On the sixteen stored minimum bisections of the level-5 ⟨b,c,d⟩-coset graph (cut 90,112), read on 2026-10-04, exactly 40,960 free cosets are independently flippable on every one, and they lie over the map to level 4 the same way on every one. Is that true of every minimum this route reaches, or were the sixteen one kind among several, as level 4 has two? 183 stored METIS partitions left the 131,072 basin and were never projected onto the coset graph or quenched there. Quench all of them and read the same counts on every minimum that results.

Sealed 2026-10-04, before any of the 183 partitions was projected onto the coset graph or quenched there (codifier). The third version of this seal: three readers changed the first and two the second before sealing, and what each found is at the end.

Where this comes from. PG-02 has compacted twenty-two runs and predicted nothing. On 4 October four readings of the sixteen stored level-5 minima (sections E to H of READING-PG02-ORBITS.md) found that the free cosets split into an exact part and a remainder: exactly 40,960 on every minimum flip independently, and under the map to level 4 they sit 64 or 128 to a fibre over 448 cosets of one bipartite class. All of it was read on minima already on disk. The codifier's three leads on why (whole fibres; five blocks of S9; the 64 left out) were each scored wrong the same day; the two leads that held were about counts. So this seal predicts counts, on minima that do not exist yet.

Known at sealing, cited as known. Sections E to H in full, and orbit_reading.json. On each of the sixteen (reproduced by the sighted pass with its own script): cut 90,112; free cosets 45,448 to 45,591; independently flippable 40,960; fibres 576, 256, 192 at 0, 64, 128; T of 448 in one class; the non-flipping cosets in 96 fibres, 32 inside T, 32 in T's class outside it, 32 in the other class; 77,824 cut edges between a fibre over T alone and a fibre over neither set. The sixteen files, their md5s and flippable counts:

For the 183: the METIS edgecut of each, from the cuts files. On the full graph (not the coset graph) all of them were rebalanced and greedy-swapped with no change (results/L5/rm_pre_*), and rm_s89, rm_s111 and rm_s169 were quenched there without leaving their METIS cut. None was projected onto the coset graph, quenched there, or read. Whether any of them projects is not recorded anywhere.

Found by the sighted pass on control members, and so known at sealing:

• A symmetry. Left multiplication by the generator a is an automorphism of both coset graphs, exchanges the two bipartite classes at each level and commutes with the level map (built on grig4.adj and grig5.adj). It carries each of the three stored minima with T in the class of layers 0, 3, 7, 11, 15 onto a bisection whose independently flippable set is exactly that of a stored minimum of the other class: rand_s24 and rm_s17 onto rm_s20's, rm_s31 onto that of rm_s43, rm_s53 and rm_s67. So the class of T is not a kind of minimum, and the "flat" three of section B are, up to this symmetry, on the same exact parts as four of the thirteen. The eight pairs that share an exact part outright leave ten distinct exact parts among the sixteen files; the a-symmetry merges two more, so there are eight up to this symmetry, and at most eight up to all the graph's symmetries.
• Stop-early states of rm_s17. After greedy swaps and no quench: cut 90,692, (a) = 40,761, T of 451 in two classes. After 10 sweeps: 90,660, 40,771, T of 450 in two classes. After 100 sweeps: 90,240, 40,902, T of 448 in one class, (d) = 141, (e) = 71,546. After 1,000 sweeps: 90,112, 40,960, and all five counts as on the stored minimum. In every state the cut is 2 × (a) plus the cut edges with no independently flippable end (9,118 after 10 sweeps, 8,436 after 100, 8,192 after 1,000).
• Reproduction. The file refine saves for rm_s17 after 1,000 sweeps on the M5 is byte-identical to barrier5/min_rm_s17.bits (md5 92593afd20852127cbb188cc897d802f): the stored minimum is the first end-of-sweep state at 90,112, and gate (iii) can be met.
• The 131,072 basin. min_kwbasin_s1.bits has (a) = 12,759, 818 fibres with none and the other 206 at 64 different counts from 1 to 123, T of 206 in two classes.

Left out, and why. rm_s348 (edgecut 90,440) was projected and quenched on the coset graph by run 031 with this draw's parameters and reached 90,112 (results/Q5/from_full/quench_rm_s348.qbits); its lifted profile is in results/structure.txt. No count of this seal has been computed on it. It is not in the population. Its stored minimum is read with the others, after the draw's commit, and reported unscored.

What the run is for. Level 4 has two kinds of minimum: one with eight free cosets, all independently flippable, and one with none (6 of the lab's 81 on disk; 14 of 279 in a blind reader's own rebuild and search from random starts, which is not on disk). If level 5 has a second kind at a rate like that, sixteen draws of one kind happen three or four times in ten (0.29 at 6 of 81, 0.44 at 14 of 279), and a hundred more will show it. So the run says whether the exact part is a fact about the minima this route reaches or about one kind of them. If A and B confirm, the conjecture's first clause is redrawn (the exact part is organised by the map between levels; the seam proper is what does not flip). If A is falsified, the readings of 4 October describe one kind, and the other is the next thing to read.

The codifier's lead, held with conviction. A and B both confirm: every drawn minimum has all five counts. At least 70 of the 75 near partitions project and quench to 90,112, as all fifteen of the sixteen that started there did. Of the 108 far ones at most 53 give a minimum, most because they do not project. The share of drawn minima with T in the class of layers 0, 3, 7, 11, 15 is between 0.05 and 0.40, as three of sixteen were. The two classes are exchanged by a symmetry of the graph (above), so a route blind to the classes would give a half, and three of sixteen has probability 0.011 under a half. The share is a fact about the route (its vertex numbering, the basin it starts in) and not about the graph, which is why it is a lead and not a clause.

What a blind reader gave as the odds. About 0.70 that A confirms, 0.20 that it is falsified. Near starts are nearly foretold (rm_s17 before any quench already has 40,761 independently flippable cosets); the far stratum and the chance of a second kind carry the risk.

Two exact restatements, used as cross-checks in the script and reported. No two independently flippable cosets are adjacent and each has two cut edges, so (a) = 40,960 is the same statement as: exactly 8,192 cut edges have no independently flippable end. The level map is a homomorphism onto a connected bipartite graph, so T lies in one class of level 4 exactly when every independently flippable coset lies in one bipartite class of the level-5 graph, which needs no map. And (b) as predicted implies (a) and the size in (c): 256 × 64 + 192 × 128 = 40,960 and 256 + 192 = 448.

The reductio this seal passed. By construction: none of the counts follows from the cut and the balance; (a) is equivalent to the 8,192 above, which is a fact about where the cut sits, not about its size. Negation: a minimum with another count; level 4 has one. Twins: the sixteen (gate i). Eight pairs of the sixteen share all 40,960 and differ by about 4,000 cosets elsewhere, so the count does not depend on where on its plateau a state was saved, as far as sixteen states show. Neighbours: level 4, where a minimum has eight free cosets, all independently flippable, over two level-3 cosets, or none at all; level 3, where none of the 2,460 minima has a flippable coset, though 2,456 have free cosets. Invariance: the counts are unchanged by exchanging the sides and by the graph's symmetries that commute with the level map, so they cannot tell a new minimum from a symmetric copy of an old one; the readings say so. A pass without the mechanism: the counts could belong to the valley and not to the minimum; on rm_s17 they are not exact 580 above it or 128 above it and are exact at it, and the stop-early states of two more tags are reported. Cost: 183 quenches of about 195 seconds each alone; eight at a time on the M5 is unmeasured, and two hours is the codifier's guess. The counts take seconds.

Reported, not scored. Per tag: whether it projects, the cut after rebalance, after greedy swaps and at the end, min_seen, the seconds; all of it by stratum. Per drawn minimum: free cosets and f; flippable cosets; the 8,192 counted directly; the class T lies in, found with and without the level map; T's layer profile; the cut per layer boundary; the distance to each of the sixteen and to each other drawn minimum outside the flippable sets, and whether any two files are equal or complements. The exact parts shared whole, directly or after the a-symmetry, among the sixteen and the drawn minima together, and the number of distinct ones. The orbits of the sets T under the automorphism group of grig4_q.adj. The share of drawn minima with T in each class. For every tag that ends above 90,112: its cut and counts (a) to (c). The stop-early states of rm_s20 and rm_s27 at 10, 100, 300 and 1,000 sweeps, with the cut and all five counts of each saved state. The stored minimum of rm_s348, with all five counts.

Readers. On the first draft, three at once. A second reader on the frontmatter alone: the bins overlapped and a confirm was never defined; the controls could void a falsifying result, and are now gates that finish before the first quench; a tag at 90,112 with unequal sides fell nowhere; "does not project" was undefined; the machine was a free choice. A blind sharpness and reductio reader: the two negative controls were far states and did not guard against "a property of the valley"; the "by construction" argument did not follow, and is replaced by the two exact restatements; the counts are invariant under the graph's symmetries, so sixteen agreeing files may be few objects; level 4 has a second kind, which is the real risk, and its odds are quoted above; the 96 and the 77,824 had the same standing as the scored counts and were left out without a reason, and are now clause B; the lead's rates were not numbers. A sighted pass with the disk (an agent launched by the codifier, read-only, on control members only): one of the 184 tags, rm_s348, had already been quenched on the coset graph by run 031, and is dropped; every cited figure reproduces on the sixteen with the pass's own script; the two negative-control values. On the second draft, two. The second reader again: min_seen needed its balance stated; the stop-early states and the orbits were promised in a reading and defined nowhere; undefined cases of (d) and (e); readings for clause B. The sighted pass again: the a-symmetry on the disk's own graphs at both levels and what it does to the sixteen; rm_s17's stop-early states; the byte-identical reproduction; project's exit codes; that the level-4 group is not stored and has to be rebuilt; that an artifact already on disk would show the run as in flight. The changed gates went back through the sighted pass a third time before the commit; it corrected three sentences and found nothing else.

Scored 2026-10-05 — confirmed (clause A confirmed; clause B confirmed)

Scored on 2026-10-05, Phoenix time, from R24_grig5_bisection/barrier5/RESULTS101.md and run101.json (commits fbc427a gates, 6a40f37 draw, 473569e results). A scoring reader, given those files, the gate files and the log, the sealed clauses and a draft of this note without the lead, recounted every tag from the JSON and checked every number present there; it found no number copied wrong, five sentences that said more than the counts or the seal's readings allow, and two reported facts the draft had left out (the 84 shared exact parts; rm_s489). Its edits are folded in. It could not check commit hashes or times, the group order, or the hibernate times; the codifier checked the commit times in git.

From: PG-02's line L1, through the readings of 4 October on the sixteen stored level-5 minima, which found counts that were the same on all sixteen and had predicted none of them. What the run did to it: the same five counts hold on every one of 126 minima that had not been quenched when the seal was written.

Before either clause. No quenched line shows min_seen below 90,112: not on any of the 183 tags' draws, and not on rm_s297's short first run. So the clauses are scored. All four gates passed and were committed at 17:44:10 on 4 October, before the first quench of a listed tag (about 17:44:27, from the log): (i) the sixteen give the five counts and their flippable counts; (ii) the off-minimum states give 12,759 and 40,761; (iii) rm_s17 quenched by the run's code path is byte-identical to the stored file; (iv) 183 partitions, no md5 mismatch.

The draw. All 183 tags project; none exits 3. Every draw's quenched line reads sweeps=20000 and beta=8. 126 are drawn minima (final line cut=90112 with equal sides, the file recomputing to it): 75 of the 75 near tags and 51 of the 108 far ones. The other 57, all far, end between 93,496 and 114,676. One tag, rm_s297, stopped at 9,032 sweeps on its first run because the wall-clock cap ran out while the M5 was hibernating (02:27 to 08:33 on 5 October); by the seal's rule it was run again alone with a larger cap (7,200 s), reached sweeps=20000 in 172 s, and that run is its draw. No other tag was run twice. The recompute of each saved file's cut and sides, which is part of what makes a tag a drawn minimum, ran as the draw ran. Every tag's lines and the md5 of all 549 files under fresh/ were committed (6a40f37, 14:36) before the five counts were computed (run101.json's counts, 14:44).

Clause A. (a) = 40,960 on all 126 drawn minima. At least 20 are drawn and none differs: confirmed.

Clause B. On all 126, (b) is 576 fibres with none, 256 with 64 and 192 with 128; (c) is 448 in one bipartite class; (d) is 96 as 32, 32 and 32; (e) is 77,824. At least 20 and none differs: confirmed.

What the confirms say, by the readings fixed in the seal. 126 tags gave 126 files; no two of the 142 files (the sixteen and the 126) are equal or complementary, so N = 126 and the exact part holds on 142 of 142 files, 16 + N of 16 + N. The route (random-matching METIS on the full graph, projected, quenched at one rng seed, beta and sweep count, on one machine) is the route of the 126. 51 of the drawn minima come from far starts, above the seal's threshold of 10, so the statement covers minima drawn from far starts; it says nothing of the 57 far tags that did not reach 90,112. On every one of the 126 the exact part and the non-flipping part lie over level 4 as on the sixteen. It does not reach minima found another way, or another level, and it does not say why.

How many objects (reported; the seal bars saying more than this). There are 32 distinct exact parts among the 142 files, 25 up to multiplication by a; the sixteen alone gave 10 and 8. 84 of the 126 new files (53 near, 31 far) carry an exact part identical to a stored file's, and nine of the sixteen's ten exact parts recur; on those 84, (a) counts a set already counted. The other 42 new files carry 22 exact parts not seen before. The set T of every one of the 142 files lies in one orbit, of size 128, under the automorphism group of the level-4 coset graph (order 16,384); there are at most 32 distinct T among them. Flippable counts run from 41,206 to 41,323 on the 126.

Cross-checks. On all 183 tags the cut edges with no independently flippable end number cut − 2(a), which is 8,192 on a minimum; the class of T comes out the same computed through the level map and without it.

Reported, not scored. - The count does not mark the minimum. rm_s489 ends above the minimum, at cut 94,208, with (a) exactly 40,960: its fibres hold 0, 64 or 128 and nothing else (592, 224, 208), its T is 432 cosets in one class, its (d) is 192 as 72, 56, 64 and its (e) is 72,704. So (a) and the shape of (b) hold there and (c), (d) and (e) do not. And rm_s226, at cut 99,932, has 43,040 independently flippable cosets, more than any of the 142 minima read here. Across the 57, (a) runs from 18,831 to 43,040 and T lies in one class on one (rm_s489). The results file's sentence "the exact part is a property of the minimum, not of its neighbourhood" is not adopted: rm_s489 contradicts it for (a). - What varies on the minima. Free cosets 45,432 to 45,587; f from 0.5042 to 0.5059; 125 distinct boundary profiles among the 126. - The class of T and the flat profile, which are one fact. 13 of the 126 have T in the class of the flat three of the sixteen (0.103); nine of them are far and four near. None of the 126 has the flat profile. All 13 have exactly 5,632 on 14 of their 16 boundaries, the other two offset by equal and opposite amounts between 4 and 138; no minimum of the other class has any boundary at 5,632 (the scoring reader's count from the per-tag profiles). - Stop-early states. The sealed counts appear where the cut has reached 90,112 and not before: rm_s20 at 1,000 sweeps and rm_s27 at 300. rm_s20 at 300 sweeps, cut 90,118, gives (a) 40,955, (d) 102 and (e) 77,001; at 10 and 100 sweeps its T is 449 cosets across two classes, so (c) fails and the split of (d) and (e) are undefined. - rm_s348, read for the first time: all five counts as predicted.

What it does not say. That the counts are a property of every level-5 minimum: the 126 came down one route, 84 of them onto exact parts already on disk, and every T is in one orbit. That the new minima are different objects from the sixteen under the graph's symmetries, beyond what the orbit of T, the shared exact parts and the flippable counts show. That (a) = 40,960 is a mark of the minimum. Anything about another level. Why.

The codifier's lead, scored part by part (the scoring reader did not see it). "A and B both confirm": held. "At least 70 of the 75 near partitions project and quench to 90,112": held, 75 of 75. "Of the 108 far ones at most 53 give a minimum": held, 51. "Most because they do not project": wrong in every instance; all 183 project, and the 57 that gave no minimum failed in the quench. "The share with T in the class of layers 0, 3, 7, 11, 15 is between 0.05 and 0.40": held, 13 of 126, 0.103. Four statements about counts, four held; the one reason attached to a count was wrong. That is the programme's seventh lead on a mechanism and the seventh wrong. The blind reader's odds were 0.70 that A confirms.

PN-008's forecasts are scored in panels/SEAT-RECORD.md (rows added 2026-10-05). On the direction, Ledger said the fraction would be 1 (p = 0.55); Framing, Hypothesis, Lateral, Bound and the House said it would be below 1.

Tools. The lab's assessment, taken as written: a quench costs 172 s alone on the M5 and 345 s beside run 100's nine-thread draw of H8 (ratio 2.01); gate (iii) had already shown the slowdown, 414 s against the seal's 195 s. The draw spanned about 18.5 hours, about 12.5 of refine and 6.1 of hibernate. The seal's guess of two hours assumed eight quenches at a time; the draw ran one at a time because run 100 held nine of the ten cores. The hibernate cost one tag and not the run because records are flushed after each tag; refine's cap counts wall clock, not work, and the seal's rerun rule covered it. rm_s297's first run survives only as a line of run101.log; its rerun's lines are in run101.json.

Pointer corrected 2026-10-05 (codifier; no sealed field of the prediction, falsifier or statistic is changed). The script: line said run101.py (to be written by the lab from this seal — …); the script exists and is committed, and the line now says (written by the lab from this seal — …). The two main() inside that remark are written main, because the record-field check reads a parenthesis as a remark only up to the first closing bracket and was taking index_reading.py (which is at R24_grig5_bisection/index_reading.py) for a missing file. Nothing else in the line is changed.