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Manuscript IV@1 · IV.tex · figure · 351cb2d7

Fixed-number spectral convergence

Model 2: spectral removal at fixed number cutoff › Finite-rank gravitational kernels · lines 2187–2217

IV@1#fig:fixed-M-convergence

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gravitationalscalaroperator / instrumenterror / restrictionsupport / scope
fixed finite MM
BR,MB_{R,M}
→
B∞,MB_{\infty,M}
δ‾M(R)=MCLR−1+2M(M−1)CPR−1/2\overline\delta_M(R)=MC_L R^{-1}+\sqrt{2M(M-1)}C_P R^{-1/2}
AR,M(2)→A∞,M(2)A^{(2)}_{R,M}\to A^{(2)}_{\infty,M}
→
UR,M→U∞,MU_{R,M}\to U_{\infty,M}
→
probabilities, uniformly over states
parallel branch BR,N,M→B∞,MB_{R,N,M}\to B_{\infty,M}
remainder δ‾M(R)+MρN+2M(M−1) σN\overline\delta_M(R)+M\rho_N+\sqrt{2M(M-1)}\,\sigma_N
ρN→0\rho_N\to 0 by compactness of LL; σN→0\sigma_N\to 0 by Hilbert-tensor approximation
STOPMeasured M→∞M\to\infty: not asserted.
ΠM\Pi_M may have infinite rank.

Model 2 convergence at fixed number cutoff. Separate operator-norm and Hilbert–Schmidt coefficient estimates give bounded-generator convergence, hence unitary and uniform probability convergence. Finite-rank gravitational projection has an additional explicit remainder. These results do not supply a measured number-cutoff removal theorem.

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