Measurement contracts, uncertainty, and invalidation

Scope

This note defines the minimum mathematics needed to turn an indication into a decision-bearing result. It connects the metrology audit, Candidate 014, and the energy model. It does not replace a domain-specific measurement procedure.

Result record

For one measured quantity, retain

where:

The tuple is incomplete when a field that can alter the downstream decision is missing. A confidence score alone cannot represent these distinct dependencies (C-519–C-524).

Measurement model and dimensional validity

Let

where input quantity $X_i$ has unit $[X_i]$ and $Y$ has unit $[Y]$. Every additive term produced by $f$ must have unit $[Y]$. Software, coefficients, numerical precision, and preprocessing that implement $f$ are part of $\mathcal P$. A coefficient of friction is therefore not portable without its interface pair, load/motion state, environment, procedure, support, and history (C-1497).

For estimates $x_i$, first-order covariance propagation gives

$c_i$ has unit $[Y]/[X_i]$ and covariance $u(x_i,x_j)$ has unit $[X_i][X_j]$, so every summand has unit $[Y]^2$. Removing off-diagonal terms asserts independence; it is not a harmless simplification when calibration, clock, environment, preprocessing, prior, or training data are shared (C-525, C-533). For nonlinear or discontinuous models, propagate sampled input distributions and report empirical interval coverage.

Error, correction, and coverage

Given reference value $y_{\mathrm{ref}}$,

Error $e$, correction $c$, and corrected estimate $\hat y_{\mathrm{corr}}$ have unit $[Y]$. The estimated correction remains uncertain. Expanded uncertainty is

where $k$ is dimensionless. A stated $k$ does not by itself establish a coverage probability; the distributional method and achieved coverage must be reported (C-524, C-526).

Repeatability and reproducibility

For replicate result $y_{lors}$ from location $l$, operator $o$, run $r$, and system $s$, a variance-component null is

$\mu$ and every random effect have unit $[Y]$; their variances have unit $[Y]^2$. The design declares which factors are fixed or random and which are confounded. A short run with one system estimates repeatability, not broad reproducibility (C-527, C-528).

Decision rules and guard bands

Let specification require $Y\le T$, where $T$ has unit $[Y]$. A simple guarded acceptance rule is

where guard multiplier $g\ge0$ is dimensionless. Increasing $g$ generally reduces false acceptance while increasing false rejection. Choose $g$ from a declared loss or risk allocation; do not hide that choice inside “confidence.” The rule version, tolerance version, uncertainty method, and cost owner are dependencies of the decision (C-531).

Drift and recalibration

Represent indication drift relative to the last accepted calibration as

where $d$, check-standard result $y_{\mathrm{check}}$, and reference value $y_{\mathrm{ref}}$ share unit $[Y]$. The review policy is a function

where $h_t$ is calibration/check history, $r_t$ is decision risk, $u_t$ is current uncertainty, $c_t$ is calibration cost, and $e_t$ is equipment and environment state. Inputs keep their native units; $\pi$ returns a categorical action such as continue, check, restrict, recalibrate, or quarantine. A calendar interval alone is not evidence of stability (C-532).

Dependency invalidation

Let a directed acyclic graph $G=(V,E)$ contain calibration, raw-data, software, model, transformation, result, and decision versions. Edge $(a,b)\in E$ means that node $b$ depends on node $a$. If dependency $a$ changes or fails review, its invalidation cone is

where $a\leadsto v$ denotes a directed path and $\mathcal I(a)$ is a set of version identifiers. Each affected node is re-evaluated, superseded, restricted, or withdrawn; provenance alone does not decide which action is correct (C-534, C-535).

Energy-measurement instantiation

For sampled power $P_m$ at time $t_m$ with interval $\Delta t_m$,

where $P_m$ is W, $\Delta t_m$ is s, and $\hat E$ is J. The model must name the electrical boundary, voltage/current/phase calibration, bandwidth, anti-aliasing, clock alignment, integration rule, missing-sample policy, warm-up, idle allocation, retries, useful outputs, and facility attribution. Uncertainty propagates through the integration model with shared meter, coefficient, and clock covariance retained.

Compare candidate $C$ and baseline $B$ only after both satisfy the same quality, risk, latency, and workload envelope. For paired run $r$,

where energies are J, $N_q$ is a qualified-event count, and $\Delta e_r$ is J/qualified event. Report absolute values, paired effect, coverage, and all failed or rejected runs. A software counter or narrow device boundary may rank systems differently from calibrated end-to-end measurement; the direction is an empirical question (C-536).

Falsification conditions

Reject the cross-layer composition as a distinct systems contribution if:

1. a complete conventional metrology, statistics, and content-addressed provenance stack matches its coverage and stale-decision frontier;
2. it improves results only by receiving extra sensors, calibration runs, compute, storage, or analyst time;
3. dependencies are recorded but do not trigger correct downstream action;
4. uncertainty intervals become narrower while empirical coverage worsens;
5. shared dependencies are counted as independent evidence; or
6. the metadata and review burden costs more than the errors or work it avoids.