Measures Reference

This page is the measure quick reference for the current PILOTS source tree. It is organized by measure family and should be updated in the same commit as the corresponding C++ files under src/measures/.

For each measure or family, the page records:

• source file(s)

• required input fields and topology dependencies

• important config keys

• output columns

• physical meaning

• typical use cases

Common conventions

Static selection and identity

Most time-correlation and displacement measures require:

• a static selection (configured by group, topo_group and combine)

• identity-consistent frames, i.e. the same canonical particle ordering across frames

Frame-window controls

Many measures support:

• frame_start / frame_end: analysis window in frame index space

• remove_drift / drift_group: subtract COM drift before evaluating displacements

• components: choose xx, yy, zz, xxyy, xxzz, yyzz, or xxyyzz

Common list-valued parameters

• q_values: wave-vector magnitudes

• a_values: overlap cutoff radii

• p_values: Rouse mode indices

• waiting_frames / lag_frames: explicit waiting-time and lag grids for two-time observables

MSD and single-particle glassy dynamics

src/measures/RegisterMSD.cpp

msd

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional static drift_group

Outputs

lag, time, tensor components, count-pairs, SEM, block statistics

Physical meaning

Mean-squared displacement (or selected Cartesian tensor block) averaged over time origins.

Typical use

Diffusion, caging, subdiffusion, anisotropic transport.

\[\mathrm{MSD}(t)=\left\langle \frac{1}{N}\sum_i |\mathbf r_i(t_0+t)-\mathbf r_i(t_0)|^2\right\rangle_{t_0}\]

src/measures/RegisterAlpha2.cpp

alpha2

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional drift removal; 1D correlator infrastructure

Outputs

lag, time, \(\langle r^2\rangle\), \(\langle r^4\rangle\), \(\alpha_2\), count-pairs, SEM

Physical meaning

Non-Gaussian parameter for self displacements.

Typical use

Detect departure from Gaussian diffusion, caging-to-hopping crossover, dynamic heterogeneity onset.

\[\alpha_2(t)=\frac{3}{5}\frac{\langle r^4(t)\rangle}{\langle r^2(t)\rangle^2}-1\]

src/measures/RegisterScattering.cpp

self_isf

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional drift removal; q_values

Outputs

lag, time, q, \(F_s(q,t)\), count-pairs, SEM

Physical meaning

Isotropic self intermediate scattering function.

\[F_s(q,t)=\left\langle\frac{1}{N}\sum_i \frac{\sin(q|\Delta\mathbf r_i|)}{q|\Delta\mathbf r_i|}\right\rangle_{t_0}\]

chi4_overlap

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional drift removal; a_values

Outputs

lag, time, a, \(Q\), \(Q^2\), \(\chi_4\), count-pairs, SEM

Physical meaning

Four-point susceptibility based on the overlap indicator.

\[Q_a(t)=\left\langle\frac{1}{N}\sum_i \Theta(a-|\Delta\mathbf r_i|)\right\rangle_{t_0}, \qquad \chi_4(a,t)=N\left[\langle Q_a^2\rangle-\langle Q_a\rangle^2\right]\]

chi4_isf

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional drift removal; q_values

Outputs

lag, time, q, \(\overline F_s\), \(\overline{F_s^2}\), \(\chi_4^{F_s}\)

Physical meaning

Four-point susceptibility built from the self ISF amplitude.

\[\chi_4^{F_s}(q,t)=N\left[\overline{F_s^2}(q,t)-\overline F_s(q,t)^2\right]\]

src/measures/RegisterTwoTime.cpp

This file implements aging / two-time observables. These are needed when the system is not stationary and the observable depends on both the waiting time \(t_w\) and the lag \(\tau\).

The family uses an explicit waiting-time grid and does not collapse over all time origins like a stationary correlator. Common keys are:

• waiting_frames or waiting_start_frame / waiting_end_frame / waiting_stride_frames

• waiting_window_frames: number of consecutive origins averaged inside each waiting-time bin

• origin_stride_frames: subsampling stride inside the waiting-time bin

• lag_frames or max_lag_frames / lag_stride_frames

• finite frame_end is required; follow mode is intentionally rejected

msd_tw

Outputs

waiting frame, waiting timestep, waiting time, lag in frames and timesteps, lag time, two-time MSD, SEM across origins inside the waiting-time bin, number of origins

\[\mathrm{MSD}(t_w,\tau)=\left\langle \frac{1}{N}\sum_i |\mathbf r_i(t_w+\tau)-\mathbf r_i(t_w)|^2\right\rangle_{t_w\text{-bin}}\]

fs_tw

Additional key

q_values

Outputs

waiting frame/time, lag frame/time, q, \(F_s(q;t_w,\tau)\), SEM, number of origins

\[F_s(q;t_w,\tau)=\left\langle \frac{1}{N}\sum_i \frac{\sin(q|\mathbf r_i(t_w+\tau)-\mathbf r_i(t_w)|)}{q|\mathbf r_i(t_w+\tau)-\mathbf r_i(t_w)|} \right\rangle_{t_w\text{-bin}}\]

q_tw

Additional key

a_values

Outputs

waiting frame/time, lag frame/time, a, \(Q(t_w,\tau)\), SEM, number of origins

\[Q_a(t_w,\tau)=\left\langle \frac{1}{N}\sum_i \Theta\!\left(a-|\mathbf r_i(t_w+\tau)-\mathbf r_i(t_w)|\right) \right\rangle_{t_w\text{-bin}}\]

chi4_tw

Additional keys

a_values and waiting_window_frames >= 2

Outputs

waiting frame/time, lag frame/time, a, \(Q\), \(Q^2\), \(\chi_4(t_w,\tau)\), SEM of \(Q\), number of origins in the waiting-time bin

Physical meaning

Waiting-time-resolved fluctuation amplitude of the overlap field.

\[\chi_4(t_w,\tau)=N\left[\langle Q_a(t_w,\tau)^2\rangle_{t_w\text{-bin}}- \langle Q_a(t_w,\tau)\rangle_{t_w\text{-bin}}^2\right]\]

waiting_time_scan

Additional key

observable = msd|fs|q|chi4

Meaning

Convenience front-end that reuses the same kernels as msd_tw, fs_tw, q_tw or chi4_tw while keeping a single config style for waiting-time scans.

Polymer dynamics and static chain measures

src/measures/RegisterPolymerDynamics.cpp

g1

Required fields

xu, yu, zu

Dependencies

static selection; identity-consistent frames; optional drift removal

\[g_1(t)=\left\langle\frac{1}{\sum_c N_c}\sum_c\sum_n |\mathbf r_{c,n}(t_0+t)-\mathbf r_{c,n}(t_0)|^2\right\rangle_{t_0}\]

g2

Additional dependencies

chain membership from chain_id_field or mol or topology bonds

\[g_2(t)=\left\langle\frac{1}{\sum_c N_c}\sum_c\sum_n |(\mathbf r_{c,n}-\mathbf R_{\mathrm{cm},c})(t_0+t)- (\mathbf r_{c,n}-\mathbf R_{\mathrm{cm},c})(t_0)|^2\right\rangle_{t_0}\]

g3

Additional dependencies

chain membership from chain_id_field or mol or topology bonds

\[g_3(t)=\left\langle\frac{1}{N_{\mathrm{chain}}}\sum_c |\mathbf R_{\mathrm{cm},c}(t_0+t)-\mathbf R_{\mathrm{cm},c}(t_0)|^2\right\rangle_{t_0}\]

rouse

Additional dependencies

chain membership plus linear chain ordering from chain_pos_field or topology bonds

Additional key

p_values

Outputs

lag, time, mode index, \(C_p(t)\), normalized \(C_p(t)/C_p(0)\)

\[ \begin{align}\begin{aligned}\mathbf X_{p,c}(t)=\frac{1}{N_c}\sum_{n=0}^{N_c-1}\mathbf r_{c,n}(t) \cos\!\left[\frac{p\pi(n+1/2)}{N_c}\right]\\C_p(t)=\left\langle\frac{1}{N_{\mathrm{chain}}}\sum_c \mathbf X_{p,c}(t_0+t)\cdot \mathbf X_{p,c}(t_0)\right\rangle_{t_0}\end{aligned}\end{align} \]

src/measures/RegisterPolymerStatics.cpp

rg_ree

Dependencies

linear chain ordering from chain_pos_field or topology bonds

Outputs

frame, time, \(\langle R_g^2\rangle\), \(\langle R_g\rangle\), \(\langle R_{ee}^2\rangle\), \(\langle R_{ee}\rangle\)

\[R_g^2=\frac{1}{N}\sum_n |\mathbf r_n-\mathbf R_{\mathrm{cm}}|^2, \qquad R_{ee}^2=|\mathbf r_N-\mathbf r_1|^2\]

Structure and van Hove family

src/measures/RegisterStructure.cpp

rdf

Required fields

xu, yu, zu

Dependencies

static selection; optional second group group_b / topo_group_b / combine_b

Outputs

r-bin, shell volume, \(g(r)\), cumulative coordination number

\[g_{AB}(r)=\frac{\langle n_{AB}(r)\rangle}{N_A\rho_B\Delta V(r)}, \qquad \Delta V(r)=\frac{4\pi}{3}(r_{hi}^3-r_{lo}^3)\]

static_structure_factor / sq_static

\[S(q)=1+\frac{2}{N}\sum_{i<j}\frac{\sin(qr_{ij})}{qr_{ij}}\]

dynamic_structure_factor / coherent_isf

Outputs

lag, time, q, coherent \(F(q,t)\), self part \(F_s(q,t)\), distinct part \(F_d(q,t)\)

\[F(q,t)=\left\langle \frac{1}{N}\sum_i\sum_j j_0\!\left(q|\mathbf r_j(t_0+t)-\mathbf r_i(t_0)|\right)\right\rangle_{t_0}\]

van_hove_self and van_hove_distinct

Outputs

lag, time, radial bin, shell-averaged \(G_s(r,t)\) or \(G_d(r,t)\)

\[ \begin{align}\begin{aligned}G_s(\mathbf r,t)=\frac{1}{N}\left\langle\sum_i \delta\!\left(\mathbf r-[\mathbf r_i(t_0+t)-\mathbf r_i(t_0)]\right)\right\rangle_{t_0}\\G_d(\mathbf r,t)=\frac{1}{N}\left\langle\sum_{i\neq j} \delta\!\left(\mathbf r-[\mathbf r_j(t_0+t)-\mathbf r_i(t_0)]\right)\right\rangle_{t_0}\end{aligned}\end{align} \]

Dielectric and orientation / relaxation family

src/measures/RegisterDielectric.cpp

bulk_dielectric

Required fields

xu, yu, zu, charge field (default q)

Dependencies

entity grouping from entity_id_field or mol or topology bonds; temperature and unit metadata

Meaning

Classic neutral-entity dipole-fluctuation dielectric. Use this when grouped entities are approximately neutral.

Outputs

per-component permittivity and isotropic average

\[\varepsilon_\alpha = 1 + \frac{\mathrm{Var}(M_\alpha)}{\varepsilon_0 V k_B T}, \qquad \mathbf M=\sum_e \boldsymbol\mu_e\]

bulk_fragment_dielectric

Additional keys

charge_model = fragment_internal (default for this alias), entity_reference = geometric|mass

Meaning

Conduction-blocked charged-mixture dielectric built from fragment-internal dipoles

\[\boldsymbol\mu_e^{\mathrm{int}} = \sum_{i\in e} q_i \left(\mathbf r_i-\mathbf R_e\right)\]

This yields a finite, origin-invariant entity-internal polarization even when grouped fragments carry net charge. It is useful as a charged-mixture surrogate/background-polarization dielectric, but should not be confused with the full zero-frequency static dielectric of a conducting electrolyte.

slab_dielectric / layered_dielectric

Additional keys

axis and n_bins

Outputs

slab coordinate, \(\varepsilon_\parallel(z)\), \(\varepsilon_\perp(z)\) and related susceptibilities

\[ \begin{align}\begin{aligned}\chi_{\parallel}(z)=\frac{C_{\parallel}(z)}{\varepsilon_0 V_{\mathrm{bin}} k_B T}, \qquad \chi_{\perp}(z)=\frac{C_{\perp}(z)}{\varepsilon_0 V_{\mathrm{bin}} k_B T}\\\varepsilon_{\parallel}(z)=1+\chi_{\parallel}(z), \qquad \varepsilon_{\perp}(z)=\frac{1}{1-\chi_{\perp}(z)}\end{aligned}\end{align} \]

slab_fragment_dielectric / layered_fragment_dielectric

Additional keys

charge_model = fragment_internal (default for this alias), entity_reference = geometric|mass and the slab-workflow keys slab_align_recenter, halfcell_fold, target_center_frac, anchor_group / anchor_topo_group / anchor_combine.

Meaning

Charged-mixture, fragment-internal layered dielectric. This is the finite, conduction-blocked local dielectric surrogate for systems where grouped entities may carry net charge.

slab_bg_dielectric / layered_bg_dielectric

Additional keys

species_groups (comma-separated static groups to resolve), background_groups (comma-separated subset whose sum defines the background polarization; defaults to species_groups), plus the same entity_reference and slab-workflow keys used by slab_fragment_dielectric.

Meaning

Species-resolved background-polarization layered dielectric inspired by local polarization-density fluctuation methods. The measure constructs species-resolved fragment-internal polarization fields, sums the requested background_groups into a total background polarization, and outputs both the total background dielectric and per-species susceptibility contributions.

Outputs

For each slab bin, one TOTAL_BG row plus one row per species group. The total row contains \(\varepsilon_\parallel^{\mathrm{bg}}(z)\) and \(\varepsilon_\perp^{\mathrm{bg}}(z)\). Species rows contain their partial covariances and susceptibility contributions with respect to TOTAL_BG.

\[ \begin{align}\begin{aligned}\boldsymbol\mu_{e,g}^{\mathrm{int}} = \sum_{i\in e\cap g} q_i \left(\mathbf r_i-\mathbf R_e\right)\\\mathbf M_g = \sum_e \boldsymbol\mu_{e,g}^{\mathrm{int}}, \qquad \mathbf M_{\mathrm{bg}} = \sum_{g\in \mathrm{background}} \mathbf M_g\\\chi_{g,\parallel}^{\mathrm{bg}}(z)= \frac{\mathrm{Cov}\!\left(M_{g,\parallel}^{\mathrm{bin}}(z),\,M_{\mathrm{bg},\parallel}\right)} {\varepsilon_0 V_{\mathrm{bin}} k_B T}\\\chi_{g,\perp}^{\mathrm{bg}}(z)= \frac{\mathrm{Cov}\!\left(M_{g,\perp}^{\mathrm{bin}}(z),\,M_{\mathrm{bg},\perp}\right)} {\varepsilon_0 V_{\mathrm{bin}} k_B T}\\\chi_{\mathrm{bg}}(z)=\sum_{g\in \mathrm{background}} \chi_g^{\mathrm{bg}}(z)\end{aligned}\end{align} \]

The TOTAL_BG row then reports \(\varepsilon_\parallel^{\mathrm{bg}}(z)=1+\chi_{\parallel}^{\mathrm{bg}}(z)\) and \(\varepsilon_\perp^{\mathrm{bg}}(z)=1/[1-\chi_{\perp}^{\mathrm{bg}}(z)]\).

src/measures/RegisterOrientation.cpp

bond_vector_acf

Dependencies

topology bonds

segment_reorientation

Dependencies

ordered chains from chain_pos_field or topology bonds

ring_normal_acf

Dependencies

ring identifiers via ring_id_field

dipole_acf and dielectric_spectrum

Dependencies

entity grouping from entity_id_field or mol or topology bonds, plus charge field

Outputs

lag-resolved orientational or dipolar ACFs; for dielectric_spectrum also frequency-domain spectrum

Physical meaning

Rotational relaxation, segmental reorientation, ring-plane relaxation, dielectric loss.

The common orientational kernel is

\[C_l(t)=\left\langle P_l[\hat{\mathbf u}(t_0+t)\cdot\hat{\mathbf u}(t_0)]\right\rangle_{t_0}\]

For the total dipole family,

\[C_\mu(t)=\langle \mathbf M(t_0+t)\cdot \mathbf M(t_0)\rangle_{t_0}\]

Transport family

src/measures/RegisterTransport.cpp

current_acf / conductivity_gk

Required fields

charge field and a vector field triplet (for example vx, vy, vz)

Outputs

lag, time, current ACF tensor block, and Green–Kubo running conductivity

\[\mathbf J(t)=\sum_i q_i\mathbf v_i(t), \qquad \sigma_{\alpha\alpha}^{\mathrm{GK}}(t)=\frac{1}{V k_B T}\int_0^t C_{\alpha\alpha}(\tau)\,d\tau\]

conductivity_einstein / ionic_conductivity_einstein / charge_msd

\[ \begin{align}\begin{aligned}\mathbf M(t;t_0)=\sum_i q_i[\mathbf r_i(t_0+t)-\mathbf r_i(t_0)]\\\sigma_E(t)=\frac{1}{6Vk_B T t}\left\langle|\mathbf M(t;t_0)|^2\right\rangle_{t_0}\end{aligned}\end{align} \]

site_hop_stats / hop_stats

Required field

discrete site_field

Outputs

committed transition counts, dwell-time statistics, hop rates

Physical meaning

State-to-state hopping statistics with persistence filtering.

src/measures/RegisterTransportSpecies.cpp

species_msd

Required field

integer-like species_field

\[MSD^{(a)}_{\alpha\beta}(t)=\left\langle\frac{1}{N_a}\sum_{i\in a} \Delta r_{i,\alpha}(t;t_0)\Delta r_{i,\beta}(t;t_0)\right\rangle_{t_0}\]

diffusion_tensor

Meaning

Einstein-form diffusion tensor inferred from the species-resolved MSD tensor.

vacf

Required fields

species field plus vector field triplet

\[C^{(a)}_{\alpha\beta}(t)=\left\langle\frac{1}{N_a}\sum_{i\in a} v_{i,\alpha}(t_0+t)v_{i,\beta}(t_0)\right\rangle_{t_0}\]

species_current_corr

\[J_a^\alpha(t)=\sum_{i\in a} q_i v_i^\alpha(t), \qquad C^{ab}_{\alpha\beta}(t)=\langle J_a^\alpha(t_0+t)J_b^\beta(t_0)\rangle_{t_0}\]

onsager_transport

\[ \begin{align}\begin{aligned}M_a^\alpha(t;t_0)=\sum_{i\in a} q_i\Delta r_{i,\alpha}(t;t_0)\\L_{ab}^{\alpha\beta}(t)=\frac{\langle M_a^\alpha M_b^\beta\rangle_{t_0}}{2Vk_B T t}\end{aligned}\end{align} \]

transference_number

Meaning

Transport-number-like quantity constructed from the Einstein-form Onsager matrix block sums. The value is reference-frame dependent and should be interpreted together with the selected drift-removal / frame convention.

Profiles, coordination, and box statistics

src/measures/RegisterOMIEC.cpp

Even though this file was introduced for OMIEC-oriented workflows, the retained measures are all general soft-matter statistics.

profile1d

Modes

number, charge, mass

Important config keys

axis, n_bins, mode, and, for slab workflows, slab_align_recenter, halfcell_fold, target_center_frac, and optional anchor_group / anchor_topo_group / anchor_combine.

Outputs

1D bin center, average weight, density in the analysis coordinate. When slab alignment or half-cell folding is enabled, the coordinate reported in the text output is the transformed slab coordinate rather than the raw box axis.

coordination

\[CN_i=\sum_{j\in B}\Theta(r_{\mathrm{cut}}-r_{ij})\]

residence_acf

\[h_{ij}(t)=\Theta(r_{\mathrm{cut}}-r_{ij}(t)), \qquad C_{\mathrm{res}}(t)=\left\langle\frac{\sum_{ij}h_{ij}(t_0)h_{ij}(t_0+t)}{\sum_{ij}h_{ij}(t_0)}\right\rangle_{t_0}\]

box_stats / swelling_stats

Outputs

frame, time, box lengths, box volume, and ratios relative to the first analyzed frame

Insertion and slab-kernel calibration family

src/measures/RegisterInsertionProfiles.cpp

These measures share the same slab-analysis transform used by profile1d:

• slab_align_recenter recenters a slab-like anchor selection to target_center_frac using a circular mean in fractional coordinate space.

• halfcell_fold folds a periodic electrolyte / OMIEC / electrolyte cell into a single reservoir-to-slab half-cell.

accessibility_profile / probe_accessibility_profile

Meaning

Probe-radius-dependent accessible-volume fraction profile \(h(z; r_p)\) from hard-core insertion geometry.

\[h(z; r_p)=\left\langle \frac{N_{\mathrm{accessible}}(z; r_p)}{N_{\mathrm{trial}}(z)}\right\rangle\]

Important config keys

probe_radius, radius_field or occluder_radius, samples_plane, samples_axis, convergence_refine_factor, slab transform keys, and reservoir-window keys reservoir_lo_frac / reservoir_hi_frac with optional right-window counterparts.

Outputs

Absolute accessibility, SEM across analyzed frames, reservoir-normalized accessibility, \(-\ln[h(z)/h_{\mathrm{res}}]\), and coarse-versus-refined voxel convergence diagnostics.

widom_profile

Meaning

Total charge-off Widom insertion profile using the configured soft backend.

\[\mu^{\mathrm{ex}}_{0}(z)=-\beta^{-1}\ln\left\langle e^{-\beta \Delta U_0}\right\rangle_z\]

Important config keys

All accessibility keys plus beta and energy_model. The current public backend supports none and full lj126 pair energies, with optional sigma_field / epsilon_field or constant probe/occluder LJ parameters.

conditional_widom_profile

Meaning

Mode-A-style decomposition into hard accessibility and conditional charge-off insertion over the accessible region.

\[\mu_{\mathrm{hard}}(z)=-\beta^{-1}\ln h(z)\]

\[\mu_{\mathrm{cond}}(z)=-\beta^{-1}\ln\left\langle e^{-\beta \Delta U_0}\right\rangle_{z,\mathrm{accessible}}\]

\[\mu_{\mathrm{full}}(z)=\mu_{\mathrm{hard}}(z)+\mu_{\mathrm{cond}}(z)\]

Outputs

Reservoir-referenced hard, conditional, and full charge-off insertion factors; corresponding free-energy profiles; SEM estimates from the per-frame distribution; and a conditional/full identity residual for audit.

Anisotropy, geometry, dynamic heterogeneity, and local structure

src/measures/RegisterAnisotropy.cpp

directional_msd

\[MSD_{\alpha\beta}(t)=\left\langle\frac{1}{N}\sum_i \Delta r_{i,\alpha}(t;t_0)\Delta r_{i,\beta}(t;t_0)\right\rangle_{t_0}\]

directional_isf

\[F_{s,\alpha}(q,t)=\left\langle\frac{1}{N}\sum_i \cos[q\,\Delta r_{i,\alpha}(t;t_0)]\right\rangle_{t_0}\]

slab_rdf

Meaning

Slab-resolved radial structure using a global density normalization.

cylindrical_profile / 2d_map / interface_excess

Meaning

Direction-resolved density / weight maps and Gibbs-style interfacial excess estimates.

src/measures/RegisterHeterogeneity.cpp

g4_rt, s4_qt, xi4_t

Meaning

Spatial amplitude and length-scale measures of dynamic heterogeneity.

\[S_4(q,t)\approx \frac{S_4(0,t)}{1+[q\xi_4(t)]^2}\]

string_motion / string_length_dist / mobile_cluster

Meaning

Cooperative motion diagnostics based on mobile-particle connectivity in real space or exchange space.

excitation_map / facilitation_acf

Meaning

Per-particle excitation propensity and facilitation diagnostics.

src/measures/RegisterLocalStructure.cpp

q4_q6_w6

Outputs

summary statistics and optional per-particle values for bond-orientational order invariants

\[q_l(i)=\sqrt{\frac{4\pi}{2l+1}\sum_{m=-l}^l |q_{lm}(i)|^2}\]

locally_favored_structure

Meaning

Threshold-based classifier over \(q_4\), \(q_6\), \(\hat w_6\), and coordination number.

voronoi_volume

Meaning

Statistics over an externally supplied Voronoi-volume field.

free_volume

\[v_i^{\mathrm{free}}=V_i^{\mathrm{Vor}}-v_i^{\mathrm{excl}}\]

softness_proxy

Meaning

Transparent linear proxy built from local structural descriptors; not a hidden ML model.

Implementation notes for maintainers

• Each measure family lives in one src/measures/*.cpp translation unit and registers itself at the bottom.

• If a new measure shares geometry, chain-ordering, entity-grouping, or integer-field logic, extend src/measures/MeasureCommon.hpp rather than duplicating helpers.

• Two-time / aging observables should stay separate from the stationary correlator path, because their output is a waiting-time/lag grid rather than a one-dimensional lag-average.