Run 040 produced seventeen level-5 minimum bisections at exactly 90,112 and run 042 sorted them into ten classes (twins at |q| ≈ 0.953 are one class — a balanced random half of the free cosets flipped), with the overlaps between classes on a ladder 0, 0.33, 0.48, 0.64, 0.80. P-044 reads the ladder as the symmetry orbit of one seam — left multiplication by G/St(5) is an automorphism of the coset graph Q₅ and carries minima to minima — and the memory of run 040 as a register whose states are that orbit. Is it? What is the orbit's size (the capacity), its overlap spectrum (the ladder, exactly), the stabiliser of the seam, and the barrier between the seam and its generator images (the write cost)?

slug: seam-orbit-as-register
state: run
created: 2026-09-28
sealed: 2026-09-28
statistic: R27_tree_groups (Lab_Tree_Model), from R24's files — grig5.adj (the level-5 Cayley graph as CSR, rows in generator order a, b, c, d; regenerable by grig_build), grig5_q.map (coset index of every full-graph vertex) and grig5_q.adj (Q₅); the seeds are the bisections in R24_grig5_bisection/barrier5/.bits whose cut on Q₅ recounts to 90,112, and the ten classes are read from patches5.json (twins collapsed). (i) The left-multiplication permutation L_h of the 4,194,304 vertices for h ∈ {a, b, c, d}, by BFS on the CSR alone — L_h(identity) = the h-neighbour of vertex 0, and L_h(g·s) = the s-neighbour of L_h(g) — and its projection π_h to the 2²⁰ cosets through grig5_q.map. Controls, all exact, any failure withdraws the run: every L_h is a bijection preserving every edge with its label; L_h² = identity; the orders of L_a L_b, L_a L_c, L_a L_d equal the orders of ab, ac, ad on the 32 leaves by R27's interpreter (two independent constructions of the same quotient); each π_h is well defined (the four members of a coset land in one coset) and is an automorphism of Q₅; every image of a seed under any π_h recounts to cut 90,112. (ii) The orbit of each seed under ⟨π_a, π_b, π_c, π_d⟩ by BFS on partitions taken up to complement (canonical sign: the side of coset 0), capped at 4,096 states; the orbit size |O|; the number of distinct orbits among the seeds; for each class representative R the maximum |q(R, x)| over the orbit x of the first seed (q on Q₅ as in run 042: |2·agreements/2²⁰ − 1|) — a class lies in the orbit if that maximum is ≥ 0.95 and lies in it exactly if it is 1. (iii) The overlap spectrum within the orbit: all pairwise |q| when |O| ≤ 512, else 10⁵ random pairs, reported as a histogram at resolution 0.01 and as the list of distinct values (clusters wider than ±0.02 count as distinct). (iv) The stabiliser: |Stab| = 4,194,304/|O| by orbit–stabiliser; the Schreier generators from the BFS tree (|O| × 4 words in a, b, c, d, each checked to fix the seed); the order of the group their leaf permutations generate on the 32 leaves (must equal |Stab|); and the alignment depth k — the least k ≤ 5 such that the stabiliser contains the level-k stabiliser St(k)/St(5), tested as |Stab| / |image of Stab in G/St(k)| = 4,194,304 / |G/St(k)| with |G/St(k)| = 2, 8, 128, 4096, 4,194,304 for k = 1…5; "none" if no k qualifies. (v) The write cost: the barrier by barrier5/interp.c from the first seed to each of its four generator images π_h(seed) (skipping images equal to the seed), with the |q| of each pair; then from the seed to one orbit element at each distinct |q| value of the ladder, nearest by BFS depth — at most twelve interp runs in all.
script: R27_tree_groups/seam_orbit/ (Lab_Tree_Model; sources only, no run number in the files)
artifacts: [R27_tree_groups/seam_orbit/orbit.json, R27_tree_groups/seam_orbit/RESULTS.md]
measured: 2026-09-28
outcome: falsified
builds_on: [040, 042, P-044]
concepts: [growth-of-groups, group-theory, information-theory]
programme: PG-02
source: ../lab/runs/048-seam-orbit-as-register.md

Prediction(1) One orbit — every one of the ten classes has maximum |q| ≥ 0.95 against the first seed's orbit, and the seeds fall into one orbit up to twin flips (the exact-match count may be less than ten; twins are free-spin choices, not symmetry images). (2) |O| ∈ [2⁴, 2⁸] — the stabiliser has order between 2¹⁴ and 2¹⁸ — and the seam is aligned with the tree: k* ≤ 4, the stabiliser contains everything that fixes the first four levels. (3) The orbit's overlap spectrum is exactly the ladder — the distinct values are 0, 0.33, 0.48, 0.64, 0.80 each within ±0.02, and nothing else below 0.95; 0.953 does not appear as an orbit overlap (twins are not images). (4) The generator image with the largest |q| is at |q| = 0.80 ± 0.02 and its barrier is 39,235 within 10 %; barriers are non-increasing in |q| down the ladder.

Falsifierany of — the ten classes span two or more orbits with some class at maximum |q| ≤ 0.85 against the first seed's orbit (the minima are not one crystal and "register" is the wrong word); |O| > 4,096 (the cap: reported as "> 2¹²"; the seam has no tree-aligned stabiliser); an orbit overlap value in (0.85, 0.95) or above 0.95 other than 1 (symmetry images at twin distance — the ladder and the free spins are not separate structures); the barrier to the largest-|q| generator image below 20,000 or above 60,000 (the symmetry path is not the quench path's cost — the write costs half, or more than 1.5 times, what run 040 measured). Any control failing in (i) withdraws the run. Between prediction and falsifier is unanticipated.

VerdictFALSIFIED on every clause, controls all passing (L_h label-preserving involutions; orders of ab, ac, ad = 16, 8, 4 by both constructions; every π_h an automorphism of Q₅; every image recounts to 90,112). (1) Not one orbit: the 16 seeds are 16 distinct orbits within the cap; four classes (s31, s34, s39, s43) reach only max |q| = 0.80 against the first seed's orbit. (2) |O| ≥ 4,096 (cap hit); the BFS depth profile is the Grigorchuk sphere series through radius 12, so no two elements of length ≤ 12 give the same state, and all 9,655 Schreier words are the identity — no stabiliser within reach, k* undetermined. (3) Twin-level overlaps appear inside the orbit (0.952–0.955, 446 pairs) and the spectrum has twelve clusters, two wider than ±0.02. (4) π_d(seed) is the seed's twin — |q| = 0.954, barrier 0 — and the generator images at 0.80 (π_b, π_c) cost 5,720, a seventh of run 040's 39,235. What holds: the barrier falls monotonically with |q| down the ladder (38,038 at q ≈ 0 to 5,720 at 0.80 to 0 at 0.954).

Sealed 2026-09-28 before any left-multiplication permutation or orbit was computed (codifier). Known at sealing: the seventeen 90,112 minima and their pairwise overlaps (run 040, run 042's patches5.json); that twins differ by ~22,000 zero-mode components inside the free set; that left multiplication is an automorphism of the coset graph; the orders of ab, ac, ad in G (16, 8, 4 — from memory, the control in (i) checks the level-5 values against the interpreter); nothing about which cosets any generator image moves. No pre-seal glimpse.

Why the BFS needs no element list. P-044 waited on the level-5 element permutations. It does not need them: in a Cayley graph with right-multiplication edges, left multiplication by h is the unique label-preserving automorphism sending the identity to h, and it is read off the adjacency by one BFS — L_h(g·s) is the s-neighbour of L_h(g). The interpreter enters only as the control on the orders, an independent construction of the same quotient.

What the outcomes mean. Confirmed: run 040's memory is a register — capacity log₂|O| bits, states the orbit, write = a generator, the ladder is the group's word metric on the orbit, and the wall of run 040 is the write cost. Falsified on the orbit count: the minima are several crystals and the quench found a union of orbits; the capacity is larger and the register reading is incomplete rather than wrong. Falsified on the cap: the seam breaks the tree's symmetry almost completely, and the ladder is not a symmetry structure. Falsified on the spectrum: the orbit carries twin-distance images, so the free spins are partly symmetry, and run 042's local/global split is not the orbit/twin split. Unanticipated on the barrier alone: the symmetry path costs a different amount from the quench path, which is a fact about the interpolator, not the register.

Scored 2026-09-28 (codifier; numbers from R27_tree_groups/seam_orbit/RESULTS.md, Lab_Tree_Model). Falsified as written, on all four clauses; nothing withdrawn. What the falsification says:

The generator d is the free-spin symmetry. π_d carries the seed to its twin at zero barrier. Run 042 read the twin difference as ~22,000 independent zero-mode flips — local flavour. At least one twin is a group image: d acts trivially on half the tree and flips exactly the free cosets. So the local/global split of run 042 is not symmetry versus non-symmetry; part of the local layer is symmetry too. The seal's clause (3) bet the opposite.
The seam breaks the tree's symmetry almost completely. No stabiliser element among 9,655 Schreier words; every element of length ≤ 12 gives a distinct minimum. The degenerate minima number at least 4,096 per seed, and plausibly |G/St(5)|/|Stab| with |Stab| small — the "register" has at least 12 bits of capacity, but it is not the small tree-aligned crystal the probe pictured.
The write cost is the distance. Barrier against |q| from the seed: 38,038 (0.0002), 35,170 (0.047), 30,090 (0.13), 26,458 (0.26), 23,116 (0.34), 20,268 (0.43), 17,840 (0.48), 11,856 (0.64), 5,720 (0.80), 0 (0.954) — monotone and close to linear in 1 − |q|. Part of that is the greedy interpolator, whose path cost tracks Hamming distance on any graph; the symmetry adds that a single generator step (b or c) moves the seam by 1 − 0.80 of the graph at a seventh of the cost of a quench-to-quench barrier.
Six of ten classes sit at twin level from the first seed's orbit (max |q| ≈ 0.953), so modulo free spins they are images of one seam; the four at 0.80 are undetermined under the cap.

The question it leaves, and the next seal. Quotient out the free spins first: canonicalise each minimum by its frozen core (the cosets that are not free) and take the orbit of the core. The seal's picture — one crystal, a small orbit, a ladder — may be true of the core and false of the full partition, which is what this run measured. That is P-044's successor, not a rescore.

generated 2026-10-25the lab index