PG-02 — The cell's finite-level behaviour is the automaton plus a seam, and the recursion that looks exact drops the seam

Rewritten 2026-10-24, after run 101 was scored (2026-10-05). Earlier versions are in git. The one saved query on this grain, lab/db/queries/cell/levels.sql, carries the per-level fractions but none of run 101's five counts, and its free_coset_share column is the share of all cosets that L1's restatement of 2026-10-03 set aside. So this section cites the run files and the status entries.

Survived. - The exact part at level 5, now predicted and not only read. Run 101 is the programme's first seal on objects not yet drawn, and both clauses were confirmed. On all 126 minima it drew, the five counts match the sixteen stored minima: (a) 40,960 independently flippable cosets; (b) 576 fibres of the level map hold none of them, 256 hold 64 and 192 hold 128; (c) the 448 level-4 cosets under them lie in one bipartite class; (d) the non-flipping cosets lie over 96 level-4 cosets, 32, 32 and 32; (e) 77,824 cut edges. The count holds on 142 of 142 files, 51 of them from far starts. 40,960 is 5/11 of the cut 90,112. - What the confirm does not reach. All 126 came down one route. 84 of them carry an exact part already on disk. There are 32 distinct exact parts among the 142 files, 25 up to the generator a. Every T lies in one orbit of size 128 under the 16,384-element group. The run says nothing of other routes, other levels, or why. - Level 4 and level 3. At level 4 the minima are 512 families of 70, each a frame and eight free cosets that do not see one another, plus one rigid orbit of 512 with no free coset. 20,000 random starts found no cut below 176, no orbit beyond the fifteen, and no count but 8 and 0. A level-4 minimum's cut is 160 single edges and 8 two-edge paths. Level 3 has no flippable coset on any of its 2,460 minima. - A trend in the exact part, and none in f. The independently flippable cosets are none of the cosets at level 3, 1/128 at level 4 and 5/128 at level 5. The free share of the cut, f, is seven values from 0 to 1 at level 3, 0 or 0.0455 at level 4, and 0.5042 to 0.5059 on the 126. Three levels give no law: level 6 has 2^40 cosets and cannot be built. - From before the programme was filed. The exact fractions: the minimum bisection 90,112 = (11/16)·|G₅|/2⁵ at level 5 and 176 at level 4 (031); λ₂ = 4 − 4cos(π/16) to seven digits (038, falsified as an identity). The two-flavour split (036, 042), the anisotropic quasicrystal (046) and class separation in transport (047). Run 088 (confirmed): the bulk signature in transport is spectral degeneracy, not intermediate growth.

Died. - Clause 2 of the conjecture, the orphans as the seam. Struck on 2026-10-04. Nothing is supported on a level-4 minimum's free cosets at any eigenvalue, and L4 is closed. "At the Schreier fibres" was struck with it: a level-5 minimum is not a lift of a level-4 bisection. - f as the seam's scaling. It has no trend over three levels (above). - The flat three as a second kind. Left multiplication by a exchanges the two bipartite classes and carries each flat minimum onto the exact part of a minimum of the other class. The class of T is not a kind of minimum. With this goes the reading that the class "separates the flat kind from the other". - A second kind of level-5 minimum, on this route. None of the 126 has another count. - The count as a mark of the minimum. rm_s489, at cut 94,208, has exactly 40,960 with fibres at 0, 64 and 128 only, and fails (c), (d) and (e). rm_s226, at 99,932, has 43,040, more than any minimum read. - The codifier's leads on mechanism. Seven made, seven wrong; the leads on counts held. Among the dead: the 40,960 as whole fibres, as blocks of S9, the trap at W_n as one connected wall, and the trap as pinned by a balance. What pins the trap at exactly W_n is not known, and no reading is ordered on a guess about it. - Earlier deaths, unchanged. Identification as measurement (032, 048, 050; 034 withdrawn); the growth class from one finite instance (035, 043); exact spectral decimation on the coset tower (064: 18 of 24 new level-4 eigenvalues are orphans); isotropic operators as transport instruments (044, 045); the seam as a small register (048, 050).

The one object. The automaton's substitution structure plus the seam, and the record now says where the seam meets the structure: over the level map Q₅ → Q₄, a 1,024-fold covering. The level map is many-to-one (the glossary's logical form of irreversibility): it merges 1,024 cosets into one. A level-5 minimum sorts level 4's cosets into four sets of fixed size: 416 under the exact part alone, 32 under both parts, 64 under the non-flipping part alone, 512 under neither. The frontmatter object: still names "the free cosets and zero-modes of the minimum bisections" as the seam. Clause 1 is not yet redrawn. Run 101's seal fixed the redraw in advance of the draw: the exact part is organised by the map between levels, and the seam proper is what does not flip (4,137 to 4,340 free cosets on the sixteen, about a twentieth of the cut). That redraw is the owner's to approve, and whatever is written has to account for rm_s489 and rm_s226. Until then it is a reading.

What the next seal is for, and why it is next. The record names the order: PN-008's continuation with the lab's report; then the redrawn clause 1; then the line that follows from it. No line carries the next seal yet, and no seal is drafted. What the seal is for is named in CASE-30's gate check: the seal that follows run 101, "where to look for another kind of ground state, and what to read to explain the count". It is next because it is the one thing run 101's confirm does not reach. The 126 came by one route at one rng seed, most onto exact parts already on disk, and every T in one orbit. The record names routes that were not used: the slow schedule, four times the quench's rate of minima at level 4 and not tried at level 5; and the House's tests 5 and 6 (a long equal-cut walk from one stored state; an attempt below 90,112 from the far starts). These are named, not adopted. L2, L5 (whose degeneracy-only null does not exist) and S2's removal run stay open and are not next.

What would end the programme. Either of two findings. An observable that orders the five groups by growth class at a fixed level after the abelian and degeneracy controls: then growth has a finite signature and the last clause is false. Or the free part of the cut going to zero with level: then the seam is a finite artefact and the object is the recursion alone, a different programme. The second cannot be read past level 5 on these graphs. Clause 1 stands under pressure as filed: an exact share of the cut, 5/11, lies inside the seam.

Vocabulary. 1. Words the programme used that the vocabulary did not have. free coset, coset kind, flippable, independently flippable, exact part, non-flipping (coset, share), the rest, frame, fibre (of the level map), layer, W_n, T (the 448), trap at W_n, dust, zero-mode pair, abelian / degeneracy control, basin, quench, near / far stratum. The candidates, which stand in run 101's sealed fields (question, statistic, prediction, falsifier) and in this Distilled: free coset, flippable, independently flippable, exact part, non-flipping, fibre, T, basin, quench, near / far stratum. 2. Words the vocabulary had that the programme could not use or circled. "kind": retired from PG-02 on 2026-10-04, yet run 101's sealed question asks whether the sixteen were "one kind among several". The identity form (rung-vocabulary §5): "are the flat three a second kind" was settled by a symmetry and never named as an identity question. Rigidity, the injectivity of an invariant (§8.2): run 101's counts are unchanged by the graph's symmetries and "cannot tell a new minimum from a symmetric copy of an old one", which is that axis, unnamed. Many-to-one (GLOSSARY, reversibility): "the part the recursion drops" circles it. 3. Axis or rung that lacked each candidate. free coset, flippable, independently flippable, non-flipping: each is a count (§1, exact), and no axis names a degree of freedom an invariant does not register; rigidity (§8.2) is nearest. exact part: a count with a place under a map; the place axis (§4) has no stated domain to put it in. fibre: the same place axis. T: a symbol; no axis is missing; it collides. basin, quench, near / far stratum: they name a route, not a quantity; no axis is missing. 4. What the vocabulary enforced. L4 as worded ("supported on the free cosets") had no object, because the group is transitive on cosets; it was restated, struck again by blind readers before sealing, and closed. Clause 2 and "at the Schreier fibres" were struck from the seam's definition. Run 101's fixed readings barred "different objects"; the scoring reader struck five sentences that said more than the counts, and the results file's "the exact part is a property of the minimum, not of its neighbourhood" was not adopted. 5. Symbols and collisions. Q_n, Q₅ → Q₄, f, (a)–(e), T, W_n, S2 and S9 (partitions of level-4 cosets), a (the generator), φ(μ), k (the House's law). Collisions: - S2 is this programme's own scaffold and, in the same file, a partition of the level-4 cosets. The block notes PG-01's scaffold and not its own. - T, seven meanings (GLOSSARY, Collisions open); PG-02 writes "the 448" where two meet. - Q: Q_n against the run-local Q of 086, 089, 095, 105. - W: W_n against PG-03's W80 and PG-04's kind W. k: the House's k against rung k and PG-03's k. - band: the conjecture's "band energies" against BANDS.md's band; write "spectral band". - kind, orbit / class: coset kind; group orbit against PG-03's periodic orbit. 6. Definitions that drifted. - seam: "at the Schreier fibres, with the orphan eigenvalues" (2026-10-01); "the free cosets of the minimum bisections, and nothing else" (2026-10-04). GLOSSARY's entry still carries the 2026-10-01 wording, and the cross-reading of today lists seam as unchanged. - f: "4.34% of all" cosets (042); free cosets over cut edges (2026-10-03). - flippable: the level-4 kind that flips (section E, 2026-10-04 evening); split into flippable and independently flippable (section F, the same night). - exact part: "the 40,960, if the same on every minimum" until run 101 scored (2026-10-05). - twin → zero-mode pair; twin controls → abelian / degeneracy control (2026-10-04). The previous Distilled still wrote "twin controls". 7. Prose-only terms. dust, trap, frame, the thirteen, flat kind; and "seam proper", which this Distilled uses only as the seal's reading. 8. Names that carry an open claim. "exact part": exact on 142 of 142 files from one route, and the redraw calls it the recursion's. "seam proper": that what does not flip is the seam. "trap": that it holds. "flat kind": that it is a kind, now dead. 9. Metaphors. seam (one line that joins; imported "at the fibres", struck). wall (connected; tested, wrong, retired). dust (scattered and unconnected; tested by section I's second addition and held: the largest piece is 4 to 16 edges on the level-4 trap states and 12 to 16 on the sixteen level-5 minima). trap (holds; tested only as a count). frame (a rigid surround; the rigidity was wrong, and the word keeps its defined sense). basin, quench (physics; the chain's own words). 10. Quantifiers that weakened since the last rewrite. "Predicted on none" became 126 of 126, on one route. "The class separates the flat kind" became one class up to a. "The kind to look for is level 4's other one" found none on this route. The count as a property of the minimum, which nothing sealed, is now contradicted by rm_s489.

Entry. No candidate enters docs/GLOSSARY.md at this rewrite. Each stands in a sealed field of a scored run (101) and in this Distilled. No second programme uses one with the same meaning: PG-01's "polynomial fibre" is another thing, and its visible and invisible flip is a back-propagation candidate, not the same word. No panel reached the object blind: PN-008 was given "independently flippable" in the case text. Ceiling. This is the first Distilled to carry the paragraph. Each candidate is at its first of two rewrites. "flat kind" died with its lead and leaves the block.