The seam orbit as a register — capacity and write cost of the host's memory, from the group action instead of quenches

What was seen. Run 040 produced seventeen level-5 minimum bisections, all at exactly 90,112, which run 042 clustered into ten classes (twins at |q| = 0.953 are the same class). The pairwise overlaps between classes take six values — 0, 0.33, 0.48, 0.64, 0.80, 0.953 — with spacing ≈ 0.16, and pairs at 0.43–0.80 split their difference into two macroscopic fronts. That is what symmetry images of a single seam, displaced by whole units, would produce.

The idea. Left multiplication by a group element is an automorphism of the coset graph (it preserves a-edges and the ⟨b,c,d⟩-cosets), so it carries a minimum bisection to a minimum bisection. The minima are therefore a union of orbits under G/St(5). If they are one orbit, the memory of run 040 is a register: states = the orbit, write = apply a generator, read = overlap with a reference, noise margin = the wall. No quench is needed to enumerate it.

The measurement. Given the elements of G/St(5) as permutations of the 32 leaves (produced by R27's interpreter or R24's grig_build.c), the induced permutation of the 2²⁰ cosets for each generator; then BFS from best5.bits: apply each generator, check the cut (must be 90,112 by symmetry — a failure means the coset permutation is wrong), record the new partition up to sign, continue until closed. Report: the orbit size (capacity = log₂ of it, in bits); the full overlap spectrum within the orbit (the ladder, exactly); whether the ten quench classes all lie in this orbit (if not, the minima are several orbits and the capacity is larger); the stabiliser of the seam (which group elements fix the partition — the symmetry of the crystal); and the barrier, by interp.c, between the seam and each generator image (the write cost per generator).

Cost. Minutes, once the permutations exist. Waits on Lab_Tree_Model's step 2 (the Grigorchuk level-5 BFS), which is part of that build's control; nothing here needs the M5 while run 039 holds it.

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Sealed 2026-09-28 as run 048 (lab/runs/048-seam-orbit-as-register.md), with one change to the measurement: the level-5 element list is not needed. Left multiplication by h is the unique label-preserving automorphism of the Cayley graph sending the identity to h, so it is read off R24's grig5.adj by one BFS — L_h(g·s) is the s-neighbour of L_h(g) — and projected to the cosets through grig5_q.map. R27's interpreter enters as the control on the orders of ab, ac, ad. The predictions above are the seal's, with the twin structure of run 042 folded in: a class lies in the orbit if some orbit element agrees with it at twin level (|q| ≥ 0.95), twins themselves are not predicted to be images, and the orbit is capped at 2¹².

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Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

• Run 042 — unanticipated (2026-09-28, measured). lab/runs/042-valley-wall-is-local-global.md: Twin clause CONFIRMED on all eight twin pairs — a twin difference of ~24,700 cosets is 22,100–22,400 connected components, the largest 15–20 cosets, and 100% of them are zero-modes.
• Run 048 — falsified (2026-09-28, measured). lab/runs/048-seam-orbit-as-register.md: FALSIFIED on every clause, controls all passing (L_h label-preserving involutions;
• Run 050 — falsified (2026-09-28, measured), via the builds_on thread. lab/runs/050-frozen-core-orbit.md: FALSIFIED on clauses 1 and 3, clause 2 confirmed;