lee 2026 reduction operator invertibility

Eunghyun Lee, Integrability of multispecies long-range swap models with species-dependent interpolation, arXiv:2604.12136v1, Remark 3.1 after Equation (28), source label 1122am331, conjectures:

These results lead us to conjecture that $\mathfrak A_k$ is invertible for all parameters $\mu_i \in [0,1]$.

LEE-A is preregistered in #14343. For every $N\ge1$, $n\ge2$, $j\ge1$, $k\ge0$ with $j+k+1\le n$ and $\mu\in[0,1]^N$, the literal recursion $\mathfrak A_0=I$, $\mathfrak A_k=I-\mathcal B_{k+1}\mathfrak A_{k-1}^{-1}\mathcal B'_k$ has invertible matrices on words $\operatorname{Fin}(n)\to\operatorname{Fin}(N)$.

The source's Proposition 2.2, “Integrability under invertibility”, uses these operators to reduce the model to two-particle interactions. The source's Yang–Baxter result then supplies integrability. The frozen motivation LeeReductionInvertibility.result proves LEE-A on the entire closed parameter cube.

The source's Section 2.3 calls invertibility for arbitrary species compositions in the interior parameter regime open. The scalar spectral-radius experiments in Remark 3.1 do not establish the general assertion. Literature readings in #14343 report only arXiv v1, no OpenAlex citations, and no MathDB matching settlement; exhaustive absence of another published proof is ASSUMED-UNVERIFIED.

Use the extended walk on block positions $0,\ldots,m+1$, with absorbing boundary rows. Each interior row has nonnegative outgoing $B,B'$ masses summing to one. Its interior restriction is substochastic. A deterministic choice of positive-weight edges reaches the boundary in at most $2Nm$ steps. The general attained-maximum principle forces every zero-boundary harmonic chain to vanish. Backward extension through the earlier invertible pivots and strong induction give invertibility of every pivot; those pivots equal the literal $\mathfrak A_k$ recursion.

A counterexample would be valid $N,n,j,k,\mu$ with $\det(\mathfrak A_k)=0$. result : claim proves that no such counterexample exists in the stated real matrix model. The finite computation below is an independent consistency check, not the proof of the unbounded statement.

Kernel evidence: D5.S3.StatisticalMechanics.LongRangeSwap.LeeReductionInvertibility.result. Its live proof uses pivots_isUnit, extend, harmonic_chain_zero, LeeCollisionExit.Path.exits, and FiniteBinaryDirichletMaximum.maximum_zero_of_two_successor_exit. The axiom closure of every public declaration is contained in ${\mathrm{propext},\mathrm{Classical.choice},\mathrm{Quot.sound}}$. Source definitions $B,B'$ are Equation (7), label 138am42, and the adjacent word-coordinate operators are Lemma 3.4, Equation (26). Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

• Proved — mechanism. Path.exits constructs an exit of length at most $2Nm$ with positive product of the chosen weights. Its private rank proof uses minimum_nonincreasing and direction_constant_minimum: the minimum collision label never increases, and a constant minimum allows at most one reversal. This applies to the closed cube, including binary parameters. harmonic_chain_zero consumes the attained-maximum principle and this path. extend and pivots_isUnit construct and certify the Gaussian elimination pivots, and result identifies them with $\mathfrak A_k$. These are kernel-checked declarations. Path.exits itself permits every real parameter vector; the nonnegative two-mass maximum principle uses the closed cube. The general maximum theorem permits any state type with an attained uniform bound, without a finiteness assumption. Invertibility outside the closed cube is open here. Finite-dimensional injectivity of the corresponding zero-boundary block system implies its invertibility (paper argument from this proved uniqueness); a separate block-matrix representation theorem is not delivered.
• Proved — consequence by paper argument. Combining LEE-A with the source's Proposition 2.2, “Integrability under invertibility”, removes its invertibility condition for arbitrary species compositions and $\mu_i\in[0,1]$. The reduction to two-particle interactions and the source's Yang–Baxter theorem therefore give general integrability. This consequence uses the published proposition and theorem; the stochastic model and Yang–Baxter conclusion are not themselves formalized here.
• Open — spectral-radius extension. Remark 3.1 reports $\rho(\mathcal B_{k+1}\mathfrak A_{k-1}^{-1}\mathcal B'_k)<1$ for $N=n\le4$. That is the source's numerical observation, not a computation reproduced here. The strict inequality for general $N,n$ remains open; invertibility is proved without this bound.
• Computed — exact-rational determinants. Command /usr/bin/time -l python3 /tmp/op-lee/exact_check.py exited 0. The scope is $N=1,2,3$, $n=2,3$, all valid $j,k$, and $\mu\in{0,1/3,1/2,1}^N$: 336 samples, all determinants nonzero. SymPy determinants agree exactly with independent Fraction Gaussian determinants. This finite grid does not certify other parameter values; the uniform extension is instead proved by result.

Exact-rational computation source

Requires Python 3 and SymPy. Save the following bytes as exact_check.py, create /tmp/op-lee, and run the command above. Script SHA-256: f53510c2c2c1e05ca0867c3349c41079545eb64f4fa8dbfdd59346fd8ddc83d9. Result-data SHA-256: 44ff07994aae4ce51d681ca26bf4dbac52cf03dffc063b668a2336468f4489ab.

from itertools import product
from fractions import Fraction
from pathlib import Path
from hashlib import sha256
import json,sympy as S

def local(pi,nu,mu,prime=False):
 if pi==nu and nu[0]==nu[1]:return 1-mu[nu[0]] if prime else mu[nu[0]]
 if pi==(nu[1],nu[0]) and (nu[1]<nu[0] if prime else nu[0]<nu[1]):return S.Integer(1)
 return S.Integer(0)
def adjacent(words,s,mu,prime=False):
 return S.Matrix([[local(pi[s:s+2],nu[s:s+2],mu,prime) if all(pi[r]==nu[r] for r in range(len(pi)) if r not in (s,s+1)) else 0 for nu in words] for pi in words])
def check():
 readings=[]
 grid=[S.Rational(0),S.Rational(1,3),S.Rational(1,2),S.Rational(1)]
 for N in range(1,4):
  for n in range(2,4):
   words=list(product(range(N),repeat=n));eye=S.eye(len(words))
   for mu in product(grid,repeat=N):
    for j in range(1,n):
     a=eye
     for k in range(n-j):
      if k:a=eye-adjacent(words,j+k-1,mu)*a.inv()*adjacent(words,j+k-2,mu,True)
      det=a.det(method='domain-ge')
      # independent Fraction Gaussian determinant, no SymPy determinant calls
      q=[[Fraction(int(x.p),int(x.q)) for x in row] for row in a.tolist()]
      independent=Fraction(1)
      for i in range(len(q)):
       pivot=next((r for r in range(i,len(q)) if q[r][i]),None)
       if pivot is None:independent=Fraction(0);break
       if pivot!=i:q[i],q[pivot]=q[pivot],q[i];independent=-independent
       d=q[i][i];independent*=d
       for r in range(i+1,len(q)):
        c=q[r][i]/d
        for s in range(i+1,len(q)):q[r][s]-=c*q[i][s]
        q[r][i]=0
      assert independent==Fraction(int(det.p),int(det.q)),(N,n,j,k,mu,det,independent)
      readings.append({'N':N,'n':n,'j':j,'k':k,'mu':list(map(str,mu)),'det':str(det),'independent_det':str(independent)})
      if det==0:raise AssertionError(readings[-1])
 p=Path('/tmp/op-lee/exact-results.json');p.write_text(json.dumps(readings,indent=2))
 print(json.dumps({'samples':len(readings),'all_nonzero':True,'grid':list(map(str,grid)),'data':str(p),'sha256':sha256(p.read_bytes()).hexdigest()}))
if __name__=='__main__':check()

The literature search does not prove exhaustive publication-level novelty. The accessible arXiv text and Crossref metadata support the source locator; no claim is made to have verified an inaccessible journal PDF. The paper-level integrability consequence is separate from the kernel-checked matrix result.