Run 030 found two metastable families of near-minimal cuts on Cay(G/St(4)) (≤ 224 edges from METIS, 270 from disorder). N-026 correction 3 says the quotient is a path of 16 fibres and guesses the families differ in how they thread through a fibre. Do they?

Sealed before the positions were computed (2026-09-27, codifier). Two clauses, two seals; each must draw. Between 0.35 and 0.60 on clause 1, or Δpos = 2, is unanticipated.

What it tests. "Likely explains" in run 030 was: the metastable cut families come from the fibre's internal structure. This makes that a measurement. If both families are crosswise and co-located, the metastability lives inside the fibres they cut — then the barrier between families is the cost of re-threading a fibre, of order (2 − c)·F·2J ≈ 230 at level 4, which is why T = 1 never crosses it. If a family is lengthwise, or they sit at different path positions, the explanation is something else and the guess is withdrawn.

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Measured, same evening

The Schreier path on level 4, in leaf order: 14, 6, 2, 10, 8, 0, 4, 12, 13, 5, 1, 9, 11, 3, 7, 15, with generators along it a, bc, a, bd, a, bc, a, cd, a, bc, a, bd, a, bc, a — a-edges at the even steps, b/c/d double edges at the odd ones. Every vertex mapped to the position of g⁻¹(leaf 0); the covering checked (every Cayley edge lands on a path edge or a loop).

Fraction of each fibre on side +1, best family: 0.47–0.53 across all sixteen; disorder family: 0.45–0.55. Agreement between the two families inside each fibre, sign-aligned: 0.55–0.67 (0.5 is random). L_min = 0.128, Δpos = 10.

The structure this reveals. A cut with no b/c/d edge is constant on the right cosets g⟨b,c,d⟩ — four elements each, a K₄ under b, c, d — so it is a partition of the 1024 cosets, and its cost is the number of a-edges between cosets on opposite sides. The Cayley graph is the K₄-blow-up of a 4-regular coset graph on N/4 vertices whose edges are the a-edges; the minimum bisection is that graph's, lifted. Bisecting the coset graph directly: level 3 gives 16 (= run 030's), level 4 gives 176 — 0.69 of a fibre, below the 198 and 224 found on the full graph. The coset graph is not path-like in the way the full graph is: normalised gap 0.076 (level 4) against the full graph's 0.0128, bisection 8.6% of its edges against 15.2% for a random 4-regular graph of the same order (25% against 19% at level 3). Frustrated under the balance constraint, half an expander at this size.

So what explains the glassy walls is the landscape of minimum bisection on that graph. Sparse graph bipartitioning is a spin glass (Fu–Anderson 1986): near-optimal bisections are exponentially many, mutually uncorrelated, and separated by barriers that grow with the graph. Every number in run 030 is that: six partitions with cuts 176–270 that agree on ~60% of each fibre; two families unreachable from each other at T = 1; a wall whose energy drifts for 10⁵ sweeps into seed-dependent plateaus. A torus has no coset graph — its minimum cut is a hyperplane whose only degeneracy is translation, with no barrier between translates — and so no glass.

What this changes upstream. N-026 correction 3's guess (threading through a fibre) is withdrawn; the path picture stands for the spectrum and fails for the cut. Run 031's mechanism paragraph is wrong for the same reason; its band stands as sealed, and the level-4 minimum it validates against is now ≤ 176. The scored number of run 030 is unaffected (the best-family free energy was measured from a 224-cut start and reached 198–216 cuts; a 176 start would lower ΔF slightly, toward the prediction band — recorded, not re-scored).

Next, if the glass is to be measured rather than named. The barrier between two bisections — the minimum over balanced interpolating paths of the maximum cut — for pairs of the six partitions above; Fu–Anderson says it grows with N. And the overlap distribution over ~50 independent local minima of the coset graph, which for a glass is broad with a peak near 0.5.