Hand-considered hypotheses, 601–700
Computation and information in science. Numerical method, simulation error,
the limits of analysis, and learning applied to physical measurement. Each entry was written on its own and
given its own logical form — the forms are printed so that structural repetition would be visible
rather than hidden. Numerical quantities offered in passing are order-of-magnitude estimates and are marked
as such; nothing here has been run, benchmarked, or cited, and no result is claimed. This is the seventh
hundred of a planned thousand.
Arithmetic, rounding, and the machine
Bound
H601. Accuracy in a long integration is bounded by the
bits the state sheds per step, not by the integrator's order: the error grows linearly in step count at a
rate proportional to machine epsilon times the state's magnitude. Doubling the word length should move that
bound by exactly a factor of two, so a run that improves more than that was never round-off-limited.
Comparing binary32, binary64 and binary128 trajectories at fixed truncation error separates the two cases.
Misattribution
H602. Divergence between two runs of the same deterministic code, same input, same version, is
usually blamed on chaos; often the seed is the compiler, which reassociates a sum and flips the last bit,
and chaos then amplifies an artefact of instruction scheduling. The separation time is computable from the
Lyapunov exponent and the position of the flipped bit. Compile at two optimisation levels and compare that
prediction with the observed split — agreement to within a factor of two would settle it (estimate). The
practical consequence is that a bitwise-identical rerun at one optimisation level is the only reproducibility
claim the build can actually support.
Unit trap
H603. Phase reported to twelve digits out of a
single-precision reduction is a unit trap: binary32 carries twenty-four mantissa bits, so near
2π the representable increment is about 1e-7 cycles (estimate), and every digit
past the seventh belongs to the formatting routine. The mantissa, not the interferometer, sets the floor.
Re-run the identical reduction in binary64; the leading seven digits should not move while the rest do.
Identity
H604. Cancellation in a difference estimator and
"loss of significance" in a variance are one event seen twice: the leading bits of two numbers agree, and
the subtraction hands back only their disagreement. That is why E[x²]−E[x]² returns a negative
variance instead of a merely noisy one when the mean is large. Recentre the data first and the two formulæ
agree to the last bit — a check that costs nothing and is almost never run.
Reversal
H605. More precision can degrade a fit when rounding had been doing quiet regularisation; solving
normal equations for a nearly singular calibration matrix is the archetype. In binary64 the unresolved pivots
are noise that happens to stabilise the solve, and in binary128 the same pivots are resolved, so the solution
walks into the null space. Hold the algorithm fixed and vary only the word length: the residual should fall
while the parameter error climbs. Precision and accuracy part company exactly where the smallest singular
values live, and no amount of arithmetic repairs that.
Hidden term
H606. Every measured order of convergence carries the error of the reference solution, so an
observed order above the scheme's theoretical one is measuring the reference rather than the scheme. The
contamination is invisible on a log–log plot because the reference's error enters the ratio of successive
differences with the same exponent as the truncation term. Recompute the reference at two tolerances; if the
apparent order shifts, the reference was the term. The check costs two extra solves and almost never appears
in a published convergence table.
Threshold
H607. An increment smaller than the representable
step at the accumulator's current magnitude is lost outright, and in a closed loop that loss wears the
costume of stiction — the actuator simply does not move. A deadband added to cure the "stiction" then hides
genuine small signals. Scale the accumulator so the increment is representable and the stiction should
vanish without anything being touched on the hardware side.
Count
H608. A binary32 pipeline fed by a sixteen-bit converter has at most 2^16 distinguishable
input states however many samples are averaged: averaging recovers precision only when the noise exceeds one
step and is independent between samples, and below that the average converges to a lattice rather than to
the truth. Hold the input constant and the averaged value will settle a fraction of a step away and stay
there.
Scale argument
H609. For a fixed word length there is an
optimal step size, not a smallest one: round-off contributes roughly eps/h while truncation
falls as h^p, so the sum bottoms out near eps^(1/(p+1)). Refining past that point
buys nothing and costs accuracy, and the turn is predictable before the run is made. An error-versus-step
curve will show a minimum that convergence studies are usually plotted to avoid.
Artefact mimicking effect
H610. A dither tone injected to
break up quantisation harmonics is itself a spectral line, and if it lands near a mechanical resonance it
will be integrated as a mode. Its giveaway is linewidth: the artefact carries the dither source's width
rather than the resonator's. Move the dither frequency by ten per cent — a real mode stays where it is and
the artefact follows.
Selection effect
H611. Runs published out of a stochastic code are disproportionately those whose seeds produced a
clean result, so the seed-to-seed variance in the literature is smaller than the code's own ensemble spread
— an editorial mechanism that corrupts the same error bar as a numerical one. Publishing every seed for one
small case would expose the gap in an afternoon.
Inversion of standard practice
H612. When a simulated
functional disagrees with experiment the reflex is to refine the mesh; when the discrepancy sits in an
integral quantity the first move should be to refine the quadrature, because functional error converges at
the quadrature's rate while pointwise field error converges at the scheme's. Refining the mesh first can
slow the functional's convergence by changing the cell count the quadrature was built on. Compute one
functional at fixed mesh with two quadrature orders and see which lever actually moves it.
Discretisation and the mesh
Misattribution
H613. Apparent viscosity in an under-resolved
flow is frequently the scheme's numerical dissipation, which has the same units, the same sign and a
magnitude set by the cell Reynolds number. It is easily mistaken for a turbulence model because it also
transports momentum down-gradient. Halve the cell size at fixed physical viscosity: real viscosity is
invariant while the numerical part should fall with the square of the step.
Reversal
H614. Refinement is not monotone when a boundary layer
is thinner than the first cell, because splitting cells can move the first node deeper into the layer's
interior and increase the error. The mechanism is sampling: each mesh reads the profile at a different
fraction of its thickness. Two refinement levels that bracket the layer thickness will show the error
rising before it falls.
Count
H615. The stiffness a finite-element model reports is
bounded by the number of independent constraints, not by the number of elements: beyond roughly one
constraint per active mode the assembled system is reporting its own Dirichlet data (estimate).
Over-refined meshes therefore appear to stiffen even when the physics is untouched. Fix the constraints,
vary only the element count, and the reported stiffness should flatten onto a plateau.
Threshold
H616. A stable advection scheme has a cell Reynolds number above which its numerical diffusion exceeds
the physical diffusivity, and that threshold belongs to the scheme rather than to the fluid: below it
refining sharpens the solution, above it refining merely redistributes an artefact. A Schmidt-number sweep
at fixed mesh locates the threshold with no experimental input at all.
Artefact mimicking effect
H617. A structured mesh aligned with one axis imposes that axis on an isotropic problem, so elastic
constants recovered by inversion come back anisotropic by a few per cent (estimate), with the fast direction
along the grid. Refinement does not remove it, because it is a symmetry of the discretisation and not a
resolution error. Rotate the mesh forty-five degrees and the apparent anisotropy should rotate with it. The
cure is grid-symmetry breaking rather than refinement, and it costs one diagonal offset.
Scale argument
H618. Numerical dispersion is incurred per
element and is roughly fixed, so once the cell falls below the wavelength the total phase error grows with
the number of elements traversed instead of shrinking. Refining past that point makes a wave packet arrive
at the wrong time while looking smoother. Track one Fourier mode's phase across three meshes; the error
should turn upward at the wavelength-to-cell ratio where the per-element error saturates.
Identity
H619. Numerical and physical dispersion produce the same phase signature at a single frequency, which
is why one dispersion measurement cannot tell a fluid from a grid. They separate only in frequency
dependence: the physical term goes as the square of the wavenumber, the numerical one as a trigonometric
function of it. Three frequencies, not one, convert the ambiguity into a fit. A measurement that cannot
separate them is a measurement of the phase, not of the medium.
Selection effect
H620. Adaptive refinement selects cells by an
indicator computed from the current solution, so the refined mesh reliably "confirms" the feature the
indicator was built to find — a closed loop with no external check anywhere in it. The bias is largest
wherever the indicator is a gradient, since gradients peak on the artefact as readily as on the physics.
Refining instead on a solution produced by a different scheme breaks the loop.
Inversion of standard practice
H621. Optimising a model for
pointwise field accuracy can degrade the integral quantity one actually needs, because pointwise errors of
opposite sign cancel in the integral and refinement reshuffles the cancellation. The usual practice —
refine until the field looks converged — therefore targets the wrong norm. Compute the functional and the
field L2 error on the same three meshes; they need not converge together.
Hidden term
H622. Observed discretisation error carries a term
from the boundary condition's own truncation, which for a one-sided implementation is one order below the
interior scheme and does not shrink when the interior is refined. Reports of "sub-second-order" convergence
in otherwise clean codes are usually this term. Swap in an exact boundary closure at fixed interior mesh
and the slope should jump by an order.
Derived quantity
H623. The effective conductivity recovered from a heterogeneous medium is a derived quantity whose
value depends on the averaging volume: below the representative volume it is not a material property at all,
and the number reported is a property of the window. Two groups using different windows will disagree while
both are right about their own windows. Vary the window at fixed physics and look for the plateau — the
plateau's existence, not its height, is the finding. An effective value quoted without its window is an
incomplete number.
Unit trap
H624. Nondimensionalising by a reference scale silently absorbs whichever coefficient shares that
scale's units, so a term that is physically active can vanish from the dimensionless equations and reappear
as a boundary condition — the trap springing when two groups choose different references and report
different physics from the same system. Recover the dimensional form from each nondimensionalisation and
compare term by term.
Bound
H625. A discrete model cannot carry energy at
wavelengths shorter than two cells, so any spectral power reported below that cutoff is the scheme's
response to the forcing and not the solution's. The bound is hard, independent of order and of mesh
quality. Refine only along the direction of the suspected shortwave and see whether the reported power
moves with the physics or stays pinned at the cutoff.
Artefact mimicking effect
H626. A staircase representation of
a smooth boundary pins the contact line at cell corners, and the resulting apparent contact angle becomes a
function of grid orientation rather than of surface chemistry. The artefact is small in angle but large in
hysteresis, and hysteresis is the quantity usually reported. Rotate the interface relative to the grid: the
hysteresis should change while the angle does not.
Solvers, tolerances, and stopping rules
Bound
H627. The absolute error a converged solve can promise is the tolerance times the condition number,
so a solver "converged to 1e-10" on a matrix of condition 1e8 has said nothing
below 1e-2. Claims of sub-tolerance physical changes are therefore claims about the
preconditioner. Compute the condition number once and multiply; the product is the real error bar. It
routinely lands two or three orders of magnitude above the tolerance the solver was set to (estimate).
Identity
H628. Failure to converge and ill-posedness are one
condition seen from two sides: the residual cannot be driven below the model's null-space component,
whether that component is called a stalled iteration or a regularisation defect. This is why adding
iterations to a bad inverse problem produces structure instead of accuracy. Project the residual onto the
smallest singular vectors and the same subspace will explain both diagnoses.
Reversal
H629. Loosening a stopping tolerance can improve a
recovered physical parameter when the forward model is misspecified, because early stopping is Tikhonov
regularisation with the iteration count as the parameter. The optimum sits at a residual comparable to the
model error, not at zero. Sweep the tolerance over four decades with the model error fixed and the
parameter error should show a minimum rather than a monotone fall.
Threshold
H630. There is a tolerance beneath which tightening changes the answer only by amplifying
discretisation error, and it sits at the discretisation error's own size rather than orders below it —
which is why most codes default far past the point where iterations can still buy accuracy. Solve one fixed
mesh at t and t/100; the difference should be at the discretisation level, not
beneath it.
Hidden term
H631. The "convergence factor" printed by many
solvers is a residual scaled by the preconditioner, so it measures the preconditioner and only partly the
solution. Changing the preconditioner can make a stalled solve look healthy without improving the answer at
all. Print the true residual from the original operator once every hundred iterations and compare the two
histories.
Misattribution
H632. Slowness blamed on ill-conditioning is
often the preconditioner's own cost profile: an incomplete factorisation whose fill is data-dependent makes
the iteration count follow the sparsity pattern rather than the spectrum. Two matrices with identical
condition numbers can then differ tenfold in run time (estimate). Time the setup and the iteration
separately; the surprise usually sits in the first.
Count
H633. A Krylov solver needs on the order of the square root of the condition number of iterations,
and any count far beyond that amplifies round-off rather than reducing error, since the excess iterations
are a random walk in the residual rather than progress. Record the true residual norm past the predicted
count; it should plateau, and the plateau's height is the accuracy actually achievable.
Unit trap
H634. An absolute tolerance of 1e-6 means different things in SI and in cgs, so a
default quoted without units changes meaning by orders of magnitude under a unit change — the reason solvers
that accept only relative tolerances are safer. Re-run a converged case in cgs and see whether the reported
error bar moves while the physics does not.
Artefact mimicking effect
H635. Early stopping in an iterative
reconstruction produces a smooth image, and smoothness gets read as evidence for a better-conditioned
physical model rather than as an artefact of where the iteration was halted. The giveaway is that the
"improvement" tracks the iteration budget and not the data. Give two different operators the same budget on
the same data; if the smoothness ranking follows the budget, it is the stopping rule.
Scale argument
H636. The optimal stopping index in a
semi-convergent iteration scales inversely with the square of the noise level, a consequence of stopping
when the residual matches the noise rather than when it stops falling. Halving the measurement noise should
therefore move the optimum by a factor of four, not two. A noise-injection sweep on a single dataset finds
the optimum without any new measurement.
Inversion of standard practice
H637. Driving every solve to
machine precision is the wrong default for noisy data: past the noise floor the extra iterations fit the
noise, and the solution's error grows while its residual falls. The correct default is a stopping rule tied
to the data's uncertainty. Run one example to full precision and to the noise floor, compare both against a
known truth, and the fully converged answer is usually the farther one.
Closure error
H638. In a partitioned multiphysics solve the
flux mismatch at the interface is not noise but a conserved quantity that accumulates as a bias, and its
sign identifies which subsystem was solved last. Codes that report this mismatch as a generic "residual"
hide the structure that would localise the defect. Reverse the solve order: the bias should change sign
while its magnitude stays fixed.
Surrogates and learned models
Misattribution
H639. A surrogate trained on coarse simulations learns the solver's error as faithfully as the
physics, so it reproduces a discretisation bias and then appears to validate against more coarse data. The
bias is invisible in the training loss because it is part of the target. Train one architecture on two
meshes, evaluate both against experiment, and the distance between the two surrogates is the learned error.
Nothing in the training procedure distinguishes a real feature from a stable bias.
Hidden term
H640. A Gaussian-process surrogate's uncertainty is
a statement about the kernel rather than about the physics, and it shrinks with proximity to training points
regardless of whether those points were right. Extrapolated confidence intervals are therefore artefacts of
length-scale choice. Hold the training set fixed and vary only the kernel's length scale: the predicted
mean should barely move while the interval changes by an order of magnitude.
Misattribution
H641. When sparse identification or symbolic
regression "discovers" a governing term it often recovers the preprocessing, because a smoothing window or a
normalisation is a linear operator that looks exactly like a physical term in the candidate library. The
discovered coefficient is then a property of the filter. Re-run the identification on unsmoothed data with
the noise handled explicitly; the term should either vanish or change identity.
Bound
H642. A model trained on one instrument's data cannot
beat that instrument's calibration accuracy on absolute quantities, but it can beat it on ratios, because
calibration error is common-mode and cancels. The bound predicts where transfer learning will and will not
work, and it needs no new measurements to check. Test one ratio quantity and one single-channel quantity;
the ratio should transfer and the absolute should not.
Selection effect
H643. Choosing hyperparameters on a validation split drawn from the same instrument biases the
reported accuracy upward by roughly the number of effective trials divided by the validation size, so with
enough trials any model beats the baseline on noise alone. Nested splitting, or a validation set from a
second instrument, separates the two explanations.
Artefact mimicking effect
H644. Much of the reported gain from
physics-informed training is the soft constraint acting as a smoother, and it disappears when the baseline
is smoothed identically. The constraint's effect on the loss is dominated by its suppression of
high-frequency components, not by its encoding of the equation. Compare against a baseline with matched
spectral content and the margin should collapse.
Reversal
H645. Adding training data can degrade a model when the data are heteroscedastic and the loss is
unweighted: the noisiest regime dominates the gradient and the fit regresses toward noise. More data from
the noisy regime is worse than less data from the quiet one. Weight the loss by inverse variance and the
degradation should reverse — a test that separates data quantity from data quality. Volume is not a monotone
good, and the inflection is measurable without new experiments.
Identity
H646. "Distribution shift" and "instrument drift" are
the same failure whenever the covariate is an instrument state: both are a change in the map the model
learned, and both are invisible if the state was never recorded. Naming them separately has produced two
literatures that do not check each other. Log the instrument state as a feature and the shift becomes a
measurable coefficient instead of an unexplained error.
Count
H647. The number of independent physical constraints learnable from a dataset is bounded by the
number of independent calibration states, not by the number of samples — a million spectra at one gain
setting teach at most one gain's worth of physics, and the bound is indifferent to model size. Hold the
sample count fixed, vary the calibration states, and generalisation should track the latter.
Hidden term
H648. An ensemble's reported uncertainty contains the ensembling procedure's own variance, which for
small ensembles is comparable to the model's, so the interval is partly a statement about how many members
were trained. Five members and fifty differ in spread without differing in error — fix the data and the
architecture, vary only the member count, and plot spread against error.
Unit trap
H649. Normalising every input to unit variance
destroys the dimensional relationships among them, so a network cannot represent a dimensionally consistent
law even in principle — it can fit one unit system. Change the units of a single input and the learned
function changes shape. A dimensionally aware normalisation, one scale per dimension shared by quantities of
that dimension, should make the fit invariant to the change.
Derived quantity
H650. Feature importance from a tree ensemble
over correlated physical covariates is a derived quantity that depends on the correlation structure rather
than on the physics: the split is arbitrary among twins and the ranking is stable only by accident. Two
seeds will disagree about which of two correlated variables matters. Permute one twin and its importance
should migrate to the other, which tells you the ranking was never about the physics.
Threshold
H651. Transfer from simulation to experiment is
gated by a ratio and not by a size: the closure model's error divided by the target's variance must fall
below one, and model capacity plays no part in it. Below that ratio better architectures buy nothing, which
is why benchmark gains fail to travel. Compute the closure error directly, by differencing two closures on
one flow, and compare it with the measured variance.
Selection effect
H652. Selecting a model by its loss on a finite test set favours models with higher variance, because
the minimum of many noisy estimates is biased downward, and the bias grows with the number of candidates
while shrinking with test-set size — so the reported winner is partly selected for luck. Report the loss on
a fresh set drawn after selection and the gap is the selection bias, measured rather than assumed.
Reversal
H653. A learned model constrained by a physical law that is only approximately true ends up worse
than the same model unconstrained, because the constraint forces a false equation that the data cannot
overrule. The practice of stacking constraints until the output "looks physical" therefore has an interior
optimum. Add them one at a time and plot error against count; the curve should turn. A constraint is
information only if it is true, which is the part the workflow tends to assume.
Bound
H654. No learned model resolves structure below the
label noise of its training data, and for physical measurements the label noise is the instrument's, which
is frequently larger than the feature that matters. This is why larger models saturate on small
experimental sets. Add synthetic labels with known noise and find the noise level at which the model's
resolution stops improving.
Artefact mimicking effect
H655. Data augmentation introduces
symmetries the physics may not possess — rotation invariance for a crystal with a preferred axis, say — and
a model will exploit the false symmetry to fit noise more easily. The exploitation shows up as an
anomalously low training loss on augmented copies and a matching failure on the true orientation. Train
with and without the augmentation and evaluate only on the physical orientation.
Data analysis and its limits
Bound
H656. The practical bandwidth of a sampled measurement
is set by the anti-alias filter's transition band, not by the sample rate: a filter that only reaches its
stopband at four times the corner leaves a usable band about a third of Nyquist wide (rough estimate).
Quoting Nyquist as the bandwidth therefore overstates resolution by a factor of two or three. Measure the
system's response to a swept tone instead of trusting the rate printed in the header.
Aliasing invisibility
H657. Aliased power is indistinguishable from genuine power at the alias frequency inside a single
record, so no amount of averaging at one rate can separate them; separation requires a second rate at an
incommensurate ratio, where the artefact moves and the signal does not. Most instruments offer such a rate,
and it is rarely used for this purpose.
Count
H658. A record of N real samples carries at most N/2
independent spectral amplitudes, so a peak found by fitting more parameters than that is fitting the
window's shape. The bound explains why "significant" narrow features proliferate as fitting freedom grows.
Reduce the model to N/2 parameters and the feature count should collapse to the number of resolvable
lines.
Threshold
H659. Believing the tallest peak in a periodogram requires knowing how many independent frequencies
were searched, because past a few thousand the maximum is more likely a sidelobe than a period, and the
threshold for belief is set by that number rather than by the peak's height. The correction is a look-elsewhere factor of order the search size. Search a
phase-scrambled copy of the same data and count how often a peak that tall appears. It is the cheapest
correction in periodogram analysis and the most often skipped.
Misattribution
H660. Low-frequency excess reported as 1/f
noise is often the instrument's gain drift filtered by its own feedback loop, whose transfer function turns
a slow exponential into a power law across two decades. The exponent then belongs to the loop, not to the
noise process. Open the loop, or record the loop's error signal, and the power law should change slope or
disappear.
Identity
H661. Baseline drift and low-frequency noise are the same process observed with and without a
high-pass filter, so detrending changes the measured exponent and not the process — a claim that a signal
"becomes white after detrending" is a statement about the filter's corner. Apply two detrending orders to
one record and compare the spectra above the corner; they should agree there.
Reversal
H662. Smoothing before differentiating increases a
derivative's noise instead of reducing it, because smoothing correlates adjacent samples and the
differencing then amplifies the correlated part. The correct order — difference, then smooth — is the
reverse of the intuitive one. Compute both on the same record and compare each against a known analytic
derivative added to the data.
Hidden term
H663. Every correlation coefficient computed on an
autocorrelated series has an effective sample size of roughly n/τ, with τ the integral
timescale, so the significance threshold is wrong by a factor of about sqrt(τ). This is why
geophysical correlations "significant at p<0.01" fail to replicate. Estimate τ from the autocorrelation
and rescale the p-value; most of the surprise will go.
Scale argument
H664. The variance of a measured mean falls as
1/n only while the record is longer than the correlation time; for shorter records it falls as
1/n² with an offset set by the correlation, so extrapolating the 1/n law badly
overstates precision. The crossover is the record length itself and is measurable. Split one long record
into segments and plot the variance of the segment means against segment length.
Artefact mimicking effect
H665. A trend fitted to a short
record has a standard error scaling as the record length to the minus three halves, so doubling the record
cuts the trend's error by nearly a factor of three, and a significant trend in a short record is usually the
endpoint's noise. A linear fit weights the ends most, which is exactly where a record is least constrained.
Fit the same slope to the first and second halves and see whether the sign survives.
Unit trap
H666. Averaging in decibels computes the geometric
mean, which for lognormal data sits a fixed distance below the arithmetic mean — a little under half a
decibel for a spread of four decibels (estimate) — and that offset is routinely mistaken for a calibration
error. The average must be formed in linear power before conversion. Average one dataset both ways and
compare against a calibrated reference.
Selection effect
H667. Published values are a truncated sample
of the noise distribution, cut at the significance threshold, so a meta-analysis recovers a variance smaller
than the true one and a mean displaced away from the threshold. The truncation point is known, which makes
the bias correctable rather than merely acknowledged. Recover the untruncated variance by maximum likelihood
on the published set and compare it with a pre-registered batch.
Derived quantity
H668. Signal-to-noise ratio defined as mean
over standard deviation is a derived quantity that depends on the assumed signal shape: a sinusoid and a
matched filter differ by a factor of about root two on the same data (estimate). Two papers can therefore
report different ratios for one measurement without either being wrong. Recompute both definitions on a
single record to see the factor appear.
Threshold
H669. Below a record length of roughly one over the linewidth, a spectral peak's measured width is
the window's, so the damping is not merely imprecise but unmeasurable, and a Lorentzian fit returns the
window shape with a plausible-looking error bar. Fit the same line in records of increasing length and watch
the width move until it stops.
Bound
H670. An inverse problem's resolution is bounded by the
singular value spectrum of its forward operator, not by the reconstruction algorithm, so no regulariser can
beat the bound — it can only trade bias against variance inside it. This is why two algorithms with very
different reputations converge to the same blurred image. Compare the operators' singular spectra directly;
the limit is where the spectrum meets the noise level.
Inversion of standard practice
H671. Imposing a constraint on an
inverse problem can increase error when the constraint is wrong, because the solution is now forced to
satisfy a false statement and the data can no longer correct it. The habit of adding constraints until the
answer looks physical therefore has an interior optimum. Add them cumulatively, plot misfit against
constraint weight, and the error should turn upward before the misfit saturates.
Count
H672. The number of parameters identifiable from a single experiment is bounded by the number of
independent instrument channels, not by the number of data points, because one channel's samples are
repeated measurements of one projection — fits that claim more are trading parameters against each other.
Fix the data, add one genuinely independent channel, and the identifiable count should rise by one rather
than by the sample count.
Artefact mimicking effect
H673. The ringing that follows a
sharp filter's step response is often reported as a physical oscillation of the system, because it occurs at
the filter's corner frequency and decays plausibly. Its giveaway is invariance: the frequency does not move
when the physical system is changed and moves only with the filter. Change the filter order at fixed corner
and the "oscillation" should change shape.
Provenance and the record
Provenance gap
H674. Two pipelines agreeing to every printed digit is not evidence that either is right, because
they can share an input convention, a rounding step or a copied constant while sharing no code at all.
Agreement tests the last common ancestor, not the physics. Inject a deliberate one-part-in-a-thousand
perturbation upstream and require both pipelines to track it; agreement that survives the perturbation is the
informative kind. Pipelines are usually compared by output, which is precisely the comparison that cannot
detect a shared ancestor.
Hidden term
H675. Every number carries the provenance of its
last formatting step: rounding at an intermediate stage is a lossy channel whose capacity is the printed
digit count, and downstream users treat those digits as weights. A quantity printed to four digits because
the table looked tidy will be averaged with four-digit confidence. Reformatted at the calibration's true
precision, the same data give a different pooled answer.
Identity
H676. Reproducibility and repeatability differ only in
which variance component is included, yet they are reported as separate virtues, so a result called
reproducible but not repeatable has simply confounded the two. The distinction is a bookkeeping choice
rather than a property of nature. Run one analysis on two copies of a dataset with different seeds and on
two datasets from one instrument; the two spreads identify the components.
Count
H677. The number of independent checks a pipeline can
perform is bounded by the number of independent data streams, so N checks on one stream carry N−1
redundancies and the pipeline's apparent robustness is largely self-agreement. Audit trails can look
thorough while a common-mode error passes through every one of them. Add one genuinely independent stream —
another instrument, another physical quantity — and count how many checks change their verdict.
Threshold
H678. Past some data volume the
provenance record grows faster than the data itself, because per-sample metadata is fixed while per-sample
information falls; beyond that point the archive's binding constraint is curation rather than storage.
The crossover is computable from the schema's size and the sample's entropy. Plot provenance bytes per
measurement byte against record length for one instrument and the knee should be visible.
Provenance gap
H679. A dataset's calibration is not recoverable
from the data alone: gain, offset and reference state are hidden variables that a reanalysis inherits
without knowing. Any reanalysis of an archived dataset therefore reproduces the original's systematic error
under a new name. Re-calibrate from the archived instrument state and the two analyses should differ
systematically rather than randomly.
Misattribution
H680. Disagreement between two laboratories is more often a unit or precision convention than a
physical discrepancy, and the convention hides because both report SI; the tell is a fixed ratio rather than
a scatter. Exchange raw files instead of reduced values and the ratio resolves into a conversion factor.
Inversion of standard practice
H681. Storing measurements at
higher precision can lose information when the accompanying metadata is truncated, because the extra digits
invite re-processing that the metadata no longer supports. Maximising numeric precision therefore trades
recoverable context for unrecoverable digits. Compare two archives of one experiment: one at full precision
with a sparse header, one rounded with a complete header, and ask which answers more questions.
Bound
H682. The information rate through a measurement chain
is bounded by its slowest stage's bandwidth, and a fast sensor behind a slow amplifier cannot resolve
transients no matter what is done to the data afterwards. The lost information was never sampled, so no
filter recovers it. Step a known fast edge through each stage and find where it broadens.
Selection effect
H683. Archives receive the results that were
significant, so the archived distribution is truncated and any variance computed from it is biased low —
a property of the archive rather than of the field's measurements. The bias is computable from the
truncation point whenever near-misses were also deposited. Compare an archived spread with a
pre-registered batch and the truncation appears as a deficit just below threshold.
Order, noise, and other cross-cuttings
Non-commutation
H684. Calibrating then averaging is not
averaging then calibrating when the instrument response is curved: the difference is second order in the
curvature and is exactly the size of the hysteresis usually attributed to the sensor. The order matters
because the mean of a nonlinear function is not the function of the mean. Apply both orders to one dataset
and the discrepancy should match a Jensen-gap prediction computed from the calibration curve.
Non-commutation
H685. Discretise-then-optimise and
optimise-then-discretise give gradients differing by the discretisation error, and that difference is
systematic along the mesh's preferred direction, so an optimiser run on the discrete gradient drifts where
the continuous problem has no gradient. The drift is small per iteration and accumulates. Run both on a
problem with an analytically known optimum and compare the converged designs.
Order of limits
H686. The limit of infinite data and the limit of infinite capacity do not commute, and the order a
study takes them in decides whether it observes overfitting or its absence. Growing capacity at fixed data
gives one curve; growing data at fixed capacity gives another, and neither is the physical limit. Vary both
on a grid and ask whether the error surface is separable — if it is not, neither one-dimensional story is
admissible. Most capacity-scaling claims are one slice of that grid reported as though it were the
surface.
Round-off amplification
H687. Subtracting two large, nearly
equal means amplifies each one's quantisation error by the ratio of the magnitude to their difference, so
two sixteen-bit readings differing by one part in a thousand carry an error of order one per cent of that
difference (estimate). The arithmetic is exact; the inputs were not. Form the difference from the raw
integers rather than the reduced floats and compare.
Conditioning
H688. The condition number of a calibration matrix is a physical quantity equal to the ratio of the
instrument's dynamic range to its resolution, so an ill-conditioned calibration is a hardware property and
not a numerical accident — it predicts which parameter combinations the instrument cannot separate before
any data are taken. Measure it beforehand and it should match the parameter covariance observed
afterwards.
Misattribution
H689. Apparent chaos in a deterministic
simulation can be the stepper sitting at its stability boundary, where the amplification factor is near one
in magnitude and the phase error per step is large; the trajectory then looks chaotic while the underlying
equation is not. The signature is dependence on step size. Halve the step and the "Lyapunov exponent" should
change, which no true exponent does.
Threshold
H690. Linearisation of a forward model is valid only when the model's curvature across the data's
uncertainty is small — a condition on the product of the second derivative and the noise variance, not on
the nonlinearity alone — so weakly nonlinear models with large noise violate it more often than strongly
nonlinear ones with small noise. Compare linearised and full forward solves across the error bars rather than
at the central value.
Identity
H691. Quantisation noise and additive white noise are indistinguishable below one step, so a
reported noise floor is the converter's until proven otherwise; they separate only through dependence on the
signal, since quantisation noise vanishes for a signal that lands on the lattice. Feed an exactly
representable constant and see whether the floor collapses.
Reversal
H692. Oversampling can increase effective noise when
the converter's noise is correlated between samples, because averaging then reinforces the correlation
instead of suppressing it; the familiar root-N improvement requires independence it may not have. Correlated
noise is common in converters with on-chip references. Compare the Allan deviation of an oversampled and a
decimated acquisition from the same device.
Reversal
H693. Adding a small amount of noise to a threshold
detector or a coarse quantiser can improve the recovered signal's fidelity, because the noise decorrelates
the error from the signal. The improvement is a window: too little does nothing, too much swamps. Sweep the
added noise and the error should fall before it rises, with the minimum at a level set by the step size.
Count
H694. A digitised record's information content is
bounded by the converter's resolution times the number of independent samples, and independence is set by
the analogue bandwidth rather than the sample rate, so a twelve-bit converter at a megasample per second
behind a one-kilohertz front end carries of order 2^12 × 2×10^3 distinguishable states per
second (estimate), not 2^12 × 10^6. Extra samples are interpolation. Decimate by a hundred and
the recoverable information should be unchanged until the decimation reaches the bandwidth.
Hidden variable
H695. The random seed is a hidden variable in pipelines that call themselves deterministic, and
because its effect is small it is rarely recorded, so two "identical" analyses can differ by more than the
effect being reported. Its influence is largest wherever the estimator is unstable, which is exactly where
the result is interesting. Vary the seed a hundred times and carry the spread as a line in the error
budget. Determinism that rests on an unrecorded seed is not determinism but a lottery with a quiet
jackpot.
Artefact mimicking effect
H696. An adaptive step-size
controller acts as a filter on the state it controls, so a simulation with tight error control can smooth a
physical oscillation whose amplitude is comparable to the local error target. The smoothing resembles
physical damping and disappears when the controller is replaced by a fixed step. Run both and compare the
oscillation's decay rate.
Bound
H697. The number of correct digits surviving an inverse
problem is set by the forward operator's condition number, so precision quoted after inversion is illusory
past that count: a condition number of 1e6 destroys six digits regardless of the arithmetic's
word length. The bound applies to the recovered parameter rather than to the residual. Perturb the data at
the tenth digit and measure how far the parameter moves.
Derived quantity
H698. An effective resolution quoted for a reconstructed image depends on the assumed point-spread
function, so two groups working from different assumptions report different resolutions without any
measurement differing — resolution here is a property of operator-plus-prior, not of the object.
Reconstruct one dataset under two PSF assumptions and compare the reported widths.
Threshold
H699. The apparent emergence of a capability with model size can be a threshold in the loss's
curvature rather than in the model's competence: a metric discontinuous in a continuous underlying error
produces a sharp step exactly where the error crosses the metric's resolution. The step's position then
moves when the metric changes and not when the model does. Plot a continuous proper score beside the
discrete metric and see whether the step survives. A capability that exists only for one metric is a
property of the metric, and the burden of proof sits with the metric.
Provenance gap
H700. A pipeline's default parameters are an
unrecorded prior: the smoothing width, the outlier rule and the convergence threshold all encode choices
that appear neither in the paper nor in the documentation. A reanalysis therefore reproduces the defaults
rather than the physics. Change one default by a factor of two, record how far the headline number moves,
and that distance is the weight of the prior.
Status
100 of 1,000 in this series; H601–H700 here. The subjects were chosen to sit on the
seam between the computation and the measurement — where a solver's tolerance becomes an experimental error
bar, where a learned model inherits a discretisation, where a provenance gap becomes a physical claim.
None has been tested. The forms are labelled so that a reader can check that the entries
differ structurally and not only in vocabulary.
If one of these is worth an afternoon, it is H627: that a converged residual and a
condition number multiply into the only honest error bar, and that most sub-tolerance claims in the
literature are claims about a preconditioner. It needs one matrix, one condition-number estimate, and no
apparatus at all.
Incoprea Research · hand-considered hypotheses 601–700 ·
computation and information · 3 November 2026 · ~/Desktop/Nov3/