Verdict

CONFIRMED. The construction is correct as the main loop reconstructed it (docs/reports/2026-10-01-kucone-partial.md). C1 holds by hand for general \(d \geq 2\), with the two symmetries. C2: my own code (h720, which imports nothing of kucone’s) reproduces the identity exactly for \(d = \pm 2\) to \(\pm 12\) in both slabs, and its multipliers are identical, row for row, to kucone’s on every sampled \(d\). C4 is written below. Remarks, none of which changes the statement:

1. The proviso in the reconstruction (“provided the star of \(t\) meets the first diamond only in its apex hole and the second only in its corner hole”) is true but not needed. Neither the identity nor the hand proof uses it, and the same goes for disjointness of the two diamonds (proved, below).
2. Only two vertices of the uniform row are used, \(v\) and \(v'\) (proved; the certificate has exactly two hypD rows).
3. h656 delete is settled (computed). Orientation on the holes of \(D_a \cup D_b\) alone is enough at every \(d = 2\) to \(6\) (slab up). It is not minimal. At \(d = 2\) the window system is infeasible with no orientation row at all. For \(d \geq 3\) some single deletions keep it infeasible: 2 of 4 holes at \(d = 3\), 6 of 12 at \(d = 4\), 12 of 28 at \(d = 5\) and 28 of 52 at \(d = 6\).
4. The cases \(d \in \{0, \pm 1\}\), which the construction leaves out, follow directly from \(D\) and the bond rule (proved, below), so the lemma covers every column vertex of both slabs.

C1. The construction, by hand

Coordinates (from h650, checked against the note). \(D\) is the row \(\{(0, 2j, 2j)\} \cup \{(1, 2j+1, 2j+1)\}\), a chain of axis \(x\) (its two bonds at \((0,2j,2j)\) are \(d_0\) and \(d_1\), which agree in \(x\)), with every vertex of moment \(+y\). Write \(u = (y+z)/2\) and \(w = (y-z)/2\). A column (family \(\{2,3\}\), bonds \(d_2\) and \(d_3\) at A) has fixed \(u\) and runs along \(w\). Slab up is the pair of grids \(x = 1\) (B) and \(x = 2\) (A), and its columns sit at odd \(u\). The translation by \((0,2,2) \in A\) preserves both sublattices and \(D\), so the column at \(u = 1\) is general. The target is \(t = (x_t, 1+d, 1-d)\), with \(x_t = 1\) (B) for even \(d\) and \(2\) (A) for odd \(d\).

The sense table (proved). At \(t\) the column bonds are \(\pm(1,-1,1)\) and \(\pm(1,1,-1)\) (B case) or \(d_2\) and \(d_3\) (A case). Sense \(-w\) means that the bond lowering \(w\) is the outgoing one, so the moment \(n = \tfrac12 \sum(\text{outgoing})\) is that bond plus one of the two non-column bonds. In both cases this gives \(n(t) \in \{+z, -y\}\), which is kucone’s NEG row \(p_z(t) + m_y(t) = 1\).

The diamonds, \(d \geq 3\). \(v = (x_t - 1, 2-d, 2-d)\), \(v' = (1,1,1)\), \(w' = (2x_t - 2, 2d, 0)\).

• \(v \in D\): an A vertex \((0, 2j, 2j)\) for even \(d\) and a B vertex \((1, 2j+1, 2j+1)\) for odd \(d\). \(v' \in D\) is a B vertex. \(w'\) is an A vertex (sum \(2d\) or \(2 + 2d\), \(\equiv 0 \bmod 4\) for the right parity of \(d\)). No hypothesis is placed on \(w'\).
• \(t - v = (1, 2d-1, -1)\) and \(w' - v' = (2x_t - 3, 2d-1, -1)\). Both pairs have separation \(s = 2d - 1 \geq 5\), odd, along \(y\), with transverse offsets \((\pm1, \pm1)\) in \((x, z)\). That is exactly the scope of Theorem 9 and Theorem 3 (offset one, \(s \geq 5\)).
• \(D_a\) is the causal \(y\)-diamond from \(h_v = v + 2e_y\) to \(t - 2e_y\), and \(D_b\) the one from \(v' + 2e_y = (1,3,1)\) to \(w' - 2e_y\). Each has \(n = 2d - 5\) steps.

\(d = 2\). \(v = (0,-2,-2)\) and \(w' = (0,6,2)\), so \(t - v = (1,5,1)\) and \(w' - v' = (-1,5,1)\), both with \(s = 5\). The general formula would give \(s = 3\), outside Theorem 9. Here \(D_b = \{(1,3,1), (0,4,2)\}\), and its \(-z\) extreme is its lower apex \((1,3,1) = t + 2e_z\), one hole with two corner roles (see C4).

Structure of a \(y\)-diamond (proved). A step from an A-hole uses \(\tau \in \{\pm1\}^3\) with \(\tau_x\tau_y\tau_z = +1\), and from a B-hole one with product \(-1\). Steps alternate between hole types. With \(\tau_y = +1\), an A-step changes \(a = x + z\) by \(\pm 2\) and fixes \(b = x - z\), and a B-step does the reverse. So the holes of a diamond at level \(j\) form a product of an interval in \(a\) and an interval in \(b\). The bounds are \(|a - a_{lo}| \leq 2\alpha(j)\) and \(|a_{hi} - a| \leq 2(\alpha_{tot} - \alpha(j))\), and the same for \(b\) with \(\beta\), where \(\alpha(j)\) and \(\beta(j)\) count the A-steps and B-steps among the first \(j\).

Fact 1: the \(-z\) extreme of \(D_b\) is unique and equals \(c = t + 2e_z\) (proved). \(D_b\) starts at the B-hole \((1,3,1)\), so \(a_{lo} = 2\) and \(b_{lo} = 0\), with \(\beta_{tot} = d - 2\) and \(\alpha_{tot} = d - 3\). The smallest \(z = (a - b)/2\) needs the least \(a\) and the greatest \(b\) on one level.

• Even \(d\) (\(a_{hi} = b_{hi} = 0\)). The least \(a\) over all levels is \(4 - d\), attained only at \(\alpha = d/2 - 1\). The greatest \(b\) is \(d - 2\), attained only at \(\beta = d/2 - 1\).
• Odd \(d\) (\(a_{hi} = b_{hi} = 2\)). The least \(a\) is \(5 - d\), only at \(\alpha = (d-3)/2\). The greatest \(b\) is \(d - 1\), only at \(\beta = (d-1)/2\).

In both cases the two are attained together only at level \(j = d - 2\) (which is at most \(n\) exactly when \(d \geq 3\)). That gives the unique hole \((x_t, d+1, 3-d)\), which is \(t + 2e_z\), a B-hole for even \(d\) and an A-hole for odd \(d\), as it must be.

Fact 2: the star of \(t\) (proved, not needed). \(D_a\) lies in the past of \(t - 2e_y\), so every hole of \(D_a\) has \(y \leq d - 1\). Every hole of the star of \(t\) other than \(t - 2e_y\) has \(y \geq d\). Every hole of \(D_b\) has \(z \geq 3 - d\). Every hole of the star other than \(t + 2e_z\) has \(z \leq 2 - d\).

Fact 3: \(D_a \cap D_b = \emptyset\) (proved, not needed). A common hole would put \(t - 2e_y\) in the causal future of \((1,3,1)\). That needs \(|\Delta z| \leq \Delta y\), but \(\Delta z = -d\) and \(\Delta y = d - 4\).

Theorem 9 without the hypothesis on the upper vertex (proved from the note). Theorem 9 is an identity in the variables: \(\Lambda + (p(v) - 1) + (m(w) - 1) = -1 - \sum_4 \text{lateral corners}\), where \(\Lambda\) is the combination of one-hot, (T2), bond and orientation rows. The hypotheses enter only as the two added rows, and the proof uses them only to cancel \(p_z(v)\) and to fix the constant. Moving them across gives \(\Lambda = 1 - \sum_6 \text{corners}\), an identity of the relaxation. Every state satisfying the rows of \(\Lambda\) then has exactly one corner variable equal to 1, which is the note’s corner corollary. So using the identity without the hypothesis on \(w\) is legitimate. h720 checks \(\Lambda = 1 - \sum_6\) directly for every diamond it builds, \(s = 5\) included.

The sum (proved). With \(\Lambda_a = 1 - p_y(v) - m_y(t) - C_a\) and \(\Lambda_b = 1 - p_y(v') - m_y(w') - p_z(t) - C'_b\) (where \(C_a\) is the four lateral corners of \(D_a\), \(C'_b\) the three lateral corners of \(D_b\) other than \(c\), and \(p_z(t)\) the \(-z\) corner of \(D_b\) by Fact 1): \[\Lambda_a + (p_y(v)-1) + \Lambda_b + (p_y(v')-1) + (p_z(t)+m_y(t)-1) = -1 - C_a - C'_b - m_y(w').\] That is \(-1\) minus eight variables, which matches the “slack 8” of every h655 row.

Symmetries (proved).

• Slab down: \(M(x,y,z) = (1-x, y-1, z-1)\). It exchanges the sublattices, maps \(D\) onto \(D\), preserves \(w\), sends \(u \mapsto u - 1\) and slab up to slab down, and acts on moments by \(n_x \mapsto -n_x\). So \(+y\) on \(D\) and the set \(\{+z, -y\}\) are invariant, and every row type maps to its own type.
• \(d < 0\): \(R(x,y,z) = (x, 2-y, 2-z)\) followed by the global reversal. \(R\) preserves the sublattices and \(D\), maps \(u = 1\) to itself and \(w\) to \(-w\), and the moment of \(R\) composed with reversal fixes \(+y\) and the set \(\{+z, -y\}\). The reversal takes “out” orientation rows to “in” rows. h654/h720 rewrite these by Law C as \(\mathrm{in}(h) = \tfrac12\sum_{4\,\mathrm{hex}} T2 - \tfrac12\sum_{\mathrm{bridgeheads}} \text{one-hot} - \mathrm{out}(h)\), an identity checked exactly in h720. So orientation in the note’s form is all that is used.

C2. Own-code exact verification (computed)

h720_kucheck.py (own lattice, hexagons from the note’s \((O, \sigma)\) definition, diamonds by forward and backward reachability, Theorem 9’s multipliers from the note’s statement, \(\nu\) counted from the bonds of the hexagons with both holes in \(D\)) checks, in Fraction arithmetic on the infinite lattice, for \(d = \pm 2\) to \(\pm 12\) in both slabs (44 rows):

• 1. for each diamond alone, \(\Lambda = 1 - \sum\) of its six corner variables exactly, with the corners computed from the geometry and their extremes asserted unique;
• the roles: \(p_y(v)\) and \(m_y(t)\) are corners of \(D_a\), and \(p_y(v')\) and \(p_z(t)\) are corners of \(D_b\);
• (ii) the \(K_u\) combination is \(-1\) minus a non-negative combination with exactly 8 negative coefficients;
• (iii) Facts 1 to 3.

All 44 pass, along with the Law C substitution (log .tmp/h720_2_12.log, 3.5 s). Row counts and \(L^1\) norms agree with kucone’s logs. For example, \(d = 2\) has 48 rows and \(L^1\) 25, \(d = 12\) has 3,832 and 2,676, and \(d = -12\) has 3,239 and 2,284.

h721_compare.py rebuilds kucone’s \(\lambda\) through h655_verify.construct and h654_family.substitute_Or, then compares dictionaries. At \(d = 2, 3, 5, 8, 12, -2, -3, -7, -12\), both slabs (18 rows), the multipliers are identical and kucone’s check is exact (log .tmp/h721.log, 2 s). A direct re-run of h655_verify.py 2,7,12,-5,-12 gave ALL EXACT on 10 rows (.tmp/h655_kucheck_sample.log, 1.7 s). The 30-row sweep was not re-run.

C3. h656 delete settled (computed, LP, floating point)

h722_minimal.py is my own version, run through the timeout wrapper h723_run.py. The window is kuwall’s cone about \(t\) (grids \(-(d+1)\) to \(d+3\), sup-radius \(d\) in \((u,w)\)), joined with every vertex of every hexagon and every methylene of the holes of \(D_a \cup D_b\), so that each diamond hole is complete. This matters: h656 used the bare cone, where some diamond holes may be incomplete. The rows are one-hot, (T2) and bond everywhere in the window, \(p_y = 1\) on every \(D\) vertex of the window, NEG, and orientation only at the holes of \(D_a \cup D_b\). Slab up:

With the window padded by 2 the pattern for \(d = 2, 3, 4\) is unchanged.

• The holes that can go. At \(d = 3\) they are the two lower apex holes \((1,1,-1)\) and \((1,3,1)\), the holes above \(v\) and \(v'\). At \(d = 4\) they are those two, plus the apex \(t - 2e_y\), the corner \(t + 2e_z\), and \((1,1,-1)\) and \((0,4,2)\).
• Greedy irreducible subsets (h724_greedy.py, order-dependent, not minimum): 0 of 4 at \(d = 2\), 2 of 4 at \(d = 3\) (keeping \(t - 2e_y\) and \(t + 2e_z\)), 7 of 12 at \(d = 4\), 19 of 28 at \(d = 5\).
• One diamond is not enough: for \(d = 3\) to \(5\) it is feasible with orientation on \(D_a\) alone, \(D_b\) alone, or none.

So the answer to h656’s question:

• Restricting orientation to the two diamonds suffices (computed here, and proved by the certificate, whose rows lie in that set).
• It is not minimal. The extra rows of the window (the whole row \(D\), and the (T2) and bond rows round the diamonds) make some diamond holes redundant, unlike Theorem 9 alone, where the note records every hole as needed.
• At \(d = 2\), orientation is not needed at all in the relaxation. This agrees with kuwall’s and check6’s “reach 2 with W0 alone”.

These are finite-window LP statements: feasibility after a deletion shows only that no certificate exists in that window.

C4. The hand lemma

Lemma (first step of \(K_u\) at one vertex). Let a frozen state on the lattice, or the lift of one on any cell, have two vertices \(v, v' \in D\) of moment \(+y\), with \(v\) and \(v'\) as above for the target \(t = (x_t, 1+d, 1-d)\), \(d \geq 2\), and suppose every hole of the causal diamonds \(D_a\) (from \(v + 2e_y\) to \(t - 2e_y\)) and \(D_b\) (from \(v' + 2e_y\) to \(w' - 2e_y\)) is oriented. Then \(n(t) \notin \{+z, -y\}\): \(t\) is a kink or a \(+\) pass-through of its column. The same holds in slab down (by \(M\)) and for \(d \leq -2\) (by \(R\) and the global reversal, with the diamonds above \(t\)).

Proof.

1. \(t\) does not have moment \(-y\). If \(t\) had type \(y\), then \(v\) and \(t\), of type \(y\) on diagonally neighbouring \(y\)-lines with separation \(s = 2d - 1 \geq 5\) (5 at \(d = 2\)), would have equal signs by Theorem 3, whose diamond is \(D_a\). So \(\varepsilon(t) = +1\). Equivalently, apply the corner law to \(D_a\): its lower-apex corner holds (\(p_y(v) = 1\)), so its upper-apex corner \(m_y(t)\) fails.
2. \(t\) does not have moment \(+z\). Apply the corner corollary of Theorem 9 to \(D_b\). Exactly one of its six corner holes has its inward rod on its own axis on the outer side. The lower apex does, since \(v'\) has moment \(+y\) and points into \(v' + 2e_y\). The \(-z\) corner of \(D_b\) is the unique hole \(c = t + 2e_z\) (Fact 1), and its inward rod on the outer side is \(t\) with moment \(+z\). So \(p_z(t) = 0\).

For \(d = 2\) (and \(d = 3\) at the upper apex) \(c\) is also an apex of \(D_b\). The corner law is still a statement about six distinct variables, and at \(d = 2\) orientation of the single hole \(c\) already forbids two inward rods, \(v'\) and \(t\). \(\square\)

The remaining distances (proved). At \(d = 0\) the target is \(v'\) itself (slab up) or \((0,0,0)\) (slab down), and is a \(+\) pass-through. At \(d = \pm 1\) the target is bonded to a vertex of \(D\):

• slab up: \((2,2,0) = (1,1,1) + (1,1,-1)\), bond \(d_3\); \((2,0,2)\), bond \(d_2\);
• slab down: \((-1,1,-1)\) (\(d = 1\)) and \((-1,-1,1)\) (\(d = -1\)) to \((0,0,0)\), bonds \(d_2\) and \(d_3\).

The bond rule \(d \cdot n(a) = d \cdot n(b)\) with \(n = +y\) at the \(D\) end gives \(d_3 \cdot n(t) = -1\) or \(d_2 \cdot n(t) = +1\). Both \(+z\) and \(-y\) give the opposite value.

Hypotheses used, and their availability.

On a cell: the state lifts to a periodic frozen state on the lattice. Every lifted hole is oriented, and the uniform chain lifts to \(D\) and its translates. Any column vertex of the two slabs has a lift at some integer \(d\), so the lemma applies with no condition on the side or on whether the diamonds wrap. With check6’s later steps, \(K_u\) given Theorem U then holds for odd frozen states on every cell. That combination is not re-checked here (out of scope).

What a reader still has to check. The coordinates of \(v\), \(v'\), \(w'\) and the two separations; the \(a/b\) product structure of a \(y\)-diamond and Fact 1 (the one non-trivial geometric fact); the identification of the corner variables \(m_y(t)\) and \(p_z(t)\); the sense table; the symmetries \(M\) and \(R\) with reversal; and that Theorem 9 and its corollary carry over to \(y\)-diamonds and a B lower vertex by the cyclic permutation of axes and the inversion, both of which preserve the system. Theorem 9 itself is not re-proved here.

Scripts and timings

All in experiments/08_frozen_structure/scratch_h/, Mac .venv, about 2.5 min of compute in all. h720_kucheck.py 2:12 (own exact check, 44 rows, 3.5 s, .tmp/h720_2_12.log, multipliers in .tmp/h720_lam_*.pkl); h721_compare.py (identical to kucone, 2 s, .tmp/h721.log); h655_verify.py 2,7,12,-5,-12 re-run (1.7 s, .tmp/h655_kucheck_sample.log); h722_minimal.py via h723_run.py (\(d = 2,3,4\) 2.4 s; \(d = 5,6\) 103 s; padded \(d = 2,3,4\) about 20 s; .tmp/h722.log); h724_greedy.py via h723 (\(d = 2,3,4\) 2.4 s; \(d = 5\) 13 s; .tmp/h724.log).

Rules kept or broken

Kept: Mac only, no Sparks; one LP process at a time, under subprocess.run(timeout=...) (h723), with h720 and h721 pure arithmetic of seconds in the foreground; no job over ten minutes; new scripts h720 to h724 only, each with its header and usage line, no existing script modified; temporary files in .tmp/; no git commits; the 30-row sweep not re-run; later steps of \(K_u\), Theorem H and Theorem U not touched. Minor: scripts written with zsh heredocs, which worked in this sandbox.