most known is
Bitcoin Core. These computers are the ones that distribute information related with Bitcoin such as blocks, transactions etc. They all
autonomously follow some consensus rules such as not exceeding the 21M coin limit and readjusting the mining difficulty every 2016 blocks. They keep a copy of the transaction history.
The coins should be imagined as unspent transaction outputs (UTXO for short) and not like actual coins. When you broadcast a transaction to the network, you essentially sign some inputs of your previous transactions and you then create outputs depending on where you want to send. For example, if you have an input of 0.05 BTC and another one of 0.08, you can create an output of 0.13 BTC. Nodes verify the inputs that you choose to spend to check if they're valid.
To answer your question, one can't create a bitcoin out of thin air, because in our example, creating an output of 1.13 BTC from 0.05 and 0.08 BTC won't be accepted as “valid” from the computers that handle this peer-to-peer and completely decentralized network.