feat(psychometric): recover Driver p.16 MANIFESTVARstd after positive Θ

[seonghobae]

Stacked on #208 (MANIFESTTRAITVARstd). Next licensed 2017-era summary.ctsemFit.R std after the measurement-side trait matrix is MANIFESTVARstd (Θ correlation; counterpart to Ψ_τ).

Driver, Oud, and Voelkle (2017, Table 2, p. 12 MANIFESTVAR Θ; Eq. 5, p. 5; p. 16 suffix std; footnote 4; JSS PDF re-opened 2026-08-23T22:40Z) name MANIFESTVAR the residual covariance of the indicators. The 2017-era source forms MANIFESTVARstd whenever verbose = TRUE:

MANIFESTVARdiag <- diag(diag(MANIFESTVAR), n.manifest) + diag(c(ridging), n.manifest)
MANIFESTVARstd <- solve(sqrt(MANIFESTVARdiag)) %&% MANIFESTVAR

OpenMx %&% is t(A) %*% B %*% A. Unlike TRAITVARstd, the 2017-era source does add diag(c(ridging), n.manifest). The default ridging = FALSE adds 0, not 0.0001; that ridge is a numerical hack and is not this exact map. The 2017-era source assigns dimnames(MANIFESTVARstd) to latentNames; the matrix is n.manifest × n.manifest. That assignment is a source bug and is not this exact map. Scalar map after strictly positive MANIFESTVAR is θ / θ = 1.

Recovery

• recover_standardised_manifest_variance(θ, clock) forms strictly positive θ, then 1/√θ, then the quadratic.
• Distinct positive θ recover the same 1 (to_bits equal).
• Zero MANIFESTVAR fails closed. Unlike TRAITVAR / MANIFESTTRAITVAR, the 2017-era source does not skip forming MANIFESTVARstd when θ = 0; solve(sqrt(0)) fails.
• Non-event clock fails closed.
• MANIFESTVAR does not require a < 0.

Claim boundary

• Unstandardised MANIFESTVAR is not MANIFESTVARstd.
• MANIFESTTRAITVARstd is not MANIFESTVARstd even when both equal 1 (indicator-level trait vs contemporaneous measurement error).
• Equation 5 Var(y) = λ² Var(η) + θ is not MANIFESTVARstd.

Meredith (1993) and Mislevy (1991) remain closed (Unpaywall re-checked this cycle: is_oa: false, 0 locations).

Do not merge without an independent APPROVE plus exact-head required-check success. Do not self-approve. Do not request Copilot.

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[seonghobae]

Current-head verification\n\n- Exact head: 22202c57c0818d3efb28438e95bdde5d0e7143b0\n- Exact base: a65d13db4c21337fe991d9e2dfe09f71ce11aa96\n- Classification: WAIT_AND_REMEDIATE (draft PR; protected merge path is unavailable).\n- Root cause fixed: rustfmt drift in the stacked psychometric tests and the final production branch uncovered edge.\n- Local evidence: workspace tests, clippy -D warnings, rustfmt, rustdoc -D warnings, Python quality coverage 100%, documentation/workspace contracts, line coverage LCOV gate, branch coverage 1834/1834 (100%), and git diff --check passed.\n- Hosted evidence: all six required checks are newly queued for this exact head; no result is reused from the predecessor head.\n- Review state: no qualifying independent approval yet; no force-merge candidate.\n\nNext action: re-fetch exact-head checks and review state, then process the next stack item while this head runs.