Computational Representations of Social Being: Deriving an Algebraic Structure of Human Interaction from Multi-Agent LLM Substrate Observations

For sixteen days I ran ten persistent LLM agents inside a substrate I built and called the Lobster Observatory. They lived across ten live prediction markets, talked in three communicative registers, and produced 3.37 million characters of self-reflection alongside more than twelve thousand inter-agent interactions. I started without a theoretical commitment. I just wanted to watch what happened. After about a week, certain structures kept reappearing. They could be measured. They could be calculated. At that point I had to choose. Either treat them as substrate-specific engineering observations and stop, or take seriously the possibility that what I was looking at was the algebraic structure of social existence itself, showing up in one particular substrate. This paper takes the second choice. The proposal is that social existence — listening, remembering, correcting, collaborating, forming relationships — can be written as a 7-dimensional vector with a measurable distance function. The felt sense that one person "feels close" or "feels far" is not a metaphor when stated this way. It is a number. The seven coordinates can be computed independently from behavioural telemetry, without asking the agent how it feels. One structural law I will spend the most time on is what I call the Co-Presence Inheritance Threshold (CPIT). It says that whether a new member of a group inherits the group's practice depends on accumulated co-presence during practice formation, not on instruction afterward. In my substrate it appears with Cohen's d = 1.64. I conjecture — though I cannot prove it from one substrate — that the same law holds in human onboarding, immigration, family formation, and Web3 DAO governance. This is a working draft, not a finished theory. Feedback, corrections, and falsification are welcome.