Calibrated Obstructions and Nuisance-Cancelling Controls for Gravity-Related Decoherence A Superconducting-Microsphere Interferometer: Regularized Dynamics, Preparation Records, and Detector Certificates

We develop a paired negative and constructive identifiability analysis for the superconducting-microsphere interferometer proposed by Pino and collaborators. For a shared residual force-diffusion amplitude independently restricted to an interval, proportional quadratic environmental and gravitational responses produce an exact complete-record match whenever the gravitational coefficient shift does not exceed the calibration width. This establishes family overlap, not containment of the entire alternative family. We distinguish calibration of the environmental source from calibration of total centre-of-mass heating, and derive a regularized-model near-obstruction bound with explicit domain and implementation errors.

A complementary three-separation contrast cancels a common preparation offset and quadratic environmental attenuation. For a continuous homogeneous sphere at separations $R/4,R/2,3R/4$, its unsmeared gravitational coefficient is $-5041/81920$, in units $G{M}^2/(\hslash R)$. We prove that the contrast remains strictly negative for every finite Gaussian regularization length, obtain a uniform quantitative lower bound for $0<r_{\mathrm{c}}\le R/100$, and give validated spectral quadrature and large-smearing bounds. A robust signed contrast margin survives only when setting-dependent preparation, environmental-shape, trajectory, and sensing errors are sufficiently small.

Log-visibility inference is connected to complete attempted-run records through confidence intervals; it is not treated as a bounded per-run statistic. A separate bounded-statistic route provides uniform sample certificates. The analytical results and embedded regressions do not claim reproduced apparatus performance or an experimental exclusion.

Figure fig:scope-En01-B, Figure fig:split-En01-B, Figure fig:chain-En01-B, Figure fig:pair-En01-B, and Figure fig:gate-En01-B name the stop. Not a proof, and not an admission of a quantum coefficient.

A reduced fringe contrast does not identify the source of decoherence. The relevant question is whether a calibrated environmental family can reproduce the same recorded outcomes as a specified additional gravity-related dynamical model.

We use the staged superconducting-microsphere proposal of Pino and collaborators Pino2018 as the experimental reference. Its long-wavelength localization model motivates a residual force-diffusion nuisance, while its preparation and sensing stages require explicit accounting of squared-position outcomes and failures. The reference is a proposal, not a completed gravitational-decoherence experiment.

The contribution has two linked parts:

1. Calibrated obstruction. A shared uncertain residual amplitude can leave an exact non-identifiable region even after an environmental calibration interval is supplied. The matching condition is imposed across all stages and settings.

2. Nuisance-cancelling control. A specified separation contrast eliminates selected constant and quadratic attenuation terms, but retains a nonquadratic gravitational response with a proved regularization-dependent margin.

These conclusions concern different models and must not be conflated. The obstruction is exact in the declared quadratic dynamical model. The positive contrast is a controlled attenuation diagnostic whose application to the full apparatus requires a residual ledger. Coefficient separation alone is not detector separation.

All analytical results are proved here. External inputs are the reference architecture, the motivation for regularized mass-density models, and the radiation constraints cited in Section sec:radiation. Neither a manuscript registry nor an unverified companion result is a mathematical premise.

En01-A En01A provides conceptual background on accessible state differences. Here the difference is between dynamically constructed outputs, not a state and its record pinching. Its copy-screening and Möbius results are not imported.

{Proposed reference values from Pino2018 (Sec. V and Appendix). The listed rounded post-cooling durations sum to $596.2$ ms. They are not demonstrated capabilities.}

The order is

The three control separations introduced below are proposed settings; their feasibility is not established by the single reference separation.

An attempted run includes its preparation record, its accept/reject flag, sensing failures, and detector bins. A quantum instrument is a countably additive family of completely positive maps whose total map is trace preserving. For a continuous outcome, instrument integrals, rather than individual density elements, are trace nonincreasing.

Let ${\Phi }_{j,h}$ be propagation maps and ${\mathcal{I}}_{4,h}(\mathrm{d}z),{\mathcal{I}}_{7,h}(\mathrm{d}y)$ the finite sensing instruments. The successful record law is schematically

Rejection and failure outcomes complete its mass to one. Later maps may depend on earlier accessible outcomes.

The $h$-dependence of a sensing instrument includes dynamics during its finite interaction time. It is not a choice of different electronics for the two hypotheses. A common terminal detector channel is applied only after completing the hypothesis-dependent sphere–sensor evolution.

For an impulsive diagnostic, a normalized squared-position instrument has Kraus density

In a resolved positive-outcome regime its packet-centre separation is approximately $2\sigma \sqrt{z/\chi }$. Replacing finite split sensing by this impulsive map requires a separate approximation bound.

Let $\mu$ be a positive finite intrinsic mass measure on ${\mathbb{R}}^3$, with total mass $M$, and let

The width $r_{\mathrm{c}}$ is the standard deviation in each Cartesian direction. A fixed-orientation rigid body moves along a unit vector $e$, with centre-of-mass position operator $\hat{x}$ on $L^2(\mathbb{R})$:

We declare the generator

The alternative adds $\lambda {\mathcal{G}}_{r_{\mathrm{c}}}$, with $\lambda \ge 0$. Unit strength means this normalization, not every convention called Díosi–Penrose in the literature. Regularization and normalization are physical model specifications Bahrami2014,TilloyDiosi2017.

Our Fourier convention is

Thus ${\hat{f}}_{r_{\mathrm{c}}}(k)=e^{-{r_{\mathrm{c}}}^2|k{|}^2/2}\hat{\mu }(k)$.

Put $k_e=k\cdot e$, $U_k=e^{i{k}_e\hat{x}}$, and

Then $Q_{r_{\mathrm{c}}}$ is a finite positive even measure and

Writing $q_{r_{\mathrm{c}}}=Q_{r_{\mathrm{c}}}({\mathbb{R}}^3)$,

Its exponential is a CPTP semigroup on trace-class operators. The position coherence rate is

Proof. Since $|\hat{\mu }(k)|\le M$,

Fourier transforming $1/|r|$ gives $4\pi /|k{|}^2$. On smooth finite-rank position kernels, substitution in (eq:DP) gives multiplication by $-{\Gamma }_{r_{\mathrm{c}}}(x-x^{\prime })$. Evenness removes the sine part in (eq:kick). The bounded integral extends the identity to trace-class operators.

For $q_{r_{\mathrm{c}}}>0$, define

Then

is a norm-convergent CPTP mixture. The zero-rate case is the identity. The norm bound follows from the norm-one unitary conjugations.

For every tested setting and nuisance value, the common Hamiltonian, environmental evolution, and apparatus couplings define a strongly continuous CPTP propagator with the composition property. Its instruments are well defined. Unbounded diffusion terms, when used, have admitted realizations and the same operation domains in matched comparisons.

Under Assumption ass:baseline, adding $\lambda {\mathcal{G}}_{r_{\mathrm{c}}}$ gives a unique strongly continuous CPTP bounded-perturbation propagator.

Proof. Interleave the baseline propagator with Poisson-distributed jumps of rate $\lambda q_{r_{\mathrm{c}}}$, applying $\mathcal{J}$ at each jump. The sum over ordered jump times converges in trace norm, with factorial bounds, and is CPTP. It solves the bounded variation-of-constants equation. Picard iteration with bound $2\lambda q_{r_{\mathrm{c}}}$ proves uniqueness.

The gravitational contribution is included during cooling, pre-split evolution, and finite sensing. Boundedness of ${\mathcal{G}}_{r_{\mathrm{c}}}$ does not make the kinetic Hamiltonian or the position operator bounded.

For two displaced intrinsic profiles,

and Fourier substitution gives $E_G(d)=\hslash {\Gamma }_{r_{\mathrm{c}}}(d)\ge 0$.

A continuous homogeneous sphere has

so

A microscopic constituent specification instead has

Its cross terms cannot be discarded without an approximation argument. Writing

its rate is

This follows by convolving two Gaussian mass profiles and using their Newtonian potential.

Finite-width centre-of-mass packets are states of $\hat{x}$, not intrinsic mass profiles. Their coherence kernel experiences ${\Gamma }_{r_{\mathrm{c}}}(x-x^{\prime })$ at every position pair. Packet width is not interchangeable with $r_{\mathrm{c}}$.

Define

For $r_{\mathrm{c}}>0$, these are finite and

The quadratic generator is

For isotropic profiles,

Proof. Integrate $0\le z^2/2-(1-\cos z)\le z^4/24$. Angular means of ${k_e}^2$ and ${k_e}^4$ are $k^2/3$ and $k^4/5$. Gaussian decay controls all moments.

Let ${\mathcal{K}}_x(\rho )=[\hat{x},[\hat{x},\rho ]]$. The narrow residual model is

Here $f_s,g_s\ge 0$ are declared functions of stage, time, and accessible history. The same amplitude $b$ is used across settings. Independent environmental characterization supplies

The reference's vibrational localization has a potential-dependent transfer function and can vanish during free motion Pino2018 (Sec. II). Thus proportionality to a persistent gravitational contribution is not inferred from the words “force noise”. It is an explicit model condition.

Suppose the two hypotheses use the same interval $I$, all known dynamics and apparatus specifications match, and

at every relevant stage, setting, and history, with one constant $c\ge 0$. Put $a_G=\lambda A_{r_{\mathrm{c}}}c$.

The coefficient intervals $I$ and $I+a_G$ overlap exactly when

Whenever (eq:obstruction) holds, at least one null–alternative pair has identical complete attempted-run laws, including preparation, heralding, finite sensing, and failures. For that pair, every test satisfies

Consequently no test has both errors uniformly below $1/2$ over the entire two families.

This is family overlap, not containment of every alternative member in the null. When $a_G>\Delta b$, the coefficient intervals are disjoint, but observable separation does not follow without a detector argument.

Proof. Matching requires $b_0=b_1+a_G$. Such values in $I$ exist precisely when $[b_{-},b_{+}]\cap [b_{-}+a_G,b_{+}+a_G]\ne \varnothing$, which for $a_G\ge 0$ is (eq:obstruction). One explicit pair is $b_1=b_{-}$, $b_0=b_{-}+a_G$.

For this pair the total generators in (eq:scalar0)–(eq:scalar1) agree. Uniqueness gives identical propagation. The same apparatus coupling and instrument give identical unnormalized outcome states. Induct over stages and accessible histories to obtain identical complete laws.

For a randomized test $\phi \in [0,1]$ declaring alternative,

The remaining qualifications follow from interval overlap alone.

If $c=0$, or all $f_s$ vanish on the effective observation domain, there is no gravitational shift in this quadratic comparison. Otherwise nonproportional $f_s,g_s$ cannot be matched by this scalar shift at the generator level. That failure is not, by itself, an identifiability theorem.

Let $C$ denote calibration data and $Z$ the interferometer record.

Write the joint laws as

They are equal if and only if their calibration marginals are equal and their conditional interferometer laws agree almost everywhere under the common marginal.

A match of interferometer laws after fixing an interval does not imply a match of the joint laws when an external calibration distinguishes the corresponding environmental amplitudes.

Proof. Equality of joint laws implies equality of marginals. Uniqueness of conditional probabilities up to marginal null sets gives the conditional equality. Conversely, integrate the common conditional law against the common marginal.

Theorem thm:width is therefore an obstruction for inference using the declared interval and interferometer records. If raw external calibration data are also used, their laws belong in the comparison. Coverage of an interval alone does not establish equality of those laws.

For a pure force-diffusion term $-D(t){\mathcal{K}}_x$, its contribution to momentum heating is

If two models have identical centre-of-mass generators throughout a heating calibration and the interferometer, all admitted centre-of-mass calibration instruments have identical laws. Total heating alone does not identify the environmental part of $D_{\mathrm{total}}$.

Proof. On a common moment domain,

Move the double commutator to the observable. This is the diffusion contribution; Hamiltonian or damping terms may also change $⟨p^2⟩$.

Identical generators give identical states before each heating instrument and hence identical recorded distributions, as in Theorem thm:width.

An external force sensor, gas characterization, or bath-resolved measurement can be an independent source calibration. A measurement of total centre-of-mass diffusion is not automatically such a calibration.

In a model with bounded self-adjoint position operator $\Vert \hat{x}\Vert \le L$,

Proof. Evenness writes the generator as $\int [\cos (k_e{\mathrm{ad}}_x)-\mathrm{id}]Q_{r_{\mathrm{c}}}(\mathrm{d}k)$. The cosine family is the average of two unitary-conjugation trace-norm isometries. Taylor's integral remainder after the quadratic term is bounded by $|k_e{|}^4\Vert {\mathrm{ad}}_x{\Vert }^4/24$. Since $\Vert {\mathrm{ad}}_x{\Vert }_{1\to 1}\le 2L$, integration proves the result.

Choose $b_0=b_1+\lambda A_{r_{\mathrm{c}}}c$ in the calibration interval. Replace the gravitational quadratic term by $\lambda g_s(t){\mathcal{G}}_{r_{\mathrm{c}}}$. Suppose bounded-aperture comparison models have $\Vert \hat{x}\Vert \le L_s(t)$, and the actual complete laws are within total variation $e_h$ of those comparison laws.

With uniform history-wise bounds when feedback is used,

where

The sums denote the stages of an attempted run, not an unweighted sum over alternative experimental designs. A bound exceeding one is replaced by the trivial bound one.

For $N$ independent attempted runs,

Thus uniformly small errors ${\alpha }_N,{\beta }_N\le \alpha <1/2$ require $N\zeta \ge 1-2\alpha$.

Proof. After matching the scalar shift, the generator difference is

The first norm is bounded by Proposition prop:aperture. The second is at most $4\lambda A_{r_{\mathrm{c}}}{L_s}^2|g_s-c{f}_s|$. Variation of constants and CPTP contractivity bound the state trace distance by half the integrated generator norms. Complete instruments and terminal processing contract the same norm. Add the two law-approximation errors.

For product laws, $\mathrm{TV}(p^{\otimes N},q^{\otimes N})\le N\mathrm{TV}(p,q)$ by telescoping. Every test has error sum at least one minus product-law total variation.

A scalar rate remainder is not a full-protocol trace-distance estimate. The $e_h$ terms must cover aperture replacement, boundary dynamics, preparation, sensing, and any other approximation. Spatial tail probabilities alone need not control the change of the Hamiltonian at a truncation boundary.

Let $\rho (t)$ be an admitted exact evolution and $\hat{\rho }(t)$ a Hermitian normalized absolutely continuous approximation on a common variation-of-constants domain. If

is integrable in trace norm, then

The statement also applies to direct sums of unnormalized instrument blocks.

Proof. Propagate the initial difference and the residual integral. The admitted propagator contracts Hermitian trace norm.

For a setting with fixed effective separation $d_i$, suppose

The common offset and quadratic coefficient can differ between fitted hypotheses, but each is shared across the settings of its hypothesis.

The constant and quadratic forms are motivated by saturated short-wavelength and long-wavelength localization, respectively Pino2018 (Sec. II). Actual gas-scattering localization and setting-dependent preparation are not assumed to obey them exactly; their departures enter $r_{h,i}$.

Equation (eq:Y) is a controlled diagnostic, not the complete seven-stage prediction.

Choose

Then

Define

Under (eq:Y),

If $|r_{h,i}|\le {ϵ}_{h,i}$, put $E_h={\sum }_i|w_i|{ϵ}_{h,i}$. The contrast intervals are

For a fixed alternative they are disjoint exactly when

with remaining gap

For an alternative family, a common oriented certificate instead requires one sign $\sigma \in \{-1,1\}$ and

An infimum of absolute values does not establish a common oriented threshold when the sign varies.

Proof. Multiply (eq:Y) by the weights and sum. The two nuisance terms vanish. The residual magnitude is bounded by the weighted sum of its component bounds. Two centred closed intervals with radii $E_0,E_1$ and centre distance $|\lambda t{\mathcal{B}}_{r_{\mathrm{c}}}|$ are disjoint precisely under the displayed strict inequality. The family statement applies the same orientation to every alternative.

For a continuous homogeneous sphere,

where

In particular $A_0=G{M}^2/(2\hslash R^3)$.

Proof. The intrinsic density autocorrelation is

It is the overlap volume of two spheres times the squared density. Its Newtonian pairing at displacement $d$ is

Integrating the polynomial separately below and above $r=d$ gives

for $u\le 2$, and $J(d)=M^2/d$ otherwise. The rate is $G[J(0)-J(d)]/\hslash$. Its quadratic coefficient follows at $d=0$.

With $h_{\ast }=5041/81920$,

Proof. The quadratic contribution cancels. The remaining moments are

Therefore

Replacing gravity by a purely quadratic kernel makes the contrast zero. Allowing an independently free preparation offset at each setting also destroys the cancellation premise.

Set

Equation (eq:spectral) gives

The constant term vanishes, and $S(u)=O(u^4)$ at the origin. Its pointwise oscillations do not determine the sign of the complete integral.

For a homogeneous sphere and $r_{\mathrm{c}}>0$, its smeared autocorrelation

is positive, smooth, radial, and strictly decreasing at every radius $r>0$.

Proof. The unsmeared $w_0$ is nonnegative, radial, and decreasing on $0<r<2R$, and vanishes outside that ball. Its layer-cake representation is a positive mixture of indicators of balls of nonzero radii.

For a ball of radius $a>0$, let

By the divergence theorem,

Pair a point with $e\cdot n_z>0$ with its reflection across the plane orthogonal to $e$. For $r>0$, the Gaussian value at the nearer point is strictly larger. The paired integral before the minus sign is positive. Thus $L_a^{\prime }(r)<0$. Integrate this strict inequality against the positive layer-cake mixture. Smoothness and positivity follow from Gaussian convolution.

For every finite $r_{\mathrm{c}}>0$, the function

is strictly concave on $z>0$. Consequently

for the three separations used here. On every compact interval of finite positive regularization lengths, its negative magnitude has a strictly positive minimum.

Proof. Let $I(d)={{\int }_0}^d{r}^2{w}_{r_{\mathrm{c}}}(r)\mathrm{d}r$. The shell theorem gives

Writing $\psi (z)={\Gamma }_{r_{\mathrm{c}}}(\sqrt{z})$, differentiation gives

Strict decrease of $w_{r_{\mathrm{c}}}$ implies $3I(d)>d^3{w}_{r_{\mathrm{c}}}(d)$, so ${\psi }^{\prime \prime }<0$.

For $z_i={d_i}^2$, strict concavity gives

Since $z_2-z_1=3R^2/16$ and $z_3-z_2=5R^2/16$, their slope difference equals $16{\mathcal{B}}_{r_{\mathrm{c}}}/(15R^2)$. This proves the sign.

Continuity in $r_{\mathrm{c}}$ follows by dominated convergence in the spectral integral on compact positive intervals. A continuous strictly positive $-{\mathcal{B}}_{r_{\mathrm{c}}}$ has a positive minimum on such an interval.

The last assertion is an existence result. The next theorem supplies an explicit usable lower bound.

For all $\alpha \ge 0$,

Hence, on $0<r_{\mathrm{c}}\le R/100$,

More generally,

Proof. Because ${\sum }_i{w}_i=0$, $|S(u)|\le {\sum }_i|w_i|=16$. Parseval's identity for the uniform unit-ball density gives

Indeed, $\int {\mu }^2=3M^2/(4\pi R^3)$; applying the stated Fourier normalization proves the identity.

Use

in (eq:b). The bound is $(2/\pi )16(3\pi /2)|{\alpha }^2-{\beta }^2|$. Set $\beta =0$ and apply Corollary cor:exactcontrast. The final rational inequality is direct.

For alternatives with $\lambda \ge {\lambda }_{\min }>0$, $t\ge t_{\min }>0$, and $0<r_{\mathrm{c}}\le R/100$, the oriented contrast $-C$ has a common deterministic gap at least

A positive value certifies separation within the declared attenuation model and residual bounds.

Proof. Use the fixed negative sign and (eq:small) in Theorem thm:contrast.

For the homogeneous sphere,

Thus its constructive margin tends to zero as $\alpha \to \infty$, despite remaining strictly signed for every finite $\alpha$.

Proof. The quadratic terms cancel and

Using $|\hat{\mu }(k)|\le M$,

The weighted fourth moment is $47R^4/32$. Substitute and express the result in units $K_G$.

Let $U\ge 1$, $\alpha >0$, and use composite Simpson quadrature with an even number $n$ of equal intervals on $[0,U]$. For $j=0,\dots ,4$, define

Let $P_0=1$ and

If $P_j(u)={\sum }_{\ell }{p}_{j,\ell }{u}^{\ell }$, put

Then the finite-interval quadrature error for $b(\alpha )$ is at most

The omitted tail is bounded by

Validated nodal intervals and an interval for $2/\pi$, combined with these errors, give a rigorous enclosure for $b(\alpha )$.

An enclosure at ${\alpha }_0$ extends to a whole regularization cell by adding $48{\sup }_{\alpha \text{ in cell}}|{\alpha }^2-{{\alpha }_0}^2|$.

Proof. The normalized sphere form factor is an average of $e^{-iuZ}$ with $|Z|\le 1$, so $|F^{(j)}|\le 1$. Leibniz's rule gives $|(F^2{)}^{(j)}|\le 2^j$. Since $\mathrm{sinc}(vu)={{\int }_0}^1\cos (vtu)\mathrm{d}t$, $|S^{(j)}|\le s_j$. The Gaussian derivative is $P_j{e}^{-{\alpha }^2{u}^2}$, bounded by $e_j$. The product derivative bound is therefore $M_4$. The standard Simpson remainder, obtained by integrating the fourth-order interpolation remainder panel by panel, gives the stated finite-interval error.

For the first tail bound, use $|F|\le 1$, $|S|\le 16$, and

For $u\ge 1$, $|F(u)|\le 6/u^2$. Integrating $16\cdot 36/u^4$ gives the second bound after multiplying by $2/\pi$. The cell extension is Theorem thm:small.

The embedded optional validated calculation uses exact rational interval Taylor evaluations, including a Machin-identity enclosure for $\pi$. It does not assume floating-point trigonometric evaluations are rigorous. Ordinary floating central values are separately labelled diagnostic.

constructive theorem requires every relevant departure to enter its residual budget. An independently free offset at each setting invalidates the cancellation rather than merely increasing precision requirements.

If $|d_i-{\bar{d}}_i|\le {\delta }_i$ and $|{\Gamma }_{r_{\mathrm{c}}}^{\prime }(d)|\le L_i$ on the corresponding windows, then

For the homogeneous sphere, Gaussian smearing gives $A_{r_{\mathrm{c}}}\le A_0$, and