Small-lift amplitude approximation and energy-scale scope

PDF p. 3: End of paragraph following Eq. (11)

Assume the Axiom of Choice, Countable Choice and Dependent Choice. Use finitely many blocks in the article's fixed downward-branch effective emitter model, with the finite integer-spin blocks and rank-one dipole family of the atomic-jumps result. Let $P_J$ be the projection onto block $J$, put $H_0={\omega }_0{\sum }_J{\eta }_J{P}_J$, and put $H^{(\mathrm{\Delta })}=H_0+H_{\mathrm{\Delta }}$ with the stated diagonal lift. Adopt all hypotheses of the spherical-jumps result for both static Hamiltonians, with the same coupling scale $\gamma >0$, the same three Hermitian Cartesian dipole operators of norm at most one, and the same bath. In particular assume the centered stationary Gaussian bath, factorized remote-past preparation, weak-Markov regime $\mathrm{\Gamma }{\tau }_B\ll 1$, integrable correlations and frequency-domain spectral square root, finite nonzero zeroth and finite first time-domain kernel moments, pointwise Fourier identities at every required Bohr frequency, and existence of the required principal values in that supplier's conventions. Adopt its regulated isotropic photon-model hypotheses and the compatible vacuum-limit hypotheses of the vacuum-density result. Require a nonnegative scalar representative that matches the ordinary ULE spectral density pointwise at every required transition frequency for each Hamiltonian, not just almost everywhere. These are applicability assumptions for the ULE coefficients; small lift alone does not establish them.

Restrict each resulting spherical jump to its $J\to J-1$ matrix entries. This is the retained effective branch, not a claim that the full lifted microscopic jump has no other entries. Keep its dipole matrix elements and eigenvectors unchanged by the phenomenological diagonal lift, and use the unmodified free-space vacuum spectral representative $\mathcal{J}(\omega )=A{\omega }^3$, $A>0$, pointwise at every unperturbed and shifted retained transition frequency. No cubic-law condition is imposed at frequencies between those sampled points. The general cutoff-bath postulate alone does not assert this pointwise condition. For ${\mathrm{\Omega }}_J={\omega }_0{h}_J$ with fixed $h_J={\eta }_J-{\eta }_{J-1}>0$, the shift is $s_{Jm\delta }=\mathrm{\Delta }[m-(J-1)\delta ]$. If $r_J=(2J-1)\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}({\omega }_0{h}_J)\le r<1$, then each nonzero downward jump coefficient changes by relative magnitude at most $(3+3r+r^2)r_J$. Zero fixed dipole coefficients remain zero. Thus the claimed $O(J\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}{\omega }_0)$ amplitude order holds on a fixed finite set of blocks with positive gaps held fixed. It is not guaranteed by small lift alone for an arbitrary spectral regulator, a zero reference spectrum, a changed dipole operator, or an omitted transition branch.

After replacing those coefficients by the unperturbed effective jumps, keep $\gamma ,f(J),\mathrm{\Delta },{\eta }_J$ and physical observation times fixed. The effective noise restricted to matrices block diagonal in $J$, in particular any encoded input supported in one $J$ manifold and its reference-system extension, is independent of ${\omega }_0$; fixed recovery operations on those outputs inherit this independence. This explains the energy-scale independence relevant to the single-manifold code comparisons. It does not imply independence of the full channel's Choi weights on arbitrary inter-manifold inputs. The finite-cascade lemma gives an exact counterexample to that extension with $J\le 2$, ${\eta }_J=J(J+1)$, zero lift and the article's normalized fixed jumps. At fixed microscopic dipoles and vacuum normalization, varying ${\omega }_0$ also varies the physical rate through the cubic spectrum; holding the effective rates fixed is a comparison of effective models, not a claim of fixed microscopic parameters.

Original attribution is preserved here: PDF p.3 says, “We retain the unperturbed jump operators because, for $J\mathrm{\Delta }\ll {\omega }_0$, corrections to their amplitudes are only of relative order $J\mathrm{\Delta }\mathrm{/}{\omega }_0$, and the results below are independent of ${\omega }_0$.” The amplitude calculation above uses the article's unmodified cubic spectrum and fixed phenomenological coupling, with the spectral law supplied by the local vacuum-spectrum deduction and the shifted coefficient formula supplied by the spherical-jumps deduction under its hypotheses for both Hamiltonians; it does not infer smoothness for an arbitrary regularized bath. The parameter specification makes explicit that the normalized effective rates and f(J), rather than all microscopic dipoles and field factors, are fixed.

The energy-scale statement is retained on the single-J input sector for which it is sound. The broad reading that includes every full-channel Choi weight is corrected in the generated theorem, with its original attribution retained: p.4 Fig.2 specifies $H_S+H_{\mathrm{\Delta }}$ and all $J\le 2$, while p.14 S4 only invokes a global phase within each J manifold. The finite-cascade counterexample uses those exact fixed effective matrices, $\mathrm{\Delta }=0$, ${\eta }_J=J(J+1)$, $\gamma =1$ and $t=1\mathrm{/}1000$. At the article's threshold ${10}^{-8}$, the local rational interval certificate in Proof steps 3.2–4.3 for the displayed Gram formula gives effective rank 9 at ${\omega }_0=1$ and 12 at ${\omega }_0=100$. This diagnostic and its original-PDF attribution are retained internally for independent assessment rather than declaring a final article defect. Removing $H_S$ by a frame change transforms the cross-block jumps into time-dependent operators; discarding their unequal phases would be an additional approximation. No such approximation was located in the cited original passages.