The Watson-Crick base-pair edge is the nucleobase contact surface used in ordinary canonical base pairing. In RNA, the familiar adenine–uracil and guanine–cytosine contacts use this edge on both partners with cis-oriented glycosidic bonds in the geometric classification.^[ref]

For adenine, this edge includes its N1 ring nitrogen and N6 amino group. These provide a hydrogen bond acceptor and a hydrogen bond donor, respectively, in the ordinary neutral chemical form. The N6 group also borders the Hoogsteen base-pair edge; the classification divides contact regions, not disjoint lists in which every atom must belong to only one edge.^[ref]

A simple canonical A–U example has the uracil N3 hydrogen directed toward adenine N1 and an adenine N6 hydrogen directed toward uracil O4. These are the two conventional hydrogen bonds. In the corresponding G–C geometry, three conventional hydrogen bonds can form. The number of bonds is a property of the chemical pairing arrangement, not the definition of the edge.

The term also does not require the second base to use its own Watson–Crick edge. A Watson–Crick–Hoogsteen contact uses the named edge on one partner and the Hoogsteen base-pair edge on the other. Even a Watson–Crick–Watson–Crick edge combination can have a trans relationship of glycosidic bonds, placing it in a different family from the ordinary cis contact.^[ref]

An original comparison separates the information being recorded. “A contacts U” supplies two identities. “Both use Watson–Crick edges” adds two contact surfaces. “The glycosidic bonds are cis” adds their relative orientation. Only the full description identifies the canonical family; sequence identity alone cannot determine the geometry. Experimental structural coordinates or restraints are needed to decide which arrangement occurs in a particular molecule.

The edge framework consequently covers both canonical and noncanonical base pairs without equating every use of a familiar contact surface with the familiar double-helix geometry. It also makes RNA base-pair isostericity a geometric question rather than a simple count of hydrogen bonds.

The Hoogsteen base-pair edge is a contact surface of a nucleobase distinct from its Watson-Crick base-pair edge and sugar base-pair edge. For a purine, it is the side containing N7 and the exocyclic group at position 6; for example, adenine presents N7 and its N6 amino group on this side. The structural classification extends the corresponding category to the CH edge of pyrimidines.^[ref]

The word describes one partner's contacting edge, not a complete pair or a mandatory rotation of the base. A complete RNA base-pair geometry identifies the second partner's edge and the cis/trans orientation of the glycosidic bonds. The primary matrices include Hoogsteen–Watson–Crick, Hoogsteen–Hoogsteen and Hoogsteen–sugar contacts.^[ref]

An original counting example makes this useful. Fix one adenine as the labeled first partner and stipulate that it uses its Hoogsteen edge. The second partner still has three edge categories available; each can appear with a cis or trans relationship. There are therefore $3\times 2=6$ labeled edge/orientation descriptions before specifying the second base's identity. This counts descriptions, not six experimentally established partners for that particular adenine.

Edge identity must also be distinguished from chemical role. N7 of ordinary neutral adenine is a hydrogen bond acceptor, while its amino group can supply a hydrogen bond donor. A favorable contact requires an appropriately positioned complementary group, not merely two bases facing sides with the same category name. Structural conditions and chemical state matter.

When annotating an experimental RNA structure, record the interacting atoms and the two glycosidic bonds, rather than classifying from the sequence alone. An A–U pair can use a different geometry from the canonical Watson–Crick arrangement. Conversely, an edge category can recur with different base identities. This is why RNA base-pair isostericity is evaluated within explicit geometric families rather than inferred only from the letters A, C, G and U.

The sugar base-pair edge is the side of a nucleobase adjoining its ribose attachment region, including the ribose $2^{\prime }$-hydroxyl group when it participates in the contact. It is one of the three RNA base-pair edges used to classify RNA base-pair geometry. The original nomenclature also calls it the shallow-groove edge.^[ref]

This definition deliberately extends beyond the base ring. In the usual anti orientation about the glycosidic bond, the ribose $2^{\prime }$-hydroxyl can act as a hydrogen bond donor or hydrogen bond acceptor in the pairing interaction. Some sugar-edge contacts therefore connect a base to its partner's ribose rather than consisting solely of direct base-to-base hydrogen bonds.^[ref]

For an original annotation example, suppose residue 1 contributes its $2^{\prime }$-hydroxyl to an inter-residue hydrogen bond. Recording only the two aromatic base rings would omit that contact. A full record must identify the ribose atom, its partner and the geometry. Calling the interaction sugar-edge does not mean that an additional sugar molecule has joined the RNA; the group belongs to an existing nucleotide.

A complete pair description still needs the other partner's edge and the cis/trans relationship of the two glycosidic bonds. Sugar–sugar and Watson–Crick–sugar contacts are different families. The words “sugar edge” alone do not specify a unique pair of bases or a unique three-dimensional arrangement.

There is also an experimental limitation. The primary structural survey did not infer the $2^{\prime }$-hydroxyl hydrogen's exact position from the available X-ray crystallography coordinates. It represented the group as a unit and used compatible heavy-atom geometry to discuss possible hydrogen bonds. A close oxygen contact is evidence to interpret with donor/acceptor chemistry, not direct observation of the hydrogen's orientation.^[ref]

This distinction is useful when comparing RNA with a modified backbone: removing or replacing a ribose group can change a contact opportunity without changing the attached nucleobase's letter. Whether the whole fold survives that alteration remains a structural and experimental question.

A right marginal branch is a branch of the right coronary artery running toward the right ventricular margin and contributing to supply of the right ventricle. More than one marginal artery can occur.^[ref] The name concerns a branching route and territory rather than a separate main coronary origin from the aorta.

The distinction between parent flow and branch flow is quantitative. In an original steady model, let a proximal right-coronary segment supply a marginal branch plus all remaining represented branches. Then conservation of mass givesChosen values $80=30+50\mathrm{mL}\mathrm{mi}{\mathrm{n}}^{-1}$ satisfy the balance. A measurement of the proximal $80$ would not establish that the marginal branch receives $80$.

Nor does the adjective marginal imply negligible importance. A branch's functional contribution depends on its supplied tissue and actual flow, while its name describes an anatomical location. Coronary dominance addresses the origin of the posterior descending artery, so it answers a different classification question from identifying a marginal branch.

An anatomical or angiographic observation should therefore specify the main vessel, the branch connection and the represented region. A visible route can guide the interpretation of a selective measurement, but branch patency, flow and coronary oxygen delivery remain distinct observables.

A jump matrix is a prescribed invertible matrix-valued function $J$ on an oriented contour in a matrix Riemann-Hilbert problem. With plus boundary on the left of the contour, the unknown holomorphic matrix function satisfiesMatrix multiplication order matters. Where $M_{-}$ is invertible, $J=M_{-}^{-1}{M}_{+}$, rather than $M_{+}{M}_{-}^{-1}$ in general. Reversing contour orientation swaps the sides and replaces $J$ by $J^{-1}$.

For the positively oriented unit circle, a constant invertible matrix $J=A$ is realized by $M=A$ inside and $M=I$ outside. The exterior solution is normalized to $I$ at infinity. More complicated jumps encode boundary values of spectral or scattering data; prescribed poles, endpoint behavior and normalization must also be specified to define the full Riemann-Hilbert problem.^[ref]

An additive Riemann-Hilbert problem seeks a holomorphic function on each side of an oriented contour $\Gamma$, with prescribed difference of boundary valuesThe plus side is on the left of the orientation. Conditions at infinity and at contour endpoints are part of the problem; the jump alone does not usually give uniqueness.

On a positively oriented smooth simple closed contour, for Hölder-continuous data $g$, the Cauchy transformhas the required jump by the Sokhotski–Plemelj theorem and vanishes at infinity. For the unit circle and $g=1$, the Cauchy integral formula gives $F=1$ inside and $F=0$ outside. This concrete solution checks both the sign of the jump and the normalization. Subtracting two solutions removes the jump, so analytic continuation and growth conditions can establish uniqueness.^[ref]

Gaussian reparameterization represents a random variable with a normal distribution using a location, a scale and parameter-independent noise:Here $\epsilon$ has the standard normal distribution. The resulting $Z$ has expected value $\mu$ and variance ${\sigma }^2$. The probability distribution is unchanged; the representation separates the random draw from the adjustable parameters. This makes reparameterization gradients available for differentiable expectations, including the Gaussian approximate posterior in a variational autoencoder.^[ref]

For a complete example, fix $\sigma$ and considerHolding the same noise value fixed while differentiating givesAt $\mu =1$, $\sigma =2$, the particular noise draw $\epsilon =1/2$ gives $Z=2$ and derivative estimate $4$, while the true derivative is $2$. An individual sample need not equal the derivative; the equality is in expected value.

Parameterizing a positive scale as $\sigma =e^{\rho }$ also gives $\partial Z/\partial \rho =e^{\rho }\epsilon$. One can differentiate through the deterministic transformation while leaving the noise law fixed. For a general integrand, exchanging derivative and expectation still requires justified differentiation under the integral sign; the representation alone is not a proof of that exchange. Reusing noise when parameters change also connects the construction to common random numbers.

A learning rate controls the size of an iterative learning update. In gradient descent, it scales a direction obtained from the gradient of a loss:A stochastic gradient descent update uses an estimated gradient in this expression. The documented SGD algorithm makes the learning rate an explicit multiplier; momentum and other modifications add separate behavior.^[ref]

A scalar example shows why size matters. For $L(\theta )={\theta }^2/2$, the gradient is $\theta$, so a constant learning rate givesFor nonzero initial value, convergence to zero occurs exactly when $|1-\eta |<1$, or $0<\eta <2$. With ${\theta }_0=4$ and $\eta =1/2$, the values are $4,2,1,1/2,\dots$. With $\eta =5/2$, they begin $4,-6,9,-13.5,\dots$: increasing the step has made the iteration diverge rather than merely learn faster.

This threshold belongs to the stated quadratic. Changing curvature changes the allowed scale: for $L(\theta )=a{\theta }^2/2$ with $a>0$, the same proof gives $0<\eta <2/a$. A numerical learning rate therefore has meaning only together with the objective and update rule.

In reinforcement learning, an incremental value estimate can instead use the update $Q\leftarrow Q+\alpha (r-Q)$. For an illustrative $Q=2$, observed target $r=4$ and $\alpha =0.1$, the new estimate is $2.2$. Here the learning rate weights new evidence against the previous estimate, showing the same step-size idea without requiring a gradient calculation.

A learning rate schedule can vary the rate with iteration. Report the schedule, objective scaling and other update settings when comparing results; the rate alone does not determine stability or final accuracy.

The effective regurgitant orifice area is a flow-equivalent area for a leaking heart valve: the instantaneous backward volume flow rate $Q_R(t)$ divided by the corresponding jet speed $u_j(t)$,It characterizes the effective stream passing through the leak. It need not equal the anatomical opening traced on an image, because flow contracts and its velocity is not uniform across every geometric surface. The vena contracta and the upstream convergence pattern provide complementary flow information.^[ref]

The proximal isovelocity surface area method illustrates how an area can be inferred without drawing the hole itself. Color Doppler identifies an upstream surface at a selected aliasing speed $v_a$. In an ideal hemispherical convergence model of radius $r$, conservation of mass givesContinuous-wave Doppler echocardiography supplies the jet speed. The convergence radius and jet speed must refer to matching times. The hemisphere is an assumption about flow geometry, not a property guaranteed by the instrument.^[ref]

For an original model calculation, let $r=0.8$ cm, $v_a=30$ cm/s and $u_j=500$ cm/s. Then $Q_R\simeq 120.64$ mL/s and $A_{\mathrm{eff}}\simeq 0.2413\mathrm{c}{\mathrm{m}}^2$. At otherwise fixed inputs a ten-percent radius overestimate multiplies the inferred area by $1.1^2=1.21$. The radius-squared dependence explains why a small geometric measurement error can matter substantially.

An instantaneous effective area and an integrated regurgitant volume answer different questions:If the effective area is constant, this becomes $V_R=A_{\mathrm{eff}}\mathrm{VTI}$ using the velocity-time integral. A stipulated VTI of 150 cm with the calculated constant area gives 36.19 mL. A single peak area generally cannot be pulled outside the integral when the leak changes during systole. Alternatively, $V_R/\mathrm{VTI}$ defines a velocity-weighted mean effective area over that interval. The technical report explicitly distinguishes instantaneous convergence measurements from temporally varying regurgitation.^[ref]

Label smoothing replaces a sharply concentrated training target with a mixture that assigns some probability to alternative labels. For a $K$-class example whose observed label is $y$, a common uniform version usesHere $p$ is the predicted probability vector and $0\le \epsilon <1$ controls smoothing. The target still sums to one. This is a regularization of the training objective, not a statement that the example is known to have several simultaneously correct labels.^[ref]

With three classes and $\epsilon =0.1$, the observed class receives $0.9333\dots$ and each other class $0.0333\dots$. This convention differs from assigning $0.9$ to the observed class and splitting $0.1$ only among the incorrect classes. State the convention when reporting a smoothing parameter.

For a softmax function applied to logits $z_k$, the categorical cross-entropy loss has gradient $\partial L/\partial z_k=p_k-q_k$. With fixed smoothed targets and unconstrained probabilities, the loss is minimized at $p=q$. The three-class example consequently has optimum true-versus-other logit gap $\log (0.9333\dots /0.0333\dots )=\log 28\approx 3.332$, rather than requiring an unbounded gap to approach a one-hot target.^[ref]

The changed objective can discourage extreme training predictions. Its effect on generalization, calibration or another downstream task is an empirical question; the derivation alone does not guarantee improvement on every dataset. Evaluation should therefore retain the distinction between a training regularizer and the true, unknown distribution of labels.

For a finite connected undirected graph, the commute time of a random walk between $a$ and $b$ is the expected time to travel from $a$ to $b$ and back:For the simple random walk and unit edge resistances, it satisfies $C_{ab}=2|E|R_{\mathrm{eff}}(a,b)$, where $R_{\mathrm{eff}}$ is effective resistance. Each additional parallel route lowers resistance and therefore lowers commute time when the total edge count is fixed.^[ref]

On a path with $m$ edges, the endpoint resistance is $m$ and the commute time is $2m^2$; each one-way hitting-time expectation is $m^2$. On a triangle, two routes give resistance $1\parallel 2=2/3$, so the commute time between any pair is $2\cdot 3\cdot (2/3)=4$. The identity applies to this reversible walk and its associated resistor network, not to arbitrary directed transition rules.

A fixed-step isotropic random flight is a random walk in continuous space with independent steps of a common length $\ell$, each direction uniformly distributed over all directions. In $d$ dimensions, write its position as $S_n=\ell {\sum }_{j=1}^n{U}_j$, where the $U_j$ are independent unit vectors. Rotational symmetry gives $\mathbb{E}{U}_j=0$ and $\mathbb{E}(U_j{U}_j^{\mathsf{T}})=I_d/d$. Independence then givesThus 100 steps of length one have total path length 100 but root mean square displacement 10. A zero mean vector does not mean that a typical walker remains at its starting point.^[ref]

An exact two-step example in three dimensions shows how the endpoint spreads. The cosine $C=U_1\cdot U_2$ is uniform on $[-1,1]$, and $R^2=\Vert S_2{\Vert }^2=2{\ell }^2(1+C)$. Changing variables gives the radial probability density functionThis is a density per unit distance; a density per unit volume includes the spherical volume factor $4\pi r^2$. Uniform polar angle would give a different model: isotropic simulation in three dimensions samples the cosine of that angle uniformly.

In three dimensions the one-step characteristic function is $\sin (\ell k)/(\ell k)$, with value one at $k=0$, so the $n$-step characteristic function is its $n$th power. For a step duration $\tau$, the diffusion limit keeps $D={\ell }^2/(6\tau )$ fixed as $\ell ,\tau$ tend to zero. It yields the Gaussian diffusion equation kernel. At finite $n$, however, the walker cannot travel farther than $n\ell$; a Gaussian distribution is a limiting approximation, not the exact finite-step law.^[ref]

Spectral-density conversion between wavelength and Raman shift preserves the integrated physical quantity while changing its horizontal coordinate. Fix the excitation wavelength ${\lambda }_0$, and let $s$ be the Raman shift. Using numerical vacuum wavelengths in nanometres and shifts in $\mathrm{c}{\mathrm{m}}^{-1}$,Let $D_{\lambda }$ denote detected counts per nanometre and $D_s$ counts per unit Raman shift. Conservation of counts under a change of variables givesThe conversion factor carries the corresponding density-unit change. The same argument applies to calibrated power densities, provided both sides describe the same quantity.

For an original narrow bin centred at $600\mathrm{nm}$, a count density of $1200$ per nanometre becomes $43.2$ per $\mathrm{c}{\mathrm{m}}^{-1}$. A one-nanometre bin there spans approximately $27.78\mathrm{c}{\mathrm{m}}^{-1}$, so its integrated count remains about $1200$. At $800\mathrm{nm}$, the same wavelength density becomes $76.8$ per $\mathrm{c}{\mathrm{m}}^{-1}$. Relabeling bin positions without adjusting the density would conceal this coordinate dependence.

For finite bins, use their exact transformed edges and preserve each bin's integral. Raman spectral rebinning with conserved integrated counts distinguishes this operation from interpolating peak heights onto a new grid. Instrument response correction is another step; the Jacobian alone does not calibrate detector sensitivity or remove background.

The primary FSRS apparatus calibrated Raman shifts using known solvent lines and dispersed the probe spectrum onto an array.^[ref] Its dimensionless Raman pump-on pump-off gain ratio is a different observable: a ratio of matched spectral densities can cancel their common coordinate factor. It should not be treated as a count density merely because it is plotted against Raman shift.

Reduced-mass ambiguity in a Raman stiffness inference means that a measured frequency ratio does not determine a stiffness ratio without a mass model. In the assigned harmonic oscillator approximation used in Raman shift as a probe of bond stiffness, the vibrational wavenumber obeys $\tilde{\nu }\propto \sqrt{k_{\mathrm{eff}}/{\mu }_{\mathrm{eff}}}$. ThusThe reduced mass of an isolated two-body oscillator is $m_1{m}_2/(m_1+m_2)$. An effective mass for a collective normal mode additionally depends on which atoms move and their relative displacements. Keeping the same peak label does not establish that this effective mass stayed fixed.

For an original example, a band moves from $1000$ to $980\mathrm{c}{\mathrm{m}}^{-1}$, giving $q=0.98$. With unchanged effective mass, the inferred stiffness ratio is $0.9604$, a $3.96\%$ reduction. With an effective-mass ratio $1.06$, the same observed shift instead gives $k_2/k_1=1.018024$: stiffness has increased by about $1.80\%$. The two interpretations satisfy the same measured frequency ratio.

This ambiguity matters when comparing compositions, isotopes or changing mode assignments. Identifiability of stiffness from isotope-resolved Raman shifts requires additional constraints on the mass model and assignment; a frequency shift alone supplies one equation for two ratios. A controlled experiment can compare a known composition under load, stabilize temperature and optical conditions, and measure the spectral reference while checking that the same normal mode is followed.

HORIBA's silicon microstructure application maps stress using a material-specific relationship between stress and the Raman peak shift.^[ref] That experiment motivates an independently calibrated response. It does not make every redshift a universal direct measurement of a reduced spring constant.

A Raman band-area map assigns each spatial position an integrated, usually background-corrected spectral intensity over a stated Raman shift range. If $I(x,y,\tilde{\nu })$ is the spectrum and $B$ is its estimated baseline,This is one way to summarize the spectra collected in Raman mapping. It is useful for following a band's spatial distribution while preserving a different interpretation from a Raman peak-position map.^[ref]

Peak height alone does not determine area. For an isolated Gaussian function band of height $H$ and width parameter $s$, integrating the complete band givesTwo hypothetical bands of equal height10 intensity units and $s=2$ or4cm⁻¹ have full areas about50.13 and100.27 intensity-units·cm⁻¹. Thus an area map can change while a height map stays constant. A finite integration window captures only part of the complete band, so the actual window must accompany the value.

Raman integration-window bias arises if shifting or broadening moves part of the chosen band outside that fixed range. Neighboring bands and an incorrectly subtracted background can also change the result. For a sampled spectrum, a quadrature uses the actual channel spacing, for example $A\approx {\sum }_i(I_i-B_i)\Delta {\tilde{\nu }}_i$. A bare sum without the spacing factor has a different numerical convention when sampling changes.

Comparing areas across positions also requires consistent exposure, illumination and focus, or an appropriate normalization and measurement calibration. A larger area is therefore not automatically a larger concentration of one material. The manufacturer's discussion of spectral analysis explicitly cautions that broad background can contaminate simple intensity-based images.^[ref]

A Raman peak-position map assigns to each sample position the fitted Raman shift of a selected spectral band. Starting from position-resolved spectra, the analysis estimates ${\tilde{\nu }}_{\mathrm{peak}}(x,y)$ and displays its spatial variation. It answers a different question from a Raman band-area map, which records an integrated spectral intensity. Manufacturer documentation uses fitted band position to reveal structural variation, including strain-related changes in a silicon-carbide wafer.^[ref]

For a simple isolated Gaussian function band above background $B$,the fitted center ${\tilde{\nu }}_0$ is the peak position. A maximum chosen from discrete channels can instead jump in steps of the channel spacing. Fitting uses the neighboring samples too, although its result still depends on the line-shape model, background and noise. Raman peak-fit uncertainty belongs with the resulting map.

For an explicitly hypothetical band, suppose a reference region is at1000.0cm⁻¹ and another region at1000.5cm⁻¹. The shift difference is0.5cm⁻¹. If an independently established local stress calibration were $\Delta \tilde{\nu }=K\sigma$ with $K=2\mathrm{c}{\mathrm{m}}^{-1}/GPa$, that model would imply0.25GPa. The coefficient is illustrative, not a material constant asserted for the manufacturer's wafer. A quantitative stress map requires the applicable measurement calibration and control of other causes of spectral change.

Overlapping bands can move the maximum of their sum when only their relative intensities change. A stable center estimate must therefore track a specified band rather than interpret every maximum shift as strain. Instrument drift, temperature and laser-induced heating can also affect position. Retaining the underlying spectra and fit results makes it possible to test those interpretations rather than rely only on the map's colours.

A radio interferometer combines voltage signals from separated antennas to measure the spatial coherence of radio waves. A correlator compares signals after correcting for the geometrical delay associated with an observing direction. The resulting complex visibility retains relative wave phase information that independent power measurements lose.

For a baseline vector $\mathbf{b}$ and source direction $\mathbf{s}$, the geometrical delay is $\tau =\mathbf{b}\cdot \mathbf{s}/c$. The angular resolution scale is roughly $\lambda /B$, where $B$ is the longest relevant baseline. Multiple baselines and the changing orientation supplied by Earth's rotation sample different spatial frequencies for aperture synthesis; a single pair does not supply a complete image.

3C 273 is a bright quasar whose star-like optical appearance initially concealed its extragalactic character. In 1963, Maarten Schmidt identified its emission lines as a familiar series shifted by a common redshift of about $z=0.158$.

The identification connected the spectrum to a large cosmological distance and hence an enormous luminosity. The lesson is methodological: an object's unresolved angular size does not establish that it is an ordinary nearby star. Interpreting spectral-line patterns can change the inferred physical scale.^[ref]

Simpson baseline integration for pulse-energy correction subtracts a three-point estimate of integrated background from a measured gate integral. For a baseline $b(t)$ on $[0,T]$, Simpson's rule givesIt is exact for a quadratic or cubic baseline, provided the three values represent background rather than pulse-contaminated samples. For a four-times differentiable baseline, the remainder is ${\int }_0^{T}bdt-B_S=-T^5{b}^{(4)}(\xi )/2880$ for some $0<\xi <T$.^[ref]

For an original chosen quadratic background, take $T=6\mu \mathrm{s}$ with beginning, midpoint and end values $2$, $3$ and $2\mathrm{mW}$. Its integrated baseline is $16\mathrm{nJ}$, whereas endpoint averaging would give $12\mathrm{nJ}$. A raw gate integral of $116\mathrm{nJ}$ therefore gives $100\mathrm{nJ}$ after the three-point correction. Overestimating only the midpoint by $0.50\mathrm{mW}$ would subtract an extra $2\mathrm{nJ}$.

The uncertainty of a three-point baseline integral is separate from its quadrature error. With baseline-sample covariance matrix $C_b$ and $\mathbf{w}=(T/6)(1,4,1{)}^T$, its variance is ${\mathbf{w}}^T{C}_b\mathbf{w}$. Independent samples of common standard uncertainty ${\sigma }_b$ give $u(B_S)=T{\sigma }_b/\sqrt{2}$; a perfectly shared additive baseline error of the same uncertainty gives $T{\sigma }_b$ instead. Covariance terms cannot be discarded.^[ref]

A matched blank acquisition or independently measured background can supply the midpoint. Historical NIST pulse-energy apparatus used before/after baselines for background and thermal-drift compensation.^[ref] The three-point example is an original extension, not a claim about that instrument's processing. Higher polynomial exactness does not compensate for an incorrectly identified baseline.

An always-taker is a unit with potential treatment receipts $D(0)=D(1)=1$: it takes the treatment under either value of binary encouragement $Z$. This $(1,1)$ pair is a principal stratum defined by both potential receipts.

For example, a unit may participate whether or not it receives an invitation. Encouraged participants combine always-takers and compliers; unencouraged participants combine always-takers and defiers. Under instrumental-variable monotonicity, the latter group consists only of always-takers. Even then, comparing observed participants across assignment arms mixes different strata and is not automatically a within-stratum causal effect. The exclusion restriction, when imposed, says assignment cannot affect an always-taker's outcome through any channel other than the treatment whose receipt is unchanged.^[ref]