# Claim-free OID method

## Scope and claim levels

- **Mathematical theorem/finite-dimensional consequence:** weighted POD orthogonality; least-squares normal equations; right-inverse identities on the generated range; LR minimum training mean-square state residual; LE minimum `G`-norm field among exact generated-range inverses.
- **Standard extension:** diagonal spatial quadrature/masks, sign-invariant MAC, principal angles, projector similarity, and delayed sample pairing.
- **Project choice:** state domain, metric `G`, observable metric `H`, POD rank, delay, truncation tolerance, and observable definition. Each changes the question.
- **Unsupported claim:** importance, causal effect, physical mechanism, controller necessity, controllability, or actuator/control energy inferred from reconstruction scores or modes.

For centered retained state coefficients `a` and observable POD coordinates `b`, the forward map is the minimum-MSE linear regression `C = Sigma_ba Sigma_aa^+`, so `b_hat=C a`. If `C` is rank deficient, active code either fails or explicitly restricts all identities to `range(C)`.

The LR inverse is `L_R = Sigma_aa C^T (C Sigma_aa C^T)^+`. On the generated range it is an exact right inverse and minimizes empirical state reconstruction MSE among exact linear inverses. “Minimum MSE linear inverse” does not mean most-probable nonlinear state.

For a declared positive-definite state-field metric `G`, the LE inverse is `L_E = G^-1 C^T (C G^-1 C^T)^+`. It minimizes the `G`-norm of the reconstructed field under the empirical mapping constraint. This is not control energy, actuator cost, controllability, or a Navier–Stokes energy principle.

`q_ctl-q_blk` is a same-parent, time-paired difference between closed-loop branches. Feedback changes future state and future action together; this difference is not an isolated action effect, autonomous perturbation, tangent state, or causal intervention estimate.

## Schlegel section and equation map

The active finite-dimensional implementation uses Schlegel et al.'s notation only where the mathematical objects match:

- Section 2.3, equations (2.8)-(2.10): the linear stochastic estimate `b = C a`, observable quadratic form, and EPOD context. Active `C = Sigma_ba Sigma_aa^+` is the fitted regression map, not raw cross-covariance.
- Section 2.4, equations (2.11)-(2.15): the delayed linear OID assumption, its finite-dimensional map, and modes obtained by applying an inverse map to observable POD directions. Active delayed pairing implements `b(t+tau) = C(tau) a(t)` without padding or crossing declared segments.
- Sections 2.5-2.6: pseudoinverse choice, LR/LE variants, snapshot algorithm, and delay selection context. This project fixes delay and rank by declared analysis choices; it does not infer causality from a fitted delay.
- Appendix, especially (A 1)-(A 4): generated-range projection/right-inverse constraint and least-residual versus least-energetic variational definitions. Active code restricts identities to `range(C)` when truncation is explicitly enabled.

Schlegel's application language about observer/control design is not inherited as a result here. In particular, this project's `G`-minimum is a declared state metric, not actuator energy or a Lyapunov-control certificate.

## Literature support matrix

| Source | Direct support | Boundary |
|---|---|---|
| Schlegel et al. (2012) | LR/LE forward/inverse construction and observable-linked modes | Does not license this project's importance, causal, or controller claims |
| Sirovich (1987) | Method of snapshots for POD | Weighting, masks, ranks, and domains here are project choices/standard extensions |
| Adrian (1994), LSE | Conditional linear minimum-mean-square estimation | A regression estimate is not causal or a most-probable nonlinear state |
| Borée (2003), EPOD | Correlated-field extension of POD and its connection to LSE; Schlegel identifies prefiltered LR-OID with EPOD | Equivalence requires the stated POD prefiltering and metric assumptions |
| Everson & Sirovich (1995), gappy POD | Least-squares recovery from incomplete snapshots in a fixed empirical basis | Active OID is not gappy POD; local sensors overlapping the state domain require explicit overlap analysis |
| Theiler et al. (1992) | Broad rationale for surrogate/null testing of time series | The project's circular-shift null, exclusions, and selection loop are not directly derived from Theiler |

Bibliographic leads do not upgrade project choices to theorems. Any new study must inspect the primary source and state its exact assumptions.

## Bibliography

- Schlegel, M., Noack, B. R., Jordan, P., Dillmann, A., Gröschel, E., Schröder, W., Wei, M., Freund, J. B., Lehmann, O., & Tadmor, G. (2012). “On least-order flow representations for aerodynamics and aeroacoustics.” *Journal of Fluid Mechanics*, 697, 367-398. DOI: 10.1017/jfm.2012.70.
- Sirovich, L. (1987). “Turbulence and the dynamics of coherent structures. I. Coherent structures.” *Quarterly of Applied Mathematics*, 45(3), 561-571. DOI: 10.1090/qam/910462.
- Adrian, R. J. (1994). “Stochastic estimation of conditional structure: a review.” *Applied Scientific Research*, 53, 291-303. DOI: 10.1007/BF00849106.
- Borée, J. (2003). “Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows.” *Experiments in Fluids*, 35, 188-192. DOI: 10.1007/s00348-003-0656-3.
- Everson, R., & Sirovich, L. (1995). “Karhunen-Loève procedure for gappy data.” *Journal of the Optical Society of America A*, 12(8), 1657-1664. DOI: 10.1364/JOSAA.12.001657.
- Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., & Farmer, J. D. (1992). “Testing for nonlinearity in time series: the method of surrogate data.” *Physica D: Nonlinear Phenomena*, 58(1-4), 77-94. DOI: 10.1016/0167-2789(92)90102-S.
