Independent check of the cone certificate: (R⁺_O) and (S) confirmed with corrections for every separation, no gap; the closed form, Lemmas 1, A, 3, B and 5, and the evaluation at the all-z state all hold, and the periodisation onto any torus is sound

Verdict. Every s: CONFIRMED WITH CORRECTIONS (no gap; the corrections are about notation and bounds, not substance). The s that were computed: CONFIRMED (re-certified independently: s = 5 … 101 odd and gaps 0 … 10, every offset, both sublattice orders).

Everything was written from the definitions. Scripts are scratch_h/h320_lib.py and h321–h326. None of them imports h290–h299 or h310–h319. Nothing tracked was edited, nothing was committed, and everything ran on this Mac. SAT went through h32_cap.

Table

Corrections

C1. The orientation of the bond row is part of the closed form. The multiplier +¼·min(4, n_E) is correct only for the row written with the A-end first: d·n(a) − d·n(a + d), where a ∈ A and d ∈ {d₀, …, d₃}. All other rows are written as (left side − right side). With the opposite orientation the combination fails at every s: 6, 30 and 66 wrong-sign columns at s = 5, 7, 9. The report’s parenthesis in Lemma 3 (‘β = −¼, in the y-sign of the scripts’) points at this clash but does not resolve it. Repair: in §0 row 1 and §1, state ‘bond row d·n(a) − d·n(a+d) = 0 with a the A end’. In that convention the identity reads Σ λ_i(A_i x − b_i) = −1 − r·x with r ≥ 0, which is the convention of §1, and the four r-entries are the corner in-rods. That is all that changes.

C2. Lemma B (iii), bound. For A-vertices the proof ends ‘2(n_a − n_b) ≥ 8 − |a_w| − |b_w| ≥ 4’. Under the hypothesis |a_w| + |b_w| = 2·max(|dx|, |dy|) ≤ 2, the right side is ≥ 6. The stated ‘≥ 4’ is true, but it is not the bound the hypothesis gives. It is also what makes the A-vertex case (iii) impossible at (2,0), where |a_w| + |b_w| = 4. The conclusion is unaffected, because n_a − n_b ≤ 1. The B-vertex version, 2(n_a − n_b) ≤ |a_w| + |b_w| − 4, re-derived: at L_in, b = −2η(J+1); at U_ε, a = a_w − 2ε(n_a − J); then 4J ≤ 2n_b − 4 + |b_w| and 4J ≥ 2n_a − |a_w|. It is correct. It is attained at (2,0) exactly when n is even, and that holds for every admissible s at (2,0), since s ≡ 2 (mod 4). The sharpness remark should read ‘from the first separation’, not ‘s = 10’.

C3. §3.1 is written for h_v ∈ HA only. From a B-hole the first step changes b, so j_a(t) = ⌊t/2⌋ and j_b(t) = ⌈t/2⌉, and likewise for n_a and n_b. The HA formula applied to B-starts disagrees with the true D in 225 of 476 test pairs. The report reaches v ∈ B only through the rotary reflection (x,y,z) ↦ (y,−x,−z) plus the global flip of the state. That route is valid: the map fixes A, maps DIRS to DIRS, acts on n as a vector (so every row is preserved), and the flip preserves orientation in real states (law C). So this is a scope note, not a gap. Lemma B and the certificate were also checked directly for v ∈ B.

Remarks (no change needed). At each s only two of the four diagonal offsets are admissible for a given sublattice of v (dx + dy + s ≡ 3 mod 4). The ‘705 925 rows at s = 101’ counts box rows; the certificate’s support is 136 537 rows (|D| = 21 450 agrees). The global flip does not map the LP to itself (‘one away’ becomes ‘one toward’), so it must be applied to states, as the report does, not to the certificate.

Periodisation (task 4)

Let a state on the (L,L,L) torus have all holes of the projected D oriented, and let v, w be as in the statement with some lift w̃ − v = (dx, dy, s), s > 0. Lift the state to a periodic state x̃ of the lattice. Every lattice row whose projection is a torus row holds at x̃: one-hot, (T2), bond, and Lemma O at the lifts of the projected holes of D. The hypotheses hold at v and w̃. The identity is a finite formal identity, so evaluating it at x̃ gives 0 = −1 − r·x̃ ≤ −1.

Distinct lattice rows with the same projection cause no problem. Every row is an equality, so multipliers of any sign may be summed. The projected combination has torus columns that are sums of lattice columns of one sign, and the same constant. Projected rows are never degenerate for L ≥ 2 (4L ≥ 8): the six hexagon vertices are distinct, and so are the six methylenes of a hole. v ≠ w on the torus except in the same-line case with s ≡ 0 (mod 4L), where the hypotheses already contradict one-hot. So D may wrap round the torus any number of times. The only input needed is orientation of the projected holes of D, and in an odd state that is every hole. Every z-difference mod 4L has a positive lift, and every transverse offset (±1,±1) mod 4L lifts to exactly (±1,±1). So the torus statement covers every pair.

Use in rphand (task 5)

• Stripe theorem (Theorem W, case 1). Needs (S) on the lines of one sublattice: two type-k vertices of one k-line, any gap, either sublattice, in an odd state. That is the offset-(0,0) certificate, every gap (row 2, and C3 for B), with Lemma O at every hole.
• Case 1 (i), (ii) and case 2. Need (R⁺) between a type-k vertex of an S′-line and a type-k vertex of a diagonally adjacent S-line: offset (±1,±1), any separation on the torus, either vertex lower, either sublattice lower. These are the four cases certified. Separation ±1 is the bond rule, ±3 is conservation, |s| ≥ 5 is the cone. Axes x and y follow by the cyclic permutation of axes, which fixes A and preserves the rows.
• RP₁ steps 2–3. Same pairs. RP₁ is not on the path of the Corollary.

So (R⁺_O) as proved is exactly what is used, with Lemma O supplied at every hole by oddness. The chain H ⇐ K_u + Lemma O ⇐ K_u + Theorem U needs no further input beyond what was already recorded as proved and independently checked: Lemma sP (cage check), LC/sLC, the two-product lemma, Lemma E, Theorem D, and corner-hole orientation. K_u keeps its certificate scope (cells with a ≥ 3 and b, c ≥ 4, and the small cells by torus verdicts).

Independent computations (this Mac, wall clock)

Logs: .tmp/h322_R_1_41.log, .tmp/h322_R_43_101.log, .tmp/h322_S.log, .tmp/h324_B_125.log. The planned brute force of Lemma B over every s ≤ 201 (about 18 minutes projected) was cut to every s ≤ 125 plus s = 199 … 201, to stay under the ten-minute foreground limit.