On the intermediate-growth host Cay(G/St(4)) (n = 4096), is the domain-wall free-energy barrier controlled by the host's local, low-dimensional structure — or does the entropy of near-minimal cuts eat it? First test of the scale-dependent-dimension speculation (N-026, P-025).

Sealed before any host was sampled (2026-09-27, codifier). Validated against is_sealed() in docs/design/2026-09-26-run-figure.py: one statistic, every boundary a closed constant.

What it tests. Baspin's obstruction to 3D self-correction is a free-energy statement: the barrier that sets memory time is f̄ = max over the cheapest logical path of (E − T·log #paths), and layer codes lose because the number of paths over their linear wall grows exponentially in the wall's height. The speculation in N-026 is that a host whose effective dimension is low at small scales (2.4 at radius 5) and high at large scales (8 at radius 24) could decouple the two terms: the entropy per unit wall is set by local structure (how many ways a wall can wiggle), while the wall's minimum size is set by the large-scale isoperimetry (the separator, 416 edges by Fiedler cut against 148 for the 2D torus at the same size). On a Euclidean lattice one D fixes both; here they might come apart.

The Ising model on the host is the cleanest first proxy: it is the k = 1 code whose logical operators are cuts, so its barrier is exactly the membrane-type barrier the speculation is about, and its free-energy profile in M is computable to high precision by window sampling. If the host's cut entropy already eats the classical barrier, the quantum question is moot; if it does not, the quantum question reduces to whether a code with cut-like logicals for both Pauli types can sit on the host (the homological route gives it for one type only — N-026).

Why these boundaries. Onsager's exact 2D interface tension at T = 1 is σ = 2 + log tanh 1 = 1.728 per unit length against energy 2, so the 2D value of ρ is 0.136. A 3D flat interface at T = 1 has per-site entropy of order e^{−8}, ρ ≈ 0. The prediction band [0.05, 0.30] is "2D-like within a factor of two"; the falsifier at 0.50 is "half the wall is gone", which is what a degenerate family of cuts — the mechanism self-similarity would supply — looks like at this temperature.

Validation, controls, sizing. The 2D 64×64 torus must return ρ within 0.03 of 0.136 and the 3D 16³ torus ρ < 0.03, or the sampler is wrong and nothing downstream counts. Controls: a random 4-regular graph (expander) at 4096; the n = 128 quotient G/St(3) and matching small tori for a two-point scaling of ΔF and ΔE. Two independent replicas per host must agree on ΔF within 5% or the run is unsettled (window sampling can trap a droplet-to-strip transition). T_c per host from a magnetisation sweep is reported as context, not scored. C, single core, minutes per host.

Risks stated up front. (1) The Fiedler separator 416 is an upper bound on the bisection; the sampler finds the true near-minimal cuts and their energy may be lower. (2) Higher large-scale dimension might raise entropy rather than lower it — more routings for a wall to take — which would argue against the speculation even inside the prediction band. (3) The Ising proxy says nothing about string-like excitations; on this host the homological code's strings have length 2^n = log n, and no membrane-only code is known to exist on it. A confirmed prediction here is necessary, not sufficient.

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Measured, same day

The instrument, twice corrected before any host was read

The reference value in the sealed text is wrong, and the validation criterion with it. "Onsager gives 0.136" is (2 − σ)/2: the wall's ground-state energy 2 per unit length, less its free energy σ = 1.728, over the ground-state energy. But ρ is defined on thermal averages, and a 2D interface at T = 1 carries thermal energy e = σ − T·dσ/dT = 2.552 per unit length, not 2. The correct Onsager value is 1 − σ/e = 0.323. The 2D control then cannot land within 0.03 of 0.136 — and it did not (0.339 ± 0.003). The seal stands as written; this paragraph is the record that the band was placed on a miscalculation, and that the control — not the object — is what showed it. Rule for OPS: compute the control's value and check it sits inside the band before sealing; a band the control fails is a statement about the instrument.

The method named at seal failed validation. Window sampling in M, walking down from the ordered state, followed the droplet branch metastably through the droplet→strip transition (barrier ≈ 20 at T = 1 on the 64×64 torus) and reported the droplet's F(0): ΔF = 286 against Onsager's 221. Walking up from a bisection crossed the transition inside a window whose histogram then mixed a metastable branch with the stable one, and reported 140–166, irreproducibly across seeds. Both are in R23_ising_barrier/ as the record. Replaced by thermodynamic integration in β at fixed M = 0 from the exactly known β = 0 value, β·F_{M=0}(β) = −log C(n, n/2) + ∫₀^β ⟨E⟩_{M=0} dβ', with ⟨E⟩_{M=0} from Kawasaki (magnetisation-conserving) Metropolis over 101 β points, and the ordered state as an exact reference (F(n) = E(n) = −m, one microstate; min_M F differs from it by < 0.5, below resolution). Cooling from disorder and heating from a near-minimum bisection agree to 1% on the tori. Validation on the corrected criterion: 64×64 torus ΔF = 223.1 ± 0.6 vs Onsager 221.2 (+0.9%), ΔE = 337 ± 1 vs 326.6 (+3%, the two-interface torus at finite size), ρ = 0.339 ± 0.003 vs 0.323; 16³ torus finds the exact slab, ΔE = 1024, ρ = 0.004. The statistic is the same quantity as sealed; the estimator is not.

Result, T = 1

Errors are seed spreads over 4–6 runs except the Grigorchuk best family, where ± is the systematic range described next. T_c from magnetisation sweeps at n = 4096: Grigorchuk ≈ 2.05, 2D torus 2.27, random 4-regular ≈ 2.9, 3D torus 4.5 — the host orders less readily than the square lattice at the same degree, as a graph with a triangle (bcd = 1) and 8-cycles should.

Why the object's number is not settled

The Grigorchuk wall is glassy. Its energy at T = 1, started from the best cut, drifts downward for 10⁵ sweeps and plateaus at seed-dependent values (435 vs 475 in blocks of 50,000; the torus is flat at 337 from the first block). Slow cooling from disorder, and parallel tempering from disorder (101 replicas), both land in a 270-edge cut family with ΔF = 346; seeded from the 224-edge METIS cut, tempering stays in that family (ΔF = 325.8, ΔE = 462.6, ρ = 0.296) and the dynamics finds cuts down to 198. The partition function is dominated by the lowest-F family (e^{20} over the other), which is entropically narrow — a golf-course landscape. So ΔF ≤ 326 is an upper bound on the equilibrium value, ΔE lies in 435–475, and ρ = 0.29 ± 0.03 straddles the sealed edge at 0.30. What would settle it: sampling of order 10⁶ sweeps in the best family, or a better minimum bisection than 198. Neither changes the comparison below. The glassiness is itself a finding: on the self-similar host, near-minimal cuts come in metastable families 25% apart in size, and the tori and the expander show nothing of the kind.

Correction, same evening (run 032): the minimum bisection at level 4 is ≤ 176, not ≤ 198 — found by bisecting the ⟨b,c,d⟩-coset graph directly; and the cut families are lengthwise (a-edges only, every fibre split), not fibre-threading. The glassiness is the spin-glass landscape of minimum bisection on that coset graph. The scored ρ was measured from a 224-cut start reaching 198–216; a 176 start would move ΔF down slightly, toward the band. Not re-scored.

What it means for the speculation

Entropy is local and it is 2D's. TΔS per cut edge: Grigorchuk 0.7–0.75, 2D torus 0.89, expander 0.59, 3D torus 0.01. The host's walls wiggle like strings in a plane, because at the smallest scale they are strings — local dimension 2.4 means codimension-1 objects are one-dimensional there. The premise of the speculation (local structure sets the entropy term) is confirmed; the hope built on it (low local dimension makes walls rigid) had the sign backwards. Rigidity is what a 3D torus has (ρ = 0.004), and it comes from the wall being a membrane at the smallest scale — high local dimension, not low.

The gain over 2D is entirely energy. ΔF = 326 vs 223 at n = 4096: ×1.46, and the minimum cut is ×1.55–1.7 (≤ 198–216 vs 128). Between the two quotients ΔF grows ×14 for ×32 in n, exponent 0.76 — against 0.59 (2D, aspect-ratio-limited), 0.81 (3D), 0.96 (expander). At this size the host's barrier scales like a lattice of dimension ≈ 2.5 by the cut, while a genuine 3D lattice beats it 3:1 in absolute barrier and has rigid walls. Not a fourth mechanism; a 2D wall on a slightly-better-than-2D cut.

Correction to N-026. The Fiedler separator (416) overestimated the bisection by ≥ 2×; the true minimum bisection at n = 4096 is ≤ 198. By the cut (|∂| ~ n^{1−1/D}, with or without the torus's factor of two faces) the host's effective dimension is 2.2–2.8, not the 5–5.7 that the spectral gap and ball growth gave. The two are different quantities: the edge expansion h = 198/(4·2048) = 0.024 is 3.8× the Cheeger floor λ₂/2 = 0.0064, where the 64×64 torus is 13× its floor and a random 4-regular graph about 2×. Corrected the same evening: that ratio is not "expander-like" — it is what a path gives. The Cayley graph covers the level-n Schreier graph, which is a path on 2ⁿ leaves, so λ₂ is at most the path's, ~1/(2ⁿ)²: the measured gaps 0.0505, 0.0128, 0.0032 fall 4× per doubling of leaves, exactly 1/(#leaves)². The quotient is spectrally a path of length 2ⁿ = (8/5)(log₂N − 2) thickened by fibres of N/2ⁿ vertices; gap ~1/log²N and separator ~N/log N are both the path's, and h/(λ₂/2) ~ #leaves is the torus's own one-axis behaviour (13 at L = 64, 3.8 at 16 leaves). The cut is what a code or a memory sees; the gap is not.

For Baspin's question. ρ is roughly constant across the two sizes (0.32 → 0.29–0.37): the free-energy barrier grows like the energy barrier, so the k = 1 classical memory on this host is self-correcting in the thermodynamic sense with barrier ~ n^{0.76}, as the 2D Ising memory already is with n^{1/2}. The entropic obstruction that kills layer codes (entropy growing faster than energy with size) is absent here at the classical level — and so is anything that would make it interesting quantumly, since no membrane-only code exists on the host (N-026: the homological strings are log n).

Next

The one thing left in the speculation is whether the cut exponent climbs toward 1 with n, as D_eff = α ln n would have it. That needs only the minimum bisection of G/St(5) — 4.2M vertices, 8.4M edges, a METIS-scale job with no sampling — against ≤ 198 at n = 4096. If it comes out near 198 × 1024^{0.76} ≈ 39,000 the host is a ~3D lattice for memories; near 198 × 1024^{0.9} ≈ 100,000 it is climbing. Sealed as run 031 the same day, with the fibre-count reasoning that makes ~100,000 the prediction.