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Each pair has the same exponentially growing normal direction and opposite tangential frequencies. Take the phase-weighted difference of the two collocated global relations for a linear boundary value problem. For example, the bottom pair gives the adjoint test

v_{B n} (x, y) = \frac{e ^{i p_{n}} v _{r_{n}} - e ^{- i p_{n}} v _{r_{n}^{- 1}}}{2 i} = e^{- ω_{n} y} ϕ_{n} (x) .

The other three phase-weighted differences give

v_{T n} = e^{ω_{n} y} ϕ_{n} (x), v_{L n} = e^{- ω_{n} x} ϕ_{n} (y), v_{R n} = e^{ω_{n} x} ϕ_{n} (y) .

Thus the four spectral sets supply four real sine-tested equations per mode. We retain these

4 N

equations; requiring every uncombined complex equation as well would instead form an overdetermined finite approximation. For real traces each retained equation is the appropriate real combination of its conjugate pair.

The key advantage is that

v_{B n}, v_{T n}

vanish on the vertical sides, while

v_{L n}, v_{R n}

vanish on the horizontal sides. Orthogonality then eliminates all off-mode unknowns. Define the known right-hand sides

R_{jn} = \int_{\partial Ω} f^{(M)} \partial_{n} v_{jn} d s .

(e^{ω_{n}} e^{- ω_{n}} e^{- ω_{n}} e^{ω_{n}}) (c_{B n} c_{T n}) = (R_{B n} R_{T n}), (e^{ω_{n}} e^{- ω_{n}} e^{- ω_{n}} e^{ω_{n}}) (c_{L n} c_{R n}) = (R_{L n} R_{R n}) .

For example, with

f_{jn} = \int_{- 1}^{1} f_{j}^{(M)} (s) ϕ_{n} (s) d s

, the bottom and top right-hand sides are explicitly

R_{B n} = ω_{n} e^{ω_{n}} f_{B n} - ω_{n} e^{- ω_{n}} f_{T n} + p_{n} \int_{- 1}^{1} e^{- ω_{n} s} [(- 1)^{n} f_{R}^{(M)} (s) - f_{L}^{(M)} (s)] d s,

R_{T n} = - ω_{n} e^{- ω_{n}} f_{B n} + ω_{n} e^{ω_{n}} f_{T n} + p_{n} \int_{- 1}^{1} e^{ω_{n} s} [(- 1)^{n} f_{R}^{(M)} (s) - f_{L}^{(M)} (s)] d s .

For completeness the left and right right-hand sides are

R_{L n} = ω_{n} e^{ω_{n}} f_{L n} - ω_{n} e^{- ω_{n}} f_{R n} + p_{n} \int_{- 1}^{1} e^{- ω_{n} s} [(- 1)^{n} f_{T}^{(M)} (s) - f_{B}^{(M)} (s)] d s,

R_{R n} = - ω_{n} e^{- ω_{n}} f_{L n} + ω_{n} e^{ω_{n}} f_{R n} + p_{n} \int_{- 1}^{1} e^{ω_{n} s} [(- 1)^{n} f_{T}^{(M)} (s) - f_{B}^{(M)} (s)] d s .

Scale each row by

e^{- ω_{n}}

. Every block becomes

(1 δ_{n} δ_{n} 1), δ_{n} = e^{- 2 ω_{n}} < e^{- π} < 1.

There is just one off-diagonal entry in each row. This proves diagonal dominance of paired square global-relation collocation. Therefore the assembled $4 N \times 4 N$ system is strictly diagonally dominant, with row margin

1 - δ_{n} > 1 - e^{- π}

. This proof applies to the explicitly paired and scaled collocation system, rather than assuming arbitrary raw spectral rows are diagonally dominant. Each block is a positive-definite matrix and have spectral condition number of a positive-definite matrix

κ_{2} = \frac{1 + δ _{n}}{1 - δ _{n}} < \frac{1 + e ^{- π}}{1 - e ^{- π}} < 1.091.

For instance, setting

ρ_{jn} = e^{- ω_{n}} R_{jn}

gives the concise answer

c_{B n} = \frac{ρ _{B n} - δ _{n} ρ _{T n}}{1 - δ _{n}^{2}}, c_{T n} = \frac{ρ _{T n} - δ _{n} ρ _{B n}}{1 - δ _{n}^{2}},

with the identical formula for the opposite vertical sides. If the known-data integrals use exact

f_{j}

, these are exactly the first

N

sine coefficients of the true normal traces: no discarded unknown mode contributes to a retained row. With a finite Legendre polynomial approximation, the error is solely that of the known-data approximation and the final normal-trace truncation. These square modified Helmholtz Dirichlet-to-Neumann coefficients determine the unknown boundary normal derivative; Green's third identity or the separated-variable construction can then recover the interior solution.