Message Cost

The idea

Information is physical. Erasing one bit — forcing it to a known state regardless of what it held — destroys a distinction, and the second law makes you pay for that destruction in heat. The floor is the Landauer bound: erasing one bit at temperature T dissipates at least T · ln 2 of energy. In PROTIUM, where the temperature unit Hf☉ is the energy unit He☉ relabeled (k_B = 1), that floor reads with no conversion constant in the way:

The information content of the message and the energy to erase it land in the same unit — the same He☉ the Decay Workbook uses for decay Q-values. That shared currency is the point of the tool below.

How the message is measured

"How much information is in this message" has more than one honest answer, and the tool shows two because the gap between them is the real lesson:

Naive bit-count assumes every character is independent and equally likely — bits = characters × log₂(alphabet size). It is the most a message of this length could carry. Shannon content measures the actual symbol distribution: repeated and common characters carry less, so this is always less than or equal to the naive count. The difference is the redundancy a compressor would remove — and it is a choice of model, the same coarse-graining choice that makes entropy observer-relative. Change what you assume about the source, and the information — and its energy cost — changes with it.

Scope. This is the Landauer floor for irreversible erasure — one bit at a time, at thermal equilibrium with a bath at temperature T. It is not the energy of running a computation (reversible operations can in principle cost nothing), nor the entropy of a gas or any ensemble quantity. PROTIUM is strong here precisely because the bound reduces to single-bit, single-temperature bookkeeping; it is not a general thermodynamic-entropy calculator, and deliberately so. The temperature axis Hf☉ is reserved, not yet formalized in the framework — used here in its commensurate-with-energy reading (k_B = 1), which is the honest case for it.