Remark. What the supplied-circle limit establishes [fdip-004K]
AGENTDRAFTED
The fixed-mode limit compares finite operators and
one Gaussian moment with a declared circle. Its length, measure,
mesh spacing, factor \(n^2\) and positive mass are supplied.
Without \(n^2\), each fixed-mode Laplacian eigenvalue tends to zero.
This explains why a finite positive operator alone does not determine
a continuum scale or geometry.
The quantifier on the frequency matters. For even \(n\) and
\(k=n/2\), the discrete eigenvalue is \(4n^2\), whereas the continuum
frequency has eigenvalue \(4\pi ^2k^2=\pi ^2n^2\). Their ratio stays
\(4/\pi ^2\); the fixed-mode proof is therefore not a relative
uniform estimate over all modes. It proves neither operator-norm
resolvent convergence nor convergence of determinants or Gaussian
measures. Removing the positive mass or adding interactions requires
additional arguments.
For comparison, Weisz distinguishes the critical-point and
phase-diagram control needed for low-energy lattice-QCD continuum
limits from the renormalization questions for composite observables
and high-energy scales
[weisz2010renormalization, Section 1.1, printed p. 2].
The circle calculation is a much smaller prescribed approximation:
it contains no interacting critical-point analysis. It establishes no
emergent geometry, reflection positivity, Lorentzian reconstruction,
Standard Model dynamics or gravitational dynamics. Those identifications
remain among the open physical interfaces.