An essay on relational invariants — the things between us that never change.
Between any two people, certain quantities are frozen at the moment they first become defined, and no amount of time, effort, or reinterpretation can alter them. If someone is twenty years older than you, they are twenty years older than you forever. If someone is your sister, she is your sister forever. These are not properties of either person alone; they are properties of the pair — and unlike almost everything else about a pair of people, they do not move.
Call these fixed gaps: relations between two entities that are invariant over time.
A twenty-year age gap is an unbridgeable chasm between a five-year-old and a twenty-five-year-old. Between a fifty-year-old and a seventy-year-old, it is nearly nothing. The absolute difference never changed — but the ratio of the two ages shrinks toward one as both grow.
This single arithmetic fact explains a great deal of family life. Parents and children can eventually become something like friends not because the gap closed, but because the proportion did. What feels like "growing closer" to an older relative is often just this ratio effect playing out on schedule.
The general principle: the invariant is the substrate; the meaning is a non-invariant function of it. What changes over a lifetime is never the gap itself, only the weight and reading we give it.
Friendships, marriages, and colleagueships are chosen — and therefore revocable, and therefore in need of maintenance. Blood relations and age gaps are given, not chosen. That is why they can bear weight that chosen relationships cannot: you cannot be fired as a brother. It is also why they can trap: there is no exit clause, only distance.
And because the structural facts between two people are frozen, everything that does change — trust, resentment, intimacy — must live entirely in interpretation and conduct. Family conflicts are rarely about the fixed facts; they are fights over what the fixed facts should be taken to mean.
The age structure of a family is rigid, so people do not change their relations — they change their location in them. You do not stop being a child; you additionally become a parent, then a grandparent. Much of what we call maturing is discovering, from the other side, what a relation you have been inside your whole life actually felt like to the other person.
The pattern is not human at all. Any two things that came into existence carry a fixed formation gap forever — two stars, two rocks, two documents, two blocks in a chain. Geology and archaeology are built on this: strata do not give absolute dates directly; they give order and intervals, and the intervals are incorruptible. Causal order is stronger still: if A caused B, no reinterpretation ever reverses it.
This is the deepest version of the observation. Potential energy has no absolute value — only energy differences are physical. Voltage is a potential difference; the zero point is a convention. Position has no absolute meaning; only distances and relative velocities are measurable.
So it is not merely that gaps between objects stay fixed while the objects change. For many physical quantities, the gap is the only measurable reality, and the "absolute value" was a fiction penciled in for convenience. (The theory document makes this precise: every gap structure admits an absolute-value assignment, but only up to an arbitrary constant — the gauge freedom is exactly the fictional part.)
Einstein showed that some gaps we assumed were fixed are not: simultaneity, elapsed time, and length all depend on the observer. But physics responded by finding the truly invariant gaps underneath — the spacetime interval, proper time, causal order. When one inventory of fixed gaps dissolved, the framework was rebuilt around a deeper inventory. The instinct that reality is anchored in invariant relations survived; only the list of which relations qualify was revised.
The lesson generalizes: which gaps are fixed depends on which transformations you admit. Enlarge the group of admissible transformations and fewer gaps survive — but the survivors are more fundamental. This is Felix Klein's Erlangen program applied to pairs of things.
If the changeable stuff (states, appearances, interpretations) is the surface, and the fixed gaps are the skeleton, one is led to a position philosophers call structuralism or relationalism — Leibniz's side of his dispute with Newton. Objects may not have intrinsic, standalone natures at all; a thing may be nothing more than the bundle of its fixed relations to everything else.
The theory document turns this into a small theorem: the complete table of gaps determines the entire configuration of objects uniquely, up to a single global translation — that is, up to precisely the information (an absolute origin) that was never real in the first place.
- Between people and between objects alike, what persists are relations, not conditions.
- The more fundamental the domain, the more the relations turn out to be all there is.
- What we experience as change is the drift of non-invariant readings over an invariant substrate — ratios shrinking, positions rotating through a fixed lattice, meanings renegotiated over facts that cannot be.
The mathematics is developed in theory.md.