The minimum depth of open-boundary Yang–Baxter integrable circuits: an exact formula and two errata for arXiv:2607.02093
Abstract
An open-boundary integrable circuit in the construction of Garc'ia Fernández, Paletta and Retore (arXiv:2607.02093) is a fixed word in two-site gates and two boundary gates, determined by the set vec n ⊆ {1,…,N} of sites carrying the inhomogeneity -κ. They ask for the minimum depth of the circuit over all vec n with |vec n|=κ₋ and conjecture 1/2(N+3)-κ₋ for odd N and 1/2(N+4)-κ₋ for even N. Both conjectures are false: at N=8, κ₋=2 the placement vec n=(6,3) has depth 3 against the conjectured 4, and at N=11, κ₋=2 the placement vec n=(8,4) has depth 4 against 5; N=8 lies inside the range N=2,…,10 the source says it studied, and both lie well inside the N ≤ 51 (odd), N ≤ 50 (even) to which it reports having checked the conjectures. The minimum is ⌈ (N+1)/(min(κ₋,N-κ₋)+1)⌉ for every N ≥ 2 and 0 ≤ κ₋ ≤ N: the lower bound is a chain argument on the word, the upper bound an explicit placement whose +κ sites are spread over κ₋+1 staircases rather than the two of the conjectured placements. The minimum was also computed exhaustively over all 2^(N) placements for 2 ≤ N ≤ 19, and the explicit placement was checked to attain the formula for every N ≤ 120. For the circuits with inhomogeneities +κ and ρ only, the inequality d ≥ N+1 that the source verified exhaustively for N=5,6,7,9 and by sampling for N=11,13,15 has a one-line proof for every N, and the minimum is verified here to be exactly N+1, exhaustively over all 2^(N-1) placements, for every 3 ≤ N ≤ 20; a product in the source's Eq. (73) is printed in the wrong order. Every finite statement here — the depth semantics, the tables, the two counterexamples, the placements to N=120, the (+κ,ρ) sweeps and the product-order check — is verified in Lean 4 without Mathlib; the closed form for all N is proved on paper.
Open review
This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.
Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-09-07 03:53 UTC
File fingerprint
0a0c6cd257a130bbb9c5f80ad9dde782137cfc500d4c0e0a9fa90c14639b35d6
Claim ledger
Stated results
YB1routine2026-09-07
Circuit depth is the height of the heap of pieces and equals the minimum number of gate layers: the ASAP fold is a layering of the word, and no layering uses fewer layers. Kernel-clean, no native_decide.
YB2known data2026-09-07
Exhaustive minimum circuit depth over all 2^N placements n-vec in 1,...,N, for 2 <= N <= 10 and every kappa₋; agrees with the source's Conjecture 1 on all odd N <= 9 and with Conjecture 2 for N = 2, 4, 6.
YB3candidate2026-09-07
Exhaustive minimum circuit depth for 11 <= N <= 16 over all 2^N placements, equal to ceil((N+1)/(min(kappa₋, N-kappa₋)+1)) at every kappa₋.
YB4candidate2026-09-07
Exhaustive minimum circuit depth for 17 <= N <= 19 (2¹9 = 524288 placements at N = 19), equal to the same closed form.
YB5correction2026-09-07
Conjecture 1 of arXiv:2607.02093v1 (odd N, d = (N+3)/2 - kappa₋) is FALSE. First counterexample N = 11, kappa₋ = 2, n-vec = (8,4), depth 4 against the conjectured 5, with its explicit four-layer schedule and the exhaustive proof that three layers are impossible; the failure recurs at every odd N in 11..19.
YB6correction2026-09-07
Conjecture 2 of arXiv:2607.02093v1 (even N, d = (N+4)/2 - kappa₋) is FALSE. First counterexample N = 8, kappa₋ = 2, n-vec = (6,3), depth 3 against the conjectured 4 – inside the range N = 2,...,10 the source names – with its explicit three-layer schedule; the failure recurs at every even N in 8..18.
YB7candidate2026-09-07
For the (+kappa, rho) circuits the minimum depth over all 2^(N-1) placements n-vecʳho in 1,...,N-1 is exactly N+1, for every 3 <= N <= 20; this settles exhaustively the three sizes N = 11, 13, 15 for which the source sampled only.
YB8candidate2026-09-07
An explicit placement attaining ceil((N+1)/(min(kappa₋, N-kappa₋)+1)) for every N <= 120 and every kappa₋ <= N: kappa₋+1 staircases with gap profile (g₀,...,g_(kappa_–1), d-2), replacing the source's Eqs. (61) and (63).
YB9routine2026-09-07
Controls for the model and the sweeps: every (kappa,-kappa) word has exactly N+1 gates; the maximum depth is N+1; the sign-flip duality kappa₋ <-> N-kappa₋ holds configuration by configuration; the kappa₋ = 0 staircase has depth N+1; no table entry is the sentinel; the too-small claims are refuted at N = 8 and N = 11 and the witnesses exhibited; the twelve gates of the N = 11 witness cannot share one layer.
YB10prose2026-09-07
THEOREM (proved in the record, not yet formalized): for every N >= 2 and every 0 <= kappa₋ <= N, the minimum depth over all placements with |n-vec| = kappa₋ is exactly ceil((N+1)/(min(kappa₋, N-kappa₋)+1)).
This ledger entry is reported in prose and is not bound to a Lean theorem.YB11correction2026-09-07
The source's Eq. (73) prints the left product of the (+kappa, rho) circuit as a rightward (increasing) product where its own Lemma 1 (Eq. (35)) and Theorem 1 (Eq. (42)) give a leftward (decreasing) one; the two are not interchangeable, and only the leftward reading reproduces the source's own d = 2N value in its Fig. 14.
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. *Open-boundary integrable quantum circuits with different geometries*, arXiv:2607.02093 v1 (submitted 2 Jul 2026, single version as of 2026-09-07; primary math-ph, cross-listed cond-mat.stat-mech, hep-th, nlin.SI, quant-ph). All section, equation and figure numbers below are read off the compiled PDF of v1 (arxiv.org/pdf/2607.02093v1), not the raw LaTeX.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7