Appendix B 9 October 2026

A small omitted source does not remove causal feedback

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23259571

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B A small omitted source does not remove causal feedback

Proposition B.1 (Retained high response and endpoint dressing)

For an admitted same-parent block evolution, suppose

i u_L'=A_Lu_L+B^\dagger u_H+r_L,\qquad i u_H'=Du_H+Bu_L+r_H .

Then, with the true high-block propagator $U_H$ ,

u_H(t)=U_H(t,0)u_H(0) -i\int_0^tU_H(t,s)\bigl(B(s)u_L(s)+r_H(s)\bigr)\,ds .

Thus the low equation retains an entrance term, the high source, and the retarded feedback kernel $-iB(t)^\dagger\int_0^tU_H(t,s)B(s)u_L(s)\,ds$ . For a fixed $D\geq gI>0$ and absolutely continuous $F=Bu_L+r_H$ ,

\begin{align} u_H(t)={}&e^{-itD}u_H(0)-D^{-1}F(t) +e^{-itD}D^{-1}F(0)\tag{B.1}\\ &+D^{-1}\int_0^te^{-i(t-s)D}F'(s)\,ds . \notag\end{align}

The last three terms have $D^{1/2}$ norm at most $g^{-1/2}(\|F(t)\|+\|F(0)\|+\int_0^t\|F'\|)$ .

Proof

The first formula is variation of constants in the true high block. Substitution into the low equation gives the kernel and both other terms. Since $\partial_se^{-i(t-s)D}=iDe^{-i(t-s)D}$ , integration by parts gives (B.1); the spectral theorem bounds $\|D^{-1/2}\|\leq g^{-1/2}$ .

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After removal of a constant real scalar frequency $\omega$ , the relevant inverse is $(D-\omega)^{-1}$ and the temporal quantity is $F'+i\omega F$ . A retained right clock must also be restored in that combination. An angular kinetic gap in a frozen model does not by itself prove the required gap of a full radial/source block.

The elementary two-level parent $H=\left(\begin{smallmatrix}0&b\\b&\Delta\end{smallmatrix}\right)$ already isolates the issue. With original state $(1,0)$ and no forcing, the high population is

\frac{4b^2}{\Delta^2+4b^2} \sin^2\!\left(\tfrac12t\sqrt{\Delta^2+4b^2}\right).

This follows by subtracting $\Delta/2$ times the identity and squaring the remaining traceless matrix. An exactly zero high source therefore does not imply a zero high response. For a forced zero-entrance solution with source $(g_0,0)$ one instead has $u_H''(0)=-bg_0$ , a separate causal counterexample.

In angular tensor applications a rank-two coupling connects both top bands $l=L-1,L$ to omitted $L+1,L+2$ unless a genuine parity restriction has been established. A concrete molecular application must identify its full tensor coupling, correlated entrance and local forcing bounds. Small forcing bounds do not replace $Bu_L$ , its propagated angular/source derivatives, or the original entrance error in the formulas above. Coherent currents can respond at first order in a high amplitude while its population is second order: the current conversion must retain the cross terms even when the high population is small.