From signal to a defensible interpretation.

METHODOLOGY · REVISED 9 OCTOBER 2026

An acquired signal, an inverted distribution and a derived molecular weight are not the same thing. DiffAtOnce connects these layers and preserves the decisions needed to move between them. This guide develops the physical foundations, numerical treatment and practical checks used in Prime and ResinAtOnce, without presenting a simulation, a good fit or a software test as experimental validation.

The scientific pathway: preparation and acquisition → signed signal and reviewed encoding → inversion and diagnostics → documented corrections → distribution and derived quantities → explicit review → reproducible project and report. When information is missing, its status remains unknown or requires review; no result is invented.

01 · A shared Core, two scientific workflows

DiffAtOnce Prime is the general-purpose diffusion workspace: signal selection, DOSY, MwOSY, local analysis, solver comparison and dedicated research workflows. ResinAtOnce organises preparations, gradient series and batches of resins and polymers, distinguishes resin, reactive diluent, additive, impurity and solvent signals, and connects results to the report. They share scientific contracts and modules; their interfaces and individual workflows are not assumed to be identical.

The Core calculates; the interface supports review and application. Advanced options expand at the user's request, but collapsing them does not change custom or imported parameters. Restoring standard values is an explicit decision for the next calculation, not a silent change to a saved result. A standard profile is a starting point, not a universal calibration for all samples.

DiffAtOnce Legacy, the online laboratory, Academy and the TopSpin tools have their own documentation. A feature's availability in one product does not establish its presence in the others. Consult the Prime guide, the ResinAtOnce guide and the narrated use cases to locate the controls in your version.

02 · What diffusion NMR measures

Translational motion during an encoding interval produces a distribution of phases. For free Gaussian diffusion, a single coefficient D, balanced encoding and no additional unmodelled effects, ideal attenuation follows the Stejskal–Tanner equation:

\[I(b)=I_0\exp(-bD).\]

D has units of m²/s and b of s/m²; the product bD is dimensionless. For an ideal rectangular PGSE pair, with duration δ, onset separation Δ and gradient g:

\[b=\gamma^2g^2\delta^2(\Delta-\delta/3).\]

This expression cannot simply be copied to bipolar trains, ramps, shaped gradients, spatial encoding or sequences with different effective signs. If γ is expressed in rad·s⁻¹·T⁻¹, that convention must be used throughout the calculation. The general one-directional expression uses the effective gradient, including RF-imposed sign changes:

\[q(t)=\gamma\int_0^t g_{\mathrm{eff}}(u)\,du,\qquad b=\int_0^{T_E}q(t)^2\,dt.\]

In anisotropic or restricted systems, a scalar description may be insufficient. Relaxation, exchange, flow, geometry and sample evolution can alter the signal. The selected equation must describe the actual experiment, not merely its appearance. Stejskal and Tanner, 1965, establish the foundations of gradient encoding.

03 · Designing a gradient series that contains information

The series should observe both the initial decay and appreciable attenuation above the noise. If all bD values are very small, an almost constant signal contains little information about D; if almost all the signal disappears in the first few points, a reliable distribution cannot be recovered either. Adding bins does not add observations.

• Define the expected D range, available time, sequence, gradient limits and required signal-to-noise ratio. Check that δ, Δ, echo times and waveforms are physically achievable.
• Retain low-b points, informative points along the decay and, when the budget permits, replicates that support estimates of noise and drift.
• Control temperature, equilibration, composition, concentration, sample height, tube and relaxation conditions. A repeated experiment is not an independent preparation.
• Record duration and order. A reproducible permuted order can help distinguish temporal drift from dependence on b, but does not make a reaction stationary.

The planner proposes designs and simulations; it neither programs the spectrometer nor verifies its safeguards. Comparing designs requires matched acquisition budgets and gradient, echo and thermal-load limits, not just the same number of points. p-DOSY, 2015, addresses acquisition order in changing samples.

04 · Detecting a probe is not calibrating its gradient

PROBHD, serial number, instrument, nucleus and compatible evidence help suggest a probe. A similar family or name does not authorise substitution of its calibration. The acquired waveform, timings and timing model must be preserved. Gradient percentage and physical amplitude are different quantities.

Calibration can come from a documented experimental profile, reviewed b values, calibrated Gmax or a reference solute under known conditions. It must record units, sequence, date, uncertainty, instrument and source. Calibration with a solute followed by an internal correction using the same solute requires checking that the same information is not counted twice.

Preserved TBO protocol: the shared TBO_HDO_20260904 calibration uses an HDO slope of 1.58627, Δ = 50 ms, δ = 1.20 ms and SMSQ10.100. It is not replaced by the nominal gradient of 5.35 G/mm. The literal Dref = 1.902 comparison corresponds to 298.15 K, not the entire actual HDO series. These values document that protocol; they are not a recipe transferable to any probe.

A downloaded profile or a correct hash establishes integrity and declared provenance, not physical gradient validity. Applying a different calibration requires review and a new calculation revision.

05 · From original files to a matrix of observations

The conventional DOSY workflow reads compatible processed spectra and their metadata. Prime also provides dedicated workflows for complex Bruker FID and ultrafast SER data; these are not presented as a universal importer for any sequence. The shared Core's Bruker and Spinsolve readers require contracts and metadata compatible with the selected workflow.

Before inversion, review title, nucleus, solvent, TE, sequence, gradients, timings, RG, NS, axis orientation, phase, baseline and processing. Apodisation, cropping, interpolation, FFT and delay correction must be documented. Do not normalise each spectrum to its own maximum: this would destroy part of the attenuation being measured.

The matrix \(Y_{ij}\) relates acquisition i to spectral region or bin j. Signed signal is preserved in workflows that support it; a noise-induced negative value is not converted to zero to manufacture a positive signal. Taking the magnitude of a complex signal changes the noise model and is not interchangeable, without review, with its correctly phased real part.

Original files are read without modification. An adaptation, aligned copy or different processing must retain its lineage back to the input rather than replace it.

06 · Narrow and broad peaks, local extraction and alignment

Double-clicking Prime's upper spectrum marks the measured maximum in the neighbourhood and opens its attenuation. The same region is extracted throughout the series; a different maximum is not chased in each acquisition. An exact comparison with the map requires matching the bin, ppm averaging, acquisitions, method, grid and factors. A global band is not a local signal.

Width helps describe a signal, but does not by itself identify a polymer or macromolecule. Overlap, exchange, couplings, resolution, relaxation and processing also broaden peaks. Regions must be selected from each sample's ppm positions and structure: indices from another resin are not transferable.

When drift is present, diagnostics compare shift, shape and area throughout the series. iCOshift-Prime is an in-house modified variant, not a complete reproduction of MATLAB iCOshift 3.0. Applying alignment requires explicit selection and justification, creates a copy and preserves the previous one. An unresolved case does not establish the absence of drift. Changing proportions of overlapping signals can resemble shifts.

Alignment does not correct convection, phase, incorrect b values or extinguished signal. Spatial SPEN/ultrafast workflows require their own reconstruction. See the original iCOshift method and the guide for the installed variant.

07 · Noise, averages and separation of training from evaluation

The forward model uses measured coordinates. For a set of D coefficients and b acquisitions, the basic kernel is \(K_{ik}=\exp(-b_iD_k)\). Standards and thermal normalisation are applied subsequently to derived quantities, rather than silently changing the measured operator.

When acquisitions are grouped through an operator H, the observation, operator and covariance must be propagated together:

\[Y'=HY,\qquad K'=HK,\qquad \Sigma'=H\Sigma H^{\mathsf T}.\]

In general, the average of \(\exp(-b_iD)\) is not \(\exp(-\bar bD)\). Only identical b values or another explicit justification allow the operator to be reduced without changing its meaning. Nor is the noise of an average automatically the individual noise divided by the square root of the number of points when there is correlation or unequal group sizes.

Normalising by a shared I₀ introduces correlation between ratios. Individual, averaged, heterogeneous and shared noise are distinguished. A small residual does not by itself provide an independent sigma. Without an adequate noise model, compatibility remains unknown.

The held-out partition is fixed before constructing masks, scales or signal-dependent averages, or selecting regularisation/rank. Replicates of a preparation must remain in the same group. Building a mask using TEST and then calling it independent validation contaminates the evaluation.

08 · Two different fits: intensity and ln(I/I₀)

A monoexponential fit in intensity minimises discrepancies in I; a straight line in the logarithm minimises a different objective and changes the error structure. Both can be useful diagnostics, but need not produce exactly the same D.

\[\ln(I_i/I_0)=c-D(b_i-b_0).\]

The linear panel uses the available positive point with the lowest b as its reference and a free intercept. It presents slope, intercept, R², measured and corrected D, and conditional OLS error. NaN/Inf, invalid b or I ≤ 0 are inadmissible for the logarithm; the reasons are recorded without deleting those data from the original inversion.

Exploratory automatic selection to improve R² retains, by default, at least six points and 80 % of the b range, removing no more than 20 % of valid points. Manual selection and restoration of all valid points are also available. Excluded points must remain visible. A higher R² after removing points can bias D; it does not establish the absence of convection, monodispersity or greater accuracy. This mask is not automatically transferred to the inversion.

09 · Inverting an attenuation is not identifying a unique mixture

For a continuous distribution, the ideal signal is a Laplace transform. Discretisation using integrated weights gives:

\[I(b)=\int_0^{\infty}e^{-bD}\,d\mu(D),\qquad \widehat Y=KX.\]

The inversion is ill-conditioned: very different distributions can explain observations that are close within the noise. The grid, support, regularisation and constraints are part of the result. A smooth curve is not necessarily more truthful, and a very narrow minimum can be a resolution artefact.

The laboratory supports controlled comparisons without automatically changing the production solver. RAI-S, DOME-S, SILT and other methods have their own contracts; sharing a selector does not make their assumptions equivalent. Local Diffusion-Inversion is an included C# library; an HTTP connection or external executable is a different interface, with version, parameters, units and response review.

10 · TRAIn-MF signed-v2.2: shared information, separate statuses

TRAIn-MF exploits structure across frequencies instead of fitting each column as though it were unrelated to the others. In a factorised representation, \(X=SA\) and \(Y\approx KSA\). S describes diffusion profiles; A describes spectral contributions under the protocol's conventions. Neither their amplitudes nor their rank automatically represent concentrations or a number of species.

The signed-v2.2 integration preserves signed signal, a null model, held-out selection, training isolation, independent sigma when available, X/S/A factors, and nominal and active ranks. The selected rank is predictive, not a chemical identification. In the absence of supported signal, the null model does not invent bands, D, Rh or molecular weights.

legacy-v1, signed-v2 and MF-AUTO are retained for compatibility. MF-NNLS256 remains a different method. An old file is not silently converted to signed-v2.2 when opened. The MATLAB reference and C# port are compared through predictions and descriptors within tolerances, not by demanding arbitrary identity of interchangeable or rescalable factors.

The provenance of signed-v2.2 is linked to train-dosy-public, commit 2f85657b50b752ca6ab3c122026b11288651fe8d. The adaptations and DiffAtOnce bridge are identified as in-house work. Execution, convergence, noise compatibility and identifiability are recorded separately; requires_review is not hidden to obtain a more attractive map.

11 · DOSY, the 2D Gaussian field and lateral lobes

The horizontal axis retains chemical shift; the vertical axis represents D or apparent molecular weight. Original, Model and Residual are different representations. Spatial reconstruction through a multiscale Gaussian dictionary describes the 2D view and preserves unfitted signal. Its atoms are not molecules.

The lateral projection integrates contributions over ppm. Many spectral signals can share a diffusion lobe. Therefore the number of 2D atoms is not the number of lateral components. The lateral-modes-v1 protocol detects lobes through prominence and local support and fits the unsmoothed active projection in log₁₀(D) or log₁₀(M).

If one large and two smaller supported lobes are observed, the aim is to describe those three, not to draw one Gaussian per spatial atom or force a broad curve across valleys. The standard limit of ten is a configurable maximum, not a requirement for ten. Total signal, fitted sum, components, signed residual and unassigned positive signal are compared on a common scale.

Residual is a signed diagnostic, not a positive mixture for counting species. The historical dictionary-marginal-v1 protocol remains distinct. Recalculating lateral components does not repeat inversion, change D or reapply standards or temperature; retaining the new decomposition requires explicit review. Width and added visual area are not physical dispersity or concentration.

12 · Internal standards, temperature and viscosity: visible factors

The user can obtain D without an internal standard, provided b and conditions have been reviewed and the absence is stated. Detecting a candidate signal does not confirm which standard was added. TTMS, TMS, TSP and silicones are not interchangeable simply because their signals lie near 0 ppm. Identity, region, overlap, attenuation and reference Rh or D are reviewed with their source.

For a confirmed standard:

\[\begin{aligned}f_{\rm ref}&=\frac{D_{{\rm ref},\rm expected}(T)}{D_{{\rm ref},\rm measured}},\\D_{x,\rm final}&=f_{\rm ref}f_TD_{x,\rm measured}.\end{aligned}\]

The subscript ref identifies the standard and x the analyte: the standard's ratio calibrates the complete axis; it does not compare the standard's expected D with each analyte's measured D.

A primary standard with controls or a consensus of compatible factors can be chosen; each analyte is not corrected with a different factor merely to match desired values. ResinAtOnce's implementation blocks consensus when disagreements exceed the documented 10 % criterion. A discordant control requires investigation, not automatic exclusion to improve the result.

Empirical linear normalisation uses \(f_T=T_{\rm out}/T_{\rm acq}\), output temperature divided by acquisition temperature. It is not a viscosity measurement or a convection correction. For a Stokes–Einstein model with temperature-dependent viscosity, ideal transfer between two conditions depends on T/η; this law must be selected and documented, not added to a result that already includes it.

Principal KK2 result: fTTMS = expected DTTMS(T) / measured DTTMS and fT = 293/T are applied jointly. fTTMS, fT and their product are retained per block; the product is applied once to the complete axis, and Rh, molecular weights and moments are recalculated. applyReferenceSummary is not used, and results are not rescaled to earlier tables.

Prime supports documented solvent-viscosity correlations or manual values; a pure/non-deuterated solvent viscosity is an approximation for a mixture. ResinAtOnce's historical native workflow uses a fixed reference η of 0.000584794653074285 Pa·s at 293 K for its ¹H/CDCl₃ model. This protocol must not be confused with an η(T) curve or a measurement of the solution.

13 · Diagnosing convection before discussing correction

Coherent flow can produce additional phase and attenuation. Detected, not detectable and inconclusive convection are different statuses. “Not detectable” establishes neither absolute absence nor a universal detection limit. A high R² does not rule out flow either.

Compare series from the same preparation with compatible diffusion times, analyte and standards separately, real and imaginary parts when available, phase, residuals and apparent D dependence on d20. Different timings, temperatures or preparations can introduce alternative causes.

A physical correction requires motion encoding/sequence moments, a noise model, controls, stability and sufficient evidence to separate diffusion from flow. Another workflow uses a convection-free control acquisition from the same preparation under reviewed conditions. These are different sources: the report must state which was applied.

Without a valid control or identifiability, no button can invent corrected D. The diagnosis is retained, an appropriate acquisition is proposed, and the entire map is not rescaled by appearance. Changes during a series can also bias inversion, even when they do not arise from convection.

14 · Rh is a hydrodynamic conversion, not a universal geometric rule

For an appropriate Stokes–Einstein model:

\[R_h=\frac{k_BT}{6\pi\eta D}.\]

T in kelvin, η in Pa·s and D in m²/s are required to obtain Rh in metres; 1 Å = 10⁻¹⁰ m. This is an effective hydrodynamic radius: it incorporates friction, solvation and shape under the model's assumptions. It is not automatically a radius of gyration, the size of a rigid sphere or a direct microscopic dimension.

A distribution supports calculation of mean Rh after transforming each bin; in general \(\langle R_h\rangle\neq k_BT/(6\pi\eta\langle D\rangle)\). The difference arises from averaging a nonlinear function. Weighting and support must be declared. Recovering the assumed Rh of a standard after calibrating with it is internal consistency, not an independent measurement of that Rh.

15 · MwOSY: D→M calibration within a domain

MwOSY transforms the axis of a diffusion result using a documented molecular relationship. A typical power law is expressed with explicit reference units:

\[\log_{10}(D/D_*)=a\log_{10}(M/M_*)+c.\]

\[M=M_*10^{[\log_{10}(D/D_*)-c]/a}.\]

The historical high-dilution PS and PPG curves are different options. The shared concentration model was developed with PS; selecting PPG does not create another validated concentration law. Its empirical form uses:

\[D\eta(C)=D\eta(0)\exp[\kappa(M)C^{\nu}].\]

C is solute mass divided by net solvent mass, in mg/mg. If title lists solute first and then solute + solvent, subtract to obtain the solvent. Arithmetic example: 20 mg and 420 mg total give 400 mg of solvent and C = 0.05, not 20/420 or 5 mg/mL. Ambiguous or missing weighed masses, or those without units, are not converted to zero.

Here κ(M) is the signed coefficient used in the code, negative in the historical branches. The published notation with exp[−κ C^ν] and positive κ is equivalent only if that convention is also changed; a second minus sign is not added. The coefficients, ν and segments of κ(M) belong to the profile; they are not fitted to reproduce an expected molecular weight, nor is interpolation performed across a gap not represented by the law. The historical model retains its segment transition at 117 000 Da. Family calibration v2 uses D9 directly and must not be confused with the historical Dη-based equation.

Solvent, temperature, concentration, family, architecture, signal and molecular-weight reference delimit applicability. If compatible conditions are missing, retain D rather than force M. Molecular weights remain apparent for stars, brushes or branched resins without specific evidence; a small reference compound calibrates D, not a universal molecular relationship.

16 · What each bin represents and when the Jacobian is needed

The contract distinguishes integrated amplitude per bin, density per linear unit and density per logarithmic unit. Variable, unit, logarithm base, centres, bounds, orientation, support, exclusions and threshold must be recorded. An integrated amplitude is not multiplied by bin width a second time.

For a bin with positive bounds L and U, amplitude a and a base-10 logarithm:

\[\rho_{\rm lin}=a/(U-L),\qquad \rho_{\log}=a/\log_{10}(U/L).\]

Transforming D→M transforms the complete axis and its bounds, preserves each cell's integrated weight and correctly reorders it if the calibration reverses orientation. The Jacobian applies to a density:

\[\rho_M(M)=\rho_D(D(M))\left|\frac{dD}{dM}\right|.\]

For a law \(D\propto M^a\), densities per logarithmic unit with the same base satisfy \(\rho_{\log M}=|a|\rho_{\log D}\) at corresponding positions. Integrated amplitude remains unchanged. Changing the logarithm base also requires its factor, not merely relabelling the axis.

Verifiable invariants are amplitude conservation and equivalence between representations with ascending/descending D and M. The historical summary with its threshold—for example, 1 %—is retained separately from the laboratory's full support, 0 %. The map's visual threshold is a different criterion and must not truncate scientific moments.

17 · Mn, Mw and dispersity require declared weighting

With normalised number fractions \(n_i\):

\[\begin{aligned}M_n&=\sum_i n_iM_i,\\M_w&=\frac{\sum_i n_iM_i^2}{\sum_i n_iM_i}.\end{aligned}\]

With normalised mass fractions \(w_i\):

\[\begin{aligned}M_n&=\left(\sum_i w_i/M_i\right)^{-1},\\M_w&=\sum_i w_iM_i.\end{aligned}\]

In both cases \(\mathit{Đ}=M_w/M_n\), only when the contract and domain allow that interpretation. If the NMR response per species is unknown, amplitudes are not automatically converted to number or mass fractions: Mn, Mw and molecular dispersity remain unavailable. A mass-response assumption produces conditional apparent descriptors, not validated absolute moments.

The molecular weight obtained from mean D, \(M(\langle D\rangle)\), is neither Mn nor Mw; the molecular weight at the maximum is another quantity. The same distinction applies to modal D, monoexponential D and the mean of a distribution. Separate labels, support and weighting are reported. Visual Gaussian areas do not replace the contract's weights.

18 · An uncertainty budget is not the same as stability

The budget combines documented sources: fitting, gradient, standard, temperature, viscosity and molecular calibration. An empty cell means unknown, not zero. Shared scale errors do not simply decrease by adding bins or signals. Covariances matter, for example when T is used simultaneously in expected D, correction and Rh.

In a first-order approximation, with Jacobian J and input covariance Σ:

\[\Sigma_{\rm out}\approx J\Sigma_{\rm in}J^{\mathsf T}.\]

Correlated Monte Carlo propagates coherent inputs and accommodates nonlinear relationships under the declared model. Scale uncertainty, shape variation and model bias are separated. An interval does not automatically cover unknown biases in sequence, flow, architecture or viscosity.

An interval for the complete procedure would require repeating data-dependent preprocessing, inversion and regularisation/model selection in each appropriate resample. If these decisions remain fixed, the interval is labelled conditional. Heterogeneous, correlated and grouped data do not by default justify an iid residual bootstrap.

Stability examines sensitivity to grid, regularisation and perturbations: it is not, by itself, a confidence interval. Coverage is evaluated only with independent simulations or appropriate controls, stating the number of trials and statistical error. The JCGM/GUM documents and Monte Carlo supplement underpin propagation; they do not automatically validate the sample's physical model.

19 · Family calibration without learning from validation data

The versioned family profile retains the literal equation, units, family and architecture, solvent, T, concentration, signal, instrument, reference molecular-weight method, domain and provenance. The v2 model uses \(\log_{10}(D_9)=a\log_{10}(M_{\rm Da})+c\), with \(D_9=D/(10^{-9}\,{\rm m^2/s})\). Its coefficients are not mixed with those of a Dη equation.

The fit accounts for uncertainty in D and reference molecular weights, parameter covariance and shared sources through an in-house errors-in-variables adaptation. With x = log₁₀(M), y = log₁₀(D9) and r = y − ax − c, residual covariance depends on the slope:

\[V(a)=C_{yy}+a^2C_{xx}-a(C_{xy}+C_{yx}).\]

\[Q=r^{\mathsf T}V(a)^{-1}r.\]

Parity with York or ODRPACK is not claimed. The first-order logarithmic approximation has relative-uncertainty guards; local parameter covariance is not turned into external validation. The preparation- or batch-based partition is fixed before fitting; held-out and external validation do not change the coefficients.

Validation errors, equation, conditions and uncertainty are exported. Unit changes must transform intercepts and domains consistently. An incompatible family, architecture or condition, or extrapolation outside the domain, blocks application. A new profile creates a derived revision; it does not rewrite historical calibrations.

The integration supports local and full-map review with explicit application. Synthetic fixtures verify contracts and calculations; they do not constitute a new series of real family standards. SEC/GPC or another orthogonal reference retains its calibration, limitations and molecular target, rather than being treated as uncertainty-free truth.

20 · Overlap: separating signal does not recover concentration without assumptions

An overlapping signal may contain contributions with different D values, but observed amplitudes also depend on equivalent protons, response, relaxation, excitation and processing. The dominant component in an algorithm need not be the most concentrated. Two species with almost indistinguishable D may not be identifiable from that attenuation.

Anchored fits and joint pilots preserve the following relationship for each acquisition:

\[A_{\rm obs}=\sum_c A_{c,\rm pred}+A_{\rm residual,signed}.\]

A negative residual is not removed to impose a false sum. Anchors, companion signals, minority recovery, false components and held-out prediction are reviewed. NMF or nonnegative factorisation does not guarantee a unique chemical solution; its factors can have scale and ordering ambiguities.

qNMR quantification requires comparable integrals, proton counts, nominal standard amount and purity, and reviewed quantitative acquisition. In the ideal case, \(n_x=n_{s,\rm nominal}p_s(A_x/N_x)/(A_s/N_s)\), with documented corrections and units. Purity p is applied once: an already corrected amount must not be used as though it were nominal. Volume and composition are needed to convert to concentration. This amount calculation does not itself include an uncertainty budget. Relaxation delay, saturation, baseline and overlap do not disappear when a standard is used. See Pauli and colleagues, 2012.

21 · SPEN and ultrafast: the observable determines the kernel

Spatial encoding can store diffusion information in a single acquisition or in rapid schemes, but requires a spatial model, chirp waveform, gradients, coil profile and readout order. A spatial map does not automatically become ppm–D DOSY by applying conventional b.

• SPEN-DOSY/EPSI: reviewed spatial encoding and readout contract, axis, phase, measured profile and kernel. Published examples retain their DOI, repository, licence and hashes; a reproduction using the authors' tools is identified as external.
• Ultrafast CPMG D–T₂: the echo axis retains time, not chemical shift. The ideal separable model uses one diffusion kernel and one relaxation kernel; T₂ is effective and may include diffusion during readout or other effects.
• TRAIn2D and FISTA 2D: different methods. The in-house TRAIn2D adaptation uses its criterion relative to the NNLS residual; FISTA uses regularisation and a KKT criterion. A failed method is not silently replaced by another.

\[Y\approx K_DXK_{T_2}^{\mathsf T},\qquad (K_{T_2})_{j\ell}=e^{-t_j/T_{2,\ell}}.\]

The local Marek 1502 example supports study of notebook processing and contracts; it has NS = 16 and is not advertised as single-scan. Its assumed uniform profile and exploratory scale do not validate quantitative D, MwOSY or a new sequence. Private data are not redistributed here.

The guided workflow requires controls against conventional DOSY from the same preparation, predeclared tolerances, residuals and edges. SPEN literature shows dependencies on SNR and separation in D; it does not establish that SCORE or any other method resolves every mixture. See Lorandel and colleagues, 2025, and the examples linked from Prime.

22 · Comparing samples and following kinetics without confusing causes

Each acquisition should record concentration, T, solvent, additives, standards, duration and order. Compare measured D of the analyte and standards first, then factors and corrected D. A residual dependence on concentration or time calls for model review; it does not automatically identify aggregation, molecular growth or curing.

Independent preparations are distinguished from repeated series. Orthogonal SEC/GPC or other references include their calibration and limitations. Compare the same target: a spectral-response mean is not compared as though it were necessarily absolute Mn from another technique.

A sample that evolves during its own series violates the stationary-signal assumption. The kinetics module retains within-series warnings and apparent rates where appropriate. Agreement between controls, invariance to acquisition order and residual analysis help decide the next experiment; they do not replace independent experimental design.

23 · A reproducible benchmark, not a contest for attractive curves

The shared comparison initially includes monoexponential, TRAIn, NNLS, Tikhonov and MaxEnt. Development, selection and TEST are separated by preparation; cases, parameters and criteria are frozen before TEST is opened. Including another method requires a verifiable implementation, licence, environment and inputs.

Tests cover null signal, monoexponentials, mixtures, minority components, broad distributions, overlap, correlated noise, drift, replicates, short trains and components outside the support. D bias, held-out prediction, minority amplitude, false components, conditional coverage, censoring, time and memory are measured.

Wasserstein distance is calculated in a declared common coordinate—for example, log₁₀(D)—after normalising shape; total-amplitude error is reported separately. Changing from D to log D changes the metric. Extending the grid does not establish that an out-of-support component is identifiable.

NMRInversions, DRECT, DREAM or other external comparators contribute results only if they have been executed with verified environment and provenance. An adapter or checkbox is not an execution. Failures, versions and unavailable methods are recorded without promoting them as validated.

24 · Experimental developments: what is investigated and what is not promised

An implementation inspired by published work is labelled as in-house; parity with DRECT, DREAM or STELT is not claimed without verifiable code, weights, domain and reproduction. These pilots retain independent revisions and do not replace accepted maps.

25 · Beatrix supports review; it does not invent numerical facts

Beatrix uses the visible selection, calculated facts and documentation to explain warnings and propose checks during the workflow. Hover, selection and queries must retain the same context; a historical card is not applied to another series. Documentary matches are hypotheses, not identity probabilities.

The tables by Gottlieb, Kotlyar and Nudelman and Fulmer and colleagues guide review of solvents and impurities. Shift compatibility depends on nucleus, medium and conditions; an isolated ppm match confirms neither phthalates, an internal standard nor any other species. Seek companion signals, multiplicity and evidence from the same preparation.

Prime can be configured with local rules, CPU GGUF models or a compatible server/API. Model size, RAM, CPU and available service determine resources and speed. A GPU is used only when the configured server provides one; choosing “large” neither downloads a model nor automatically activates a GPU. In ResinAtOnce, the local Light/Advanced profiles are CPU budgets for the included assistant, not Prime's API connection.

Local inference requires a compatible model/corpus/manifest and integrity checks. If it fails, rule-based guidance remains available and is identified as such. A remote provider requires HTTPS, configuration and explicit permission to send context. Keys, protected identities and credentials do not belong in the project or scientific report.

Useful questions include: “What still needs review in this series?”, “Why does this method not converge?”, “Am I comparing a local signal with a global band?” or “What supporting evidence do I need for this hypothesis?”. COSY/HSQC/HMBC actually observed from the same preparation can support review. Beatrix does not turn a compatible fragment into a confirmed species or modify calibrations to meet an expectation.

26 · Saving, reproducing and communicating the correct revision

Traceability combines original files and their hashes, sample, sequence, nucleus, gradients and timings, TE, weighed masses, processing, noise, calibration, solver, parameters, distribution and statuses. The cache depends on inputs, protocol and solver; a change or corruption invalidates retrieval. Recovering a complete calculation does not mean resuming an internal iteration.

Versioned packages support exchange and replay from preserved attenuations and, in compatible workflows, extraction from included processed originals. Universal FID reprocessing is not claimed. Every derivative retains its relationship to its source; accepting and applying a revision are explicit actions distinct from simply viewing it.

“Result and edits do not match” indicates that the saved result belongs to different conditions. Restore its configuration or calculate a new revision, without renaming an earlier result as though it were new. Cancellation preserves previous complete results and the attempt's status; it does not turn a partial file into an accepted result.

Supported outputs include project and contract JSON, CSV tables, HTML/DOCX reports and figures. High-resolution PNG preserves view, zoom, theme and labels; 4800 pixels/300 dpi is a presentation proposal, not increased scientific resolution. An image does not replace data or manifests. See Prime export and ResinAtOnce export.

27 · Worked example: the same signal, different questions

A purely synthetic arithmetic example. It does not represent a validated sample, standard or calibration. Assume measured D = 0.500 × 10⁻⁹ m²/s, standard factor 0.900, acquisition T of 295 K and linear normalisation to 293 K.

\[f_T=293/295,\quad f_{\rm total}=0.900(293/295).\]

\[D_{\rm final}=0.446949\times10^{-9}\;{\rm m^2/s}.\]

This product is applied once to all D centres and bounds in the distribution; it is not multiplied again during export or conversion to M. Rh additionally requires η and a consistent thermal state. The central value alone does not report width, components or molecular weight.

Now consider a synthetic calibration \(M=10\,000/D_9^2\) and two bins D9 = 1 and 0.5 with equal integrated amplitudes. Their molecular weights are 10 000 and 40 000 Da. Each bin's weight remains 0.5 after the axis is transformed:

By contrast, transforming mean D9 = 0.75 gives M(mean D) ≈ 17 777.8 Da: it is none of the moments above. This is the practical reason for preserving labels, weighting, units and support on screen, in the project and in the report.

28 · What it means for a feature to be available

Five checks are distinguished: execution—the process finished; convergence—it met a numerical criterion; noise compatibility—the model and declared noise explain the data according to the test; identifiability—the data distinguish the intended quantity or components; and experimental validation—independent controls support physical use within that domain. None automatically implies the next.

Final checklist before communicating results

1. Identify the preparation, series, region, signal role and result actually saved.
2. Check b, units, sequence, calibration, per-block factors and their single application to the axis.
3. Review attenuation/prediction, residuals, correlation, support and signal censored at the edges.
4. Declare the measure, weighting, threshold, modal/mean/monoexponential D, Rh and available moments.
5. State the molecular domain, concentration, uncertainty and what remains unknown or unconverged.
6. Preserve the previous revision, manifests and authorised inputs; open the exported report and check its figures and labels.

If an error occurs, retain its message, version, method and steps, and consult the FAQ or support. Do not delete contracts or alter hashes to silence a verification. Installation registration, researcher accounts and support requests are different processes.

References, implementations and further reading

Sources underpin methods and limitations; they do not automatically certify an implementation or a sample. Project papers, DOIs and provenance-tracked examples are collected in Publications and Prime sources. In-house adaptations—including iCOshift-Prime, the TRAIn-MF port, the laboratory's MaxEnt and the pilots—retain that status.

1. Stejskal, E. O.; Tanner, J. E. (1965). Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient. Journal of Chemical Physics 42, 288–292. 10.1063/1.1695690. Foundations of gradient encoding.
2. Holz, M.; Heil, S. R.; Sacco, A. (2000). Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate ¹H NMR PFG measurements. PCCP 2, 4740–4742. 10.1039/B005319H. Diffusion calibration and temperature dependence.
3. Lawson, C. L.; Hanson, R. J. (1974; SIAM edition 1995). Solving Least Squares Problems. 10.1137/1.9781611971217. NNLS foundations, not chemical identification.
4. Day, I. J. (2011). On the inversion of diffusion NMR data: Tikhonov regularization and optimal choice of the regularization parameter. JMR 211, 178–185. 10.1016/j.jmr.2011.05.014. Regularisation and parameter selection.
5. Xu, K.; Zhang, S. (2014). Trust-Region Algorithm for the Inversion of Molecular Diffusion NMR Data. Analytical Chemistry 86, 592–599. 10.1021/ac402698h. Original TRAIn; TRAIn-MF and the C# port are distinct later extensions.
6. Arrabal-Campos, F. M. train-dosy / TRAIn-MF, signed-v2.2. Author's code and provenance. Software reference pinned to the commit stated in this guide; not equivalent to experimental validation.
7. Cherni, A.; Chouzenoux, E.; Delsuc, M.-A. (2017). PALMA, an improved algorithm for DOSY signal processing. Analyst 142, 772–779. 10.1039/C6AN01902A. Hybrid entropy–ℓ₁ regularisation; not the independent in-house MaxEnt.
8. Nilsson, M.; Morris, G. A. (2008). Speedy Component Resolution: An Improved Tool for Processing Diffusion-Ordered Spectroscopy Data. Analytical Chemistry 80, 3777–3782. 10.1021/ac7025833. SCORE and joint component resolution under its model.
9. Savorani, F.; Tomasi, G.; Engelsen, S. B. (2010). icoshift: A versatile tool for the rapid alignment of 1D NMR spectra. JMR 202, 190–202. 10.1016/j.jmr.2009.11.012. Original method; iCOshift-Prime is a modified variant.
10. Oikonomou, M.; Asencio-Hernández, J.; Velders, A. H.; Delsuc, M.-A. (2015). Accurate DOSY measure for out-of-equilibrium systems using permutated DOSY (p-DOSY). JMR 258, 12–16. 10.1016/j.jmr.2015.06.002. Acquisition order and temporal evolution.
11. Shrot, Y.; Frydman, L. (2008). Single-scan 2D DOSY NMR spectroscopy. JMR 195, 226–231. 10.1016/j.jmr.2008.09.011. Published foundations of spatially encoded DOSY, not validation of the private prototype.
12. Ahola, S. and colleagues (2015). Ultrafast multidimensional Laplace NMR for a rapid and sensitive chemical analysis. Nature Communications 6, 8363. 10.1038/ncomms9363. Ultrafast multidimensional Laplace NMR and D–T₂.
13. Lorandel, B. and colleagues (2023). Accounting for gradient non-uniformity in spatially-encoded diffusion-ordered NMR spectroscopy. JMR 355, 107543. 10.1016/j.jmr.2023.107543. Actual spatial fields and SPEN accuracy.
14. Lorandel, B. and colleagues (2025; online 2024). Speedy Component Resolution Using Spatially Encoded Diffusion NMR Data. Magnetic Resonance in Chemistry 63, 49–61. 10.1002/mrc.5488. SCORE/OUTSCORE with separation and SNR limitations.
15. JCGM (2008). Evaluation of measurement data — Guide to the expression of uncertainty in measurement. JCGM 100:2008. 10.59161/JCGM100-2008E. Sources and correlations; also consult the amendments in the current official catalogue.
16. JCGM (2008). Supplement 1 to the GUM — Propagation of distributions using a Monte Carlo method. JCGM 101:2008. 10.59161/JCGM101-2008. Propagation under the joint input model.
17. Malz, F.; Jancke, H. (2005). Validation of quantitative NMR. Journal of Pharmaceutical and Biomedical Analysis 38, 813–823. 10.1016/j.jpba.2005.01.043. qNMR acquisition, processing and referencing.
18. Pauli, G. F. and colleagues (2012). Quantitative ¹H NMR: Development and Potential of an Analytical Method—an Update. Journal of Natural Products. Full article. Development, applicability and quantitative review.
19. Arrabal-Campos, F. M.; González-Lázaro, M.; Pérez, J. M.; Martínez Lao, J. A.; Fernández, I. (2025). Concentration-independent molecular weight determination of polymers via diffusion NMR: A universal approach across solvents. European Polymer Journal 226, 113710. 10.1016/j.eurpolymj.2024.113710. Published concentration/viscosity model for the systems studied; the title does not extend its domain to every architecture.
20. Ruzicka, E.; Pellechia, P.; Benicewicz, B. C. (2023). Polymer Molecular Weights via DOSY NMR. Analytical Chemistry 95, 7849–7854. 10.1021/acs.analchem.2c05531. Calibration, preparation and limitations of molecular-weight estimation.
21. Gottlieb, H. E.; Kotlyar, V.; Nudelman, A. (1997). NMR Chemical Shifts of Common Laboratory Solvents as Trace Impurities. Journal of Organic Chemistry 62, 7512–7515. 10.1021/jo971176v. Chemical-shift tables; not a sample identification by themselves.
22. Fulmer, G. R. and colleagues (2010). NMR Chemical Shifts of Trace Impurities: Common Laboratory Solvents, Organics, and Gases in Deuterated Solvents Relevant to the Organometallic Chemist. Organometallics 29, 2176–2179. 10.1021/om100106e. Complementary documentary reference for solvents and impurities.

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