Quantum Probability as Accessible-Event Statistics on Admissible Effect Domains: Residual Event Sharpness and a Recoverability Gap

Quantum theory separates imported globally admissible propagation structure from the accessible-event statistics assigned at a finite physical interface. We write $\Gadm$TeX\Gadm for the imported globally admissible propagation structure and $\Ievt$TeX\Ievt for the accessible event layer. The probability rule is assigned to $\Ievt$TeX\Ievt, while detector microtheory and full laboratory protocols remain external to the present event-domain analysis. No detector opening, threshold, durable-readout, or shared-record condition is derived here.

The analysis has two parts. First, once accessible events are represented on an admissible sharp/effect domain, imported projector-domain and effect-domain representation theorems imply the Born--trace form. Second, if a declared calibration family assigns realized event interfaces a family-relative sharpness coordinate $r$TeXr, then a nonzero residual floor $r_{\min}>0$TeXr_{\min}>0 blocks complete recovery of the family's no-record reference visibility.

Current measurement-characterization protocols fix event operators relative to a declared comparison frame rather than redefining them after inspection of the data. Measurement calibration, gate set tomography, gauge-free tomography, self-consistent measurement tomography, simultaneous state-and-measurement tomography, detector and instrument characterization, recent mid-circuit measurement benchmarking and readout-correction protocols, and recent precision-bound analyses for detector estimation all operate at that level [citation]. Recent detector-evaluation and photocounting analyses also emphasize that detector metrics, dead-time/afterpulse corrections, and reporting conventions must be declared before comparison, which is exactly the level at which the family-internal comparison below is formulated [citation].

Accessible event identities

definition: Accessible event identity. An accessible event identity is a stable outcome class represented by an operator on the declared event domain. In the realistic branch of the analysis that operator is an effect $E$TeXE with $0\le E\le \Id$TeX0\le E\le \Id; in the sharp limit it is a projector $P=P^2=P^\dagger$TeXP=P^2=P^\dagger. Two realizations instantiate the same accessible event identity only when they induce the same operator on the declared event domain.

The event-identity condition is imposed only on the declared event domain. It is not a universal noncontextuality postulate over arbitrary preparation, transformation, and readout contexts.

Minimal admissibility conditions

definition: Admissible event probabilities. A probability assignment $p$TeXp on the accessible event domain is admissible when it satisfies:

- Normalization: $p(\Id)=1$TeXp(\Id)=1. - Positivity: $0\le p(X)\le 1$TeX0\le p(X)\le 1 for every event operator $X$TeXX in the domain. - Orthogonal additivity on sharp events: for mutually orthogonal projectors $\{P_i\}$TeX\{P_i\} with $\sum_i P_i\le \Id$TeX\sum_i P_i\le \Id, equation p (\sum_i P_i)=\sum_i p(P_i). equation - Event-identity stability: the same event operator carries the same probability. - Operational continuity: small perturbations of the event operator induce small perturbations of the assigned probability.

These conditions suffice for the present event-domain analysis. Normalization expresses completeness of an event cycle, positivity expresses frequency interpretability, orthogonal additivity encodes mutual exclusivity on the sharp domain, event-identity stability prevents redundant realization changes from altering the assigned statistics, and operational continuity excludes wildly unstable assignments.

Representation theorems on the sharp/effect domain

The next two propositions record imported sharp-domain and effect-domain representation-theorem premises used at the admissible event level. The family-relative recoverability statements below are formulated on top of those imported premises and are not themselves derivations of the Born--trace rule.

proposition: Imported projector-domain Born--trace premise on the declared admissible projector domain. Let $\Khil$TeX\Khil be a complex Hilbert space with $\dim\Khil\ge 3$TeX\dim\Khil\ge 3. Let $p$TeXp be a normalized countably additive probability assignment on the projector lattice of $\Khil$TeX\Khil. Then there exists a unique density operator $\rho$TeX\rho such that

p(P)=\Tr(\rho P)

for every projector $P$TeXP.

This is the standard projector-domain uniqueness content of Gleason's theorem [citation]. This exact projector-domain identity is read only on the declared admissible projector domain and is not a family-relative recoverability or benchmark statement. The usual qubit caveat remains: projector data alone do not force Born form in dimension two without additional structure.

proposition: Imported effect-domain Born--trace premise on the declared admissible effect domain. Let $\Khil$TeX\Khil be a complex Hilbert space with $\dim\Khil\ge 2$TeX\dim\Khil\ge 2. Let $p$TeXp be a normalized generalized probability measure on the effect domain, additive on finite POVM decompositions and satisfying $\sum_k p(E_k)=1$TeX\sum_k p(E_k)=1 for every finite POVM $\{E_k\}$TeX\{E_k\}. Then there exists a unique density operator $\rho$TeX\rho such that

p(E)=\Tr(\rho E)

for every effect $E$TeXE in the domain.

This is the effect-domain route associated with Busch's generalized Gleason theorem [citation]; recent probability-based generalizations extend the same effect-domain logic [citation]. This exact effect-domain identity is read only on the declared admissible effect domain and is not a family-relative recoverability or benchmark statement. Once the accessible event layer is represented on an admissible sharp/effect domain, the stable probability rule on that domain is Born--trace. Realistic event interfaces are then parameterized only through declared calibration families and family-relative sharpness without deforming that rule.

definition: Declared calibration family. A declared calibration family is a tuple

\mathcal F=(\mathcal P,\mathcal O_V,\mathcal U,\{P_k\},\{N_k\},\{E_k(r)\}_{r\in[0,1]},\mathsf{Est}),

where:

- $\mathcal P$TeX\mathcal P is a fixed platform class and geometry, - $\mathcal O_V$TeX\mathcal O_V is a fixed visibility observable and extraction rule, - $\mathcal U$TeX\mathcal U is a declared suppression-control axis, - $\{P_k\}$TeX\{P_k\} is a fixed sharp target projector-valued measurement (PVM) on the same outcome set as $\{N_k\}$TeX\{N_k\}, with $P_jP_k=\delta_{jk}P_k$TeXP_jP_k=\delta_{jk}P_k and $\sum_k P_k=\Id$TeX\sum_k P_k=\Id, - $\{N_k\}$TeX\{N_k\} is a fixed no-record reference POVM family for the declared comparison, with $\sum_k N_k=\Id$TeX\sum_k N_k=\Id, - $\{E_k(r)\}$TeX\{E_k(r)\} is a nested realized-effect family with $E_k(1)=P_k$TeXE_k(1)=P_k and $E_k(0)=N_k$TeXE_k(0)=N_k, - and $\mathsf{Est}$TeX\mathsf{Est} is a fixed estimator, uncertainty budget, and analysis-window rule.

All downstream family-relative sharpness, recoverability-gap, and reported lower-bound claims are read only relative to this declared calibration family.

All comparisons below are internal to a family fixed before recovery analysis. The reference POVM family, visibility observable, and estimator remain fixed throughout the recovery comparison.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

Associated with each declared family is the fixed analysis convention

\mathcal C_{\mathcal F}:=(\mathcal O_V,\{N_k\},\mathsf{Est},\delta_V,\delta_r,\delta_m,\delta_\Delta,W_{\mathcal F},1-\alpha),

where $W_{\mathcal F}$TeXW_{\mathcal F} denotes the family fit window, $1-\alpha$TeX1-\alpha the declared one-sided coverage convention, $\delta_V,\delta_r,\delta_m$TeX\delta_V,\delta_r,\delta_m the associated visibility, sharpness, and local-slope radii, and $\delta_\Delta$TeX\delta_\Delta any additional one-sided report radius used for published gap statements on that same convention. All downstream quantitative recovery and reported lower-bound claims are read only on this fixed analysis convention, including the declared operating window $W_{\mathcal F}$TeXW_{\mathcal F} and one-sided coverage convention $1-\alpha$TeX1-\alpha.

A declared affine family

The affine family

E_k(r)=rP_k+(1-r)N_k,
  \qquad 0\le r\le 1,

provides a declared comparison path from the no-record reference POVM to the sharp target family. Because $\{P_k\}$TeX\{P_k\} is a PVM and $\{N_k\}$TeX\{N_k\} is a POVM on the same outcome set, each $E_k(r)$TeXE_k(r) is an effect and

\sum_k E_k(r)=r\sum_k P_k+(1-r)\sum_k N_k=\Id.

Hence $\{E_k(r)\}_k$TeX\{E_k(r)\}_k is a POVM for every $r\in[0,1]$TeXr\in[0,1]. The parameter $r$TeXr is a family-relative sharpness coordinate: $r=1$TeXr=1 is the sharp target limit and $r=0$TeXr=0 is the declared no-record reference limit. Generalized-measurement realizations and detector-characterization programs give practical examples in which nonprojective event operators are the natural level of description [citation].

proposition: Family-relative identifiability in the affine family. Fix $\{P_k\}$TeX\{P_k\} and $\{N_k\}$TeX\{N_k\} in reference and assume that $P_j\neq N_j$TeXP_j\neq N_j for at least one index $j$TeXj. If

E_k(r)=E_k(r')
  \qquad \text{for all } k,

then $r=r'$TeXr=r'.

proof. For any index $j$TeXj with $P_j\neq N_j$TeXP_j\neq N_j,

E_j(r)-E_j(r')=(r-r')(P_j-N_j).

If $E_j(r)=E_j(r')$TeXE_j(r)=E_j(r'), then $(r-r')(P_j-N_j)=0$TeX(r-r')(P_j-N_j)=0. Since $P_j-N_j\neq 0$TeXP_j-N_j\neq 0, one has $r=r'$TeXr=r'.

Once the comparison family is fixed, the sharpness coordinate cannot be absorbed into a silent replacement of the no-record reference POVM while remaining inside the same declared family.

proposition: Family-fixity and no-retuning. Let $\mathcal F$TeX\mathcal F be a declared calibration family with fixed analysis convention $\mathcal C_{\mathcal F}$TeX\mathcal C_{\mathcal F} from reference. Any change of platform class, visibility observable, suppression axis, no-record reference POVM family, realized-effect family, estimator, one-sided coverage convention, report radii, or fit window defines a new family rather than a refinement within the same family.

proof. These objects are part of the data fixed in reference and reference. Changing any of them changes the comparison problem and therefore leaves the declared family.

Residual sharpness profiles

Let $u\in\mathcal U$TeXu\in\mathcal U denote the declared suppression parameter. A realized event interface is assigned a family-relative profile $r_{\mathcal F}(u)\in[0,1]$TeXr_{\mathcal F}(u)\in[0,1] such that more aggressive suppression does not increase sharpness:

u_1<u_2 \quad \Longrightarrow \quad r_{\mathcal F}(u_1)\ge r_{\mathcal F}(u_2).

A residual sharpness floor exists when there are $u_0$TeXu_0 and $r_{\min}>0$TeXr_{\min}>0 such that

r_{\mathcal F}(u)\ge r_{\min}
  \qquad \text{for all } u\ge u_0.

Because simultaneous state and measurement estimation can carry unavoidable gauge freedom in general [citation], the coordinate $r_{\mathcal F}$TeXr_{\mathcal F} is not presented as a universal detector invariant. It is a calibrated coordinate relative to the declared family $\mathcal F$TeX\mathcal F, which is the level at which contemporary characterization methods actually operate.

Illustrative fixed-family interferometric benchmark

This subsection records one fixed-family interferometric benchmark. Its explicit estimator, coverage convention, and support conditions belong to the reporting convention of this family only; the general recoverability theorem below uses them only through the declared one-sided radii.

A concrete worked family is obtained from a two-outcome interference platform with fixed phase scan $\Theta=\{\theta_j\}_{j=1}^n$TeX\Theta=\{\theta_j\}_{j=1}^n, fixed fringe observable

I(\theta)=A\,[1+V\cos(\theta-\phi_0)]+B,

and fixed sharp target projectors $\{P_+,P_-\}$TeX\{P_+,P_-\} on the declared event space. For the no-record reference we choose

N_+=N_-:=\frac{\Id}{2},

so that the realized family is the affine path

E_\pm(r)=rP_\pm+(1-r)\frac{\Id}{2}.

This same symmetric family is held fixed throughout the benchmark comparison.

For a tomographically reconstructed realized effect pair $\{\widehat E_+,\widehat E_-\}$TeX\{\widehat E_+,\widehat E_-\}, let $\widehat r_{\mathcal F}$TeX\widehat r_{\mathcal F} be the fixed-family sharpness estimator and let $\widehat m_{\mathcal F}$TeX\widehat m_{\mathcal F} be the fitted local visibility slope on the declared near-reference interval; the explicit estimator and component-budget formulas used by the benchmark are recorded in reference. Let $1-\alpha$TeX1-\alpha denote the declared one-sided coverage convention. The benchmark is admissible only together with a declared coverage event

\Omega_{1-\alpha}^{\mathcal F}
  :=
  \left\{
    r_{\min}\ge \widehat r_{\min}-\delta_r
    \quad\text{and}\quad
    m_{\mathcal F}\ge \widehat m_{\mathcal F}-\delta_m
  \right\},
  \qquad
  \Pr\!\bigl(\Omega_{1-\alpha}^{\mathcal F}\bigr)\ge 1-\alpha,

where $\delta_r$TeX\delta_r and $\delta_m$TeX\delta_m are the declared one-sided coverage radii for the sharpness floor and local slope under $\mathcal C_{\mathcal F}$TeX\mathcal C_{\mathcal F}. If a gap floor is published on that same convention, the benchmark also declares a one-sided report radius

\delta_\Delta=
  
    \delta_V,  \text{if }V_{0,\mathcal F}\text{ is analytic and family-fixed},
[3pt]
    \delta_{V_0}+\delta_V\ \text{or a covariance-aware equivalent},  \text{if }V_{0,\mathcal F}\text{ is estimated}.

Support for a nonzero residual floor and for local monotone separation is then reported only when

\widehat r_{\min}-\delta_r>0,
  \qquad
  \widehat m_{\mathcal F}-\delta_m>0.

These inequalities are the fixed-family support conditions used below for a reported recoverability-gap floor.

For a fixed declared family $\mathcal F$TeX\mathcal F, let $V_{\mathcal F}(r)$TeXV_{\mathcal F}(r) denote the visibility assigned by the fixed observable $\mathcal O_V$TeX\mathcal O_V when the realized event interface has family-relative sharpness $r$TeXr. The family no-record reference visibility is

V_{0,\mathcal F}=V_{\mathcal F}(0).

It is fixed by the declared platform, visibility observable, no-record reference POVM family, and estimator.

theorem: Residual sharpness implies a positive recoverability gap within one declared calibration family. Let $\mathcal F$TeX\mathcal F be a declared calibration family. Assume:

- $V_{\mathcal F}:[0,1]\to\mathbb R$TeXV_{\mathcal F}:[0,1]\to\mathbb R is continuous and nonincreasing, - there exists $r_* >0$TeXr_* >0 such that $V_{\mathcal F}$TeXV_{\mathcal F} is strictly decreasing on $[0,r_*]$TeX[0,r_*], - and a residual sharpness floor $r_{\min}>0$TeXr_{\min}>0 exists with $r_{\min}\le r_*$TeXr_{\min}\le r_*.

Then for every sufficiently aggressive suppression setting $u\ge u_0$TeXu\ge u_0,

V_{\mathcal F}(r_{\mathcal F}(u))\le V_{\mathcal F}(r_{\min})<V_{0,\mathcal F}.

Equivalently, with the recovery gap

\Delta_V(u):=V_{0,\mathcal F}-V_{\mathcal F}(r_{\mathcal F}(u)),

one has

\Delta_V(u)\ge \Delta^{\min}_{V,\mathcal F}:=V_{0,\mathcal F}-V_{\mathcal F}(r_{\min})>0.

This exact recoverability-gap statement is internal to one declared calibration family and is not a comparison across altered platform, reference, or estimator conventions.

proof. By reference, one has $r_{\mathcal F}(u)\ge r_{\min}$TeXr_{\mathcal F}(u)\ge r_{\min} for all $u\ge u_0$TeXu\ge u_0. Because $V_{\mathcal F}$TeXV_{\mathcal F} is nonincreasing,

V_{\mathcal F}(r_{\mathcal F}(u))\le V_{\mathcal F}(r_{\min}).

Because $V_{\mathcal F}$TeXV_{\mathcal F} is strictly decreasing on $[0,r_*]$TeX[0,r_*] and $0<r_{\min}\le r_*$TeX0<r_{\min}\le r_*,

V_{\mathcal F}(r_{\min})<V_{\mathcal F}(0)=V_{0,\mathcal F}.

Subtracting $V_{\mathcal F}(r_{\mathcal F}(u))$TeXV_{\mathcal F}(r_{\mathcal F}(u)) from $V_{0,\mathcal F}$TeXV_{0,\mathcal F} gives reference.

corollary: Quantitative lower bound within one declared calibration family. Under the assumptions of reference, suppose in addition that $V_{\mathcal F}$TeXV_{\mathcal F} is differentiable on $[0,r_{\min}]$TeX[0,r_{\min}] and that

-V'_{\mathcal F}(r)\ge m_{\mathcal F}>0
  \qquad \text{for all } r\in[0,r_{\min}].

Then

\Delta_V(u)\ge m_{\mathcal F}r_{\min}
  \qquad \text{for all } u\ge u_0.

proof. By the fundamental theorem of calculus,

V_{0,\mathcal F}-V_{\mathcal F}(r_{\min})
  =
  \int_0^{r_{\min}}\!\bigl(-V'_{\mathcal F}(r)\bigr)\,dr
  \ge
  \int_0^{r_{\min}}\! m_{\mathcal F}\,dr
  =m_{\mathcal F}r_{\min}.

The claim then follows from reference.

remark. If one is willing to assume only that $V_{\mathcal F}$TeXV_{\mathcal F} is nonincreasing, the conclusion weakens to $\Delta_V(u)\ge V_{0,\mathcal F}-V_{\mathcal F}(r_{\min})\ge 0$TeX\Delta_V(u)\ge V_{0,\mathcal F}-V_{\mathcal F}(r_{\min})\ge 0. The strict positivity in reference requires strict decrease near the no-record limit or an equivalent local separation assumption.

proposition: Convention-backed reported family-internal lower bound on the fixed analysis window. Assume the hypotheses of reference. Suppose the fixed-family estimates $\widehat r_{\min}$TeX\widehat r_{\min} and $\widehat m_{\mathcal F}$TeX\widehat m_{\mathcal F} satisfy the support conditions in reference, and let $\Omega_{1-\alpha}^{\mathcal F}$TeX\Omega_{1-\alpha}^{\mathcal F} and $\delta_\Delta$TeX\delta_\Delta be the declared coverage event and gap-report radius from reference and reference. Assume moreover that $\delta_\Delta$TeX\delta_\Delta is valid as a one-sided report radius for the published gap statement on the same fixed analysis convention. Then, on $\Omega_{1-\alpha}^{\mathcal F}$TeX\Omega_{1-\alpha}^{\mathcal F},

\Delta_V(u)\ge
  \Delta^{\mathrm{rep}}_{V,\mathcal F}:=\max\!\left\{0,(\widehat m_{\mathcal F}-\delta_m)(\widehat r_{\min}-\delta_r)-\delta_\Delta\right\}

for all $u\ge u_0$TeXu\ge u_0.

proof. On the declared coverage event $\Omega_{1-\alpha}^{\mathcal F}$TeX\Omega_{1-\alpha}^{\mathcal F}, one has $m_{\mathcal F}\ge \widehat m_{\mathcal F}-\delta_m>0$TeXm_{\mathcal F}\ge \widehat m_{\mathcal F}-\delta_m>0 and $r_{\min}\ge \widehat r_{\min}-\delta_r>0$TeXr_{\min}\ge \widehat r_{\min}-\delta_r>0. By reference,

\Delta_V(u)\ge m_{\mathcal F}r_{\min}\ge (\widehat m_{\mathcal F}-\delta_m)(\widehat r_{\min}-\delta_r).

By the stated validity assumption on the declared one-sided report radius $\delta_\Delta$TeX\delta_\Delta, subtracting $\delta_\Delta$TeX\delta_\Delta preserves a conservative lower bound on the same coverage convention, and the positive part enforces nonnegativity.

The comparison is performed within a single declared family. An apparent recovery to $V_{0,\mathcal F}$TeXV_{0,\mathcal F} obtained by changing the platform class, by changing $\{N_k\}$TeX\{N_k\}, by replacing the visibility observable, or by altering the estimator changes the comparison problem rather than testing reference.

Calibration and tomography

Measurement characterization already treats event operators as objects to be reconstructed or benchmarked relative to specified protocols. Quantum calibration, gate set tomography, self-consistent measurement tomography, gauge-free tomography, simultaneous state-and-measurement tomography, detector tomography, instrument characterization, mid-circuit benchmarking and readout-correction protocols, and recent precision-bound analyses for detector estimation provide concrete instances [citation]. The recoverability result assumes such a declared family together with a fixed reporting convention and derives a consequence for family-internal recovery on that convention. The visibility observable, no-record reference POVM, estimator, coverage rule, and fit window are held fixed throughout the analysis.

Generalized observables and complementarity

Effects and their sharpness properties are well established in the generalized-observable literature [citation]. Wave-particle duality and joint-measurement analyses relate visibility, which-way information, and generalized observables through inequalities and compatibility conditions [citation]. Once the platform class, the visibility observable, the sharp target family, and the no-record reference POVM family are fixed, a nonzero residual event-sharpness floor blocks full recovery of the corresponding no-record reference visibility within that same declared family.

Accessible-event probabilities on an admissible sharp/effect domain take Born--trace form. Within a declared calibration family, a nonzero residual sharpness floor and strict local decrease of the family visibility near the no-record limit imply a positive recoverability gap. A local slope bound yields the quantitative estimate $\Delta_V(u)\ge m_{\mathcal F}r_{\min}$TeX\Delta_V(u)\ge m_{\mathcal F}r_{\min} for all $u\ge u_0$TeXu\ge u_0, and a declared one-sided coverage convention yields the family-internal reported floor $\Delta^{\mathrm{rep}}_{V,\mathcal F}$TeX\Delta^{\mathrm{rep}}_{V,\mathcal F} in reference on the corresponding coverage event.

Nonideality enters through realized effects rather than through a deformation of the probability rule. The recoverability gap is internal to the declared family: it does not compare different platform classes, visibility definitions, no-record reference POVM families, or estimator rules. The illustrative interferometric benchmark records one admissible fixed-family realization of the sharpness coordinate and of the associated reporting convention. The present result remains an event-side probability and family-internal recoverability statement only. No detector microtheory, detector opening law, operational threshold law, durable-readout law, or shared-record semantics is derived here. The positive family-internal recoverability-gap claim is unavailable if the comparison family is changed or if the residual-floor or local-decrease assumptions fail on the stated event domain; the reported lower bound is unavailable if the declared-coverage assumptions fail on the stated convention.

For the symmetric two-outcome benchmark in reference, the family-relative sharpness estimator is obtained by projection onto the declared affine family,

\widehat r_{\mathcal F}:=\arg\min_{r\in[0,1]}\sum_{\sigma\in\{+,-\}}\norm{\widehat E_\sigma-E_\sigma(r)}_2^2.

The visibility estimator $\widehat V_{\mathcal F}$TeX\widehat V_{\mathcal F} is the amplitude returned by the fixed least-squares fit of reference on the declared phase window $\Theta$TeX\Theta, and $\widehat m_{\mathcal F}$TeX\widehat m_{\mathcal F} is the fitted local slope of the resulting family visibility on the declared near-reference interval.

A benchmark implementation may build candidate component budgets from the fixed analysis pipeline as

s_V^2:=\sigma_{\mathrm{stat},V}^2+\sigma_{\mathrm{drift},V}^2+\sigma_{\mathrm{fit},V}^2+\sigma_{\mathrm{det},V}^2,
  

  s_r^2:=\sigma_{\mathrm{stat},r}^2+\sigma_{\mathrm{tom},r}^2+\sigma_{\mathrm{SPAM},r}^2,
  

  s_m^2:=\sigma_{\mathrm{fit},m}^2+\sigma_{\mathrm{window},m}^2+\sigma_{\mathrm{drift},m}^2.

A declared one-sided coverage factor $c_\alpha>0$TeXc_\alpha>0 then defines the report radii

\delta_V:=c_\alpha s_V,\qquad
  \delta_r:=c_\alpha s_r,\qquad
  \delta_m:=c_\alpha s_m.

The role of reference is to specify one fixed reporting convention for the benchmark family; the theorem in the main text uses only the resulting radii and not the particular component decomposition. Recent detector-evaluation, photocounting, and benchmark analyses motivate exactly this insistence on fixed reporting conventions and explicit component accounting [citation].

When the no-record reference visibility $V_{0,\mathcal F}$TeXV_{0,\mathcal F} is analytic inside the family, a direct gap report may take $\delta_\Delta=\delta_V$TeX\delta_\Delta=\delta_V. When $V_{0,\mathcal F}$TeXV_{0,\mathcal F} is inferred from a fitted reference scan, one may instead use $\delta_\Delta=\delta_{V_0}+\delta_V$TeX\delta_\Delta=\delta_{V_0}+\delta_V or a covariance-aware equivalent one-sided radius on the same coverage convention.

The effect-level description used here is compatible with the standard instrument viewpoint. A POVM $\{E_k\}$TeX\{E_k\} may be represented by measurement operators $\{M_k\}$TeX\{M_k\} such that

E_k=M_k^\dagger M_k,
  \qquad
  \sum_k M_k^\dagger M_k=\Id.

Operational quantum probability and quantum measuring processes therefore permit one to formulate the event layer at the level of effects without committing the analysis to a unique microscopic detector model [citation].

For a two-outcome target projector family $\{P_{\pm}\}$TeX\{P_{\pm}\}, a symmetric reference choice is

E_{\pm}(r)=rP_{\pm}+(1-r)\frac{\Id}{2}.

Then

p_{\pm}(r)=\Tr(\rho E_{\pm}(r))
  =r\,\Tr(\rho P_{\pm})+(1-r)\frac{1}{2}.

If a declared family further satisfies a local linear law

V_{\mathcal F}(r)=V_{0,\mathcal F}-\alpha_{\mathcal F}r+o(r)
  \qquad (\alpha_{\mathcal F}>0),

then the leading-order recovery-gap floor is

\Delta^{\min}_{V,\mathcal F}=\alpha_{\mathcal F}r_{\min}+o(r_{\min}).

The worked family in the main text does not assume a universal microscopic visibility law; it uses only the declared affine comparison family, the fixed fringe observable, and the fixed reporting convention.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.