run 034Settled
../lab/runs/034-glass-barrier-coset-graph.mdPredictionΔ₄ ∈ [10, 300] — leaving the best bisection costs at least ten kT of free energy before any other family is reached — and Δ₄ − Δ₃ ≥ 6 — the barrier grows with the graph
FalsifierΔ₄ ≤ 3 — the families are connected by paths that cost nothing at T = 1, as a torus's translated strips are, and the glass identification is wrong — or Δ₄ − Δ₃ ≤ 1, no growth
VerdictWITHDRAWN — the instrument failed its own positive control, as the seal said would void it. The random 4-regular graph at N = 1024 gave Δ = 0.00 on both seeds (F falls monotonically from the reference to the bulk, by 39–45 kT), so the Franz–Parisi profile at T = 1 with this sampler does not see a glass; and on the coset graph the two seeds gave Δ₄ = 3.0 and 28.4, outside the 25% agreement the seal required. The torus control passed (Δ ≤ 2.8 at N = 1024). Not scorable. What the run did establish, in the second look it also named: the T = 0 landscape. Greedy balanced pair-swap interpolation between the best minimum and each of the eight next-best, recording the extra cut edges above the endpoints — coset graph N = 1024: 54, 58, 56, 44, 42, 48, 58, 52 (28% of the 176-edge cut, every pair); random 4-regular N = 1024: 0, 2, 10, 2, 8, 0, 2, 30 (median 2 on a 274-edge cut); 32×32 torus: 4 for translating a strip (six pairs), 60 for rotating it (two); coset graph N = 32: 2, 0, 0, 4, 0, 2, 0, 0. The coset graph's minima are separated by energy barriers of order 100 J that the random graph's are not, at this size, and the barrier grew from ~0 to ~50 edges between N = 32 and N = 1024. Overlaps among the 50 anneals: coset graph mean |q| = 0.18 (histogram in tenths 415, 339, 225, 144, 67, 17, 10, 3, 4, 1 of 1225 pairs) — uncorrelated minima, more so than the random graph's 0.29. Twelve distinct cut values 176–256. So the glass is in the energy landscape and this run's instrument was the wrong temperature to see it; the T = 0 measure that worked was not sealed and is reported, not scored.
Sealed before any anneal or profile was run (2026-09-27, codifier). Between 3 and 10 on clause 1, or a growth between 1 and 6, is unanticipated. Two seeds must agree on Δ₄ within 25% or the run is unsettled.
Validation and controls, fixed in advance. The 32×32 torus (N = 1024, 4-regular) must give Δ ≤ 3: translating a strip is free, so its profile is flat, and a profile that is not flat there is a bias of the estimator, voiding the run. A random 4-regular graph on 1024 vertices is the positive control — sparse-graph bisection is the Fu–Anderson glass — and should give Δ ≥ 10; if it does not, the statistic cannot see a glass and the run is void either way. The same at N = 32 (4×8 torus, random 4-regular on 32) for the growth clause.
Also reported, not scored. The overlap distribution P(|q|) over the 50 anneals' pairwise overlaps, per graph — expected broad with most weight near 0 on the coset graph and the random graph; on the torus it is broad too (translates of a strip overlap at every value), which is why overlap alone was not sealed. The distribution of the 50 cut sizes. And a T = 0 upper bound on the energy barrier between pairs of minima by greedy pair-swap interpolation, as a cruder second look.
Why the Franz–Parisi potential. It is the standard object for exactly this question: the free energy of configurations constrained to overlap Q with an equilibrium reference. A glass shows a rise away from the reference before other states are reached; a landscape with a connected family of minima shows none. It is also the quantity that governs the memory: the rate at which a wall in one family becomes a wall in another is e^{−Δ/T}.
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| graph | N | 50 anneals: cuts min/median/max | mean pairwise \ | q\ | FP Δ (seeds 1, 2) | T = 0 greedy barriers to the 8 next-best (extra cut edges) | |
|---|---|---|---|---|---|---|---|
| 4×8 torus | 32 | 8 / 8 / 8 | 0.50 | 6.5, 7.2 | 4 4 0 4 0 4 0 0 | ||
| random 4-regular | 32 | 12 / 12 / 14 | 0.89 | 1.8, 2.2 | 0 0 0 0 0 0 0 0 | ||
| coset graph, level 3 | 32 | 16 / 16 / 16 | 0.27 | 0.0, 0.0 | 2 0 0 4 0 2 0 0 | ||
| 32×32 torus | 1024 | 64 / 68 / 128 | 0.25 | 0.0, 2.8 | 4 60 4 4 4 4 4 60 | ||
| random 4-regular | 1024 | 274 / 284 / 290 | 0.29 | 0.0, 0.0 | 0 2 10 2 8 0 2 30 | ||
| coset graph, level 4 | 1024 | 176 / 184 / 256 | 0.18 | 3.0, 28.4 | 54 58 56 44 42 48 58 52 |
Two things the failed instrument taught. (1) At T = 1 the constrained free energy in the overlap is dominated by the entropy of the bulk: on every graph F(Q ≤ 0) sits far below the single reference configuration (−40 kT on the random graph), so the profile has no barrier to show even where the energy landscape is rugged; the Franz–Parisi diagnostic needs a temperature below the glass's own, which for a T = 0 bisection glass means T → 0, where this sampler cannot go. (2) The 4×8 torus gave Δ ≈ 7 where the seal expected ≤ 3: translating a strip under pair-swap dynamics is not free — it costs a constant ~4 edges (the notch and the bump), 8 J at T = 0 — and only at N = 1024 does entropy at T = 1 wash that to ≤ 3. The validation expectation was set on the wrong dynamics; it happened to hold at the scored size.
What stands. The T = 0 interpolation barriers are the measurement the question wanted, and they are unambiguous on the same footing for all six graphs: the coset graph's minima are walled off from one another by ~50 extra cut edges at N = 1024 (~100 J), against a median of 2 on the random regular graph and the torus's constant 4. Between N = 32 and N = 1024 the coset graph's barrier went from ~0 to ~50 while the random graph's stayed near 0. That is the growth the sealed clause 2 asked for, seen in the energy rather than the free energy. It also revises the identification: at this size the coset graph is more rugged than the random regular graph, not "half an expander" in its landscape — the random graph's near-optimal bisections at N = 1024 lie in a connected valley (a known finite-size feature of sparse bisection), the coset graph's do not.
To seal next, if it is to be scored. The T = 0 greedy barrier itself — a named procedure, hence a sealable statistic — at levels 3, 4, 5, with the random regular graph and the torus as controls: prediction, the coset graph's barrier grows with N and the random graph's does not at these sizes.
generated 2026-10-25the lab index