The coset graph is a union of squares, and squares kill unique-neighbour expansion — no expander code on the intermediate-growth host at any level

The fact. Let Q_n be the ⟨b,c,d⟩-coset graph of G/St(n) (vertices: right cosets g⟨b,c,d⟩; edges: a-edges). Q_n is bipartite (the a-count mod 2 kills b, c, d). The relator (ad)⁴ = 1 is an 8-cycle in the Cayley graph alternating a- and d-edges; contracting cosets keeps its four a-edges and identifies the endpoints of each d-edge, so it becomes a 4-cycle of Q_n. Checked at levels 3 and 4: every a-edge lies on exactly one such square — 2048 edges, 512 squares at level 4 — so the edge set of Q_n is a disjoint union of squares.

The consequence. Each vertex has four edges, each square uses two at every corner, so every vertex lies on exactly two squares; the incidence between left vertices and squares is 2-regular on both sides, hence a disjoint union of cycles. Checked: every cycle has length 4 at level 3 and 8 at level 4 (64 cycles covering all 512 left vertices) — length 2ⁿ⁻¹, doubling with the level. Take S = the left vertices of one cycle. Every right neighbour of S lies on one of the cycle's squares, and both left corners of that square are in S, so its count is 2 (4 if both its squares are in the cycle). U(S) = ∅ with |S| = 2ⁿ⁻¹, and unions of cycles give U = ∅ at every multiple of 2ⁿ⁻¹. Unique-neighbour expansion of Q_n is identically zero at those sizes, at every level.

What it means. Sipser–Spielman's bit-flipping guarantee, Tanner's distance bound, every expander-code argument that consumes unique neighbours — none applies to a code on Q_n, at any N, and not by a factor: the hypothesis is false outright. This has nothing to do with amenability or intermediate growth. It is the relators. An expander whose edges decompose into squares would fail the same way. Run 036's edge expansion (0.78 of random at N/8) never saw it, because an edge counted twice into the same vertex is still two boundary edges.

So the code ceiling of the intermediate cell read in P-026 — distance ~ expansion × N — is not wrong about the expansion; it is wrong about the machinery. Whatever code the host can carry is not a Tanner code on Q_n. The square structure is the obstruction and the hint: squaring the graph (P-036) turns each square into a pair of distance-2 relations and the obstruction to a local code disappears while the host's expansion stays.

Measured alongside (run 041, withdrawn as sealed). The annealer found U = 0 at s = 8, 16, 32, 64, 128, 256 on level 4 — every multiple of 8, as the theorem says. The random (4,4)-biregular bipartite control at the same N: U = 8, 10, 19, 18, 10, 0 at those sizes. So a random degree-4 bipartite graph has weak unique-neighbour expansion at small sizes and none at a quarter of the class — the known reason expander codes want degree well above 4 — and the host has none at all.

Cross-links, collected. The same (ad)⁴ squares are where run 035's Kuramoto windings got trapped (vortex glasses on the relators, (ad)⁴ first), and they are the shortest primitive cycles in run 033's cogrowth count — the host's smallest primes, length 4, the first term of its Ihara zeta. Run 032 found the minimum bisections are lengthwise a-edge cuts; whether the cut lines run between square-cycles or through them is not recorded. One probe would tie the four: the relator squares as the host's local geometry — trap, obstruction, prime, seam. Candidate for the next probe in this thread's block; not filed tonight.

Corrected 2026-09-28 (run 043). The theorem stands; the exclusivity does not. Basilica — torsion-free, exponential growth, no Klein-four subgroup, no squares — has unique-neighbour minimum 0 at N/8 on its level-4 Cayley graph (N = 4096) at best-of-40, against 94 for a random 4-regular graph. So "it is the relators, and squares are how" is one route of at least two, and "no expander code on the host, and the reason is the squares" must be read as "no expander code on Grigorchuk's coset graph, for which the squares are the proof". Whether Basilica's route is also relator-driven (its Cayley graph has the commutation [a, a^b] = 1 as an 8-cycle) or something else is open. IMG(z² + i), which has the relator (ac)⁴ = 1 but no contraction, sits at 0.38 of its control — reduced, not zero — which is what the corrected reading in run 043's seal predicted for it. Second route named (N-036, same day). Basilica's and Aleshin's zeros are unions of cosets of ⟨g²⟩, the subgroup generated by the squares of the generators — v·g ∈ S forces v·g⁻¹ = v·g·g⁻² ∈ S when g² ≠ 1. Both routes are about g²: this note's is g² = 1 with a contraction; N-036's is g² ≠ 1 with ⟨g²⟩ proper. IMG(z² + i) has neither.

Back-links (2026-09-29)

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

• Run 036 — confirmed (2026-09-27, measured). lab/runs/036-expansion-profile-coset-graph.md: CONFIRMED — ρ_Φ = 104/134 = 0.776.