Admissibility of lagged state space models through the ARIMA form of the whole model · C1

D4 per-lag check: The test applied by smoothEigens() of smooth 4.6.0 and of its repository until commit 4a44c1c (src/headers/eigenCalc.h) to models without adaptive regressors: every eigenvalue of every diagonal block of $D$, one block per unique lag, has modulus below 1; reimplemented in numpy as per_lag_max().

D5 PBH reference: The largest modulus among the eigenvalues $\lambda$ of the expanded discount matrix for which the smallest singular values of $[\lambda I - \tilde D;\ \tilde w']$ and of $[\lambda I - \tilde D,\ \tilde g]$ both exceed $10^{-7}(\|\tilde D\|_2 + \|\tilde g\| + \|\tilde w\|)$ (Popov-Belevitch-Hautus rank tests), computed with numpy as pbh_max().

D6 cyclotomic cancellation: The procedure that computes the coefficients of $$\begin{aligned} \theta(B) &= \det(I - D L(B)) \\ \phi(B) &= \det(I - F L(B)) \end{aligned}$$ by an inverse discrete Fourier transform of their values at the $N$-th roots of unity ($N$ the smallest power of 2 above $\sum_j l_j$), and divides both by every cyclotomic polynomial $\Phi_n(B)$, $n$ dividing one of the lags, for as long as it divides both with a remainder at most $10^{-9}$ times the sum of absolute coefficients; the quotient of $\theta$ is the cancelled MA polynomial $\theta^*(B)$.

D7 step-down test: The Levinson-Durbin step-down (Schur-Cohn) recursion on $1 + c_1 B + \dots + c_n B^n$: for $j = n, \dots, 1$, $$\kappa_j = c_j, \qquad c_i \leftarrow \frac{c_i - \kappa_j c_{j-i}}{1 - \kappa_j^2};$$ the test passes when every $|\kappa_j| < 1$, which holds if and only if all roots lie outside the unit circle (the $\kappa_j$ are the partial autocorrelations of the autoregression with this polynomial). Source: Barndorff-Nielsen and Schou (1973).

D8 GUM: Generalised Univariate Model of the smooth package: a lagged SSOE model whose $F$, $g$ and $w$ are all estimated. Source: Svetunkov (2016).

D9 ETS(A,N,A): Exponential smoothing with additive error, no trend and additive seasonality of period $m$; here in the lagged form with a level of lag 1 and one seasonal state of lag $m$, $F = I$, $w = (1, 1)'$, $g = (\alpha, \gamma)'$. Source: Hyndman et al. (2008).

D10 check of the whole model: Admissibility of a lagged SSOE model from the MA polynomial $$\theta(B) = \det(I - D L(B))$$ of its ARIMA form: $\theta$ is divided by its common factors with $\phi(B) = \det(I - F L(B))$ by cyclotomic cancellation, and the model is accepted when the cancelled polynomial $\theta^*$ has no root in $|B| \le 1$ and, for a model with ARIMA components, the AR polynomial of its ARIMA part is stationary; its largest mode modulus is $1/\min|B|$ over the roots of $\theta^*$. In the numpy prototype (admissibility.py, confirm.py, ets_arima.py) the roots come from numpy; in smooth the step-down test decides and bisection gives the modulus.

D41 exact oracle: Admissibility decided in exact rational arithmetic with certified root enclosures (independent/oracle.py, python-flint 0.9.0), written by the contributor P5 from the definitions of this claimset without the prototype: each float of $F$, $g$, $w$ is converted exactly to a rational, $D = F - g w'$, and $$\theta(B) = \det(I - D L(B)), \qquad \phi(B) = \det(I - F L(B))$$ by fraction-free elimination over $\mathbb{Q}[B]$; $G = \gcd(\theta, \phi)$ exactly and $\theta_G = \theta / G$; $C$ is the product of the cyclotomic polynomials $\Phi_n$, $n$ dividing a lag, at the multiplicity dividing both, and $\theta_C = \theta / C$; the roots of $\theta_G$, $\theta_C$ and $G / C$ are enclosed in Arb balls of 128 bits, raised to 512 and 2048 bits while a modulus ball contains 1, and a root counts as on the unit circle only when it is proved a root of unity whose $\Phi_n$ divides the polynomial. The largest mode modulus is $1/\min|B|$ over the roots; a model is admissible when it is below 1.

D17 ETS(A,N,A) grid: The 3042 ETS(A,N,A) models with $m = 4$ and $m = 12$ and $\alpha$ and $\gamma$ on the grid 0.05, 0.10, ..., 1.95 (confirm.py, plan PL1).

D18 random GUM models with lags (1, 12): 200 GUM models with lags (1, 12): entries of $F$ normal with mean 0 and standard deviation 0.35, of $g$ with mean 0 and standard deviation 0.175, of $w$ with mean 1 and standard deviation 0.175 (confirm.py, seed 2026).

D20 random additive ETS and GUM models with long seasonal lags: The 575 models of plan PL4 (confirm.py --test stepdown --seed 2026 --n 200): the 500 models of plan PL3 generated afresh, 50 additive ETS models with seasonal lags from {24, 168, 365} and 25 GUM models with lags (1, 24, 168) and normal $F$ (standard deviation 0.3). The generator allows MA polynomial degrees up to 536 (level, trend, seasonal lags 168 and 365, constant).

D42 models of the planned tests: The models of the planned tests PL1-PL6 and the two examples of C10 and C11, drawn again by export_models.py exactly as confirm.py and cpp_check.py draw them (the counts of the original tests are reproduced), with the prototype's values for each: 3242 models of PL1 and PL2, 500 of PL3, 575 of PL4 and PL5 (MA degree up to 535), 800 of PL6 and 2 examples; 5119 in all.

D29 admissibility decision: Whether a method accepts a model as admissible (decaying weights); compared between a method and a reference as the number of models on which the two differ.

D30 largest mode modulus: The largest modulus of the modes that decide admissibility: for the PBH reference the largest $|\lambda|$ of the reachable and observable eigenvalues of the expanded discount matrix, for the check of the whole model $1/\min|B|$ over the roots $B$ of the cancelled MA polynomial, for the per-lag check the largest $|\lambda|$ of its blocks; a model is admissible when it is below 1.

D44 non-cyclotomic shared factor in the closed unit disc: Whether $G / C$, the factor shared by $\theta$ and $\phi$ after removing the cyclotomic polynomials $\Phi_n$ ($n$ dividing a lag), has a root in $|B| \le 1$, certified by the exact oracle; counted over models.

D26 comparison with a reference: Each model's admissibility decision, and its largest mode modulus where both find it inadmissible, computed by the method studied and by a reference on the same model (the PBH reference, the roots by numpy, the numpy prototype of the check of the whole model or an exact oracle); the result is the number of models whose decisions differ and the largest absolute difference of the modulus. In the planned tests, models whose reference modulus lies within 1e-8 of 1 are left out and counted.

D43 comparison with the exact oracle: For each model, the admissibility decision and largest mode modulus of a method of the study against those of the exact oracle: the PBH reference against $\theta_G$; the prototype's cancellation, step-down test and bisection against $\theta_C$. Counted are the models whose decisions differ and the largest absolute difference of the modulus on the models both find inadmissible (for the bisection, on every model with a value). Models the oracle cannot decide, and models whose certified largest modulus lies within $10^{-8}$ of 1, are left out and counted.

D2 expanded discount matrix: The transition matrix $\tilde D$ of size $\sum_j l_j$ of the lag-1 system equivalent to $v_t = D v_{t-l}$, in which component $j$ carries itself and its $l_j - 1$ lagged copies; $\tilde F$, $\tilde g$ and $\tilde w$ are expanded in the same way, so that $\tilde D = \tilde F - \tilde g \tilde w'$.

D3 decaying weights: A lagged SSOE model has decaying weights when the weights $\tilde w' \tilde D^j \tilde g$ of the past observations in its fitted values tend to zero, that is, when every eigenvalue $\lambda$ of its expanded discount matrix that is reachable from $\tilde g$ and observable through $\tilde w$ has $|\lambda| < 1$; the largest such $|\lambda|$ is its largest mode modulus. A mode that $\tilde g$ does not reach but $\tilde w$ sees carries an initial state whose effect is not covered: with $|\lambda| > 1$ it grows.