The Fibre of a World: Matter, space, and the physics of comparison

We develop a structural theory of worlds, observers and the information retained between them. Matter is a configuration equipped with an agreement with a visible vacuum; space is given by locality, interfaces and gluing. Relative algebraic K-theory records the distinctions and relations erased by an observation. For square-zero rational worlds, we determine the blind-sector filtration and the first layer that faithfully records every internal symmetry. Matrix-valued probes recover every nonzero odd cyclic class, while the required memory capacity is unbounded. Pinched projective lines realize matter as a seam: their dark shadows are precisely degree-zero line bundles, with branch-lattice torsors of lifts. At complex hypersurfaces, the singularity sector admits a B-type D-brane interpretation through matrix factorizations. Structured experiments produce infinitely many independent degree-zero classes absent from the point-shaped ledger. Finally, links between hidden sectors transport every rational symmetry without necessarily identifying the underlying worlds; a fifth-power residue and its graph cohomology class measure the obstruction. The resulting universe is organized by what can be assembled, observed and transported across a comparison.