- slug
- frozen-core-orbit
- state
- run
- created
- 2026-09-28
- sealed
- 2026-09-28
- statistic
- R27_tree_groups/seam_orbit_core/ (Lab_Tree_Model), reusing seam_orbit/'s L_h and π_h (same controls, re-run). For a minimum S on Q₅, a coset is free if exactly two of its four neighbours lie on each side (flipping it alone costs 0); the core of S is the partial assignment core(S) = {(v, S_v) : v not free in S}, taken up to global complement (canonical sign: the side of the lowest-index non-free coset). Because π_h is an automorphism, core(π_h S) = π_h(core(S)); the orbit BFS runs on cores exactly as run 048's did on partitions, same generators, cap 4,096. Reported: (i) for each of the 16 seeds, the number of free cosets and |core|; (ii) whether π_d(core(S)) = core(S) exactly for each seed, and for π_a, π_b, π_c the fraction of core cosets changed; (iii) the core-orbit size |O_c| of the first seed (min_rand_s130) and of every other seed, the BFS depth profile, and the number of distinct core orbits among the 16 seeds; (iv) the stabiliser of the core within the cap — Schreier words that fix it, the order of the group they generate on the 32 leaves, and its images m_k in G/St(k), k = 1…5; (v) for each of run 042's ten classes, the maximum core overlap against the first seed's core orbit, where core overlap between two cores = |2·agree/n − 1| over the n cosets non-free in both. Unsealed: the overlap spectrum of the core orbit (10⁵ pairs if |O_c| > 512).
- script
- R27_tree_groups/seam_orbit_core/ (Lab_Tree_Model)
- artifacts
- [R27_tree_groups/seam_orbit_core/core.json, R27_tree_groups/seam_orbit_core/RESULTS.md]
- builds_on
- [048]
- concepts
- [growth-of-groups, group-theory, information-theory]
- programme
- PG-02
- measured
- 2026-09-28
- outcome
- falsified
- next
- Count the distinct minima without the cap: find the stabiliser of one core directly (level-by-level search in G/St(5) for elements fixing it) and get |O| by orbit–stabiliser — is the register the whole group, 4,194,304 states per seed? | builds_on: 050
- source
../lab/runs/050-frozen-core-orbit.md
Prediction(1) d is exactly the free-spin symmetry: π_d(core(S)) = core(S) for all 16 seeds, while π_a, π_b, π_c each change ≥ 5 % of the core. (2) The quotient does not rescue the crystal: |O_c| still hits the 4,096 cap for the first seed, and the 16 seeds give ≥ 8 distinct core orbits — the seam's symmetry breaking is in the core, not in the free layer. (3) The core stabiliser within the cap is non-trivial (it contains at least d), with leaf-group order ≥ 2 and m₁ = 1 (it fixes the first level — d acts trivially on the first letter).
Falsifierany of — π_d changes more than 1 % of the core on any seed (the twin map is not a free-spin symmetry and run 048's zero barrier is a coincidence of the interpolator); |O_c| ≤ 256 for the first seed (the crystal picture holds on the core — P-044 was right about the core and wrong only about the free layer); the core stabiliser within the cap is trivial (then d does not fix the core and clause 1 is already false). Controls as in run 048 (all must pass, or withdrawn). Between prediction and falsifier is unanticipated.
VerdictFALSIFIED on clauses 1 and 3, clause 2 confirmed; controls and equivariance pass (64/64). d is not exactly the free-spin symmetry: π_d changes 0.31–0.36 % of the core on 14 seeds and 18 % on the twin pair s24/s17 (falsifier: > 1 % on any seed). The core stabiliser within the cap is trivial — all 9,655 Schreier words are the identity, m_k = 1 at every level (falsifier: trivial stabiliser). Clause 2 holds: every seed's core orbit hits the 4,096 cap with the Grigorchuk sphere series, and the 16 seeds give 16 distinct core orbits — the symmetry breaking is in the core. Six of ten classes sit at core overlap 0.9987 against the first seed's core orbit; the other four at 0.849.
Sealed 2026-09-28 after run 048's numbers and before any core was computed (codifier). Known at sealing: everything in run 048's RESULTS — the orbits, the Grigorchuk sphere series in the depth profile, the trivial Schreier words, π_d at |q| = 0.954 with barrier 0, π_b and π_c at 0.80 with barrier ~5,720, π_a at |q| ≈ 0; run 042's free-coset census (~45,500 per minimum, 4.34 %, all 2+2); that d = (1, b) acts trivially on the first letter. Not known: which cosets π_d moves.
What the outcomes mean. Confirmed: the minima are one seam per orbit of a large group action, decorated by a free layer that d permutes; the register has ≥ 12 bits and no tree-aligned crystal, and the free spins are the only symmetric part. Falsified on the orbit size: the core is a small crystal and 048's large orbits came from the free layer — P-044's register lives on the core. Falsified on d: the twin relation is not a symmetry, and 048's barrier-0 image needs another reading.
Scored 2026-09-28 (codifier; R27_tree_groups/seam_orbit_core/RESULTS.md, Lab_Tree_Model). Falsified as written on clauses 1 and 3; clause 2 confirmed.
- d is almost, not exactly, the free-spin symmetry. On fourteen seeds π_d moves 0.3–0.4 % of the core (~3,500 of ~1,003,000 non-free cosets) — a thin frozen layer next to the free spins, which is why run 048's barrier to π_d(seed) was zero: the moved frozen cosets are free in the image. On the twin pair s24/s17, π_d moves 18 %, exactly as much as π_b does there — on those two seeds d acts like a generator, not a decoration. The seal's "d fixes the core exactly" was too clean.
- The quotient does not rescue the crystal. Every core orbit reaches the cap along the Grigorchuk sphere series, no Schreier word is non-trivial, and the 16 seeds are 16 core orbits. The symmetry breaking is in the frozen core, not in the free layer; P-044's small tree-aligned crystal is gone at both levels.
- Six classes are one seam modulo free spins. s24, s20, s27, s52, s59 reach core overlap 0.9987 with an element of s130's core orbit; s31, s34, s39, s43 stop at 0.849. The ladder of run 042 survives in the core spectrum (clusters at 0.52, 0.69, 0.85, 0.999).
generated 2026-10-25the lab index