Does the minimum bisection of Cay(G/St(n)) track the Schreier-fibre count |G_n|/2^n — a separator of order n/log n whose exponent climbs toward 1 — or does the exponent 0.76 measured between n = 128 and n = 4096 (run 030) hold? Decides whether the intermediate-growth host beats a 3D lattice as a classical memory beyond n ~ 10^5–10^6.

Sealed before the graph was built (2026-09-27, codifier). One statistic, closed constants. Between 40,000 and 65,000 is unanticipated. The Cheeger floor from the measured gap (λ₂/deg = 0.00321 at this quotient) is b5 ≥ λ₂·vol/4 ≈ 13,500; anything below that is an error.

Why the fibre count is the prediction, held with conviction. The Schreier graph of the Grigorchuk group on level n is a path on 2^n leaves — a-edges alternating with b/c/d double edges — and the Cayley graph of G/St(n) covers it, with |G_n|/2^n vertices over every leaf. Cutting the path in the middle costs one or two fibres of edges: |G_n|/2^n or twice that. That is an upper bound on the bisection, exact at level 3 (16) and beaten by 23% at level 4 (≤ 198 against 256), where the optimal cut is not fibre-aligned. So b_n ≈ c·|G_n|/2^n with c ∈ [0.5, 1], and since log₂|G_n| = (5/8)·2^n + 2, the separator is b_n ≈ c·5n/(8(log₂ n − 2)) — of order n/log n, with a local exponent 0.8 (level 3→4), 0.9 (4→5), climbing to 1. Run 030's 0.76 was the level 3→4 step read off two tiny quotients. This is the concrete form of N-026's "n/polylog", and it is why the number matters: the k = 1 memory's barrier is ≈ 0.7·b_n (2D wall entropy, run 030), so against the 3D torus's 2n^{2/3} the host loses at n = 4096 (198 vs 512), and wins by 2× at n = 4.2M if the prediction holds (≈ 100,000 vs 52,000). The 4D torus (2n^{3/4} ≈ 184,000) still wins at this size.

Noted under the seal, not in it (same evening). The Schreier path's a-edges are a perfect matching, so the middle edge of a 32-leaf path is a b/c/d double edge and the balanced fibre-aligned cut costs two fibres, 262,144; one fibre (131,072) is the a-edge cut one fibre off balance. Levels 3 and 4 nevertheless reached 1.00× and 0.77× of one fibre with exact balance, by cuts that thread through the fibre. The band's top, 1.07× one fibre, is therefore a bet that threading keeps working at level 5; a balanced minimum between one and two fibres would score unanticipated, and would say the threading was a small-quotient accident.

Second note under the seal (run 032, same evening). The mechanism paragraph above is wrong: the level-3 and level-4 minima are not fibre-aligned at all — they cut a-edges only, spread over every a-step of the path, every fibre split in half: bisections of the ⟨b,c,d⟩-coset graph (N/4 vertices, 4-regular). Bisecting that graph directly gives 16 at level 3 and 176 at level 4 (the validation threshold "≤ 224" stands; the best known level-4 value is now 176, c₄ = 0.69). The empirical scaling b_n ≈ c·|G_n|/2ⁿ with c ∈ [0.5, 1] still fits both levels, so the band stands as sealed — but the codifier's expectation has moved to its lower half: the a-edge fraction cut fell 25% → 8.6% between levels 3 and 4, and if it falls by the same factor again b₅ ≈ 62,000, just under the band; if it tracks 2/2ⁿ, b₅ ≈ 131,000. The Cheeger floor from the coset graph's own gap (path-like, ~0.019 at level 5) is b₅ ≥ ~20,000. Procedure unchanged, with one addition: also bisect the coset graph directly (N/4 = 1,048,576 vertices), which is where the minimum was found at level 4.

Third note under the seal — a prediction on the exact value, made 2026-09-27 18:15 MST, before the level-5 quenches finish (Claude Code's summary at this moment: METIS floor 90,144; quenches running). Bisecting the level-4 coset graph gives 176 = (11/16)·256 exactly, and 90,144 = (11/16)·131,072 + 32. The codifier predicts the level-5 minimum is 90,112 = (11/16)·|G₅|/2⁵ exactly, i.e. the a-edge fraction cut is 11/2^{n+3} for n ≥ 4 and the constant c = 11/16 is structural, not a glass. The level-4 optimum has the structure to match: it cuts 24, 20, 20, 24, 24, 20, 20, 24 a-edges at the eight a-steps of the Schreier path, and the fraction of each K-coset block on the + side runs 11/16, 11/16, 41/64, 19/32, 1/2, 13/32, 23/64, 5/16, 5/16 from one end of the path to the other — a tilted cut, neither crosswise nor lengthwise, symmetric about the middle. If the quench lands on 90,112, N-028's last standing sentence ("the error term is a glass, not a zeta function") falls on its first half: the minimum is crystalline; only the landscape above it is glassy. If it lands elsewhere, the 11/16 is a level-4 accident. Not a seal — a dated prediction recorded so that the result cannot be read back into it.

Validation. Level 3 must return 16 and level 4 must return ≤ 224 (METIS, run 030); if the procedure finds below 198 at level 4, report it — it revises run 030's "≤ 198" and the c above.

Sizing. Build the Cayley graph in C: BFS over G/St(5) with elements stored as their permutation of the 32 leaves (32 bytes), right-multiplication by a generator = index permutation, a hash table of 4.2M keys; 8.4M edges to a METIS graph file (~100 MB). gpmetis or pymetis at that size is minutes and ~2 GB. Port the action from R22_grigorchuk_cell/grig.py (a flips the first bit; b = (a,c), c = (a,d), d = (1,b) on the tail). Fibre-aligned seeds: partition by the position of g's image of leaf 0 on the Schreier path (positions 0–15 vs 16–31, and the a-edge-shifted 0–14 vs 15–31 rebalanced by moving the fewest vertices).

Measured, same day

b5 = 90,112 = 0.6875 × |G_5|/2^5 = 131,072 — inside the sealed band, 2.25× the falsifier's 40,000, 6.7× the Cheeger floor 13,464. Level 4 → 5 exponent log(b5/b4)/log 1024 = 0.900 with b4 = 176; level 3 → 4, log(176/16)/log 32 = 0.692. Unscored — outcome: is the codifier's. Everything below is in R24_grig5_bisection/ (README, results/summary.md, the best partition per level as a bit vector).

What was built

Cay(G/St(n)) for n = 3, 4, 5 in C (grig_build.c; the action ported from R22_grigorchuk_cell/grig.py, elements as permutations of the 2^n leaves, BFS from the identity in generator order a, b, c, d, a hash table of 4.2M keys): 128, 4,096 and 4,194,304 elements, each equal to the closed form 2^(5·2^(n−3)+2); the level-5 BFS takes 1.1 s and 295 MB. The graph is 4-regular and simple at all three levels — zero loops and zero duplicate edges dropped — with 256, 8,192 and 8,388,608 edges. The Schreier graph on the leaves is a path with the a-edges at even steps and b/c/d double edges at odd steps (position 0 = the leaf 1^n); the covering used for the fibre seeds is g ↦ g⁻¹(0), the map that intertwines right multiplication with the Schreier action ("g's image of leaf 0" in the left-Cayley-graph convention), and every fibre has exactly |G_n|/2^n vertices. Per the second note under the seal, also the ⟨b,c,d⟩-coset graph Q (coset_build.c; N/4 vertices, 4-regular, simple at all three levels): a balanced bisection of Q is a balanced a-edge-only bisection of the full graph with the same cut, and every Q result was lifted and recounted on the full graph (lift.c, analyze.c).

The procedure, stage by stage

Validation

Level 3 returns 16 by every method. Level 4 is ≤ 224 as required, and the procedure finds 180 on the full graph and 176 on the coset graph — below run 030's "≤ 198", which is revised: b4 ≤ 176, c4 = 176/256 = 0.6875, and the prediction line's "0.77× at level 4" reads 0.69×. The Kawasaki port was checked against R23_ising_barrier/ising_fe.c (kaw mode) on the level-3 graph at β = 0.3, 0.6, 1.0: mean cuts 75.46 / 75.47, 30.51 / 30.41, 16.80 / 16.90 over 2×10⁵ sweeps. Every saved partition's cut and balance were recounted from scratch (refine verifies before writing; analyze recounts a bit vector independently).

What the numbers say, without scoring

• Every minimum found, at every level, cuts a-edges only — b, c, d generate a Klein four-group whose cosets are K4's, and a K4 costs three edges to cut where an a-edge costs one. The 131,072 basin that default METIS always lands in is a-edge-only and splits every fibre 50/50; it is not the fibre-aligned cut (262,144) and not one fibre's worth of fibre-aligned edges. The 90k family is a ramp in the fibre profile (side-0 share climbing 0.22 → 0.78, or 0.69 → 0.31, along the path).
• 90,112 = 11/16 × 131,072 and 176 = 11/16 × 256. The same fraction of the fibre count at levels 4 and 5, reached from three unrelated starting cuts at level 5, with the level-4 176-cut's fibre profile (0.688 … 0.312) reappearing in one of the level-5 90,112 cuts. The fraction of a-edges cut is 25% (level 3), 8.6% (level 4), 4.3% (level 5) — it fell by 2.0× between levels 4 and 5 after 2.9× between 3 and 4. Between the two expectations in the second note (≈ 62,000 if it fell by the same factor again, ≈ 131,000 if it tracked 2/2ⁿ) the measurement sits at 90,112.
• Where the refinements act. On the full graph, neither named refinement moves a METIS cut: greedy pair swaps find nothing at any level, and the β = 8 quench is a zero-temperature walk on a plateau (it does bring the fibre-aligned seeds down, to 133,138 and 172,202, far above the minimum). On the coset graph the same two stages do act — a pair swap there moves two whole cosets, eight vertices, a move the full-graph chain cannot make in one step — and take 90,144 to 90,112. METIS itself is the reverse: strong on the full graph with random matching (17% of seeds below one fibre), weak on Q (0.6%). The route that found the minimum is therefore METIS on the full graph, refinement on Q.
• Time. The level-5 BFS is seconds, each METIS run 3.5 s and 1.0 GB, the full-graph quenches 3000–3300 s each (11 runs in parallel, every one past 10⁴ sweeps), the coset-graph quenches 3 min each.