Actual source graphs and finite-time complete currents

11 Actual source graphs and finite-time complete currents

Write $m_0=\|f_R\|^2$ and $m_1=\|(h+1)^{1/2}f_R\|^2$. The form tail gives $m_0\leq1$ and, at $R=40$, $m_1<(1+4\cdot10^{-14})^2 <1+10^{-12}$. The source ladder operators give the ordered bounds

Indeed $H_s=\Omega(N+1/2)$ and $Y=(a+a^\dagger)/\sqrt2$. The downward and upward coefficients of $aH_s^{-1}$ and $a^\dagger H_s^{-1}$ are bounded by $2/(3\Omega)$ and $2/\Omega$, respectively. For the weighted versions, their squared ratios are

The first inequality is direct and the second ratio decreases from its value $6$ at $n=0$. Their sum in $Y$ proves (11.1). These are full-source operator bounds, not a bound on a selected actual occupation.

Consequently

The two terms follow by splitting $K_0=(h+1)+H_s$ on the forcing vector, with the ordering in (11.1) preserved.

Proposition 11.1 (All-frequency response with a physical source graph)

For $\theta<1$ and every $E\in\mathbb R$, $\eta>0$,

$$\|T_{\rm ph}^{1/2}(A-E-i\eta)^{-1}LH_s^{-1}\|^2 \leq \frac{22}{7}\frac{1+\theta}{1-\theta} \frac{g^2C_\Omega^2}{\eta^2}.$$(11.3)

For the actual input $p$ of (10.6), with $q(0)=0$, let $q_T$ agree with its forced $Q$ equation on $[0,T]$ and then evolve homogeneously with $A$. If $H_sp\in L^2(0,T)$, then

$$\int_0^\infty e^{-2\eta t}\|T_{\rm ph}^{1/2}q_T(t)\|^2dt \leq \frac{22}{7}\frac{1+\theta}{1-\theta} \frac{g^2C_\Omega^2}{\eta^2} \int_0^T e^{-2\eta s}\|H_sp(s)\|^2ds .$$(11.4)

Also, whenever $H_sp\in L^1(0,t)$,

$$Q(t):=\|T_{\rm ph}^{1/2}q(t)\| \leq |g|\sqrt{\frac{22}{7}\frac{1+\theta}{1-\theta}}\, C_\Omega\int_0^t\|H_sp(s)\|\,ds .$$(11.5)

Proof

By (10.8) and (10.10), $T_{\rm ph}\leq22K/[7(1-\theta)]$. The form $K=A+1$ commutes with the actual $A$ resolvent, whose unweighted norm is at most $1/\eta$. Its input satisfies $\|K^{1/2}LH_s^{-1}\| \leq\sqrt{1+\theta}|g|C_\Omega$. Their ordered product proves (11.3). The finite lower bound on $K$ is positive; rescaling by that bound gives the equivalent unit-coercive form convention if needed. No commutation of $T_{\rm ph}$ with $A$ is used.

Extend $1_{[0,T]}H_sp$ by zero. The causal exponentially damped Fourier transform of the Duhamel equation is the resolvent multiplier applied to this complete source vector. Plancherel with (11.3) proves (11.4), as in Theorem 6.1. There is no derivative of the time cutoff in this argument and no frequency-band assumption. Finally Duhamel in the stationary positive $K$ form and $\|K^{1/2}e^{-itA}v\|=\|K^{1/2}v\|$ give (11.5).

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The $1/\eta^2$ cost includes all actual bound poles and continuum. It cannot be discarded because the fixed-reference continuum-only calculation happened to be $\eta$ independent. The source weights in (10.12) and (11.3) are also different.

Proposition 11.2 (Source graphs for the actual feedback evolution)

Let the normalized entrance be $\Psi(0)=\phi\otimes|1\rangle\otimes\zeta$, with any normalized passive finite internal vector $\zeta$. Under the actual static parent (10.1), put $a=|g|d_0$. Then

$$\begin{align}H_1(t):=\|H_s^{1/2}\Psi(t)\| &\leq\sqrt{3\Omega/2}+a\sqrt{\Omega/2}\,t, \tag{11.6}\\ \|H_s\Psi(t)\| &\leq\Omega\{3/2+\sqrt3\,at+a^2t^2/2\}, \tag{11.7}\\ e(t):=\|q(t)\| &\leq |g|\{\sqrt3\,t+at^2/2\}. \tag{11.8}\end{align}$$

Since $P$ commutes with $H_s$, the first two bounds also control the corresponding graphs of the actual $p$.

Proof

The commutator is $[H_s,H]=-ig\Omega d_Rp_Y$, with $p_Y=-i\partial_Y$. The expectation identity and $\|p_Y\Psi\|\leq\sqrt{2/\Omega}\|H_s^{1/2}\Psi\|$ give

$$\left|\frac d{dt}H_1^2\right| \leq a\sqrt{2\Omega}\,H_1 .$$

Regularizing the square root if necessary and integrating gives (11.6). The norm derivative for $H_s\Psi$ obeys

$$\frac d{dt}\|H_s\Psi\| \leq\|[H_s,H]\Psi\| \leq a\sqrt{2\Omega}\,H_1(t).$$

Its initial norm is $3\Omega/2$; integrating the preceding linear bound gives (11.7).

These computations can be justified without assuming the future source stays Gaussian. Let $P_N$ project onto source occupations $0,\ldots,N-1$, and use the full-space operators

$$H_N=h+H_s+g d_RP_NYP_N.$$

They are self-adjoint on $D(h)\cap D(H_s)$ with uniform infinitesimal relative bounds and the coercivity (10.2). For every vector in this common domain, $H_Nu\to Hu$, because $P_N$ commutes with $H_s$ and converges in its form norm. The resolvent identity, with the uniformly bounded nonreal resolvents on its left, gives strong resolvent convergence; the associated unitary groups converge strongly on compact time intervals. For $N\geq2$ the entrance lies in $P_N\mathcal H$ and remains there, so both source graphs are bounded operators on that invariant subspace. The commutator is $-ig\Omega d_RP_Np_YP_N$ and obeys the same uniform estimates as above. Hence the displayed graph bounds hold for $e^{-itH_N}\Psi(0)$. At each time, strong wave convergence and weak lower semicontinuity of the closed graphs of $H_s^{1/2}$ and $H_s$ pass these bounds to the actual wave. No finite source truncation remains in the result.

Lastly, from the exact equation for $q$ and its zero entrance,

$$\|q(t)\|\leq |g|\|f_R\|\int_0^t\|Yp(s)\|\,ds \leq |g|\sqrt{2/\Omega}\int_0^tH_1(s)\,ds .$$

Here $\|f_R\|\leq1$ and $P$ commutes with $H_s$. This is (11.8). The feedback term in (10.6) has not been set to zero anywhere.

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11.1 All coherent terms in the original physical currents

The nominated current constitution is the nonrelativistic canonical one:

Define differences relative to the actual $P$ wave,

The vector $\phi p$ is generally unnormalized and does not solve an autonomous closed evolution. It is a wave comparison, not a separately assigned physical flow. To make its continuity defect explicit, set $\mathcal F=gY\langle\phi,d_Rq\rangle_r$ and $M(t)=\|p(t)\|^2$. The feedback equation gives, in distributions,

The products are integrable under the earned source graphs: for example $\|\mathcal F\|\leq |g|d_0\|Yq\|$. Even if $M>0$ and the comparison is normalized, writing $\rho_P=|\phi p|^2$, its density $\rho_P/M$ and current $j[\phi p]/M$ have source $s_P/M-M'\rho_P/M^2$. Normalization therefore does not create conservative equivariance. The estimates below retain the original unnormalized comparison and need neither a second flow nor a continuity-defect allowance.

Lemma 11.3 (Full electron and source current differences)

With $d_r=\|\nabla_rq\|$ and $d_Y=\|\partial_Yq\|$,

$$\begin{align}\|\delta\rho\|_1&\leq2e+e^2,\tag{11.10}\\ \|\delta j_r\|_1&\leq d_r+e+ed_r,\tag{11.11}\\ \|\delta j_Y\|_1 &\leq\Omega\{(1+e)d_Y+ e\sqrt{2/\Omega}\,H_1(t)\}, \tag{11.12}\end{align}$$

where $d_r\leq\sqrt2\,Q(t)$ and $d_Y\leq\sqrt{2/\Omega}\,Q(t)$.

Proof

For any derivative $D$, expand before estimating:

$$j_D[\phi p+q]-j_D[\phi p] =c_D\operatorname{Im}\{ (\phi p)^\dagger Dq+q^\dagger D(\phi p)+q^\dagger Dq\}.$$

Thus every interference term and the quadratic error current remain. Cauchy–Schwarz, $\|p\|\leq1$ and $\|\nabla\phi\|=1$ give the electron estimate. For the source, $\|\partial_Yp\|\leq\sqrt{2/\Omega}\,H_1$ because $P$ commutes with the source kinetic form. Multiplication by $c_Y=\Omega$ gives (11.12). The density expansion gives (11.10). Finally the electron and source kinetic terms are included with their stated coefficients in $T_{\rm ph}$, giving the two derivative bounds. Spectral orthogonality of $P$ and $Q$ has not been used to discard a pointwise cross term.

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Corollary 11.4 (An explicit finite-time current bound)

For the same static parent with $\Omega=.01$, $g=10^{-6}$, $R=40$, $d_0\leq41$ and the entrance of Proposition 11.2,

$$\begin{align}Q(100)&<.00051,\qquad e(100)<.000176,\tag{11.13}\\ \int_0^{100}\|\delta j_r(t)\|_1dt &<.045033131<.046,\tag{11.14}\\ \int_0^{100}\|\delta j_Y(t)\|_1dt &<.003970830<.004. \tag{11.15}\end{align}$$

Proof

Equations (10.4), (10.9) and (11.2) give the rational enclosure

$$\frac{22}{7}\frac{1+.00062}{1-.00062} \left\{\frac{32(1+10^{-12})}{9(.01)^2} +\frac{25}{4(.01)}\right\}<338^2 .$$

Here $(\sqrt3+1/\sqrt2)^2=7/2+\sqrt6<25/4$. Let $a=41\cdot10^{-6}$. Equations (11.5)–(11.8) consequently admit the polynomial upper bounds

$$\begin{aligned}\overline Q(t)&=338\cdot10^{-8} \{(3/2)t+(7/8)at^2+a^2t^3/6\},\\ \overline e(t)&=10^{-6}\{(7/4)t+at^2/2\},\\ \overline H_1(t)&=.123+.071at. \end{aligned}$$

All roundings are outward: $\sqrt3<7/4$, $\sqrt{.015}<.123$ and $\sqrt{.005}<.071$. Substitute $d_r\leq(10/7)\overline Q$ and $d_Y\leq15\overline Q$, using $\sqrt2<10/7$, $\sqrt{200}<15$. The two current integrands are bounded by

$$\frac{10}{7}\overline Q+\overline e+ \frac{10}{7}\overline e\,\overline Q, \qquad .01\{15(1+\overline e)\overline Q+ 15\overline e\,\overline H_1\}.$$

These are explicit nonnegative rational polynomials. Exact integration on $[0,100]$ gives respectively $0.045033130836\ldots$ and $0.003970829749\ldots$. Direct evaluation of the same polynomials gives (11.13). The exact integrals are

$$\frac{113483489708747242649}{2520000000000000000000}, \qquad \frac{95299913979542242649}{24000000000000000000000},$$

so integer comparison verifies both printed outward ceilings. No continuum quadrature enters this bound.

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The electron integral has units of relative-coordinate length, and the source integral of source-coordinate length in the chosen dimensionless $Y$ convention. They bound $j[\Psi]-j[\phi p]$, not the entire absolute currents. They are not event probabilities. A detector or tube claim requires a corresponding surface/guard and an original-law transport interface with its own constants. Comparing to a freely propagated intended source instead of the actual $p$ would additionally require the feedback phase and current cost of $p-p_{\rm free}$.

The compact scalar and oscillator constitute a definite mathematical example with actual source feedback, full Coulomb poles and continuum, and complete canonical currents. A finite-mass source/COM reduction, source field energy, retardation, loading, clock and a microscopic Pauli or Dirac current identification remain separate physical questions. None is inferred from the exact Coulomb calibration or from the small forcing tail.

11.2 Flow and companion-paper interfaces

For the entrance of Proposition 11.2, the actual parent (10.1) is the constant-$g$ case of the Full Coulomb–source flow theorem in the revised companion paper [3]. The reduced-mass units, the smooth compact dipole and its full collar, the first excited source entrance, the passive internal factor, and both canonical currents agree. Constant $g$ satisfies that theorem's bounded absolutely continuous control hypothesis on every finite horizon, including $[0,100]$. Its common first-domain, tangent-regularity and logarithmic-current arguments consequently supply a deterministic reference-compatible flow for this actual wave, with reference-almost-sure collision and node avoidance. An original joint law absolutely continuous with respect to the entrance wave density is transported by reweighting those same paths once. A quantitative domination factor for (2.7) is a further premise, not a consequence of absolute continuity.

The current bounds above do not apply that flow theorem to $\phi p$. Conservative two-flow stability in the revised P3 requires two admitted conservative flows, a common positive tube, and a bound for the probability of leaving it. Its separate signed-source appendix requires a normalized reference, classical local regularity, and finite source and divergence costs. Neither conclusion follows by analogy from (11.9). No variable-coefficient spin-curl current is used here; adding one would require its own derivative regularity and current estimates.

The Gaussian preparation and instrument results of P1 [1] and the finite repeated-record protocol of P2 [2] concern their own effective parents, complete entrance laws and decoders. The present estimates establish neither those entrance premises nor fresh independence, renewed readiness, or repeated unknown-input measurement. Conversely, wave reset or preparation of an actual law does not supply a missing derivative, source graph or flow hypothesis for the present parent. Any composition must verify the same complete configuration and every retained correlation on the same time interval.