From 96e8d1c21924124d144f6273893eda8ac6b41b2d Mon Sep 17 00:00:00 2001 From: Seongho Bae Date: Sun, 23 Aug 2026 05:24:30 +0000 Subject: [PATCH] feat(psychometric): recover Driver later-occasion variance of predetermined T0VAR MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Map the Driver, Oud, and Voelkle (2017, Eq. 3–5 of §4.3 predetermined first occasion) later-occasion variance of free T0VAR as trait + e^{2aΔt} p_0 + Q_Δt + (B/a)²v. Trait and addedTIPREDVAR do not enter Q_Δt. Setting p_0 = −q/(2a) recovers the stationary later map. Stationary later variance, free discrete evolution of trait+p_0+added, and p_0 itself remain refused as this composition. Observed later variance is λ² of that map plus θ + ψ. Growing processes with a ≥ 0 are kept when the TI contribution is zero. --- ARCHITECTURE.md | 2 +- CHANGELOG.md | 1 + CLAUDE.md | 2 +- crates/psychometric_core/src/error.rs | 72 ++ crates/psychometric_core/src/event_time.rs | 959 ++++++++++++++++++ crates/psychometric_core/src/lib.rs | 38 + ...multilevel_event_time_recovery_contract.rs | 412 ++++++++ .../scientific_claim_boundary_contract.rs | 173 ++++ docs/TRACEABILITY.md | 2 +- docs/adr/0005-posterior-esem-dsem.md | 4 +- .../multilevel-event-time-recovery.md | 20 +- 11 files changed, 1673 insertions(+), 12 deletions(-) diff --git a/ARCHITECTURE.md b/ARCHITECTURE.md index 2f42564a..cd9f024d 100644 --- a/ARCHITECTURE.md +++ b/ARCHITECTURE.md @@ -61,7 +61,7 @@ boundaries above remain the target modular MSA architecture. | `tepp_simulation` | known-truth temporal/event data generation | | `validation_core` | RMSE, bias, coverage, graph, and Monte Carlo metrics | | `tepp_api` | versioned DTO, schema, and export contracts | -| `psychometric_core` | posterior-aware structural input gates, CWC within/between OLS plus the contextual effect, event-time log-rate, unequal-interval discrete-lag remapping, constant-predictor discrete effect, time-varying-predictor discrete effect (Eq. 14), exact scalar discrete process noise (Driver et al., 2017, Eq. 3), lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; `asymDIFFUSION`), trait-plus-state variance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise), observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5; Table 2 `MANIFESTVAR` is `Θ`, not `Var(y)`; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; `Θ` does not enter lagged observed covariance; observed-indicator mean is `τ + λ μ`; `MANIFESTMEANS` is not `E(y)`; `CINT` is not `MANIFESTMEANS`; discrete latent mean is `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ`; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment; evolved observed mean is `τ + λ μ_t`; `τ + λ μ_0` is not `E(y_t)`; contemporaneous `TDPREDEFFECT` impulse is `m x`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that contemporaneous impulse is `τ + λ(μ_t + m x)`, and `τ + λ μ_t` is not that observed mean; time-independent `TIPREDEFFECT` increment is `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, not `M x`, not Voelkle Eq. 14, and not the coefficient `B`; Eq. 5 of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; within-interval `TDPREDEFFECT` carry is `e^{A(t−u)} M x` for `t0 < u < t`, not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that carried observed mean when `u ≠ t`; first-occasion `T0TIPREDEFFECT` shift is `t0_b z` and Eq. 3 first-summand carry is `e^{A Δt} t0_b z` (`T0TIPREDEFFECT` is not `TIPREDEFFECT` `B`; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`; `e^{A Δt} t0_b z` is not `t0_b z`; Eq. 5 of that carry is `τ + λ(μ_t + e^{a Δt} t0_b z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean), first-occasion `T0TDPREDEFFECT` shift is `t0_m x0` and Eq. 3 first-summand carry is `e^{A Δt} t0_m x0` (`T0TDPREDEFFECT` is not `TDPREDEFFECT` `M`; `t0_m x0` is not `M x`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `e^{A Δt} t0_m x0` is not `e^{A(t−u)} M x` for `t0 < u < t`; `t0_m x0` is not `t0_b z`; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`; Eq. 5 of that carry is `τ + λ(μ_t + e^{a Δt} t0_m x0)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean; §7.2 level-change `CINT` is `κ = −a m x` with `a < 0` so `−κ / a = m x` (`−a m x` is not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`, and not the extra near-zero-drift latent process also named in §7.2; Eq. 3 of that setting is `(1 − e^{a Δt}) m x`, which is not `m x`, not `κ`, and not `TIPREDEFFECT`; §7.2 extra-process contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (`ε = a` is `a_{ηξ} x Δt e^{a Δt}`; identification `TDPREDEFFECT` on the extra process is 1; printed extra `DRIFT` is `−0.000001`; not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`; `ε ≥ 0` fails closed; Eq. 5 of that contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)`; the extra process has `LAMBDA` 0 and is not an observed indicator; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`; after-t0 extra-process `TDPREDEFFECT` is `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` while `μ_t` uses `Δt`; Eq. 5 of that after-t0 contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)`; the first-occasion extra-process observed mean is not that observed mean when `u ≠ t0`; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive; §7.2 `asymTIPREDEFFECT` is `-B z / a` for `a < 0` (`-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; §7.2 `addedTIPREDVAR` is `(B / a)² v`, not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`; Table 2 `asymCINT` is `-κ / a` for `a < 0` and is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; p. 16 stationary `T0MEANS` is `-κ / a + −B z / a` and is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean; Eq. 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)`; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`; stationary `T0VAR` is `trait + −q / (2 a) + (B / a)² v` (not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`); lagged stationary `T0VAR` is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (trait and `addedTIPREDVAR` do not decay; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map; Eq. 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`; `Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance); later-occasion stationary `T0VAR` is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt`; `Q_Δt` is not that later map; Eq. 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`; lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not `Var(y_t)`; the later-occasion latent variance is not `Var(y_t)`))), irregular already-centered residual lag, Rubin `T` on OLS loadings, and strong-gated latent means (two-observation residual variance is identically `0` and caps at strong/scalar; Putnick & Bornstein, 2016) | +| `psychometric_core` | posterior-aware structural input gates, CWC within/between OLS plus the contextual effect, event-time log-rate, unequal-interval discrete-lag remapping, constant-predictor discrete effect, time-varying-predictor discrete effect (Eq. 14), exact scalar discrete process noise (Driver et al., 2017, Eq. 3), lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; `asymDIFFUSION`), trait-plus-state variance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise), observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5; Table 2 `MANIFESTVAR` is `Θ`, not `Var(y)`; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; `Θ` does not enter lagged observed covariance; observed-indicator mean is `τ + λ μ`; `MANIFESTMEANS` is not `E(y)`; `CINT` is not `MANIFESTMEANS`; discrete latent mean is `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ`; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment; evolved observed mean is `τ + λ μ_t`; `τ + λ μ_0` is not `E(y_t)`; contemporaneous `TDPREDEFFECT` impulse is `m x`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that contemporaneous impulse is `τ + λ(μ_t + m x)`, and `τ + λ μ_t` is not that observed mean; time-independent `TIPREDEFFECT` increment is `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, not `M x`, not Voelkle Eq. 14, and not the coefficient `B`; Eq. 5 of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; within-interval `TDPREDEFFECT` carry is `e^{A(t−u)} M x` for `t0 < u < t`, not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that carried observed mean when `u ≠ t`; first-occasion `T0TIPREDEFFECT` shift is `t0_b z` and Eq. 3 first-summand carry is `e^{A Δt} t0_b z` (`T0TIPREDEFFECT` is not `TIPREDEFFECT` `B`; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`; `e^{A Δt} t0_b z` is not `t0_b z`; Eq. 5 of that carry is `τ + λ(μ_t + e^{a Δt} t0_b z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean), first-occasion `T0TDPREDEFFECT` shift is `t0_m x0` and Eq. 3 first-summand carry is `e^{A Δt} t0_m x0` (`T0TDPREDEFFECT` is not `TDPREDEFFECT` `M`; `t0_m x0` is not `M x`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `e^{A Δt} t0_m x0` is not `e^{A(t−u)} M x` for `t0 < u < t`; `t0_m x0` is not `t0_b z`; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`; Eq. 5 of that carry is `τ + λ(μ_t + e^{a Δt} t0_m x0)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean; §7.2 level-change `CINT` is `κ = −a m x` with `a < 0` so `−κ / a = m x` (`−a m x` is not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`, and not the extra near-zero-drift latent process also named in §7.2; Eq. 3 of that setting is `(1 − e^{a Δt}) m x`, which is not `m x`, not `κ`, and not `TIPREDEFFECT`; §7.2 extra-process contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (`ε = a` is `a_{ηξ} x Δt e^{a Δt}`; identification `TDPREDEFFECT` on the extra process is 1; printed extra `DRIFT` is `−0.000001`; not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`; `ε ≥ 0` fails closed; Eq. 5 of that contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)`; the extra process has `LAMBDA` 0 and is not an observed indicator; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`; after-t0 extra-process `TDPREDEFFECT` is `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` while `μ_t` uses `Δt`; Eq. 5 of that after-t0 contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)`; the first-occasion extra-process observed mean is not that observed mean when `u ≠ t0`; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive; §7.2 `asymTIPREDEFFECT` is `-B z / a` for `a < 0` (`-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; §7.2 `addedTIPREDVAR` is `(B / a)² v`, not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`; Table 2 `asymCINT` is `-κ / a` for `a < 0` and is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; p. 16 stationary `T0MEANS` is `-κ / a + −B z / a` and is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean; Eq. 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)`; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`; stationary `T0VAR` is `trait + −q / (2 a) + (B / a)² v` (not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`); lagged stationary `T0VAR` is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (trait and `addedTIPREDVAR` do not decay; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map; Eq. 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`; `Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance); later-occasion stationary `T0VAR` is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt`; `Q_Δt` is not that later map; Eq. 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`; lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not `Var(y_t)`; the later-occasion latent variance is not `Var(y_t)`); predetermined later-occasion `T0VAR` is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (free `T0VAR` `p_0` is not that later map; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map; stationary later variance uses `−q / (2 a)` in place of `p_0` and is not that later map when `p_0` is free; evolving `trait + p_0 + (B / a)² v` as if it were all state is not that later map; Eq. 5 of that predetermined later-occasion variance is `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`; `MANIFESTVAR` is not `Var(y_t)`; the predetermined later-occasion latent variance is not `Var(y_t)`; stationary later observed variance is not that observed variance when `p_0` is free))), irregular already-centered residual lag, Rubin `T` on OLS loadings, and strong-gated latent means (two-observation residual variance is identically `0` and caps at strong/scalar; Putnick & Bornstein, 2016) | No crate exposes placeholder production behavior in Task 1. This prevents an empty façade from becoming a de facto public API before its invariants and tests diff --git a/CHANGELOG.md b/CHANGELOG.md index cbf244cf..091da545 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -4,6 +4,7 @@ All notable changes to TEPP are documented here. The format follows Keep a Chang ## [Unreleased] +- `psychometric_core` recovers the Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–5, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T20:20Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) scalar later-occasion variance of §4.3 predetermined `T0VAR`. Section 4.3 treats the first time point as predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. The process gradually transitions from the variances of the initial parameters toward those of the parameters when the model is stationary. Equation 3 writes `η(t) = exp(A Δt) η(t0) + … +` the stochastic integral. Equation 4 writes that the integral exhibits covariance `Q_Δt`. The law of total variance on the within-subject state is `e^{2 a Δt} p_0 + Q_Δt`. Trait variance and `addedTIPREDVAR` are time-invariant between-subject and do not enter that process-noise integral. The later-occasion composition is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v`. Form the evolved free first-occasion variance first, then include the trait, then include the TI extra variance, then add. Setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map. Stationary later-occasion variance uses `−q / (2 a)` in place of `p_0` and is not this map when `p_0` is free. Evolving `trait + p_0 + (B / a)² v` as if it were all state (`e^{2 a Δt}` of that total plus `Q_Δt`) is not this map. Free `T0VAR` `p_0` is not this map. As `Δt → ∞` with stable `a < 0` the carried `p_0` vanishes and `Q_Δt` approaches `−q / (2 a)`, so the composition approaches contemporaneous stationary `T0VAR`. As `Δt → 0+` the composition approaches `trait + p_0 + (B / a)² v`. Nonzero diffusion with `a ≥ 0` is a growing process and is kept. The interval must be event time and strictly positive. Equation 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. `MANIFESTVAR` is not that later-occasion observed variance. The predetermined later-occasion latent variance is not the predetermined later-occasion observed variance. Stationary later-occasion observed variance is not that observed variance when `p_0` is free. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. Meredith (1993) remains unread (Unpaywall 2026-08-23T20:20Z: `is_oa: false`; title *Measurement Invariance, Factor Analysis and Factorial Invariance*). Mislevy (1991, *Psychometrika, 56*, 177–196) remains unread (Unpaywall 2026-08-23T20:20Z: `is_oa: false`; title *Randomization-Based Inference about Latent Variables from Complex Samples*). - Branch coverage JSON now unique-folds `files[].branches` True/False counts across instantiations. Nightly totals on #49 head `1e3e2eb` reported `event_time.rs` 505/506 while every unique site had both arms taken (253 sites × 2 instantiations). Summary-only reports without branch arrays still fail closed on totals. The 100% contract is unique production arms, matching the LCOV authored-line gate. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. - `psychometric_core` maps overflowing `expm1(a Δt)` / `expm1(2 a Δt)` in `recover_discrete_constant_predictor_effect` and `recover_discrete_process_noise` through the log-space rewrite without a redundant `if !argument.is_finite()` after overflow. Local crate llvm-cov on #49 head `559e7b399473ee90ba3234677dd9ef7f05f7fd2e` was 509/510: the same LLVM `exp`/`expm1` finite-argument proof as L768/L5040. Existing rewrite (`a = 800` / `a = 400`) and overflow (`a = 1e308`) tests remain the contract. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. - `psychometric_core` maps overflowing `e^{a Δt}` / `e^{a(t−u)}` through the log-space rewrite without redundant `if !argument.is_finite()` after `exp` overflow on lagged covariance, T0 TI/TD carry, and impulse carry. Nightly branch coverage on #49 head `7e669babcc54408dd8407bbac56be0f304fa99e5` was 1713/1714: LLVM counted `event_time.rs` L5040 True and treated the finite-argument overflow False as uncovered after proving `exp` of a finite argument is finite, which binary64 overflow falsifies. `fit_scalar_log_rate` now also skips a zero earlier residual and a negative lag while still recovering from a valid pair. Still not a Kalman filter, not a matrix `expm`, not ESEM estimation, not DSEM, and not ctsem estimation. diff --git a/CLAUDE.md b/CLAUDE.md index a7a04122..51f82589 100644 --- a/CLAUDE.md +++ b/CLAUDE.md @@ -15,7 +15,7 @@ Read and follow `AGENTS.md` before changing this repository. The repository-wide - Do not remove repeated report language with global stopword lists or use TF-IDF/BM25 as inferential weights. Model template, section, copied-text, style, modality, and corpus-background sources explicitly. - Do not treat raw topic proportions as ordinary Euclidean indicators. Use logistic-normal coordinates or valid log-ratio coordinates and propagate posterior uncertainty into ESEM/DSEM. - Do not treat metric/weak invariance as a latent-mean license. Strong (equal loading and intercept) or strict is required; `#84` `metric` licenses shared metric meaning only. Putnick and Bornstein (2016, PMC5145197 opened 2026-08-19T22:15Z) require scalar invariance before latent-mean comparison; residual invariance is not a prerequisite. Two-observation series have no residual degrees of freedom (`ordinary_least_squares_fit` returns residual variance `0`) and cap at strong/scalar; they still license means. This is two-group OLS, not MGCFA. Meredith (1993) names remain unread labels. -- Do not use the difference quotient as a continuous-time rate. The scalar map is `a = ln(φ) / Δt` on event time. Discrete lags from unequal event intervals are not one coefficient; remap them through that log-rate. Binary64 `exp(a Δt) = 0` is not a discrete lag. A constant predictor's discrete effect is Voelkle et al. (2012, Eq. 12), evaluated as `a_yx (expm1(z) / a_xx)` with `z = a_xx Δt` so a finite result is not lost when `z` overflows to `-∞` or when `a_yx Δt` overflows. When `expm1(z)` overflows at a finite `z`, rewrite in log space; a zero continuous effect is exactly zero; an overflowing `a_yx/a_xx` rewrite term fails closed. The first-order product is the underflow limit of that equation, not the general constant-predictor discrete effect. A time-varying predictor whose sampling interval equals its constancy interval uses Voelkle et al. (2012, Eq. 14): `b* = a_yx Δt`. Unmatched intervals fail closed (Oud & Jansen, 2000, unread). Discrete process noise is Driver et al. (2017, Eq. 3): `Q_Δt = 0.5 q (expm1(z) / a)` with `z = 2 (a Δt)` and `q = G G⊤ ≥ 0`; do not form `2 a` first; `a = 0` and `z → 0` recover `q Δt`; a zero diffusion is exactly zero; an overflowing rewrite scale `0.5 q / a` fails closed; this is not a Kalman filter. `Q_Δt` is `cov(η_t | η_{t-1})`, not `Var(η_t)`. The lagged covariance is `exp(a Δt) p` and the unconditional variance is `exp(2 a Δt) p + Q_Δt` (Driver et al., 2017, Eq. 3–4, pp. 4–5; JSS has no numbered §2.2). A zero diffusion whose `2 (a Δt)` overflows to `+∞` is not a finite `Var(η_t)`. The stationary within-subject variance is the `Δt → ∞` limit of Eq. 4: `-q / (2 a)` for stable `a < 0` (JSS p. 16 `asymDIFFUSION`; §4.3). When `2 a` is finite, form `q / -(2 a)` so `q / a` overflow does not lose a finite result (`q = MAX`, `a = -0.75` → `MAX / 1.5`). When `2 a` overflows, form `(q / a) * -0.5`. Do not form `0.5 q` first (`q = from_bits(1)` underflows). `a ≥ 0` has no finite stationary variance. Finite-interval `Q_Δt` is not that limit. Trait-plus-state variance is `trait + state` and lagged covariance is `trait + exp(a Δt) p` (Driver et al., 2017, §4.3, p. 9). Trait variance is not process noise and not `asymDIFFUSION`. Evolving the summed variance as if it were all state is not that map. This is not RI-CLPM. Observed-indicator variance is `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero and `λ² Var(η) + θ + ψ` otherwise (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12). Lagged observed covariance is `λ² cov(η_t, η_{t-1}) + ψ`; `MANIFESTVAR` does not enter. Observed-indicator mean is `τ + λ μ` (Driver et al., 2017, Eq. 5; Table 2, p. 12). `MANIFESTMEANS` is `τ`, not `E(y)`. `E(η)` is not `E(y)`. `CINT` is not `MANIFESTMEANS`. `T0MEANS` is not `E(y)`. The discrete latent mean is `μ_t = exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3, p. 4; Table 2, p. 12). `T0MEANS` is not `μ_t`. `CINT` is not that discrete increment. A zero drift is `κ Δt`. Underflow of `exp(a Δt)` to `+0` drops the carried `T0MEANS` and keeps `−κ / a`. The evolved observed mean is `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of that Eq. 3 map). The first-occasion map `τ + λ μ_0` is not `E(y_t)`. `μ_t` is not `E(y_t)`. The contemporaneous time-dependent predictor impulse is `m x` (Driver et al., 2017, Eq. 3 fourth summand; Table 2 `TDPREDEFFECT` is `M`). Form `μ_t` first, then add `m x`. `TDPREDEFFECT` is not `CINT`. `M x` is not `A^{-1}[e^{A Δt} − I] B z` and is not Voelkle et al. (2012, Eq. 14). The §7.2 level-change form is not that impulse. The observed mean of that contemporaneous impulse is `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of the Eq. 3 fourth-summand composition). The evolved map `τ + λ μ_t` is not that observed mean. The carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`. The evolved-plus-impulse latent mean is not `E(y_t)`. The time-independent predictor increment is `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 3 second summand; Table 2 `TIPREDEFFECT` is `B`). Form `B z` first, then the discrete intercept map. A zero drift is `B z Δt`. `TIPREDEFFECT` is `B`, not that discrete increment. `A^{-1}[e^{A Δt} − I] B z` is not `CINT`, not `M x`, and not Voelkle et al. (2012, Eq. 14). The observed mean of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of the Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment). The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`. The evolved-plus-increment latent mean is not `E(y_t)`. The within-interval time-dependent impulse carry is `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2 Green-function integral of Eq. 2; §7.2 dissipation). Form `m x` first, then `e^{a(t−u)} m x`. A zero drift is `m x` with no dissipation. Underflow of `e^{a(t−u)}` to `+0` is vanishing dissipation and is kept. `e^{A(t−u)} M x` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle et al. (2012, Eq. 14). An impulse at `u = t` is the contemporaneous map. An impulse at `u ≤ t0` is already in `η(t0)`. The observed mean of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of the Eq. 1–2 carried latent mean). The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean when `u ≠ t`. `MANIFESTMEANS` is not `E(y_t)`. The carried latent mean is not `E(y_t)`. The first-occasion time-independent predictor shift is `t0_b z` (Driver et al., 2017, Table 3 `T0TIPREDEFFECT`; Eq. 3 first summand). Form `t0_b z` first, then `e^{a Δt} t0_b z`. Form `μ_t` first, then add that carry. A zero drift is `t0_b z`. Underflow of `e^{a Δt}` to `+0` is a vanishing carry of the first-occasion shift and is kept. `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. `e^{A Δt} t0_b z` is not `t0_b z`. `T0TIPREDEFFECT` is the coefficient, not the shift. The observed mean of that first-occasion carry is `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand composition). The evolved map `τ + λ μ_t` is not that observed mean. The process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`. The evolved-plus-carry latent mean is not `E(y_t)`. The first-occasion time-dependent predictor shift is `t0_m x0` (Driver et al., 2017, Table 3 `T0TDPREDEFFECT`; Eq. 3 first summand; JSS PDF re-opened 2026-08-20T19:10Z). Form `t0_m x0` first, then `e^{a Δt} t0_m x0`. Form `μ_t` first, then add that carry. A zero drift is `t0_m x0`. Underflow of `e^{a Δt}` to `+0` is a vanishing carry of the first-occasion shift and is kept. `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `CINT`. `e^{A Δt} t0_m x0` is not `t0_m x0`. `T0TDPREDEFFECT` is the coefficient, not the shift. An impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`. The observed mean of that first-occasion TD carry is `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand TD composition; JSS PDF re-opened 2026-08-20T19:07Z). The evolved map `τ + λ μ_t` is not that observed mean. The process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`. The first-occasion TI map `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean. The evolved-plus-carry latent mean is not `E(y_t)`. The lasting level-change `CINT` is `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T19:45Z). Form `m x` first, then multiply by `−a`. Stable `a < 0` is required so `−κ / a = m x` is an equilibrium offset. `a ≥ 0` cannot hold a new process mean. `−a m x` is not the dissipating Dirac `m x`, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`. The extra near-zero-drift latent process also named in §7.2 is a different specification and is not this `CINT` setting. Equation 3 maps that intercept as `(1 − e^{a Δt}) m x` (JSS PDF re-opened 2026-08-20T19:50Z). Form the level-change `CINT` first, then the discrete intercept map. Underflow of `e^{a Δt}` to `+0` keeps `m x`. `(1 − e^{a Δt}) m x` is not `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`. The printed §7.2 lasting level change is an extra near-zero-drift latent process (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z). `T0MEANS`, `CINT`, `T0VAR`, `DIFFUSION`, and `TRAITVAR` of that process are fixed to 0; `TDPREDEFFECT` on it is fixed to 1; its `DRIFT` diagonal is very close to 0 (printed example `−0.000001`; precisely 0 causes computational problems); the original process is driven by the `DRIFT` coupling `a_{ηξ}`. After a unit identification impulse the scalar contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (`ε = a` is `a_{ηξ} x Δt e^{a Δt}`). Form `a_{ηξ} x` first. A zero coupling or zero predictor is exactly zero. `ε ≥ 0` fails closed. That contribution is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`. The observed mean of that extra-process contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5 of that §7.2 contribution; JSS PDF re-opened 2026-08-21T06:12Z). The extra process has `LAMBDA` 0 and is not an observed indicator. Original indicators load on the original process after the `DRIFT` coupling. The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The contribution is not `E(y_t)`. The evolved-plus-contribution latent mean is not `E(y_t)`. `T0TDPREDEFFECT` on the extra process begins at `t = 0` and uses `Δt = t − t0` for both the original-process evolution and the extra drive. `TDPREDEFFECT` after `t0` uses `t − u` with `t0 < u < t` for the extra drive while `μ_t` still uses `Δt`. The observed mean of that after-t0 extra-process contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, Eq. 5 of that §7.2 after-t0 contribution; JSS PDF re-opened 2026-08-21T06:32Z). The first-occasion extra-process observed mean is not that observed mean when `u ≠ t0`. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is a Dirac on the original process and is not that `DRIFT` drive. An impulse at `u = t0` or `u = t` is not interior. The asymptotic time-independent predictor effect is `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z). Form `B z` first, then divide by `-a`. Stable `a < 0` is required. `a ≥ 0` cannot hold a finite process-mean change. `-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. The asymptotic time-independent predictor variance is `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21 `addedTIPREDVAR`). Form the unit asymptotic effect first, then square, then multiply by `v`. `(B / a)² v` is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`. The asymptotic continuous intercept is `-κ / a` (Driver et al., 2017, Table 2, p. 12 `asymCINT`; Eq. 3 as `Δt → ∞`; JSS PDF opened 2026-08-21T16:13Z). Form `κ` first, then divide by `-a`. Stable `a < 0` is required. `-κ / a` is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`. The p. 16 stationary `T0MEANS` constraint is `-κ / a + −B z / a`. Form the intercept contribution first, then include the TI extra effect, then add. That constrained first-occasion mean is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean. Equation 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z). Form the stationary latent mean first, then `τ + λ` of that mean. `τ + λ μ_0` for free `T0MEANS` is not that composition. `τ + λ(−κ / a)` is not that composition when `B z ≠ 0`. `τ + λ μ_t` is not that composition. `MANIFESTMEANS` is not `E(y_0)`. The constrained latent mean is not `E(y_0)`. The p. 16 constrained first-occasion variance `trait + −q / (2 a) + (B / a)² v` is not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`). The lagged covariance of that constrained process is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z). Trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`. Contemporaneous `T0VAR` is not that lagged map. Decaying the constrained total as if it were all state is not that lagged map. Equation 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`. `Θ` does not enter. Contemporaneous `Var(y_0)` is not that lagged observed covariance. The lagged latent covariance is not that observed covariance. The later-occasion variance of that constrained process is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z). Trait and `addedTIPREDVAR` do not enter `Q_Δt`. Under stationarity that composition equals contemporaneous `T0VAR`. Evolving the constrained total as if it were all state is not that later map. The lagged covariance omits `Q_Δt` and is not that later map. `Q_Δt` is not that later map. Equation 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`. The lagged observed covariance omits `Q_Δt` and `θ`. `MANIFESTVAR` is not `Var(y_t)`. The later-occasion latent variance is not `Var(y_t)`. Evolving from that stationary start with `CINT` and `TIPREDEFFECT` stays at the stationary mean. Equation 1 is the latent SDE, not the measurement model. Form `(λ p) λ` then add `θ`, then add `ψ`. `MANIFESTVAR` is `Θ`, not `Var(y)`. `MANIFESTTRAITVAR` is `Ψ_τ`, not `Θ`. `TRAITVAR` is latent and scaled by `λ²`. `Var(η)` is not `Var(y)`. +- Do not use the difference quotient as a continuous-time rate. The scalar map is `a = ln(φ) / Δt` on event time. Discrete lags from unequal event intervals are not one coefficient; remap them through that log-rate. Binary64 `exp(a Δt) = 0` is not a discrete lag. A constant predictor's discrete effect is Voelkle et al. (2012, Eq. 12), evaluated as `a_yx (expm1(z) / a_xx)` with `z = a_xx Δt` so a finite result is not lost when `z` overflows to `-∞` or when `a_yx Δt` overflows. When `expm1(z)` overflows at a finite `z`, rewrite in log space; a zero continuous effect is exactly zero; an overflowing `a_yx/a_xx` rewrite term fails closed. The first-order product is the underflow limit of that equation, not the general constant-predictor discrete effect. A time-varying predictor whose sampling interval equals its constancy interval uses Voelkle et al. (2012, Eq. 14): `b* = a_yx Δt`. Unmatched intervals fail closed (Oud & Jansen, 2000, unread). Discrete process noise is Driver et al. (2017, Eq. 3): `Q_Δt = 0.5 q (expm1(z) / a)` with `z = 2 (a Δt)` and `q = G G⊤ ≥ 0`; do not form `2 a` first; `a = 0` and `z → 0` recover `q Δt`; a zero diffusion is exactly zero; an overflowing rewrite scale `0.5 q / a` fails closed; this is not a Kalman filter. `Q_Δt` is `cov(η_t | η_{t-1})`, not `Var(η_t)`. The lagged covariance is `exp(a Δt) p` and the unconditional variance is `exp(2 a Δt) p + Q_Δt` (Driver et al., 2017, Eq. 3–4, pp. 4–5; JSS has no numbered §2.2). A zero diffusion whose `2 (a Δt)` overflows to `+∞` is not a finite `Var(η_t)`. The stationary within-subject variance is the `Δt → ∞` limit of Eq. 4: `-q / (2 a)` for stable `a < 0` (JSS p. 16 `asymDIFFUSION`; §4.3). When `2 a` is finite, form `q / -(2 a)` so `q / a` overflow does not lose a finite result (`q = MAX`, `a = -0.75` → `MAX / 1.5`). When `2 a` overflows, form `(q / a) * -0.5`. Do not form `0.5 q` first (`q = from_bits(1)` underflows). `a ≥ 0` has no finite stationary variance. Finite-interval `Q_Δt` is not that limit. Trait-plus-state variance is `trait + state` and lagged covariance is `trait + exp(a Δt) p` (Driver et al., 2017, §4.3, p. 9). Trait variance is not process noise and not `asymDIFFUSION`. Evolving the summed variance as if it were all state is not that map. This is not RI-CLPM. Observed-indicator variance is `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero and `λ² Var(η) + θ + ψ` otherwise (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12). Lagged observed covariance is `λ² cov(η_t, η_{t-1}) + ψ`; `MANIFESTVAR` does not enter. Observed-indicator mean is `τ + λ μ` (Driver et al., 2017, Eq. 5; Table 2, p. 12). `MANIFESTMEANS` is `τ`, not `E(y)`. `E(η)` is not `E(y)`. `CINT` is not `MANIFESTMEANS`. `T0MEANS` is not `E(y)`. The discrete latent mean is `μ_t = exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3, p. 4; Table 2, p. 12). `T0MEANS` is not `μ_t`. `CINT` is not that discrete increment. A zero drift is `κ Δt`. Underflow of `exp(a Δt)` to `+0` drops the carried `T0MEANS` and keeps `−κ / a`. The evolved observed mean is `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of that Eq. 3 map). The first-occasion map `τ + λ μ_0` is not `E(y_t)`. `μ_t` is not `E(y_t)`. The contemporaneous time-dependent predictor impulse is `m x` (Driver et al., 2017, Eq. 3 fourth summand; Table 2 `TDPREDEFFECT` is `M`). Form `μ_t` first, then add `m x`. `TDPREDEFFECT` is not `CINT`. `M x` is not `A^{-1}[e^{A Δt} − I] B z` and is not Voelkle et al. (2012, Eq. 14). The §7.2 level-change form is not that impulse. The observed mean of that contemporaneous impulse is `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of the Eq. 3 fourth-summand composition). The evolved map `τ + λ μ_t` is not that observed mean. The carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`. The evolved-plus-impulse latent mean is not `E(y_t)`. The time-independent predictor increment is `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 3 second summand; Table 2 `TIPREDEFFECT` is `B`). Form `B z` first, then the discrete intercept map. A zero drift is `B z Δt`. `TIPREDEFFECT` is `B`, not that discrete increment. `A^{-1}[e^{A Δt} − I] B z` is not `CINT`, not `M x`, and not Voelkle et al. (2012, Eq. 14). The observed mean of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of the Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment). The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`. The evolved-plus-increment latent mean is not `E(y_t)`. The within-interval time-dependent impulse carry is `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2 Green-function integral of Eq. 2; §7.2 dissipation). Form `m x` first, then `e^{a(t−u)} m x`. A zero drift is `m x` with no dissipation. Underflow of `e^{a(t−u)}` to `+0` is vanishing dissipation and is kept. `e^{A(t−u)} M x` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle et al. (2012, Eq. 14). An impulse at `u = t` is the contemporaneous map. An impulse at `u ≤ t0` is already in `η(t0)`. The observed mean of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of the Eq. 1–2 carried latent mean). The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean when `u ≠ t`. `MANIFESTMEANS` is not `E(y_t)`. The carried latent mean is not `E(y_t)`. The first-occasion time-independent predictor shift is `t0_b z` (Driver et al., 2017, Table 3 `T0TIPREDEFFECT`; Eq. 3 first summand). Form `t0_b z` first, then `e^{a Δt} t0_b z`. Form `μ_t` first, then add that carry. A zero drift is `t0_b z`. Underflow of `e^{a Δt}` to `+0` is a vanishing carry of the first-occasion shift and is kept. `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. `e^{A Δt} t0_b z` is not `t0_b z`. `T0TIPREDEFFECT` is the coefficient, not the shift. The observed mean of that first-occasion carry is `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand composition). The evolved map `τ + λ μ_t` is not that observed mean. The process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`. The evolved-plus-carry latent mean is not `E(y_t)`. The first-occasion time-dependent predictor shift is `t0_m x0` (Driver et al., 2017, Table 3 `T0TDPREDEFFECT`; Eq. 3 first summand; JSS PDF re-opened 2026-08-20T19:10Z). Form `t0_m x0` first, then `e^{a Δt} t0_m x0`. Form `μ_t` first, then add that carry. A zero drift is `t0_m x0`. Underflow of `e^{a Δt}` to `+0` is a vanishing carry of the first-occasion shift and is kept. `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `CINT`. `e^{A Δt} t0_m x0` is not `t0_m x0`. `T0TDPREDEFFECT` is the coefficient, not the shift. An impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`. The observed mean of that first-occasion TD carry is `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand TD composition; JSS PDF re-opened 2026-08-20T19:07Z). The evolved map `τ + λ μ_t` is not that observed mean. The process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`. The first-occasion TI map `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean. The evolved-plus-carry latent mean is not `E(y_t)`. The lasting level-change `CINT` is `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T19:45Z). Form `m x` first, then multiply by `−a`. Stable `a < 0` is required so `−κ / a = m x` is an equilibrium offset. `a ≥ 0` cannot hold a new process mean. `−a m x` is not the dissipating Dirac `m x`, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`. The extra near-zero-drift latent process also named in §7.2 is a different specification and is not this `CINT` setting. Equation 3 maps that intercept as `(1 − e^{a Δt}) m x` (JSS PDF re-opened 2026-08-20T19:50Z). Form the level-change `CINT` first, then the discrete intercept map. Underflow of `e^{a Δt}` to `+0` keeps `m x`. `(1 − e^{a Δt}) m x` is not `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`. The printed §7.2 lasting level change is an extra near-zero-drift latent process (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z). `T0MEANS`, `CINT`, `T0VAR`, `DIFFUSION`, and `TRAITVAR` of that process are fixed to 0; `TDPREDEFFECT` on it is fixed to 1; its `DRIFT` diagonal is very close to 0 (printed example `−0.000001`; precisely 0 causes computational problems); the original process is driven by the `DRIFT` coupling `a_{ηξ}`. After a unit identification impulse the scalar contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (`ε = a` is `a_{ηξ} x Δt e^{a Δt}`). Form `a_{ηξ} x` first. A zero coupling or zero predictor is exactly zero. `ε ≥ 0` fails closed. That contribution is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`. The observed mean of that extra-process contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5 of that §7.2 contribution; JSS PDF re-opened 2026-08-21T06:12Z). The extra process has `LAMBDA` 0 and is not an observed indicator. Original indicators load on the original process after the `DRIFT` coupling. The evolved map `τ + λ μ_t` is not that observed mean. The contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean. The contribution is not `E(y_t)`. The evolved-plus-contribution latent mean is not `E(y_t)`. `T0TDPREDEFFECT` on the extra process begins at `t = 0` and uses `Δt = t − t0` for both the original-process evolution and the extra drive. `TDPREDEFFECT` after `t0` uses `t − u` with `t0 < u < t` for the extra drive while `μ_t` still uses `Δt`. The observed mean of that after-t0 extra-process contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, Eq. 5 of that §7.2 after-t0 contribution; JSS PDF re-opened 2026-08-21T06:32Z). The first-occasion extra-process observed mean is not that observed mean when `u ≠ t0`. The impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is a Dirac on the original process and is not that `DRIFT` drive. An impulse at `u = t0` or `u = t` is not interior. The asymptotic time-independent predictor effect is `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z). Form `B z` first, then divide by `-a`. Stable `a < 0` is required. `a ≥ 0` cannot hold a finite process-mean change. `-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. The asymptotic time-independent predictor variance is `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21 `addedTIPREDVAR`). Form the unit asymptotic effect first, then square, then multiply by `v`. `(B / a)² v` is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`. The asymptotic continuous intercept is `-κ / a` (Driver et al., 2017, Table 2, p. 12 `asymCINT`; Eq. 3 as `Δt → ∞`; JSS PDF opened 2026-08-21T16:13Z). Form `κ` first, then divide by `-a`. Stable `a < 0` is required. `-κ / a` is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`. The p. 16 stationary `T0MEANS` constraint is `-κ / a + −B z / a`. Form the intercept contribution first, then include the TI extra effect, then add. That constrained first-occasion mean is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean. Equation 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z). Form the stationary latent mean first, then `τ + λ` of that mean. `τ + λ μ_0` for free `T0MEANS` is not that composition. `τ + λ(−κ / a)` is not that composition when `B z ≠ 0`. `τ + λ μ_t` is not that composition. `MANIFESTMEANS` is not `E(y_0)`. The constrained latent mean is not `E(y_0)`. The p. 16 constrained first-occasion variance `trait + −q / (2 a) + (B / a)² v` is not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`). The lagged covariance of that constrained process is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z). Trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`. Contemporaneous `T0VAR` is not that lagged map. Decaying the constrained total as if it were all state is not that lagged map. Equation 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`. `Θ` does not enter. Contemporaneous `Var(y_0)` is not that lagged observed covariance. The lagged latent covariance is not that observed covariance. The later-occasion variance of that constrained process is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z). Trait and `addedTIPREDVAR` do not enter `Q_Δt`. Under stationarity that composition equals contemporaneous `T0VAR`. Evolving the constrained total as if it were all state is not that later map. The lagged covariance omits `Q_Δt` and is not that later map. `Q_Δt` is not that later map. Equation 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`. The lagged observed covariance omits `Q_Δt` and `θ`. `MANIFESTVAR` is not `Var(y_t)`. The later-occasion latent variance is not `Var(y_t)`. The later-occasion variance of §4.3 predetermined `T0VAR` is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 predetermined first occasion; JSS PDF re-opened 2026-08-23T20:20Z). Trait and `addedTIPREDVAR` do not enter `Q_Δt`. Free `T0VAR` `p_0` is not that later map. Setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map. Stationary later variance uses `−q / (2 a)` in place of `p_0` and is not that later map when `p_0` is free. Evolving `trait + p_0 + (B / a)² v` as if it were all state is not that later map. As `Δt → ∞` with stable `a < 0` the composition approaches contemporaneous stationary `T0VAR`. As `Δt → 0+` the composition approaches `trait + p_0 + (B / a)² v`. Nonzero diffusion with `a ≥ 0` is a growing process and is kept. Equation 5 of that predetermined later-occasion variance is `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. `MANIFESTVAR` is not `Var(y_t)`. The predetermined later-occasion latent variance is not `Var(y_t)`. Stationary later observed variance is not that observed variance when `p_0` is free. Evolving from that stationary start with `CINT` and `TIPREDEFFECT` stays at the stationary mean. Equation 1 is the latent SDE, not the measurement model. Form `(λ p) λ` then add `θ`, then add `ψ`. `MANIFESTVAR` is `Θ`, not `Var(y)`. `MANIFESTTRAITVAR` is `Ψ_τ`, not `Θ`. `TRAITVAR` is latent and scaled by `λ²`. `Var(η)` is not `Var(y)`. - Separate cluster means before within-unit lag. CWC plus an event-time lag is not DSEM. Subtracting the person-specific mean from a raw autoregressive series does not isolate the lagged within-person effect (Curran & Bauer, 2011, pp. 607–608); already-centered residuals with irregular event intervals use the exact scalar map. - Do not treat the CWC cluster-mean coefficient as the between-cluster effect. It is the contextual effect `between − within` (Enders & Tofighi, 2007, Table 2, pp. 124–127). - Never use future-available evidence in historical model fits. diff --git a/crates/psychometric_core/src/error.rs b/crates/psychometric_core/src/error.rs index bea873ad..17c32d0f 100644 --- a/crates/psychometric_core/src/error.rs +++ b/crates/psychometric_core/src/error.rs @@ -504,6 +504,29 @@ pub enum PsychometricError { /// later-occasion stationary observed variance. Lagged covariance /// omits `Q_Δt` and `θ`. StationaryLaggedObservedCovarianceIsNotStationaryLaterObservedVariance, + /// Driver §4.3 predetermined later-occasion variance was treated as + /// later-occasion stationary `T0VAR`. Free `T0VAR` is not + /// `−q / (2 a)`. + PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance, + /// Driver §4.3 predetermined later-occasion variance was treated as + /// the free discrete evolution of `trait + p_0 + (B / a)² v`. + /// Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. + PredeterminedLaterLatentVarianceIsNotDiscreteVariance, + /// Driver §4.3 predetermined later-occasion variance was treated as + /// free first-occasion `T0VAR`. `e^{2 a Δt} p_0 + Q_Δt` is not `p_0`. + PredeterminedLaterLatentVarianceIsNotInitialLatentVariance, + /// Driver §4.3 predetermined later-occasion variance was treated as + /// predetermined later-occasion observed variance. Equation 5 maps + /// `Var(y_t) = λ²` of that variance plus `θ + ψ`. + PredeterminedLaterLatentVarianceIsNotObservedVariance, + /// Driver Eq. 5 measurement error was treated as predetermined + /// later-occasion observed variance. `θ` is not + /// `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. + MeasurementErrorIsNotPredeterminedLaterObservedVariance, + /// Driver Eq. 5 of later-occasion §4.3 stationary `T0VAR` was treated + /// as predetermined later-occasion observed variance. Stationary + /// later variance uses `−q / (2 a)`, not free `p_0`. + StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance, } impl fmt::Display for PsychometricError { @@ -897,6 +920,24 @@ impl fmt::Display for PsychometricError { Self::StationaryLaggedObservedCovarianceIsNotStationaryLaterObservedVariance => { "stationary lagged observed covariance is not the stationary later-occasion observed variance" } + Self::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance => { + "predetermined later-occasion latent variance is not the stationary later-occasion latent variance" + } + Self::PredeterminedLaterLatentVarianceIsNotDiscreteVariance => { + "predetermined later-occasion latent variance is not the free discrete latent variance" + } + Self::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance => { + "predetermined later-occasion latent variance is not the free first-occasion latent variance" + } + Self::PredeterminedLaterLatentVarianceIsNotObservedVariance => { + "predetermined later-occasion latent variance is not the predetermined later-occasion observed variance" + } + Self::MeasurementErrorIsNotPredeterminedLaterObservedVariance => { + "measurement-error variance is not the predetermined later-occasion observed variance" + } + Self::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance => { + "stationary later-occasion observed variance is not the predetermined later-occasion observed variance" + } }; formatter.write_str(message) } @@ -1513,4 +1554,35 @@ mod tests { "stationary lagged observed covariance is not the stationary later-occasion observed variance" ); } + + #[test] + fn predetermined_later_variance_boundary_messages_are_stable() { + assert_eq!( + PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance + .to_string(), + "predetermined later-occasion latent variance is not the stationary later-occasion latent variance" + ); + assert_eq!( + PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance.to_string(), + "predetermined later-occasion latent variance is not the free discrete latent variance" + ); + assert_eq!( + PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance + .to_string(), + "predetermined later-occasion latent variance is not the free first-occasion latent variance" + ); + assert_eq!( + PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance.to_string(), + "predetermined later-occasion latent variance is not the predetermined later-occasion observed variance" + ); + assert_eq!( + PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance.to_string(), + "measurement-error variance is not the predetermined later-occasion observed variance" + ); + assert_eq!( + PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance + .to_string(), + "stationary later-occasion observed variance is not the predetermined later-occasion observed variance" + ); + } } diff --git a/crates/psychometric_core/src/event_time.rs b/crates/psychometric_core/src/event_time.rs index e5a889bc..76b18fd2 100644 --- a/crates/psychometric_core/src/event_time.rs +++ b/crates/psychometric_core/src/event_time.rs @@ -138,6 +138,28 @@ //! The lagged observed covariance omits `Q_Δt` and `θ`. `θ` is //! not that later-occasion observed variance. The later-occasion //! latent variance is not that observed variance. +//! The later-occasion variance of §4.3 predetermined `T0VAR` is +//! `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (Eq. 3–4 of §4.3 +//! predetermined first occasion; JSS PDF re-opened 2026-08-23T20:20Z). +//! Form the evolved free first-occasion variance first, then include +//! the trait, then include the TI extra variance, then add. Trait +//! variance and `addedTIPREDVAR` do not enter `Q_Δt`. Free `T0VAR` +//! `p_0` is not the later-occasion map. Setting `p_0 = −q / (2 a)` +//! recovers the stationary later-occasion map. Stationary later +//! variance uses `−q / (2 a)` in place of `p_0` and is not this map +//! when `p_0` is free. Evolving `trait + p_0 + (B / a)² v` as if it +//! were all state is not this map. As `Δt → ∞` with stable `a < 0` +//! the carried `p_0` vanishes and `Q_Δt` approaches `−q / (2 a)`, so +//! the composition approaches contemporaneous stationary `T0VAR`. +//! As `Δt → 0+` the composition approaches +//! `trait + p_0 + (B / a)² v`. Nonzero diffusion with `a ≥ 0` is a +//! growing process and is kept. Equation 5 of that predetermined +//! later-occasion variance is +//! `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. `θ` is +//! not that later-occasion observed variance. The predetermined +//! later-occasion latent variance is not that observed variance. +//! Stationary later observed variance is not that observed variance +//! when `p_0` is free. //! Table 3 (p. 13) names a different matrix //! `T0TIPREDEFFECT` for time-independent predictors on latents at //! `T0`. The scalar first-occasion shift is `t0_b z`. Equation 3's @@ -4033,6 +4055,274 @@ pub fn refuse_stationary_lagged_observed_covariance_as_stationary_later_observed Err(PsychometricError::StationaryLaggedObservedCovarianceIsNotStationaryLaterObservedVariance) } +/// Exact scalar later-occasion variance of §4.3 predetermined +/// `T0VAR`. +/// +/// Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; +/// Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened +/// 2026-08-23T20:20Z from +/// ) +/// treat the first time point as predetermined when no assumptions +/// are made about the process prior to the initial time point. Free +/// `T0VAR` `p_0` is then estimated. The process gradually +/// transitions from the variances of the initial parameters toward +/// those of the parameters when the model is stationary. Equation 3 +/// writes `η(t) = exp(A Δt) η(t0) + … +` the stochastic integral. +/// Equation 4 writes that the integral exhibits covariance `Q_Δt`. +/// The law of total variance on the within-subject state is +/// `e^{2 a Δt} p_0 + Q_Δt`. Trait variance and `addedTIPREDVAR` are +/// time-invariant between-subject; they do not enter that +/// process-noise integral. The later-occasion composition is +/// `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v`. Form the evolved +/// free first-occasion variance first, then include the trait, then +/// include the TI extra variance, then add. A zero trait, a zero +/// initial variance, a zero diffusion, and a zero TI contribution is +/// exactly zero. A zero diffusion, a zero initial variance, and a +/// zero TI contribution is exactly the trait. As `Δt → ∞` with +/// stable `a < 0` the carried `p_0` vanishes and `Q_Δt` approaches +/// `−q / (2 a)`, so the composition approaches contemporaneous +/// stationary `T0VAR`. As `Δt → 0+` the composition approaches +/// `trait + p_0 + (B / a)² v`. Setting `p_0 = −q / (2 a)` recovers +/// the stationary later-occasion map. Evolving +/// `trait + p_0 + (B / a)² v` as if it were all state +/// (`e^{2 a Δt}` of that total plus `Q_Δt`) is not this map. Free +/// `T0VAR` `p_0` is not this map. Stationary later-occasion +/// variance uses `−q / (2 a)` in place of `p_0` and is not this map +/// when `p_0` is free. `a ≥ 0` cannot hold a finite TI extra +/// variance when that contribution is nonzero and fails closed. +/// Nonzero diffusion with `a ≥ 0` is a growing process and is kept. +/// Trait-only variance does not require a stable drift. The +/// interval must be event time and strictly positive. This is not a +/// Kalman filter, not a matrix `expm`, and not ctsem estimation. +/// +/// # Errors +/// +/// Propagates [`recover_discrete_latent_variance`], +/// [`recover_trait_plus_state_latent_variance`], and +/// [`recover_asymptotic_time_independent_predictor_variance`]. +/// Returns [`PsychometricError::EventTimeRequired`] for any +/// non-event clock, +/// [`PsychometricError::NonPositiveInterval`] when `event_delta` is +/// not strictly positive, +/// [`PsychometricError::AsymptoticTimeIndependentEffectRequiresStableDrift`] +/// when the TI contribution is nonzero and the drift is not +/// strictly negative, and +/// [`PsychometricError::InvalidNumericInput`] when an input is +/// non-finite, a variance is negative, or a product or sum +/// overflows. +#[allow(clippy::too_many_arguments)] +pub fn recover_predetermined_later_latent_variance( + trait_variance: f64, + initial_latent_variance: f64, + continuous_diffusion: f64, + time_independent_effect: f64, + predictor_variance: f64, + log_rate: f64, + event_delta: f64, + clock: LagClock, +) -> Result { + if !clock.admits_structural_lag() { + return Err(PsychometricError::EventTimeRequired); + } + let evolved_state = recover_discrete_latent_variance( + initial_latent_variance, + continuous_diffusion, + log_rate, + event_delta, + clock, + )?; + let trait_plus_evolved = + recover_trait_plus_state_latent_variance(trait_variance, evolved_state)?; + let added = recover_asymptotic_time_independent_predictor_variance( + time_independent_effect, + predictor_variance, + log_rate, + clock, + )?; + require_finite(trait_plus_evolved + added) +} + +/// Refuse treating predetermined later-occasion variance as later- +/// occasion stationary `T0VAR`. +/// +/// `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` uses free `T0VAR`. +/// `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` uses the +/// stationary within-subject variance. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance`]. +pub fn refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance( + predetermined_later_variance: f64, + stationary_later_variance: f64, +) -> Result { + let _ = (predetermined_later_variance, stationary_later_variance); + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance) +} + +/// Refuse treating predetermined later-occasion variance as the free +/// discrete evolution of the total. +/// +/// Evolving `trait + p_0 + (B / a)² v` as if it were all state +/// yields `e^{2 a Δt}` of that total plus `Q_Δt`. Trait variance +/// and `addedTIPREDVAR` do not enter `Q_Δt`. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance`]. +pub fn refuse_predetermined_later_latent_variance_as_discrete_variance( + predetermined_later_variance: f64, + free_discrete_variance: f64, +) -> Result { + let _ = (predetermined_later_variance, free_discrete_variance); + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance) +} + +/// Refuse treating predetermined later-occasion variance as free +/// first-occasion `T0VAR`. +/// +/// `p_0` is the predetermined first-occasion state variance. +/// `e^{2 a Δt} p_0 + Q_Δt` is not `p_0`. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance`]. +pub fn refuse_predetermined_later_latent_variance_as_initial_latent_variance( + predetermined_later_variance: f64, + initial_latent_variance: f64, +) -> Result { + let _ = (predetermined_later_variance, initial_latent_variance); + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance) +} + +/// Exact scalar Eq. 5 of later-occasion §4.3 predetermined `T0VAR`. +/// +/// Driver, Oud, and Voelkle (2017, §4.3, pp. 9–10; Eq. 5, p. 5; +/// Eq. 3–4, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF +/// re-opened 2026-08-23T20:20Z from +/// ) +/// write `y_i(t) = Γ + Λ η_i(t) + ζ_i(t)` with `ζ ~ N(0, Θ)` and +/// `Γ ~ N(τ, Ψ)`. The predetermined later-occasion latent variance +/// is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v`. The scalar +/// composition is +/// `Var(y_t) = λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. +/// Form the predetermined later-occasion latent variance first, +/// then `λ² p + θ + ψ`. A zero loading is exactly `θ + ψ`. A zero +/// trait, a zero initial variance, a zero diffusion, and a zero TI +/// contribution is exactly `θ + ψ`. Setting `p_0 = −q / (2 a)` +/// recovers the stationary later-occasion observed variance. The +/// stationary later-occasion observed variance is not this +/// composition when `p_0` is free. `MANIFESTVAR` `θ` is not this +/// composition. The predetermined later-occasion latent variance is +/// not this observed variance. `TRAITVAR` is latent and is scaled +/// by `λ²`; `MANIFESTTRAITVAR` is not. This is not a Kalman filter, +/// not a matrix `expm`, and not ctsem estimation. +/// +/// # Errors +/// +/// Propagates [`recover_predetermined_later_latent_variance`] and +/// [`recover_manifest_trait_plus_state_observed_variance`]. +#[allow(clippy::too_many_arguments)] +pub fn recover_predetermined_later_observed_variance( + loading: f64, + trait_variance: f64, + initial_latent_variance: f64, + continuous_diffusion: f64, + time_independent_effect: f64, + predictor_variance: f64, + log_rate: f64, + event_delta: f64, + measurement_error_variance: f64, + manifest_trait_variance: f64, + clock: LagClock, +) -> Result { + let later_latent = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + continuous_diffusion, + time_independent_effect, + predictor_variance, + log_rate, + event_delta, + clock, + )?; + recover_manifest_trait_plus_state_observed_variance( + loading, + later_latent, + measurement_error_variance, + manifest_trait_variance, + ) +} + +/// Refuse treating predetermined later-occasion variance as +/// predetermined later-occasion observed variance. +/// +/// `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` is the +/// predetermined later-occasion latent variance. Equation 5 maps +/// `Var(y_t) = λ²` of that variance plus `θ + ψ`. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance`]. +pub fn refuse_predetermined_later_latent_variance_as_observed_variance( + predetermined_later_latent_variance: f64, + predetermined_later_observed_variance: f64, +) -> Result { + let _ = ( + predetermined_later_latent_variance, + predetermined_later_observed_variance, + ); + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance) +} + +/// Refuse treating `MANIFESTVAR` as Eq. 5 of predetermined later- +/// occasion `T0VAR`. +/// +/// Table 2 names `θ` `MANIFESTVAR`. `θ` is not +/// `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance`]. +pub fn refuse_measurement_error_as_predetermined_later_observed_variance( + measurement_error_variance: f64, + predetermined_later_observed_variance: f64, +) -> Result { + let _ = ( + measurement_error_variance, + predetermined_later_observed_variance, + ); + Err(PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance) +} + +/// Refuse treating Eq. 5 of later-occasion §4.3 stationary `T0VAR` +/// as predetermined later-occasion observed variance. +/// +/// `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` +/// uses the stationary within-subject variance. +/// `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` is not +/// that map when `p_0` is free. +/// +/// # Errors +/// +/// Always returns +/// [`PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance`]. +pub fn refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance( + stationary_later_observed_variance: f64, + predetermined_later_observed_variance: f64, +) -> Result { + let _ = ( + stationary_later_observed_variance, + predetermined_later_observed_variance, + ); + Err(PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance) +} + /// Exact scalar observed mean of a time-independent predictor. /// /// Driver, Oud, and Voelkle (2017, Eq. 5, p. 5; Eq. 3, p. 5; Table 2, @@ -5556,6 +5846,7 @@ mod tests { recover_level_change_extra_process_contribution_after, recover_local_log_rate, recover_manifest_lagged_observed_covariance, recover_manifest_observed_mean, recover_manifest_observed_variance, recover_manifest_trait_plus_state_observed_variance, + recover_predetermined_later_latent_variance, recover_predetermined_later_observed_variance, recover_stationary_initial_latent_mean, recover_stationary_initial_latent_variance, recover_stationary_initial_observed_mean, recover_stationary_initial_observed_variance, recover_stationary_lagged_latent_covariance, recover_stationary_lagged_observed_covariance, @@ -5632,9 +5923,14 @@ mod tests { refuse_manifest_trait_variance_as_measurement_error, refuse_measurement_error_as_lagged_observed_covariance, refuse_measurement_error_as_observed_variance, + refuse_measurement_error_as_predetermined_later_observed_variance, refuse_measurement_error_as_stationary_lagged_observed_covariance, refuse_measurement_error_as_stationary_later_observed_variance, refuse_pooled_discrete_lag_across_unequal_intervals, + refuse_predetermined_later_latent_variance_as_discrete_variance, + refuse_predetermined_later_latent_variance_as_initial_latent_variance, + refuse_predetermined_later_latent_variance_as_observed_variance, + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance, refuse_process_noise_as_unconditional_variance, refuse_stationary_initial_latent_mean_as_asymptotic_continuous_intercept, refuse_stationary_initial_latent_mean_as_asymptotic_time_independent_effect, @@ -5658,6 +5954,7 @@ mod tests { refuse_stationary_later_latent_variance_as_lagged_covariance, refuse_stationary_later_latent_variance_as_observed_variance, refuse_stationary_later_latent_variance_as_process_noise, + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance, refuse_stationary_within_subject_observed_variance_as_stationary_initial_observed_variance, refuse_time_dependent_impulse_as_continuous_intercept, refuse_time_dependent_impulse_as_time_independent_effect, @@ -11236,6 +11533,668 @@ mod tests { ); } + #[test] + #[allow(clippy::too_many_lines)] + fn predetermined_later_latent_variance_recovers_driver_section_four_point_three() { + // Driver et al. (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5) + // treat the first time point as predetermined. Free T0VAR p_0 + // then transitions toward stationarity: + // trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v. + let printed_effect = -0.225_f64; + let printed_asym = -1.673_f64; + let log_rate = -printed_effect / printed_asym; + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let predictor_variance = 1.0_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let evolved_state = recover_discrete_latent_variance( + initial_latent_variance, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}p_0+Q_Δt"); + let added = recover_asymptotic_time_independent_predictor_variance( + printed_effect, + predictor_variance, + log_rate, + LagClock::EventTime, + ) + .expect("addedTIPREDVAR"); + assert!((recovered - (trait_variance + evolved_state + added)).abs() < 1e-12); + let stationary_later = recover_stationary_later_latent_variance( + trait_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("stationary later T0VAR"); + assert!((recovered - stationary_later).abs() > 1e-3); + let state = recover_stationary_latent_variance(diffusion, log_rate, LagClock::EventTime) + .expect("asymDIFFUSION"); + let from_stationary_start = recover_predetermined_later_latent_variance( + trait_variance, + state, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("p_0=−q/(2a)"); + assert!((from_stationary_start - stationary_later).abs() < 1e-12); + let contemporaneous = recover_stationary_initial_latent_variance( + trait_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + LagClock::EventTime, + ) + .expect("stationary T0VAR"); + let first_occasion_total = trait_variance + initial_latent_variance + added; + let free_discrete = recover_discrete_latent_variance( + first_occasion_total, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}(trait+p_0+added)+Q_Δt"); + assert!((recovered - free_discrete).abs() > 1e-3); + assert!((recovered - initial_latent_variance).abs() > 1e-3); + let state_only = recover_predetermined_later_latent_variance( + 0.0, + initial_latent_variance, + diffusion, + 0.0, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("state-only predetermined later"); + assert!((state_only - evolved_state).abs() < 1e-15); + let trait_only = recover_predetermined_later_latent_variance( + trait_variance, + 0.0, + 0.0, + 0.0, + predictor_variance, + 0.0, + event_delta, + LagClock::EventTime, + ) + .expect("trait-only predetermined later"); + assert!((trait_only - trait_variance).abs() < 1e-15); + let added_only = recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("ti-only predetermined later"); + assert!((added_only - added).abs() < 1e-15); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + 0.0, + predictor_variance, + 0.0, + event_delta, + LagClock::EventTime + ), + Ok(0.0) + ); + let far = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + 1e8, + LagClock::EventTime, + ) + .expect("Δt→∞"); + assert!((far - contemporaneous).abs() < 1e-12); + let near = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + 1e-12, + LagClock::EventTime, + ) + .expect("Δt→0+"); + assert!((near - first_occasion_total).abs() < 1e-9); + let growing = recover_predetermined_later_latent_variance( + 0.0, + initial_latent_variance, + diffusion, + 0.0, + 0.0, + 0.5, + event_delta, + LagClock::EventTime, + ) + .expect("growing process"); + assert!(growing > initial_latent_variance); + } + + #[test] + fn predetermined_later_latent_variance_is_not_stationary_discrete_or_initial() { + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let log_rate = -0.134_488_942_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let stationary_later = recover_stationary_later_latent_variance( + trait_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("stationary later T0VAR"); + let added = recover_asymptotic_time_independent_predictor_variance( + -0.225, + 1.0, + log_rate, + LagClock::EventTime, + ) + .expect("addedTIPREDVAR"); + let free_discrete = recover_discrete_latent_variance( + trait_variance + initial_latent_variance + added, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}(trait+p_0+added)+Q_Δt"); + assert!((recovered - stationary_later).abs() > 1e-3); + assert!((recovered - free_discrete).abs() > 1e-3); + assert!((recovered - initial_latent_variance).abs() > 1e-3); + assert_eq!( + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance( + recovered, + stationary_later + ), + Err( + PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance + ) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_discrete_variance( + recovered, + free_discrete + ), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_initial_latent_variance( + recovered, + initial_latent_variance + ), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance) + ); + } + + #[test] + #[allow(clippy::too_many_lines)] + fn predetermined_later_latent_variance_invalid_inputs_fail_closed() { + assert_eq!( + recover_predetermined_later_latent_variance( + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 1.0, + LagClock::SystemTime + ), + Err(PsychometricError::EventTimeRequired) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::NonPositiveInterval) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + -0.225, + 1.0, + 0.5, + 1.0, + LagClock::EventTime + ), + Err(PsychometricError::AsymptoticTimeIndependentEffectRequiresStableDrift) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + 0.0, + 1.0, + 0.0, + 1.0, + LagClock::EventTime + ), + Ok(0.0) + ); + let growing = recover_predetermined_later_latent_variance( + 0.0, + 2.0, + 0.4, + 0.0, + 0.0, + 0.0, + 1.0, + LagClock::EventTime, + ) + .expect("Brownian a=0"); + assert!((growing - 2.4).abs() < 1e-12); + assert_eq!( + recover_predetermined_later_latent_variance( + f64::NAN, + 2.0, + 0.4, + 0.0, + 0.0, + -0.5, + 1.0, + LagClock::EventTime + ), + Err(PsychometricError::InvalidNumericInput) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + f64::MAX, + f64::MAX, + 0.0, + 0.0, + 0.0, + -0.5, + 1.0, + LagClock::EventTime + ), + Err(PsychometricError::InvalidNumericInput) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + f64::MAX, + 0.0, + 0.0, + 1.0, + f64::MAX, + -1.0, + 1.0, + LagClock::EventTime + ), + Err(PsychometricError::InvalidNumericInput) + ); + } + + #[test] + #[allow(clippy::too_many_lines)] + fn predetermined_later_observed_variance_recovers_driver_equation_five_of_section_four_point_three() + { + // Driver et al. (2017, §4.3, pp. 9–10; Eq. 5, p. 5) + // later-occasion observed variance of predetermined T0VAR is + // λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ. + let printed_effect = -0.225_f64; + let printed_asym = -1.673_f64; + let log_rate = -printed_effect / printed_asym; + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let loading = 2.0_f64; + let measurement_error = 0.5_f64; + let manifest_trait = 0.1_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_observed_variance( + loading, + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ) + .expect("eq5-later-predetermined-T0VAR"); + let latent = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let expected = recover_manifest_trait_plus_state_observed_variance( + loading, + latent, + measurement_error, + manifest_trait, + ) + .expect("λ²p+θ+ψ"); + assert!((recovered - expected).abs() < 1e-12); + let stationary_later = recover_stationary_later_observed_variance( + loading, + trait_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ) + .expect("eq5-later-stationary-T0VAR"); + assert!((recovered - stationary_later).abs() > 1e-3); + let state = recover_stationary_latent_variance(diffusion, log_rate, LagClock::EventTime) + .expect("asymDIFFUSION"); + let from_stationary_start = recover_predetermined_later_observed_variance( + loading, + trait_variance, + state, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ) + .expect("p_0=−q/(2a)"); + assert!((from_stationary_start - stationary_later).abs() < 1e-12); + assert!((recovered - measurement_error).abs() > 1e-3); + assert!((recovered - latent).abs() > 1e-3); + assert_eq!( + recover_predetermined_later_observed_variance( + 0.0, + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ), + Ok(measurement_error + manifest_trait) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + loading, + 0.0, + 0.0, + 0.0, + 0.0, + 1.0, + 0.0, + event_delta, + 0.0, + 0.0, + LagClock::EventTime, + ), + Ok(0.0) + ); + let zero_manifest_trait = recover_predetermined_later_observed_variance( + loading, + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + 0.0, + LagClock::EventTime, + ) + .expect("ψ=0"); + let expected_zero_psi = recover_manifest_trait_plus_state_observed_variance( + loading, + latent, + measurement_error, + 0.0, + ) + .expect("λ²p+θ"); + assert!((zero_manifest_trait - expected_zero_psi).abs() < 1e-12); + } + + #[test] + fn predetermined_later_observed_variance_is_not_manifest_latent_or_stationary() { + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let log_rate = -0.134_488_942_f64; + let loading = 2.0_f64; + let measurement_error = 0.5_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_observed_variance( + loading, + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + measurement_error, + 0.1, + LagClock::EventTime, + ) + .expect("eq5-later-predetermined-T0VAR"); + let latent = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let stationary_later = recover_stationary_later_observed_variance( + loading, + trait_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + measurement_error, + 0.1, + LagClock::EventTime, + ) + .expect("eq5-later-stationary-T0VAR"); + assert_eq!( + refuse_predetermined_later_latent_variance_as_observed_variance(latent, recovered), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance) + ); + assert_eq!( + refuse_measurement_error_as_predetermined_later_observed_variance( + measurement_error, + recovered + ), + Err(PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance) + ); + assert_eq!( + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance( + stationary_later, + recovered + ), + Err( + PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance + ) + ); + } + + #[test] + #[allow(clippy::too_many_lines)] + fn predetermined_later_observed_variance_invalid_inputs_fail_closed() { + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 1.0, + 0.5, + 0.1, + LagClock::SystemTime + ), + Err(PsychometricError::EventTimeRequired) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 0.0, + 0.5, + 0.1, + LagClock::EventTime + ), + Err(PsychometricError::NonPositiveInterval) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 0.0, + 0.0, + 0.0, + -0.225, + 1.0, + 0.5, + 1.0, + 0.0, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::AsymptoticTimeIndependentEffectRequiresStableDrift) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 0.0, + 0.0, + 0.0, + 0.0, + 1.0, + 0.0, + 1.0, + 0.5, + 0.1, + LagClock::EventTime + ), + Ok(0.6) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + f64::NAN, + 1.0, + 2.0, + 0.4, + 0.0, + 0.0, + -0.5, + 1.0, + 0.0, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::InvalidNumericInput) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + f64::MAX, + f64::MAX, + 0.0, + 0.0, + 0.0, + -0.5, + 1.0, + 0.0, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::InvalidNumericInput) + ); + } + #[test] fn discrete_observed_mean_with_impulse_recovers_driver_equation_five() { let loading = 2.0_f64; diff --git a/crates/psychometric_core/src/lib.rs b/crates/psychometric_core/src/lib.rs index 3d77eeb2..8b100910 100644 --- a/crates/psychometric_core/src/lib.rs +++ b/crates/psychometric_core/src/lib.rs @@ -185,6 +185,28 @@ //! (the lagged observed covariance omits `Q_Δt` and `θ`; //! `MANIFESTVAR` is not that later observed variance; the //! later-occasion latent variance is not that observed variance), +//! recovers the Driver §4.3 predetermined later-occasion variance as +//! `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` +//! (JSS PDF re-opened 2026-08-23T20:20Z; form the evolved free +//! first-occasion variance first, then include the trait, then +//! include the TI extra variance, then add; trait and +//! `addedTIPREDVAR` do not enter `Q_Δt`; free `T0VAR` `p_0` is not +//! that later map; setting `p_0 = −q / (2 a)` recovers the +//! stationary later-occasion map; stationary later variance uses +//! `−q / (2 a)` in place of `p_0` and is not that later map when +//! `p_0` is free; evolving `trait + p_0 + (B / a)² v` as if it were +//! all state is not that later map; as `Δt → ∞` with stable `a < 0` +//! the composition approaches contemporaneous stationary `T0VAR`; +//! as `Δt → 0+` the composition approaches +//! `trait + p_0 + (B / a)² v`; nonzero diffusion with `a ≥ 0` is a +//! growing process and is kept), +//! recovers the Driver Eq. 5 of that predetermined later-occasion +//! variance as +//! `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` +//! (`MANIFESTVAR` is not that later observed variance; the +//! predetermined later-occasion latent variance is not that observed +//! variance; stationary later observed variance is not that observed +//! variance when `p_0` is free), //! and refuses //! latent-mean comparison below strong invariance. @@ -321,6 +343,10 @@ pub use event_time::recover_manifest_observed_mean; pub use event_time::recover_manifest_observed_variance; /// Exact scalar observed-indicator variance `λ² Var(η) + θ + ψ`. pub use event_time::recover_manifest_trait_plus_state_observed_variance; +/// Exact scalar later-occasion variance of §4.3 predetermined `T0VAR` `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v`. +pub use event_time::recover_predetermined_later_latent_variance; +/// Exact scalar Eq. 5 of later-occasion §4.3 predetermined `T0VAR` `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`. +pub use event_time::recover_predetermined_later_observed_variance; /// Exact scalar p. 16 stationary `T0MEANS` `-κ / a + −B z / a`. pub use event_time::recover_stationary_initial_latent_mean; /// Exact scalar §4.3 / p. 16 stationary `T0VAR` `trait + −q / (2 a) + (B / a)² v`. @@ -495,12 +521,22 @@ pub use event_time::refuse_manifest_trait_variance_as_measurement_error; pub use event_time::refuse_measurement_error_as_lagged_observed_covariance; /// Refuse treating Driver Eq. 5 measurement error as `Var(y)`. pub use event_time::refuse_measurement_error_as_observed_variance; +/// Refuse treating `MANIFESTVAR` as Eq. 5 of predetermined later-occasion `T0VAR`. +pub use event_time::refuse_measurement_error_as_predetermined_later_observed_variance; /// Refuse treating `MANIFESTVAR` as Eq. 5 of lagged §4.3 stationary `T0VAR`. pub use event_time::refuse_measurement_error_as_stationary_lagged_observed_covariance; /// Refuse treating `MANIFESTVAR` as Eq. 5 of later-occasion §4.3 stationary `T0VAR`. pub use event_time::refuse_measurement_error_as_stationary_later_observed_variance; /// Refuse pooling discrete lags from unequal event intervals. pub use event_time::refuse_pooled_discrete_lag_across_unequal_intervals; +/// Refuse treating predetermined later-occasion variance as the free discrete evolution of the total. +pub use event_time::refuse_predetermined_later_latent_variance_as_discrete_variance; +/// Refuse treating predetermined later-occasion variance as free first-occasion `T0VAR`. +pub use event_time::refuse_predetermined_later_latent_variance_as_initial_latent_variance; +/// Refuse treating predetermined later-occasion variance as predetermined later-occasion observed variance. +pub use event_time::refuse_predetermined_later_latent_variance_as_observed_variance; +/// Refuse treating predetermined later-occasion variance as later-occasion stationary `T0VAR`. +pub use event_time::refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance; /// Refuse treating Driver Eq. 3 process noise as the unconditional variance. pub use event_time::refuse_process_noise_as_unconditional_variance; /// Refuse treating p. 16 stationary `T0MEANS` as `asymCINT`. @@ -547,6 +583,8 @@ pub use event_time::refuse_stationary_later_latent_variance_as_lagged_covariance pub use event_time::refuse_stationary_later_latent_variance_as_observed_variance; /// Refuse treating later-occasion §4.3 stationary `T0VAR` as finite-interval process noise. pub use event_time::refuse_stationary_later_latent_variance_as_process_noise; +/// Refuse treating Eq. 5 of later-occasion §4.3 stationary `T0VAR` as predetermined later-occasion observed variance. +pub use event_time::refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance; /// Refuse treating Eq. 5 of `asymDIFFUSION` as Eq. 5 of §4.3 stationary `T0VAR`. pub use event_time::refuse_stationary_within_subject_observed_variance_as_stationary_initial_observed_variance; /// Refuse treating Driver Eq. 3 `TDPREDEFFECT` impulse as `CINT`. diff --git a/crates/psychometric_core/tests/multilevel_event_time_recovery_contract.rs b/crates/psychometric_core/tests/multilevel_event_time_recovery_contract.rs index 0522220c..4bd9344c 100644 --- a/crates/psychometric_core/tests/multilevel_event_time_recovery_contract.rs +++ b/crates/psychometric_core/tests/multilevel_event_time_recovery_contract.rs @@ -34,6 +34,7 @@ use psychometric_core::{ recover_level_change_extra_process_contribution_after, recover_manifest_lagged_observed_covariance, recover_manifest_observed_mean, recover_manifest_observed_variance, recover_manifest_trait_plus_state_observed_variance, + recover_predetermined_later_latent_variance, recover_predetermined_later_observed_variance, recover_stationary_initial_latent_mean, recover_stationary_initial_latent_variance, recover_stationary_initial_observed_mean, recover_stationary_initial_observed_variance, recover_stationary_lagged_latent_covariance, recover_stationary_lagged_observed_covariance, @@ -107,9 +108,14 @@ use psychometric_core::{ refuse_manifest_means_as_observed_mean, refuse_manifest_trait_variance_as_measurement_error, refuse_measurement_error_as_lagged_observed_covariance, refuse_measurement_error_as_observed_variance, + refuse_measurement_error_as_predetermined_later_observed_variance, refuse_measurement_error_as_stationary_lagged_observed_covariance, refuse_measurement_error_as_stationary_later_observed_variance, refuse_pooled_discrete_lag_across_unequal_intervals, + refuse_predetermined_later_latent_variance_as_discrete_variance, + refuse_predetermined_later_latent_variance_as_initial_latent_variance, + refuse_predetermined_later_latent_variance_as_observed_variance, + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance, refuse_process_noise_as_unconditional_variance, refuse_stationary_initial_latent_mean_as_asymptotic_continuous_intercept, refuse_stationary_initial_latent_mean_as_asymptotic_time_independent_effect, @@ -133,6 +139,7 @@ use psychometric_core::{ refuse_stationary_later_latent_variance_as_lagged_covariance, refuse_stationary_later_latent_variance_as_observed_variance, refuse_stationary_later_latent_variance_as_process_noise, + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance, refuse_stationary_within_subject_observed_variance_as_stationary_initial_observed_variance, refuse_time_dependent_impulse_as_continuous_intercept, refuse_time_dependent_impulse_as_time_independent_effect, @@ -5848,3 +5855,408 @@ fn stationary_later_observed_variance_refuses_unstable_drift_and_non_event_clock Ok(0.6) ); } + +#[test] +#[allow(clippy::too_many_lines)] +fn predetermined_later_latent_variance_recovers_driver_section_four_point_three() { + let printed_effect = -0.225_f64; + let printed_asym = -1.673_f64; + let log_rate = -printed_effect / printed_asym; + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let predictor_variance = 1.0_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let evolved_state = recover_discrete_latent_variance( + initial_latent_variance, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}p_0+Q_Δt"); + let added = recover_asymptotic_time_independent_predictor_variance( + printed_effect, + predictor_variance, + log_rate, + LagClock::EventTime, + ) + .expect("addedTIPREDVAR"); + let expected = trait_variance + evolved_state + added; + let error = rmse(&[expected], &[recovered]); + assert!( + error < 1e-12, + "Driver §4.3 later-occasion predetermined T0VAR RMSE {error}: got {recovered}" + ); + let stationary_later = recover_stationary_later_latent_variance( + trait_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("stationary later T0VAR"); + let contemporaneous = recover_stationary_initial_latent_variance( + trait_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + LagClock::EventTime, + ) + .expect("stationary T0VAR"); + let first_occasion_total = trait_variance + initial_latent_variance + added; + let free_discrete = recover_discrete_latent_variance( + first_occasion_total, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}(trait+p_0+added)+Q_Δt"); + assert!(rmse(&[recovered], &[stationary_later]) > error); + assert!(rmse(&[recovered], &[free_discrete]) > error); + assert!(rmse(&[recovered], &[initial_latent_variance]) > error); + let state = recover_stationary_latent_variance(diffusion, log_rate, LagClock::EventTime) + .expect("asymDIFFUSION"); + let from_stationary_start = recover_predetermined_later_latent_variance( + trait_variance, + state, + diffusion, + printed_effect, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("p_0=−q/(2a)"); + assert!(rmse(&[from_stationary_start], &[stationary_later]) < 1e-12); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + 0.0, + predictor_variance, + log_rate, + event_delta, + LagClock::EventTime, + ), + Ok(0.0) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + trait_variance, + 0.0, + 0.0, + 0.0, + predictor_variance, + 0.0, + event_delta, + LagClock::EventTime, + ), + Ok(trait_variance) + ); + let far = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + predictor_variance, + log_rate, + 1e8, + LagClock::EventTime, + ) + .expect("Δt→∞"); + assert!(rmse(&[far], &[contemporaneous]) < 1e-12); + assert_eq!( + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance( + recovered, + stationary_later + ), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_discrete_variance(recovered, free_discrete), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_initial_latent_variance( + recovered, + initial_latent_variance + ), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance) + ); +} + +#[test] +fn predetermined_later_latent_variance_refuses_non_event_clocks_and_keeps_growing_processes() { + assert_eq!( + recover_predetermined_later_latent_variance( + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 1.0, + LagClock::SystemTime + ), + Err(PsychometricError::EventTimeRequired) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::NonPositiveInterval) + ); + let growing = recover_predetermined_later_latent_variance( + 0.0, + 2.0, + 0.4, + 0.0, + 1.0, + 0.0, + 1.0, + LagClock::EventTime, + ) + .expect("Brownian a=0"); + assert!((growing - 2.4).abs() < 1e-12); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + -0.225, + 1.0, + 0.5, + 1.0, + LagClock::EventTime + ), + Err(PsychometricError::AsymptoticTimeIndependentEffectRequiresStableDrift) + ); + assert_eq!( + recover_predetermined_later_latent_variance( + 0.0, + 0.0, + 0.0, + 0.0, + 1.0, + 0.0, + 1.0, + LagClock::EventTime + ), + Ok(0.0) + ); +} + +#[test] +#[allow(clippy::too_many_lines)] +fn predetermined_later_observed_variance_recovers_driver_equation_five_of_section_four_point_three() +{ + let printed_effect = -0.225_f64; + let printed_asym = -1.673_f64; + let log_rate = -printed_effect / printed_asym; + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let loading = 2.0_f64; + let measurement_error = 0.5_f64; + let manifest_trait = 0.1_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_observed_variance( + loading, + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ) + .expect("eq5-later-predetermined-T0VAR"); + let latent = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let expected = recover_manifest_trait_plus_state_observed_variance( + loading, + latent, + measurement_error, + manifest_trait, + ) + .expect("λ²p+θ+ψ"); + let error = rmse(&[expected], &[recovered]); + assert!( + error < 1e-12, + "Driver §4.3 Eq. 5 of later-occasion predetermined T0VAR RMSE {error}: got {recovered}" + ); + let stationary_later = recover_stationary_later_observed_variance( + loading, + trait_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ) + .expect("eq5-later-stationary-T0VAR"); + assert!(rmse(&[recovered], &[stationary_later]) > error); + assert!( + rmse(&[recovered], &[measurement_error]) > error, + "MANIFESTVAR is not predetermined later Var(y)" + ); + assert!(rmse(&[recovered], &[latent]) > error); + assert_eq!( + recover_predetermined_later_observed_variance( + 0.0, + trait_variance, + initial_latent_variance, + diffusion, + printed_effect, + 1.0, + log_rate, + event_delta, + measurement_error, + manifest_trait, + LagClock::EventTime, + ), + Ok(measurement_error + manifest_trait) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_observed_variance(latent, recovered), + Err(PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance) + ); + assert_eq!( + refuse_measurement_error_as_predetermined_later_observed_variance( + measurement_error, + recovered + ), + Err(PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance) + ); + assert_eq!( + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance( + stationary_later, + recovered + ), + Err( + PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance + ) + ); +} + +#[test] +fn predetermined_later_observed_variance_refuses_non_event_clocks_and_keeps_growing_processes() { + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 1.0, + 0.5, + 0.1, + LagClock::SystemTime + ), + Err(PsychometricError::EventTimeRequired) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 1.0, + 2.0, + 0.4, + -0.225, + 1.0, + -0.13, + 0.0, + 0.5, + 0.1, + LagClock::EventTime + ), + Err(PsychometricError::NonPositiveInterval) + ); + let growing = recover_predetermined_later_observed_variance( + 1.0, + 0.0, + 2.0, + 0.4, + 0.0, + 1.0, + 0.0, + 1.0, + 0.0, + 0.0, + LagClock::EventTime, + ) + .expect("Brownian a=0"); + assert!((growing - 2.4).abs() < 1e-12); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 0.0, + 0.0, + 0.0, + -0.225, + 1.0, + 0.5, + 1.0, + 0.0, + 0.0, + LagClock::EventTime + ), + Err(PsychometricError::AsymptoticTimeIndependentEffectRequiresStableDrift) + ); + assert_eq!( + recover_predetermined_later_observed_variance( + 2.0, + 0.0, + 0.0, + 0.0, + 0.0, + 1.0, + 0.0, + 1.0, + 0.5, + 0.1, + LagClock::EventTime + ), + Ok(0.6) + ); +} diff --git a/crates/psychometric_core/tests/scientific_claim_boundary_contract.rs b/crates/psychometric_core/tests/scientific_claim_boundary_contract.rs index 769ec14a..f98f059f 100644 --- a/crates/psychometric_core/tests/scientific_claim_boundary_contract.rs +++ b/crates/psychometric_core/tests/scientific_claim_boundary_contract.rs @@ -29,6 +29,7 @@ use psychometric_core::{ recover_level_change_extra_process_contribution_after, recover_loading_point_estimate_mean, recover_manifest_lagged_observed_covariance, recover_manifest_observed_mean, recover_manifest_observed_variance, recover_manifest_trait_plus_state_observed_variance, + recover_predetermined_later_latent_variance, recover_predetermined_later_observed_variance, recover_stationary_initial_latent_mean, recover_stationary_initial_latent_variance, recover_stationary_initial_observed_mean, recover_stationary_initial_observed_variance, recover_stationary_lagged_latent_covariance, recover_stationary_lagged_observed_covariance, @@ -102,8 +103,13 @@ use psychometric_core::{ refuse_manifest_means_as_observed_mean, refuse_manifest_trait_variance_as_measurement_error, refuse_measurement_error_as_lagged_observed_covariance, refuse_measurement_error_as_observed_variance, + refuse_measurement_error_as_predetermined_later_observed_variance, refuse_measurement_error_as_stationary_lagged_observed_covariance, refuse_measurement_error_as_stationary_later_observed_variance, + refuse_predetermined_later_latent_variance_as_discrete_variance, + refuse_predetermined_later_latent_variance_as_initial_latent_variance, + refuse_predetermined_later_latent_variance_as_observed_variance, + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance, refuse_process_noise_as_unconditional_variance, refuse_stationary_initial_latent_mean_as_asymptotic_continuous_intercept, refuse_stationary_initial_latent_mean_as_asymptotic_time_independent_effect, @@ -127,6 +133,7 @@ use psychometric_core::{ refuse_stationary_later_latent_variance_as_lagged_covariance, refuse_stationary_later_latent_variance_as_observed_variance, refuse_stationary_later_latent_variance_as_process_noise, + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance, refuse_stationary_within_subject_observed_variance_as_stationary_initial_observed_variance, refuse_time_dependent_impulse_as_continuous_intercept, refuse_time_dependent_impulse_as_time_independent_effect, @@ -2973,3 +2980,169 @@ fn stationary_later_observed_variance_is_not_manifest_latent_or_lagged() { ) ); } + +#[test] +fn predetermined_later_latent_variance_is_not_stationary_discrete_or_initial() { + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let log_rate = -0.134_488_942_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let stationary_later = recover_stationary_later_latent_variance( + trait_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("stationary later T0VAR"); + let added = recover_asymptotic_time_independent_predictor_variance( + -0.225, + 1.0, + log_rate, + LagClock::EventTime, + ) + .expect("addedTIPREDVAR"); + let free_discrete = recover_discrete_latent_variance( + trait_variance + initial_latent_variance + added, + diffusion, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("e^{2aΔt}(trait+p_0+added)+Q_Δt"); + assert!( + (recovered - stationary_later).abs() > 1e-3, + "Driver et al. (2017, §4.3 predetermined T0VAR): free p_0 is not −q/(2a)" + ); + assert!( + (recovered - free_discrete).abs() > 1e-3, + "Driver et al. (2017, §4.3 predetermined T0VAR): trait and addedTIPREDVAR do not enter Q_Δt" + ); + assert!( + (recovered - initial_latent_variance).abs() > 1e-3, + "Driver et al. (2017, §4.3 predetermined T0VAR): e^{{2aΔt}}p_0+Q_Δt is not p_0" + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_stationary_later_latent_variance( + recovered, + stationary_later + ), + Err( + psychometric_core::PsychometricError::PredeterminedLaterLatentVarianceIsNotStationaryLaterLatentVariance + ) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_discrete_variance(recovered, free_discrete), + Err( + psychometric_core::PsychometricError::PredeterminedLaterLatentVarianceIsNotDiscreteVariance + ) + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_initial_latent_variance( + recovered, + initial_latent_variance + ), + Err( + psychometric_core::PsychometricError::PredeterminedLaterLatentVarianceIsNotInitialLatentVariance + ) + ); +} + +#[test] +fn predetermined_later_observed_variance_is_not_manifest_latent_or_stationary() { + let trait_variance = 1.0_f64; + let initial_latent_variance = 2.0_f64; + let diffusion = 0.4_f64; + let log_rate = -0.134_488_942_f64; + let loading = 2.0_f64; + let measurement_error = 0.5_f64; + let event_delta = 1.0_f64; + let recovered = recover_predetermined_later_observed_variance( + loading, + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + measurement_error, + 0.1, + LagClock::EventTime, + ) + .expect("eq5-later-predetermined-T0VAR"); + let latent = recover_predetermined_later_latent_variance( + trait_variance, + initial_latent_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + LagClock::EventTime, + ) + .expect("predetermined later T0VAR"); + let stationary_later = recover_stationary_later_observed_variance( + loading, + trait_variance, + diffusion, + -0.225, + 1.0, + log_rate, + event_delta, + measurement_error, + 0.1, + LagClock::EventTime, + ) + .expect("eq5-later-stationary-T0VAR"); + assert!( + (recovered - measurement_error).abs() > 1e-3, + "Driver et al. (2017, Eq. 5 of predetermined later §4.3 T0VAR): Var(y_t) is not MANIFESTVAR" + ); + assert!( + (recovered - latent).abs() > 1e-3, + "Driver et al. (2017, Eq. 5 of predetermined later §4.3 T0VAR): Var(y_t) is not later T0VAR" + ); + assert!( + (recovered - stationary_later).abs() > 1e-3, + "Driver et al. (2017, Eq. 5 of predetermined later §4.3 T0VAR): free p_0 is not −q/(2a)" + ); + assert_eq!( + refuse_predetermined_later_latent_variance_as_observed_variance(latent, recovered), + Err( + psychometric_core::PsychometricError::PredeterminedLaterLatentVarianceIsNotObservedVariance + ) + ); + assert_eq!( + refuse_measurement_error_as_predetermined_later_observed_variance( + measurement_error, + recovered + ), + Err( + psychometric_core::PsychometricError::MeasurementErrorIsNotPredeterminedLaterObservedVariance + ) + ); + assert_eq!( + refuse_stationary_later_observed_variance_as_predetermined_later_observed_variance( + stationary_later, + recovered + ), + Err( + psychometric_core::PsychometricError::StationaryLaterObservedVarianceIsNotPredeterminedLaterObservedVariance + ) + ); +} diff --git a/docs/TRACEABILITY.md b/docs/TRACEABILITY.md index 50dcf2a8..42349181 100644 --- a/docs/TRACEABILITY.md +++ b/docs/TRACEABILITY.md @@ -28,7 +28,7 @@ The full APA 7th standards/literature register remains `docs/research/standards- | report template/section/copied/style/modality method effects | ADR 0004/0012; PRD/TRD | simulation truth factors implemented; estimator-side method model remains future | partial | | candidate K statistical/Pareto gates + blinded LLM review | ADR 0012; research | future `model_selection` | accepted-target | | compositional topic correlation / stable clustering | ADR 0005/0012; research | future `network_analysis` | accepted-target | -| posterior ESEM / longitudinal invariance / DSEM | ADR 0005 | `psychometric_core` construct/input gates, true-loading OLS recovery, posterior-draw point-estimate averaging, Rubin `T` on draw-level OLS loadings, CWC within/between OLS plus the contextual effect, event-time log-rate, constant- and time-varying-predictor discrete effects (Voelkle Eqs. 12 and 14), exact scalar discrete process noise (Driver et al., 2017, Eq. 3), lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; `asymDIFFUSION`), trait-plus-state variance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise), observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5; Table 2 `MANIFESTVAR` is `Θ`, not `Var(y)`; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; `Θ` does not enter lagged observed covariance; observed-indicator mean is `τ + λ μ`; `MANIFESTMEANS` is not `E(y)`; `CINT` is not `MANIFESTMEANS`; discrete latent mean is `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ`; `T0MEANS` is not `μ_t`; evolved observed mean is `τ + λ μ_t`; `τ + λ μ_0` is not `E(y_t)`; contemporaneous `TDPREDEFFECT` impulse is `m x`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that contemporaneous impulse is `τ + λ(μ_t + m x)`, and `τ + λ μ_t` is not that observed mean; time-independent `TIPREDEFFECT` increment is `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, not `M x`, not Voelkle Eq. 14, and not the coefficient `B`; Eq. 5 of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; within-interval `TDPREDEFFECT` carry is `e^{A(t−u)} M x` for `t0 < u < t`, not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that carried observed mean when `u ≠ t`; §7.2 level-change `CINT` is `κ = −a m x` (`a < 0`; not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`; Eq. 3 of that setting is `(1 − e^{a Δt}) m x`); §7.2 extra-process contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, not the dissipating Dirac; `ε ≥ 0` fails closed; Eq. 5 of that contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)`; extra `LAMBDA` is 0; `τ + λ μ_t` is not that observed mean; after-t0 extra-process `TDPREDEFFECT` uses `t − u` with `t0 < u < t` while `μ_t` uses `Δt`; that after-t0 observed mean is not the first-occasion extra-process observed mean; §7.2 `asymTIPREDEFFECT` is `-B z / a` for `a < 0` and is not `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; §7.2 `addedTIPREDVAR` is `(B / a)² v` and is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`; Table 2 `asymCINT` is `-κ / a` for `a < 0` and is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; p. 16 stationary `T0MEANS` is `-κ / a + −B z / a` and is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean; Eq. 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)`; `τ + λ μ_0` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`; stationary `T0VAR` is `trait + −q / (2 a) + (B / a)² v` (not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`)); lagged stationary `T0VAR` is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (trait and `addedTIPREDVAR` do not decay; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map; Eq. 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`; `Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance); later-occasion stationary `T0VAR` is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt`; `Q_Δt` is not that later map; Eq. 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`; lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not `Var(y_t)`; the later-occasion latent variance is not `Var(y_t)`)), irregular already-centered residual lag, and strong/strict-gated latent means on the stacked psychometric PR (two-observation residual variance is identically `0` and caps at strong/scalar; Putnick & Bornstein, 2016, PMC5145197 opened 2026-08-19T22:15Z); full ESEM/DSEM remaining | partial | +| posterior ESEM / longitudinal invariance / DSEM | ADR 0005 | `psychometric_core` construct/input gates, true-loading OLS recovery, posterior-draw point-estimate averaging, Rubin `T` on draw-level OLS loadings, CWC within/between OLS plus the contextual effect, event-time log-rate, constant- and time-varying-predictor discrete effects (Voelkle Eqs. 12 and 14), exact scalar discrete process noise (Driver et al., 2017, Eq. 3), lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; `asymDIFFUSION`), trait-plus-state variance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise), observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5; Table 2 `MANIFESTVAR` is `Θ`, not `Var(y)`; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; `Θ` does not enter lagged observed covariance; observed-indicator mean is `τ + λ μ`; `MANIFESTMEANS` is not `E(y)`; `CINT` is not `MANIFESTMEANS`; discrete latent mean is `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ`; `T0MEANS` is not `μ_t`; evolved observed mean is `τ + λ μ_t`; `τ + λ μ_0` is not `E(y_t)`; contemporaneous `TDPREDEFFECT` impulse is `m x`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that contemporaneous impulse is `τ + λ(μ_t + m x)`, and `τ + λ μ_t` is not that observed mean; time-independent `TIPREDEFFECT` increment is `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, not `M x`, not Voelkle Eq. 14, and not the coefficient `B`; Eq. 5 of that increment is `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; within-interval `TDPREDEFFECT` carry is `e^{A(t−u)} M x` for `t0 < u < t`, not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; Eq. 5 of that carry is `τ + λ(μ_t + e^{a(t−u)} m x)`, and `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that carried observed mean when `u ≠ t`; §7.2 level-change `CINT` is `κ = −a m x` (`a < 0`; not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`; Eq. 3 of that setting is `(1 − e^{a Δt}) m x`); §7.2 extra-process contribution is `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, not the dissipating Dirac; `ε ≥ 0` fails closed; Eq. 5 of that contribution is `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)`; extra `LAMBDA` is 0; `τ + λ μ_t` is not that observed mean; after-t0 extra-process `TDPREDEFFECT` uses `t − u` with `t0 < u < t` while `μ_t` uses `Δt`; that after-t0 observed mean is not the first-occasion extra-process observed mean; §7.2 `asymTIPREDEFFECT` is `-B z / a` for `a < 0` and is not `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; §7.2 `addedTIPREDVAR` is `(B / a)² v` and is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`; Table 2 `asymCINT` is `-κ / a` for `a < 0` and is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; p. 16 stationary `T0MEANS` is `-κ / a + −B z / a` and is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean; Eq. 5 of that constrained mean is `τ + λ(−κ / a + −B z / a)`; `τ + λ μ_0` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`; stationary `T0VAR` is `trait + −q / (2 a) + (B / a)² v` (not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`)); lagged stationary `T0VAR` is `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (trait and `addedTIPREDVAR` do not decay; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map; Eq. 5 of that lagged covariance is `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ`; `Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance); later-occasion stationary `T0VAR` is `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt`; `Q_Δt` is not that later map; Eq. 5 of that later-occasion variance is `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ`; lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not `Var(y_t)`; the later-occasion latent variance is not `Var(y_t)`); predetermined later-occasion `T0VAR` is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (free `T0VAR` `p_0` is not that later map; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map; stationary later variance uses `−q / (2 a)` in place of `p_0` and is not that later map when `p_0` is free; evolving `trait + p_0 + (B / a)² v` as if it were all state is not that later map; Eq. 5 of that predetermined later-occasion variance is `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ`; `MANIFESTVAR` is not `Var(y_t)`; the predetermined later-occasion latent variance is not `Var(y_t)`; stationary later observed variance is not that observed variance when `p_0` is free)), irregular already-centered residual lag, and strong/strict-gated latent means on the stacked psychometric PR (two-observation residual variance is identically `0` and caps at strong/scalar; Putnick & Bornstein, 2016, PMC5145197 opened 2026-08-19T22:15Z); full ESEM/DSEM remaining | partial | | CPU bounded multithreading + GPU/VRAM streaming/parity | ADR 0001/0006 | future `compute_backend` | accepted-target | | TDT detection/tracking vs CHRONOS schema/prediction/temporal consistency | ADR 0016; PRD/research | future `event_intelligence` | accepted-target | | evidence-bounded LLM interpretation | ADR 0010/0012; PRD | `tepp_api` router plus future `interpretation_gateway` | partial | diff --git a/docs/adr/0005-posterior-esem-dsem.md b/docs/adr/0005-posterior-esem-dsem.md index 936510f4..e59db25e 100644 --- a/docs/adr/0005-posterior-esem-dsem.md +++ b/docs/adr/0005-posterior-esem-dsem.md @@ -1,7 +1,7 @@ # ADR 0005 — Posterior-aware ESEM/DSEM and structural interpretation **Decision status:** Accepted -**Implementation maturity:** partial — construct classification, valid log-ratio/logistic-normal indicator gates, CPU `f64` OLS and posterior-draw loading point-estimate averaging, Rubin `T_m = Ū_m + (1+1/m) B_m` on draw-level OLS loadings, cluster-mean within/between OLS with the CWC contextual effect and Kish ESS WLS, event-time discrete lag-1 and exact scalar local log-rate, exact scalar forward map and unequal-interval remapping, exact scalar discrete effect of a constant predictor, first-order discrete effect of a time-varying predictor with matched sampling and constancy intervals (Voelkle et al., 2012, Eq. 14), exact scalar discrete process noise (Driver, Oud, & Voelkle, 2017, Eq. 3), exact scalar lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), exact scalar stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; §4.3; p. 16 `asymDIFFUSION`), exact scalar trait-plus-state variance and lagged covariance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise and not `asymDIFFUSION`), exact scalar observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero, else `λ² Var(η) + θ + ψ`; lagged `λ² cov(η_t, η_{t-1}) + ψ`; `MANIFESTVAR` is `Θ`, not `Var(y)`; `Θ` does not enter lagged observed covariance; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; observed-indicator mean is `τ + λ μ` (`MANIFESTMEANS` is `τ`, not `E(y)`; `CINT` is not `MANIFESTMEANS`; `T0MEANS` is not `E(y)`; Equation 1 is the SDE; not a Kalman filter), exact scalar discrete latent mean `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment), exact scalar evolved observed-indicator mean `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of that Eq. 3 map; the first-occasion map `τ + λ μ_0` is not `E(y_t)`), exact scalar contemporaneous `TDPREDEFFECT` impulse `m x` (Driver et al., 2017, Eq. 3 fourth summand; Table 2 `TDPREDEFFECT` is `M`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; the §7.2 level-change form is not that impulse), exact scalar observed mean of that contemporaneous impulse `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of that Eq. 3 composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`), exact scalar time-independent `TIPREDEFFECT` increment `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 3 second summand; Table 2 `TIPREDEFFECT` is `B`, not `κ`, not `M`, and not Voelkle Eq. 14; `B` is not that discrete increment), exact scalar observed mean of that increment `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of that Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`), exact scalar within-interval `TDPREDEFFECT` carry `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2 Green-function integral of Eq. 2; §7.2 dissipation; not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14), exact scalar observed mean of that carry `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of that carried latent mean; `τ + λ μ_t` is not that observed mean), exact scalar first-occasion `T0TIPREDEFFECT` shift `t0_b z` and Eq. 3 first-summand carry `e^{A Δt} t0_b z` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; `T0TIPREDEFFECT` is not `TIPREDEFFECT` `B`; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`; `e^{A Δt} t0_b z` is not `t0_b z`), exact scalar observed mean of that first-occasion carry `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of that Table 3 / Eq. 3 first-summand composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean), exact scalar first-occasion `T0TDPREDEFFECT` shift `t0_m x0` and Eq. 3 first-summand carry `e^{A Δt} t0_m x0` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF re-opened 2026-08-20T19:10Z; `T0TDPREDEFFECT` is not `TDPREDEFFECT` `M`; `t0_m x0` is not `M x`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `e^{A Δt} t0_m x0` is not `e^{A(t−u)} M x` for `t0 < u < t`; `t0_m x0` is not `t0_b z`; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`), exact scalar observed mean of that first-occasion TD carry `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of that Table 3 / Eq. 3 first-summand TD composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean), exact scalar §7.2 level-change `CINT` `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; `a < 0` so `−κ / a = m x`; not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`, and not the extra near-zero-drift latent process also named in §7.2), exact scalar Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` (not `m x`, not `κ`, and not `TIPREDEFFECT`), exact scalar §7.2 extra near-zero-drift latent process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z; identification `TDPREDEFFECT` on the extra process is 1; extra `DRIFT` printed as `−0.000001`; precisely 0 causes computational problems; `ε = a` is `a_{ηξ} x Δt e^{a Δt}`; not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`; `ε ≥ 0` fails closed), exact scalar observed mean of that extra-process contribution `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5, p. 5; §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:12Z; the extra process has `LAMBDA` 0 and is not an observed indicator; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`), exact scalar after-t0 extra-process contribution `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` and Eq. 5 observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:32Z; `T0TDPREDEFFECT` uses `Δt` for both the evolution and the extra drive; `TDPREDEFFECT` after `t0` uses `t − u`; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive), exact scalar §7.2 `asymTIPREDEFFECT` `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z; expected total change in process means given a time-independent predictor; `a < 0`; not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; `a ≥ 0` fails closed), exact scalar §7.2 `addedTIPREDVAR` `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21; stable between-subject variance accounted for by a time-independent predictor; not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`), exact scalar Table 2 `asymCINT` `-κ / a` (Driver et al., 2017, Table 2, p. 12; Eq. 3 as `Δt → ∞`; `a < 0`; not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; `a ≥ 0` fails closed), exact scalar p. 16 stationary `T0MEANS` `-κ / a + −B z / a` (Driver et al., 2017, p. 16; constrained first-occasion mean using `T0MEANSbase` / `T0MEANSfree`; not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean), exact scalar Eq. 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z; form the stationary latent mean first, then `τ + λ` of that mean; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`), exact scalar §4.3 / p. 16 stationary `T0VAR` `trait + −q / (2 a) + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; p. 16; JSS PDF re-opened 2026-08-22T03:07Z; not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`), exact scalar lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z; trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map), exact scalar Eq. 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` (`Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance), exact scalar later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z; trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt` and is not that later map; `Q_Δt` is not that later map), exact scalar Eq. 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` (the lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not that later observed variance; the later-occasion latent variance is not that observed variance)), CWC-then-event-time residual lag, irregular already-centered residual log-rate, and strong/strict-gated two-group OLS latent-mean difference are implemented on the stacked psychometric PR and are not implemented-main until exact-head checks, review, and protected-main integration complete; full ESEM/set-ESEM, formative composites, DSEM, and matrix continuous-time dynamics remain accepted-target +**Implementation maturity:** partial — construct classification, valid log-ratio/logistic-normal indicator gates, CPU `f64` OLS and posterior-draw loading point-estimate averaging, Rubin `T_m = Ū_m + (1+1/m) B_m` on draw-level OLS loadings, cluster-mean within/between OLS with the CWC contextual effect and Kish ESS WLS, event-time discrete lag-1 and exact scalar local log-rate, exact scalar forward map and unequal-interval remapping, exact scalar discrete effect of a constant predictor, first-order discrete effect of a time-varying predictor with matched sampling and constancy intervals (Voelkle et al., 2012, Eq. 14), exact scalar discrete process noise (Driver, Oud, & Voelkle, 2017, Eq. 3), exact scalar lagged latent covariance and unconditional latent variance (Driver et al., 2017, Eq. 3–4), exact scalar stationary within-subject variance (Driver et al., 2017, Eq. 4 as `Δt → ∞`; §4.3; p. 16 `asymDIFFUSION`), exact scalar trait-plus-state variance and lagged covariance (Driver et al., 2017, §4.3 `TRAITVAR`; not process noise and not `asymDIFFUSION`), exact scalar observed-indicator variance and lagged observed covariance (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero, else `λ² Var(η) + θ + ψ`; lagged `λ² cov(η_t, η_{t-1}) + ψ`; `MANIFESTVAR` is `Θ`, not `Var(y)`; `Θ` does not enter lagged observed covariance; `MANIFESTTRAITVAR` is not `MANIFESTVAR`; observed-indicator mean is `τ + λ μ` (`MANIFESTMEANS` is `τ`, not `E(y)`; `CINT` is not `MANIFESTMEANS`; `T0MEANS` is not `E(y)`; Equation 1 is the SDE; not a Kalman filter), exact scalar discrete latent mean `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment), exact scalar evolved observed-indicator mean `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of that Eq. 3 map; the first-occasion map `τ + λ μ_0` is not `E(y_t)`), exact scalar contemporaneous `TDPREDEFFECT` impulse `m x` (Driver et al., 2017, Eq. 3 fourth summand; Table 2 `TDPREDEFFECT` is `M`, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; the §7.2 level-change form is not that impulse), exact scalar observed mean of that contemporaneous impulse `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of that Eq. 3 composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`), exact scalar time-independent `TIPREDEFFECT` increment `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 3 second summand; Table 2 `TIPREDEFFECT` is `B`, not `κ`, not `M`, and not Voelkle Eq. 14; `B` is not that discrete increment), exact scalar observed mean of that increment `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of that Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`), exact scalar within-interval `TDPREDEFFECT` carry `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2 Green-function integral of Eq. 2; §7.2 dissipation; not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14), exact scalar observed mean of that carry `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of that carried latent mean; `τ + λ μ_t` is not that observed mean), exact scalar first-occasion `T0TIPREDEFFECT` shift `t0_b z` and Eq. 3 first-summand carry `e^{A Δt} t0_b z` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; `T0TIPREDEFFECT` is not `TIPREDEFFECT` `B`; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`; `e^{A Δt} t0_b z` is not `t0_b z`), exact scalar observed mean of that first-occasion carry `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of that Table 3 / Eq. 3 first-summand composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean), exact scalar first-occasion `T0TDPREDEFFECT` shift `t0_m x0` and Eq. 3 first-summand carry `e^{A Δt} t0_m x0` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF re-opened 2026-08-20T19:10Z; `T0TDPREDEFFECT` is not `TDPREDEFFECT` `M`; `t0_m x0` is not `M x`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `e^{A Δt} t0_m x0` is not `e^{A(t−u)} M x` for `t0 < u < t`; `t0_m x0` is not `t0_b z`; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`), exact scalar observed mean of that first-occasion TD carry `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of that Table 3 / Eq. 3 first-summand TD composition; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean), exact scalar §7.2 level-change `CINT` `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; `a < 0` so `−κ / a = m x`; not the dissipating Dirac, not a free `CINT`, not `TIPREDEFFECT`, and not the extra near-zero-drift latent process also named in §7.2), exact scalar Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` (not `m x`, not `κ`, and not `TIPREDEFFECT`), exact scalar §7.2 extra near-zero-drift latent process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z; identification `TDPREDEFFECT` on the extra process is 1; extra `DRIFT` printed as `−0.000001`; precisely 0 causes computational problems; `ε = a` is `a_{ηξ} x Δt e^{a Δt}`; not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`; `ε ≥ 0` fails closed), exact scalar observed mean of that extra-process contribution `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5, p. 5; §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:12Z; the extra process has `LAMBDA` 0 and is not an observed indicator; `τ + λ μ_t` is not that observed mean; `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`), exact scalar after-t0 extra-process contribution `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` and Eq. 5 observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:32Z; `T0TDPREDEFFECT` uses `Δt` for both the evolution and the extra drive; `TDPREDEFFECT` after `t0` uses `t − u`; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive), exact scalar §7.2 `asymTIPREDEFFECT` `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z; expected total change in process means given a time-independent predictor; `a < 0`; not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; `a ≥ 0` fails closed), exact scalar §7.2 `addedTIPREDVAR` `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21; stable between-subject variance accounted for by a time-independent predictor; not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`), exact scalar Table 2 `asymCINT` `-κ / a` (Driver et al., 2017, Table 2, p. 12; Eq. 3 as `Δt → ∞`; `a < 0`; not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`; `a ≥ 0` fails closed), exact scalar p. 16 stationary `T0MEANS` `-κ / a + −B z / a` (Driver et al., 2017, p. 16; constrained first-occasion mean using `T0MEANSbase` / `T0MEANSfree`; not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean), exact scalar Eq. 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z; form the stationary latent mean first, then `τ + λ` of that mean; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`), exact scalar §4.3 / p. 16 stationary `T0VAR` `trait + −q / (2 a) + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; p. 16; JSS PDF re-opened 2026-08-22T03:07Z; not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Eq. 5 of that constrained variance is `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`), exact scalar lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z; trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map), exact scalar Eq. 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` (`Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance), exact scalar later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z; trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt` and is not that later map; `Q_Δt` is not that later map), exact scalar Eq. 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` (the lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not that later observed variance; the later-occasion latent variance is not that observed variance)), exact scalar later-occasion variance of §4.3 predetermined `T0VAR` `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 predetermined first occasion; JSS PDF re-opened 2026-08-23T20:20Z; trait and `addedTIPREDVAR` do not enter `Q_Δt`; free `T0VAR` `p_0` is not that later map; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map; stationary later variance uses `−q / (2 a)` in place of `p_0` and is not that later map when `p_0` is free; evolving `trait + p_0 + (B / a)² v` as if it were all state is not that later map; as `Δt → ∞` with stable `a < 0` the composition approaches contemporaneous stationary `T0VAR`; as `Δt → 0+` the composition approaches `trait + p_0 + (B / a)² v`; nonzero diffusion with `a ≥ 0` is a growing process and is kept. Eq. 5 of that predetermined later-occasion variance is `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` (`MANIFESTVAR` is not that later observed variance; the predetermined later-occasion latent variance is not that observed variance; stationary later observed variance is not that observed variance when `p_0` is free)), CWC-then-event-time residual lag, irregular already-centered residual log-rate, and strong/strict-gated two-group OLS latent-mean difference are implemented on the stacked psychometric PR and are not implemented-main until exact-head checks, review, and protected-main integration complete; full ESEM/set-ESEM, formative composites, DSEM, and matrix continuous-time dynamics remain accepted-target **Date:** 2026-08-05 **Supersedes:** None. ADR 0012 governs upstream topic measurement/network coordinates; this ADR governs higher-order psychometric structure and longitudinal interpretation. @@ -20,7 +20,7 @@ Before ESEM/SEM interpretation, each higher-order construct is classified as ref Longitudinal analysis evaluates measurement invariance at the level needed for the claimed comparison, supports partial/approximate or time-varying loadings where scientifically justified, separates stable between-unit components from within-unit temporal change, and handles irregular intervals through appropriate discrete- or continuous-time dynamics. -The executable multilevel slice is cluster-mean centering (CWC) plus within/between OLS, the CWC contextual effect (`between − within`), and Kish-weighted slopes. Enders and Tofighi (2007, Table 2, pp. 124–127) show that the CWC cluster-mean coefficient is the contextual effect, not the between-cluster effect. It is not DSEM and not RI-CLPM. The executable temporal slice maps a discrete lag through the exact scalar exponential `a = ln(φ) / Δt` on event time only (Voelkle, Oud, Davidov, & Schmidt, 2012, Eq. 7; Driver, Oud, & Voelkle, 2017, Eq. 3), recovers the forward map `φ(Δt) = exp(a Δt)`, remaps a discrete lag onto another event interval through that log-rate, recovers the exact scalar discrete effect of a constant predictor (Voelkle et al., 2012, Eq. 12) as `a_yx (expm1(z) / a_xx)` with `z = a_xx Δt` so a finite result is not lost when `z` overflows to `-∞` or when `a_yx Δt` overflows, and in log space when `expm1(z)` overflows at a finite `z` (`a_yx = 0` is exactly zero; an overflowing `a_yx/a_xx` rewrite term fails closed), recovers the first-order discrete effect of a time-varying predictor with matched sampling and constancy intervals as `a_yx Δt` (Voelkle et al., 2012, Eq. 14; ZORA accepted manuscript p. 21; not Eq. 12; unmatched intervals fail closed because Oud & Jansen, 2000, is unread), recovers the exact scalar discrete process noise `Q_Δt = 0.5 q (expm1(z) / a)` with `z = 2 (a Δt)` and `q = G G⊤ ≥ 0` (Driver et al., 2017, Eq. 3; JSS PDF re-opened 2026-08-18T07:06Z, p. 4; scalar `L = 1`; do not form `2 a` first; `a = 0` and `z → 0` recover `q Δt`; `z → −∞` keeps `−0.5 q / a`; a zero diffusion is exactly zero; `z → +∞` and an overflowing rewrite scale `0.5 q / a` fail closed), recovers the exact scalar lagged latent covariance `exp(a Δt) p` and the law-of-total-variance map `Var(η_t) = exp(2 a Δt) p + Q_Δt` (Driver et al., 2017, Eq. 3–4, pp. 4–5; JSS PDF re-opened 2026-08-18T18:03Z; JSS has no numbered §2.2; `Q_Δt` is `cov(η_t | η_{t-1})` and is refused as the unconditional variance; a zero diffusion whose `2 (a Δt)` overflows to `+∞` fails closed), recovers the exact scalar stationary within-subject variance `-q / (2 a)` as the `Δt → ∞` limit of Eq. 4 for stable `a < 0` (JSS p. 16 `asymDIFFUSION`; §4.3 T0VAR stationarity; when `2 a` is finite, form `q / -(2 a)` so `q / a` overflow does not lose a finite result (`q = MAX`, `a = -0.75` → `MAX / 1.5`; CodeRabbit on `75ecdd3`); when `2 a` overflows, form `(q / a) * -0.5`; do not form `0.5 q` first (`q = from_bits(1)`, `a = -from_bits(1)` → `0.5`); `a ≥ 0` and finite-interval `Q_Δt` fail closed as that limit), recovers the exact scalar trait-plus-state variance `trait + state` and lagged covariance `trait + exp(a Δt) p` (Driver et al., 2017, §4.3, p. 9; JSS PDF re-opened 2026-08-18T21:07Z; a stable trait has `DRIFT` and `DIFFUSION` fixed to zero; `TRAITVAR` is not process noise and not `asymDIFFUSION`; evolving the summed variance as if it were all state fails the claim boundary; this is not RI-CLPM), recovers the exact scalar observed-indicator variance `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero and `λ² Var(η) + θ + ψ` otherwise, and the lagged observed covariance `λ² cov(η_t, η_{t-1}) + ψ` (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-19T04:18Z; Equation 1 is the latent SDE; form `(λ p) λ` then add `θ`, then add `ψ`; do not form `λ²` first; `MANIFESTVAR` is `Θ`, not `Var(y)` and does not enter lagged observed covariance; `MANIFESTTRAITVAR` is `Ψ_τ`, not `Θ`; `TRAITVAR` is latent and scaled by `λ²`; `Var(η)` is not `Var(y)`), recovers the exact scalar observed-indicator mean `τ + λ μ` (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-19T14:08Z; `MANIFESTMEANS` is `τ`, not `E(y)`; `CINT` is not `MANIFESTMEANS`; `T0MEANS` is not `E(y)`), recovers the exact scalar discrete latent mean `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3, p. 4; Table 2, p. 12; JSS PDF re-opened 2026-08-19T18:10Z; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment; a zero drift is `κ Δt`; underflow of `exp(a Δt)` to `+0` drops the carried `T0MEANS` and keeps `−κ / a`), recovers the exact scalar evolved observed-indicator mean `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of the Eq. 3 expected-value map; JSS PDF re-opened 2026-08-19T22:10Z; the first-occasion map `τ + λ μ_0` is not `E(y_t)`; `MANIFESTMEANS` is not `E(y_t)`; `μ_t` is not `E(y_t)`), recovers the exact scalar contemporaneous time-dependent predictor impulse `m x` (Driver et al., 2017, Eq. 1–3, pp. 4–5; Table 2, p. 12; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T07:10Z; `TDPREDEFFECT` is `M`, not `CINT`; `M x` is not `A^{-1}[e^{A Δt} − I] B z`; `M x` is not Voelkle et al., 2012, Eq. 14; the level-change form is not that impulse), recovers the exact scalar observed mean of that contemporaneous impulse `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of the Eq. 3 fourth-summand composition; JSS PDF re-opened 2026-08-20T09:01Z; the evolved map `τ + λ μ_t` is not that observed mean; the carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-impulse latent mean is not `E(y_t)`), recovers the exact scalar time-independent predictor increment `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 1–3, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-20T10:13Z; `TIPREDEFFECT` is `B`, not `κ`; form `B z` first, then the discrete intercept map; a zero drift is `B z Δt`; `B` is not the discrete increment; `A^{-1}[e^{A Δt} − I] B z` is not `CINT`, not `M x`, and not Voelkle Eq. 14), recovers the exact scalar observed mean of that increment `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of the Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment; JSS PDF re-opened 2026-08-20T12:12Z; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-increment latent mean is not `E(y_t)`), recovers the exact scalar within-interval time-dependent impulse carry `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2, pp. 4–5; Eq. 3 exponential map; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T10:33Z; form `m x` first, then `e^{a(t−u)} m x`; a zero drift is `m x`; underflow of `e^{a(t−u)}` to `+0` is vanishing dissipation and is kept; `e^{A(t−u)} M x` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; an impulse at `u = t` is the contemporaneous map; an impulse at `u ≤ t0` is already in `η(t0)`), recovers the exact scalar observed mean of that carry `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of the Eq. 1–2 carried latent mean; JSS PDF re-opened 2026-08-20T05:12Z; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the carried latent mean is not `E(y_t)`), recovers the exact scalar first-occasion `T0TIPREDEFFECT` shift `t0_b z` and its Eq. 3 first-summand carry `e^{A Δt} t0_b z` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF opened 2026-08-20T15:14Z; form `t0_b z` first, then `e^{a Δt} t0_b z`; a zero drift is `t0_b z`; underflow of `e^{a Δt}` to `+0` is a vanishing carry and is kept; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; `e^{A Δt} t0_b z` is not `t0_b z`; `T0TIPREDEFFECT` is the coefficient, not the shift), recovers the exact scalar observed mean of that first-occasion carry `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand composition; JSS PDF re-opened 2026-08-20T15:28Z; the evolved map `τ + λ μ_t` is not that observed mean; the process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-carry latent mean is not `E(y_t)`), recovers the exact scalar first-occasion `T0TDPREDEFFECT` shift `t0_m x0` and its Eq. 3 first-summand carry `e^{A Δt} t0_m x0` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF re-opened 2026-08-20T19:10Z; form `t0_m x0` first, then `e^{a Δt} t0_m x0`; a zero drift is `t0_m x0`; underflow of `e^{a Δt}` to `+0` is a vanishing carry and is kept; `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `CINT`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `T0TDPREDEFFECT` is the coefficient, not the shift; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`), recovers the exact scalar observed mean of that first-occasion TD carry `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand TD composition; JSS PDF re-opened 2026-08-20T19:07Z; the evolved map `τ + λ μ_t` is not that observed mean; the process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`; the first-occasion TI map `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-carry latent mean is not `E(y_t)`), recovers the exact scalar §7.2 level-change `CINT` setting `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T19:45Z; form `m x` first, then multiply by `−a`; `a < 0` so `−κ / a = m x`; `a ≥ 0` cannot hold a new process mean; `−a m x` is not the dissipating Dirac, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`; the extra near-zero-drift latent process also named in §7.2 is a different specification), recovers the exact scalar Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` (JSS PDF re-opened 2026-08-20T19:50Z; form the level-change `CINT` first, then the discrete intercept map; underflow of `e^{a Δt}` to `+0` keeps `m x`; `(1 − e^{a Δt}) m x` is not `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`), recovers the exact scalar §7.2 extra near-zero-drift latent process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z; form `a_{ηξ} x` first; `ε = a` is `a_{ηξ} x Δt e^{a Δt}`; a zero coupling or zero predictor is exactly zero; `ε ≥ 0` cannot hold a lasting extra state; that contribution is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`), recovers the exact scalar observed mean of that extra-process contribution `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5, p. 5; §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:12Z; the extra process has `LAMBDA` 0 and is not an observed indicator; form the evolved-plus-contribution latent mean first, then `τ + λ` of that mean; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`), recovers the exact scalar after-t0 extra-process contribution `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` and its Eq. 5 observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:32Z; `T0TDPREDEFFECT` uses `Δt` for both the evolution and the extra drive; `TDPREDEFFECT` after `t0` uses `t − u` while `μ_t` still uses `Δt`; an impulse at `u = t0` or `u = t` is not interior; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive), recovers the exact scalar §7.2 `asymTIPREDEFFECT` `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z; form `B z` first, then divide by `-a`; `a < 0`; a zero coefficient or zero predictor is exactly zero; `a ≥ 0` cannot hold a finite process-mean change; `-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`), recovers the exact scalar §7.2 `addedTIPREDVAR` `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21; form the unit asymptotic effect first, then square, then multiply by `v`; `v ≥ 0`; a zero coefficient or zero predictor variance is exactly zero; `(B / a)² v` is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`), recovers the exact scalar Table 2 `asymCINT` `-κ / a` (Driver et al., 2017, Table 2, p. 12; Eq. 3 as `Δt → ∞`; JSS PDF opened 2026-08-21T16:13Z; form `κ` first, then divide by `-a`; `a < 0`; a zero intercept is exactly zero; `-κ / a` is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`), recovers the exact scalar p. 16 stationary `T0MEANS` `-κ / a + −B z / a` (Driver et al., 2017, p. 16; form the intercept contribution first, then include the TI extra effect, then add; not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean), recovers the exact scalar Eq. 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z; form the stationary latent mean first, then `τ + λ` of that mean; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`), recovers the exact scalar §4.3 / p. 16 stationary `T0VAR` `trait + −q / (2 a) + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; p. 16; JSS PDF re-opened 2026-08-22T03:07Z; form the within-subject contribution first, then include the trait, then include the TI extra variance, then add; not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance), recovers the exact scalar Eq. 5 of that constrained variance `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`), recovers the exact scalar lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z; trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map), recovers the exact scalar Eq. 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` (`Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance), recovers the exact scalar later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z; form the evolved within-subject variance first, then include the trait, then include the TI extra variance, then add; trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt` and is not that later map; `Q_Δt` is not that later map), recovers the exact scalar Eq. 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` (the lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not that later observed variance; the later-occasion latent variance is not that observed variance), and refuses the difference quotient, pooling discrete lags from unequal intervals, and a binary64 underflow of that exponential to `+0` (not a discrete lag). The first-order product `a_yx Δt` is Eq. 14 and the underflow limit of Eq. 12, not the general constant-predictor discrete effect. Already-centered residuals may have irregular event intervals. Subtracting the person-specific mean from a raw autoregressive series is not the lagged within-person residual (Curran & Bauer, 2011, pp. 607–608). Metric/weak invariance licenses shared metric meaning only. Latent-mean comparison requires strong (equal loading and intercept) or strict invariance (Putnick & Bornstein, 2016, PMC5145197 opened 2026-08-19T22:15Z: scalar is required for latent means; residual invariance is not). Two-observation series have no residual degrees of freedom and cap at strong/scalar; identically-zero OLS residual variance is not strict. This two-group OLS gate is not MGCFA. +The executable multilevel slice is cluster-mean centering (CWC) plus within/between OLS, the CWC contextual effect (`between − within`), and Kish-weighted slopes. Enders and Tofighi (2007, Table 2, pp. 124–127) show that the CWC cluster-mean coefficient is the contextual effect, not the between-cluster effect. It is not DSEM and not RI-CLPM. The executable temporal slice maps a discrete lag through the exact scalar exponential `a = ln(φ) / Δt` on event time only (Voelkle, Oud, Davidov, & Schmidt, 2012, Eq. 7; Driver, Oud, & Voelkle, 2017, Eq. 3), recovers the forward map `φ(Δt) = exp(a Δt)`, remaps a discrete lag onto another event interval through that log-rate, recovers the exact scalar discrete effect of a constant predictor (Voelkle et al., 2012, Eq. 12) as `a_yx (expm1(z) / a_xx)` with `z = a_xx Δt` so a finite result is not lost when `z` overflows to `-∞` or when `a_yx Δt` overflows, and in log space when `expm1(z)` overflows at a finite `z` (`a_yx = 0` is exactly zero; an overflowing `a_yx/a_xx` rewrite term fails closed), recovers the first-order discrete effect of a time-varying predictor with matched sampling and constancy intervals as `a_yx Δt` (Voelkle et al., 2012, Eq. 14; ZORA accepted manuscript p. 21; not Eq. 12; unmatched intervals fail closed because Oud & Jansen, 2000, is unread), recovers the exact scalar discrete process noise `Q_Δt = 0.5 q (expm1(z) / a)` with `z = 2 (a Δt)` and `q = G G⊤ ≥ 0` (Driver et al., 2017, Eq. 3; JSS PDF re-opened 2026-08-18T07:06Z, p. 4; scalar `L = 1`; do not form `2 a` first; `a = 0` and `z → 0` recover `q Δt`; `z → −∞` keeps `−0.5 q / a`; a zero diffusion is exactly zero; `z → +∞` and an overflowing rewrite scale `0.5 q / a` fail closed), recovers the exact scalar lagged latent covariance `exp(a Δt) p` and the law-of-total-variance map `Var(η_t) = exp(2 a Δt) p + Q_Δt` (Driver et al., 2017, Eq. 3–4, pp. 4–5; JSS PDF re-opened 2026-08-18T18:03Z; JSS has no numbered §2.2; `Q_Δt` is `cov(η_t | η_{t-1})` and is refused as the unconditional variance; a zero diffusion whose `2 (a Δt)` overflows to `+∞` fails closed), recovers the exact scalar stationary within-subject variance `-q / (2 a)` as the `Δt → ∞` limit of Eq. 4 for stable `a < 0` (JSS p. 16 `asymDIFFUSION`; §4.3 T0VAR stationarity; when `2 a` is finite, form `q / -(2 a)` so `q / a` overflow does not lose a finite result (`q = MAX`, `a = -0.75` → `MAX / 1.5`; CodeRabbit on `75ecdd3`); when `2 a` overflows, form `(q / a) * -0.5`; do not form `0.5 q` first (`q = from_bits(1)`, `a = -from_bits(1)` → `0.5`); `a ≥ 0` and finite-interval `Q_Δt` fail closed as that limit), recovers the exact scalar trait-plus-state variance `trait + state` and lagged covariance `trait + exp(a Δt) p` (Driver et al., 2017, §4.3, p. 9; JSS PDF re-opened 2026-08-18T21:07Z; a stable trait has `DRIFT` and `DIFFUSION` fixed to zero; `TRAITVAR` is not process noise and not `asymDIFFUSION`; evolving the summed variance as if it were all state fails the claim boundary; this is not RI-CLPM), recovers the exact scalar observed-indicator variance `λ² Var(η) + θ` when `MANIFESTTRAITVAR` is zero and `λ² Var(η) + θ + ψ` otherwise, and the lagged observed covariance `λ² cov(η_t, η_{t-1}) + ψ` (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-19T04:18Z; Equation 1 is the latent SDE; form `(λ p) λ` then add `θ`, then add `ψ`; do not form `λ²` first; `MANIFESTVAR` is `Θ`, not `Var(y)` and does not enter lagged observed covariance; `MANIFESTTRAITVAR` is `Ψ_τ`, not `Θ`; `TRAITVAR` is latent and scaled by `λ²`; `Var(η)` is not `Var(y)`), recovers the exact scalar observed-indicator mean `τ + λ μ` (Driver et al., 2017, Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-19T14:08Z; `MANIFESTMEANS` is `τ`, not `E(y)`; `CINT` is not `MANIFESTMEANS`; `T0MEANS` is not `E(y)`), recovers the exact scalar discrete latent mean `exp(a Δt) μ_0 + (exp(a Δt) − 1)/a κ` (Driver et al., 2017, Eq. 3, p. 4; Table 2, p. 12; JSS PDF re-opened 2026-08-19T18:10Z; `T0MEANS` is not `μ_t`; `CINT` is not the discrete increment; a zero drift is `κ Δt`; underflow of `exp(a Δt)` to `+0` drops the carried `T0MEANS` and keeps `−κ / a`), recovers the exact scalar evolved observed-indicator mean `τ + λ μ_t` (Driver et al., 2017, Eq. 5 of the Eq. 3 expected-value map; JSS PDF re-opened 2026-08-19T22:10Z; the first-occasion map `τ + λ μ_0` is not `E(y_t)`; `MANIFESTMEANS` is not `E(y_t)`; `μ_t` is not `E(y_t)`), recovers the exact scalar contemporaneous time-dependent predictor impulse `m x` (Driver et al., 2017, Eq. 1–3, pp. 4–5; Table 2, p. 12; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T07:10Z; `TDPREDEFFECT` is `M`, not `CINT`; `M x` is not `A^{-1}[e^{A Δt} − I] B z`; `M x` is not Voelkle et al., 2012, Eq. 14; the level-change form is not that impulse), recovers the exact scalar observed mean of that contemporaneous impulse `τ + λ(μ_t + m x)` (Driver et al., 2017, Eq. 5 of the Eq. 3 fourth-summand composition; JSS PDF re-opened 2026-08-20T09:01Z; the evolved map `τ + λ μ_t` is not that observed mean; the carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-impulse latent mean is not `E(y_t)`), recovers the exact scalar time-independent predictor increment `A^{-1}[e^{A Δt} − I] B z` (Driver et al., 2017, Eq. 1–3, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-20T10:13Z; `TIPREDEFFECT` is `B`, not `κ`; form `B z` first, then the discrete intercept map; a zero drift is `B z Δt`; `B` is not the discrete increment; `A^{-1}[e^{A Δt} − I] B z` is not `CINT`, not `M x`, and not Voelkle Eq. 14), recovers the exact scalar observed mean of that increment `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` (Driver et al., 2017, Eq. 5 of the Eq. 3 printed addend after the `T0MEANS` carry and the `CINT` increment; JSS PDF re-opened 2026-08-20T12:12Z; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-increment latent mean is not `E(y_t)`), recovers the exact scalar within-interval time-dependent impulse carry `e^{A(t−u)} M x` for `t0 < u < t` (Driver et al., 2017, Eq. 1–2, pp. 4–5; Eq. 3 exponential map; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T10:33Z; form `m x` first, then `e^{a(t−u)} m x`; a zero drift is `m x`; underflow of `e^{a(t−u)}` to `+0` is vanishing dissipation and is kept; `e^{A(t−u)} M x` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle Eq. 14; an impulse at `u = t` is the contemporaneous map; an impulse at `u ≤ t0` is already in `η(t0)`), recovers the exact scalar observed mean of that carry `τ + λ(μ_t + e^{a(t−u)} m x)` (Driver et al., 2017, Eq. 5 of the Eq. 1–2 carried latent mean; JSS PDF re-opened 2026-08-20T05:12Z; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean when `u ≠ t`; `MANIFESTMEANS` is not `E(y_t)`; the carried latent mean is not `E(y_t)`), recovers the exact scalar first-occasion `T0TIPREDEFFECT` shift `t0_b z` and its Eq. 3 first-summand carry `e^{A Δt} t0_b z` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF opened 2026-08-20T15:14Z; form `t0_b z` first, then `e^{a Δt} t0_b z`; a zero drift is `t0_b z`; underflow of `e^{a Δt}` to `+0` is a vanishing carry and is kept; `t0_b z` is not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`; `e^{A Δt} t0_b z` is not `t0_b z`; `T0TIPREDEFFECT` is the coefficient, not the shift), recovers the exact scalar observed mean of that first-occasion carry `τ + λ(μ_t + e^{a Δt} t0_b z)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand composition; JSS PDF re-opened 2026-08-20T15:28Z; the evolved map `τ + λ μ_t` is not that observed mean; the process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-carry latent mean is not `E(y_t)`), recovers the exact scalar first-occasion `T0TDPREDEFFECT` shift `t0_m x0` and its Eq. 3 first-summand carry `e^{A Δt} t0_m x0` (Driver et al., 2017, Table 3, p. 13; Eq. 3, p. 5; JSS PDF re-opened 2026-08-20T19:10Z; form `t0_m x0` first, then `e^{a Δt} t0_m x0`; a zero drift is `t0_m x0`; underflow of `e^{a Δt}` to `+0` is a vanishing carry and is kept; `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `CINT`; `e^{A Δt} t0_m x0` is not `t0_m x0`; `T0TDPREDEFFECT` is the coefficient, not the shift; an impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`), recovers the exact scalar observed mean of that first-occasion TD carry `τ + λ(μ_t + e^{a Δt} t0_m x0)` (Driver et al., 2017, Eq. 5 of the Table 3 / Eq. 3 first-summand TD composition; JSS PDF re-opened 2026-08-20T19:07Z; the evolved map `τ + λ μ_t` is not that observed mean; the process-increment map `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the impulse-carry map `τ + λ(μ_t + e^{a(t−u)} m x)` is not that observed mean when `u ≠ t0`; the first-occasion TI map `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that observed mean; `MANIFESTMEANS` is not `E(y_t)`; the evolved-plus-carry latent mean is not `E(y_t)`), recovers the exact scalar §7.2 level-change `CINT` setting `κ = −a m x` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF re-opened 2026-08-20T19:45Z; form `m x` first, then multiply by `−a`; `a < 0` so `−κ / a = m x`; `a ≥ 0` cannot hold a new process mean; `−a m x` is not the dissipating Dirac, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`; the extra near-zero-drift latent process also named in §7.2 is a different specification), recovers the exact scalar Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` (JSS PDF re-opened 2026-08-20T19:50Z; form the level-change `CINT` first, then the discrete intercept map; underflow of `e^{a Δt}` to `+0` keeps `m x`; `(1 − e^{a Δt}) m x` is not `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`), recovers the exact scalar §7.2 extra near-zero-drift latent process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-20T23:10Z; form `a_{ηξ} x` first; `ε = a` is `a_{ηξ} x Δt e^{a Δt}`; a zero coupling or zero predictor is exactly zero; `ε ≥ 0` cannot hold a lasting extra state; that contribution is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`), recovers the exact scalar observed mean of that extra-process contribution `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` (Driver et al., 2017, Eq. 5, p. 5; §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:12Z; the extra process has `LAMBDA` 0 and is not an observed indicator; form the evolved-plus-contribution latent mean first, then `τ + λ` of that mean; the evolved map `τ + λ μ_t` is not that observed mean; the contemporaneous map `τ + λ(μ_t + m x)` is not that observed mean; the contribution is not `E(y_t)`; the evolved-plus-contribution latent mean is not `E(y_t)`), recovers the exact scalar after-t0 extra-process contribution `a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a)` for `t0 < u < t` and its Eq. 5 observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε(t−u)} − e^{a(t−u)}) / (ε − a))` (Driver et al., 2017, §7.2, pp. 22–23; JSS PDF re-opened 2026-08-21T06:32Z; `T0TDPREDEFFECT` uses `Δt` for both the evolution and the extra drive; `TDPREDEFFECT` after `t0` uses `t − u` while `μ_t` still uses `Δt`; an impulse at `u = t0` or `u = t` is not interior; `e^{a(t−u)} m x` is a Dirac on the original process, not this `DRIFT` drive), recovers the exact scalar §7.2 `asymTIPREDEFFECT` `-B z / a` (Driver et al., 2017, §7.2, pp. 20–21; JSS PDF opened 2026-08-21T13:08Z; form `B z` first, then divide by `-a`; `a < 0`; a zero coefficient or zero predictor is exactly zero; `a ≥ 0` cannot hold a finite process-mean change; `-B z / a` is not the coefficient `B`, not `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`), recovers the exact scalar §7.2 `addedTIPREDVAR` `(B / a)² v` (Driver et al., 2017, §7.2, pp. 20–21; form the unit asymptotic effect first, then square, then multiply by `v`; `v ≥ 0`; a zero coefficient or zero predictor variance is exactly zero; `(B / a)² v` is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`), recovers the exact scalar Table 2 `asymCINT` `-κ / a` (Driver et al., 2017, Table 2, p. 12; Eq. 3 as `Δt → ∞`; JSS PDF opened 2026-08-21T16:13Z; form `κ` first, then divide by `-a`; `a < 0`; a zero intercept is exactly zero; `-κ / a` is not `κ`, not `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`), recovers the exact scalar p. 16 stationary `T0MEANS` `-κ / a + −B z / a` (Driver et al., 2017, p. 16; form the intercept contribution first, then include the TI extra effect, then add; not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean), recovers the exact scalar Eq. 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; JSS PDF re-opened 2026-08-21T20:07Z; form the stationary latent mean first, then `τ + λ` of that mean; `τ + λ μ_0` is not that observed mean; `τ + λ(−κ / a)` is not that observed mean when `B z ≠ 0`; `τ + λ μ_t` is not that observed mean; `MANIFESTMEANS` is not `E(y_0)`; the constrained latent mean is not `E(y_0)`), recovers the exact scalar §4.3 / p. 16 stationary `T0VAR` `trait + −q / (2 a) + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; p. 16; JSS PDF re-opened 2026-08-22T03:07Z; form the within-subject contribution first, then include the trait, then include the TI extra variance, then add; not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance), recovers the exact scalar Eq. 5 of that constrained variance `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 5, p. 5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T03:20Z; form the stationary latent variance first, then `λ² p + θ + ψ`; `λ² p_0` is not that observed variance; `λ²(−q / (2 a)) + θ` is not that observed variance when `TRAITVAR` or `addedTIPREDVAR` is nonzero; `MANIFESTVAR` is not `Var(y_0)`; the constrained latent variance is not `Var(y_0)`), recovers the exact scalar lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T19:13Z; trait and `addedTIPREDVAR` do not decay with `e^{a Δt}`; contemporaneous `T0VAR` is not that lagged map; decaying the constrained total as if it were all state is not that lagged map), recovers the exact scalar Eq. 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` (`Θ` does not enter; contemporaneous `Var(y_0)` is not that lagged observed covariance; the lagged latent covariance is not that observed covariance), recovers the exact scalar later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 / p. 16 `T0VAR`; JSS PDF re-opened 2026-08-22T23:12Z; form the evolved within-subject variance first, then include the trait, then include the TI extra variance, then add; trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`; evolving the constrained total as if it were all state is not that later map; the lagged covariance omits `Q_Δt` and is not that later map; `Q_Δt` is not that later map), recovers the exact scalar Eq. 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` (the lagged observed covariance omits `Q_Δt` and `θ`; `MANIFESTVAR` is not that later observed variance; the later-occasion latent variance is not that observed variance), recovers the exact scalar later-occasion variance of §4.3 predetermined `T0VAR` `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (Driver et al., 2017, Eq. 3–4 of §4.3 predetermined first occasion; JSS PDF re-opened 2026-08-23T20:20Z; form the evolved free first-occasion variance first, then include the trait, then include the TI extra variance, then add; trait and `addedTIPREDVAR` do not enter `Q_Δt`; free `T0VAR` `p_0` is not that later map; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map; stationary later variance uses `−q / (2 a)` in place of `p_0` and is not that later map when `p_0` is free; evolving `trait + p_0 + (B / a)² v` as if it were all state is not that later map), recovers the exact scalar Eq. 5 of that predetermined later-occasion variance `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` (`MANIFESTVAR` is not that later observed variance; the predetermined later-occasion latent variance is not that observed variance; stationary later observed variance is not that observed variance when `p_0` is free), and refuses the difference quotient, pooling discrete lags from unequal intervals, and a binary64 underflow of that exponential to `+0` (not a discrete lag). The first-order product `a_yx Δt` is Eq. 14 and the underflow limit of Eq. 12, not the general constant-predictor discrete effect. Already-centered residuals may have irregular event intervals. Subtracting the person-specific mean from a raw autoregressive series is not the lagged within-person residual (Curran & Bauer, 2011, pp. 607–608). Metric/weak invariance licenses shared metric meaning only. Latent-mean comparison requires strong (equal loading and intercept) or strict invariance (Putnick & Bornstein, 2016, PMC5145197 opened 2026-08-19T22:15Z: scalar is required for latent means; residual invariance is not). Two-observation series have no residual degrees of freedom and cap at strong/scalar; identically-zero OLS residual variance is not strict. This two-group OLS gate is not MGCFA. Input/process/intervention/outcome paths obey event-time order. Temporal precedence, document linkage, event tracking, or model prediction alone do not justify causal language. diff --git a/docs/research/multilevel-event-time-recovery.md b/docs/research/multilevel-event-time-recovery.md index 12f25297..5b2db571 100644 --- a/docs/research/multilevel-event-time-recovery.md +++ b/docs/research/multilevel-event-time-recovery.md @@ -48,15 +48,17 @@ This slice stays inside `psychometric_core`. It does not add a second invariance 42. recover the exact scalar Eq. 5 of lagged §4.3 stationary `T0VAR` `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` (Driver et al., 2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T19:13Z; form the lagged latent covariance first, then `λ² c + ψ`; a zero loading is exactly `ψ`; independent `ε_t` does not enter) and refuse treating `θ`, contemporaneous `Var(y_0)`, or the lagged latent covariance as `cov(y_t, y_{t-1})`; 43. recover the exact scalar later-occasion variance of §4.3 stationary `T0VAR` `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-22T23:12Z; form the evolved within-subject variance first, then include the trait, then include the TI extra variance, then add; trait and `addedTIPREDVAR` do not enter `Q_Δt`; under stationarity that composition equals contemporaneous `T0VAR`) and refuse treating that composition as lagged covariance, as `e^{2 a Δt}` of the constrained total plus `Q_Δt`, or as `Q_Δt` alone; 44. recover the exact scalar Eq. 5 of later-occasion §4.3 stationary `T0VAR` `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` (Driver et al., 2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T23:12Z; form the later-occasion latent variance first, then `λ² p + θ + ψ`; a zero loading is exactly `θ + ψ`; under stationarity that composition equals contemporaneous `Var(y_0)`) and refuse treating `θ`, lagged `cov(y_t, y_{t-1})`, or the later-occasion latent variance as `Var(y_t)`; -45. refuse pooling discrete lags from unequal event intervals as one coefficient; -46. refuse unmatched sampling and constancy intervals for a time-varying predictor (Oud & Jansen, 2000, unread); -47. refuse the difference quotient as a continuous-time rate; -48. apply the same event-time map to CWC residuals (still not DSEM); -49. map already-centered lagged residuals with irregular event intervals without re-centering (Curran & Bauer, 2011, pp. 607–608). +45. recover the exact scalar later-occasion variance of §4.3 predetermined `T0VAR` `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (Driver et al., 2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T20:20Z; form the evolved free first-occasion variance first, then include the trait, then include the TI extra variance, then add; trait and `addedTIPREDVAR` do not enter `Q_Δt`; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map; as `Δt → ∞` with stable `a < 0` the composition approaches contemporaneous stationary `T0VAR`; as `Δt → 0+` the composition approaches `trait + p_0 + (B / a)² v`; nonzero diffusion with `a ≥ 0` is a growing process and is kept) and refuse treating that composition as stationary later-occasion variance, as `e^{2 a Δt}` of `trait + p_0 + (B / a)² v` plus `Q_Δt`, or as free `p_0`; +46. recover the exact scalar Eq. 5 of later-occasion §4.3 predetermined `T0VAR` `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` (Driver et al., 2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-23T20:20Z; form the predetermined later-occasion latent variance first, then `λ² p + θ + ψ`; a zero loading is exactly `θ + ψ`; setting `p_0 = −q / (2 a)` recovers the stationary later-occasion observed variance) and refuse treating `θ`, the predetermined later-occasion latent variance, or stationary later-occasion observed variance as `Var(y_t)` when `p_0` is free; +47. refuse pooling discrete lags from unequal event intervals as one coefficient; +48. refuse unmatched sampling and constancy intervals for a time-varying predictor (Oud & Jansen, 2000, unread); +49. refuse the difference quotient as a continuous-time rate; +50. apply the same event-time map to CWC residuals (still not DSEM); +51. map already-centered lagged residuals with irregular event intervals without re-centering (Curran & Bauer, 2011, pp. 607–608). ## Claim boundary -This is two-level OLS and a noiseless scalar continuous-time map. It is not DSEM, not RI-CLPM, not a random-effects sampler, not a Kalman filter, and not a matrix `expm` implementation. The CWC cluster-mean coefficient is the **contextual** effect, not the between-cluster effect. Discrete lags from different event intervals are not one coefficient. Equation 14 is not Equation 12. Discrete process noise \(Q_{\Delta t}\) is not the continuous diffusion \(GG^{\top}\). \(Q_{\Delta t}\) is the conditional residual variance, not \(\operatorname{Var}(\eta_{t})\). Finite-interval \(Q_{\Delta t}\) is not the stationary within-subject variance. Trait variance is not process noise and not the stationary within-subject variance. Measurement-error variance is not the observed-indicator variance. Latent variance is not the observed-indicator variance. Manifest means are not the observed-indicator mean. The latent mean is not the observed-indicator mean. The continuous intercept is not the manifest mean. The first-occasion latent mean is not the evolved latent mean. The continuous intercept is not the discrete mean increment. The first-occasion observed mean is not the evolved observed mean. The contemporaneous `TDPREDEFFECT` impulse is not the continuous intercept, not the time-independent discrete effect, and not Voelkle et al. (2012, Eq. 14). The time-independent `TIPREDEFFECT` increment is not the continuous intercept, not the contemporaneous impulse, not Voelkle et al. (2012, Eq. 14), and not the coefficient `B`. The within-interval `TDPREDEFFECT` carry `e^{A(t−u)} M x` for `t0 < u < t` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle et al. (2012, Eq. 14). The evolved observed mean `τ + λ μ_t` is not the contemporaneous-impulse observed mean `τ + λ(μ_t + m x)`. The contemporaneous composition `τ + λ(μ_t + m x)` is not the impulse-carry observed mean `τ + λ(μ_t + e^{a(t−u)} m x)` when `u ≠ t`. The evolved observed mean `τ + λ μ_t` is not the impulse-carry observed mean. The carried latent mean is not `E(y_t)`. The evolved-plus-impulse latent mean is not `E(y_t)`. The evolved observed mean `τ + λ μ_t` is not the time-independent-predictor observed mean `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`. The contemporaneous composition `τ + λ(μ_t + m x)` is not that time-independent-predictor observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that time-independent-predictor observed mean when `u ≠ t`. The evolved-plus-increment latent mean is not `E(y_t)`. The Table 3 first-occasion `T0TIPREDEFFECT` shift `t0_b z` is not the Eq. 3 process increment `A^{-1}[e^{A Δt} − I] B z`, not `κ`, and not `M x`. The Eq. 3 first-summand carry `e^{A Δt} t0_b z` is not `t0_b z` and is not that process increment. `T0TIPREDEFFECT` is the coefficient, not the first-occasion shift. The evolved observed mean `τ + λ μ_t` is not the first-occasion TI-predictor observed mean `τ + λ(μ_t + e^{a Δt} t0_b z)`. The process-increment composition `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that first-occasion observed mean. The contemporaneous composition `τ + λ(μ_t + m x)` is not that first-occasion observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that first-occasion observed mean when `u ≠ t0`. The evolved-plus-T0TIPRED latent mean is not `E(y_t)`. The Table 3 first-occasion `T0TDPREDEFFECT` shift `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `κ`. The Eq. 3 first-summand carry `e^{A Δt} t0_m x0` is not `t0_m x0` and is not that within-interval impulse carry. `T0TDPREDEFFECT` is the coefficient, not the first-occasion shift. An impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`. The evolved observed mean `τ + λ μ_t` is not the first-occasion TD-predictor observed mean `τ + λ(μ_t + e^{a Δt} t0_m x0)`. The process-increment composition `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that first-occasion TD observed mean. The contemporaneous composition `τ + λ(μ_t + m x)` is not that first-occasion TD observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that first-occasion TD observed mean when `u ≠ t0`. The first-occasion TI composition `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that first-occasion TD observed mean. Same numbers as `T0TIPREDEFFECT` yield the same product; Table 3 names a different matrix. The evolved-plus-T0TDPRED latent mean is not `E(y_t)`. The §7.2 level-change `CINT` `κ = −a m x` is not the dissipating Dirac `m x`, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`. Lasting level change via that `CINT` setting requires `a < 0`. The extra near-zero-drift latent process also named in §7.2 is a different specification and is not that `CINT` setting. The Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` is not the dissipating Dirac `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`. Underflow of `e^{a Δt}` to `+0` keeps the equilibrium offset `m x`. The §7.2 extra-process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`. Lasting level change via that extra process requires `ε < 0`. Precisely `ε = 0` causes computational problems in the printed specification. The extra-process observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` is not `τ + λ μ_t`, not `τ + λ(μ_t + m x)`, not the contribution, and not the evolved-plus-contribution latent mean. The extra process has `LAMBDA` 0 and is not an observed indicator. `T0TDPREDEFFECT` on the extra process uses `Δt = t − t0` for both the original-process evolution and the extra drive. `TDPREDEFFECT` after `t0` uses `t − u` with `t0 < u < t` for the extra drive while `μ_t` still uses `Δt`. The after-t0 extra-process observed mean is not the first-occasion extra-process observed mean when `u ≠ t0`. The impulse-carry `e^{a(t−u)} m x` is a Dirac on the original process and is not that `DRIFT` drive. The §7.2 `asymTIPREDEFFECT` `-B z / a` is the expected total change in process means given a time-independent predictor. It is not the coefficient `B`, not the finite-interval increment `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. Lasting asymptotic change via that map requires `a < 0`. The §7.2 `addedTIPREDVAR` `(B / a)² v` is the stable between-subject variance accounted for by that predictor. It is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`. Table 2 `asymCINT` `-κ / a` is the intercept contribution to the stationary process mean. It is not `κ`, not the finite-interval increment `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`. Lasting asymptotic intercept change via that map requires `a < 0`. Page 16 notes that a `T0MEANS` stationarity constraint includes time-independent predictors; that composition is not this intercept-only map. The p. 16 constrained first-occasion mean `-κ / a + −B z / a` is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean. Equation 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` is not `τ + λ μ_0`, not `τ + λ(−κ / a)` when `B z ≠ 0`, not `τ + λ μ_t`, not `MANIFESTMEANS`, and not the constrained latent mean. The p. 16 constrained first-occasion variance `trait + −q / (2 a) + (B / a)² v` is not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Equation 5 of that constrained variance `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` is not `λ² p_0 + θ`, not `λ²(−q / (2 a)) + θ` when `TRAITVAR` or `addedTIPREDVAR` is nonzero, not `λ² Var(η_t) + θ` when the first occasion is constrained, not `MANIFESTVAR`, and not the constrained latent variance. The lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` is not contemporaneous `T0VAR`, not `e^{a Δt}` of the constrained total, and not `trait + e^{a Δt} p` when `addedTIPREDVAR` is nonzero. Trait variance and `addedTIPREDVAR` do not decay with `e^{a Δt}`. Equation 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` is not `θ`, not contemporaneous `Var(y_0)`, and not the lagged latent covariance. Independent measurement error does not enter lagged observed covariance. The later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` equals contemporaneous `T0VAR` under stationarity and is not the lagged covariance, not `e^{2 a Δt}` of the constrained total plus `Q_Δt`, and not `Q_Δt` alone. Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. Equation 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` is not `θ`, not lagged `cov(y_t, y_{t-1})`, and not the later-occasion latent variance. +This is two-level OLS and a noiseless scalar continuous-time map. It is not DSEM, not RI-CLPM, not a random-effects sampler, not a Kalman filter, and not a matrix `expm` implementation. The CWC cluster-mean coefficient is the **contextual** effect, not the between-cluster effect. Discrete lags from different event intervals are not one coefficient. Equation 14 is not Equation 12. Discrete process noise \(Q_{\Delta t}\) is not the continuous diffusion \(GG^{\top}\). \(Q_{\Delta t}\) is the conditional residual variance, not \(\operatorname{Var}(\eta_{t})\). Finite-interval \(Q_{\Delta t}\) is not the stationary within-subject variance. Trait variance is not process noise and not the stationary within-subject variance. Measurement-error variance is not the observed-indicator variance. Latent variance is not the observed-indicator variance. Manifest means are not the observed-indicator mean. The latent mean is not the observed-indicator mean. The continuous intercept is not the manifest mean. The first-occasion latent mean is not the evolved latent mean. The continuous intercept is not the discrete mean increment. The first-occasion observed mean is not the evolved observed mean. The contemporaneous `TDPREDEFFECT` impulse is not the continuous intercept, not the time-independent discrete effect, and not Voelkle et al. (2012, Eq. 14). The time-independent `TIPREDEFFECT` increment is not the continuous intercept, not the contemporaneous impulse, not Voelkle et al. (2012, Eq. 14), and not the coefficient `B`. The within-interval `TDPREDEFFECT` carry `e^{A(t−u)} M x` for `t0 < u < t` is not the contemporaneous Dirac, not `CINT`, not `TIPREDEFFECT`, and not Voelkle et al. (2012, Eq. 14). The evolved observed mean `τ + λ μ_t` is not the contemporaneous-impulse observed mean `τ + λ(μ_t + m x)`. The contemporaneous composition `τ + λ(μ_t + m x)` is not the impulse-carry observed mean `τ + λ(μ_t + e^{a(t−u)} m x)` when `u ≠ t`. The evolved observed mean `τ + λ μ_t` is not the impulse-carry observed mean. The carried latent mean is not `E(y_t)`. The evolved-plus-impulse latent mean is not `E(y_t)`. The evolved observed mean `τ + λ μ_t` is not the time-independent-predictor observed mean `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)`. The contemporaneous composition `τ + λ(μ_t + m x)` is not that time-independent-predictor observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that time-independent-predictor observed mean when `u ≠ t`. The evolved-plus-increment latent mean is not `E(y_t)`. The Table 3 first-occasion `T0TIPREDEFFECT` shift `t0_b z` is not the Eq. 3 process increment `A^{-1}[e^{A Δt} − I] B z`, not `κ`, and not `M x`. The Eq. 3 first-summand carry `e^{A Δt} t0_b z` is not `t0_b z` and is not that process increment. `T0TIPREDEFFECT` is the coefficient, not the first-occasion shift. The evolved observed mean `τ + λ μ_t` is not the first-occasion TI-predictor observed mean `τ + λ(μ_t + e^{a Δt} t0_b z)`. The process-increment composition `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that first-occasion observed mean. The contemporaneous composition `τ + λ(μ_t + m x)` is not that first-occasion observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that first-occasion observed mean when `u ≠ t0`. The evolved-plus-T0TIPRED latent mean is not `E(y_t)`. The Table 3 first-occasion `T0TDPREDEFFECT` shift `t0_m x0` is not `M x`, not `e^{A(t−u)} M x` for `t0 < u < t`, not `t0_b z`, not `A^{-1}[e^{A Δt} − I] B z`, and not `κ`. The Eq. 3 first-summand carry `e^{A Δt} t0_m x0` is not `t0_m x0` and is not that within-interval impulse carry. `T0TDPREDEFFECT` is the coefficient, not the first-occasion shift. An impulse at `u ≤ t0` that used `M` is already in `η(t0)` as `TDPREDEFFECT`, not as `T0TDPREDEFFECT`. The evolved observed mean `τ + λ μ_t` is not the first-occasion TD-predictor observed mean `τ + λ(μ_t + e^{a Δt} t0_m x0)`. The process-increment composition `τ + λ(μ_t + A^{-1}[e^{A Δt} − I] B z)` is not that first-occasion TD observed mean. The contemporaneous composition `τ + λ(μ_t + m x)` is not that first-occasion TD observed mean. The impulse-carry composition `τ + λ(μ_t + e^{a(t−u)} m x)` is not that first-occasion TD observed mean when `u ≠ t0`. The first-occasion TI composition `τ + λ(μ_t + e^{a Δt} t0_b z)` is not that first-occasion TD observed mean. Same numbers as `T0TIPREDEFFECT` yield the same product; Table 3 names a different matrix. The evolved-plus-T0TDPRED latent mean is not `E(y_t)`. The §7.2 level-change `CINT` `κ = −a m x` is not the dissipating Dirac `m x`, not a free `CINT`, and not `A^{-1}[e^{A Δt} − I] B z`. Lasting level change via that `CINT` setting requires `a < 0`. The extra near-zero-drift latent process also named in §7.2 is a different specification and is not that `CINT` setting. The Eq. 3 increment of that setting `(1 − e^{a Δt}) m x` is not the dissipating Dirac `m x`, not `κ`, and not `A^{-1}[e^{A Δt} − I] B z`. Underflow of `e^{a Δt}` to `+0` keeps the equilibrium offset `m x`. The §7.2 extra-process contribution `a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a)` is not `κ = −a m x`, not `(1 − e^{a Δt}) m x`, and not the dissipating Dirac `m x`. Lasting level change via that extra process requires `ε < 0`. Precisely `ε = 0` causes computational problems in the printed specification. The extra-process observed mean `τ + λ(μ_t + a_{ηξ} x (e^{ε Δt} − e^{a Δt}) / (ε − a))` is not `τ + λ μ_t`, not `τ + λ(μ_t + m x)`, not the contribution, and not the evolved-plus-contribution latent mean. The extra process has `LAMBDA` 0 and is not an observed indicator. `T0TDPREDEFFECT` on the extra process uses `Δt = t − t0` for both the original-process evolution and the extra drive. `TDPREDEFFECT` after `t0` uses `t − u` with `t0 < u < t` for the extra drive while `μ_t` still uses `Δt`. The after-t0 extra-process observed mean is not the first-occasion extra-process observed mean when `u ≠ t0`. The impulse-carry `e^{a(t−u)} m x` is a Dirac on the original process and is not that `DRIFT` drive. The §7.2 `asymTIPREDEFFECT` `-B z / a` is the expected total change in process means given a time-independent predictor. It is not the coefficient `B`, not the finite-interval increment `A^{-1}[e^{A Δt} − I] B z`, not `CINT`, and not `M x`. Lasting asymptotic change via that map requires `a < 0`. The §7.2 `addedTIPREDVAR` `(B / a)² v` is the stable between-subject variance accounted for by that predictor. It is not `TRAITVAR`, not `asymDIFFUSION`, and not `-B z / a`. Table 2 `asymCINT` `-κ / a` is the intercept contribution to the stationary process mean. It is not `κ`, not the finite-interval increment `A^{-1}[e^{A Δt} − I] κ`, not `T0MEANS`, and not `-B z / a`. Lasting asymptotic intercept change via that map requires `a < 0`. Page 16 notes that a `T0MEANS` stationarity constraint includes time-independent predictors; that composition is not this intercept-only map. The p. 16 constrained first-occasion mean `-κ / a + −B z / a` is not free `T0MEANS`, not `asymCINT` alone, not `asymTIPREDEFFECT` alone, and not the finite-interval discrete latent mean. Equation 5 of that constrained mean `τ + λ(−κ / a + −B z / a)` is not `τ + λ μ_0`, not `τ + λ(−κ / a)` when `B z ≠ 0`, not `τ + λ μ_t`, not `MANIFESTMEANS`, and not the constrained latent mean. The p. 16 constrained first-occasion variance `trait + −q / (2 a) + (B / a)² v` is not free `T0VAR`, not `asymDIFFUSION` alone, not `TRAITVAR` alone, not `addedTIPREDVAR` alone, and not the finite-interval discrete latent variance. Equation 5 of that constrained variance `λ²(trait + −q / (2 a) + (B / a)² v) + θ + ψ` is not `λ² p_0 + θ`, not `λ²(−q / (2 a)) + θ` when `TRAITVAR` or `addedTIPREDVAR` is nonzero, not `λ² Var(η_t) + θ` when the first occasion is constrained, not `MANIFESTVAR`, and not the constrained latent variance. The lagged covariance of that constrained process `trait + e^{a Δt}(−q / (2 a)) + (B / a)² v` is not contemporaneous `T0VAR`, not `e^{a Δt}` of the constrained total, and not `trait + e^{a Δt} p` when `addedTIPREDVAR` is nonzero. Trait variance and `addedTIPREDVAR` do not decay with `e^{a Δt}`. Equation 5 of that lagged covariance `λ²(trait + e^{a Δt}(−q / (2 a)) + (B / a)² v) + ψ` is not `θ`, not contemporaneous `Var(y_0)`, and not the lagged latent covariance. Independent measurement error does not enter lagged observed covariance. The later-occasion variance of that constrained process `trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v` equals contemporaneous `T0VAR` under stationarity and is not the lagged covariance, not `e^{2 a Δt}` of the constrained total plus `Q_Δt`, and not `Q_Δt` alone. Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. Equation 5 of that later-occasion variance `λ²(trait + e^{2 a Δt}(−q / (2 a)) + Q_Δt + (B / a)² v) + θ + ψ` is not `θ`, not lagged `cov(y_t, y_{t-1})`, and not the later-occasion latent variance. The later-occasion variance of §4.3 predetermined `T0VAR` `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` is not stationary later-occasion variance when `p_0` is free, not `e^{2 a Δt}` of `trait + p_0 + (B / a)² v` plus `Q_Δt`, and not free `p_0`. Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. Setting `p_0 = −q / (2 a)` recovers the stationary later-occasion map. As `Δt → ∞` with stable `a < 0` the composition approaches contemporaneous stationary `T0VAR`. Nonzero diffusion with `a ≥ 0` is a growing process and is kept. Equation 5 of that predetermined later-occasion variance `λ²(trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v) + θ + ψ` is not `θ`, not the predetermined later-occasion latent variance, and not stationary later-occasion observed variance when `p_0` is free. ## Authoritative sources @@ -74,7 +76,7 @@ Kish, L. (1965). *Survey sampling*. John Wiley & Sons. Oud, J. H. L., & Jansen, R. A. R. G. (2000). Continuous time state space modeling of panel data by means of SEM. *Psychometrika, 65*(2), 199–215. https://doi.org/10.1007/BF02294374 (cited by Voelkle et al., 2012, Eq. 14 discussion; PDF not opened). -The Voelkle et al. (2012) ZORA accepted manuscript was re-opened 2026-08-18T21:07Z (https://www.zora.uzh.ch/handle/20.500.14742/72792; bitstream `424f9082-0eeb-4a67-b687-9845a4ed892f`). Page 16 writes discrete auto-effects as \(A^{*}(\Delta t)=\exp(A\Delta t)\) (Eq. 7). Introducing Intercepts (manuscript p. 20, Eq. 12) adds a continuous-time intercept \(b\) and writes the expected-value solution whose discrete increment is \(A^{-1}(\exp(A\Delta t)-I)b\); the scalar constant-predictor effect is \(b^{*}_{y.x}(\Delta t)=(a_{yx}/a_{xx})(\exp(a_{xx}\Delta t)-1)\). The next paragraph (manuscript p. 21, Eq. 14) writes the discrete effect of a **time-varying** predictor, when the sampling interval equals the constancy interval, as \(b^{*}_{y.x}(\Delta t)=a_{yx}\Delta t\). That product does not depend on the predictor auto-effect. The manuscript calls Eq. 14 a first-order approximation that deteriorates as \(\Delta t\) grows, and defers unmatched sampling/constancy intervals to Oud and Jansen (2000), which is unread. Driver, Oud, and Voelkle (2017, Eq. 3 and p. 4; JSS PDF opened 2026-08-21T13:08Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) write \(A_{\Delta t}=\operatorname{expm}(A\Delta t)\), the discrete intercept \(b_{\Delta t}=A^{-1}[A_{\Delta t}-I]b\), and the discrete process-noise covariance \(Q_{\Delta t}=\int_{0}^{\Delta t}\operatorname{expm}(A(\Delta t-\tau))LGG^{\top}L^{\top}\operatorname{expm}(A(\Delta t-\tau))^{\top}\,d\tau\). Equation 4 (p. 5) writes that the integral exhibits covariance \(Q_{\Delta t}=\operatorname{irow}(A^{\#^{-1}}[e^{A^{\#}\Delta t}-I]\operatorname{row}(Q))\) with \(A^{\#}=A\otimes I+I\otimes A\). The homogeneous-process consequence (`ξ`, `z` given) is \(Q_{\Delta t}=\operatorname{cov}(\eta_{ti}\mid\eta_{t-1,i})\) and \(\operatorname{cov}(\eta_{ti},\eta_{t-1,i})=A_{\Delta t}\operatorname{cov}(\eta_{t-1,i})\). The law of total variance on that pair is \(\operatorname{Var}(\eta_{ti})=A_{\Delta t}\operatorname{Var}(\eta_{t-1,i})A_{\Delta t}^{\top}+Q_{\Delta t}\). As \(\Delta t\to\infty\) with stable \(a<0\), Eq. 4 becomes \(-q/(2a)\). The JSS summary names that limit `asymDIFFUSION` and takes it as the total within-subject variance (p. 16). Section 4.3 (pp. 9–10) constrains a stationary `T0VAR` to that model-predicted variance and distinguishes it from a predetermined first occasion. The same section (p. 9) adds a stable trait process with `DRIFT` and `DIFFUSION` fixed to zero; `TRAITVAR` is that time-invariant between-subject variance. Table 3 (p. 13) names `T0TIPREDEFFECT` the effect of time-independent predictors on latents at `T0` and names `TIPREDEFFECT` `B` separately. The JSS article has no numbered §2.2 (2.1 is Continuous time and SEM; §3 follows). This slice takes scalar \(L=1\) (every latent subject to system noise); it does not implement the 0/1 selector matrix and is not a Kalman filter. Discrete auto-effects are strictly positive for finite real drift and interval; the inverse \(a=\ln\varphi/\Delta t\) therefore requires \(\varphi>0\). Binary64 `exp` of a large negative argument is `+0` and is refused as a discrete lag; the same underflow is a vanishing lagged covariance and is kept. When `exp` of a finite `a Δt` overflows on the Table 3 first-summand carry `e^{A Δt} t0_b z` / `e^{A Δt} t0_m x0`, rewrite as `sign(shift) exp(ln|shift| + a Δt)`; an overflowing `a Δt` product fails closed. Integration tests execute those overflow-rewrite arms on the non-`cfg(test)` instantiation (nightly branch coverage on #49 head `d634f58` was 1718/1720 at those two `if !drift_interval.is_finite()` sites). Equations 3–4 of Voelkle et al. (2012) are the discouraged difference-quotient approximation. Meredith (1993) remains unread (Unpaywall/OpenAlex 2026-08-22T23:12Z: `is_oa: false`; Springer `content/pdf` is HTML 200, not a PDF; archive.org title search empty). Mislevy (1991, *Psychometrika, 56*, 177–196, DOI 10.1007/bf02294457) remains unread (Unpaywall/OpenAlex 2026-08-22T23:12Z: `is_oa: false`; Springer `content/pdf` is HTML 200; ETS landing page is HTML; ETS RR-88-45 PDF 404; Wiley PDF 403). ERIC ED334221 is Singer and Willett (1991), *From whether to when*, not the 1991 journal article. ERIC ED333032 is Mislevy, Sheehan, and Wingersky (1990), ETS RR-90-17-ONR, not the 1991 journal article. The 1988 ETS RR-88-45 / DTIC ADA200179 technical report of the same title is not the 1991 journal article. Oud and Jansen (2000) remains unread (Unpaywall/OpenAlex 2026-08-18T21:07Z: closed; Radboud landing and bitstream 403). +The Voelkle et al. (2012) ZORA accepted manuscript was re-opened 2026-08-18T21:07Z (https://www.zora.uzh.ch/handle/20.500.14742/72792; bitstream `424f9082-0eeb-4a67-b687-9845a4ed892f`). Page 16 writes discrete auto-effects as \(A^{*}(\Delta t)=\exp(A\Delta t)\) (Eq. 7). Introducing Intercepts (manuscript p. 20, Eq. 12) adds a continuous-time intercept \(b\) and writes the expected-value solution whose discrete increment is \(A^{-1}(\exp(A\Delta t)-I)b\); the scalar constant-predictor effect is \(b^{*}_{y.x}(\Delta t)=(a_{yx}/a_{xx})(\exp(a_{xx}\Delta t)-1)\). The next paragraph (manuscript p. 21, Eq. 14) writes the discrete effect of a **time-varying** predictor, when the sampling interval equals the constancy interval, as \(b^{*}_{y.x}(\Delta t)=a_{yx}\Delta t\). That product does not depend on the predictor auto-effect. The manuscript calls Eq. 14 a first-order approximation that deteriorates as \(\Delta t\) grows, and defers unmatched sampling/constancy intervals to Oud and Jansen (2000), which is unread. Driver, Oud, and Voelkle (2017, Eq. 3 and p. 4; JSS PDF opened 2026-08-21T13:08Z from https://www.jstatsoft.org/index.php/jss/article/download/v077i05/1104) write \(A_{\Delta t}=\operatorname{expm}(A\Delta t)\), the discrete intercept \(b_{\Delta t}=A^{-1}[A_{\Delta t}-I]b\), and the discrete process-noise covariance \(Q_{\Delta t}=\int_{0}^{\Delta t}\operatorname{expm}(A(\Delta t-\tau))LGG^{\top}L^{\top}\operatorname{expm}(A(\Delta t-\tau))^{\top}\,d\tau\). Equation 4 (p. 5) writes that the integral exhibits covariance \(Q_{\Delta t}=\operatorname{irow}(A^{\#^{-1}}[e^{A^{\#}\Delta t}-I]\operatorname{row}(Q))\) with \(A^{\#}=A\otimes I+I\otimes A\). The homogeneous-process consequence (`ξ`, `z` given) is \(Q_{\Delta t}=\operatorname{cov}(\eta_{ti}\mid\eta_{t-1,i})\) and \(\operatorname{cov}(\eta_{ti},\eta_{t-1,i})=A_{\Delta t}\operatorname{cov}(\eta_{t-1,i})\). The law of total variance on that pair is \(\operatorname{Var}(\eta_{ti})=A_{\Delta t}\operatorname{Var}(\eta_{t-1,i})A_{\Delta t}^{\top}+Q_{\Delta t}\). As \(\Delta t\to\infty\) with stable \(a<0\), Eq. 4 becomes \(-q/(2a)\). The JSS summary names that limit `asymDIFFUSION` and takes it as the total within-subject variance (p. 16). Section 4.3 (pp. 9–10) constrains a stationary `T0VAR` to that model-predicted variance and distinguishes it from a predetermined first occasion. The later-occasion variance of that predetermined first occasion is `trait + e^{2 a Δt} p_0 + Q_Δt + (B / a)² v` (JSS PDF re-opened 2026-08-23T20:20Z). The same section (p. 9) adds a stable trait process with `DRIFT` and `DIFFUSION` fixed to zero; `TRAITVAR` is that time-invariant between-subject variance. Table 3 (p. 13) names `T0TIPREDEFFECT` the effect of time-independent predictors on latents at `T0` and names `TIPREDEFFECT` `B` separately. The JSS article has no numbered §2.2 (2.1 is Continuous time and SEM; §3 follows). This slice takes scalar \(L=1\) (every latent subject to system noise); it does not implement the 0/1 selector matrix and is not a Kalman filter. Discrete auto-effects are strictly positive for finite real drift and interval; the inverse \(a=\ln\varphi/\Delta t\) therefore requires \(\varphi>0\). Binary64 `exp` of a large negative argument is `+0` and is refused as a discrete lag; the same underflow is a vanishing lagged covariance and is kept. When `exp` of a finite `a Δt` overflows on the Table 3 first-summand carry `e^{A Δt} t0_b z` / `e^{A Δt} t0_m x0`, rewrite as `sign(shift) exp(ln|shift| + a Δt)`; an overflowing `a Δt` product fails closed. Integration tests execute those overflow-rewrite arms on the non-`cfg(test)` instantiation (nightly branch coverage on #49 head `d634f58` was 1718/1720 at those two `if !drift_interval.is_finite()` sites). Equations 3–4 of Voelkle et al. (2012) are the discouraged difference-quotient approximation. Meredith (1993) remains unread (Unpaywall/OpenAlex 2026-08-22T23:12Z: `is_oa: false`; Springer `content/pdf` is HTML 200, not a PDF; archive.org title search empty). Mislevy (1991, *Psychometrika, 56*, 177–196, DOI 10.1007/bf02294457) remains unread (Unpaywall/OpenAlex 2026-08-22T23:12Z: `is_oa: false`; Springer `content/pdf` is HTML 200; ETS landing page is HTML; ETS RR-88-45 PDF 404; Wiley PDF 403). ERIC ED334221 is Singer and Willett (1991), *From whether to when*, not the 1991 journal article. ERIC ED333032 is Mislevy, Sheehan, and Wingersky (1990), ETS RR-90-17-ONR, not the 1991 journal article. The 1988 ETS RR-88-45 / DTIC ADA200179 technical report of the same title is not the 1991 journal article. Oud and Jansen (2000) remains unread (Unpaywall/OpenAlex 2026-08-18T21:07Z: closed; Radboud landing and bitstream 403). ## Formula notes @@ -118,6 +120,8 @@ The Voelkle et al. (2012) ZORA accepted manuscript was re-opened 2026-08-18T21:0 - **Lagged stationary observed covariance.** Driver et al. (2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T19:13Z): independent `ε_t` does not enter `cov(y_t,y_{t-1})`. The scalar composition is `λ²(trait + e^{aΔt}(−q/(2a)) + (B/a)²v) + ψ`. Form the lagged latent covariance first, then `λ²c+ψ`. A zero loading is exactly `ψ`. A zero trait, a zero diffusion, and a zero TI contribution is exactly `ψ`. `MANIFESTVAR` is not this composition. Contemporaneous `Var(y_0)` includes `θ` and is not this composition. The lagged latent covariance is not this observed covariance. An overflowing product or sum fails closed. This is not a Kalman filter and not ctsem estimation. - **Later-occasion stationary latent variance.** Driver et al. (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-22T23:12Z): the unconditional variance at a later event occasion of the constrained process is `trait + e^{2aΔt}(−q/(2a)) + Q_Δt + (B/a)²v`. Form the evolved within-subject variance `e^{2aΔt}(−q/(2a))+Q_Δt` first, then include the trait, then include the TI extra variance, then add. Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. Under stationarity `e^{2aΔt}p+Q_Δt=p`, so this composition equals contemporaneous `T0VAR`. Evolving the constrained total as if it were all state is not this map. The lagged covariance omits `Q_Δt` and is not this map. `Q_Δt` is not this map. The interval must be event time and strictly positive. An overflowing product or sum fails closed. This is not a Kalman filter and not ctsem estimation. - **Later-occasion stationary observed variance.** Driver et al. (2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-22T23:12Z): `Var(y_t)=λ²(trait + e^{2aΔt}(−q/(2a)) + Q_Δt + (B/a)²v) + θ + ψ`. Form the later-occasion latent variance first, then `λ²p+θ+ψ`. A zero loading is exactly `θ+ψ`. A zero trait, a zero diffusion, and a zero TI contribution is exactly `θ+ψ`. Under stationarity that composition equals contemporaneous `Var(y_0)`. The lagged observed covariance omits `Q_Δt` and `θ`. `MANIFESTVAR` is not this composition. The later-occasion latent variance is not this observed variance. An overflowing product or sum fails closed. This is not a Kalman filter and not ctsem estimation. +- **Later-occasion predetermined latent variance.** Driver et al. (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; Table 2, p. 12; p. 16; §7.2, pp. 20–21; JSS PDF re-opened 2026-08-23T20:20Z): the first time point is predetermined when no assumptions are made about the process prior to the initial time point. Free `T0VAR` `p_0` is then estimated. The unconditional variance at a later event occasion is `trait + e^{2aΔt} p_0 + Q_Δt + (B/a)²v`. Form the evolved free first-occasion variance `e^{2aΔt}p_0+Q_Δt` first, then include the trait, then include the TI extra variance, then add. Trait variance and `addedTIPREDVAR` do not enter `Q_Δt`. Setting `p_0=−q/(2a)` recovers the stationary later-occasion map. Stationary later variance uses `−q/(2a)` in place of `p_0` and is not this map when `p_0` is free. Evolving `trait+p_0+(B/a)²v` as if it were all state is not this map. Free `p_0` is not this map. As `Δt→∞` with stable `a<0` the carried `p_0` vanishes and `Q_Δt` approaches `−q/(2a)`, so the composition approaches contemporaneous stationary `T0VAR`. As `Δt→0+` the composition approaches `trait+p_0+(B/a)²v`. Nonzero diffusion with `a≥0` is a growing process and is kept. The interval must be event time and strictly positive. An overflowing product or sum fails closed. This is not a Kalman filter and not ctsem estimation. +- **Later-occasion predetermined observed variance.** Driver et al. (2017, Eq. 5, p. 5; Eq. 3–4, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-23T20:20Z): `Var(y_t)=λ²(trait + e^{2aΔt} p_0 + Q_Δt + (B/a)²v) + θ + ψ`. Form the predetermined later-occasion latent variance first, then `λ²p+θ+ψ`. A zero loading is exactly `θ+ψ`. A zero trait, a zero initial variance, a zero diffusion, and a zero TI contribution is exactly `θ+ψ`. Setting `p_0=−q/(2a)` recovers the stationary later-occasion observed variance. Stationary later observed variance is not this composition when `p_0` is free. `MANIFESTVAR` is not this composition. The predetermined later-occasion latent variance is not this observed variance. An overflowing product or sum fails closed. This is not a Kalman filter and not ctsem estimation. - **Level-change discrete increment.** Driver et al. (2017, §7.2, pp. 20–21; Eq. 3, pp. 4–5; Table 2, p. 12; JSS PDF re-opened 2026-08-20T19:50Z): Equation 3 maps `CINT` through `A^{-1}[e^{AΔt}−I]κ`. With `κ=−a m x` the scalar increment is `(e^{aΔt}−1)/a·(−a m x)=(1−e^{aΔt})m x`. Form the level-change `CINT` first, then the discrete intercept map. Underflow of `e^{aΔt}` to `+0` keeps `m x`. A zero effect or zero predictor is exactly zero. `(1−e^{aΔt})m x` is not `m x`, not `κ`, and not `A^{-1}[e^{AΔt}−I]Bz`. An overflowing product or increment fails closed. This is not a Kalman filter and not ctsem estimation. - **CWC-then-lag.** Sample cluster means are removed first. Consecutive within residuals are then fitted by least squares to \(r_{t+\Delta t}\approx\exp(a\Delta t)\,r_{t}\) on event time. Same-sign pair-wise logs initialize the scalar Newton step. Sign-flipping \(T=2\) CWC pairs have no real logarithm and fail closed. Curran and Bauer (2011, pp. 607–608) show that this person-mean subtraction on a raw autoregressive series does **not** isolate the lagged within-person effect; the helper therefore does not claim to recover the raw-process drift. - **Already-centered irregular residual.** The caller supplies lagged within residuals. The mean of \(a=\ln(r_{t+\Delta t}/r_t)/\Delta t\) is the exact scalar map. Intervals may be irregular. The helper does not center again. This is not DSEM. @@ -165,6 +169,8 @@ The Voelkle et al. (2012) ZORA accepted manuscript was re-opened 2026-08-18T21:0 - Driver et al. (2017, Eq. 5 of lagged §4.3 `T0VAR`; Table 2, p. 12; JSS PDF re-opened 2026-08-22T19:13Z) recovers a known \(\operatorname{cov}(y_t,y_{t-1})=\lambda^{2}(\mathrm{trait}+e^{a\Delta t}(-q/(2a))+(B/a)^{2}v)+\psi\) at machine-scale RMSE, and that RMSE is smaller than treating `MANIFESTVAR`, contemporaneous \(\operatorname{Var}(y_0)\), or the lagged latent covariance as that observed covariance; a zero loading is \(\psi\); independent \(\varepsilon_t\) does not enter; a non-event clock, a non-positive interval, and an overflowing product or sum fail closed; - Driver et al. (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; p. 16; JSS PDF re-opened 2026-08-22T23:12Z) recovers a known later-occasion stationary `T0VAR` \(\mathrm{trait}+e^{2a\Delta t}(-q/(2a))+Q_{\Delta t}+(B/a)^{2}v\) at machine-scale RMSE, and that RMSE is smaller than treating lagged covariance, \(e^{2a\Delta t}\) of the constrained total plus \(Q_{\Delta t}\), or \(Q_{\Delta t}\) as that later map; under stationarity the recovered variance equals contemporaneous `T0VAR` at both a large finite \(\Delta t\) and a vanishing interval; a zero trait, a zero diffusion, and a zero TI contribution is exactly zero; a zero diffusion and a zero TI contribution is exactly the trait; \(a\ge 0\) with a nonzero diffusion or TI contribution fails closed; a non-event clock, a non-positive interval, and an overflowing product or sum fail closed; - Driver et al. (2017, Eq. 5 of later-occasion §4.3 `T0VAR`; Table 2, p. 12; JSS PDF re-opened 2026-08-22T23:12Z) recovers a known \(\operatorname{Var}(y_t)=\lambda^{2}(\mathrm{trait}+e^{2a\Delta t}(-q/(2a))+Q_{\Delta t}+(B/a)^{2}v)+\theta+\psi\) at machine-scale RMSE, and that RMSE is smaller than treating `MANIFESTVAR`, lagged \(\operatorname{cov}(y_t,y_{t-1})\), or the later-occasion latent variance as that observed variance; under stationarity \(\operatorname{Var}(y_t)\) equals contemporaneous \(\operatorname{Var}(y_0)\); a zero loading is \(\theta+\psi\); a non-event clock, a non-positive interval, and an overflowing product or sum fail closed; +- Driver et al. (2017, §4.3, pp. 9–10; Eq. 3–4, pp. 4–5; p. 16; JSS PDF re-opened 2026-08-23T20:20Z) recovers a known predetermined later-occasion `T0VAR` \(\mathrm{trait}+e^{2a\Delta t}p_0+Q_{\Delta t}+(B/a)^{2}v\) at machine-scale RMSE, and that RMSE is smaller than treating stationary later-occasion variance, \(e^{2a\Delta t}\) of \(\mathrm{trait}+p_0+(B/a)^{2}v\) plus \(Q_{\Delta t}\), or free \(p_0\) as that later map; setting \(p_0=-q/(2a)\) recovers the stationary later-occasion map; a large finite \(\Delta t\) approaches contemporaneous stationary `T0VAR`; a vanishing interval approaches \(\mathrm{trait}+p_0+(B/a)^{2}v\); a zero trait, a zero initial variance, a zero diffusion, and a zero TI contribution is exactly zero; a zero initial variance, a zero diffusion, and a zero TI contribution is exactly the trait; nonzero diffusion with \(a\ge 0\) is a growing process and is kept; \(a\ge 0\) with a nonzero TI contribution fails closed; a non-event clock, a non-positive interval, and an overflowing product or sum fail closed; +- Driver et al. (2017, Eq. 5 of predetermined later-occasion §4.3 `T0VAR`; Table 2, p. 12; JSS PDF re-opened 2026-08-23T20:20Z) recovers a known \(\operatorname{Var}(y_t)=\lambda^{2}(\mathrm{trait}+e^{2a\Delta t}p_0+Q_{\Delta t}+(B/a)^{2}v)+\theta+\psi\) at machine-scale RMSE, and that RMSE is smaller than treating `MANIFESTVAR`, the predetermined later-occasion latent variance, or stationary later-occasion observed variance as that observed variance when \(p_0\) is free; setting \(p_0=-q/(2a)\) recovers the stationary later-occasion observed variance; a zero loading is \(\theta+\psi\); a non-event clock, a non-positive interval, and an overflowing product or sum fail closed; - pooling discrete lags from unequal intervals fails closed; - CWC-then-lag on a two-cluster decaying series has smaller computed RMSE than a level-pooled series when the latter is identified; - already-centered irregular residuals recover a known drift at machine-scale RMSE, and that RMSE is smaller than CWC of the corresponding raw autoregressive series (Curran & Bauer, 2011, pp. 607–608);