slug: posske-2026-a392387-xx-chain-nullspace-2p bibkey: posske2026a392387 doi: null url: https://oeis.org/A392387 triage: theorem motivation_gids:
- D5/S3/Quantum/SpinChains/XXChainNullspaceCount.result
The nullspace of the periodic XX chain on twice a prime number of sites
Problem
OEIS A392387 (Thore Posske, 2026) is the nullspace dimension of the periodic
spin-1/2 XX Heisenberg chain on n > 1 sites. Its COMMENTS give the count
Also, the number of subsets K of {1,…,n} such that the sum of cosines of the angles in {(2j + (1 + (-1)^|K|)/2 )*Pi/n | j in K} is zero.
and its FORMULA section records
Conjecture: a(2p) = 2(6^((p-1)/2)+1) for odd prime p.
Issue #10063 fixes the readings: the formal object is this subset count, the
combinatorial description common to the entry and to Hu, Gerken and Posske
(arXiv:2602.15098, appendix on the XX model); its identification with the
dimension of the zero-energy subspace is the source’s Jordan–Wigner reading and
is not formalized; the empty set is counted; p ranges over odd primes and
n = 2p.
Motivation
The appendix of arXiv:2602.15098 rephrases the nullspace degeneracy of the XX
chain as the number of choices of distinct momenta with vanishing total energy
and calls the counting of such vanishing sums of roots of unity “an unsolved
mathematical problem”. The frozen declaration
D5/S3/Quantum/SpinChains/XXChainNullspaceCount.result settles it for chains
of 2p sites: the zero-energy subspace has dimension 2(6^((p−1)/2) + 1) for
every odd prime p, two states from odd fermion number and 2·6^((p−1)/2)
from even fermion number.
Gap
Issue #10063 preregisters the conjecture and its literature check. The entry
(revision 52, 2026-02-20) lists this formula as a conjecture next to the proved
characterization a(p) = 2 exactly for primes p; the source paper gives only
a brute-force table for n ≤ 22; MathDB returns no entry for “A392387”, and
the repository had no declaration or dossier for it. These readings are
not-found-in-searched-scope; they do not establish an exhaustive worldwide
literature search or priority.
Route
Write p = 2m + 1. A subset K ⊆ {1, …, 2p} is determined by its class-state
function on the residues r modulo p: which of the two elements of
{1, …, 2p} congruent to r, one even and one odd, lie in K. Let z(r) be
the number of taken even elements minus the number of taken odd elements of
the class r, η the primitive p-th root of unity with −η = e^{iπ/p}, and
ω = e^{iπ/(2p)}. The cosine sum of K is the real part of
Z = Σ_r z(r) η^r when |K| is odd and of ωZ when |K| is even.
The only rational linear relation among 1, η, …, η^{p−1} is that their sum
vanishes, because the cyclotomic polynomial Φ_p is the minimal polynomial of
η. Hence Re Z = 0 exactly when z(r) + z(−r) = 2z(0) for all r, and
Re(ωZ) = 0 exactly when z(−r) − z(r − 1) = z(0) − z(−1) for all r; since
r ↦ −1 − r fixes m, the latter is the symmetry z(r) = z(−1 − r).
For odd |K| the first relation forces z to be constantly 1 or constantly
−1 (if z(0) = 0, then z is odd and |K| is even): the even elements and
the odd elements of {1, …, 2p}, two subsets. For even |K|, count the
symmetric class-state functions over the fundamental domain 0, …, m − 1 of
r ↦ −1 − r: each of the m pairs of classes has 6 states with equal z,
and the parity of |K| forces the fixed class m to be empty or full, 2
states. The total is 2 + 2·6^m.
Falsifier
A zero-sum subset outside the two classes above, or a missing one, would change
the count; the equivalences with the relations on z exclude both. A rational
relation among 1, η, …, η^{p−2} would break the characterization; the degree
p − 1 of Φ_p excludes it. A proof that a(2p) differs from
2(6^((p−1)/2) + 1) for some odd prime would contradict the kernel-checked
theorem.
Evidence
Enumeration of all subsets with floating-point cosine sums (zero test
|Σ| < 10⁻⁹, no sums in [10⁻⁹, 10⁻⁶)) reproduces the entry’s DATA for
n = 2, …, 20 and gives a(6) = 14, a(10) = 74, a(14) = 434 and
a(22) = 15554, each split as 2 odd-size and 2·6^m even-size subsets.
The canonical source is
D5/S3/Quantum/SpinChains/XXChainNullspaceCount.lean. Its public declarations
are nullspaceCount, claim, and result. The frozen module state has
statement identity
sha256:8c612fc929792c7b8f53472c4bed5c5bdc4eae4a24cc913ae117958da5dd1565.
The result declaration has statement identity
sha256:4cfdd0cdb4dbd00412e214229b820c9258ec4566b5cbbcdfa7bd735919c308f0.
The Freeze event is
sha256:edc55918d549053a26fe09fcdf107c9d474ea4d8635eedd6f73c86c299c9ba90
and has no project-level frozen prerequisites. The proof uses only the
standard axioms propext, Classical.choice and Quot.sound; no sorry,
native_decide, or new axiom.
Triage
theorem; resolution proved for the quoted conjecture. The public theorem
has proof_shape: content; its local steps characterize the zero-sum subsets
by the relations on z and count them through an explicit bijection with
pairs of class states over a fundamental domain. admission_basis: open-problem-resolution under preregistration issue #10063. There is no atom
and no digestion coverage edge. The result is a uniform theorem for every odd
prime, not a bounded enumeration, checker, numeric reduction or certified
instance, so utility: none applies.
ASSUMED-UNVERIFIED
The identification of the subset count with the dimension of the zero-energy subspace of the XX chain is the source’s Jordan–Wigner reading; the Hamiltonian, the transformation and the fermion-parity-dependent momentum quantization are not formalized. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof.