Spectral-gap certificates and signed response for the two-dimensional O(3) model

We study the complete transfer-operator spectrum of the nearest-neighbour classical O(3)

model on the square lattice. A finite-purity criterion gives gap certificates at β = 0.82,

0.85, 0.86 and 0.865, and on two explicit intervals, under stated comparison and arithmetic

premises. Extracting the exact relative rotor and graph determinant, complete finite estimates

imply a uniformly vanishing rectangle–square error for all seam strengths and integer heights

4 ≤ H ≤ βα, for each fixed α < 1/2. This is a region excluding high-purity landing at large

coupling. Exact spherical elimination retains the full internal response and gives a boundary-

order signed residual bound, which is not summable over growing sizes at fixed coupling. For

a four-torus logarithmic comparison, every Taylor coefficient below order H cancels and the

first winding coefficient is 4H/3H−1. A complete elementary remainder proves positivity only

on an explicit, rapidly shrinking small-coupling interval. Positive raw comparison does not

imply a positive physical increment: rotor, determinant and absolute normalization remain

essential. Internal analytic review and finite exact checks support these scoped statements.

Fixed-large-coupling winding-tail control, sufficient anchored crossing, all-coupling reach and

continuum or Yang–Mills conclusions remain open.