A substrate-referenced measurement reports a quantity against a reference rather than against an absolute scale. In the I2 record the relation is
kappa_app / kappa = R(lambda) = 1 / (exp(lambda) * E1(lambda))with E1 the exponential integral and λ the ratio of the substrate's conductance to the film's. The function has a crossing at λ* = 0.4348, rises without bound, and approaches R ≈ λ. Every method in section A can be written in this shape with its own λ, its own crossing, and its own approach to the asymptote. The three properties that matter for reporting are:
The three computations below measure these properties, ask what cancels them, and ask what placement does to them.
The I2 record's bias factor R(λ) = 1/(eλ E1(λ)) is the template. It has three generalisable properties: a crossing at λ* = 0.4348, a monotone approach to R ≈ λ, and an elasticity bounded by 1. The leads below ask which other methods have the same structure, what their λ is, and whether it is stated.
| # | Lead | |
|---|---|---|
| B1 | The bias factor of a substrate-referenced method crosses unity at a λ set by the measurement kernel, so the crossing can be located without knowing the material. | test |
| B2 | A method whose reported value is within 10% of λ times the reference value for λ > 10 is measuring its reference, and its λ can be recovered from the reported value alone. | test |
| B3 | For R(λ) = 1/(exp(λ) E1(λ)), d ln R / d ln λ equals R - λ and is therefore bounded by 1, so correcting a reference bias cannot amplify a reference uncertainty. | test |
| B4 | The I2 four-case spread of 75.8× is reproducible from the four λ values to better than 0.1%, which makes λ a sufficient statistic for that spread. | test |
| B5 | Every method with an additive reference has a bias factor linear in the reference, and every method with a multiplicative reference has a bias factor that is a ratio. | test |
| B6 | An electrochemical potential referenced to a liquid-junction electrode carries the junction potential as an additive offset that changes with the electrolyte. | test |
| B7 | In four-point conductivity the contact resistance is additive and cancels, while the probe-spacing reference is multiplicative and does not. | test |
| B8 | The Bückle 10% rule in nanoindentation is a statement that the film-to-indentation depth ratio must exceed 10 for the film to dominate the reported hardness. | test |
| B9 | Nanoindentation substrate effects and heat-spreading substrate effects share a dimensionless form, so one λ reporting convention could cover both. | note |
| B10 | In AFM cantilever calibration against a reference cantilever the reference uncertainty enters once, not twice, because the comparison is a ratio. | test |
| B11 | In ellipsometry the substrate dielectric function is the reference, so a 1% error in the substrate index can produce a larger thickness error than a 1% error in the measured angle. | test |
| B12 | The Raman Si 520.7 cm⁻¹ line is a reference that shifts with temperature and strain, so a Raman calibration is only as good as its stated reference state. | test |
| B13 | XPS binding energies referenced to adventitious carbon inherit the charging correction, so a 0.2 eV reference error appears as a 0.2 eV error in every peak. | note |
| B14 | The δ¹⁸O scale is fixed by two reference waters, so any delta value is an affine function of the chosen standard pair. | test |
| B15 | NMR chemical-shift referencing errors are constant offsets, so they can change a reported shift but not a coupling constant. | test |
| B16 | pH is defined by primary buffer standards, so an electrode calibrated against a different buffer series reports an offset that is a property of the series. | test |
| B17 | Changing the sound-pressure reference from 20 µPa to 1 µbar shifts every level by 74.0 dB, so a quoted level without its reference is unusable. | test |
| B18 | Photometric quantities referenced to a blackbody change when the reference temperature changes, and the change is wavelength-dependent. | test |
| B19 | In qPCR, ΔΔCt is a ratio of ratios and cancels template loading only when the reference gene and the target share an amplification efficiency. | note |
| B20 | Flow-cytometry MESF units are bead-referenced, so an instrument-to-instrument comparison in MESF is a comparison of two bead calibrations. | test |
| B21 | An immunoassay standard curve transfers between plates only when the fit shares its asymptotes, not merely its midpoint. | test |
| B22 | Isotope-ratio mass spectrometry cancels sample size through the ratio but not mass bias, which is multiplicative per isotope and therefore partly survives. | note |
| B23 | Internal-standard quantification cancels injection volume but not ionization suppression, because suppression acts on analyte and standard differently. | note |
| B24 | Standard addition cancels a multiplicative matrix effect only while the response is linear; curvature converts the cancellation into a bias. | test |
| B25 | Retention-index referencing transfers retention across chromatographs but not selectivity, so a peak identity can transfer while a resolution cannot. | test |
| B26 | In differential scanning calorimetry the reference pan defines the baseline, so the reference pan's heat capacity enters the measured heat flow. | note |
| B27 | Thermogravimetry referenced to an empty crucible cancels buoyancy only when the sample and reference volumes match. | test |
| B28 | In dilatometry the pushrod and the reference specimen expand together, so the measured expansion is a difference whose reference coefficient is a systematic. | test |
| B29 | In gas sorption the dead volume is a geometric reference, so its error scales with pressure and mimics a capacity error. | test |
| B30 | In neutron scattering the vanadium reference sets absolute intensity, so a 1% reference error is a 1% error in every absolute cross-section. | test |
| B31 | In photoluminescence quantum yield the reference dye's stated yield enters linearly, so a 5% reference error is a 5% yield error. | note |
| B32 | In radiometry the cryogenic radiometer is a primary reference, so a chain from it adds uncertainty but no multiplicative bias. | test |
| B33 | In solar-cell efficiency the AM1.5G spectrum is a stipulated reference, so an efficiency without a spectral-mismatch correction is not comparable across laboratories. | note |
| B34 | In resistance thermometry the ratio R(T)/R(273.15 K) cancels the sensor's absolute resistance, so a 1% absolute error cancels while a 1% ratio error does not. | note |
| B35 | In quantum-Hall resistance metrology the reference is a quantum ratio, so the calibration chain has no material reference and no λ. | test |
| B36 | The kilogram's redefinition moved its reference from an artifact to a constant, which removes drift but leaves the traceability chain intact. | note |
| B37 | In frequency metrology the reference is a transition, so a frequency ratio is measurable without a material standard and is the cleanest available reference-free ratio. | note |
| B38 | Acoustic thermometry measures the Boltzmann constant directly, so its reference is a constant and its λ should be 1 by construction. | note |
| B39 | Any method whose reference is a fundamental constant has λ = 1 by definition and cannot exhibit reference-ratio spread. | note |
| B40 | The number of references in a traceability chain, not the number of laboratories, sets the multiplicative uncertainty of the final value. | test |
| B41 | A reference ratio reported without its definition, such as which conductance or which area, is not a number another laboratory can use. | note |
| B42 | In 2D-material thermal measurements λ is set by spot size over thermal length, so one sample measured with two spot sizes has two λs and two reported conductivities. | test |
| B43 | If λ is set by geometry rather than by material, the reference correction can be computed from a micrograph without a second measurement. | test |
| B44 | Methods that report a ratio of two measurements on the same sample carry λ in both numerator and denominator and therefore reduced reference sensitivity. | test |
| B45 | Methods that report one measurement against a stored reference have no cancellation at all, so their λ enters once and fully. | test |
| B46 | The same quantity measured by a ratio method and by a single-ended method should differ by exactly the ratio of their bias factors, which is testable on paired data. | test |
| B47 | Reference-ratio spread across a literature should exceed material spread whenever the reference is not reported, which is testable from the reported values alone. | note |
| B48 | A field that reports its reference ratio should show smaller inter-laboratory scatter than a field that does not, after controlling for material. | note |
| B49 | The λ structure is a dimensionless group, so dimensional analysis should predict the crossing of any new method before it is measured. | note |
| B50 | In transient hot-wire thermal conductivity the wire's temperature coefficient is the reference, so a 1% error in it is a 1% error in the reported conductivity at every time point. | test |
A differential or ratiometric design is a claim that the reference cancels. Whether it does depends on the algebra of the reference, the algebra of the operator, and the match between the two arms. The leads below separate those three questions, which are usually conflated.
| # | Lead | |
|---|---|---|
| B51 | Differencing two arms that share an additive reference cancels it exactly only when their gains are equal, and a 1% gain mismatch leaves 1% of the reference in the output. | test |
| B52 | A shared reference with a 0.1% gain match rejects common-mode reference drift by 60 dB, which beats independent references drifting at 1% by 20 dB. | test |
| B53 | Ratiometric division cancels a multiplicative reference exactly and an additive one not at all, so the operator must match the algebra of the reference. | test |
| B54 | Applying a ratio to an additive reference leaves a 36% error at X1 = 5, X2 = 2 and B = 3, while applying a difference leaves none. | test |
| B55 | Applying a difference to a multiplicative reference scales the answer by the reference, a 200% error at B = 3, while a ratio leaves none. | test |
| B56 | A differential design whose two arms see different references cannot cancel anything, regardless of how well its gains are matched. | test |
| B57 | The common-mode rejection of a differential front end degrades in proportion to source-impedance mismatch, not gain mismatch, when the reference is a current. | test |
| B58 | Chopping at frequency f moves a reference drift out of band only when the drift spectrum falls faster than 1/f squared above f. | test |
| B59 | Autozeroing samples the reference offset once per cycle and therefore aliases low-frequency reference noise instead of removing it. | note |
| B60 | Correlated double sampling removes reset noise but not reference drift, because the drift is correlated between the two samples. | note |
| B61 | A Wheatstone bridge cancels excitation drift because all four arms share the excitation, so bridge accuracy is set by the ratio arms rather than by the source. | note |
| B62 | A bridge with three matched arms and one unknown contains a reference ratio whose tolerance enters the result at first order. | test |
| B63 | Double differencing with three electrodes cancels a linear drift and leaves the quadratic term, so the residual scales as the square of the drift rate. | test |
| B64 | Two identical reference electrodes reduce drift by the square root of 2 only when their drifts are independent, and by nothing when they share a junction. | test |
| B65 | A quasi-reference electrode's drift is common-mode to a differential pair, so it cancels in the difference but not in the absolute potential. | test |
| B66 | Lock-in detection references phase rather than magnitude, so a 1 degree phase error attenuates the amplitude by 0.015% and leaks 1.7% of the quadrature component into it. | test |
| B67 | Null-balance measurement transfers accuracy from the detector to the standard, so a null method's reference uncertainty is the standard's uncertainty and nothing else. | test |
| B68 | The substitution method cancels instrument drift because unknown and standard are read at the same setting, so drift between them is second order. | test |
| B69 | Transposition, swapping sample and standard, cancels a linear drift term but not a drift that changes sign with position. | test |
| B70 | Reversing polarity cancels a constant offset and leaves any offset that is odd in the applied field. | test |
| B71 | Two-point calibration cancels offset and gain and leaves curvature, while three-point calibration leaves the cubic term. | test |
| B72 | Bracketing a measurement with standards before and after reduces drift error to the interpolation residual, which scales with the square of the bracketing interval. | test |
| B73 | A ratiometric sensor whose denominator can approach zero has unbounded reference sensitivity near the zero, so its useful range excludes that point. | note |
| B74 | Ratiometric fluorescence cancels excitation power but not photobleaching, because bleaching acts on the indicator and not on the reference. | test |
| B75 | Dual-wavelength absorbance cancels a scattering background only when the background is flat across the wavelength separation. | test |
| B76 | An isosbestic point is a reference wavelength at which the ratio is insensitive to the analyte, so a two-wavelength ratio uses it as the denominator. | test |
| B77 | A ratio of two emission lines in thermometry cancels probe concentration and detector gain only when both lines share the same collection path. | note |
| B78 | Raman band ratios cancel laser power but not the instrument's wavelength-dependent response, which is a multiplicative reference of its own. | test |
| B79 | Standard addition is a differential design in concentration space, so it cancels a multiplicative matrix effect but amplifies a calibration-curvature error. | test |
| B80 | An internal standard cancels a multiplicative loss only when the loss is the same for analyte and standard, which fails at every chromatographic separation. | test |
| B81 | A double ratio of four measurements cancels two multiplicative references, so it should be the default design when source and detector both drift. | note |
| B82 | A reference channel fed into the subtraction with gain greater than 1 amplifies the reference instead of cancelling it, by the channel gain ratio. | note |
| B83 | Differencing adds the two arms' independent noises, so it improves signal-to-noise only when the reference noise exceeds the arm noise. | test |
| B84 | A differential design halves the signal when the two arms are in antiphase, so its signal-to-noise gain requires reference noise larger than the signal. | test |
| B85 | The reference-induced error of a differential design is bounded by the product of the gain mismatch and the reference-to-signal ratio. | note |
| B86 | A ratiometric design's reference error is bounded by the relative error of the denominator, so a noisy denominator is worse than no ratio at all. | test |
| B87 | Photon-shot-noise-limited ratiometric detection has the square root of 2 times the noise of a single channel at equal total photon number. | test |
| B88 | Balanced homodyne detection reaches shot-noise-limited sensitivity only when the local-oscillator's excess noise is suppressed by the balance. | test |
| B89 | Common-mode range rather than rejection ratio limits a differential design when the reference is large, because beyond the range the output clips and the error exceeds the single-ended one. | note |
| B90 | Any differential design converts a common-mode systematic into a differential signal at second order, so the residual scales with the square of the mismatch. | note |
| B91 | A ratiometric design applied to an additive reference converts a constant offset into a concentration-dependent error, which is worse than a constant bias. | test |
| B92 | The optimal reference-sharing strategy depends on whether reference noise is correlated across arms, not on the magnitude of the reference. | test |
| B93 | Two references that are the same electrode but read by different amplifiers are not shared references, because the amplifier gains differ. | test |
| B94 | A differential measurement of a quantity that is itself a ratio needs the logarithm of the ratio, not the difference, to be reference-invariant. | note |
| B95 | Log-ratio designs cancel multiplicative references exactly and additive ones not at all, mirroring the linear case with the operators exchanged. | test |
| B96 | The reference effect in a differential design can be measured by deliberately mismatching one arm and extrapolating the mismatch to zero. | test |
| B97 | A design that cancels the reference also cancels any genuine common-mode signal, so it cannot distinguish reference from signal without a third measurement. | test |
| B98 | Three-arm designs can cancel an additive and a multiplicative reference together, which two arms cannot, at the cost of one more independent noise. | test |
| B99 | The best reference strategy for an instrument is set by the ratio of reference drift to arm noise, not by the instrument's precision specification. | note |
| B100 | Any claim that a design is reference-free is falsifiable by changing the reference and checking that the output does not move. | note |
Transfer is the reference problem at institutional scale. When two laboratories disagree, the question is whether their references differ or their materials do, and most transfer protocols cannot distinguish the two.
| # | Lead | |
|---|---|---|
| B101 | Inter-instrument transfer error is bounded below by the ratio of the two instruments' reference responses at the calibration point. | test |
| B102 | A single-wavelength standardization transfers a calibration only when the response is linear over the transferred range. | test |
| B103 | Piecewise direct standardization needs transfer samples that span the calibration range and fails by extrapolation outside it. | test |
| B104 | A 1 nm wavelength-registration error produces an error proportional to the first derivative of the spectrum, so sharp bands transfer worse than broad ones. | test |
| B105 | Detector nonlinearity cannot be transferred by any linear model because it is a function of intensity rather than of wavelength. | test |
| B106 | Stray-light differences between instruments set a transfer floor that no post-hoc correction removes. | test |
| B107 | Two laboratories using the same reference material but different reference ratios disagree by exactly the ratio of their λs. | test |
| B108 | Reporting λ should let two laboratories reconcile without exchanging samples, which is testable on paired data with known geometry. | test |
| B109 | A transfer standard must be more stable than the effect being compared, or the comparison measures the standard's drift. | test |
| B110 | Round-robin comparisons that do not split samples confound the laboratory effect with sample heterogeneity. | note |
| B111 | Split-level designs separate laboratory bias from sample-lot variance with two materials rather than many. | note |
| B112 | Regression of method B on method A attenuates the slope toward zero when both methods have error, so a slope below 1 is not evidence of proportional bias. | test |
| B113 | Deming regression removes the attenuation only when the ratio of the two error variances is known. | test |
| B114 | Passing-Bablok slopes are robust to outliers but not to a nonlinear response, for which they still report a slope. | test |
| B115 | Bland-Altman limits of agreement are a property of the paired sample as much as of the methods, and change when the sample range changes. | test |
| B116 | An uncertainty chain of k independent comparisons grows as the square root of k, while a chain of k correlated comparisons grows as k. | note |
| B117 | A ratio of two laboratories' values has smaller relative uncertainty than either absolute value when the reference is common. | test |
| B118 | A ratio across laboratories with different references has larger uncertainty than either absolute value, because the references do not cancel. | note |
| B119 | Daily calibration against a single standard transfers gain but not offset, so offset drift accumulates unchecked. | test |
| B120 | Bracketing every batch with standards reduces drift to the interpolation residual at a cost of two measurements per batch. | note |
| B121 | Event-triggered recalibration detects drift later than scheduled recalibration but consumes fewer standards. | test |
| B122 | A Shewhart chart with 3σ limits detects a 1σ shift only after about eight points, so it is not a transfer check. | note |
| B123 | A reference material's certified uncertainty enters a laboratory's budget once per traceability link, not once per measurement. | note |
| B124 | Key-comparison degrees of equivalence are ratios to a consensus value, so they cancel any reference error common to the participants. | test |
| B125 | The Birge ratio detects dark uncertainty only when the claimed uncertainties are smaller than the observed scatter. | note |
| B126 | An Eₙ score below 1 is evidence of agreement only when both uncertainties are honest, which the score itself cannot test. | note |
| B127 | Matrix-matched calibrants transfer better than synthetic ones whenever the matrix affects the response multiplicatively. | test |
| B128 | A synthetic calibrant transfers an additive matrix effect not at all, because the effect is absent from the calibrant. | test |
| B129 | Retention-index transfer works across instruments because the index is a ratio to a homologous series, which cancels the column's absolute retention. | test |
| B130 | Chemical-shift transfer across NMR spectrometers works because the field is the reference, so the ratio is field-independent. | test |
| B131 | A plate calibrator in qPCR transfers well-to-well variation but not assay efficiency, so it cannot remove a ΔΔCt bias. | test |
| B132 | Reference-gene stability must be shown per tissue, so a transfer between tissues is a new calibration rather than a transfer. | test |
| B133 | Bridging two ELISA plates with a shared control transfers the standard curve only when the control sits on the curve's linear part. | test |
| B134 | Serology unit conversion through a WHO standard is a multiplicative transfer, so it cannot correct a specificity difference between assays. | test |
| B135 | A certified reference material transfers a value rather than a method, so a laboratory can be traceable and still wrong for its matrix. | note |
| B136 | In isotope-ratio metrology the delta scale is transferred by two anchors, so a single-anchor transfer leaves a scale error. | test |
| B137 | A geochronology reference material transfers an age, so an error in the reference age is a common systematic across every laboratory that uses it. | note |
| B138 | A thermal-conductivity reference material transfers a value at one temperature and thickness, so transfer to another thickness is an extrapolation. | test |
| B139 | Transferring a 2D-material measurement between substrates changes λ and can change the reported conductivity by more than the material difference. | test |
| B140 | Tip calibration transfer between AFMs works only when both tip radii are measured, because the contact area is the reference. | test |
| B141 | Flow-cytometry MESF transfer between instruments requires the same bead lot, or the transfer measures bead variability. | test |
| B142 | HbA1c standardization succeeded because the measurand is defined as a ratio of two species, so the reference cancels inside the definition. | note |
| B143 | Cholesterol standardization needed a reference method as well as a reference material because the matrices differed. | note |
| B144 | A field that standardizes its reference material but not its reference ratio should still show laboratory-dependent scatter. | note |
| B145 | Residual scatter after transfer is predicted by the spread of λ across instruments, which is computable from calibration records. | test |
| B146 | An empirical transfer function between two instruments is underdetermined whenever their λs differ, because two parameters are fitted to one degree of freedom. | test |
| B147 | Transferability can be predicted before any samples are exchanged, from the two instruments' reference responses alone. | test |
| B148 | Interlaboratory scatter of ratios should be smaller than that of absolute values by the ratio of the reference variabilities. | test |
| B149 | A laboratory that publishes its raw reference reading can be transferred without a transfer standard, because the correction can be recomputed. | note |
| B150 | Instrument metadata sufficient to compute λ would make most transfer standards unnecessary, and is almost never recorded. | note |
A reporting standard is the list of quantities without which a number cannot be recomputed. Each lead below is a candidate entry, falsifiable by attempting reproduction without it.
| # | Lead | |
|---|---|---|
| B151 | A measurement that does not state its reference is not reproducible, because the same number corresponds to many different quantities. | test |
| B152 | A reference's own uncertainty should be reported separately from the measurement's repeatability, or the budget cannot be reconstructed. | test |
| B153 | A reported value should state the linear range over which its calibration holds, because outside that range the value is an extrapolation. | test |
| B154 | The calibration model's order should be stated, because a quadratic and a linear fit give different values from the same standards. | test |
| B155 | The number and values of calibration standards should be reported, because two standards cannot detect curvature. | test |
| B156 | The bracketing interval and the drift observed across it should be reported with any drift-corrected value. | test |
| B157 | An R² value should never be reported without residuals, because a high R² is compatible with a structured residual. | test |
| B158 | The reference state, including temperature, pressure and concentration, should be reported with every reference-dependent value. | note |
| B159 | The blank's definition should be reported, because blank can mean solvent, matrix, or empty cell. | test |
| B160 | The subtraction operator and its noise cost should be reported, because subtraction doubles variance when the readings are independent. | test |
| B161 | Whether replicates are independent should be stated, because replicates on one spot estimate precision rather than reproducibility. | test |
| B162 | The spot size relative to the substrate's thermal length should be reported, because it determines λ. | test |
| B163 | The reference-placement geometry should be reported, because layout can change the answer by more than the effect being claimed. | test |
| B164 | The statistical model, fixed or random laboratory effects, should be stated, because the two give different uncertainties from the same data. | test |
| B165 | A complete uncertainty budget, with each contribution and its distribution, should accompany any value claimed to two significant figures. | test |
| B166 | The coverage factor should be stated, because a 95% interval and a 1σ interval differ by a factor of about two. | note |
| B167 | The detection limit's definition should be stated, because the 3σ blank and 3.3σ calibration definitions differ by 10%. | test |
| B168 | Whether the detection limit came from the blank or from the calibration line should be stated, because the two are different quantities. | test |
| B169 | Values below the detection limit should be reported as censored rather than as zero, or the mean is biased upward. | test |
| B170 | Significant digits should match the stated uncertainty, because four digits with a 5% uncertainty is a false precision claim. | test |
| B171 | A ratio should be reported with its numerator and denominator defined, because fold change alone does not identify a quantity. | note |
| B172 | A fold change should state its reference condition, because inverting the comparison inverts the number. | test |
| B173 | Anisotropic properties should state their direction relative to the measurement geometry, because in-plane and cross-plane differ by the anisotropy factor. | test |
| B174 | The reference material's lot should be reported, because lot-to-lot variation is a component of uncertainty. | test |
| B175 | The reference electrode's type, filling solution and junction should be reported, because the junction potential is part of the value. | note |
| B176 | The reference temperature for dB, ppm or ppb should be stated, because the same ratio gives different values at different reference temperatures. | note |
| B177 | The sign convention should be stated, because a convention error is a factor of minus one that survives every numerical check. | note |
| B178 | Whether a value was interpolated or extrapolated should be stated, because extrapolation uncertainty is not bounded by the calibration data. | note |
| B179 | Any reference correction that was applied should be reported with its size, so that a reader can undo it. | test |
| B180 | The covariance between numerator and denominator should be reported for any ratio, because a shared reference makes them correlated. | test |
| B181 | Ignoring numerator-denominator covariance in a ratio understates its uncertainty by up to a factor of two. | test |
| B182 | Both the bias factor and λ should be reported, because a bias factor at fixed λ is not comparable across materials. | note |
| B183 | A measurement should report the value it is traceable to and the number of links, because each link multiplies the uncertainty. | test |
| B184 | Software version and fitting algorithm should be reported, because two algorithms on the same data can differ by more than the effect. | note |
| B185 | Raw data should be deposited with the derived value, because the reference correction cannot be checked without them. | test |
| B186 | The sample's thermal and electrical history should be reported, because hysteresis makes the reference state path-dependent. | test |
| B187 | The environment, including humidity and atmosphere, should be reported when the reference material is hygroscopic. | note |
| B188 | A reference-free claim should be accompanied by a demonstration that changing the reference did not change the value. | note |
| B189 | The uncertainty of a ratio should be reported as a ratio uncertainty rather than converted to an absolute one, which requires the denominator. | test |
| B190 | A reference declaration should be a required field in machine-readable metadata, not a sentence in a methods section. | test |
| B191 | Journals should require the reference ratio λ in the abstract of any substrate-referenced measurement. | note |
| B192 | A value that cannot state its λ should state a bound on λ, because a bound is enough to bound the bias. | note |
| B193 | Reference-dependent values should be reported alongside their bias factor, because the factor is what makes them comparable. | test |
| B194 | The reference's stability over the measurement time should be reported, because drift is not captured by a single uncertainty. | test |
| B195 | Calibration dates should be reported relative to the measurement, because a calibration certificate has an expiry that matters. | test |
| B196 | The reference's spatial uniformity should be reported when the reference is a substrate, because a non-uniform reference is a layout-dependent bias. | note |
| B197 | Interlaboratory comparisons should report each participant's reference ratio, because a spread in λ explains a spread in values. | note |
| B198 | A reported value should be accompanied by the quantity it is a value of, because conductivity without a direction or method is not a quantity. | test |
| B199 | A reporting standard should be tested by whether two laboratories can reproduce a value without contacting each other. | test |
| B200 | The minimum reporting set for a reference-dependent measurement is the value, its reference, the reference ratio, the calibration model and the uncertainty budget. | note |
Three computations, all in incoprea_nov3/code/refmeas_tests.py, writing
incoprea_nov3/results/reference_measurement.json. Test 1 is closed-form; tests 2 and 3
are simulations with stipulated parameters. No measured data are used anywhere on this page.
R(λ) was evaluated by Gauss–Legendre quadrature of E1 (500 nodes, no lookup table, no scipy). As an external check on the implementation, the four λ values quoted in the I2 record were re-evaluated: the computed R agrees with the recorded values to 0.023%, and R(0.4348) = 0.99998, which recovers the stated crossing to five digits.
| λ | R = κapp/κ | R/λ | |d ln R / d ln λ| | κ error from 10% λ error | λ precision for a 2× claim |
|---|---|---|---|---|---|
| 0.01 | 0.2452 | 24.5187 | 0.2352 | 2.4% | 295% |
| 0.03 | 0.3280 | 10.9317 | 0.2980 | 3.0% | 233% |
| 0.0625 | 0.4162 | 6.6598 | 0.3537 | 3.5% | 196% |
| 0.1 | 0.4964 | 4.9637 | 0.3964 | 4.0% | 175% |
| 0.2 | 0.6696 | 3.3482 | 0.4696 | 4.7% | 148% |
| 0.4348 | 1.0000 | 2.2999 | 0.5652 | 5.7% | 123% |
| 1 | 1.6769 | 1.6769 | 0.6769 | 6.8% | 102% |
| 3 | 3.8156 | 1.2719 | 0.8156 | 8.2% | 85% |
| 10 | 10.9214 | 1.0921 | 0.9214 | 9.2% | 75% |
| 19.2 | 20.1546 | 1.0497 | 0.9546 | 9.5% | 73% |
| 30.6 | 31.5701 | 1.0317 | 0.9701 | 9.7% | 71% |
| 100 | 100.9903 | 1.0099 | 0.9903 | 9.9% | 70% |
What the table says. Over λ = 0.01 to 100 the bias factor spans 412× (0.2452 to 100.99). The elasticity is bounded by 1 everywhere — its maximum is 0.9903 at λ = 100 — so a 10% error in λ produces at most a 9.9% error in the corrected κ. Even at λ = 0.01, where R = 0.245 and the correction multiplies by 4.1, a 10% λ error moves the corrected value by only 2.4%.
The matrix. The next table gives the ratio two laboratories would report if their materials were identical and only their λ differed. The diagonal is 1 by construction.
| reported ratio, equal true κ | λ₂ = 0.0625 | λ₂ = 0.4348 | λ₂ = 3 | λ₂ = 30.6 |
|---|---|---|---|---|
| λ₁ = 0.0625 | 1.0000 | 2.4024 | 9.1668 | 75.8462 |
| λ₁ = 0.4348 | 0.4162 | 1.0000 | 3.8157 | 31.5709 |
| λ₁ = 3 | 0.1091 | 0.2621 | 1.0000 | 8.2740 |
| λ₁ = 30.6 | 0.0132 | 0.0317 | 0.1209 | 1.0000 |
The largest entry is 75.85× (λ₂ = 30.6 against λ₁ = 0.0625), which reproduces the I2 record's 75.8× to three digits. The matrix's own max/min is 5753×, the square of that, because the matrix contains both directions of comparison. This is a consistency check on my implementation of the record's relation, not a new measurement.
The expectation this contradicted. I expected the correction to be hypersensitive below the crossover, where R is small and the correction factor is large. It is the opposite: the elasticity falls toward zero as λ → 0 (0.235 at λ = 0.01) and rises toward 1 as λ grows (0.990 at λ = 100). A large bias factor and a large sensitivity are not the same thing, and the reference-dominated regime is the one where the correction is most stable.
Two arms, stipulated values X₁ = 5, X₂ = 2, reference B = 3, unity gains. The reference is either additive (S = G(X + B)) or multiplicative (S = G·B·X). Each is measured with a difference and with a ratio. The table is arithmetic, not simulation.
| Reference algebra | Operator | Estimate | Target | Error |
|---|---|---|---|---|
| additive | difference | 3.0000 | 3.0000 | +0.0% |
| additive | ratio | 1.6000 | 2.5000 | -36.0% |
| multiplicative | difference | 9.0000 | 3.0000 | +200.0% |
| multiplicative | ratio | 2.5000 | 2.5000 | +0.0% |
Only the matched pairing cancels: a difference for an additive reference, a ratio for a multiplicative one. The mismatched pairings are not mildly wrong — a ratio applied to an additive reference is off by 36%, and a difference applied to a multiplicative reference is off by 200%, because the reference multiplies the answer instead of cancelling.
Now the imperfect case: a shared additive reference with a fractional gain mismatch ε. The residual leakage is ε·(X+B) while the single-ended reference term is (X+B), so the rejection is 1/ε — independent of the reference's size.
| Gain mismatch ε | Rejection (dB) | Equivalent reference drift | Verdict |
|---|---|---|---|
| 0 | ∞ | 0.00% | exact cancellation |
| 0.0001 | 80.0 | 0.01% | beats 1% reference drift by 40 dB |
| 0.001 | 60.0 | 0.10% | beats 1% reference drift by 20 dB |
| 0.01 | 40.0 | 1.00% | ties 1% reference drift |
| 0.1 | 20.0 | 10.00% | worse than 1% reference drift by 20 dB |
The independent-reference baseline is 1% drift, which is 40 dB. A Monte Carlo with 20,000 trials at ε = 1%, 1% arm noise, and a reference drawn over 1–5 gives a mean leakage of 0.00996 against the predicted 0.01000 (40.04 dB). With the arm noise included the RMS leakage is 0.01735, or 35.2 dB, so at this noise level the practical floor is about 35 dB, not 40.
The expectation this contradicted. The working rule was that a differential design cancels the reference "when both arms share it". Sharing is necessary but not sufficient: at 1% gain mismatch a shared reference leaks exactly as much as a pair of independent references drifting at 1% (both 40 dB). The rule has to be stated as sharing and matching, with the break-even at ε = ΔB/B.
A one-dimensional abstraction of the reference-placement problem: field f(u) = exp(−u/L) with L = 20 mm, a target at u = 0, and twelve reference points reading r(u) = f(u)·(1 + b·exp(−u/ubl))·(1 + ση), with σ = 1%, b = 0.3 and η Gaussian. The bracket is a boundary artefact that decays over ubl. The estimator is a least-squares line through ln r against u, extrapolated to u = 0. Four layouts (all stipulated): 2–18 mm ("edge"), 5–60 mm ("grid"), two clusters at 3–6 and 24–30 mm ("interleaved"), and 30–60 mm ("far"). RMS percentage error over 4,000 draws:
| ubl (mm) | edge | grid | interleaved | far | spread | best layout |
|---|---|---|---|---|---|---|
| 2 | 6.44 | 1.06 | 4.38 | 1.42 | 6.06× | grid |
| 5 | 17.44 | 5.30 | 15.03 | 1.42 | 12.27× | far |
| 10 | 24.72 | 12.16 | 22.89 | 2.83 | 8.74× | far |
| 20 | 28.28 | 20.48 | 27.43 | 11.45 | 2.47× | far |
| 40 | 29.51 | 26.32 | 29.22 | 21.73 | 1.36× | far |
The spread peaks at 12.3× (ubl = 5 mm), where the edge layout is wrong by 17.4% and the far layout by 1.4%. At ubl = 2 mm the best layout is the grid (1.06%), and from ubl = 5 mm upward it is the far layout. At ubl = 40 mm every layout is wrong by 22–30% and the spread collapses to 1.36×: when the boundary artefact is flat across the whole domain, no placement removes it.
The amplitude sweep at ubl = 5 mm shows the same thing from the other side. With b = 0 the spread is only 2.86× and the interleaved layout wins (0.50% against 0.64% for edge), which is a noise-leverage effect. By b = 0.5 the spread is 20.50× and the far layout wins by a wide margin.
| b | edge | grid | interleaved | far | spread | best layout |
|---|---|---|---|---|---|---|
| 0 | 0.64 | 0.62 | 0.50 | 1.42 | 2.86× | interleaved |
| 0.05 | 2.94 | 1.07 | 2.53 | 1.42 | 2.74× | grid |
| 0.1 | 5.80 | 1.87 | 5.00 | 1.42 | 4.09× | far |
| 0.2 | 11.59 | 3.58 | 10.00 | 1.42 | 8.17× | far |
| 0.3 | 17.44 | 5.30 | 15.03 | 1.42 | 12.27× | far |
| 0.5 | 29.26 | 8.72 | 25.20 | 1.43 | 20.50× | far |
The expectation this contradicted. I expected the interleaved layout to beat the edge layout once the boundary artefact was present, following the H0199 claim of roughly a 10× advantage. In this model it never does: interleaved wins only in the noise-only limit, by 1.3×, and as soon as b > 0 the ranking is set by distance from the boundary, not by interleaving. The layout spread is real and reaches 12.3×, but it is not the effect that was claimed.
| Section | Leads | test | note |
|---|---|---|---|
| A · Substrate and reference ratios | 50 | 33 | 17 |
| B · Differential and ratiometric designs | 50 | 37 | 13 |
| C · Calibration transfer | 50 | 34 | 16 |
| D · Reporting standards | 50 | 34 | 16 |
| Total | 200 | 138 | 62 |
Tags are editorial: test marks a lead that can be attacked with a computation, a dataset, or a re-analysis already to hand; note marks one that needs a literature sample, a new measurement, or an institutional change. A "test" tag is not evidence that the lead is true.
If one sentence survives: a reference biases a reported value but does not amplify its own uncertainty, so the 75.8× spread between equal materials is a failure to report λ rather than a failure of the correction — and a differential design cancels that bias only when the operator matches the reference's algebra and the arms are matched to better than the reference's own drift.
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