Branch-Weight Gravitational Witnesses: Cumulants, Coherence Recovery, and Classical-Record Bounds

1. Introduction

A gravitational probe can have the same mean response to two very different sources: a single averaged mass distribution, or a source that produces different responses in different runs. Preparing a superposition with adjustable branch probabilities allows these possibilities to be compared systematically. The experiment varies the probability of a mass configuration while holding its position, shape, and probe coupling fixed. The resulting change in probe fluctuations can then be compared with the source coherence that survives the interaction or can be recovered afterward.

For a two-branch source, write

$$|{\psi }_p\rangle =\sqrt{p}|L\rangle +e^{i\theta }\sqrt{1-p}|R\rangle ,0\le p\le 1.$$

(1)

If the two configurations produce fixed responses $q_L$ and $q_R$, their between-branch variance is $p(1-p){(q_L-q_R)}^2$. This familiar probability law is equally compatible with coherent source–probe entanglement and with a classical mixture of branch responses. Its usefulness lies in the controlled dependence on p: a single calibrated response assignment must explain the complete scan, including its conditional noise and higher moments.

The physical interpretation requires care. Deterministic expectation-value sourcing, stochastic semiclassical dynamics, collapse models, and coherent Newtonian interactions are different kinds of models, but those names do not specify disjoint sets of observable statistics. A classical channel can reproduce some interference revivals, and access to a coherent probe record can restore source interference without reversing the gravitational interaction itself [4]. Conversely, a coherent interaction need not remain recoverable if information escapes into an uncontrolled environment. The comparison must therefore specify the dynamical model and the operations used to recover coherence.

The analysis below combines branch-conditioned probe statistics with such an operational recovery test. Its quantitative exclusion result concerns a source measurement that preserves the branches, produces a classical record, and uses that record to drive the probe. Record distributions sufficiently different to produce different probe means also restrict the coherence that survives their acquisition. We express this restriction first through the overlap of the distributions, then through either a calibrated feedback range or measured conditional variances. The variance form gives a precise meaning to the noise cost of sustaining a large response separation with high recovered coherence.

Several lessons from protected-code and noise-spectroscopy calculations guide this construction [2]. Recombination can close motional trajectories while leaving an accumulated stochastic phase. Local visibility data can omit spatial noise correlations. Additional preparations can improve reconstruction without separating a signal from a background with the same response dependence. A shared calibration error can preserve every redundant measurement. The response and recovery tests below therefore include explicit trajectory, calibration, and identifiability conditions.

2. Relation to Gravitational and Coherence Tests

Unequal branch amplitudes and gravitational force fluctuations already occur in analyses of gravitational cat states [1]. Page and Geilker’s experiment provides an important macroscopic constraint on a particular expectation-value sourcing prescription [3]; it does not settle the predictions of every stochastic or hybrid theory. The present work uses the controlled branch probabilities to organize comparisons of response distributions and recovery, without claiming the underlying probability identities as new.

The proposals of Bose and collaborators and of Marletto and Vedral focus on gravitationally induced entanglement between masses [5,6]. Analyses of field information and matter coherence clarify how these measurements depend on the accessible systems and the assumed interaction map [7,8]. More recent approaches also test specified classical communication models through inequalities that do not require final-state entanglement [9]. The branch-weight tests considered here concern a complementary set of calibrated response and coherence measurements.

Measurement-and-feedback constructions demonstrate why a stochastic gravitational response cannot be identified with a coherent quantum mediator from its mean attraction alone [10,11,12,13]. Their compulsory noise depends on the model. In particular, the spatially correlated monitoring in the least-decoherence Tilloy–Diósi construction relaxes the restriction to local operations and classical communication (LOCC) [13]. The Tilloy–Diósi and Kafri–Taylor–Milburn models are not interchangeable constituent-level descriptions [14]. Published examples of gravity-induced entanglement in a Diósi–Penrose classical-gravity model make the danger of a categorical nonentanglement assumption explicit [15]. Broader classical–quantum theories likewise require their own decoherence and diffusion analysis [16,17,18].

The record-overlap inequalities below are applications of measurement disturbance and coherence accounting, closely related to which-way complementarity and fidelity–distinguishability bounds [19,20,21]. Their role is to translate these ingredients into a branch-response test with experimentally specified controls. Applying them to a particular gravitational theory requires showing that its reduced channel meets the assumptions. Even a measured entanglement signal has a theory-dependent gravitational interpretation when the mediator assumptions are enlarged [22].

3. Source Configurations and Measured Probe Statistics

Let ${\left\{|a\rangle \right\}}_{a=1}^m$ be orthogonal, sufficiently stable source configurations, with

$$|\psi \rangle ={\displaystyle \sum _{a=1}^m}\sqrt{p_a}{e}^{i{\theta }_a}|a\rangle ,p_a\ge 0,\sum _a{p}_a=1.$$

(2)

The probabilities belong to the simplex ${\Delta }_{m-1}$. The branch phases, source trajectories, and interaction time are held fixed or recorded during a probability scan. Preparing a different p must not silently change the branch-dependent mass profiles or probe coupling.

In a branch-preserving coherent interaction, a pure initial probe evolves as

$$U_{SP}|a\rangle |P_0\rangle =|a\rangle |P_a\rangle ,{\rho }_P^{\mathrm{out}}=\sum _a{p}_a|P_a\rangle \langle P_a|.$$

(3)

The reduced probe mixture follows from tracing orthogonal source branches; it is independent of the ${\theta }_a$. A classical mixture of source configurations produces the same reduced probe state in this model. Probe cumulants alone consequently contain no evidence that the initial source was coherent.

Let $\mathit{Q}\in {\mathbb{R}}^d$ be a vector of compatible measured probe outputs. Different quadratures requiring incompatible measurements are estimated on separate ensembles. Define the endpoint response law, measured with branch a prepared alone, by

$${\mathit{q}}_a=\mathbb{E}[\mathit{Q}\mid a],V_a=Cov(\mathit{Q}\mid a).$$

(4)

When the endpoint laws remain fixed across the scan, a branch-resolved response has

$$P(\mathit{Q}\mid p)=\sum _a{p}_a{P}_a\left(\mathit{Q}\right).$$

(5)

This equation is an assumption about the complete measurement distribution, not merely about its mean. It holds for Equation (3) with a fixed probe measurement and for a classical sampler of the same endpoints.

The law of total covariance gives

$$\begin{array}{cc}\hfill \mathbb{E}[\mathit{Q}\mid p]& =\overline{\mathit{q}}:=\sum _a{p}_a{\mathit{q}}_a,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\Sigma }_{\mathrm{obs}}\left(p\right)& =\sum _a{p}_a{V}_a+{\Sigma }_{\mathrm{br}}\left(p\right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\Sigma }_{\mathrm{br}}\left(p\right)& =\sum _a{p}_a({\mathit{q}}_a-\overline{\mathit{q}}){({\mathit{q}}_a-\overline{\mathit{q}})}^T.\hfill \end{array}$$

(8)

The first term in Equation (6) includes intrinsic probe fluctuations and any conditional channel noise. It is not generally a constant instrumental covariance. For example, two branches with the same mean can have different widths, producing a probability-dependent measured variance even though ${\Sigma }_{\mathrm{br}}=0$.

Cumulants can be subtracted individually only under stronger conditions. If the scalar response is $Q=q_A+ϵ$, where A has probabilities $p_a$ and $ϵ$ is a common independent additive noise, then

$${\kappa }_r\left(Q\right)={\kappa }_r\left(q_A\right)+{\kappa }_r\left(ϵ\right).$$

(9)

With branch-dependent noise, higher cumulants must instead be computed from the mixture in Equation (5). Endpoint noise measurements and checks for drift are therefore part of the witness.

A physical mass-density model must also justify the branch projection. For a code projector $P_C$, a projected density $P_C\widehat{\mu }\left(\mathit{x}\right)P_C$ is sufficient for exact reduced dynamics only when the relevant subspace is invariant under the retained density operators and evolution. In general $P_C{\widehat{\mu }}^2{P}_C\ne {\left(P_C\widehat{\mu }{P}_C\right)}^2$. Finite wave-packet width, leakage, and controller noise cannot be discarded merely because the logical configurations have convenient nominal densities. These qualifications matter when branch labels refer to codes containing several constituents.

4. Cumulants and Covariance Geometry

For a scalar fixed response $q_a$, the cumulant-generating function of the latent branch response is

$$K_p\left(t\right)=log\sum _a{p}_a{e}^{t{q}_a}.$$

(10)

With $q_L-q_R=\Delta q$, the binary cumulants are

$$\begin{array}{cc}\hfill {\kappa }_1^{\mathrm{br}}& =p{q}_L+(1-p)q_R,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\kappa }_2^{\mathrm{br}}& =p(1-p)\Delta q^2,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\kappa }_3^{\mathrm{br}}& =p(1-p)(1-2p)\Delta q^3,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\kappa }_4^{\mathrm{br}}& =p(1-p)\left[1-6p(1-p)\right]\Delta q^4.\hfill \end{array}$$

(14)

Exchanging the weights $p\leftrightarrow 1-p$ with the physical responses held fixed reverses the third cumulant and preserves the second and fourth. Relabeling the branches exchanges both p and $\Delta q$, leaving every physical cumulant unchanged. At equal weights the two-point distribution is symmetric and all odd central cumulants vanish. Unequal weights thus expose asymmetry hidden by the equal-weight preparation, provided the conditional measurement noise is accounted for.

For m branches the covariance has the pairwise form

$${\Sigma }_{\mathrm{br}}\left(p\right)={\displaystyle \frac{1}{2}}\sum _{a,b}{p}_a{p}_b({\mathit{q}}_a-{\mathit{q}}_b){({\mathit{q}}_a-{\mathit{q}}_b)}^T.$$

(15)

Expanding the right-hand side gives ${\sum }_a{p}_a{\mathit{q}}_a{\mathit{q}}_a^T-\overline{\mathit{q}}{\overline{\mathit{q}}}^T$, which is Equation (8). For three scalar responses,

$${\kappa }_2^{\mathrm{br}}=p_1{p}_2{(q_1-q_2)}^2+p_1{p}_3{(q_1-q_3)}^2+p_2{p}_3{(q_2-q_3)}^2.$$

(16)

An edge scan measures a pair of responses; interior settings test whether those endpoint laws still describe the preparation and interaction when all three branches are occupied. Once fixed endpoint laws are known, the interior predictions are determined. Their value is a consistency check on that common model.

Proposition 1 

(Between-branch covariance rank). If k branches have positive probability, then

$$rank{\Sigma }_{\mathrm{br}}\left(p\right)\le min(d,k-1).$$

(17)

Proof. 

Put the occupied vectors $\sqrt{p_a}({\mathit{q}}_a-\overline{\mathit{q}})$ in the columns of a matrix A. Then ${\Sigma }_{\mathrm{br}}=A{A}^T$ and $A{(\sqrt{p_1},\dots ,\sqrt{p_k})}^T=0$. Hence $rankA\le k-1$.    □

This rank restriction applies to the isolated between-branch term. A two-branch experiment can have a full-rank measured covariance because its conditional probe noise is full rank. A violation of Equation (17) becomes a model test only after the $V_a$ have been measured with adequate uncertainty bounds.

The scalar higher cumulants are

$$\begin{array}{cc}\hfill {\kappa }_3^{\mathrm{br}}& =\sum _a{p}_a{(q_a-\overline{q})}^3,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\kappa }_4^{\mathrm{br}}& =\sum _a{p}_a{(q_a-\overline{q})}^4-3{\left({\kappa }_2^{\mathrm{br}}\right)}^2.\hfill \end{array}$$

(19)

They add constraints on the mixture without changing its classical–quantum ambiguity. For the restricted deterministic alternative $Q=\overline{q}\left(p\right)+ϵ$, with the same independent calibrated noise $ϵ$ at every p, all cumulants beyond the first equal those of $ϵ$. An observed scan incompatible with that prediction excludes this deterministic affine-response model. A general expectation-value gravitational theory can also change the probe dynamics or its quantum width as p changes; excluding that theory requires its full readout prediction.

5. Ensemble-Size Scans

Suppose n distinguishable or individually addressable source units each contribute $q_{A_j}$ to an additive response $Q_n={\sum }_j{q}_{A_j}$. Treating indistinguishable particles this way additionally requires a justified mode description. Let a single unit have mean $\overline{q}$ and variance ${\sigma }^2$. Independent branch preparations give

$$\mathbb{E}\left[Q_n\right]=n\overline{q},Var\left(Q_n\right)=n{\sigma }^2.$$

(20)

For a coherent collectively correlated state ${\sum }_a\sqrt{p_a}{e}^{i{\theta }_a}{|a\rangle }^{\otimes n}$, the branch response is $n{q}_a$, so

$$Var\left(Q_n\right)=n^2{\sigma }^2.$$

(21)

Exactly the same statistics follow from the incoherent preparation

$${\rho }_{\mathrm{mix}}^{\left(n\right)}=\sum _a{p}_a{\left(|a\rangle \langle a|\right)}^{\otimes n}.$$

(22)

The quadratic scaling identifies collective branch correlations under the stated additivity assumption. It cannot establish entanglement or recoverable coherence.

Exchangeable scalar branch responses with common pair correlation ${\rho }_n$ satisfy

$$Var\left(Q_n\right)=n{\sigma }^2\left[1+(n-1){\rho }_n\right].$$

(23)

Covariance positivity requires $-1/(n-1)\le {\rho }_n\le 1$ for $n>1$, although a specified finite-alphabet branch distribution can impose stronger restrictions. A common correlation parameter across different ensemble sizes is an additional preparation assumption. Changing n can alter geometry, conditional probe noise, controller errors, and collective backgrounds; these changes must be included before Equation (23) is used as an exclusion test.

6. Coherence Recovery with Specified Controls

The source coherence is quantified in its fixed branch basis by

$$C_{{\ell }_1}\left({\rho }_S\right)={\displaystyle \frac{1}{m-1}}\sum _{a\ne b}\left|{\left({\rho }_S\right)}_{ab}\right|.$$

(24)

For Equation (2), its input value is $C_{{\ell }_1}\left({\rho }_S^{\mathrm{in}}\right)={(m-1)}^{-1}{\sum }_{a\ne b}\sqrt{p_a{p}_b}$. It is zero at a simplex vertex and is generally less than one. Recovery ratios are defined only when it is positive. In the binary case, the corresponding ideal input interference visibility is $V_{\mathrm{in}}\left(p\right)=2\sqrt{p(1-p)}$.

A trace-preserving operation on the probe alone leaves the reduced source unchanged. Recovery therefore uses a probe measurement followed by an outcome-conditioned source correction, or coherent joint control. For an instrument $\left\{{\mathcal{M}}_y\right\}$ on the accessible probe,

$${\tilde{\rho }}_{S|y}={Tr}_P\left[(I\otimes {\mathcal{M}}_y)\left({\rho }_{SP}^{\mathrm{out}}\right)\right],{\rho }_S^{\mathrm{rec}}=\sum _y{D}_y{\tilde{\rho }}_{S|y}{D}_y^{†},$$

(25)

where $D_y={\sum }_a{e}^{i{\phi }_a\left(y\right)}|a\rangle \langle a|$ is a permitted diagonal phase correction. All outcomes are retained. Discarding failed outcomes and quoting only the successful visibility defines a different, probabilistic task.

Let ${\mathfrak{A}}_{\mathrm{phase}}$ be the set of these protocols allowed by the apparatus, fixed before the data are analyzed. Define

$${\mathcal{R}}_{\mathrm{ad}}=\underset{{\mathfrak{A}}_{\mathrm{phase}}}{sup}{\displaystyle \frac{C_{{\ell }_1}\left({\rho }_S^{\mathrm{rec}}\right)}{C_{{\ell }_1}\left({\rho }_S^{\mathrm{in}}\right)}}.$$

(26)

The diagonal controls preserve branch populations and cannot prepare source superpositions. Their ordinary classical control circuitry is allowed. A device that mixes the source branches or supplies an independent coherent source state falls outside this recovery test. A measured protocol gives an achieved recovery ratio; an experimentally inaccessible supremum is not itself an observed quantity.

For a branch-preserving interaction with initially independent probe and environment, $0\le {\mathcal{R}}_{\mathrm{ad}}\le 1$ follows from the channel structure. To see this, purify the initial probe and write its full output as ${\sum }_a\sqrt{p_a}{e}^{i{\theta }_a}|a\rangle {|v_a\rangle }_{PE}$. For probe outcome effect $E_y$,

$${\left({\tilde{\rho }}_{S|y}\right)}_{ab}={\left({\rho }_S^{\mathrm{in}}\right)}_{ab}\langle v_b|(E_y\otimes I_E)|v_a\rangle .$$

(27)

Cauchy–Schwarz and ${\sum }_y{E}_y=I$ give

$$\sum _y\left|\langle v_b|(E_y\otimes I_E)|v_a\rangle \right|\le \sum _y\sqrt{w_a\left(y\right)w_b\left(y\right)}\le 1,$$

(28)

where $w_a\left(y\right)=\langle v_a|(E_y\otimes I_E)|v_a\rangle$. Summing the off-diagonal magnitudes proves the bound. There is no need to impose recovery less than one as a data-dependent acceptance rule.

Joint recovery uses a separately specified family ${\mathfrak{J}}_{\mathrm{ctrl}}$ of controlled operations, with dilations of the form ${\sum }_a|a\rangle \langle a|\otimes W_a$ on the accessible probe and initially independent auxiliary systems. It preserves the source branches while allowing their existing correlations to be consumed. Define ${\mathcal{R}}_{\mathrm{joint}}$ by replacing Equation (25) with the final source state under these operations. These operations preserve the populations $p_a$. Positivity of the final source state gives $|{\rho }_{ab}|\le \sqrt{p_a{p}_b}$, hence ${\mathcal{R}}_{\mathrm{joint}}\le 1$ for the pure input family in Equation (2). If the known controlled interaction and all its records remain accessible and its inverse belongs to ${\mathfrak{J}}_{\mathrm{ctrl}}$, then ${\mathcal{R}}_{\mathrm{joint}}=1$. This is a statement about access and control, rather than a criterion for the nature of gravity.

If ${\rho }_{SP}^{\mathrm{out}}={\sum }_a{p}_a|a\rangle \langle a|\otimes {\rho }_P^{\left(a\right)}$, every probe outcome leaves a diagonal source state, and both recovery classes just specified have zero recovered source coherence. Partially dephased states require a quantitative analysis; loss of reduced visibility alone does not imply this block-diagonal form.

7. Classical Records and Response–Recovery Bounds

Consider now a more restricted channel. The source undergoes a branch-preserving measurement with diagonal Kraus operators

$$K_{y\nu }=\sum _a{k}_{a,y\nu }|a\rangle \langle a|,\sum _{y,\nu }{\left|k_{a,y\nu }\right|}^2=1.$$

(29)

Its recorded outcome $Y=y$ is irreversibly classical, with

$$r_a\left(y\right)=\sum _{\nu }{\left|k_{a,y\nu }\right|}^2,h_{ab}\left(y\right)=\sum _{\nu }{k}_{a,y\nu }{k}_{b,y\nu }^{*}.$$

(30)

The probe is prepared or driven conditionally on y, with state ${\tau }_P^y$. The accessible output is

$${\rho }_{SYP}^{\mathrm{out}}=\sum _y{\mathcal{K}}_y\left({\rho }_S^{\mathrm{in}}\right)\otimes |y\rangle \langle y|\otimes {\tau }_P^y,{\mathcal{K}}_y\left(\rho \right)=\sum _{\nu }{K}_{y\nu }\rho K_{y\nu }^{†}.$$

(31)

The instrument, conditional probe states, feedback rule, and accessible-record structure are fixed across endpoint calibration, probability scans, and coherent recovery runs. Preparation-setting metadata is not used to change this channel. A purification of the outcome dephasing is inaccessible. No coherent source–probe interaction or other signal path is added after Equation (31). This assumption distinguishes a classical-record channel from a coherent recording device that may subsequently be measured in an eraser basis.

Define the classical record overlap

$$B_{ab}=\sum _y\sqrt{r_a\left(y\right)r_b\left(y\right)}.$$

(32)

It lies between zero and one. The more separated two record distributions are, the smaller their overlap can be. Giving the complete classical outcome to the recovery apparatus can only increase its options, so a bound obtained with access to y also applies when only a noisy copy or the driven probe is accessible.

Theorem 1 

(Classical-record recovery bound). For Equation (31), Equation (2), and deterministic adaptive recovery with diagonal source phase corrections,

$${\mathcal{R}}_{\mathrm{ad}}^{\mathrm{cl}}\left(p\right)\le {\displaystyle \frac{{\sum }_{a\ne b}\sqrt{p_a{p}_b}{B}_{ab}}{{\sum }_{a\ne b}\sqrt{p_a{p}_b}}}.$$

(33)

The same upper bound holds if source corrections are any fixed completely positive, trace-preserving maps that are nonincreasing for $C_{{\ell }_1}$ on every source state.

Proof. 

Cauchy–Schwarz gives $|h_{ab}\left(y\right)|\le \sqrt{r_a\left(y\right)r_b\left(y\right)}$. For an input pure state, the unnormalized conditional source coherence therefore obeys

$$C_{{\ell }_1}\left({\mathcal{K}}_y\left({\rho }_S^{\mathrm{in}}\right)\right)\le {\displaystyle \frac{1}{m-1}}\sum _{a\ne b}\sqrt{p_a{p}_b}\sqrt{r_a\left(y\right)r_b\left(y\right)}.$$

(34)

A diagonal phase correction preserves this value. A more general correction of the stated kind cannot increase it, by homogeneity on positive unnormalized operators. Convexity bounds the coherence of the sum of corrected outcomes by the sum of their coherences. Summing over y and dividing by the input coherence yields Equation (33). A probe measurement after the classical record can only refine or randomize each y using a branch-independent state ${\tau }_P^y$; it does not supply a coherent record or evade this average bound.    □

The distributions $r_a\left(y\right)$ alone do not specify the disturbance of a general instrument. Equation (29) supplies the required branch-preserving structure; extra inaccessible records can lower recovery further. In the efficient positive diagonal case $K_y={\sum }_a\sqrt{r_a\left(y\right)}|a\rangle \langle a|$, identity source correction attains the overlap average in Equation (33). The theorem is thus sharp within its channel class.

7.1. A Calibrated Response Diameter

Let the same record drive a vector response with conditional mean $\mathit{f}\left(y\right)$, independent of the source branch given y. Put

$${\mathit{\mu }}_a=\sum _y{r}_a\left(y\right)\mathit{f}\left(y\right),D_f=\underset{y,y^{\prime }}{sup}\parallel \mathit{f}\left(y\right)-\mathit{f}\left(y^{\prime }\right)\parallel .$$

(35)

The norm and units are fixed by calibration; for heterogeneous probe coordinates this requires a specified scaling. A finite-range test uses an independently justified $0<D_f<\infty$ for all allowed outcomes, rather than the largest value seen in a finite data sample.

Lemma 1 

(Response separation). For every branch pair,

$$\parallel {\mathit{\mu }}_a-{\mathit{\mu }}_b\parallel \le D_{f}TV(r_a,r_b),TV(r_a,r_b)\le \sqrt{1-B_{ab}^2}.$$

(36)

Proof. 

A unit-vector projection of $\mathit{f}$ has range width at most $D_f$. The difference of its expectations is at most that width times total variation. Projecting along ${\mathit{\mu }}_a-{\mathit{\mu }}_b$ proves the first inequality. For the second, write $|r_a-r_b|=|\sqrt{r_a}-\sqrt{r_b}|(\sqrt{r_a}+\sqrt{r_b})$ and apply Cauchy–Schwarz. The two squared norms are $2(1-B_{ab})$ and $2(1+B_{ab})$.    □

Consequently, with $d_{ab}=\parallel {\mathit{\mu }}_a-{\mathit{\mu }}_b\parallel$,

$${\mathcal{R}}_{\mathrm{ad}}^{\mathrm{cl}}\left(p\right)\le {\displaystyle \frac{{\sum }_{a\ne b}\sqrt{p_a{p}_b}\sqrt{1-d_{ab}^2/D_f^2}}{{\sum }_{a\ne b}\sqrt{p_a{p}_b}}}.$$

(37)

A separation exceeding $D_f$ is incompatible with the assumed response map. For two branches,

$${\mathcal{R}}_{\mathrm{ad}}^{\mathrm{cl}}\le \sqrt{1-{\delta }^2/D_f^2},\delta =|{\mu }_L-{\mu }_R|.$$

(38)

The numerical ratio $\delta /D_f$ determines the exclusion threshold. A merely nonzero covariance together with a qualitative claim of high recovery is insufficient: when $\delta /D_f$ is small, the allowed recovery can be arbitrarily close to one.

7.2. Conditional Variance without a Bounded Range

For a scalar response f, let ${\mu }_a={\mathbb{E}}_a\left[f\right]$ and ${\sigma }_a^2={Var}_a\left(f\right)$ be finite. The following inequality is the Bhattacharyya-overlap form of the sharp moment-constrained Hellinger bound established by Nishiyama [24]. We give a short proof for the record distributions and their common response function. It also applies to continuous records, with sums replaced by integrals.

Theorem 2 

(Response variance and record overlap). For ${\delta }_{ab}=|{\mu }_a-{\mu }_b|$,

$$B_{ab}{\delta }_{ab}\le \sqrt{1-B_{ab}^2}({\sigma }_a+{\sigma }_b).$$

(39)

When ${\delta }_{ab}>0$, this implies

$$B_{ab}\le {\displaystyle \frac{{\sigma }_a+{\sigma }_b}{\sqrt{{\delta }_{ab}^2+{({\sigma }_a+{\sigma }_b)}^2}}}.$$

(40)

Proof. 

In the record space, let $|s\rangle$ and $|t\rangle$ have components $\sqrt{r_a\left(y\right)}$ and $\sqrt{r_b\left(y\right)}$, so $\langle s|t\rangle =B_{ab}$, and let F multiply by $f\left(y\right)$. The vector $(F-{\mu }_b)|t\rangle$ is orthogonal to $|t\rangle$ and has norm ${\sigma }_b$. Thus $|\langle s|(F-{\mu }_b)|t\rangle |\le \sqrt{1-B_{ab}^2}{\sigma }_b$. Similarly, $|\langle s|(F-{\mu }_a)|t\rangle |\le \sqrt{1-B_{ab}^2}{\sigma }_a$. Subtracting the two matrix elements gives $({\mu }_a-{\mu }_b)B_{ab}$; the triangle inequality proves Equation (39).    □

Combining Equation (40) with Equation (33) gives a multibranch variance bound. In the binary case, an achieved recovery R compatible with this classical-record model must satisfy

$${\sigma }_L+{\sigma }_R\ge {\displaystyle \frac{\delta R}{\sqrt{1-R^2}}},0\le R<1.$$

(41)

Perfect deterministic recovery and nonzero separation are incompatible with finite conditional variance in this class.

The variances in this result are conditional variances within the endpoint preparations, not the between-branch variance $p(1-p){\delta }^2$. If the observed endpoint output is a noisy Q with $\mathbb{E}[Q\mid y]=f\left(y\right)$, the law of total variance gives $Var(Q\mid a)\ge {Var}_a\left(f\right)$. Using the larger observed standard deviations produces a conservative bound without subtracting the conditional apparatus noise. Correlations or a direct source-dependent path that violate $Q\perp a\mid y$ require a different model. Finite second moments also require a justified treatment of tails; finite samples cannot by themselves certify that all unobserved rare events have been bounded.

7.3. A Saturating Binary Channel

Take two record values $y=\pm$, an efficient positive diagonal measurement, and

$$r_L(\pm )={\displaystyle \frac{1\pm t}{2}},r_R(\pm )={\displaystyle \frac{1\mp t}{2}},f(\pm )=\pm {\displaystyle \frac{D_f}{2}},0\le t\le 1.$$

(42)

Then

$$\delta =t{D}_f,B_{LR}=\sqrt{1-t^2},{\sigma }_L={\sigma }_R={\displaystyle \frac{D_f}{2}}\sqrt{1-t^2},{\mathcal{R}}_{\mathrm{ad}}^{\mathrm{cl}}=B_{LR}.$$

(43)

Both the diameter and variance bounds are attained. Weak record acquisition ($t\ll 1$) preserves nearly all coherence, but a fixed nonzero separation then requires a large feedback range and large within-branch variance. At $t=1$, the record determines the branch, endpoint variance vanishes, and the source coherence is lost. The unconditional probe variance is

$$Var(Q\mid p)={\displaystyle \frac{D_f^2}{4}}(1-t^2)+p(1-p)t^2{D}_f^2.$$

(44)

At $p=1/2$ it is $D_f^2/4$ for every t. A single equal-weight variance measurement would therefore miss the distinction between conditional measurement noise and branch separation in this example.

8. Coherent and Irreversible Response Examples

A coherent momentum-transfer model makes the recovery distinction explicit. Let X and P be probe position and momentum, with $[X,P]=i\hslash$, and take

$$U_{SP}=\sum _a|a\rangle \langle a|\otimes e^{i{q}_{a}X/\hslash }.$$

(45)

For an initial probe wave function ${\psi }_0\left(x\right)$, branch a shifts its momentum distribution by $q_a$. The momentum readout is a mixture of these translated distributions, so Equation (9) holds with the common initial momentum noise. For a Gaussian position distribution with variance $s_X^2$, the reduced source coherence is multiplied in magnitude by

$$\left|\langle e^{i(q_a-q_b)X/\hslash }\rangle \right|=exp\left[-{\displaystyle \frac{{(q_a-q_b)}^2{s}_X^2}{2{\hslash }^2}}\right].$$

(46)

Reduced visibility can be small even though the global evolution is coherent.

Measuring the probe position instead of its momentum gives a conditional source amplitude proportional to $\sqrt{p_a}{e}^{i{\theta }_a}{e}^{i{q}_{a}x/\hslash }{\psi }_0\left(x\right)$. The diagonal correction $D_x={\sum }_a{e}^{-i{q}_{a}x/\hslash }|a\rangle \langle a|$ restores the input source state for every outcome. Hence ${\mathcal{R}}_{\mathrm{ad}}=1$ in the ideal position-eraser protocol, for any number of branches. The inverse controlled kick also gives ${\mathcal{R}}_{\mathrm{joint}}=1$ when available. Finite readout resolution and imperfect phase compensation reduce this recovery and must be included in a laboratory calculation. Momentum statistics and position erasure are measured on separate identically prepared ensembles; they are not two simultaneously obtainable records of the same run.

This example also explains why a coherent recording isometry cannot be substituted for Equation (31). For a binary source, the isometry $|L\rangle |0\rangle \mapsto |L\rangle |0\rangle$, $|R\rangle |0\rangle \mapsto |R\rangle |1\rangle$ has perfectly distinguishable record distributions in the computational basis, with $B_{LR}=0$. If the record remains coherent, measuring it in the $(|0\rangle \pm |1\rangle )/\sqrt{2}$ basis and applying a conditional source phase flip restores the initial state. If it has already been irreversibly dephased into a classical record, that option is absent. The accessible channel, rather than the notation used for a record register, determines the recovery bound.

For comparison, apply a binary inaccessible dephasing channel after the controlled kick, multiplying source off-diagonal blocks by $e^{-\gamma T}$. Position erasure removes the coherent probe phases but leaves that factor, giving exactly ${\mathcal{R}}_{\mathrm{ad}}=e^{-\gamma T}$ under the stated ideal controls. For multibranch dephasing, arbitrary pairwise positive rates are not sufficient: the correlation matrix multiplying the density-matrix entries must define a completely positive channel. One permissible Markovian example is

$$\dot{\rho }=-{\displaystyle \frac{1}{2}}\sum _j{\lambda }_j[A_j,[A_j,\rho ]],A_j=\sum _a{a}_{ja}|a\rangle \langle a|,{\lambda }_j\ge 0,$$

(47)

which gives ${\gamma }_{ab}=\frac{1}{2}{\sum }_j{\lambda }_j{(a_{ja}-a_{jb})}^2$. This is an effective channel example, not a derivation of a particular objective-collapse theory.

A gravitational realization of Equation (45) would require branch-dependent force profiles and a regime in which its prescribed impulse approximation is valid. The example establishes the operational distinction and does not supply an experimental force strength, achievable noise floor, or evidence that the channel is gravitational.

9. Trajectories, Noise, and Independent Calibration

Recovery must be assessed against the actual path history. For the localization master equation $\dot{\rho }=\dots -\Lambda \left(t\right)[X,[X,\rho ]]$, common quadratic motion and ideal relative closure give the normalized recovered visibility

$$v\left(T\right)=exp\left[-{\int }_0^T\Lambda \left(t\right)\Delta x{\left(t\right)}^2\mathrm{d}t\right].$$

(48)

A classical-force representation has $\langle \xi \left(t\right)\xi \left(t^{\prime }\right)\rangle =2{\hslash }^2\Lambda \left(t\right)\delta (t-t^{\prime })$ and relative phase $\varphi ={\hslash }^{-1}\int \xi \left(t\right)\Delta x\left(t\right)\mathrm{d}t$. Gaussian averaging yields Equation (48). Closing the relative trajectory removes final motional mismatch but does not reverse this random phase. Changing a branch weight, constituent, or controller can therefore alter recovery through changed paths or noise exposure. Equation (48) is exact for the stated quadratic, branch-independent localization model with ideal closure; general wave-packet or nonquadratic dynamics need additional analysis.

A signed visibility residual can be reported as

$$\delta {\Gamma }_{\varphi }\left(T\right)={\displaystyle \frac{1}{T}}log{\displaystyle \frac{v_{\mathrm{ref}}(T,p)}{v_{\mathrm{obs}}^{\mathrm{rec}}(T,p)}},v_{\mathrm{obs}}^{\mathrm{rec}}=V_{\mathrm{obs}}^{\mathrm{rec}}/V_{\mathrm{in}}\left(p\right).$$

(49)

Here $v_{\mathrm{ref}}$ is the predicted normalized visibility under an independently calibrated reference channel and the same recovery. It can factor into background and coherent-recovery contributions only if their independent composition has been justified. Negative residuals are retained as evidence of uncertainty or reference-model mismatch. This finite-time logarithmic loss is not necessarily a constant Markovian dephasing rate. It is derived from the recovery measurement and should not be counted as a statistically independent witness coordinate.

For momentum readout one can likewise report a signed variance residual relative to the full conditional-noise model. Dividing a variance increase by T gives a variance-growth rate; a convention $\dot{Var}\left(P\right)=2D_{PP}$ assigns half of it to $D_{PP}$. Neither this residual nor Equation (49) gives a universal noise floor for all stochastic classical-gravity theories. The classical-record model has the explicit conditional-variance constraint in Equation (41).

Two-site coherence measurements may supply information missing from local visibilities. For zero-mean Gaussian phase gaps with $Cov({\varphi }_L,{\varphi }_R)/2=\left(\begin{array}{cc}{\chi }_L& c\\ c& {\chi }_R\end{array}\right)$, the magnitudes of the simultaneous-flip coherences, normalized to their prepared input values, are

$$W_{+}=e^{-{\chi }_L-{\chi }_R-2c},W_{-}=e^{-{\chi }_L-{\chi }_R+2c},c={\displaystyle \frac{1}{4}}log{\displaystyle \frac{W_{-}}{W_{+}}}.$$

(50)

Local exponents alone do not determine c. Unknown deterministic phases require both quadratures of the relevant coherences. Spatial quantum-noise spectroscopy provides established methods for such measurements [23,25].

A remaining ambiguity survives even unlimited counts. Suppose a preparation has calibrated response vector ${\mathit{v}}_s$ to several noise channels, whose covariance is $S_{st}$. The measured quadratic or cross response is ${\mathit{v}}_s^T{S}_{st}{\mathit{v}}_t$. An invertible shared basis change

$${\mathit{v}}_s\mapsto A_s{\mathit{v}}_s,S_{st}\mapsto A_s^{-T}{S}_{st}{A}_t^{-1}$$

(51)

preserves every such measurement and the positivity of the full covariance. Redundant preparations test the assumed response dependence, but they cannot fix this shared response-basis ambiguity. Independent force, trajectory, probe, and noise-response calibrations are needed to interpret the reconstructed entries physically.

For commuting Gaussian dephasing, phase magnitudes also have an exact classical representation. With phase coefficients ${\mathit{v}}_a$ and a real positive covariance S,

$${\displaystyle \frac{|{\rho }_{ab}^{\mathrm{out}}|}{|{\rho }_{ab}^{\mathrm{in}}|}}=exp\left[-{\displaystyle \frac{1}{2}}{({\mathit{v}}_a-{\mathit{v}}_b)}^{T}S({\mathit{v}}_a-{\mathit{v}}_b)\right].$$

(52)

A classical Gaussian random phase with covariance S reproduces these magnitudes for every tested response vector. Additional branch-weight or composition settings may constrain S more accurately without breaking that equivalence. Observables sensitive to coherent induced interactions, noncommuting dynamics, or antisymmetric quantum correlations require a separate calculation.

10. Identifiability and a Usable Exclusion Protocol

Let the settings s specify branch weights, trajectories, measurement choices, and recovery controls. For model M, let ${\pi }_{M,s,k}\left({\eta }_M\right)$ be the probability of count outcome k, with nuisance parameters ${\eta }_M$ constrained by independent calibration data. The experimentally relevant prediction set is

$${\mathcal{P}}_M=\{{\left({\pi }_{M,s,k}\left({\eta }_M\right)\right)}_{s,k}:{\eta }_M\in {\mathcal{B}}_M\},$$

(53)

where ${\mathcal{B}}_M$ includes the justified calibration uncertainties. Overlapping prediction sets cannot be separated uniformly by increasing the number of repetitions at those settings. A particular observation may still exclude one model if it lies outside that model’s calibrated set.

At a regular interior point, a local diagnostic is the derivative matrix of the outcome probabilities, including rows for independent calibration measurements in a joint fit. Where the permitted nuisance variations are locally unrestricted, if a proposed signal parameter g satisfies

$${\partial }_g\mathit{\pi }\in span\left\{{\partial }_{{\eta }_j}\mathit{\pi }\right\},$$

(54)

then its first-order effect is degenerate with nuisance changes. At constrained boundaries, only the permitted tangent directions count. An added science or calibration setting helps when it supplies an independent probability dependence after preparation, calibration, and model parameters are included. A rank increase is a local result and does not prove global prediction-set separation.

Within a specified mass-only gravitational model, two code implementations give identical predictions when their retained mass-density operators, reduced matter dynamics, and preparation, control, and readout maps are identical. This equality is independent of like or unlike constituent labels; equal nominal density profiles alone are insufficient when density fluctuations, leakage, or motional dynamics differ. Physical changes in profiles or trajectories can change the prediction, but their utility depends on whether the new dependence survives the allowed background and calibration fits. In a regularized feedback model, the same regulator must be included consistently in the force and compulsory-noise calculation; it cannot be chosen to suppress noise while the unsmoothed force is retained. No intrinsic or equal-resource advantage of changing constituent composition follows from the present bounds.

For the binary classical-record test, endpoint preparations establish a nonzero separation $\delta$, and separate coherent preparations establish an achieved deterministic recovery R. A sufficient diameter-based incompatibility is

$$R_{-}>\sqrt{1-{({\delta }_{-}/D_{f,+})}^2},$$

(55)

where $R_{-}$ and ${\delta }_{-}$ are simultaneous lower confidence bounds and $D_{f,+}$ is a justified upper calibration bound, with $0<{\delta }_{-}\le D_{f,+}$. If ${\delta }_{-}>D_{f,+}$, the endpoint response alone is incompatible. Alternatively, using an upper bound $s_{+}$ on the sum of observed endpoint standard deviations, a sufficient variance-based incompatibility is

$$R_{-}>{\displaystyle \frac{s_{+}}{\sqrt{{\delta }_{-}^2+s_{+}^2}}}.$$

(56)

These are conditional model tests. The uncertainty bounds must be joint, include drift and readout errors, and account for the sampling assumptions used to control response tails. Separate uncertainty intervals casually inserted into a nonlinear inequality are not automatically a valid significance calculation.

The fixed-channel assumption must also be tested or independently justified. A device informed of the preparation probabilities could sample a matching probe distribution without interrogating the source and leave its coherence intact; agreement with a probe-only weight scan would not constrain that setting-dependent simulator. A practical experiment uses endpoint and interior probability settings, coherent and deliberately incoherent source controls, and separate probe-statistics and recovery ensembles under the common channel assumptions. Endpoint data measure $P_a$, rather than only nominal $q_a$. Interior data test Equation (5) and the inferred covariance. Recovery data retain all outcomes and use a fixed phase-control family. The likelihood for the raw observations is then fitted jointly with the independent calibration data, permitting each rival its best allowed nuisance fit. Simulations under the constrained rival can assess finite-sample rejection probabilities. Equal-resource comparisons must charge each strategy for preparation, switching, calibration, and acquisition. None of these precision or platform forecasts is supplied by the elementary covariance law.

11. Discussion and Conclusions

Branch probabilities are useful experimental controls because they impose relations among measurements that a model must satisfy with one fixed set of responses. The simplex covariance, its rank, and the higher cumulants describe those relations. Ensemble-size scans add information about branch correlations. Their statistical agreement establishes a branch-resolved response model under the calibrated conditions; it does not distinguish a coherent source from an incoherent mixture with the same branch probabilities.

Coherence recovery tests a different part of the dynamics. For a branch-preserving instrument with an irreversible classical record, recoverable coherence is bounded by the overlaps of the record distributions. A response diameter or conditional variance turns that statement into an observable quantitative restriction. The binary example shows that these restrictions are sharp and that nearly complete recovery can coexist with a small calibrated response separation. A credible exclusion therefore compares measured values with the bound, rather than describing recovery as high and noise as low without a numerical threshold.

The coherent kick and eraser example supplies a contrasting accessible channel with nonzero branch separation and perfect ideal recovery. Its distinction from the classical-record model depends on the coherent availability of the probe record. Uncontrolled dephasing, finite measurement resolution, or restricted control can obscure that distinction; a gravitational interpretation further depends on the interaction and competing backgrounds. General stochastic or hybrid theories cannot be assigned a universal position in this response–recovery space merely from their names.

The original witness organization can be expressed by

$${\mathcal{W}}_{n,m}\left(p\right)=\left({\kappa }_1,{\Sigma }_{\mathrm{br}},{\kappa }_3,{\kappa }_4,\dots ,R_{\mathrm{ad}}^{\mathrm{ach}},R_{\mathrm{joint}}^{\mathrm{ach}}\right),$$

(57)

accompanied by the conditional probe laws and independent noise, trajectory, and calibration constraints. The superscript denotes achieved recovery in the stated controls; a visibility-loss residual is derived from those data and adds no independent measurement. Their strongest present use is to test explicitly derived models. Additional branches or constituent compositions can be valuable when they make predictions identifiable, but redundant settings cannot remove an exact signal–background degeneracy. Establishing a gravitational discriminator requires the full rival dynamics, a justified apparatus model, and a separation that survives their allowed uncertainty sets.