FORMULATION CHECK FOR REFERENCE [36]
Henry Shin - Refacte - 4 November 2026

Paper: Yucheng Liu, Tak Shing Au Yeung, Eric T. Chung, and Simon See,
Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level
Schwarz Methods, arXiv:2603.26149v1, 27 March 2026.
https://arxiv.org/html/2603.26149v1

Scope

This note checks two printed formulations. It does not identify the formula
used in the implementation, invalidate the numerical experiments, or disprove
an intended corrected theorem. The projection issue has a plausible repair.

1. Symmetry of the printed local preconditioner

Equation (2) defines
  M1_inverse = sum_i transpose(R_i) D_i inverse(A_i) R_i,
  A_i = R_i A transpose(R_i),
with positive diagonal weights satisfying
  sum_i transpose(R_i) D_i R_i = identity.
The introduction claims that the resulting two-level preconditioner is
symmetric positive definite. In general the printed local formula is not
symmetric, because D_i need not commute with inverse(A_i).

An exact example on a two-dimensional grid follows. Let
  T4 = [[2,-1,0,0],[-1,2,-1,0],[0,-1,2,-1],[0,0,-1,2]],
  T2 = [[2,-1],[-1,2]],
  A = T4 tensor I2 + I4 tensor T2.
This is the positive-definite Dirichlet matrix on a 2-by-4 interior grid.
Partition into the first and last two columns. One layer of overlap makes
R1 select coordinates 1,...,6 and R2 select coordinates 3,...,8.
The reciprocal-overlap weights are
  D1 = diag(1,1,1/2,1/2,1/2,1/2),
  D2 = diag(1/2,1/2,1/2,1/2,1,1).
They satisfy the partition-of-unity identity. Direct rational calculation
using equation (2) gives
  M1_inverse[1,3] = 15/161,
  M1_inverse[3,1] = 15/322.
Thus M1_inverse is not symmetric. The coarse term in equation (4) is symmetric;
adding it as in equation (5) cannot cancel the antisymmetric part.

A smaller example uses A=T4, R1 selecting coordinates 1,2,3, R2 selecting
2,3,4, D1=diag(1,1/2,1/2), and D2=diag(1/2,1/2,1). Equation (2) then gives
  [[3/4, 1/2, 1/4,   0],
   [1/4, 7/8, 1/2, 1/8],
   [1/8, 1/2, 7/8, 1/4],
   [  0, 1/4, 1/2, 3/4]].

2. The metric of the projection in Lemma 3.3

The lemma introduces Pi as an orthogonal projection onto K direct-sum I,
but Pi_I as a Btilde-orthogonal projection onto I. If the first projection
means Euclidean orthogonal projection, the stated equivalence is false.
Choose
  B = I2, Btilde = [[1,1/2],[1/2,1]], K={0}, I=span((1,0)), tau=3/2.
Both matrices are positive definite, so the kernel conditions hold.
The Euclidean and weighted projectors are respectively
  P = [[1,0],[0,0]], Q = [[1,1/2],[0,0]].
For every u=(x,y), the first inequality holds since
  tau*u^T*Btilde*u - (u-Pu)^T*B*(u-Pu)
    = (3/2)*(x+y/2)^2 + y^2/8 >= 0.
For v=(-1/2,1), however, the second inequality reads 5/4 <= 9/8.

This objection is repairable by specifying a different metric for Pi:
use Btilde on its image and the Euclidean metric on its kernel, equivalently
Btilde plus the Euclidean projector onto its kernel. Pi then agrees with Pi_I
on the image. The surrounding weighted-inner-product context and the later
theorem make this a plausible intended interpretation.

Consequence for the essay

The reported computational work remains an example worth examining. Its
printed guarantee requires reconciliation of the local operator and projection
definitions before it can serve as an established guarantee in the essay.
These observations make no claim about a human advantage over AI.