slug: pronko-2025-fredkin-xi-hamiltonian-commutation bibkey: pronko2025fredkin doi: 10.1088/1751-8121/ae1644 url: https://doi.org/10.1088/1751-8121/ae1644 triage: theorem motivation_gids:

• D5/S3/Quantum/Dynamics/PronkoFredkinXiCommutes.result

Pronko’s commutation of the Dyck-class operator Ξ with the periodic Fredkin Hamiltonian

Problem

Pronko reads spin words as lattice paths in section 2.2, printed p. 4:

A Dyck path of length N starts at (x,y)=(a,0) and ends at (x,y)=(b,N), at each step Δx = 1 and Δy = ±1, with the condition that at least once the path hits the x-axis, i.e., min y = 0 along the path. … we will call all such paths as belonging to the equivalence class C_{a,b}(N).

A letter ↑ is a step up and a letter ↓ a step down. Section 4.1, printed pp. 9-10, defines for even N

Ξ = Σ_{k=0}^{N/2} (−1)^k Σ_{ℓ₁ℓ₂…ℓ_N ∈ C_{k,k}(N)} n_1^{ℓ₁} n_2^{ℓ₂} ⋯ n_N^{ℓ_N}, (4.1)

with n^{ℓ} the spin-up and spin-down projections, and states on printed p. 10:

It commutes with the Hamiltonian and anti-commutes with the cyclic shift operator: [Ξ, H] = 0, C Ξ C⁻¹ = −Ξ. The first relation may require a proof though we find it satisfied in all checks; the second one is obvious from (4.1).

H is the periodic Fredkin Hamiltonian (2.3) with density (2.2), F_{j,j+1,j+2} = n↑_j Π_{j+1,j+2} + Π_{j,j+1} n↓_{j+2}, sites modulo N.

Issue #9996 fixes the readings: (i) ℓ ∈ C_{a,b}(N) when the path started at height a stays at height ≥ 0, touches 0, and ends at b; (ii) the projections and H are those of the frozen module D5/S3/Quantum/Dynamics/PronkoFredkinNonCyclicAnnihilation; (iii) N is even and N ≥ 3; (iv) only the first relation [Ξ, H] = 0 is settled.

Motivation

The frozen declaration D5/S3/Quantum/Dynamics/PronkoFredkinXiCommutes.result proves [Ξ, H] = 0 for every even N ≥ 3. The paper uses this relation, with C Ξ C⁻¹ = −Ξ, to explain the double degeneracy it observes numerically in the S^z = 0 sector of the spectrum.

Gap

Issue #9996 preregisters this printed, explicitly unproven relation and its literature check. Crossref reports one citing work; Semantic Scholar lists exactly one, arXiv:2509.04838, whose text does not discuss Ξ or the spectral doubling. MathDB has entries for Conjectures 1-4 of the paper (/p/369467-/p/369470) and none for this relation.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent proof.

Route

Ξ is diagonal. A word is balanced when it has as many up letters as down letters; a balanced word lies in exactly one class C_{a,a}(N), with a = −min_{0 ≤ i ≤ N} s_i for the prefix heights s_i, and Ξ takes the value (−1)^a there; Ξ vanishes on unbalanced words. The off-diagonal part of F_{j,j+1,j+2} exchanges the letters at j + 1, j + 2 when the letter at j is up, or at j, j + 1 when the letter at j + 2 is down, indices modulo N. An exchange keeps the number of up letters. An exchange that does not wrap changes one prefix height by two, and the control letter forces the smaller value to occur at another index, so a is unchanged. An exchange of the last and the first letter moves every interior height by the same amount, and the control letter forces an interior height ≤ 0 in both words, so a changes by zero or two. Hence Ξ commutes with every density and with H. The argument uses only N ≥ 3; for odd N there are no balanced words.

Falsifier

A refutation would be an even N ≥ 3 and two basis words joined by a nonzero off-diagonal entry of H whose depths a have different parities. The depth-parity invariance proved for every periodic Fredkin exchange rules this out.

Evidence

H was built term by term from (2.2) and (2.3), and Ξ from (4.1) by class membership, for N = 2, 4, …, 12. For N = 4..12 the largest entry of [Ξ, H] is zero and no nonzero off-diagonal entry of H joins balanced words of different depth parity. For N = 2 the three-site terms overlap and max|[Ξ, H]| = 2; that case is outside the statement. The Lean proof has only the standard axiom closure propext, Classical.choice, and Quot.sound. These finite checks support the reading but do not establish the universal theorem.

Triage

A printed relation stated as needing proof in Pronko, Journal of Physics A: Mathematical and Theoretical 58 (2025) 445204, section 4.1, preregistered in issue #9996. Resolution: proved.

The public surface is exactly pathHeight, inClass, Xi, claim, and result. This is a uniform theorem, not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies. The relation C Ξ C⁻¹ = −Ξ and the spectral doubling are not asserted.

ASSUMED-UNVERIFIED

OpenAlex was rate limited and is ASSUMED-UNVERIFIED. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The Lean kernel does not authenticate the external PDF, its printed pagination, the literature-check coverage, or publication history. The finite checks for small N do not establish the universal theorem.