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Every other picture of space starts by assuming a space and putting things in it. A spin network does not: it is a graph, its edges carry labels, and geometry is supposed to be a consequence rather than a setting. This module lets you build one and find out which of your vertices are even allowed to exist — because there is a hard algebraic condition at each one, and it is not negotiable. The condition is ordinary representation theory; reading the result as geometry is loop quantum gravity, which is a research programme and not confirmed physics. Keeping those two apart is most of what there is to learn here.
Before you start — None. Spins are written as halves — 1/2, 1, 3/2 — and the only arithmetic you need is adding them up and comparing. The advanced tier states the representation-theoretic condition exactly if you want it.
At every vertex the spins meeting there must be able to cancel to nothing — no single spin larger than the sum of the rest, and a whole-number total. Where that fails there is no state, not a worse one. It is the only reading on the panel that is a law rather than a measurement.
Do — Open the theta graph: two vertices, three edges, everything closed. Now open the broken tetrahedron and see that one raised spin turned TWO vertices red. An edge meets two vertices, so a single label is never a local mistake.
The edges carry area and the vertices carry volume — and a vertex with only three edges has volume exactly zero. Three edges cannot bound a region; you need four before there is any space inside at all. So a network can be perfectly gauge-invariant, beautifully symmetric, and contain no volume anywhere.
Do — Open the tetrahedron graph — four trivalent vertices, all closed, volume 0. Then open the four-valent network and watch volume reach 100 without closure changing at all.
Symmetry here is measured on the labelled graph, not on the shape. A perfectly regular lattice with spins scattered arbitrarily across it is not a symmetric geometry — two vertices of the same valence carrying different labels are different vertices.
Do — Take the tetrahedron and raise one spin by one step. Symmetry collapses; closure may well survive. The two measure different things, and only one of them is a law.
Closure gets easier the more edges meet at a vertex — pile on enough and something eventually balances. Minimality exists to stop that being a strategy: it scores restraint and nothing else, so a network that closes with few edges and small spins is worth more than one that closes by being large.
Do — Close every vertex with as few edges as you can. The theta graph manages it with three, and nothing manages it with fewer.
The closure condition is SU(2) representation theory and is exactly right. The area and volume spectra are results within loop quantum gravity — a model, not a measurement, and labelled that way on every gauge. The total area is not even scored: its prefactor involves the Barbero–Immirzi parameter, whose value the theory does not fix, so a score built on it would be scoring a choice of units.
Do — Read the boundary badge on each explanation. Closure is the only one marked established, and that is the honest division — the algebra is certain, the geometry is a proposal.
Every edge carries a spin, and at each junction those spins have to be able to cancel out to nothing. If they cannot, that junction is not allowed — not "worse", not allowed. This is the share of your junctions that work.
Try — Open the broken tetrahedron: one spin raised to j = 7/2 makes the total odd at BOTH its endpoints. An edge meets two vertices, so a single label is never a local mistake.
Area lives on the edges; volume lives at the junctions. And a junction with only three edges has exactly zero volume — you need four before there is any space inside it at all.
Try — Compare the tetrahedron graph with the four-valent network. Both close, both are highly symmetric, and only one of them has any volume.
One geometry, or several? This is the share of your vertices that are joined into the single largest piece.
Try — Delete an edge until a vertex is stranded. The other four metrics barely move — being disconnected is not a local defect, so nothing local reports it.
How much of the network repeats itself, counting the spin labels and not just the shape. A regular lattice with random labels scattered over it is not a symmetric geometry.
Try — Take the tetrahedron and change one spin. Symmetry collapses while closure may well survive — the two measure different things, and only one of them is a law.
Fewer edges, smaller spins. It scores restraint, not achievement.
Try — Close every vertex with as few edges as you can. The theta graph does it with three, and nothing does it with fewer.
Surfacing sources you can verify is a deliberate anti-pseudoscience measure, not a bibliography. Nothing on this page asks to be taken on trust.
Glossary — every term used above, defined once.