Regime-qualified flow inference and control contract

This note defines the dimensional and inferential boundary for Fixture F-005. It operationalizes the residual identified by the fluid-dynamics and turbulence audit for Candidates 002, 003, 006, 007, 012, and 014.

The equations may be known while the state, boundary, forcing, constitutive response, unresolved scales, and future regime are not. The benchmark therefore binds every result to a physical/numerical identity, observation identity, regime/history identity, target, and complete resource boundary.

Immutable episode identity

For paired episode $e$, register

where the hashes identify geometry $\mathcal G$, governing equations and constitutive assumptions $\mathcal E$, initial and boundary conditions $\mathcal B$, forcing process $\mathcal F$, numerical solver/grid/time-step configuration $\mathcal N$, observation and calibration operator $\mathcal O$, data and split lineage $\mathcal D$, actuator/plant interface $\mathcal A$, and declared quantities of interest $\mathcal Q$. Hashes are byte strings. Identity of bytes does not establish physical adequacy, convergence, or calibration.

The hidden regime/history record is

where $Re$, $Pe$, $Sc$, and Rossby number $Ro$ are dimensionless; $\mathbf g$ is geometry; $\mathbf b$ and $\mathbf f$ are boundary and forcing histories; $\mathbf d$ is disturbance shape and amplitude in native units; $r_{0:t}$ is ramp direction, rate, dwell, and prior regime occupancy; and $\tau_e$ is the episode horizon in seconds. Every dimensioned element of $\mathbf g$, $\mathbf b$, $\mathbf f$, $\mathbf d$, and $r_{0:t}$ carries a unit in the episode schema.

State, closure, and numerical discrepancy

For incompressible Newtonian flow,

where velocity $\mathbf u$ is in $\mathrm{m\,s^{-1}}$, time $t$ in seconds, pressure $p$ in pascals, density $\rho$ in $\mathrm{kg\,m^{-3}}$, kinematic viscosity $\nu$ in $\mathrm{m^2\,s^{-1}}$, and body acceleration $\mathbf f$ in $\mathrm{m\,s^{-2}}$.

At resolution $\Delta$, represent the resolved evolution as

where $\mathcal N_{\Delta}$ is the declared discretized resolved operator, $\mathcal C_{\theta,\Delta}$ is a physical or learned closure, $\boldsymbol\delta_{\Delta}$ is model-form discrepancy, and $\boldsymbol\epsilon_{\Delta}$ is numerical error. All four right-hand terms have state-units per second. A fitted residual can contain any mixture of the last three terms; portability is tested rather than inferred from training loss.

Let $s(z;\mathcal S_{\mathrm{train}})$ be a preregistered dimensionless support distance from test condition $z$ to the training support. Closure risk is reported as a function rather than a pooled mean:

$q$ is a declared quantity of interest and loss $\ell_q$ retains the square or absolute unit induced by $q$. Out-of-support evaluation crosses solver, grid, order, geometry, boundary treatment, forcing band, $Re$, and regime rather than using randomly held-out neighboring snapshots.

Detector, filter, operator, and support identities

Every reported field or event is bound to

where $\mathcal V$ is the physical variable, $\mathcal K_{\ell}$ is a filter and width $\ell$ in metres, $\mathcal H$ is the measurement operator, $\mathcal D_{\gamma}$ is an event detector with threshold/parameter vector $\gamma$, $\mathcal S_x$ and $\mathcal S_t$ are spatial and temporal support, and $\mathcal F_r$ is the reference frame. Two results with different $J$ are different estimands until a registered transfer map is validated.

For observation time $t_k$,

where $\mathbf y_k$ retains sensor units, $R_k$ has squared sensor units, and $\mathcal H_{J,k}$ includes averaging kernel, exposure time, transfer function, latency, synchronization, missingness, probe intrusion, and calibration state. Aggregate supply and recipient-level delivery are different estimands unless a registered support map identifies them (C-1488). Nominal contact area and aggregate load likewise do not identify real-contact support or the local-pressure tail (C-1498). The estimator never receives hidden truth through filenames, simulator metadata, shared random state, or a truth-identical forward model.

Signed scale transfer and event fidelity

For filter width $\ell$,

where stress $\tau_{ij}^{(\ell)}$ is in $\mathrm{m^2\,s^{-2}}$, filtered strain $\widetilde S_{ij}$ in $\mathrm{s^{-1}}$, and signed transfer $\Pi_{\ell}$ in $\mathrm{m^2\,s^{-3}}$. The flux score preserves direction:

where $w(\ell)$ is dimensionless and normalized over $d\log\ell$; $L_\Pi$ is in $\mathrm{m^2\,s^{-3}}$. Report forward-transfer and backscatter event precision, recall, calibration, amplitude, spatial support, and duration separately. Matching a spectrum cannot substitute for $L_\Pi$.

For coherent event $a$ extracted by $\mathcal D_{\gamma}$, retain

where onset and offset are seconds, $\Omega_a$ is spatial support in $\mathrm{m^3}$ (or the declared lower-dimensional measure), $\Gamma_a$ is a circulation-like attribute in $\mathrm{m^2\,s^{-1}}$ when applicable, $\mathbf c_a$ is position in metres, and $\Delta q_a$ is the event-conditioned change in target $q$. Detector sweeps over reasonable filters, thresholds, frames, and planes are mandatory; event identity is never silently fixed.

Reduced state and adaptive allocation

For basis $\Phi_r$ and reduced state $\mathbf a_t$,

where rank $r$ is a count, $F_r$ is the projected or inferred dynamics, and $\mathbf c_r$ closes discarded modes. Basis normalization determines the units of $\mathbf a_t$. Reconstruction, autonomous rollout, forcing response, control, transition, and extreme fidelity are distinct losses.

Let available allocation units $i\in\mathcal I_t$ represent mesh cells, sensors, samples, model capacity, or compute quanta. An adaptive policy chooses $a_{i,t}\in\{0,1,\ldots\}$ under

where cost $c_{i,t}$ and budget $B_t$ use the same declared unit: cell-steps, bytes, seconds, or joules. Its goal-oriented efficiency is

where both methods attain the same target-error tolerance $\varepsilon_q$ in the native unit of $q$ and work uses the same ledger. $\eta_q$ is dimensionless. Regrids, subcycles, rejected steps, data transfers, synchronization, load imbalance, sensor movement/calibration, and allocator inference are included.

Observability, assimilation, and sensor value

For a scaled linearization $\mathbf x_{k+1}=A_k\mathbf x_k$ and $\mathbf y_k=C_k\mathbf x_k+\boldsymbol\eta_k$, define

$W_o$ is dimensionless only after state and sensor scaling is fixed. Report its rank and conditioning locally by trajectory, regime, sensor set, noise model, and horizon; do not promote one linearization to global observability.

For posterior mean $\widehat{\mathbf x}_k$ and covariance $P_k$, the normalized estimation error squared is

which is dimensionless. Calibration requires coverage and rank-aware tests in observable and unobservable subspaces, innovation whiteness, and recovery after dropout. A narrow posterior with structural bias is failure.

For sensor set $S$ and target $q$, use the costed decision value

where $L_q$ is expressed in a declared utility unit, energy $E_S$ in joules, traffic $B_S$ in bytes, and human maintenance $H_S$ in person-hours. Conversion weights carry reciprocal units and are preregistered. Also report every raw term. Placement must remain physically feasible under regime shift, correlated failure, latency, bandwidth, calibration drift, and probe intrusion.

Mixing, transition, and extreme-event contracts

For passive scalar $c$ with molecular diffusivity $\kappa$,

If $c$ is concentration in $\mathrm{kg\,m^{-3}}$, source $s_c$ is in $\mathrm{kg\,m^{-3}\,s^{-1}}$ and scalar dissipation $\chi$ is in $\mathrm{kg^2\,m^{-6}\,s^{-1}}$. Mixing outcomes retain scalar variance, negative-Sobolev mix norm, $\chi$, reaction completion, residence-time distribution, and remnant concentration at operational support. Visual filamentation is not a substitute. A countercurrent label likewise cannot replace finite conductance, capacity-rate, residence, leakage, boundary, and pump terms (C-1490).

For transition class $c$ with history $r_{0:t}$, define hazard

with units $\mathrm{s^{-1}}$. Report event-time likelihood, class-conditional calibration, lead time in seconds, false-alarm burden, abstention, and competing hazards. Ramp direction/rate, disturbance amplitude/shape, dwell, censoring, domain, and observation history are part of the conditioning state.

For extreme observable $Q$ and preregistered threshold $q_*$,

where $p_*$ is dimensionless and return period $T_R$ is seconds when $\Delta t_{\mathrm{eff}}$ is the effective independent sampling interval in seconds. Weighted rare-event estimators publish weights, effective sample size, degeneracy, variance, and validation on an untouched natural-distribution stream. Enriched event frequency without reweighting is invalid.

Control stability and complete energy

For plant state $\mathbf x$, command $\mathbf u_c$, disturbance $\mathbf w$, and delayed observation $\mathbf y_{t-d}$,

where delay $d$ is seconds and command components retain actuator units. Sweep delay, bandwidth, saturation, noise, forcing, regime, plant drift, failure, and fallback. Report constraint violations, gain/phase margins where defined, closed-loop poles for linearized controllers, bounded-input response, recovery, and runtime-monitor interventions. A task gain with unstable or unsafe strata does not enter an aggregate score. Equal mean flow also cannot establish phase margin or absence of coupled-delay oscillation (C-1492). Rate/state stick--slip supplies the same mean-versus-dynamics warning for interface state, compliance, inertia, damping, bandwidth, and control delay (C-1501).

Over service interval $[0,T]$, operational net energy is

where every power is watts, $T$ is seconds, and $E_{\mathrm{net}}$ is joules. Report all terms separately. Installation, embodied energy, calibration, maintenance, retraining, replacement, and end-of-life costs are additional joule rows amortized only over a declared service life and duty cycle.

The complete run record is

Counts are dimensionless, bytes are bytes, times are seconds, peak memory is bytes, human work is person-hours, and every energy row is joules. Failed runs, search, data generation, tuning, verification, standby, and unused reservations remain in the ledger.

Outcome vector and decision

Keep the confirmatory outcome as a typed vector

Each component retains its literal units and uncertainty. No weighted fluid, physics, fidelity, or efficiency score may replace it. The proposed composition survives only if it beats the strongest complete mature null at equal information and lifecycle budget on preregistered components, remains non-inferior on protected stability, calibration, conservation, and tail components, transfers across hidden regimes and model/hardware families, and has an isolating ablation. Otherwise retain this contract and retire the architectural explanation.

Editable system diagram: regime-qualified-flow-inference-control.mmd.