feat(psychometric): recover Eq. 5 of analog addedT0TIPREDVAR extra observed TD variance

[seonghobae]

Stacked on #206 (45c55f3) / #204 (b8b266d) / #202 (a7ce1a0) / #200 (8f42947) / #199 (7a476ed) / #198 (52a9b4c) / #196 (1cb91b4) / #194 (471b115) / #192 (1690c7b) / #190 (adb0bc6) / #189 (0836c35) / #188 (444304d) / #187 (6afd048) / #185 (69ffec6) / #184 (6b93147) / #183 (c10097be) / #182 (2d4d6bf) / #181 (542806b) / #49 (6f95142). ADR 0005 executable slice stays inside psychometric_core. This is not a second invariance crate and does not recreate #78 or #80.

psychometric_core recovers Equation 5 of the scalar analog of Driver, Oud, and Voelkle (2017, Eq. 5, p. 5; Table 2, p. 12 TDPREDVAR / T0TDPREDCOV; Table 3, p. 13 T0TIPREDEFFECT; p. 16; §7.2, pp. 20–21; 2017-era ctsem summary.ctsemFit.R; JSS PDF re-opened 2026-08-23T22:26Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) extra first-occasion time-dependent predictor variance. Equation 5 writes y_i(t) = Γ + Λ η_i(t) + ζ_i(t) with ζ ~ N(0, Θ) and Γ ~ N(τ, Ψ). The 2017-era summary.ctsemFit.R forms the latent extra addedT0TIPREDVAR as T0TIPREDEFFECT %*% TIPREDVAR %*% t(T0TIPREDEFFECT) immediately after T0TIPREDEFFECTstd. That file comments out TDPREDVAR and does not form addedT0TDPREDVAR. The scalar analog of that quadratic form using the stack's first-occasion TD coefficient t0_m and Table 2 TDPREDVAR v is t0_m² v. Equation 5 of that analog extra, with θ = 0 and ψ = 0, is λ² t0_m² v. Form the analog extra first, then (λ extra) λ. Do not form λ² first.

• A zero loading or zero extra is exactly zero. v < 0 fails closed.
• T0 is an event-time occasion, so a non-event clock fails closed. Free t0_m does not require stable a < 0.
• t0_m² v is the latent extra, not this observed extra.
• λ² p_0 + θ is first-occasion observed variance, not this extra.
• λ² t0_b² v is Eq. 5 of addedT0TIPREDVAR and is not this extra even when t0_m = t0_b.
• MANIFESTVAR θ is measurement error, not this extra.

Still not a Kalman filter, not a matrix expm, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T22:26Z: is_oa: false; title Measurement Invariance, Factor Analysis and Factorial Invariance). Mislevy (1991, Psychometrika, 56, 177–196) remains unread (Unpaywall 2026-08-23T22:26Z: is_oa: false; title Randomization-Based Inference about Latent Variables from Complex Samples).

Do not merge, self-approve, or request Copilot.

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[seonghobae]

Folded into the consolidation vehicle #231; this draft stays open until the vehicle merges, then closes as superseded-by-consolidation.

[seonghobae]

Superseded by consolidation: this slice's Driver et al. (2017) standardization recovery landed on protected main through the integration vehicles (#231/#232), and its contract coverage is carried by the reconciled suites (multilevel_event_time_recovery_contract, rubin_and_mean_gate_contract) with test-signature repair tracked in #234. The stacked-draft form is retired to keep the delivery queue at review-ready work only; no capability is lost — the exact-head provenance remains in the vehicle PR descriptions and CHANGELOG.