One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Body below is the tool's export of 2026-10-04, 18:20 Phoenix, unedited beyond the id; the heading's date is the tool's UTC stamp. Record loaded (S3; PN-007 ran without): the seat record as a generated view of every seat row (82 rows; panels/SEAT-RECORD.md itself is read by the tool only as far as its first table, OPS 2026-10-04), the record packet shared. Two tool faults were worked around before the round and are in MANUAL: the audit's refusal of the words "the map", and the frozen roster that kept Consult disabled until a placeholder row was written to the artifact's database. The House's FORCED 3 and forecast F4 rest on a misreading of one sentence of the case, which the case's note of the same day corrects: 12,759 is the count on the off-basin state, not the cut edges without such an end (105,554), and the identity holds there.

One round and a House, closed after House 1 (docs/v5-plan.md §12.2, S2). Body below is the tool's export of 2026-10-04, 18:20 Phoenix, unedited beyond the id; the heading's date is the tool's UTC stamp. Record loaded (S3; PN-007 ran without): the seat record as a generated view of every seat row (82 rows; panels/SEAT-RECORD.md itself is read by the tool only as far as its first table, OPS 2026-10-04), the record packet shared. Two tool faults were worked around before the round and are in MANUAL: the audit's refusal of the words "the map", and the frozen roster that kept Consult disabled until a placeholder row was written to the artifact's database. The House's FORCED 3 and forecast F4 rest on a misreading of one sentence of the case, which the case's note of the same day corrects: 12,759 is the count on the off-basin state, not the cut edges without such an end (105,554), and the identity holds there.

PN-008 — Every lowest-cut bisection found on a self-similar graph carries exactly the same number of independently flippable vertices — what fixes that number, and is there a ground state without it? · run 2026-10-05, lab roster (physics) · one round, closed after House 1, no lab step · resolved: open

Case id. CASE-30

Case

Type. lab / physics

Presenting object. Let G_n be the Grigorchuk group acting on the binary tree of depth n (the quotient of the group
by the stabiliser of level n), with its generators a, b, c, d; its order is 2^7, 2^12, 2^22 for n = 3, 4, 5.
The elements 1, b, c, d form a Klein four-group H. The graph Q_n has one vertex for each coset gH and joins
gH to g·h·a·H for each of the four h in H. It is 4-regular with no loops or repeated edges, connected and
bipartite, on 32, 1,024 and 1,048,576 vertices for n = 3, 4, 5.

Put an Ising spin on each vertex, with total magnetisation fixed at zero and energy equal to the number of
edges joining opposite spins (the cut). A ground state is a balanced bisection of smallest cut. In a
bisection a vertex is free if exactly two of its four edges are cut, so that flipping it alone leaves
the cut unchanged. A free vertex is flippable if flipping it alone also leaves the set of free vertices
unchanged, and independently flippable if in addition none of its neighbours is free and each of its
four neighbours is adjacent to exactly one free vertex. Independently flippable vertices do not interact:
any subset of them can be flipped and the cut and the free set stay the same.

The count of independently flippable vertices has been read on every smallest-cut bisection found:

Friction. At n = 5 the sixteen bisections differ from one another in most other counts (free vertices
45,448 to 45,591; flippable 41,214 to 41,330), yet the independently flippable count is 40,960 on all
sixteen, which is exactly 5/11 of the cut. One level down it is 8, which is 1/22 of the cut, or zero; two
levels down it is zero. No recursion between the levels has been found that produces these numbers. A
level-5 ground state is not a lift of a level-4 one: pulling a level-4 ground state back through the
projection from level 5 to level 4 gives cut 180,224, twice the minimum. The 40,960 are not whole fibres of
that projection either. And the count
is not explained by the sixteen being one object seen sixteen times: they carry ten different sets of
independently flippable vertices. What fixes the number, at each level, and does level 5 have a ground state
with a different number, as level 4 does?

Background. The layering and its unit. Fix one leaf of the tree. The vertices of Q_n fall into layers according to
where g⁻¹ sends that leaf along the tree's Schreier path; edges join consecutive layers only, and the
boundary between any two consecutive layers carries exactly W_n = |G_n| / 2^n edges: 16, 256 and 131,072 for
n = 3, 4, 5. The smallest cut is W_3 at n = 3 and exactly 11/16 of W_n at n = 4 and 5. The symmetries of the
graph do not preserve the layering.

The cut splits in two parts, and the parts change places. No two independently flippable vertices are
adjacent and each has exactly two cut edges, so the cut edges with an independently flippable end number
twice the count.

At both levels the cut is 10/16 plus 1/16 of W_n. At level 4 the small part touches the independently
flippable vertices; at level 5 the large part does. Nobody knows why 11, why the split is 10 and 1, or why
the parts change places. (Unexplained.)

Level 3. All 601,080,390 balanced bisections were enumerated. 4,920 have the smallest cut, 16; that is
2,460 up to exchanging the sides. Their free vertices number 0, 4, 6, 8, 10, 12 or 16. None has a flippable
vertex.

Level 4. A search (320 heuristic partitions, each refined) ended at cut 176 in 81 distinct bisections.
Under the graph's automorphism group, of order 16,384 and transitive on vertices and on edges, they fall
into 15 orbits: seven of 4,096 members, seven of 1,024, and one of 512. In the fourteen larger orbits a
bisection has exactly eight free vertices, all eight independently flippable; all 256 assignments of those
eight keep the cut at 176. In the orbit of 512 a bisection has no free vertex. In the 75 with free vertices,
the 160 cut edges without a free end each join two vertices that have exactly one cut edge. The cut 176 is
the smallest found, not proven.

Level 5, the sixteen. Each came from an independent heuristic partition of the Cayley graph of G_5
(4,194,304 vertices) that splits no coset, read as a bisection of Q_5 and refined by a Metropolis walk that
swaps one vertex from each side, at inverse temperature 8. Fifteen of the sixteen starting partitions were
within 1% of the final cut (90,144 to 90,966); one started at 95,584. The cut 90,112
is the smallest found, not proven. On every one of the sixteen:

• only vertices with 0, 1 or 2 cut edges occur;
• the cut edges are, by the kinds of their ends: free to one-cut-edge 87,302 to 87,466; free to free 1,730 to 1,940; one-cut-edge to one-cut-edge 843 to 954;
• independently flippable vertices: exactly 40,960; the 8,192 cut edges with no such end include every free-to-free cut edge and every edge joining two one-cut-edge vertices;
• free vertices that are not flippable: 4,137 to 4,340.

Level 5 over level 4. The homomorphism G_5 → G_4 induces a projection Q_5 → Q_4 that carries edges to
edges and whose 1,024 fibres have 1,024 vertices each. On every one of the sixteen, identically:

• 576 fibres hold no independently flippable vertex, 256 hold 64 and 192 hold 128 (no fibre is whole);
• the 448 level-4 vertices under those fibres all lie in one of the two bipartite classes of Q_4 (512 each), and form whole blocks of one invariant partition of Q_4 into blocks of eight;
• the free vertices that are not flippable lie in 96 fibres: 32 of them among the 448, 32 in the same bipartite class outside the 448, and 32 in the other class;
• exactly 77,824 cut edges (19/22 of the cut) join a fibre over the 448 that holds no non-flippable free vertex to a fibre over none of these sets, and no cut edge has both ends outside the 448 and the 96;
• every fibre is split nearly in half between the two sides (0.479 to 0.519).

Level 4 over level 3. Under the same kind of projection Q_4 → Q_3 (fibres of 32), the eight free vertices of a level-4
ground state lie over exactly two level-3 vertices, four in each, on all 75.

A symmetry, and what it explains. Left multiplication by the generator a is an automorphism of Q_4 and of
Q_5, exchanges the two bipartite classes at each level and commutes with the projection between the levels. Of the sixteen, three
have their 448 in one class and thirteen in the other; the symmetry carries each of the three onto the
independently flippable set of one of the thirteen. So which class the 448 lie in is not a kind of ground
state. (Explained.) The sixteen files carry ten different independently flippable sets, eight up to this
symmetry. Two different sets share a multiple of 1,024 vertices (0, 2, 4, 6, 8, 11, 12, 18, 20 or 28 times
1,024). The 448 are whole blocks of eight, and a block's fibres hold 512 or 1,024 of the vertices; why
these blocks, and why never an odd multiple of 512, is not known. (Unexplained.)

On the way down to the cut. One of the sixteen starts was read along its refinement:

In every state the cut is twice the count plus the cut edges with no independently flippable end. A
bisection far from the minimum (cut 131,072, a different basin) has 12,759.

Instrument notes. - Objects. Every number is an exact count on a finite graph at n = 3, 4 or 5. None is a limit, and there
is no level 6 to compute: Q_6 has 2^40 vertices.
- Ground state. At n = 3 the smallest cut is proven by enumeration. At n = 4 and 5 it is the smallest
found by heuristic search. A lower cut would change every row.
- Which state was read. The walk at inverse temperature 8 moves freely among states of equal cut. A
stored bisection is the first state at the lowest cut that the walk reached at the end of a sweep, so it
is one point of a plateau. The free and flippable counts differ from point to point; the independently
flippable count was 40,960 on all sixteen points read. Eight pairs of the sixteen share all 40,960 and
differ in about 4,000 vertices elsewhere.
- Selection. All sixteen came from one route, and fifteen from starts within 1% of the minimum. At level
4 the kind with no free vertex was 6 of 81 found.
- Symmetry. Every count here is unchanged by exchanging the two sides and by automorphisms of the graph
that commute with the projection between the levels, so equal counts on two files do not show the files are different objects.
- Direction. A state above the minimum reads a count below 40,960 on the one path read (40,761 at 580
above, 40,902 at 128 above).
- The pending run. It reports only bisections that end at exactly 90,112 after the same walk; starts that
cannot be projected, or that end higher, are listed and not counted.

Must predict. A run already fixed and not yet drawn refines 183 further independent starting partitions
by the same route (75 of them with a starting cut from 90,440 to 91,964, and 108 from 92,048 to 130,804)
and reads the independently flippable count on every one that ends at cut 90,112.
1. What fixes the count? Name a structure that can be computed from G_n or Q_n without searching over
bisections, and say what it gives at n = 3, 4 and 5. It should give 0; 8, with room for 0; and 40,960.
2. The run. The fraction of the bisections it reaches at cut 90,112 whose count is exactly 40,960: say
whether it is 1, with a probability, and give an interval for the fraction with stated coverage. If a
level-5 bisection at cut 90,112 with another count exists, give that count as a value or an interval, and
say what "no second kind" predicts.
3. One computation. Name one exact computation, at most a few hours on a laptop, whose result your
account and its strongest rival predict differently.

Provenance. Exact integer counts on stored bisections, read on 2026-10-04; nothing is sampled. Level 3
by exhaustive enumeration. The smallest cuts are from runs 031 and 040; free vertices were first counted in
run 042. The refinement states, the symmetry and the off-minimum count are from the seal of run 101.

Premise ledger

• [derived, load-bearing] Q_n is a 4-regular, loopless, simple, connected, bipartite graph on 32, 1,024 and 1,048,576 vertices for n = 3, 4, 5, built from the cosets gH of the Klein four-group H in G_n
• [derived, load-bearing] The layering by Schreier position puts edges only between consecutive layers, and each layer boundary carries exactly W_n = |G_n|/2^n edges (16, 256, 131,072)
• [assumed, load-bearing] The ground-state energy at n = 4 and n = 5 is 176 and 90,112 — i.e. the smallest cut found by heuristic search is the true minimum (explicitly unproven; a lower cut would change every row)
• [derived, load-bearing] Independently flippable vertices are pairwise non-adjacent and each carries exactly two cut edges, so the cut edges with such an end number exactly twice the count (cut = 2·count + rest)
• [measured, load-bearing] Every number reported is an exact integer count on a stored finite bisection, with no sampling error
• [assumed, load-bearing] The sixteen stored level-5 bisections are representative of the whole plateau of cut-90,112 bisections, so that '40,960 on all sixteen' licenses a statement about bisections not yet drawn
• [assumed, load-bearing] The Metropolis walk at inverse temperature 8, which moves freely among equal-cut states, selects a plateau point without bias toward particular independently-flippable sets
• [assumed, load-bearing] The single search route (heuristic coset-respecting start plus refinement), and the fact that fifteen of sixteen starts lay within 1% of the minimum, does not restrict which kind of ground state is reached
• [assumed, load-bearing] No recursion between levels produces the counts: the absence of a found level-4-to-level-5 relation is treated as absence of such a relation (supported by the lift of a level-4 ground state having cut 180,224, twice the minimum, and by the 40,960 not being whole fibres)
• [assumed, load-bearing] The two accounts — 'the count is fixed by structure, hence always 40,960 at level 5' and 'a second kind of level-5 ground state with a different count exists, as at level 4' — are exclusive and between them exhaust the possibilities
• [assumed] The level-4 pattern (14 orbits with 8 independently flippable, 1 orbit of 512 with none) transfers as a template to level 5, so 'room for 0' at level 4 makes a second level-5 kind a live option
• [measured, load-bearing] Aut(Q_4) has order 16,384 and is transitive on vertices and on edges, and the 81 level-4 bisections fall into 15 orbits (7×4,096, 7×1,024, 1×512)
• [derived, load-bearing] Left multiplication by a is an automorphism of Q_4 and Q_5 that exchanges bipartite classes and commutes with the level projection, so which class the 448 lie in is not a distinguishing feature of a ground state
• [assumed] Below-minimum states read counts below 40,960 (40,761 at 580 above, 40,902 at 128 above), and this one refinement path gives the direction an unconverged estimator reads
• [assumed, load-bearing] The 183 pending starting partitions are independent of the sixteen and are processed by exactly the same route, with only exact-90,112 endpoints counted

Number kinds

• [unclassifiable, target] 40,960 independently flippable vertices on every level-5 bisection at cut 90,112 — no route named: equals 5/11 of the cut, 10·W_5/16 halved, and W_5 − cut, but the case states no structure produces it
• [count] 8 independently flippable vertices at level 4 (0 on 6 of the 81) — flagged: a count with non-zero spread — 8 on 75 bisections, 0 on 6
• [count] 0 independently flippable vertices at level 3 — exhaustive over all 2,460 smallest-cut bisections
• [count] W_n = |G_n|/2^n = 16, 256, 131,072 edges across a layer boundary
• [count] group orders 2^7, 2^12, 2^22 for G_3, G_4, G_5
• [count] vertex counts 32, 1,024, 1,048,576 of Q_n (and 2^40 for Q_6)
• [count] 4 (degree of Q_n; order of the Klein four-group H)
• [count] smallest cut 16 at n = 3 — equals W_3; proven by exhaustive enumeration
• [unclassifiable] smallest cut 176 at n = 4 — stated as 11/16 of W_4 but the factor 11/16 is unexplained; minimality not proven
• [unclassifiable] smallest cut 90,112 at n = 5 — same 11/16 of W_n with no named route; minimality not proven
• [unclassifiable] the ratio 11/16 of W_n, split as 10/16 plus 1/16 — the case explicitly says nobody knows why 11, or why the split is 10 and 1, or why the parts change places
• [unclassifiable] 5/11 of the cut (level 5) and 1/22 of the cut (level 4) — flagged: ratios of one observed count to another observed count, presented as if structural
• [unclassifiable] W_n − cut = 0, 80, 40,960 — flagged: a difference of two absolutes set against the independently flippable count; coincidence at n = 5 only
• [count] 81,920 cut edges with an independently flippable end (10·W_5/16) and 8,192 without (W_5/16) — derived by identity cut = 2·count + rest
• [count] 16 cut edges with such an end at level 4 (W_4/16) and 160 without (10·W_4/16)
• [count] |Aut(Q_4)| = 16,384; orbit sizes 4,096, 1,024, 512; 15 orbits
• [count] 1,024 fibres of 1,024 vertices each in the projection Q_5 → Q_4 — fixed by |G_5|/|G_4| structure
• [count] 576 fibres with none, 256 holding 64, 192 holding 128 — identical on all sixteen; route to these multiplicities not named
• [count] 448 level-4 vertices forming whole blocks of eight, in one bipartite class of 512 — the case states why these blocks is not known
• [count] 96 fibres holding non-flippable free vertices, 32 + 32 + 32
• [unclassifiable] 77,824 cut edges = 19/22 of the cut — exact count, but the fraction 19/22 has no named route
• [count] 601,080,390 balanced bisections at n = 3; 4,920 of smallest cut; 2,460 up to side exchange — binomial C(32,16)/2 fixes the first
• [count] ten different independently flippable sets, eight up to the a-symmetry, among the sixteen
• [unclassifiable] shared-vertex multiples 0, 2, 4, 6, 8, 11, 12, 18, 20, 28 times 1,024 — the case states why never an odd multiple of 512 is not known
• [free] inverse temperature 8 of the Metropolis walk — a parameter choice of the run, no structure produces it
• [free] 320 heuristic partitions at level 4; 81 bisections found; 16 level-5 bisections; 183 pending starts (75 + 108) — sample sizes fixed by run design, not by the graph
• [free] the 1% window on fifteen of sixteen starting cuts (90,144 to 90,966) — a selection criterion of the route
• [unclassifiable] fibre split fraction 0.479 to 0.519 — measured spread, no route named for the near-half value
• [unclassifiable] free vertices 45,448–45,591 and flippable 41,214–41,330 at level 5 — counts with large spread, used only as contrast to the invariant 40,960
• [unclassifiable] 12,759 cut edges with no independently flippable end in a cut-131,072 bisection — single off-basin reading

Target form. number — The fraction of the 183 pending refinements ending at cut 90,112 whose independently flippable count is exactly 40,960, with a probability and a stated-coverage interval, plus the value or interval for any second count if another kind exists.. Bound: fired.

Bound note. Bound seat added — no known route for the target (unclassifiable).

Seat versions. Ledger v1 · Framing v1 · Hypothesis v1 · Lateral v1 · Bound v1

Seat record. shown to each seat (its own rows only) · SEAT-RECORD-all-seat-rows.md · 82 seat rows · SHA-256 acdcb3ca1d10c296e81312ef9344fc3239e793cc066c19b6ab9e74c49c6d55fb

Record packet. mode: shared (every seat gets the full packet) · full: packet-full.txt · 6 facts, cap 40 · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929

Record packets by seat (one hash per seat)

• Ledger v1: 6 facts · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929
• Framing v1: 6 facts · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929
• Hypothesis v1: 6 facts · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929
• Lateral v1: 6 facts · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929
• Bound v1: 6 facts · SHA-256 5778b73565cf72d86bc50871e12c2d227a34ffee9cd0789bf85863697db68929

Record packet — full packet — sent to every seat, House and checker (RECORD, separate from the case)

`
F01 [scored] outcome confirmed, scored 2026-09-27 — run 031, 031-grig5-bisection.md, commit 1586aaa
F02 [scored] outcome unanticipated, scored 2026-09-28 — run 040, 040-t0-barrier-extensive.md, commit 1586aaa
F03 [scored] outcome unanticipated, scored 2026-09-28 — run 042, 042-valley-wall-is-local-global.md, commit 1586aaa
F04 [scored] outcome unanticipated, scored 2026-09-27 — run 030, 030-ising-barrier-intermediate-growth.md, commit 1586aaa
F05 [scored] outcome withdrawn, scored 2026-09-27 — run 034, 034-glass-barrier-coset-graph.md, commit 1586aaa
F06 [scored] outcome confirmed, scored 2026-09-27 — run 036, 036-expansion-profile-coset-graph.md, commit 1586aaa
`

Roster

_The seats that sat, with the axis text the House receives for each and the stance each was given (what it refuses)._
- Ledger (Ledger) v1
- axis: What is exactly true here by identity or definition, before any model or heuristic is applied — and what quantity can be computed two independent ways?
- refuses: to interpret. Writes the bookkeeping exactly — identities, counts, what is pinned and what is free — and stops.
- Framing (Framing) v1
- axis: Is the named object the actual object? Is the question as posed the right question — and has the target changed category (a count that varies with a parameter is not a count)?
- refuses: to answer the question as asked when the framing is wrong. Names the object the case should have been about.
- Hypothesis (Hypothesis) v1
- axis: What theorem, tool, identity or approximation does the case (or a seat) invoke — what does each one require, and is it satisfied here? Name the tool, state its preconditions, check each against the case.
- refuses: to let a result stand on an unchecked precondition. Does not accept that a tool applies because it is standard in this area.
- Lateral (Lateral) v1
- axis: Is this a known structure under a different name — and what theorem do the two share?
- refuses: to offer a resemblance without naming the shared theorem, and to present a cross-reference as if it were a result.
- Bound (Bound) v1
- axis: The target is a free magnitude — no structure produces its value. What does consistency forbid? Give the interval that is forced, and say what forces it (unitarity, positivity, a sum rule, a stability or existence requirement).
- refuses: to stay silent because no mechanism is available. A forced interval is a result; "unknown" is not.

Round 1

Ledger (Ledger)
TAKE
The only hard identity here is cut = 2·(independently‑flippable count) + rest, because each such vertex carries exactly two cut edges and no two are adjacent — that is definitional, not a result of search. It fixes a relation, not a value: given cut = 90,112, the identity only forces 0 ≤ count ≤ 45,056 with rest = 90,112 − 2·count ≥ 0 and integral. Nothing in the record computes count independently of a found bisection; "40,960" is read off stored states, never derived from G_5 or Q_5 directly. The one place a second independent route to the same number exists is W_n − cut: at n = 5 this equals 40,960, matching the count exactly, but at n = 4 it does not (80 vs. 8), so this is not a generally valid second computation — it is a single coincident equality at one level, correctly flagged as unexplained rather than structural. Likewise |Aut(Q_4)| = 16,384 and its orbit sizes (4,096/1,024/512) are exact counts that partition the 81 level‑4 ground states, but no analogous orbit-counting object for Q_5 is given, so there is no second route to 40,960 at n = 5. By identity alone, my axis has almost no purchase on what fixes the number; it only bounds it and certifies the bookkeeping consistency of each reading.

PREDICTION
fraction of the 183 pending bisections ending at cut 90,112 whose independently-flippable count equals exactly 40,960: value = X
rest (cut edges with no independently-flippable end) on any newly found cut-90,112 bisection with count = k: value = 90,112 − 2k (exact, by identity, for whatever k is observed)
SIGN: fraction equal to 1 (no second kind found among the pending batch), p=0.55
VALUE: 0.80 to 1.0, 80% interval
NULL: no effect predicts the fraction tracks the level-4 analog base rate (75/81 ≈ 0.93) rather than 1, i.e. a second kind should appear at roughly the same relative frequency as at n=4

ASSUMES
That "no effect" is modeled on the level-4 ratio of free-vertex-having to free-vertex-free ground states (75:6) as the only prior analogy available; the case does not state this transfers.

CHANGES MY MIND
A single bisection in the 183 that ends at cut 90,112 with independently-flippable count ≠ 40,960 and ≠ 0 — any third value — would show the identity bound is the only constraint and the invariant is not forced by structure at all.

CITES
F01, F02, F03

Framing (Framing)
TAKE
The question "what fixes the count?" presupposes that "the independently flippable count" is one object across n=3,4,5 — a single invariant waiting for a formula. It isn't. At n=4 the count is already two-valued on the same graph at the same cut (8 on 75, 0 on 6): that alone proves "the count" is not a property of (G_n, ground-state energy) but a property of which bisection, i.e. a function on the solution set, not a number attached to the graph. The case's own framing — "name a structure computable from G_n or Q_n without searching over bisections" that "gives 0; 8, with room for 0; and 40,960" — asks for a single-valued function to reproduce a quantity that is demonstrably multi-valued at n=4. That's a category error: you can't fix a non-constant by naming a constant-producing structure. The object this case should be about is not "the independently flippable count of Q_n" but "the orbit structure of Aut(Q_n) acting on the ground-state set, and the distribution of the count across orbits." At n=5, with only 16 of presumably many ground states sampled from one route, "40,960 on every one found" is weak evidence for constancy and strong evidence only that the route hasn't yet hit the n=4-style sparse orbit (6/81 ≈ 7%, so ~1/16 chance of missing it by sampling 16). The right question is about orbit-size distribution, not a formula for a scalar.

PREDICTION
VALUE: 3 to 20 distinct values (orbit classes of the count) found among the 183 pending runs that land at cut 90,112, 70% interval
SIGN: not-all-40,960, p=0.75
NULL: if the count really were graph-fixed, the pending 183 would show count=40,960 on 100% of cut-90,112 outcomes

CHANGES MY MIND
If the full 183-run batch returns 40,960 on literally every bisection reaching cut 90,112 (zero exceptions), that would falsify "the count is orbit-dependent, not graph-fixed" and I'd concede the framing (a single invariant) was right after all.

ASSUMES
That the level-4 behavior (two orbit-types, one with count 0) is the right analogy for what "ground state" means at level 5, rather than a level-4 peculiarity with no level-5 counterpart.

CITES
F02, F03

Hypothesis (Hypothesis)
TOOL
Inference from a finite sample of stored bisections to a statement about the whole ground-state plateau ("the sixteen are representative," used to license "the count is fixed by structure at 40,960")
REQUIRES
- the sample must be drawn by a process unbiased with respect to the statistic being generalized (independently-flippable count)
- the sample must be large enough, or the population structured enough, to rule out a minority second kind (cf. level 4, where 6 of 81 — about 7% — had a different count)
- the sampling process must not be correlated with the mechanism that produces different counts
SATISFIED
- unbiasedness: no — all sixteen come from one route (coset-respecting heuristic start + Metropolis at β=8), and fifteen of sixteen starts lie within a 1% window of the minimum; this is explicitly flagged as a possible restriction, not shown absent
- sample size vs. minority-kind detection: no — at level 4 the minority kind (0 flippable) was found in 6/81 (~7%); a kind at that frequency has roughly (0.93)^16 ≈ 32% chance of being entirely missed by sixteen draws from one route, so absence in the sixteen is not strong evidence of absence
- independence from mechanism: not checkable — no mechanism is named, so there's nothing to check correlation against

TOOL
"No recursion between levels found" used as a premise that no such recursion exists (supports treating level-4 and level-5 counts as governed by independent, level-specific structure)
REQUIRES
- the search for a recursion must have covered a space whose scope is stated, so that non-discovery is informative
- the counterexamples offered (lift gives cut 180,224; the 40,960 are not whole fibres) must rule out recursion in general, not just the specific forms tried
SATISFIED
- scope of search: no — no space of candidate recursions is defined in the case; "not found" is reported with no bound on what was tried
- generality of the two counterexamples: no — they rule out two specific naive relations (direct lift, fibre-wise correspondence), not recursion as a category

TOOL
cut = 2·(independently flippable count) + (cut edges with no such end) — used throughout as an exact bookkeeping identity
REQUIRES
- independently flippable vertices pairwise non-adjacent
- each independently flippable vertex has exactly two cut edges
SATISFIED
- pairwise non-adjacency: yes — stated as derived, and used consistently (e.g., matches the "on the way down" table exactly at every refinement step)
- exactly two cut edges each: yes — follows directly from the definition of "free" plus "independently flippable," which is given as definitional, not inferred

TOOL
Treating simple fractions of W_n or of the cut (11/16, 5/11, 10·W_n/16, 19/22, W_n − cut) as evidence of underlying structure, used implicitly to motivate "what fixes the number" as answerable
REQUIRES
- a derivation connecting the fraction to a named algebraic or combinatorial object (e.g. an index, an eigenvalue multiplicity, a block size)
- the fraction to be stable under conditions that would distinguish structure from coincidence (e.g., predicted in advance rather than fit after the fact)
SATISFIED
- derivation: no — the case states explicitly, for every one of these ratios, that no route is named ("nobody knows why 11," "the fraction 19/22 has no named route," "flagged: a difference of two absolutes... coincidence at n=5 only")
- advance prediction vs. post-hoc fit: no — all ratios were computed after the counts were read; none was predicted from G_n before the search

TAKE
This case's central claim — that a structure exists which fixes the independently-flippable count without bisection search — rests on exactly one tool: generalizing from sixteen bisections, all produced by one route with a biased start distribution, to a universal invariant. That tool's precondition (unbiased, sufficiently powered sampling) is not satisfied, and the case itself supplies the counterexample pattern: at level 4 the same kind of search found a minority class (0 on 6 of 81, ~7%) that a sample of sixteen from a single route would very plausibly miss entirely (~32% miss probability under naive iid assumptions, likely higher given correlated starts). Separately, every numerical "explanation" offered for the target quantity (5/11, 10·W/16, W_n−cut) is explicitly flagged by the case's own audit as route-less — these are fits, not derivations, and I refuse to let "exactly 5/11 of the cut" stand in as a mechanism. The "no recursion found" premise has the same defect: the search space was never bounded, so its negative result carries little weight. My axis gives no purchase on whether 40,960 is "really" structural — only on the fact that no tool in the case has checked preconditions sufficient to assert it is.

PREDICTION
SIGN: a second kind of count appears among the 183 pending starts, p=0.6
VALUE: 0.70 to 0.97, 80% interval (fraction of cut-90,112 endpoints with count exactly 40,960)
NULL: if the count really is structurally fixed, "no effect" (no second kind) predicts the fraction equal to 1.000 exactly

ASSUMES
That the level-4 minority-kind frequency (~7%) is a reasonable base rate analogue for level 5 risk of a missed second kind — not stated in the case.

CHANGES MY MIND
A derivation, from G_n or Aut(Q_n) alone (no bisection search), that produces 40,960 — or a demonstration that the sixteen starts, despite common route, sample the plateau with provably small bias (e.g., via a mixing-time bound on the β=8 walk) — would remove my objection to the representativeness tool.

CITES
F01, F02, F03

Lateral (Lateral)
TAKE
This is a coset/Schreier graph on a self-similar group, and the "independently flippable" condition is exactly the condition for a GF(2) stabilizer-code logical-qubit count: each independently flippable vertex is a spin whose flip commutes with the cut function and with every other flippable vertex's flip (pairwise non-adjacent, no shared neighbour-constraint). That is the defining property of a basis for the kernel of a linear "parity-check" map on GF(2) — the same structure as the logical-operator count in a stabilizer code, or equivalently the nullity term in the rank–nullity theorem applied to the incidence matrix restricted to the free-vertex induced subgraph. The shared theorem is rank–nullity (GF(2) linear algebra): independently-flippable count = dim ker(M) where M is the local adjacency/constraint matrix on the free set, and 2^(that count) is the exact size of a degenerate ground-state orbit under coordinatewise flips. This explains why 8 toggles at level 4 (dim ker = 3, giving 2^3=8 flip-combinations... actually 8 is the count itself, not 2^8 — the count is the dimension, not the degeneracy) and why some orbits have kernel dimension 0 (the 512-orbit at level 4). It does NOT explain the specific numbers 8 or 40,960 — rank–nullity only guarantees the count is an invariant of the induced free-subgraph's rank, not what that rank is. So: known structure (GF(2) nullity of a stabilizer-like constraint matrix), shared theorem named (rank–nullity), but the theorem alone does not derive 40,960 from G_n without computing the matrix — it only certifies that different ground states in the same orbit-type must agree, and that a second type (rank 0, matching level 4's 512-orbit) is the live alternative.

PREDICTION
quantity: fraction of the 183 pending cut-90,112 bisections with independently-flippable count exactly 40,960 (not already reported)
SIGN: fraction < 1 (a second, lower-rank kind exists among the 183, mirroring level 4's 6-of-81), p=0.6
VALUE: 0.85 to 1.00, 80% interval
NULL: if the count were a free/search-contingent artifact unrelated to graph rank, the fraction would scatter broadly (e.g. 0.3–1.0) rather than cluster near 1

ASSUMES
That the free-vertex induced subgraph's GF(2) rank takes only a small number of discrete values across the ground-state manifold (as observed at level 4: rank corresponding to 8 or 0), and that this is the correct name for the invariant — not independently verified against the actual incidence matrix.

CHANGES MY MIND
A direct computation of the rank of the free-vertex constraint matrix on a level-4 or level-5 ground state that does NOT equal the reported independently-flippable count would kill the rank–nullity identification outright.

CITES
none

Bound (Bound)
TAKE
My axis is simple: the case treats 40,960 as a free magnitude ("no structure produces it... not explained"). Consistency does not stay silent just because no mechanism is named. The one identity the case itself states — cut = 2·(independently flippable count) + (cut edges with no such end), and the rest-term is a count of edges, hence ≥ 0 — forces, for any bisection at the claimed minimum cut 90,112, that the independently flippable count n satisfies 0 ≤ n ≤ 45,056. That's the only interval consistency hands you without a mechanism, and it is wide: 40,960 sits well inside it, not forced to its boundary. This matters for the open question (is there a second kind at level 5, as at level 4): consistency alone permits any integer in [0, 45,056], so it cannot forbid a second value appearing in the 183 pending starts. The level-4 precedent (8 vs 0, i.e. an orbit of 512 with zero) is exactly what "not forced to a single value" predicts — a structureless target is allowed to vary, and at level 4 it did. The absence of a found recursion is not evidence of impossibility; it's exactly what you expect when the only hard constraint is the identity bound, which both 8 and 0 (level 4) and 40,960 and some unseen alternative (level 5) satisfy with room to spare.

PREDICTION
SIGN: at least one of the 183 pending cut-90,112 bisections has independently flippable count ≠ 40,960, p=0.6
VALUE: 0 to 44,000, 80% interval (the alternate count, if one appears, conditional on the identity bound n ≤ 45,056 and n < 40,960 by analogy with level 4's 8→0 drop)
NULL: no second kind predicts fraction exactly 40,960 = 1.00 on all 183, i.e. the identity bound is saturated at the same point every time with no exception

SIGN: fraction of pending bisections at cut 90,112 with count exactly 40,960 is below 1, p=0.6
VALUE: 0.70 to 0.98, 80% interval
NULL: no effect (count is fixed) predicts the fraction = 1.00 exactly, no exceptions

ASSUMES
That the identity cut = 2·count + rest, with rest a nonnegative edge count, is the only structural constraint in force (I introduce no tighter bound such as an independence-number or eigenvalue bound, which the case does not supply); that level 4's zero-count orbit is a legitimate analogy for "room" at level 5.

CHANGES MY MIND
A proof (or a tighter stated structural bound, e.g. from the graph's spectral gap or an explicit independent-set bound on Q_5) that pins n to exactly 45,056·(5/11) or any single value as a consequence of G_5's algebra rather than of search outcomes — that would convert my wide consistency interval into a genuinely forced point, and I'd retract the "room for a second kind" claim.

CITES
none

House 1

CHECKS
none flagged — the checker table extracted no checkable claims. One row the checker should have carried, and did not, I raise myself: the off-basin reading (cut 131,072, "cut edges with no independently flippable end" 12,759) is parity-inconsistent with the identity the case says holds "in every state": 131,072 − 12,759 = 118,313, which is odd, while 2·count is even by construction. Either that rest-count, or the cut attached to it, is wrong. This number is used only as a contrast with the refinement path, so it is uncontested and ranks below the real disagreement — but it is a live arithmetic error in a sealed run (run 101) and must be rechecked before any direction-of-estimator argument leans on that series. The four refinement rows (90,692/40,761/9,170; 90,660/40,771/9,118; 90,240/40,902/8,436; 90,112/40,960/8,192) all satisfy the identity exactly, so no verdict here rests on the broken row.

VERDICT
The strongest objection is not about mechanism but about a precondition. [object: sampled lattice at the sizes measured] All sixteen level-5 bisections come from one route, fifteen from starts inside a 1% window; the tool that turns "40,960 on sixteen" into "40,960 always" is generalisation from an unbiased, adequately powered sample, and neither precondition is checked. At level 4 the same route found a minority kind at 6/81; sixteen draws miss a kind of that frequency about 31% of the time (base rate inherited from one level — fit on 1 point). [object: exact one-block] At n = 4 the count is two-valued on one graph at one cut, so "the count" is a function on the ground-state set, not an invariant of G_n. [object: limit] Therefore question 1 as posed asks for a single-valued structure to reproduce a demonstrably multi-valued quantity, and no account that answers it can be right. What survives is the identity, and the observation that the count is pinned while free and flippable counts spread by ~150.

DISAGREEMENT
The real conflict is between the panel and the case's framing. The case asks "what fixes the count?", presupposing one number per level. Framing and Bound deny the presupposition: at n = 4 the count is 8 or 0 on the same graph at the same cut, so the target is a function on the solution set. Hypothesis names the unmet precondition — single-route, narrow-start sampling of sixteen states cannot license "always 40,960" — and that precondition is load-bearing for exactly the claim in dispute (the pending run's fraction), so it outranks the secondary mechanism dispute, which is Lateral's naming of the invariant as a GF(2) kernel dimension (rank–nullity) versus Ledger's and Bound's position that only the identity bound is in force. Lateral's identification is testable and probably right as a name, but it derives no value and so does not settle the dispute.

FORCED
1. [object: exact one-block] At cut 90,112 the independently flippable count n obeys 0 ≤ n ≤ 45,056 with rest = 90,112 − 2n ≥ 0 — DERIVATION: each such vertex carries exactly two cut edges and no two are adjacent, so their cut-edge sets are disjoint; the remaining cut edges are a non-negative count.
2. [object: exact one-block] The count is not a function of (G_n, minimum cut) — DERIVATION: at n = 4, cut 176 carries both 8 (75 bisections, 14 orbits) and 0 (6 bisections, orbit of 512); a function of the graph alone is single-valued.
3. [object: exact one-block] The off-basin row of run 101 is internally inconsistent — DERIVATION: cut − rest must be even; 131,072 − 12,759 = 118,313.
4. [object: sampled lattice at the sizes measured] The level-5 invariance is not an artefact of reading one object sixteen times — DERIVATION: ten distinct independently-flippable sets, eight modulo the a-automorphism, pairwise overlaps varying by multiples of 1,024.
5. [object: sampled lattice at the sizes measured] FITTED — the law count = (k/32)·W_n with k ∈ {0, 1, 10}: k = 1 gives 8 at n = 4, k = 10 gives 40,960 at n = 5, and at n = 3, W_3/32 = 0.5 ∉ ℤ, leaving only 0 — fit on 3 points; W_n = |G_n|/2^n is computable from G_n without any bisection search.
6. [object: sampled lattice at the sizes measured] INHERITED FROM MEASUREMENT — 0 at n = 3 (exhaustive over 2,460), 8-or-0 at n = 4, 40,960 at n = 5.
7. [object: limit] MODEL ARTEFACT — the claim that the two accounts (fixed invariant / exactly one second kind) exhaust the possibilities; a third value, or several, is permitted by FORCED 1.

ASSERTED
1. [object: limit] The count is pinned on the plateau component the route reaches, not over all cut-90,112 bisections; the invariance of 40,960 against free-count spread of 143 is real structure within that component.
2. [object: limit] If a second level-5 kind exists at cut 90,112, its count is a multiple of 1,024 (the fibre granularity seen in the pairwise overlaps), and most plausibly 4,096 = W_5/32 — the level-4 value of k.
3. [object: sampled lattice at the sizes measured] The 183 pending starts, 108 of them far outside the 1% window, have materially higher power to reach a second basin than the original sixteen.
4. [object: limit] The minimum cut at n = 5 is 90,112; nothing in the record proves it.

ROAD NOT TAKEN
none — seats did not converge.

FORECASTS
F1.
RESTS ON: FORCED 2
SIGN: the fraction of 183-run endpoints at cut 90,112 with count exactly 40,960 is below 1 (p=0.58)
VALUE: 0.82 to 1.00, 80% interval
NULL: if the count is structurally fixed, the fraction is exactly 1.000, with zero exceptions on every endpoint.

F2.
RESTS ON: ASSERTED 2
SIGN: any second count observed is a multiple of 1,024 and lies below 40,960 (p=0.7 conditional on a second kind appearing)
VALUE: point 4,096; interval 0 to 20,480, 70% coverage
NULL: "no second kind" predicts the set of observed counts is the singleton {40,960}.

F3.
RESTS ON: none
SIGN: fewer than all 183 starts reach 90,112; the 108 far starts fail disproportionately (p=0.85)
VALUE: 70 to 165 endpoints at exactly 90,112, 80% interval
NULL: route-insensitivity predicts the success rate is flat in starting cut.

F4.
RESTS ON: FORCED 3
SIGN: recomputation changes the 12,759 or its attached cut (p=0.9)
VALUE: rest ∈ {12,758, 12,760} if the count is right, 80% interval
NULL: a correct record predicts the identity holds and the parity check passes as written.

TESTS
1. [lab] [cost: free — the state is stored in the run-101 seal] Recompute cut, count and rest on the off-basin bisection — predicts parity restored if a transcription error, predicts the definition of "rest" differs off-basin if both numbers stand. Resolves FORCED 3; the "direction an unconverged estimator reads" premise depends on this series.
2. [lab] [cost: cheap — 81 + 16 stored states, sparse GF(2) rank] Compute dim ker of the free-vertex constraint matrix on each stored bisection — predicts the dimension equals the reported count (8, 0, 40,960) if Lateral's rank–nullity naming is right, predicts a mismatch if the count is a local-pattern count with no linear-algebraic meaning.
3. [lab] [cost: cheap — 1,024 vertices, thousands of restarts in minutes] Re-search level 4 from 10,000 starts drawn from a different generator (random balanced, spectral, no coset-respecting bias) — predicts the 0-kind frequency moves far from 6/81 and possibly a count-80 (= 10·W_4/32) kind appears if the single-route premise fails; predicts frequencies stable and no new count if the route is neutral. Directly tests ledger premises "sixteen representative" and "single route does not restrict kind" at the level where exhaustive-ish search is affordable.
4. [lab] [cost: cheap — 183 result files already fixed] On the 183 run, report the full distribution of counts at 90,112, not the fraction equal to 40,960 — predicts a singleton under the structure account, predicts multiples of 1,024 under ASSERTED 2; also verify seeds are distinct and the route byte-identical (tests the "183 independent, same route" premise).
5. [lab] [cost: hours — long β=8 walk at fixed energy on 10^6 vertices] From one stored level-5 state, walk 10^7 equal-cut moves recording the count every sweep — predicts the count never leaves 40,960 if it is a component invariant, predicts drift if β=8 selection, not structure, pins it. Tests the walk-unbias premise.
6. [lab] [cost: hours — Kernighan–Lin / simulated annealing below 90,112] Attempt to beat 90,112 from the 108 far starts with slower cooling — predicts no improvement if 90,112 is the minimum; any lower cut voids every row (tests the load-bearing minimality premise).
7. [seat] [cost: cheap — analytic, no run] Task Hypothesis: state a bound on the β=8 walk's bias over the plateau, or declare none available — predicts the representativeness tool is repairable if a mixing bound exists, predicts the "always 40,960" claim stays unlicensed if not.
8. [seat] [cost: cheap — numerology check against W_n] Task Ledger: test count = (k/32)·W_n (fit on 3 points) against every stored state, including the refinement path and any new kind — predicts k ∈ {0,1,10} exactly; a count not of this form kills the law.
9. [lab] [cost: infeasible here — Q_6 has 2^40 vertices, and exhaustive enumeration of level-4 ground states needs C(1024,512) balanced partitions] The level-6 reading and the exhaustive level-4 ground-state set, which would settle the recursion and orbit questions outright, cannot be run.

RESIDUE
The sharper case drops "what fixes 40,960" and asks: on Q_4, where exhaustive-enough search is affordable, what is the full distribution of the independently flippable count over the whole cut-176 ground-state set, and does it take only values (k/32)·W_4 with k ∈ {0,1,10}? That is a question about an object (the ground-state set modulo Aut(Q_4)) that can actually be enumerated, it tests the fitted law at the one level where the law can be falsified cheaply, and it replaces a scalar with the thing the n = 4 data already shows the target to be: a distribution over orbits.

_Contract check: House adopted a flagged claim: H1-20 (from Hypothesis), H1-25 (from Lateral); House FORCED (J1): 5: no derivation sketch — it belongs under ASSERTED; 6: no derivation sketch — it belongs under ASSERTED; 7: no derivation sketch — it belongs under ASSERTED; House claims without an object (J2): VERDICT: 2 claim sentence(s) without an object tag, e.g. “At level 4 the same route found a minority kind at 6/81; sixteen draws miss a ki…”; House FORECASTS (F1/F3): F4: no VALUE interval._

BACKPROP (House, closed after House 1 — one short extra call)

open. Write back: FORCED 1–3 and the fitted law count=(k/32)·W_n (k∈{0,1,10}, fit on 3 points) as the edge into the ledger, flagged unverified; attach the parity correction owed to run 101's off-basin row (131,072−12,759 is odd) as a required recheck before that series backs any claim. Probe to carry forward: TEST 2 (GF(2) kernel-dimension computation on stored states) and TEST 3 (level-4 re-search from unbiased starts), since these are cheap and directly test the two live premises (rank–nullity naming, single-route bias).

Next case: RESIDUE — the full distribution of the independently-flippable count over Q_4's entire cut-176 ground-state set, tested against (k/32)·W_4.

Independence — record citations

• round 1: Ledger F01, F02, F03 · Framing F02, F03 · Hypothesis F01, F02, F03 · Lateral none · Bound none

Forecasts (parsed, for scoring)