Every section converges on one rule, stated here once and used everywhere after: a number that does not carry its cutoff, its fit window, its weighting and its sample size is not a measurement of a material — it is a measurement of the procedure. The three computations in section 5 are the arithmetic behind that rule, and each returns a number that a careful worker would publish without noticing.
The leads are numbered E1–E200. test marks a statement with a stated falsifying observation; note marks a definitional or bookkeeping statement whose falsification is that someone has conflated the two quantities it separates. 187 are tagged test and 13 note. None has been checked against the literature, and none is evidence.
A point defect in an infinite solid has a strain field that decays as the inverse square of distance, so its elastic energy density decays as the inverse sixth power and the energy inside a radius R converges. The energy outside R does not go away: it falls only as R−3, and the total is dominated by whatever core cutoff is chosen. A periodic supercell replaces the infinite tail with an image lattice whose leading error falls as L−3. Both statements are about the same missing integral, and neither is a small correction if the quantity being reported is an energy difference of a few tens of millielectronvolts.
The leads below are all of one form: a quantity that looks like a material constant is in fact a functional of the model's cutoff, its reference state, or its treatment of the far field. Section 5, test 3, computes the sharpest of them.
| # | Lead | Tag |
|---|---|---|
| E1 | Defect formation energies from a periodic supercell should converge as the inverse cube of the cell length when the error is the elastic image interaction; a fit against L^-3 that does not extrapolate to the dilute limit falsifies the elastic-only correction. | test |
| E2 | A vacancy's elastic dipole tensor is not zero — relaxation leaves a finite misfit volume — so a model that treats the vacancy as an empty lattice site should mispredict the sign of the lattice-parameter change per vacancy. | test |
| E3 | Relaxation volume and formation volume are different quantities and differ by a reference-lattice term; a paper that reports "the vacancy volume" without naming which is reporting an undefined number. | note |
| E4 | A pair potential necessarily satisfies the Cauchy relation, so any defect energy computed with one should disagree with a many-body result in every quantity that depends on the shear constants, and the sign of the disagreement should be predictable from the Cauchy violation. | test |
| E5 | Dislocation core energy fitted from line tension at two cutoffs should differ by (mu b^2 / 4 pi (1-nu)) ln(r2/r1); a measured difference outside that band means the core model, not the cutoff, is wrong. | test |
| E6 | Peierls stress should fall exponentially with core width, so two potentials with the same stacking-fault energy and different core widths should predict Peierls stresses differing by more than an order of magnitude. | test |
| E7 | The stacking-fault energy from a supercell should depend on the number of faulted layers if the fault's elastic tail is being resolved; a value that is independent of the layer count is a truncated integral, not an energy. | test |
| E8 | A gamma-surface computed with the cell fixed and one computed with the cell relaxed are different functions, so a Peierls stress quoted without saying which one was used is not reproducible. | note |
| E9 | The interaction energy of two point defects should fall as r^-3 with an angular factor set by their dipole symmetries, so an isotropic r^-1 or r^-2 fit to the same data indicates that the periodic images, not the pair, dominate. | test |
| E10 | A di-vacancy binding energy computed in a cell smaller than the pair separation is measuring the image interaction, so a binding energy that changes sign with cell size is an artefact rather than a result. | test |
| E11 | Two misfitting precipitates should align along elastically soft directions, so an isotropic elasticity model should predict a random pair correlation function and the measured one should show the alignment. | test |
| E12 | A defect near a free surface should feel an image attraction scaling as 1/d^3; a simulation that returns a different exponent has a boundary condition leaking into the result. | test |
| E13 | Solute-defect binding should depend on both size and modulus misfit, so a model including only size misfit should fail for solutes whose modulus contrast is large and size contrast small. | test |
| E14 | The dipole tensor of a defect computed by the Kanzaki-force route and by the stress route should agree to within the cell-size error; a disagreement of the order of the formation energy means the reference state is unconverged. | test |
| E15 | Elastic anisotropy changes only the prefactor of the 1/r^3 interaction, so a cubic crystal with a large anisotropy factor should show a pair interaction varying by more than a factor of two with direction at fixed separation. | test |
| E16 | "Core radius" is a fitting parameter with no independent definition, so a core energy quoted without the cutoff it was fitted at cannot be compared with any other core energy. | note |
| E17 | Dislocation line energy should diverge logarithmically, so doubling the box should change the total line energy by a fixed additive amount; a multiplicative change means the core is being cut rather than the box extended. | test |
| E18 | The critical radius for dislocation nucleation at a stress concentration should depend on the cutoff used to define the nucleus, so a predicted nucleation stress that is cutoff-independent is not really core-sensitive and is probably an elastic criterion. | test |
| E19 | Thermal expansion and the temperature slope of the formation energy draw on the same anharmonicity, so a potential fitted only to zero-temperature elastic constants should mispredict the slope in proportion to its expansion error. | test |
| E20 | The pressure derivative of a defect energy should equal the negative of its formation volume, so a calculation that returns one formation volume from the pressure slope and a different one from the dipole trace is internally inconsistent. | test |
| E21 | A phase-field or finite-element model must smear the defect core over at least the mesh spacing, so the peak stress at a defect should converge only when the core radius exceeds two elements. | test |
| E22 | The Nye tensor of a dislocation arrangement should be cutoff-independent while the elastic energy is not, so a Nye tensor that changes with the smoothing radius means the smoothing is deleting real dislocations. | test |
| E23 | A disclination should carry an energy quadratic in its strength, so a model linearised in the defect strength should fail at the rotation angle where the second-order term becomes comparable to the first. | test |
| E24 | A vacancy-solute complex should not have additive binding energies: the binding of a second solute to a bound pair differs from its binding to an isolated vacancy, so an additive model should show a measurable discrepancy at high solute loadings. | test |
| E25 | A defect in a nanoparticle should have a formation energy shifted by the image interaction with the curved surface, scaling as 1/R, so bulk supercell values should not be transferred below the radius where that shift exceeds kT. | test |
| E26 | Migration barriers should shift under applied strain at a rate given by the activation strain tensor, so a zero-strain calculation should mispredict the barrier at 2% strain by an amount proportional to the defect's formation volume. | test |
| E27 | A defect energy computed with a fixed charge state and one computed with a neutralising background are different calculations; the difference is an electrostatic correction and it is not small in an ionic material. | note |
| E28 | Elastic constants from stress-strain and from energy-strain should agree only while the quadratic term dominates, so the strain at which they diverge by one per cent is a measurable limit of the harmonic model. | test |
| E29 | In a metal, a cell smaller than the screening length should give a defect energy that oscillates with cell size through Friedel oscillations, so a monotone convergence plot in a metal indicates the wrong boundary treatment. | test |
| E30 | The self-interstitial crowdion should be lower in energy than the dumbbell by an amount that correlates with the potential's stacking-fault energy, so a family of potentials should show that correlation across the family. | test |
| E31 | A stacking-fault tetrahedron should be stable only above a critical size set by the stacking-fault energy, so a cell below that size should predict instability for reasons of size rather than physics. | test |
| E32 | Point-defect bias factors in cascade damage should depend on the cutoff radius used for recombination, so a swelling model that does not report its recombination radius is not reproducible. | test |
| E33 | The thermal spike in a displacement cascade should be resolution-dependent until the cell contains the spike volume, so the surviving defect count should scale with cell size below a threshold and saturate above it. | test |
| E34 | A defect energy is converged only for the property named: a value converged for formation energy is not necessarily converged for the migration barrier, the dipole tensor or the elastic constants. | note |
| E35 | Zener ordering should scale with the square of the dipole tensor and with concentration, so two alloys with the same misfit volume but different dipole symmetry should order at different temperatures. | test |
| E36 | The Snoek peak height should be proportional to solute concentration and to the dipole anisotropy, so a concentration inferred from internal friction without the anisotropy factor should be wrong by exactly that factor. | test |
| E37 | A finite cutoff removes the tail of the strain field, so the elastic part of a defect energy should be recoverable as the difference between two cutoffs and that difference should be independent of the core parameter. | test |
| E38 | The formation energy of a Schottky defect in an ionic crystal should depend on the surface charge boundary condition, so two slabs with different boundary conditions should differ by more than the defect's own energy scale below a critical thickness. | test |
| E39 | The dislocation-solute interaction should decay as 1/r for size misfit and 1/r^2 for modulus misfit, so a measured interaction that follows one exponent over three decades is mixing two mechanisms rather than revealing a new one. | test |
| E40 | The Orowan bypass stress should scale as the inverse spacing for impenetrable particles and more weakly for shearable ones, so a single-exponent fit across the shearing-to-bypassing transition should match neither. | test |
| E41 | A pile-up truncated at N dislocations concentrates stress by a factor N, so a simulation with fewer than the equilibrium number should underpredict the local stress by exactly the missing count. | test |
| E42 | The plastic zone at a crack tip has a size set by (K/sigma_y)^2, so a model whose fixed cutoff is smaller than that zone should return a toughness that depends on the cutoff. | test |
| E43 | The energy of a coherent interface should depend on the misfit strain, so two systems with the same chemistry and different misfit should differ in interface energy by an elastic term that is calculable. | test |
| E44 | An interface energy from a slab calculation contains two surfaces and one interface, so the decomposition needs an independent surface energy and the error in that surface energy propagates one-for-one into the interface value. | note |
| E45 | The equilibrium shape of a precipitate should be set by the anisotropy of interface energy, so an isotropic interface model should predict shapes that differ measurably from the Wulff construction. | test |
| E46 | The elastic self-energy of a coherent precipitate should scale with the square of the misfit and with the volume, so a model with a linear misfit term should fail as the misfit grows. | test |
| E47 | Hydrogen trapping energies should depend on local elastic strain, so the same boundary should show a different binding energy when the strain state changes, by an amount set by the strain and not by the chemistry. | test |
| E48 | The Debye temperature inferred from low-temperature heat capacity should shift with defect concentration in proportion to the defect formation volume, so a measured shift is an independent check on the dipole tensor. | test |
| E49 | An interstitial's elastic dipole should exceed a vacancy's by roughly the ratio of their formation volumes, so a model assigning the same misfit to both should mispredict their relative strengthening. | test |
| E50 | Every item in this section is a hypothesis about a model and not a measurement; the only thing that makes one testable is an explicit cutoff and an explicit reference state. | note |
A power law is a claim that one mechanism operates over the whole range. Almost every accepted scaling law in mechanics is instead the visible part of two mechanisms with different exponents, and the fit that reports "the" exponent is reporting the arithmetic of the window. Section 5, test 1, shows that for a symmetric window the returned exponent is exactly the mean of the two mechanism exponents — independent of window width — and that shifting the window moves it across the whole span between them.
The practical consequence is not that power laws are wrong. It is that an exponent without its range cannot be compared with another exponent, and a crossover is demonstrated only by showing that the exponent changes with the range.
| # | Lead | Tag |
|---|---|---|
| E51 | A single power law fitted symmetrically across a two-mechanism crossover should return exactly the mean of the two mechanism exponents, whatever the window width — so the wrong exponent is not detectable by re-fitting on a wider or narrower range. | test |
| E52 | Shifting the fit window at fixed width should move the apparent exponent across the full span between the two mechanism values, so an exponent quoted without its stress range is undefined. | test |
| E53 | A diffusion-to-dislocation creep transition should make the apparent stress exponent drift continuously with stress, so any quoted n without the range is a statement about the test, not the alloy. | test |
| E54 | Hall–Petch should give way to inverse Hall–Petch at the grain size where the dislocation source size equals the grain, so the crossover should move with the initial dislocation density of the sample. | test |
| E55 | A single power law fitted across a strength peak in grain size should return a near-zero exponent — "grain size does not matter" — for a material whose strength changes by a factor of three across the same window. | test |
| E56 | A fit restricted below the strength peak should return a positive exponent, so a paper reporting a positive grain-size exponent has fitted the inverse branch and mislabelled it. | test |
| E57 | The strain-rate sensitivity should rise sharply when dislocation drag replaces thermally activated glide, so a log stress against log rate plot should show a slope change at a rate set by the drag coefficient. | test |
| E58 | The Hall–Petch coefficient should be temperature-independent if pile-up controlled and temperature-dependent if source controlled, so the measured temperature dependence identifies the mechanism. | test |
| E59 | Fatigue life should follow a Basquin law at low strain amplitude and a Coffin–Manson law at high amplitude, and the crossover amplitude should be calculable from the two exponents plus one constant. | test |
| E60 | The Paris exponent should be low near threshold and rise steeply in the fast-growth regime, so a single exponent quoted for an alloy is an average over a range in which the mechanism changes. | test |
| E61 | The Kitagawa–Takahashi diagram should have a horizontal branch below a critical crack size and a Griffith branch above it, and the knee is the only place where both mechanisms can be tested at once. | test |
| E62 | The brittle size effect should follow Weibull statistics for large specimens and a deterministic limit for small ones, so one size exponent fitted across the transition reports a Weibull modulus that no specimen possesses. | test |
| E63 | Indentation hardness should scale as the inverse depth below a characteristic depth and be constant above it, so a Meyer's-law exponent fitted across the transition is depth-range dependent. | test |
| E64 | Grain growth should follow d^2 proportional to t in the ideal case and a higher exponent under boundary or solute drag, so an exponent between two and four signals a transition rather than a mechanism. | test |
| E65 | The grain-growth exponent should drift upward with annealing time as the driving force falls, so an exponent fitted over one decade should be smaller than one fitted over three. | test |
| E66 | Precipitation hardening should rise as the square root of particle size while particles are shearable and fall as the inverse once they are bypassed, so the peak-aged condition is a crossover whose position shifts with volume fraction. | test |
| E67 | Solid-solution strengthening should follow one concentration exponent under Labusch statistics and another under Fleischer statistics, and the crossover between them should be predictable from the ratio of size to modulus misfit. | test |
| E68 | Forest hardening and grain-boundary hardening should superpose additively in stress at low dislocation density and in quadrature at high, so any single superposition rule should fail on one side. | test |
| E69 | Stage II strain hardening should be linear in strain while stage III saturates, so a single hardening law fitted across both should return a saturation stress that depends on the fit range. | test |
| E70 | The Bauschinger effect should vanish for small reversed strain and saturate once the reversed strain exceeds the back-stress scale, so one back-stress parameter fitted across both regimes is range-dependent. | test |
| E71 | The brittle-to-ductile transition temperature should shift with strain rate in an Arrhenius manner below a critical rate and become rate-insensitive above it, so an activation energy fitted across the whole range should be wrong. | test |
| E72 | Twinning should take over from slip above a critical strain rate or below a critical temperature, and the transition should be predictable from stacking-fault energy and strain rate together. | test |
| E73 | Superplastic elongation should require a strain-rate sensitivity above one half, so a material with m below that should fail the elongation test regardless of its grain size. | test |
| E74 | Diffusion creep should dominate below a stress set by the grain size and power-law creep above it, so the transition stress should scale with grain size with an exponent set by the creep exponent, and a grain-size-independent transition falsifies the map. | test |
| E75 | Coble creep should overtake Nabarro–Herring creep at fine grain size because the boundary path shortens, and the crossover grain size should scale with the ratio of boundary to lattice diffusivity. | test |
| E76 | Sintering shrinkage should follow the mechanism-assigned exponents in the initial stage and switch to a single exponent once the pores close, so an exponent fitted across both stages should match neither. | test |
| E77 | The sintering shrinkage exponent should change with atmosphere if surface diffusion dominates, because surface diffusion redistributes matter without densifying; an exponent unchanged by atmosphere rules out surface-diffusion control. | test |
| E78 | Grain-boundary-diffusion-controlled sintering should scale with the inverse cube of the grain size, so a sintering model that omits grain size should underpredict the densification rate of a fine powder. | test |
| E79 | Ostwald ripening should follow a one-third power of time when diffusion limited and one-half when interface limited, so an exponent between them indicates mixed control and should depend on volume fraction. | test |
| E80 | The coarsening rate should increase with volume fraction in the diffusion-limited regime, so a theory that ignores volume fraction should fail at high fractions while succeeding at low ones. | test |
| E81 | Contact mechanics should follow a 3/2 load–depth law until yield and a nearly linear relation after it, so a single power law fitted across both should return an exponent between 1 and 1.5. | test |
| E82 | Wear should be linear in load and sliding distance in the mild regime and depart from linearity when the mechanism changes, so a wear coefficient quoted without the regime is not a material constant. | test |
| E83 | Friction should be rate-independent at low sliding velocity and rate-weakening above a critical velocity, so a single rate parameter fitted across both should be wrong on both sides. | test |
| E84 | The plastic-zone correction to linear elastic fracture mechanics should be negligible when the zone is small against all specimen dimensions and dominant when it is not, so a toughness value without the specimen size is incomplete. | test |
| E85 | Plane-stress and plane-strain toughness should differ, with the transition thickness scaling as (K/sigma_y)^2, so a thickness series should show a plateau rather than a monotone trend. | test |
| E86 | Short cracks should grow faster than long cracks at the same stress intensity because shielding has not developed, and the difference should vanish at a length set by the microstructure. | test |
| E87 | The fatigue threshold should depend on crack size below a critical length and be a material constant above it, so a threshold quoted without a size refers to one branch only. | test |
| E88 | The apparent toughness from a chevron-notched specimen should differ from a straight-notched one only through notch-tip constraint, so a larger difference indicates a size effect misread as a geometry effect. | note |
| E89 | The ductile-to-brittle transition should be set by the cleavage fracture stress rather than the yield stress, so raising the yield stress while holding the cleavage stress fixed should not move the transition temperature. | test |
| E90 | The strain-hardening exponent from a tensile test should depend on the strain range used to fit it, so a value quoted without the range should not reproduce. | test |
| E91 | The Considere condition should predict necking only for the strain path in which the hardening curve was measured, so a hardening law from torsion should not predict the tensile necking strain. | test |
| E92 | Dynamic recrystallisation should switch on above a critical strain that scales with the Zener–Hollomon parameter, so a model with a fixed critical strain should fail across strain rates. | test |
| E93 | Grain-boundary sliding should contribute to creep only when the boundary is weaker than the grain, so a single constitutive law cannot cover both strong and weak boundaries. | test |
| E94 | The creep–fatigue interaction should be governed by whether the dwell time exceeds the relaxation time, so a life model without a time criterion should fail on one side of that ratio. | test |
| E95 | The Palmgren–Miner sum should be conservative for one load sequence and non-conservative for the reverse, so a single linear damage rule cannot hold in both directions. | note |
| E96 | The yield stress should follow an Arrhenius form only within the thermally activated regime, so an activation energy fitted across the athermal plateau is an artefact of the fit range. | test |
| E97 | The strain-rate sensitivity at fixed temperature should peak where the activation volume equals the obstacle spacing, so a monotone sensitivity against temperature rules out a single-obstacle model. | test |
| E98 | The hardness-to-yield ratio should be constant in the fully plastic regime and size-dependent below it, so a Tabor factor quoted without the indentation depth is not a material constant. | test |
| E99 | Toughness inferred from indentation cracking should agree with the single-edge-notch value only when the crack is long against the indentation plastic zone. | test |
| E100 | Every item in this section is a crossover claim: if the exponent does not change with the window, the crossover has not been demonstrated and the single law is adequate there. | note |
The volume fraction of grain boundary in an equiaxed microstructure is of order three boundary thicknesses over the grain size. At a grain size of ten nanometres and a boundary thickness of one nanometre that is thirty per cent of the material; at a micrometre it is three tenths of one per cent. Any property measured on the fine microstructure is therefore a composite measurement, and any "bulk" value extracted from it without a grain-size series is an extrapolation with no independent check.
The leads single out the cases where the interface contribution is in series with the bulk — boundary resistance, boundary diffusivity, boundary capacitance — because those are the cases where a single-element fit assigns the interface to the bulk and reports a material constant that is a geometry.
| # | Lead | Tag |
|---|---|---|
| E101 | The grain-boundary volume fraction should scale as the boundary thickness over the grain size, so a property that depends on it should appear grain-size dependent even when the bulk property is constant. | test |
| E102 | The thermal conductivity of a polycrystal should lie below the single-crystal value at every grain size and approach it only as the grain size diverges, with the deficit set by the boundary conductance. | test |
| E103 | The boundary conductance should be extractable from a grain-size series only if the bulk conductivity is independently known, so the two fitted together from one series are degenerate. | test |
| E104 | Electrical resistivity should carry a grain-size-dependent term from boundary scattering, so a resistivity quoted without the grain size is not a bulk property. | test |
| E105 | The boundary reflection coefficient should be the only free parameter of the standard size-dependent resistivity form, so a two-parameter fit is overfitting unless the data span two decades of grain size. | test |
| E106 | Tracer diffusion in a polycrystal should follow effective-bulk kinetics at high temperature, boundary-assisted kinetics at intermediate temperature and boundary-only kinetics at low temperature, so a single Arrhenius fit across all three describes none of them. | test |
| E107 | The diffusivity extracted from a boundary-assisted penetration profile should depend on the boundary width assumed in the model, so a value quoted without that width is not a material property. | test |
| E108 | The ionic conductivity of a polycrystal should contain a boundary contribution in series with the bulk, so an impedance spectrum fitted with a single element assigns the boundary resistance to the bulk. | test |
| E109 | Space-charge layers should make the boundary conductivity depend on dopant concentration in a way the bulk does not, so a conductivity that tracks dopant level should not be called bulk. | test |
| E110 | The permittivity of a polycrystal should be lowered by boundary layers in series, so an apparent permittivity that rises as the sample is thinned is an interface artefact. | test |
| E111 | Coercivity in a soft magnet should be set by boundary pinning, so it should scale with the inverse grain size while the saturation magnetisation does not. | test |
| E112 | Boundary weak links should suppress critical current density differently from intragranular pinning, so a critical current that depends on grain size identifies which mechanism is limiting. | test |
| E113 | Hydrogen diffusivity along boundaries should show a knee where the boundary path saturates, so a single effective diffusivity fitted across the knee should depend on specimen thickness. | test |
| E114 | The corrosion rate at boundaries should scale with boundary area per unit volume, so two specimens of the same alloy and different grain size should corrode at different rates per unit area. | test |
| E115 | The strain-rate sensitivity of boundary sliding should be near one half, so a measured value near one indicates diffusion creep and the label superplastic is then unjustified. | test |
| E116 | Surface energy should dominate the sintering driving force once the surface-to-volume ratio exceeds a threshold, so a sintering model with a size-independent driving force should fail for nanopowders. | test |
| E117 | The lattice parameter of a nanoparticle should contract with size because of surface stress, so a bulk lattice parameter should not be used below the size at which the contraction exceeds the diffractometer resolution. | test |
| E118 | The melting temperature should fall with particle size roughly as the inverse radius, so a size-independent melting point should fail below a size that is measurable. | test |
| E119 | Solubility should rise with interface curvature through the Gibbs–Thomson relation, so a phase diagram computed for flat interfaces should misplace the solvus for a fine microstructure. | test |
| E120 | Nucleation at grain boundaries should occur at a lower driving force than homogeneous nucleation, so a measured nucleation rate should exceed the homogeneous prediction by a factor set by boundary geometry. | test |
| E121 | Segregation should follow a McLean isotherm, so a boundary composition that saturates with bulk concentration at a level set by boundary site density confirms the mechanism while a linear dependence refutes it. | test |
| E122 | Temper embrittlement should track the boundary concentration of the segregant rather than the bulk concentration, so two heats at the same bulk level and different boundary levels should embrittle differently. | test |
| E123 | A grain-boundary complexion transition should produce discontinuous changes in boundary mobility and energy at a fixed temperature, so a continuous property change across a supposed transition falsifies it. | test |
| E124 | Amorphous intergranular films should thicken with temperature in steps set by a disjoining pressure, so a monotone thickness–temperature curve indicates a different mechanism. | test |
| E125 | Zener pinning should set a limiting grain size that scales as the particle radius over volume fraction, so a microstructure that does not show that scaling is not pinned. | test |
| E126 | Abnormal grain growth should require a boundary whose mobility exceeds the matrix by a factor set by the pinning force, so a uniform mobility distribution should not produce abnormal grains. | test |
| E127 | Grain-boundary engineering should change properties in proportion to the fraction of special boundaries, so a property change without a measured change in boundary character is not evidence for the mechanism. | test |
| E128 | The triple-junction volume fraction should scale as the inverse square of grain size, so it should be negligible above a grain size where its contribution falls below resolution and dominant well below it. | test |
| E129 | Interfacial fracture toughness should depend on the mode mix, so a single interfacial toughness value is valid only for the loading phase at which it was measured. | test |
| E130 | Delamination should initiate where the interface energy is lowest rather than where the stress is highest, so the measured initiation site discriminates between the two predictions. | test |
| E131 | The rule of mixtures should bound a composite's modulus but not its strength, so a strength prediction from the rule of mixtures is an upper bound that is never attained. | test |
| E132 | The percolation threshold should mark the switch from matrix-dominated to network-dominated behaviour, so a property that varies smoothly through the threshold has a different controlling mechanism. | test |
| E133 | The thermal boundary conductance of a metal–ceramic interface should be dominated by electron–phonon coupling rather than by phonon mismatch, so a diffuse-mismatch model should fail for metallic contacts. | test |
| E134 | Contact resistance should scale with real contact area rather than apparent area, so a resistance that tracks the nominal area indicates a different mechanism. | test |
| E135 | Interface states should pin the Fermi level at a position set by interface chemistry, so a Schottky barrier that follows the metal work function is the exception and not the rule. | test |
| E136 | The strength of a thin film on a substrate should exceed the free-standing value because the substrate suppresses yielding, so a bulk strength inferred from a film should depend on film thickness. | test |
| E137 | The modulus measured by nanoindentation of a film should converge on the film value only while the plastic zone stays below roughly a tenth of the film thickness, so a modulus that drifts with depth is measuring the substrate. | test |
| E138 | Oxide-scale growth should be interface controlled while thin and diffusion controlled while thick, so an apparent parabolic constant should depend on the thickness at which it was fitted. | test |
| E139 | The sintering stress should scale with surface energy over pore radius, so a model with a constant sintering stress should fail across pore sizes. | test |
| E140 | The electrical double layer at a solid–liquid interface should dominate the measured impedance at low frequency, so a bulk conductivity extracted without a double-layer element should be wrong at low frequency. | test |
| E141 | Catalytic activity should scale with exposed surface area until the surface is poisoned, so an activity quoted per gram without a surface-area measurement is not comparable with any other. | test |
| E142 | Interfacial energy anisotropy should control ripening kinetics, so two alloys with the same mean precipitate size and different shape distributions should coarsen at different rates. | test |
| E143 | The yield strength of a nanoporous metal should scale with ligament size and not with relative density alone, so a Gibson–Ashby fit that omits ligament size should fail below a critical size. | test |
| E144 | The thermal conductivity of a nanoporous solid should follow the solid fraction only while the pore size exceeds the phonon mean free path, so below that size an effective-medium model should overpredict. | test |
| E145 | Grain-boundary energy should depend on the boundary plane, so an energy averaged over a boundary distribution should not reproduce a single-boundary experiment. | test |
| E146 | Boundary migration rate should be proportional to driving force only below a critical force, so a measured mobility that falls as the driving force rises indicates a different mechanism. | test |
| E147 | A phase-field interface has a finite width, so the predicted interfacial energy should depend on that width unless the model is calibrated at the same width it is used at. | test |
| E148 | The elastic modulus of a nanocrystalline solid should be grain-size independent while its strength is not, so a modulus that changes with grain size indicates an interface contribution to the measurement. | test |
| E149 | The interface-dominated regime should be identifiable by a property scaling with interface area per unit volume, so any claim of a bulk property in that regime must show that the scaling is absent. | test |
| E150 | Every item here predicts a specific scaling with interface area per volume; the test is always whether the property follows that scaling and the bulk value does not. | note |
A mean is a sufficient statistic only for a property linear in the microstructural variable. For a lognormal grain-size distribution and a property scaling as dm, the mean property and the property at the mean size differ by a factor exp((m2−m)s2/2), which vanishes at m = 1 and grows with the square of the distribution width. Section 5, test 2, tabulates it. At m = 3 and s = 0.8 the factor is 6.82.
The second point is that the tail, not the mean, sets failure. A lognormal with a given mean has a tail that depends on its width, so two microstructures identical in the mean can differ by orders of magnitude in the probability of the worst grain, the worst pore, or the worst flaw.
| # | Lead | Tag |
|---|---|---|
| E151 | A property scaling as a power m of grain size should have a mean equal to the property at the mean size times exp((m^2 - m) s^2 / 2), so a mean-based prediction is exact only for m equal to one. | test |
| E152 | Two microstructures with the same mean grain size and different widths should differ in every property with m not equal to one, so a mean quoted without a width is insufficient. | test |
| E153 | A bimodal distribution should be indistinguishable from a monodisperse one by its mean alone, so a mean-based model should predict the same property for both and a measured difference refutes the sufficiency of the mean. | test |
| E154 | Number-weighted and volume-weighted mean grain sizes should differ by a factor set by the distribution width, so a value quoted without its weighting is ambiguous. | test |
| E155 | A two-dimensional section distribution should differ from the three-dimensional distribution in both mean and width because sectioning probability scales with size, so a micrograph size should be transformed before comparison with a volume-based model. | test |
| E156 | The mean intercept length through a monodisperse sphere population should equal four thirds of the radius, so a calibration that assumes equality with the diameter carries a large systematic bias. | test |
| E157 | The standard error of the mean of log grain size should equal s over the square root of N, so a mean quoted from fewer than (s/0.05)^2 grains cannot support a five per cent claim. | test |
| E158 | The number of grains needed for a stable property average should grow as the square of the property exponent, so a property with m equal to three needs nine times the sample of one with m equal to one. | test |
| E159 | A distribution truncated by the measurement step size should bias the mean upward by an amount calculable from the truncation point, so a mean from a coarse scan is not the population mean. | test |
| E160 | The Weibull modulus should be recoverable from strength scatter at one specimen size only if the flaw population is unimodal, so a curved Weibull plot indicates a mixture rather than a new mechanism. | test |
| E161 | A Weibull modulus estimated from ten specimens should carry a confidence interval wide enough to overlap the value from a hundred, so the two should not be compared without their intervals. | test |
| E162 | The design strength should be a lower percentile and not the mean, so a safety factor applied to the mean is not equivalent to a reliability target. | test |
| E163 | The largest flaw in a volume should follow extreme-value statistics, so the strength of a large component should be predictable from the flaw-size distribution measured on a small one. | test |
| E164 | Strength should be set by the weakest link, so the mean strength of many small specimens should exceed the strength of one large specimen of the same total volume by a factor set by the Weibull modulus. | test |
| E165 | Poisson statistics should make the largest defect in a volume grow as the logarithm of the volume, so strength should fall logarithmically with size rather than as a power of it. | test |
| E166 | Defect populations should be spatially clustered, so a Poisson model should underpredict the variance of local density and therefore the scatter in strength. | test |
| E167 | The correlation length of the microstructure should set the specimen size above which property variance falls as one over volume rather than one over the number of grains. | test |
| E168 | The Hall–Petch slope from a regression across a grain-size distribution should not equal the slope that applies at a single grain size, because the regression weights the distribution differently from the mean. | test |
| E169 | Texture should be described by the full orientation distribution because a mean orientation is undefined for most textures, so a scalar texture strength should not predict a property governed by the tail. | test |
| E170 | The misorientation distribution of a twinned microstructure should be bimodal, so a mean misorientation describes neither population. | test |
| E171 | The boundary-character distribution should control properties at fixed grain size, so two microstructures with the same grain size and different character should differ. | test |
| E172 | Precipitate number density should fall as the inverse of time while the mean radius rises as the one-third power, so a mean radius alone cannot distinguish coarsening from dissolution. | test |
| E173 | Precipitate spacing should depend on volume fraction as well as radius, so a spacing inferred from mean radius alone should be wrong by a factor that grows with the fraction. | test |
| E174 | Porosity should be described by its connectivity and not its mean fraction, so two samples with equal porosity and different pore morphology should have different permeability. | test |
| E175 | Permeability should scale with the square of a characteristic pore size that is not the mean, so a Kozeny–Carman estimate using the mean pore size should be biased. | test |
| E176 | The pore-size distribution should control the capillary pressure curve, so a mean pore size cannot reproduce the entry pressure. | test |
| E177 | The grain-size distribution should stay self-similar during normal growth while its width is inherited from the initial condition, so two starting distributions should keep different widths at the same mean. | test |
| E178 | The distribution width should grow during abnormal growth because the tail becomes the population, so a width that stays constant during growth rules out the abnormal mechanism. | test |
| E179 | The recrystallised grain size should be set by the ratio of nucleation rate to growth rate, so a mean grain size without both rates does not identify the mechanism. | test |
| E180 | The JMAK exponent should diagnose nucleation and growth geometry, so a non-integer exponent should indicate mixed mechanisms rather than a new one. | test |
| E181 | Transformation kinetics should depend on the spatial distribution of nucleation sites, so two samples with equal site density and different clustering should transform at different rates. | test |
| E182 | The local volume fraction of a second phase should fluctuate more than Poisson because of clustering, so a measurement on one field of view should not represent the sample. | test |
| E183 | A two-phase property should be bounded by the Voigt and Reuss averages but not determined by volume fraction alone, so two microstructures at the same fraction should differ. | test |
| E184 | A property's standard deviation should be reported with its mean, because a ten per cent change in the mean against thirty per cent scatter is not a detectable effect. | test |
| E185 | The grain-size distribution should be lognormal after recrystallisation and depart from lognormal after abnormal growth, so the distribution shape is itself a process fingerprint. | test |
| E186 | The reported average grain size should differ between the intercept method and the equivalent-circle method by a systematic factor for the same microstructure. | test |
| E187 | Twin boundaries should be counted or excluded, so a mean grain size that does not state the convention is not reproducible. | test |
| E188 | The width of a grain-size distribution should be larger in two-dimensional sections than in three dimensions because of the sectioning transform, so a width measured on a micrograph should not enter a three-dimensional model uncorrected. | test |
| E189 | The number of grains per field should be large enough that field-to-field variance is smaller than between-sample variance, so a single field cannot support a sample-level claim. | test |
| E190 | A property distribution should be lognormal when the underlying microstructure is multiplicative, so a normal fit to a property spanning a decade indicates the wrong model. | test |
| E191 | A property's coefficient of variation should equal the microstructural coefficient of variation times the property exponent, so property scatter is predictable from microstructural scatter. | test |
| E192 | The extreme-value tail of a defect distribution should be measured directly, because the mean of a lognormal does not determine its tail, so two distributions with the same mean can have tails differing by orders of magnitude. | test |
| E193 | The fatigue limit should be a distribution and not a number, because run-outs and failures coexist at the same stress, so a single value should not be quoted without its probability. | test |
| E194 | Scatter in fatigue life should grow with life, so a constant standard deviation in log life should fail at long lives. | test |
| E195 | The fatigue size effect should be predictable from the volume of highly stressed material, so a small specimen should overpredict a large component's life by a factor set by the defect tail. | test |
| E196 | The measured grain size should depend on the EBSD step size once the step approaches the grain size, so a value from a coarse step is biased upward. | test |
| E197 | Boundary density measured by line intercept should be unbiased while the size distribution is biased, so the two cannot both be reported as the microstructure statistic. | test |
| E198 | A polycrystal's property should be the average over grains weighted by each grain's own contribution, so a number-weighted average is wrong whenever the property is nonlinear. | test |
| E199 | A mean without a width, a weighting, a truncation point and a sample size is not a measurement of a distribution but a single number of unknown relation to the population. | note |
| E200 | No microstructure in this section was measured: the statements are arithmetic consequences of stated distributions and should be treated as such. | note |
Three computations, all in code/materials_defects.py, all values in
results/materials_defects.json. No experimental data are used. Every number in the tables below is
printed by that script. The models are labelled where they are schematic.
Claim tested. An exponent extracted from a log–log fit is a property of the material.
Model. Two mechanisms acting in parallel, each an exact power law, with rates adding:
y(u) = u^p + u^q, and the crossover placed at u = 1 so that both terms equal one there. p = 1 and
q = 4 stand for diffusion creep and power-law creep. Nothing is fitted to data; the "data" are evaluations of
this function, which is the point — the arithmetic can be checked exactly.
At the crossover the two terms are equal and the local slope of log y against log u is exactly
(T1["a"]["slope_at_crossover"]). This is not a numerical accident: for a window symmetric in log u the
least-squares exponent of u^p + u^q is exactly (p+q)/2 = 2.5, for every
window width. The identity follows from writing
ln y = p ln u + ln(1 + u^(q−p)): the second term is antisymmetric about the crossover plus an even
part, and an even function is orthogonal to ln u over a symmetric window, so it contributes exactly (q−p)/2 to
the covariance.
| Fit window | Fitted n | R² | Worst residual | n error vs mean |
|---|---|---|---|---|
| ±0.2 decade | 2.5000 | 0.9900 | 15.6% | +0.0000000 |
| ±0.5 decade | 2.5000 | 0.9623 | 94.2% | +0.0000000 |
| ±1.0 decade | 2.5000 | 0.9370 | 398% | +0.0000000 |
| ±1.5 decade | 2.5000 | 0.9277 | 1124% | +0.0000000 |
| ±2.0 decade | 2.5000 | 0.9237 | 2854% | +0.0000000 |
Result (a). The fitted exponent is 2.5000 at every window width from ±0.2 to ±2 decades — the largest departure from the mean of 1 and 4 is 8.9e-16, which is rounding. Neither mechanism has that exponent. The fit quality is the only witness and it is a poor one: at ±0.2 decades R² is 0.9900 with a worst point 15.6% off, and at the crossover itself the fitted line predicts 35.0 where the true rate is 2.0 — a factor 17.5.
Result (b) — the case that looks perfect. When the two exponents are close, as they are for the two accepted solid-solution strengthening exponents 1/2 and 2/3, the crossover is nearly invisible. The fitted exponent is 0.5833, which is neither mechanism value, with R² = 0.999950 and a worst point 1.22% off. A published exponent of 0.583 would be defensible on every normal diagnostic and wrong for both mechanisms.
| Fit window | Fitted n | R² | Worst residual |
|---|---|---|---|
| ±0.5 decade | 0.5833 | 0.999987 | 0.31% |
| ±1.0 decade | 0.5833 | 0.999950 | 1.22% |
| ±2.0 decade | 0.5833 | 0.999807 | 4.87% |
Result (c) — the window is the variable. Holding the width fixed at ±0.5 decade and moving the centre, the exponent runs from 1.000 to 4.000. The same model, the same function, seven exponents. This is the only reliable diagnostic: refit on a shifted sub-range and ask whether the exponent moves.
| Window centre (decades from crossover) | Fitted n | R² | Worst residual |
|---|---|---|---|
| -1.5 | 1.0003 | 1.000000 | 0.05% |
| -1.0 | 1.0084 | 0.999957 | 1.70% |
| -0.5 | 1.2129 | 0.9855 | 38.9% |
| +0.0 | 2.5000 | 0.9623 | 94.2% |
| +0.5 | 3.7871 | 0.998497 | 38.9% |
| +1.0 | 3.9916 | 0.999997 | 1.70% |
| +1.5 | 3.9997 | 1.000000 | 0.05% |
Result (d) — a non-monotone crossover. A schematic two-branch strength model, each branch an exact ±1/2 power law meeting at d* = 15.0 nm with a peak of 154.9 MPa, and strength 52.2 MPa at 1.7 nm against 46.0 MPa at 170 nm. Fitted across the peak the exponent is -0.040 with R² = 0.0267 — "grain size does not matter" for a material whose strength triples inside the window. Fitted below the peak it is +0.497 with R² = 0.999918: an excellent fit reporting that strength rises with grain size, the opposite sign to Hall–Petch, because the fit has landed on the inverse branch. Fitted above the peak it is -0.496, which is the correct answer and indistinguishable in quality from the wrong one.
| Range (nm) | Fitted m | R² | Worst residual | Placement |
|---|---|---|---|---|
| 1.7 - 170 | -0.0402 | 0.0267 | 63.7% | spans the peak (1 decade each side) |
| 1.5 - 12 | +0.4969 | 0.999918 | 1.48% | below the peak |
| 5 - 50 | -0.0329 | 0.0204 | 24.9% | spans the peak, narrow |
| 15 - 1.5e+03 | -0.4965 | 0.999758 | 7.28% | above the peak (Hall-Petch branch) |
What this contradicts. I expected a single power law fitted across a crossover to be detectable by its residuals. It is — when the mechanisms are far apart, where R² collapses to 0.9370 and the worst point is 398% off. It is not when they are close: R² = 0.999950 and a 1.22% worst residual, which no referee would flag. Fit quality is therefore not the diagnostic; window sensitivity is.
Claim tested. A property predicted from the mean grain size is the mean property.
Model. Grain size lognormal with median 10.0 µm and logarithmic standard deviation s; property P ∝ dm. Then E[P] = dmedm exp(m²s²/2) while P(E[d]) = dmedm exp(m s²/2), so the ratio is exp((m²−m)s²/2) — exact, not simulated. The entries are confirmed by Monte Carlo at m = 2, s = 0.6: 1.433861 against 1.433329 on 2,000,000 draws.
| m in P ∝ d^m | s = 0.2 | s = 0.4 | s = 0.6 | s = 0.8 | P90/P10 at s = 0.6 |
|---|---|---|---|---|---|
| -1.0 | 1.0408 | 1.1735 | 1.4333 | 1.8965 | 4.65 |
| -0.5 | 1.0151 | 1.0618 | 1.1445 | 1.2712 | 2.16 |
| +0.5 | 0.9950 | 0.9802 | 0.9560 | 0.9231 | 2.16 |
| +1.0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 4.65 |
| +1.5 | 1.0151 | 1.0618 | 1.1445 | 1.2712 | 10.04 |
| +2.0 | 1.0408 | 1.1735 | 1.4333 | 1.8965 | 21.67 |
| +3.0 | 1.1275 | 1.6161 | 2.9447 | 6.8210 | 100.85 |
Result. The error vanishes at m = 1 and nowhere else, changes sign between m = 0 and m = 1 (0.923 at m = 0.5, s = 0.8), and grows quadratically with the distribution width. For Hall–Petch (m = −1/2) at s = 0.6 the mean-size prediction is low by 14.5%; for a coarsening or sintering rate (m = 3) at s = 0.6 it is low by 194% and at s = 0.8 by a factor 6.8. A mean-based model is not biased low or high in general — it is biased in a direction fixed by whether the property is convex or concave in d, which is why the error is so easy to miss.
The distribution also decides the scatter, not just the mean. For m = 3 and s = 0.6 the 90th percentile of the property is 101 times the 10th: a single mean value describes no grain in the sample.
Same mean, different material. A 50/50 mixture of 10 µm and 30 µm grains has the same arithmetic mean as a monodisperse 20 µm material. The properties do not agree, and the disagreement grows with the exponent:
| m | Bimodal 10/30 µm | Monodisperse 20 µm | Ratio |
|---|---|---|---|
| -0.5 | 0.249 | 0.224 | 1.115 |
| 1.0 | 20.000 | 20.000 | 1.000 |
| 2.0 | 500.000 | 400.000 | 1.250 |
| 3.0 | 14000.000 | 8000.000 | 1.750 |
How many grains. The standard error of the mean of ln d is s/√N, so a five per cent claim needs (s/0.05)² grains. For a property the relevant spread is m·s, so the requirement grows as m²:
| s | Grains for 5% on mean d | Grains for 5% on m = 2 | Grains for 5% on m = 3 |
|---|---|---|---|
| 0.2 | 16 | 64 | 145 |
| 0.4 | 64 | 256 | 577 |
| 0.6 | 144 | 576 | 1296 |
| 0.8 | 256 | 1024 | 2305 |
What this contradicts. I expected the mean-size error to be largest for the strongest size dependence and to run in one direction. It is largest for the strongest dependence, but its sign changes at m = 1, so a model calibrated on one property can be wrong in the opposite direction on another property of the same microstructure, with the same mean grain size.
Claim tested. How much of a defect's elastic energy lies outside a cutoff, and whether that fraction depends on the strength of the defect.
Model. A point dilatation centre in an infinite isotropic solid, displacement ur = A/r² with A = εa³ for a misfit strain ε and core radius a. The strain field is pure shear — its trace is zero — so the energy density is 6μA²/r⁶ and
E_out(R) = 4 pi * integral_R^inf (6 mu A^2 / r^6) r^2 dr = 8 pi mu A^2 / R^3
fraction outside R = (a/R)^3 -- the A cancels exactlywith μ = 50 GPa, a = 0.2 nm and T = 300 K (kT = 0.02585 eV). The radial integral was also done numerically on 2×10⁶ log-spaced points and agrees with the closed form to a relative error of 3.6e-11.
| Cutoff R/a | Fraction outside | E/kT at ε = 0.02 | E/kT at ε = 0.05 | E/kT at ε = 0.10 |
|---|---|---|---|---|
| 1.5 | 0.296296 | 0.2877 | 1.7979 | 7.1915 |
| 2.0 | 0.125000 | 0.1214 | 0.7585 | 3.0339 |
| 3.0 | 0.037037 | 0.0360 | 0.2247 | 0.8989 |
| 5.0 | 0.008000 | 0.0078 | 0.0485 | 0.1942 |
| 10.0 | 0.001000 | 0.0010 | 0.0061 | 0.0243 |
| 20.0 | 0.000125 | 0.0001 | 0.0008 | 0.0030 |
Result (a). The fraction outside the cutoff is identical for all three dipole strengths — the largest relative spread across ε = 0.02, 0.05 and 0.10 is 0.0e+00, i.e. zero to machine precision. Increasing the cutoff from 2a to 5a reduces the neglected fraction from 0.1250 to 0.0080; increasing the dipole strength by a factor of five changes it not at all. What does scale with the dipole is the absolute energy: at R = 1.5a the tail is 7.19 kT at ε = 0.10 against 0.288 kT at ε = 0.02, a factor of 25 for a factor of five in strain, as ε² requires.
Result (b) — the radius that matters grows slowly. Define RkT as the radius at which the remaining elastic energy equals one kT. It rises as ε2/3: a least-squares fit to the seven evaluations gives an exponent of 0.6666 against the exact 2/3. Over the whole range from ε = 0.005 to ε = 0.5 — a factor of a hundred in energy — RkT moves only from 0.39a to 8.47a. A cell that is large enough for the weakest dipole is nearly large enough for the strongest.
| Misfit strain ε | R where tail = 1 kT (in core radii) | Total elastic energy at the core (kT) |
|---|---|---|
| 0.005 | 0.393 | 0.061 |
| 0.010 | 0.624 | 0.243 |
| 0.020 | 0.990 | 0.971 |
| 0.050 | 1.824 | 6.068 |
| 0.100 | 2.895 | 24.271 |
| 0.200 | 4.596 | 97.086 |
| 0.500 | 8.466 | 606.786 |
Result (c) — the absolute number is set by the core cutoff, by the cube. Because E ∝ 1/rc³, choosing rc = 0.5a instead of 2a changes the computed energy by a factor of 64. This is the finite-cutoff problem in its sharpest form: the fraction outside a cutoff is pure geometry, while the energy itself is a statement about where the core was placed.
| Core cutoff rc / a | Computed energy, relative to rc = a |
|---|---|
| 0.5 | 8.000 |
| 0.7 | 2.915 |
| 1.0 | 1.000 |
| 1.5 | 0.296 |
| 2.0 | 0.125 |
Result (d) — the dislocation is worse, not better. A line defect has a strain field decaying as 1/r, so its energy is logarithmic and the fraction outside a cutoff depends on the outer cutoff as well. With a box of 10³ core radii, 0.67 of the line energy lies outside ten core radii; at 10⁴ it is 0.75. There is no converged line energy, only a converged energy difference between two configurations computed in the same box.
| Outer cutoff Rmax/r0 | Fraction outside 2 r0 | Fraction outside 10 r0 |
|---|---|---|
| 100 | 0.849 | 0.500 |
| 1000 | 0.900 | 0.667 |
| 10000 | 0.925 | 0.750 |
| 1000000 | 0.950 | 0.833 |
What this contradicts. I expected the elastic-energy fraction outside a cutoff to grow with defect strength — stronger dipoles, longer tails. It does not: the fraction is (a/R)³ exactly, for every strength, because the amplitude cancels. Only the absolute energy and the radius at which the tail falls below kT depend on the dipole, and that radius grows as the two-thirds power of the strain, which is slower than almost anyone would guess.
Provenance
Computation only. Code: incoprea_nov3/code/materials_defects.py.
Results: incoprea_nov3/results/materials_defects.json. Leads E1–E200 authored for this page; every
table value is printed by the script. Incoprea Research · 3 November 2026.