Preparation-Aware Identifiability of Gravity-Related Decoherence in a Superconducting-Microsphere Interferometer Physical Force Noise, Outcome-Conditioned Controls, and Complete-Record Certificates

We examine whether an analytical nuisance-cancelling signature remains useful after Gaussian preparation, outcome-dependent splitting, and finite squared-position sensing in a superconducting-microsphere interferometer. The first result is a limitation: a three-separation contrast that cancels quadratic attenuation also cancels every pre-split hypothesis difference encoded solely in the normalized point-pair coherence of a centred Gaussian state. A common diagonal filter preserves that diagnostic, but finite packet projections and unrecorded outcome averaging require additional amplitude and phase bounds.

We derive the stage transfer of one physical nuisance, displacement of the support of a quadratic potential. Its white-noise localization coefficient is proportional to curvature squared and vanishes in an exactly flat stage. Consequently, this nuisance cannot absorb a persistent gravitational localization contribution at the generator level across a protocol containing a nonzero flat interval. This alone does not prove detector identifiability.

A constructive complete-record result uses four free-expansion times, the same preparation before those intervals, and the same finite squared-position sensor afterwards. An exact third finite difference of the mean split record removes all shared pre-expansion second moments, including preparation changes caused by gravity, and retains the cubic contribution from diffusion during the flat interval. The identity holds for the full regularized kick model at the level of second moments, without assuming Gaussianity. Clipping the recorded outcome produces a bounded statistic with an explicit uniform gap, calibration-width condition, and attempted-run sample certificate. The resulting experiment measures total flat-stage momentum diffusion; source identification requires independent bounds on other sources.

The paper proves these results within declared dynamical and instrument models. An embedded regression artifact evaluates selected reference preparation formulas and finite-sensor identities. No executed apparatus reproduction, validated full-protocol forecast, or experimental exclusion is asserted.

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

En01-B En01B proved calibrated obstructions and a nonquadratic three-separation attenuation contrast. The issue addressed here is whether that contrast identifies the hypothesis difference actually prepared by the staged reference architecture, rather than a different fixed-separation experiment.

The reference proposal uses the sequence

Its preparation stages generate a coherence-length difference that later stages reveal; both splitting and final sensing use squared-position measurement Pino2018. We do not identify this matter-wave proposal with optical interferometric motion readout.

The contribution consists of three results.

1. Preparation-aware cancellation. A contrast cancelling $d^2$ removes the normalized Gaussian preparation-coherence signal, irrespective of whether the changed coefficient arose from environment or gravity.

2. Structured-force distinction. Support-displacement noise has a derived curvature-dependent transfer. Its generator response is not proportional to persistent gravity during a flat stage.

3. Complete-record certificate. A four-time variation of the same staged architecture admits a global bounded-statistic certificate based on the retained split record, including finite sensing and measurement backaction.

The four-time settings change the free-stage duration. They retain the order and types of stages but are not the single fixed design in the reference paper. No additional post-split interrogation interval is introduced.

The central positive theorem is conditional on common physical controls, a flat-stage model, calibrated competing diffusion, and moment/error bounds. It is not merely a local derivative-rank statement. Nor is it evidence that the proposed certificate has useful apparatus-specific power: its retained second-moment signal can be much smaller than a coherence signal.

All mathematical prerequisites used below are stated and proved. En01-A En01A supplies conceptual context on accessible records, not finite-dimensional premises for continuous-variable propagation. Unrelated field-theory and manuscript-registry constructions are not assumptions of this paper.

The retained system is a fixed-orientation rigid body moving along one coordinate $x$, with conjugate momentum $p$ and mass $M$. On a stage with curvature $\kappa (t)$,

Positive, zero, and negative curvature correspond to harmonic, flat, and inverted quadratic stages.

A common environmental localization term is

Cooling may instead have an admitted linear Gaussian drift and diffusion model. We allow its output to depend on the hypotheses, provided it is the same preparation procedure across the four time settings of each hypothesis.

All propagators and sensing instruments used in a comparison are CPTP when their complete outcomes are retained. The unbounded operators have admitted realizations on the required moment domains. The moments differentiated below exist and justify the displayed adjoint-generator identities. Continuous record spaces are standard Borel spaces. Any instrument approximation is accompanied by an explicit complete-law error or a separately derived moment-error bound.

This assumption does not assert that arbitrary unbounded formal master equations define instruments. It fixes the domain requirements of the theorems.

For intrinsic mass measure ${\mu }_{\mathrm{m}}$ of total mass $M$, Gaussian smearing of Cartesian standard deviation $r_{\mathrm{c}}>0$ gives

We use the decoherence-only generator

Here $k_e$ is the component along the retained motion. This is the normalization corresponding to

The alternative adds $\lambda {\mathcal{G}}_{r_{\mathrm{c}}}$, $\lambda \ge 0$.

Since $|{\hat{\mu }}_{\mathrm{m}}|\le M$,

The kick generator is bounded on trace class, with induced norm at most twice that rate. Its exponential is the Poisson mixture of the normalized random unitary kick channel. Thus its decoherence dynamics is well defined independently of the unbounded Hamiltonian; adding it to an admitted baseline is the ordinary bounded perturbation problem.

Define

All kick moments needed here are finite for $r_{\mathrm{c}}>0$. The quadratic comparison is

The full regularized generator and its quadratic comparison have the same action on $x^2,(xp+px)/2,p^2$:

Proof. Conjugation of $p$ by a position kick shifts it by $\hslash k_e$, up to a sign convention irrelevant after integration. Evenness of $Q_{r_{\mathrm{c}}}$ cancels the linear term. The quadratic term integrates to $2{\hslash }^2{A}_{r_{\mathrm{c}}}$. The position is unchanged. The same commutator calculation proves the quadratic-model identities.

The equality is about second moments, not equality of states. Regularized kick evolution is generally non-Gaussian.

A physical split filter below contains the coefficient

with units of inverse length squared. Changing the bookkeeping length $\sigma$ requires $\chi \mapsto \eta {\sigma }^2$ to keep the same apparatus. Holding $\chi$ fixed while setting $\sigma ={\xi }_h$ separately for the hypotheses changes the physical measurement.

The same principle applies to sensor strength, curvature, duration, acceptance windows, phase compensation, and electronic response. They are setting specifications, not quantities that may be silently redefined using the unknown hypothesis.

Let $u(t)$ be a centred classical support displacement. For a quadratic potential,

The last term is scalar and contributes no commutator. For an exactly quadratic translated potential this expansion is exact. For a general potential, its neglected terms require a local approximation bound.

We use the two-sided angular-frequency convention

The white convention is

so the two-sided spectrum is the constant $S_u$. One-sided spectra require their corresponding conversion; no one-sided factor is inserted silently.

In the white Gaussian displacement model, averaging the stochastic Hamiltonian gives

For $\kappa =\pm M{\omega }^2$, its coefficient scales as $M^2{\omega }^4{S}_u/(2{\hslash }^2)$. It is zero when $\kappa =0$.

Proof. Represent the force term by $-\kappa x\sqrt{S_u}\circ \mathrm{d}{W}_t$ in the stochastic Hamiltonian. The Stratonovich density equation has stochastic superoperator

Its Itô correction is $\frac{1}{2}{B}^2\rho =-{\kappa }^2{S}_u[x,[x,\rho ]]/(2{\hslash }^2)$. The stochastic integral has zero mean. The moment-domain or random-unitary realization in Assumption ass:admitted justifies the averaging.

This transfer agrees with the curvature scaling discussed for harmonic vibration in the reference proposal Pino2018. A resonant colored-noise formula is not a substitute for (eq:vib) in every stage.

Suppose the only uncertain force source is the white support amplitude $S_u$, shared across stages. Suppose gravity contributes $-\lambda A_{r_{\mathrm{c}}}[x,[x,\cdot ]]$ with $\lambda A_{r_{\mathrm{c}}}>0$ throughout a nonzero flat interval. Changing $S_u$ cannot match the two total generators on that interval, and hence cannot match them across the complete protocol by the scalar absorption construction of En01-B.

Proof. On the flat interval, the support-noise coefficient is zero for every value of $S_u$, while the gravity coefficient is positive. The double commutator is a nonzero superoperator on its admitted domain. Thus the generators cannot coincide there.

Failure of this matching construction does not exclude accidental agreement after a restricted detector. The constructive theorem below supplies a separate observation-level argument.

For deterministic quadratic dynamics let $T(t,u)$ be its two-dimensional phase-space transition matrix, and let $e_p=(0,1{)}^{\top }$. The noise-induced displacement of $R=(x,p{)}^{\top }$ is

where the repeated letter $u$ in the argument of the stochastic process denotes time, not a second independent noise. Equivalently, writing the time variables as $a,b$, its covariance is

Equation (eq:colored) is the exact added covariance for classical Gaussian support noise independent of the initial quantum state, under the stated quadratic dynamics. It includes correlations between different curved stages. The white limit reduces to

Proof. The force-driven linear Hamilton equations give the stated $\delta R$. Its mean is zero. Take the expectation of $\delta R\delta R^{\top }$ and use the displacement covariance. Independence removes initial-state/noise cross terms. The delta correlation reduces the double integral to the single one.

No force from this mechanism acts during an exactly flat interval. Nevertheless, colored noise can correlate forces before and after that interval. The finite-sensing theorem below uses independent white increments; a colored extension must carry these cross-stage terms or bound their departure from the white model.

For a centred Gaussian state write

For quadratic Hamiltonian and localization coefficient $D(t)$,

Hence

These equations follow by moving the Hamiltonian and double commutators to the three quadratic observables. They preserve Gaussianity in the quadratic localization model. They also hold for second moments of non-Gaussian states under the regularized kick model, by Lemma lem:momentsG.

For an inverted stage with curvature $-M{\omega }^2$,

The curvature contains $M$, not $M^2$. The latter would have the wrong dimensions in the Hamiltonian.

For a flat interval of duration $t$, constant coefficient $D$ gives

Cooling can be represented by its own linear Gaussian covariance equation with drift $B_{\mathrm{c}}$ and diffusion $N_{\mathrm{c}}$. It is not reproduced by substituting a thermal output state into a closed-system harmonic stage. For the restricted reference diagnostic we use the declared post-cooling thermal covariance

This is an input conditional on the preparation characterization, not a new cooling-performance claim.

Fourier inversion of the Gaussian Wigner function gives

Define

These are purity and the reference Gaussian coherence length, respectively Pino2018.

For the kernel (eq:kernel),

where

Consequently, every weight vector satisfying ${\sum }_i{w}_i{d}_i^2=0$ obeys

for each hypothesis $h$.

In particular, $w=(5,-8,3)$ at $d_i=R(1/4,1/2,3/4)$ cancels all hypothesis changes encoded only in $a_{\mathrm{pre},h}$, including gravitational changes. It does not cancel a separately justified nonquadratic post-split response merely because that response is gravitational.

Proof. The off-diagonal numerator is $(2\pi v_x{)}^{-1/2}{e}^{-d^2/{\xi }^2}$. Both diagonal entries are $(2\pi v_x{)}^{-1/2}{e}^{-d^2/(8v_x)}$. Their ratio gives (eq:apre). The uncertainty relation makes its coefficient nonnegative. The weighted logarithm is $a_{\mathrm{pre},h}{\sum }_i{w}_i{d}_i^2$.

For a pure Gaussian, ${\mathcal{V}}_{\mathrm{pair}}=1$. This does not imply high-probability preparation at every separation, nor resolved fringes after a real detector.

A decomposition $Y=Y_{\mathrm{pre}}+Y_{\mathrm{post}}+Y_{\mathrm{readout}}$ requires multiplicative visibility factorization or a controlled approximation. It is not a general identity for successive quantum instruments.

Let $\eta >0$ be fixed in physical coordinates and let

Then

For hypothesis $h$,

For a Gaussian position marginal,

Thus the outcome law depends on position variance, not directly on purity or point-pair coherence.

The normalized likelihood is the classical representation

with $X$ distributed according to the input position density. It is the physical outcome law, not a claim that physical outcomes are the reference Wiener innovations.

For any continuous positive kernel and a diagonal filter $g_z(x)$, the normalized point-pair coherence magnitude is unchanged wherever $g_z(x)g_z(y)\ne 0$ and both input diagonal entries are positive:

The identity is unaffected by normalization of the selected state.

Proof. The numerator acquires $|g_z(x)g_z(y)|$, and the denominator acquires the same factor. A common trace normalization also cancels.

This is a pointwise identity. It is not an assertion that diagonal filtering preserves every two-mode visibility or every measured fringe contrast.

For the diagonal density in ${\tau }_h(z)$, nonzero maxima exist when $z>1/(8\eta v_{x,h})$. Their separation is

This follows by differentiating $-2(\eta x^2-z{)}^2-x^2/(2v_{x,h})$. The inverse square root of the negative log-density curvature at a peak is

This explicitly defined local RMS-width approximation is not assigned an ambiguous “packet width” convention.

Even at the same apparatus outcome $z$, the conditional peak locations can differ between hypotheses. A window selected through a hypothesis-dependent $d_h(z)$ is not one common acceptance rule. Use fixed $z$-windows or calibrated hypothesis-independent bounds.

Choose fixed normalized packet modes $u_{+},u_{-}$ with disjoint supports within distance $\ell$ of centres $d/2,-d/2$. Their real envelopes are nonnegative after one declared common phase compensation. For an unnormalized block $\tau$, define

Positivity gives $0\le {\mathcal{V}}_{\mathrm{mode}}\le 1$. This is a projected two-mode coherence, still not automatically the final fringe visibility.

Suppose the filtered Gaussian kernel on these supports has the form

where, after the common compensation,

on the union of the packet supports. Assume $P_{\pm }>0$. Then

Proof. Put $A_{\pm }=\int u_{\pm }(x)\sqrt{n(x)}\mathrm{d}x>0$. Within either support, $|x-y|\le 2\ell$. Across the supports,

Taking real parts for a lower bound and absolute values for an upper bound yields

Divide by the bounds on $\sqrt{P_{+}{P}_{-}}$ and take logarithms. For $d\ge 2\ell$ the lower logarithmic endpoint differs from $a{d}^2$ by at most $4ad\ell -\log (\cos \theta )$. For $d<2\ell$, the displayed larger bound remains valid. The upper endpoint contributes $4ad\ell +4a{\ell }^2$.

The phase condition must hold for the same physical compensation under both hypotheses. Choosing a separate exact chirp cancellation for each unknown state would not produce a common measurement. The log bound is useful only if the resulting coherence remains strictly positive.

For $z\in W$, project the corresponding block into the same two output modes, possibly after a fixed record-conditioned routing map. Write

Define

Cauchy–Schwarz gives $0\le H_W\le 1$.

If $V(z)\in [V_{-},V_{+}]$, $V_{-}>0$, and all $\alpha (z)$ lie within ${\theta }_W<\pi /2$ of one common phase, then the coherence after forgetting $z$ satisfies

For $H_W>0$,

If parity gives $P_{+}(z)=P_{-}(z)$ almost everywhere, then $H_W=1$.

Proof. Integrate the positive amplitude $\sqrt{P_{+}{P}_{-}}V$. Its projection along the common phase is at least the same integral times $\cos {\theta }_W$; its modulus is at most the integral. Use the visibility bounds and divide by the integrated populations. Parity makes the numerator defining $H_W$ equal either population.

Without the phase bound, two equal-amplitude outcome blocks with phases zero and $\pi$ can cancel completely. Keeping $z$ avoids this particular loss of phase information.

For a Gaussian preparation and a window whose peak separation lies within ${\delta }_d$ of $\bar{d}$,

Combining this bound, Proposition prop:width, and Proposition prop:window supplies a preparation residual for a nominal attenuation contrast. It does not justify replacing that preparation by one free common offset.

During either sensing stage use the continuous measurement model

with record

Here $\mu$ has units ${\mathrm{m}}^{-4}{\mathrm{s}}^{-1}$, and $0\le {ϵ}_{\mathrm{m}}\le 1$ is monitoring efficiency. The dynamics includes the common mean sensor curvature, force noise, and gravity during sensing.

Equation (eq:SME) is a specified reduced sensing model, consistent in form with continuous squared-position monitoring. It is not a proof of cavity elimination for every apparatus. Assumption ass:admitted requires its instrument realization. The reference uses additional approximations to obtain its split filter Pino2018; we do not use those approximations to declare finite sensing impulsive.

For a duration $\tau$, define the recorded number

In the frozen-position efficient limit this gives $Z={\eta }_{\tau }{X}^2+N$, $N\sim N(0,1/4)$, and the earlier filter. For finite motion, it has the exact mean

The expectation is unconditional, so it uses the complete sensing instrument, not an accepted normalized branch.

With constant sensor curvature ${\kappa }_{\mathrm{s}}$ and strength $\mu$, let $m=(v_x,c,v_p{)}^{\top }$. For either quadratic localization or full regularized kicks,

The measurement term contributes

It contributes zero to the other two moment equations.

Define the sensor row

Then

where $b_{\mathrm{s},h}$ is the integrated source term. It can depend on the hypothesis and support-noise amplitude, but is common across settings with identical sensing specifications.

If $\mu >0$, ${ϵ}_{\mathrm{m}}>0$, and $\tau >0$, then $r_x>0$. If ${ϵ}_{\mathrm{m}}=0$, then $r=0$.

Proof. The unconditional double commutator with $x^2$ is averaging over a classical white stochastic quadratic Hamiltonian. Its phase-space dynamics is a random linear symplectic evolution. The homogeneous response of $v_x(u)$ to an initial increment in $v_x$, with the other second moments unchanged, is $\mathbb{E}[T_{11}(u{)}^2]\ge 0$. This is also the $11$-entry of $e^{B_{\mathrm{s}}u}$, by the same closed moment equations. It equals one at $u=0$ and is continuous. Its integral is strictly positive. Multiplication by ${\eta }_{\tau }/\tau >0$ proves the claim. Zero monitoring efficiency makes ${\eta }_{\tau }=0$.

The quadratic-Hamiltonian random-unitary argument is an unconditional moment construction. It does not replace the conditioned measurement instrument.

Let ${\Phi }_{1:3,h,i}$ denote preparation including the setting's free-stage duration, ${\mathcal{I}}_{4,h}(\mathrm{d}z)$ the finite split instrument, and ${\Phi }_{5:6,h,z}$ later record-conditioned propagation. Let ${\mathcal{I}}_{7,h,z}(\mathrm{d}y)$ be finite final sensing and let a common electronic map produce a detector outcome $b$. The complete law includes

together with cooling flags, rejected split records, and failures.

Accepted blocks are propagated unnormalized. A rejected record is not removed from the attempted-run sample. Later trace-preserving dynamics cannot change the already recorded split marginal. The constructive statistic below may therefore ignore final outcomes without pretending that the final sensing instrument is common at the input of its finite interaction.

Choose

and use weights

They satisfy

The cooling and boost procedure is identical across these settings. Its output second moments may depend arbitrarily on the hypothesis, support amplitude, and other shared nuisance parameters. The free interval is exactly flat. Forces from white support displacement consequently vanish there.

Other admitted flat-stage sources have a constant aggregate second-moment coefficient $D_{\mathrm{f}}\ge 0$. Under the two hypotheses,

This notation includes any ordinary kick source with the same constant second-moment heating action, not only Gaussian force diffusion.

The four sensing stages have identical duration, strength, curvature, and electronic normalization. White increments after the free interval are independent of earlier ones. Their integrated source offset is common across the four settings of each hypothesis.

Under the preceding specifications and moment-domain assumptions,

This identity holds for the full regularized kick model, not just its Gaussian quadratic approximation.

All shared preparation second moments cancel, including their gravitational and support-noise changes. The sensor-source offset also cancels. The retained gravitational contribution is its diffusion during the flat interval.

Proof. Insert (eq:free) into (eq:recordlinear). Every contribution from $v_{x,B},c_B,v_{p,B}$ is a polynomial in $t_i$ of degree at most two. The sensor-source offset is constant. The $r_{c}c(t_i)$ diffusion term is quadratic, and the $r_p{v}_p(t_i)$ diffusion term is linear. Only the cubic position term survives:

Lemma lem:momentsG proves that the full kick model has the same second-moment evolution. Lemma lem:rx proves positivity.

The theorem is compatible with En01-B's time-only obstruction. That obstruction assumed proportional environmental and gravitational attenuation templates. Here the selected nuisance has zero coupling in the flat stage, while the added diffusion persists there.

Suppose $D_{\mathrm{f}}\in [d_{-},d_{+}]$ and $\lambda A_{r_{\mathrm{c}}}\ge {A_{\min }}^{\mathrm{add}}>0$ over the tested alternative family. For known $\mathcal{R}$, the untruncated mean ranges have gap

A positive value gives global mean separation, not just local rank.

If sensor or timing calibration makes $\mathcal{R}$ uncertain, use the joint-family extrema

with any certified mean departures subtracted. Different free-time experiments cannot be assigned independent arbitrary preparation covariances within these families.

For complete-record density or mass function $p_i$, differentiation of (eq:fourtime) yields, when differentiation under the moment integral is justified,

Thus gravity is transverse to the support-amplitude derivative at this observation boundary when controls and competing diffusion are fixed. This calculation supplies a separating functional for those two derivatives; it does not remove other nuisance columns automatically.

If the null admits an additional free-stage momentum-diffusion coefficient that can absorb $\lambda A_{r_{\mathrm{c}}}$, the four-time second-moment signature is matched exactly. In the quadratic gravity model, admitting this same shift throughout all required stages can also match the entire dynamics.

Therefore the certificate identifies gravity only relative to independently restricted competing sources. It is not source identification from heating alone.

Proof. The first assertion follows from equality of $D_{\mathrm{f}}+h\lambda A_{r_{\mathrm{c}}}$ in (eq:fourtime). For the second, matched quadratic coefficients give identical total generators and therefore identical complete instrument laws.

A small nonzero free curvature couples support displacement back into that stage. Its response must be calculated rather than called flat. Likewise, nonconstant competing diffusion, setting-dependent reset states, colored stage correlations, or a setting-dependent sensing coefficient can create cubic or higher-order nuisance responses. Their mean or complete-law departures enter the next theorem.

The mean signature involves an unbounded record. It cannot be inserted directly into Hoeffding's inequality. For $L>0$, let

If ${\mathbb{E}}_h{Z_i}^2\le B_{h,i}$, then

Indeed, $|z-c_L(z)|\le z^2/L$.

Put

These bounds must be uniform on the tested joint families.

A finite-sensor record bound is available directly from (eq:recordY). Writing $m_t=\mathrm{tr}(x^2{\rho }_{\mathrm{c},t})$,

Using $(a+b{)}^2\le 2a^2+2b^2$, Jensen's inequality, and ${m_t}^2\le \mathrm{tr}(x^4{\rho }_{\mathrm{c},t})$, we obtain

No independence between the innovation and the conditional state was assumed in this estimate.

Randomize the four settings independently with probability $1/4$. Suppose their ideal mean ranges have gap $G_T>0$, including any certified mean-model errors. Suppose the uniform record second-moment bounds above hold.

If the split record is always available, define the predeclared bounded statistic

It belongs to $[0,1]$. If instead the split record is independently erased with common known probability $1-\gamma$, $0<\gamma \le 1$, replace the second term by $a_i{I}_{\mathrm{observed}}{c}_L(z)/(6L)$, scoring an erased record by $1/2$.

Let actual complete attempted-run laws be within total variation $e_h$ of the corresponding specified laws, uniformly over the joint families. Then the actual oriented expectation gap is at least

where $\gamma =1$ in the always-recorded case.

If $g_{\ast }>0$, the midpoint of the enlarged null and alternative expectation ranges gives a test with each error at most

after $N$ independent attempted runs from a fixed actual family member. Thus

suffices for each error to be at most $\alpha$. A simultaneous calibration coverage probability at least $1-\beta$ gives unconditional error at most $\alpha +\beta$.

Proof. Since ${\max }_i|a_i|=3$ and $|c_L|\le L$, the statistic is bounded. Randomization gives

The independent-erasure condition gives the factor $\gamma$. Equation (eq:clipping) reduces the ideal gap by at most $B/L$. For a statistic in $[0,1]$, total variation changes its expectation by at most $e_h$, giving (eq:g).

The midpoint is at least $g_{\ast }/2$ from each allowed expectation. Hoeffding's one-sided inequality gives $\exp [-2N(g_{\ast }/2{)}^2]$ Hoeffding1963. Uniformity follows from the same expectation bounds for every family member. Finally, combine test and calibration failure events by a union bound.

The independent-erasure model is a specific loss model. State-dependent cooling failures, split-record loss, or selection cannot be assigned the factor $\gamma$ without validation. They must be retained in the law and handled by a different response model or error bound. Split rejections after recording $z$, and subsequent terminal detector failures, do not erase this statistic's information.

For fixed $G_T>0$, $B>0$, and $\gamma >0$, the nominal gap before complete-law errors is maximized at

with value

Consequently a positive certificate of this form exists exactly when

provided the clipping scale is unconstrained. If hardware limits the record range, optimize only over admissible $L$.

Proof. Maximize $\gamma (G_T/L-B/L^2)/24$ over $L>0$. Its derivative vanishes at $2B/G_T$, where the unique positive maximum has the stated value.

For known transfer and diffusion intervals, $G_T>0$ first requires $\lambda A_{r_{\mathrm{c}}}>\Delta D_{\mathrm{f}}$ uniformly. Even then the bounded record certificate can be extremely weak. Large shared support noise cancels in the mean identity but increases the moment bound $B$, and hence increases the sample cost.

A valid alternative is a simultaneous confidence region for the complete finite histogram. For $m$ bins and $N$ independent attempted runs,

holds for all bins with probability at least $1-\alpha$. The bins include setting labels, preparation flags, split outcomes, and terminal failures.

A model family is rejected only if no admissible joint nuisance value, with its declared complete-law error, intersects this region. This is a calibrated procedure even when no common linear statistic has been found. It is not an asserted apparatus-specific rejection.

Equations (eq:covsolution) and (eq:free) give an independent derivation of the quadratic Gaussian preparation evolution. Equation (eq:splitlaw) reproduces the normalized diagonal-filter outcome calculation in physical coordinates. These formula identities do not require numerical execution.

A useful restricted numerical comparison uses rounded design inputs:

For this diagnostic, thermal radiation is omitted and support noise acts during the boost but not the exactly flat free stage. Gravity is compared at the unsmeared continuous-sphere quadratic coefficient $A_0=G{M}^2/(2\hslash R^3)$. This is not a microscopic unsmeared collapse prescription.

The reference's rounded comparison values are $\sqrt{v_x}\approx 239$ nm at the split input, ${\xi }_0\approx 607$ nm, and ${\xi }_1\approx 121$ nm Pino2018 (Appendix A.5). The embedded artifact prints calculated values and discrepancies, rather than inventing a completed comparison record.

The subtraction $v_x{v}_p-c^2$ is severely ill-conditioned after boosting. The reference-formula diagnostic therefore uses 70-digit decimal arithmetic, not a determinant formed by subtracting large double-precision numbers. Higher arithmetic precision is not a physical error bound.

in a lower layer is not validation of a higher layer. The cooling output used in the diagnostic is a declared input.

During quadratic Hamiltonian evolution, linear Gaussian diffusion, and regularized kicks, polynomial moments through degree four form a finite triangular system. For efficient computation during sensing, use Weyl-ordered moments