Operator-qualified optical inference contract

This note defines the quantitative boundary for Fixture F-007. It operationalizes the measurement-operator-aware residual from the optics, photonics, and inverse-sensing audit. The contract binds every decoded output to a versioned physical operator, separates measurement information from prior selection, and compares optical, digital, and hybrid routes at equal task information and lifecycle budget.

Episode and operator identity

For episode $e$ and acquisition $t$, seal

where $X_e$ identifies the physical scene or latent-state generator, $A_{e,t}$ identifies the acquisition action, $\nu_{e,t}$ is an immutable operator-version identifier, $C_{e,t}$ is the calibration record, $R_{e,t}$ is the hidden regime record, $Q_e$ is the registered downstream-query set, and $B_e$ is the resource-budget record. Identifiers and hashes are byte strings; timestamps within the records are seconds [s] from a declared clock origin.

Let $\mathcal X$ be the admissible latent-state space and let $x_e\in\mathcal X$ be the latent physical state. Each component of $x_e$ retains its native unit, such as radiance [W sr$^{-1}$ m$^{-2}$], range [m], or concentration [mol m$^{-3}$]. The observation is

where $a_{e,t}$ is the acquisition action, $c_{e,t}$ is the calibrated parameter vector, $\mathcal H_{\nu_{e,t}}$ is the versioned physical forward operator, $g_{e,t}$ is the detector response including conversion and clipping, $n_{e,t}$ is additive or conditionally specified noise, and $y_{e,t}$ is the raw observation in detector counts [count] or another declared sensor unit. The operator carries the conversion units required to map components of $x_e$ to the input unit of $g_{e,t}$. Any non-additive, coherent, correlated, or signal-dependent noise is part of the likelihood rather than being hidden in $n_{e,t}$.

The submitted observation contract is

where $\Sigma^{c}_{e,t}$ is calibration-parameter covariance in the squared native parameter units, $M^{\mathrm{sat}}_{e,t}$ is a binary saturation or dead-time mask, $\tau_{e,t}$ is acquisition time [s], and $\mathcal V_{e,t}$ is the declared validity envelope. The contract is invalid outside $\mathcal V_{e,t}$ until recalibrated or explicitly downgraded.

Aperture, diffraction, and recoverable modes

For wavelength $\lambda$ [m] and numerical aperture $\mathrm{NA}$ [dimensionless], the conventional Rayleigh lateral scale for two incoherent point sources is

$d_{\mathrm R}$ is a criterion under stated imaging assumptions, not a universal task-resolution value. Aperture diameter $D_{\mathrm{ap}}$ [m], focal length $f$ [m], field of view $\Omega$ [sr], coherence, sampling pitch $p_{\mathrm{samp}}$ [m], exposure, and noise must be reported separately. A reconstructed pixel pitch below $d_{\mathrm R}$ does not by itself establish additional measured information.

For a linearized forward operator $H_{e,t}$ with singular-value decomposition

$U_{e,t}$ and $V_{e,t}$ are unitary bases, $V_{e,t}^{*}$ is the conjugate transpose, and diagonal entry $\sigma_{e,t,j}$ of $\Sigma_{e,t}$ has the units of $H_{e,t}$. A state perturbation $v_{e,t,j}$ in the corresponding right singular direction is unidentifiable from that acquisition when $\sigma_{e,t,j}=0$. For a preregistered tolerance $\epsilon_H$ with the same units as a singular value, define the effective measured rank

where $\mathbb 1[\cdot]$ is the indicator function and $j$ indexes singular modes. Report the full singular spectrum or a validated task-relevant summary; $r_{\epsilon_H}$ is threshold-qualified.

For two task-distinct states $x_1$ and $x_2$, likelihood separation is

where $D_{\mathrm{KL}}$ is Kullback--Leibler divergence in nats. The null-space honesty track treats $x_1$ and $x_2$ as measurement-indistinguishable when $D_{12}$ lies below a preregistered discrimination threshold supported by a power calculation. A method must then return calibrated alternatives, a bound, or abstention unless it acquires additional evidence.

The Fixture F-007 analytical likelihood plot visualizes one such indistinguishable base operator and a separating added measurement. It is not an empirical performance result.

Photons, detector response, and dynamic range

For detector element $i$, use the photon-counting model when its assumptions hold:

where $k_i$ is detected count [count], $\mu_i$ is expected count [count], $\eta_i$ is quantum or detection efficiency [dimensionless], $\Phi_i$ is incident photon rate [photon/s], $\tau_i$ is exposure [s], and $b_i$ is expected background plus dark count [count]. For the ideal background-free Poisson case,

where $N_\gamma$ is expected detected photon count [count] and the signal-to- noise ratio is dimensionless. Read noise [electron rms], fixed-pattern error, coherent receiver noise, afterpulsing, pile-up, and dead time receive explicit terms whenever present.

For full-well or count-rate limit $K_i^{\max}$ [count], a simplified clipped detector output is

where $y_i$ is recorded count [count] and $s_i$ is a dimensionless saturation indicator. The saturation fraction is

where $N_{\mathrm{det}}$ is detector-element count [element] and $F_{\mathrm{sat}}$ is dimensionless. Report full well, count-rate ceiling, read noise, dark signal, analog-to-digital converter range, and dead-time or pile-up policy independently; nominal bit depth is not dynamic range.

Phase ambiguity and prior-qualified reconstruction

For coherent intensity measurement,

where $A$ is a declared complex-valued propagation and sampling operator, $x$ is a complex field amplitude in a declared native unit, $y$ is intensity or detector count in its native unit, $|\cdot|^2$ is elementwise squared magnitude, and $n$ is measurement noise in the same unit as $y$. The ambiguity class is

where $\mathcal E$ is a set of physically admissible fields. Global phase, translation, conjugate inversion, and geometry-specific ambiguities are scored as separate equivalence relations when applicable.

For reconstruction method $m$ with prior $\pi_m(x)$ and likelihood $p_m(y\mid x,O)$, the posterior is

where $O$ is the observation contract, $x'$ is an integration variable with the same native units as $x$, and the posterior density carries the reciprocal units implied by the measure $\mathrm dx'$. Method $m$ must label information origin as measurement, prior, calibration, or active intervention.

For hidden truth $x_e$ and a nominal $(1-\alpha)$ credible set $\mathcal C_{m,e,1-\alpha}$, empirical coverage over $N$ independent episodes is

where $\alpha$ and coverage are dimensionless and $N$ is episode count [episode]. Report coverage after source-family, texture, sparsity, positivity, motion, and noise-model shifts. Perceptual quality and truth fidelity remain separate outcomes.

Active sensing and illumination safety

Let $b_t$ be the belief state before action $a$, $\theta$ the uncertain task state, $d$ a downstream decision, $U(d,\theta)$ task utility in a declared native unit, and $y$ the prospective observation. Expected value of information is

$\operatorname{EVI}$ has the same unit as $U$. Action admissibility is the componentwise condition

where $N_\gamma$ is incident or detected photon count [photon] as explicitly labeled, $E_a$ is energy [J], $L_a$ is latency [s], $D_a$ is dose in the task-native safety unit, $W_a$ is actuator wear [cycle], and $R_a$ is risk on a declared scale. Superscript $\max$ denotes the preregistered ceiling in the same unit, and $\preceq$ means every component is within its ceiling. A scalarized cost may guide a policy only after the component ceilings are enforced and its weights are published.

Multiplex, fusion, and calibration crossover

For route $r$ in regime $\rho$, define the protected outcome vector

where $L_{\mathrm{task}}$ is task loss in its native unit, $U_{\mathrm{cal}}$ is dimensionless uncertainty-calibration error, $N_\gamma$ is photon count [photon], $E_{\mathrm{life}}$ is lifecycle energy [J], $T_{\mathrm{wall}}$ is wall latency [s], $F_{\mathrm{sat}}$ is saturation fraction [dimensionless], and $C_{\mathrm{cross}}$ is dimensionless crosstalk. Direct and multiplexed routes are compared componentwise across photon flux, background, detector noise, occupancy, crosstalk, and saturation regimes; no single favorable point establishes an advantage.

For two sensor estimates $\hat x_1$ and $\hat x_2$ with errors $e_1=\hat x_1-x$ and $e_2=\hat x_2-x$, retain

where $P_{12}$ is cross-covariance in squared native state units and $e_2^{\top}$ is transpose. Setting $P_{12}=0$ is a tested assumption, not a default. Fused estimates must report marginal covariance, cross-covariance or a justified bound, alignment error in native spatial and temporal units, and failure under common-mode perturbations.

For reference measurement $r_t^{\mathrm{ref}}$ and calibrated prediction $\widehat r^{\mathrm{ref}}(c_t)$ in the same observation unit, define

where $S_t$ is residual covariance in squared observation units and $z_t^2$ is dimensionless. The monitor declares a threshold $z_{\max}^2$, a window length $w$ [sample], a false-alarm target [dimensionless probability], and a fallback. Detection delay is seconds [s], recovery time is seconds [s], calibration cost is samples [sample] and joules [J], and pre-detection task loss is reported in the task's native unit. Scene shift, source drift, alignment drift, detector gain, and thermal drift are separate hidden causes.

Optical, digital, and hybrid route accounting

For route $r$, end-to-end latency is

where every term is seconds [s]. Optical propagation time cannot replace $T_r$. The route identity fixes input origin, operator shape, batch, sparsity, effective precision, output dimension, programming frequency, operator reuse, utilization, and duty cycle.

Let the intended transform be $u=Wv$, where $v$ is an input vector in declared native units, $W$ is a linear operator with corresponding conversion units, and $u$ is the desired output vector. Device $d$ at temperature $\vartheta$ [K] and age $\ell$ [s] realizes

where $\Delta W_{d,\vartheta,\ell}$ has the units of $W$, $\epsilon^{\mathrm{analog}}$ is analog transform error in output units, and $\epsilon^{\mathrm{read}}$ is detector, conversion, and readout error in output units. Report bias, covariance, tails, effective precision [bit], and task loss separately across fan-in, depth, device, temperature, age, and workload.

For device population $\mathcal D$ with $N_D$ fabricated devices, fabrication yield is

where $Y_{\mathrm{fab}}$ is dimensionless and $N_D$ is device count [device]. The envelope includes transfer-function error, task quality, trimming time [s], tuning energy [J], steady thermal power [W], thermal crosstalk [dimensionless or a declared transfer unit], and stability over the declared interval. Failed dies remain in the denominator.

Query-registered physical compaction

Let encoder $h$ transform raw observation record $O$ into retained artifact $z=h(O)$ with size $S_z$ [byte]. After the encoder is frozen, query $q$ from registered set $Q$ produces answer $f_q(O)$ from raw data and reconstructed answer $\widehat f_q(z)$ from the retained artifact. Query recovery is

where $|Q|$ is query count [query], $d_q$ is error in the native unit of query $q$, $\epsilon_q$ is a tolerance in the same unit, and $R_Q$ is dimensionless. The confirmatory evaluator adds sealed future queries and an incident- investigation query after route and retention policies are frozen. Failure is reported as lost query classes, not only as an average score.

Lifecycle, labor, and matched budgets

For one accepted output, complete lifecycle energy is

where every term is joules [J] and analog-to-digital and digital-to-analog conversion are denoted ADC and DAC. Powered propagation elements are charged to $E_{\mathrm{propagate}}$; passive loss appears through increased source or amplifier demand. Amortized embodied energy is

where numerator terms are joules [J] and $N_{\mathrm{accepted,life}}$ is the accepted-output count [output] over measured or conservatively modeled service life. Negative recovery credit must be independently substantiated.

The equal-budget vector for method $m$ is

where $N_{\mathrm{train}}$ is training-example count [example], $N_{\mathrm{scene}}$ is physical-scene count [scene], $N_\gamma$ is photon count [photon], $D_{\mathrm{dose}}$ is dose in a declared task-native unit, $N_{\mathrm{act}}$ is actuator-cycle count [cycle], $N_{\mathrm{cal}}$ is calibration-sample count [sample], $N_{\mathrm{fab}}$ is fabricated-device count [device], $N_{\mathrm{query}}$ is evaluator-query count [query], $S_{\mathrm{state}}$ is retained and working storage [byte], $T_{\mathrm{wall}}$ is wall time [s], $T_{\mathrm{human}}$ is labor [person-hour], and $E_{\mathrm{life}}$ is lifecycle energy [J]. Human design, alignment, labeling, tuning, calibration, inspection, safety review, maintenance, and incident response remain role-stratified entries.

An arm is budget-matched only when every binding component is within its preregistered tolerance or when the comparison is explicitly a Pareto frontier. Freed budget from an ablation remains unused.

Protected outcome vector and retirement estimand

Keep the following outcome families separate:

where the components respectively contain aperture/diffraction results, photon/shot-noise results, phase-ambiguity results, prior-mismatch results, calibration/drift results, saturation/dynamic-range results, fusion-covariance results, optical-transform results, conversion/readout results, analog-error results, fabrication results, thermal-work results, active-illumination safety results, and full lifecycle energy and labor results. Each $Z$ is a structured record with the native units defined above; the vector is not scalarized for acceptance.

For protected component $j$, the paired effect of mechanism $k$ is

where $m_{-k}$ removes only mechanism $k$ and receives no replacement resource. Report $\Delta_{k,j}$ in the native unit of outcome $j$ with a 95% uncertainty interval over independent scene, operator, regime, device, site, and seed strata.

Retire the residual when an equal-budget mature-null composition matches the protected vector; when a gain depends on hidden prior, operator, calibration, or query leakage; when it fails held-out devices or regimes; or when no preregistered ablation isolates value beyond the registered inverse, uncertainty, sensing, control, acceleration, optical, hybrid-design, and calibration baselines. The editable source for the system diagram is operator-qualified-physical-inference.mmd.