[Psychometric Gap] Implement posterior-aware multilevel longitudinal ESEM and DSEM estimation

[seonghobae]

Buyer problem

TEPP's core differentiation requires topic/event measurements to support valid higher-order psychometric inference across time, languages, templates, organizations, projects, and multiple memberships. Existing PRs provide bounded input gates, OLS recovery slices, invariance labels, event-time transforms, and scalar equations. They do not yet provide a production posterior-aware ESEM/DSEM estimator or a validated longitudinal decision surface.

Required scientific vertical

Implement a coherent psychometric engine that consumes validated posterior topic coordinates and explicit measurement design—not raw topic proportions—and estimates the bounded first-release multilevel longitudinal model.

The implementation must distinguish:

• reflective, formative, network, and unresolved construct roles;
• stable between-unit differences from within-unit change;
• cross-classified and multiple-membership effects;
• event-time irregular lags from system/document-time spacing;
• measurement, prevalence, semantic, lexical, and method drift;
• posterior measurement uncertainty from ordinary sampling uncertainty;
• temporal precedence from causal identification.

Existing related work

• feat(psychometric): posterior ESEM input gates with true-parameter RMSE #49 provides posterior-coordinate and invariance admission gates.
• feat(longitudinal): keep unit means out of within-unit change #78/feat(temporal): space longitudinal lags on event time #80 provide within-between and irregular-time gates.
• feat(invariance): replay shared-meaning gate on current main #84 provides a bounded invariance replay.
• feat(membership): recover nested ICC and refuse cross-classified collapse #117 provides a nested ICC and rejects inappropriate collapsed designs.
• feat(psychometric): recover ESEM loadings and refuse reverse DSEM lags #119 provides OLS loading recovery and forward lag gates.
• feat(psychometric): recover multilevel event-time structure with Rubin T and strong means #144 contains many scalar continuous-time/evolved-mean recovery slices.

These are ingredients, not a complete ESEM/DSEM product. Prefer consolidation into the existing psychometric_core boundary over one-crate-per-equation fragmentation.

Acceptance criteria

• The exact first-release estimand, identification constraints, rotation/target structure, continuous/discrete-time assumptions, and uncertainty method are fixed in ADR/TRD equations with primary-source traceability.
• Inputs are posterior plausible values or a justified joint model; point topic estimates are never treated as error-free observations.
• The engine supports explicit cross-loadings and does not silently reclassify formative/network constructs as reflective because fit improves.
• Configural, metric, scalar where means are compared, residual/method, partial, and time-varying invariance statuses are tested and tied to the comparisons they license.
• Cross-classified and weighted multiple-membership structures are fitted rather than collapsed into a single hierarchy.
• Event-time lags are strictly forward; irregular intervals are modeled explicitly. System, document, assertion, or availability time cannot substitute for event time.
• Stable between-unit components and within-unit change are separated; atomistic pooled regressions are included as a negative comparison.
• Process noise, trait variance, residual variance, diffusion, manifest variance, and topic-posterior uncertainty remain distinct parameters.
• Known-truth simulations recover loadings, intercepts, factor means/variances, lag/drift parameters, membership effects, and uncertainty with computed RMSE, bias, interval coverage, convergence, and false-path rates.
• Monte Carlo gates use Monte Carlo standard error or intervals rather than a mechanically exact nominal pass rate.
• Causal wording is unavailable unless an explicit experimental, quasi-experimental, or defensible observational design contract is supplied.
• Results emit versioned factor/structural artifacts with model/data/cutoff/seed/backend/provenance identities and clear non-identification warnings.
• Production arithmetic is Rust CPU f64; parallel execution is bounded and deterministic where required. Python/R packages may be verification oracles only.
• Production statement/branch coverage and public docs remain 100%; exact-head scientific, security, migration, and supply-chain gates pass.

Product proof

Provide at least one realistic anonymized longitudinal scenario with repeated reports, multiple projects/organizations, irregular event times, language/template variation, and known input→process→outcome truth. Demonstrate that the production estimator recovers the declared parameters and that a pooled/atomistic alternative performs materially worse on the predefined metrics.

Scope boundary

This issue does not authorize causal claims from chronology, OLS fragments labeled as full DSEM, or a dashboard that hides failed invariance/coverage. A bounded scientifically honest first model is preferable to an unvalidated all-purpose SEM engine.

Authority

Follow the approved PRD §§5, 10, 14, ADR 0005, Test Strategy, and research register.