run 042Settled
../lab/runs/042-valley-wall-is-local-global.mdPredictionTwins (every pair with |q| ≥ 0.95) — the largest component of D has at most 1,024 cosets, D has at least 50 components, and at least 80% of the components are zero-modes: a twin is a set of local, independent, cut-neutral rearrangements, each far below the graph's scale. Valley pairs (every pair among the nine with |q| ≤ 0.85) — the largest component of D holds at least 50% of D: valley-to-valley is one macroscopic front. If both hold, the energetic split of run 040 is the geometric split of run 036 seen in the landscape — valleys are the local flavor, walls are the global one.
FalsifierTwins — any twin pair whose largest component has at least 16,384 cosets (two thirds of the difference in one piece: the twin is a shifted seam, global in extent, and "valley" is the wrong word), or whose zero-modes are at most 20% of components (the pieces are not independent). Valley pairs — any pair whose largest component holds at most 10% of D (valley-to-valley differences are also local patches, and the wall is not a geometric front). Between prediction and falsifier is unanticipated.
VerdictTwin clause CONFIRMED on all eight twin pairs — a twin difference of ~24,700 cosets is 22,100–22,400 connected components, the largest 15–20 cosets, and 100% of them are zero-modes. Each minimum carries ~45,500 free cosets (4.34% of all; exactly two neighbours on each side of the cut) and a twin is a balanced random half of them flipped (92% of every twin difference lies inside the partner's free set; the remainder are 2–20-coset clusters freed by a neighbour's flip). Valley clause UNANTICIPATED: 24 of 36 pairs have one component holding 98–100% of D, but 12 pairs — those at |q| ≈ 0.43–0.49, 0.64 and 0.80 — split D into two macroscopic fronts of 43–49% each, one per sign class, below the sealed 50% and far above the 10% falsifier. The identification the run asked for holds in substance: the flat valley is the local flavor in energy (independent, cut-neutral, sub-crossover patches), the walls are macroscopic fronts; what the seal did not anticipate is that a front comes in two halves. Recorded and not scored: the overlaps among the nine distinct minima are quantized — 0, 0.33, 0.48, 0.64, 0.80, 0.953 in steps of ≈ 0.16 — as symmetry images of one seam displaced by whole units would be.
Sealed before any twin's difference set was examined (2026-09-28, codifier). What was known at sealing, all from run 040's report: the twin overlap 0.953, the 2.35% figure, the barrier of zero along a greedy twin-to-twin path, and that twins appear in every batch of quenches. The component structure of a difference set had not been computed by anyone.
Why the prediction goes local. Zero barrier along a greedy path means every balanced pair swap on the way is cut-neutral. A shifted seam — the alternative — would also be cut-neutral step by step (translating a strip on the torus cost 4 in run 034), so the barrier alone does not decide. What decides it is the square structure of N-034: the graph is a union of (ad)⁴ squares, and a square straddling the cut can be re-seated on either side at equal cost — a local zero-mode of size 2 to 8. Twins appearing in every batch is what a large population of independent local zero-modes produces (each quench freezes a random subset of them) and what a single global shift does not. The falsifier is the seam.
What each outcome would mean. Confirmed: the flat valley is the small-set regime of run 036 made visible in energy — the patches live below the crossover scale and each is a degenerate choice the cut does not feel — and the walls are fronts that cross the amenable regime, which is why they are extensive. Falsified on the twin clause: the valley is a translation degeneracy along the seam, a symmetry of the minimum rather than a local flavor, and the two splits are different things. Falsified on the valley clause: the walls are not geometric, and run 040's 43% has to be explained some other way.
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| twins (8 pairs, \ | q\ | = 0.9525–0.9532) | valley pairs (36, \ | q\ | ≤ 0.80) | |
|---|---|---|---|---|---|---|
| \ | D\ | 24,522–24,894 | 104,948–524,262 | |||
| components of D | 22,088–22,402 | 1–15,154 | ||||
| largest component | 15–20 cosets (0.06–0.08% of D) | 43–100% of D | ||||
| component sizes | ~21,300 singletons, ~400 pairs, ~410 of 3–5, ~145 of 6–20 | one or two giant fronts plus dust | ||||
| zero-modes | 100% in every pair | (every component of every pair) |
Largest-component fraction by overlap band among the nine: |q| ≈ 0 → 1.00 (16 pairs, D is half the graph, one piece); 0.29–0.34 → 0.74 (4 pairs, mixed); 0.43–0.49 → 0.82 (9 pairs; four at 0.98, five at 0.49); 0.64 → 0.47 (3 pairs, two fronts); 0.80 → 0.43 (4 pairs, two fronts of 45,000 plus 15,000 dust components).
Free spins. Every one of the seventeen minima has 45,448–45,591 cosets with exactly two neighbours on each side of its cut — 4.34% of the graph — of which only ~2,600 are adjacent to another such coset. Flipping any independent balanced subset of them is free. A twin is such a flip of about 54% of the set: 92% of each twin difference lies inside the partner's free set, the other 8% are small clusters freed once a neighbour has flipped. That is why twins appear in every batch of quenches, why their overlap is pinned at 0.953 = 1 − 2 × 0.0236, and why the barrier between them is zero: the valley is a ~45,000-dimensional degenerate manifold of local choices.
Scored as written. Twin clause: confirmed on all eight pairs, every threshold cleared by two to three orders of magnitude. Valley clause: 12 of 36 pairs below 50%, none below 10% — unanticipated. Outcome: unanticipated, with the identification the run asked for holding in substance (local zero-modes are the valley; macroscopic fronts are the walls); what the seal missed is that a front can come as two halves, one per sign class of the difference.
Recorded, not scored — the quantized overlaps. The nine representatives' pairwise overlaps take six values, 0.000, 0.33, 0.48, 0.64, 0.80 and (to twins) 0.953, with spacing ≈ 0.16. Minima that are images of a single seam under the graph's symmetry group, displaced by whole units along the Schreier direction, would give exactly a ladder like this; and the two-front pairs are the ones where the displacement leaves the old seam and the new one both visible. This is the crystalline minimum seen directly. It is the next probe in this thread's block after G_ω.
Pointers corrected 2026-10-05 (codifier; no sealed field, score or number is changed). The artifacts: line read artifacts: [R24_grig5_bisection/patches5.json]. The file is at R24_grig5_bisection/barrier5/patches5.json (on disk on the M5, untracked); the line now gives that path.
generated 2026-10-25the lab index