Dark Matter as a Frozen Defect Network: A Consolidated, Self-Contained Account of K04 Crystallisation Debris in the Finite-QEC Substrate

This note collects, in one place and in plain language, a chain of small but exact results about what dark matter could be in the finite quantum-error-correction (QEC) substrate programme. The programme models space as a discrete, self-repairing lattice whose ordered ground state tiles space with eight-element register cells — geometrically oblate square bipyramids, whose face-adjacency graph is the combinatorial three-cube Q3; its dark-matter candidate is not a new particle but crystallisation debris — defects left behind when the lattice ordered too quickly to heal. We answer four questions that a graduate scientist can follow without prior exposure to the programme. (1) Taxonomy: which defects are permanent? Using a standard topological argument we show that a defect is permanently locked only if it winds the system, so the smallest indestructible object is a one-dimensional string, not a point particle; every finite (local) defect can in principle heal, and explicit enumeration confirms that it does. (2) Mobility: the locked objects are extended and frozen — consistent with the separately-established result that such debris cannot be pushed by gravity, which rules it out as a free, collisionless particle gas but is exactly what a frozen large-scale structure should do. (3) Energy scale: we compute the two defect tensions exactly — a string costs w4 + 4w6 per lattice step and a frustrated domain wall costs 2w4 + 12w6 per cut cell — and find, as a by-product, that the cell crystal is hugely degenerate, so most domain walls cost nothing. (4) Abundance: combining a Kibble–Zurek picture of defect formation with the above, the relic density takes the clean, falsifiable form 4 ρdark ∼ 7 σwall/ξ(R), where ξ(R) is the correlation length set by the cooling rate. The fraction 4/7 — the share of domain walls that actually cost energy — is exact. The remaining unknowns reduce to two pre-existing open problems: the cooling law that fixes ξ(R), and the conversion of lattice energy units to physical units. Every numerical claim is reproduced by a short self- checking program; the programs are listed in an appendix. Revision (2026-06-15). Added an equation-of-state note: a frozen p-dimensional defect network has w = −p/3, so the dominant frustrated walls are w = −2/3 (strings −1/3) — a pinned, negative-pressure relic, NOT the w = 0 clustering cold dark matter; the cold / CMB-anchoring role belongs to the sterile-neutrino relic. The CMB third peak is a parameter-free open no-go (matter–radiation equality z_eq ≈ z_rec), closeable only by an external AeST-class field.