Conditional preparation by reversible source transport

The required apparatus law is a conditional resource statement. A pointer with a correct marginal distribution can remain correlated with a controller that predicts its future detector output. This chapter proves the finite nodal extraction result of [M19], retaining its physical archive, and separates it from global equilibration. The next chapter gives two different statistical alternatives and the complete detector integration.

32.1 The resource to be prepared

At handoff, let $U$ be the ready configuration and let $A$ contain every old preparation record, controller coordinate, exported cell label, and future-active memory. Internal quantum memories and an inaccessible reference remain in the wave. For a known normalized ready packet $\phi$, the target is

The archive has its actual marginal $P_A$, which may be very far from the wave norm distribution. Equation (32.1) is therefore a conditional ready-subsystem target, not a global equilibrium measure.

All comparisons use $\operatorname{TV}(P,Q)=\sup_E|P(E)-Q(E)|=\frac12\|P-Q\|_1$. For standard Borel conditional laws with the same archive marginal,

This equality is an average conditional guarantee. It supplies no uniform statement on arbitrary rare archive values.

Proposition 32.1 (Reversible fine-grained obstruction)

Let $S_t$ be a common invertible measurable guidance flow with measurable inverse, and let its wave reference law be equivariant: $Q_t=(S_t)_*Q_0$. For any actual law $P_t=(S_t)_*P_0$,

$$\operatorname{TV}(P_t,Q_t)=\operatorname{TV}(P_0,Q_0).$$

If $P_0=fQ_0$, then

$$\frac{dP_t}{dQ_t}=f\circ S_t^{-1},\qquad \int\alpha\!\left(\frac{dP_t}{dQ_t}\right)dQ_t =\int\alpha(f)dQ_0$$

for every defined integral on either side. Singular components are preserved.

Proof

For every measurable event $E$, $P_t(E)=P_0(S_t^{-1}E)$ and likewise for $Q$. The measurable bijection carries the full collection of events onto itself, so taking suprema proves the total-variation identity without any absolute-continuity assumption. In the density case, change variables in $\int_{S_t^{-1}E}f\,dQ_0$ to obtain the displayed Radon–Nikodym derivative. A further change of variables proves the integral identity. Null sets and their inverse images preserve singularity.

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Coarse-grained relaxation, including established pilot-wave relaxation studies such as [VW], does not contradict this statement. Fine-grained information may move to unresolved scales or exported coordinates while a restricted class of observables becomes insensitive.

Lemma 32.2 (Same-wave complete future comparison)

Two preparations with the same complete wave and control programme, and configuration-law distance at most $\delta$, have final complete-output distance at most $\delta$ under any common measurable source evolution and record map. The same holds for a common stochastic kernel. Returning waves, copies, nulls, physical timestamps, and adaptive records may be included.

Proof

A deterministic output event is a preimage under the common map. For a kernel, its probability for an output event is a measurable function in $[0,1]$; integrating that function against $P-Q$ has absolute value at most $\operatorname{TV}(P,Q)$. The complete actual path itself can be the output whenever the evolution assigns a measurable path.

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This lemma compares to the same-wave hybrid law (32.1). It does not establish that this hybrid law predicts Born records after arbitrary activation of a nonequilibrium archive.

Lemma 32.3 (Normalization cost)

If $\operatorname{TV}(P,Q)\le\delta$, $p=P(E)$, $q=Q(E)$ and $p,q>0$, then

$$\operatorname{TV}(P(\cdot|E),Q(\cdot|E)) \le\min\{1,\delta/\max(p,q)\}.$$

Proof

Assume $p\ge q$. The measures have common mass at least $1-\delta$. Their common mass outside $E$ is at most $1-p$, so the common mass inside $E$ is at least $p-\delta$. Dividing both measures by their own probabilities leaves common conditional mass at least $(p-\delta)/p$, since $q\le p$. Subtraction from one proves the bound. Interchange $P$ and $Q$ for the other ordering.

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32.2 Explicit nodal resource and admitted initial laws

Fix $N\ge2$ and $0<\eta<1/2$. Define on $(0,1)$

extended by zero. It is normalized, symmetric about $1/2$, and belongs to $H^1(\mathbb R)$. Direct integration gives

For the second inequality, $\min(|\phi_\eta|^2,1)\ge a_\eta^2$, whose integral is $1-\eta$.

The known seed wave is an array of $N$ identical cells:

It has norm one, exact nodes at cell boundaries, and $\|\varphi_{N,\eta}'\|_2^2=N^2\|\phi_\eta'\|_2^2$. Supplying this coherent wave is a preparation resource. The construction does not claim to shape an arbitrary unknown wave into it without cost.

Add an internal tag with ready state $|r\rangle$ and $N$ labels $|k\rangle$. Add an archive coordinate $y$ with known $H^1$ wave $\chi$, supported on $[0,1]$ and positive in its interior. Its actual configuration distribution need not be equilibrium. Let $z$ collect its actual initial value, every old coordinate and classical history, with a standard Borel reference measure $\mu$. The complete initial wave is $\varphi_{N,\eta}(x)\chi(y)|r\rangle\Xi(z_{\rm old})$, with the unknown input and inaccessible reference inside $\Xi$. Assume the actual law has density

All actual values lie on the nonzero-wave supports needed for the stated flow. No target wave density appears in (32.5). Multimodal distributions, steps, and correlations are allowed. An exact old copy $Z=X$ is excluded because its conditional law is singular. Regularity of the unconditional $X$ marginal would not suffice.

32.3 The source interaction and exact actual routing

First tag the occupied cell coherently, for time $\tau_1$:

It is a bounded multiplication operator, generates no configuration current, and sends $|r\rangle$ to $-i|k\rangle$ on cell $k$. The sharp coefficients are applied at exact nodes, so the wave remains $H^1$. Next use

Only the tag $k$ is present at actual $x\in I_k$, and the actual archive position becomes $Y'=Y+dk$. The archive supports are now disjoint.

For time $\tau_3$, apply the conditional dilation

The scalar transport law

follows by differentiating along characteristics. It multiplies $x-k/N$ by $N$. A final conditional translation by $-k/N$ aligns the dilated cells. Thus the exact actual configuration map is

Although the ready coordinate packets overlap after alignment, the disjoint $Y'$ supports identify the local tag component and hence its actual velocity. There is no discontinuous cutting of a connected nonzero-wave flow.

The complete final field is, up to a common phase,

The factor $N^{-1/2}$ is the dilation Jacobian. All unoccupied archive packets survive. On its support the retained archive $A=(Y',z_{\rm old})$ determines $K$ and the old $Y$; this is why correlations with $K$ cannot be discarded.

Theorem 32.4 (Conditional nodal extraction)

The finite interaction above, applied to the initial class (32.5), succeeds with probability one and obeys

$$\begin{align}\operatorname{TV}(P_{U,A},du\,P_A)&\le\frac{M}{4N},\tag{32.9}\\ \operatorname{TV}(P_{U,A},|\phi_\eta(u)|^2du\,P_A) &\le\frac{M}{4N}+d_\eta \le\frac{M}{4N}+\eta. \tag{32.10}\end{align}$$

The comparison retains the actual archive marginal and holds uniformly for unknown carried inputs and inaccessible references on which the controls act as identity.

Proof

The interaction calculation proves (32.7) and (32.8). Retaining $A$ is equivalent to retaining $(K,z)$. Their joint density after routing is

$$g(u,k,z)=N^{-1}f((k+u)/N,z),\qquad m(k,z)=\int_0^1g(u,k,z)\,du.$$

For a bounded-variation function $h$ on $(0,1)$ with mean $\bar h$, Jensen's inequality and its variation measure give

$$\begin{aligned}\int_0^1|h(u)-\bar h|\,du &\le\int_0^1\int_0^1|h(u)-h(v)|\,du\,dv\\ &\le\int_{(0,1)}2t(1-t)\,|Dh|(dt) \le\tfrac12\operatorname{Var}(h). \end{aligned}$$

The middle inequality follows by integrating the variation along intervals between $u$ and $v$; for fixed $t$ the ordered pairs whose interval crosses $t$ have measure $2t(1-t)$. Apply it to $h(u)=f((k+u)/N,z)$, integrate over $z$, and sum over open cells. Their interior variation sums to at most the total variation. The Jacobian $1/N$ and the factor one half in total variation yield $M/(4N)$. Replacing uniform density by $|\phi_\eta|^2$ costs $d_\eta$ with unchanged $P_A$, proving the second inequality.

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32.4 Resources, reproducibility, and sharp failure tests

The resource costs are $N+1$ tag states, archive extent $O(dN)$, cell resolution $1/N$, edge resolution $\eta/N$, and integrated dilation strain $\log N$. The exact seed gradient cost is

Translations and dilations are unbounded selfadjoint transport generators on their admitted domains. These statements give finite resources on the specified support and finite-gradient fields, not a bounded operator norm or a lower-bounded microscopic energy model. Exact nodes and the sharp tag coefficient are ideal spatial controls. A finite-bandwidth smooth replacement needs its own configuration-law estimate; a small wave norm error alone does not establish that estimate for nonequilibrium inputs.

A deterministic clock can drive these pulses without an equilibrium seed. Take a coordinate $S$ with generator $P_S$ and a known compact wave entirely upstream of ordered pulse regions. Let $H=P_S+\sum_rw_r(S)G_r$, with disjoint regions and fixed operators $G_r$. Then $\dot S=1$, and each admitted initial clock position crosses the same complete pulse integrals by a common finite deadline. The ordered unitary is independent of that initial position. The final clock factors and belongs to $A$. An incomplete clock traversal is a genuine failure branch, not the ready output.

For $m$ seed coordinates with product known waves, define integrated coordinate variations $M_i$ of their joint actual density, conditioning on all other coordinates and old archives. Apply the preceding interaction separately to each coordinate. Then

To prove this, successively average the density over each rescaled $U_i$. Each averaging is an $L^1$ contraction and does not increase the integrated variation in another coordinate. Telescope the one-coordinate inequality, then change each uniform factor to its ready density. The result concerns a joint finite stock, not separate correct marginals.

For the required deformation $f(x)=1+\epsilon(2x-1)$, $|\epsilon|\le1$, the exact routed marginal and complete archive comparison are

Indeed $g(u,k)=N^{-1}[1+\epsilon(2(k+u)/N-1)]$, and subtracting its $u$ average leaves $\epsilon(2u-1)/N^2$ in each cell. Integration proves the TV identity. For a detector prepared in $\phi_\eta$, let $G_\eta(u)=\int_0^u|\phi_\eta|^2$ and $u_p=G_\eta^{-1}(1-p)$. Its terminal plus probability is

At $p=1/2$, symmetry gives exactly $1/2+\epsilon/(4N)$; in general its Born error is at most $d_\eta+|\epsilon|/(4N)$.

The admissible oscillatory rival $f_N(x)=1+\epsilon\sin(2\pi Nx)$ has $M=4|\epsilon|N$. Its extracted marginal is $1+\epsilon\sin(2\pi u)$, so the balanced terminal error remains $-\epsilon/\pi$ for every $N$. This respects (32.10) and disproves a uniform statement over arbitrary absolutely continuous initial laws. Wave smoothness alone does not constrain the independent actual density variation.

For even $N$, the retained archive also satisfies

Reversing all the preparation interactions restores the original wave and actual law exactly. A balanced reader on the returned seed therefore recovers the unsuppressed deviation $\epsilon/4$. The resource was prepared by exporting nonequilibrium, not by destroying it. This return is an explicitly allowed inverse, and it identifies the precise limit of any global equilibrium interpretation.