Admissibility of lagged state space models through the ARIMA form of the whole model: claim C16

The claim

88C16 · methodological · planned test · novelty: new, by the authors · no close match on EntryLib

On 800 random additive ETS and GUM models in four families, the implementation of the check in smooth (smoothEigensCpp() through smoothEigensR()) gave the same decision as the numpy prototype on every model, a largest root modulus within the bisection tolerance of it on the models both found inadmissible, and the same modulus and MA polynomial on three consecutive calls with each model.

Scope: population: pure additive lagged SSOE models; conditions: additive ETS: level; no, additive or damped trend; 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; optional constant; some persistence fixed at 0, additive ETS with 0 to 2 seasonal lags from {24, 168, 365}, GUM with F = I and lags from {1, 4, 12}, GUM with random F and lags from {1, 4, 12, 24}, R 4.3.3, -O0 build of pkgload; not for: ARIMA components, CES, adaptive regressors, multiplicative models, seasonal harmonics with non-integer periods (TBATS) · Simulation: E13 (T1.r1, T1.r2, T1.r3, T1.r4), artefacts A5, A8 · Uses: C3, C4, C18, C20 · Assumes: C19 · Limitations: L2, L3, L4, L8 · Supersedes: Svetunkov (2016), smoothEigensCpp(), smooth 4.6.0 · Planned: lab plan PL6, planned 22 October 2026 · Outcome: computed from the recorded results

95C3 · methodological · planned test · novelty: new, by the authors · no close match on EntryLib

On 500 random additive ETS and GUM models, the admissibility decision from the roots of the MA polynomial after cyclotomic cancellation differed from the PBH reference on 0 models, and on the 172 models that both found inadmissible the largest mode modulus differed from the reference by at most 4.4e-13.

Scope: population: pure additive lagged SSOE models; conditions: additive ETS: level; no, additive or damped trend; 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; optional constant; some persistence fixed at 0, GUM with F = I and lags from {1, 4, 12}, GUM with random F and lags from {1, 4, 12, 24}, numpy prototype, models within 1e-8 of the unit circle excluded (none occurred); not for: ARIMA components, CES, adaptive regressors, multiplicative models, seasonal harmonics with non-integer periods (TBATS), the largest mode modulus of admissible models, where a common factor that is not cyclotomic can remain in the cancelled MA polynomial, an eigenvalue of modulus above 1 shared by a reachable and observable mode and an unreachable or unobservable one · Simulation: E3 (R1, R2), artefacts A1, A3 · Uses: C6, C8, C17 · Assumes: C14 · Limitations: L2, L3 · Extends: Hyndman et al. (2008), Chapter 11 · Planned: lab plan PL3, planned 21 October 2026 · Outcome: computed from the recorded results

88C4 · methodological · novelty: new, by the authors · no close match on EntryLib

On 575 random additive ETS and GUM models, including ETS models with up to two seasonal lags from {24, 168, 365} whose generator allows MA polynomial degrees up to 536, the step-down test on the MA polynomial after cyclotomic cancellation gave an admissibility decision different from the roots of that polynomial on 0 models.

Scope: population: pure additive lagged SSOE models; conditions: additive ETS: level; no, additive or damped trend; 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; optional constant; some persistence fixed at 0, GUM with F = I and lags from {1, 4, 12}, GUM with random F and lags from {1, 4, 12, 24}, additive ETS with up to two seasonal lags from {24, 168, 365}, GUM with random F and lags (1, 24, 168), numpy prototype; not for: ARIMA components, CES, adaptive regressors, multiplicative models, seasonal harmonics with non-integer periods (TBATS) · Simulation: E4 (R1, R3), artefacts A1, A3 · Cited: Barndorff-Nielsen and Schou (1973) · Uses: C3 · Assumes: C19 · Limitations: L3, L4, L5, L8 · Extends: Barndorff-Nielsen and Schou (1973) · Planned: lab plan PL4, planned 21 October 2026 · Outcome: computed from the recorded results

Claims it rests on

100C6 · formal · extends earlier work · novelty: extends earlier work, by the authors

For a lagged SSOE model, ignoring the initial states, $\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t,$ so the model has the ARIMA form with AR polynomial $\phi(B) = \det(I - F L(B))$ and MA polynomial $\theta(B) = \det(I - D L(B))$ .

Scope: population: pure additive lagged SSOE models; conditions: identity of lag-operator polynomials, initial states ignored · Proof: E6 · Consensus · Simulation: E10, artefacts A1, A2 · Extends: Hyndman et al. (2008), Chapter 11 · Verification: checked by an agent · Used by: C3, C17

100C8 · formal · novelty: new, by the authors

An eigenvalue $\lambda \neq 0$ of the expanded transition matrix $\tilde F$ with an eigenvector unobservable through $\tilde w$ or a left eigenvector unreachable from $\tilde g$ is an eigenvalue of the expanded discount matrix, and $1/\lambda$ is a common root of $\phi(B) = \det(I - F L(B))$ and $\theta(B) = \det(I - D L(B))$ .

Scope: population: pure additive lagged SSOE models · Proof: E8 · Uses: C15 · Verification: checked by an agent · Used by: C3

100C15 · formal · novelty: new, by the authors

For the lag-1 expansion $\tilde M$ of $v_t = M v_{t-l}$ in a lagged SSOE model, $\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M),$ which equals $\lambda^{\sum_j l_j} \det(I - M L(1/\lambda))$ for $\lambda \neq 0$ .

Scope: population: pure additive lagged SSOE models; conditions: any square matrix $M$ of size $k$ · Proof: E11 · Verification: checked by an agent · Used by: C8, C17

100C17 · formal · novelty: new, by the authors

For a lagged SSOE model with $\theta(B) = \det(I - D L(B))$ and $\phi(B) = \det(I - F L(B))$ , the roots of $\theta(B) / \gcd(\theta(B), \phi(B))$ are exactly the reciprocals $1/\lambda$ of the nonzero eigenvalues $\lambda$ of the reachable and observable part of the expanded discount matrix (its Kalman decomposition), with their multiplicities.

Scope: population: pure additive lagged SSOE models; conditions: the definition D3 agrees with this part unless an eigenvalue is shared by the reachable and observable part and the rest · Proof: E14 · Consensus · Uses: C6, C15 · Verification: checked by an agent · Used by: C3

100C18 · formal · novelty: new, by the authors

For a polynomial $c$ with $c(0) \neq 0$ and $r > 0$ , the step-down test passes on $c(B/r)$ if and only if $1/|B| < r$ for every root $B$ of $c$ .

Scope: population: real polynomials with a nonzero constant term · Proof: E15 · Cited: Barndorff-Nielsen and Schou (1973) · Verification: checked by an agent · Used by: C16

100C20 · formal · novelty: new, by the authors

For a square matrix $M$ of size $k$ and lags $l_1, \dots, l_k$ , $\det(I - M L(B)) = \sum_{S \subseteq \{1, \dots, k\}} (-1)^{|S|} \det(M_{SS})\, B^{\sum_{j \in S} l_j},$ with $\det(M_{\emptyset\emptyset}) = 1$ .

Scope: population: square matrices and positive integer lags · Proof: E16 · Consensus · Verification: checked by an agent · Used by: C16

Evidence

E3 · simulation · cited by C3

Planned test of the cyclotomic cancellation (lab plan PL3, run RN9): confirm.py --test cancel --seed 2026 --n 200 generated 200 additive ETS models (level; no, additive or damped trend with phi in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; beta = 0 with probability 0.1 and each gamma = 0 with probability 0.15; alpha in (0.01, 1.9)), 100 GUM models with F = I, lags from {1, 4, 12}, g uniform on (-0.3, 0.6) and w normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal F (standard deviation 0.35); for each it compared the largest root modulus of the cancelled MA polynomial with the PBH reference.

Setup: studied check of the whole model · cases random additive ETS and GUM models · procedure comparison with a reference, PBH reference · lens admissibility decision, largest mode modulus

R1 admissibility decisions that differ from the PBH reference = 0
R2 max abs difference of max |lambda| from the PBH reference, inadmissible models = 0.0000 (inadmissible_models: 172)
R3 models within 1e-8 of the unit circle, excluded = 0
R4 models compared = 500

E4 · simulation · cited by C4

Planned test of the step-down test (lab plan PL4, run RN10): confirm.py --test stepdown --seed 2026 --n 200 generated the 500 models of plan PL3 afresh plus 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), and compared the step-down decision on the cancelled MA polynomial with the decision from its roots computed by numpy. The generator allows MA polynomial degrees up to 536 (level, trend, seasonal lags 168 and 365, constant); the realised maximum was not recorded as a result.

Setup: studied step-down test · cases random additive ETS and GUM models with long seasonal lags · procedure comparison with a reference, roots by numpy · lens admissibility decision

R1 step-down decisions that differ from the decision by the roots = 0
R2 models within 1e-8 of the unit circle, excluded = 0
R3 models compared = 575

E13 · simulation · cited by C16

Planned test of the implementation in smooth (lab plan PL6, run RN43): cpp_check.py --seed 2027 --n 200 generated 200 models in each of four families (additive ETS with 0-2 seasonal lags from {4, 7, 12, 24, 52}; additive ETS with 0-2 seasonal lags from {24, 168, 365}; GUM with F = I and lags from {1, 4, 12}; GUM with normal F, standard deviation 0.35, and lags from {1, 4, 12, 24}), called smoothEigensR() of the repository (through pkgload::load_all) three times on each, and compared its decision and modulus with the numpy prototype (roots of the cancelled MA polynomial).

Setup: studied smoothEigensCpp() · cases four model families of the implementation test · procedure comparison with a reference, check of the whole model · lens admissibility decision, largest mode modulus

E13.T1 · Comparison of the C++ core (smoothEigensR()) with the numpy prototype on 200 models in each of four families: admissibility decisions that differ, largest absolute difference of the root modulus on the models found inadmissible, models whose cancelled polynomials differ in degree, largest absolute difference between three calls on the same model, and the number of models found inadmissible (models: 200)
Row	family	inadmissible
r1	ETS	86
r2	ETS high degree	95
r3	GUM identity	84
r4	GUM random	27

E10 · simulation · cited by C6

Exploratory run RN2 of explore.py --seed 0: the ARIMA-form polynomials, their cyclotomic cancellation and the PBH reference for ETS(A,N,A,A) with seasonal lags 24 and 168 (alpha = 0.3, gamma = (0.1, 0.05)) and for GUM with orders (2, 2), lags (1, 12), F = I, g = 0.1 and w = 1 (the default starting point of gum()); the determinant identity at random complex B on 100 random GUM models; the expanded discount matrix of a block-diagonal D; and mean times over 50 repeats of three computations for a random GUM with lags (1, 24, 168). The result labelled 'block-diagonal D, lags (1,12)' comes from D = [[0.7, 0], [-0.1, 0.3]], which is lower-triangular rather than block-diagonal.

Setup: studied lagged SSOE model, cyclotomic cancellation, check of the whole model · cases ETS(A,N,A,A) with seasonal lags 24 and 168, starting point of gum() · procedure comparison with a reference, PBH reference · lens largest mode modulus

R11 max relative error of the determinant identity = 0.0000 (models: 100)
R12 max |lambda|, per-lag check = 0.7 (model: block-diagonal D, lags (1,12))
R13 max |lambda|, all modes of the expanded discount matrix = 0.9045 (model: block-diagonal D, lags (1,12))
R14 mean time, ms = 4.6963 (task: coefficients by DFT, degree: 193)
R15 mean time, ms = 0.0069 (task: step-down test, degree: 193)
R16 mean time, ms = 22.2257 (task: all roots, numpy companion eigenvalues, degree: 193)

E10.T1 · Degree of the MA polynomial theta and of its shared cyclotomic factor, and the largest mode modulus from the roots of theta before and after cancellation and from the PBH reference, for ETS(A,N,A,A) with seasonal lags 24 and 168 and for the default starting point of gum() with orders (2, 2) and lags (1, 12)
Row	model	degree of theta	degree of the shared cyclotomic factor	max \|lambda\|, roots of theta before cancellation	max \|lambda\|, roots of theta after cancellation	max \|lambda\|, controllable and observable modes (PBH)
r1	ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05)	193	25	1	1	1
r2	GUM start, orders (2,2), lags (1,12)	26	14	1	0.9829	0.9829

E6 · formal · cited by C6

Derivation of the ARIMA form of a lagged SSOE model by the matrix determinant lemma.

Foundation: linear algebra over the field of rational functions in $B$

$\epsilon_t = y_t - w' v_{t-l} \Rightarrow v_t = D v_{t-l} + g y_t,\ D = F - g w'$ (substitution; from D1)
$v_{t-l} = L(B) v_t \Rightarrow (I - D L(B)) v_t = g y_t$ (definition of $L(B)$; from S1, D1)
$\det(I - D L(B))|_{B=0} = 1 \Rightarrow I - D L(B) \text{ is invertible over rational functions}$ (evaluation at $B = 0$; from S2)
$\epsilon_t = \left(1 - w' L(B) (I - D L(B))^{-1} g\right) y_t$ (substitution; from S1, S2, S3)
$I - D L(B) - g\, w' L(B) = I - F L(B)$ (definition of $D$; from D1)
$\det(I - F L(B)) = \det(I - D L(B))\left(1 - w' L(B)(I - D L(B))^{-1} g\right)$ (matrix determinant lemma $\det(A - g u') = \det(A)(1 - u' A^{-1} g)$; from S3, S5)
$\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t$ (substitution; from S4, S6)

E8 · formal · cited by C8

Unobservable or unreachable modes of the expanded transition matrix are modes of the expanded discount matrix and give common roots of the two ARIMA-form polynomials.

Foundation: linear algebra over the complex numbers

$\tilde F u = \lambda u,\ \tilde w' u = 0 \Rightarrow \tilde D u = \tilde F u - \tilde g (\tilde w' u) = \lambda u$ (substitution of $\tilde D = \tilde F - \tilde g \tilde w'$; from D2)
$u' \tilde F = \lambda u',\ u' \tilde g = 0 \Rightarrow u' \tilde D = u' \tilde F - (u' \tilde g) \tilde w' = \lambda u'$ (substitution of $\tilde D = \tilde F - \tilde g \tilde w'$; from D2)
$\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0$ (identity of the lag-1 expansion; from C15)
$\phi(1/\lambda) = \det(I - F L(1/\lambda)) = 0,\ \theta(1/\lambda) = \det(I - D L(1/\lambda)) = 0$ (substitution; from S1, S2, S3)

E11 · formal · cited by C15

Schur-complement derivation of the characteristic polynomial of the lag-1 expansion.

Foundation: linear algebra over the complex numbers

$\lambda I - \tilde M = \begin{pmatrix} \lambda I_k - A & -B \\ -C & \lambda I - S \end{pmatrix}$ (partition of the states into the $k$ current values (heads) and the $\sum_j (l_j - 1)$ lagged copies (tails); $S$ shifts each tail chain, $C$ feeds each head into its first copy, $A$ and $B$ hold the entries of $M$ that act on heads with lag 1 and on last copies; from D2)
$\det(\lambda I - S) = \lambda^{\sum_j (l_j - 1)}$ (determinant of a lower-triangular matrix with diagonal $\lambda$; from S1)
$A + B (\lambda I - S)^{-1} C = M\, \mathrm{diag}(\lambda^{1 - l_1}, \dots, \lambda^{1 - l_k}),\ \lambda \neq 0$ ($(\lambda I - S)^{-1} C$ maps head $j$ to its copies $\lambda^{-s}$, $s = 1, \dots, l_j - 1$, and $B$ reads copy $l_i - 1$ of component $i$; from S1)
$\det(\lambda I - \tilde M) = \lambda^{\sum_j (l_j - 1)} \det(\lambda I_k - M\, \mathrm{diag}(\lambda^{1 - l_j})) = \det(\Lambda(\lambda) - M),\ \lambda \neq 0$ (Schur complement, then factoring $\lambda\, \Lambda(\lambda)^{-1}$ out on the right; from S2, S3)
$\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M) \text{ for all } \lambda$ (two polynomials equal on $\lambda \neq 0$ are equal; from S4)
$\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0$ (factoring $\Lambda(\lambda)$ out on the right: $\Lambda(\lambda) - M = (I - M \Lambda(\lambda)^{-1}) \Lambda(\lambda)$; from S5, D1)

E14 · formal · cited by C17

The reduced form of the ARIMA polynomials through the Kalman decomposition of the expanded system.

Foundation: linear systems theory over the complex numbers

$\exists T: T^{-1}\tilde F T \text{ block-triangular with a reachable and observable block } (F_{co}, g_{co}, w_{co}),\ \tilde w'(zI - \tilde F)^{-1}\tilde g = w_{co}'(zI - F_{co})^{-1} g_{co}$ (Kalman canonical decomposition; from D2)
$\tilde D = \tilde F - \tilde g \tilde w' \text{ keeps the reachable and the unobservable subspaces, so } D_{co} = F_{co} - g_{co} w_{co}'$ (reachability is invariant under state feedback and observability under output injection; from S1, D2)
$1 + w_{co}'(zI - F_{co})^{-1} g_{co} = \frac{\det(zI - D_{co})}{\det(zI - F_{co})},\ \gcd(\det(zI - D_{co}), \det(zI - F_{co})) = 1$ (matrix determinant lemma; a single-input single-output realisation is minimal if and only if the numerator and denominator of its transfer function are coprime; from S2)
$1 + \tilde w'(zI - \tilde F)^{-1}\tilde g = \frac{\det(zI - \tilde D)}{\det(zI - \tilde F)} = \frac{\theta(1/z)}{\phi(1/z)}$ (matrix determinant lemma and the identity of the lag-1 expansion; from C15, D1)
$\frac{\theta(B)}{\gcd(\theta, \phi)}(B) = 0 \iff B = 1/\lambda,\ \lambda \in \mathrm{spec}(D_{co}) \setminus \{0\} \text{ (with multiplicity)}$ (uniqueness of the reduced fraction, with z = 1/B: the eigenvalues at zero give no root, theta(0) = phi(0) = 1 makes theta / gcd the reduced numerator, and the reversal keeps coprimality away from B = 0; from S3, S4)

E15 · formal · cited by C18

Scaling the argument of a polynomial scales its roots.

Foundation: algebra of complex polynomials

$c(B_i) = 0 \iff c((r B_i) / r) = 0$ (substitution)
$|r B_i| > 1\ \forall i \iff 1/|B_i| < r\ \forall i$ (r > 0; from S1)
$c(B/r) / c(0) = 1 + \sum_i (c_i / c_0)\, r^{-i} B^i \text{ has the roots of } c(B/r)$ ($c(0) \neq 0$; from S1)
$\text{the step-down passes on } c(B/r) \iff |r B_i| > 1\ \forall i$ (the step-down test passes exactly when all roots lie outside the unit circle; from D7, S1, S3)

E16 · formal · cited by C20

Expansion of det(I - M L(B)) by multilinearity in the columns.

Foundation: linear algebra over the field of rational functions in $B$

$\text{column } j \text{ of } I - M L(B) = e_j - B^{l_j} m_j$ (definition of $L(B)$; from D1)
$\det(I - M L(B)) = \sum_{S} \det\left[\, e_j\ (j \notin S),\ -B^{l_j} m_j\ (j \in S) \right]$ (multilinearity of the determinant in the columns; from S1)
$\det\left[\, e_j\ (j \notin S),\ -B^{l_j} m_j\ (j \in S) \right] = (-1)^{|S|} B^{\sum_{j \in S} l_j} \det(M_{SS})$ (Laplace expansion along the unit columns, and pulling the scalars out of the other columns; from S2)

E25 · simulation

Planned test of the contributor P5 (lab plan AC5.PL1, run AC5.RN8, with the breakdown run AC5.RN9): the exact oracle, written without the prototype, on the models of the planned tests exported by runs RN57-RN61, against the prototype's values stored with each model. The oracle was tried beforehand only on models of other seeds (AC5.RN2, AC5.RN6, AC5.RN7) and on hand-checkable models. AC5.RN8 took 30 s.

Setup: studied PBH reference, check of the whole model, step-down test, bisection for the largest root modulus, per-lag check · cases models of the planned tests, ETS(A,N,A) grid, random GUM models with lags (1, 12), random additive ETS and GUM models with long seasonal lags · procedure comparison with the exact oracle, exact oracle · lens admissibility decision, largest mode modulus, non-cyclotomic shared factor in the closed unit disc

E25.T1 · Each set of models of the planned tests (D42) given to the exact oracle (D41), and over all sets: models given, left out by the edge band and undecided; decisions of the PBH reference and of the prototype's cancellation that differ from the oracle, and the largest absolute difference of the largest mode modulus on the models both find inadmissible; models with a non-cyclotomic shared factor in the closed unit disc (D44); models whose cancelled MA degree differs from the prototype's. Empty cells: the set has no such value of the prototype (perlag holds the PBH reference only; stepdown and cpp the cancellation only).
Row	set	models given to the oracle	models compared	models with a shared non-cyclotomic factor, all roots in \|B\| > 1	models excluded by edge	admissibility decisions of the PBH reference that differ from the exact oracle	max abs difference of the largest mode modulus, PBH reference against the exact oracle, models both find inadmissible	admissibility decisions of the prototype's cancellation that differ from the exact oracle	max abs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracle, models both find inadmissible	models whose cancelled MA degree differs from the prototype's
r1	perlag	3242	3164	0	78	0	0.0000
r2	cancel	500	500	8	0	0	0.0000	0	0.0000	0
r3	stepdown	575	575	8	0			0	0.0000	0
r4	cpp	800	797	8	3			0	0.0000	0
r5	examples	2	2	0	0	0	0	0	0	0
r6	all	5119	5038	24	81	0	0.0000	0	0.0000	0

E25.T2 · The 575 models of PL4 and PL5 (MA degree up to 535 before cancellation): step-down decisions of the prototype that differ from the exact oracle; the largest absolute difference of the bisection value from the certified largest mode modulus, over the models that have one; and the degrees. (set: stepdown)
Row	set	step-down decisions of the prototype that differ from the exact oracle	max abs difference of the largest mode modulus, prototype's bisection against the exact oracle	models with a bisection value	max degree of theta, exact	max degree of the cancelled MA polynomial, exact
r1	stepdown	0	0.0000	198	535	533

E25.T3 · The models of PL1 and PL2 (3042 ETS(A,N,A) grid models, 200 GUM models with lags (1, 12)): the counts of C1 and C2 with the exact oracle in place of the PBH reference and the band of 1e-8. (set: perlag)
Row	family	models accepted by the per-lag check that do not forget their initial states, exact oracle	models rejected by the per-lag check that forget their initial states, exact oracle	models accepted by the per-lag check	models of the family
r1	ETS(A,N,A)	1482	0	3042	3042
r2	GUM	10	1	193	200

E25.T4 · The models in the edge band, by set and family (run AC5.RN9): their number, the largest distance of the certified largest mode modulus from 1, those with roots proved on the unit circle, and those whose decision by the method of the original test (the PBH reference for perlag, the prototype's cancellation for cpp) differs from the oracle's, with the largest modulus difference.
Row	set	family	models in the edge band	max distance of the certified largest mode modulus from 1, edge models	edge models with roots proved on the unit circle	edge models whose decision differs from the exact oracle	max abs difference of the largest mode modulus against the exact oracle, edge models
r1	perlag	ETS(A,N,A)	78	0.0000	2	6	0.0000
r2	cpp	ETS high degree	3	0.0000	0	0	0.0000

E25.T5 · The examples of C10 and C11 (run AC5.RN9): the certified largest mode modulus, its difference from the PBH reference and from the prototype's roots after cancellation, and the exact degrees of theta, of the shared cyclotomic factor and of the gcd of theta and phi.
Row	model	largest mode modulus, exact oracle (theta_G)	abs difference of the largest mode modulus, PBH reference against the exact oracle	abs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracle	degree of theta, exact	degree of the shared cyclotomic factor, exact	degree of the exact gcd of theta and phi
r1	ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05)	1	0.0000	0.0000	193	25	25
r2	GUM start, orders (2,2), lags (1,12)	0.9829	0.0000	0.0000	26	14	14

Limitations

L2 · measurement · could reverse · affects C1, C2, C3, C10, C11, C16 · mitigation: addressed

The PBH reference and the cyclotomic cancellation were written by the same agent and share the lag-1 expansion code and numpy's LAPACK eigenvalue routines.

L3 · sampling · reduces transferability · affects C2, C3, C4, C5, C16 · mitigation: none

The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data.

L4 · measurement · could reverse · affects C4, C5, C16 · mitigation: addressed

The reference decisions use numpy's polynomial roots (companion-matrix eigenvalues) for polynomials of degree up to 535 before cancellation with many roots near the unit circle.

L5 · deviation · reduces transferability · affects C4, C5 · mitigation: none

The plans of the tests behind the step-down and bisection claims stated MA polynomial degrees up to 366; the generator code named in the plans, run unchanged, produced degrees up to 535 before cancellation.

L8 · model specification · reduces transferability · affects C4, C16 · mitigation: none

For GUM models with structured parameters, such as a state with |F_ii| > 1 and zero persistence, theta and phi can share a factor with a root in |B| <= 1 that is not cyclotomic; the check keeps it and rejects the model although its observation weights decay, because the initial state of that mode has a growing effect.

Assumptions

C14 · methodological · tested

The PBH reference with relative rank tolerance 1e-7 classifies every eigenvalue of modulus at least 1 of the expanded discount matrix of the tested models correctly as both reachable and observable or not, outside the band of 1e-8 around the unit circle that the tests leave out. Rested on by C3.

C19 · theoretical · tested

In additive ETS models, and in GUM models with an identity transition or with transition, persistence and measurement in general position, every factor shared by $\theta(B)$ and $\phi(B)$ with a root in $|B| \le 1$ is a product of cyclotomic polynomials $\Phi_n(B)$ with $n$ dividing a lag of the model. Rested on by C16, C4.

Definitions

Methods 12

D1 lagged SSOE model: A pure additive single-source-of-error state space model with $k$ components, $\begin{aligned} y_t &= w' v_{t-l} + \epsilon_t \\ v_t &= F v_{t-l} + g \epsilon_t, \end{aligned}$ in which component $j$ of $v_{t-l}$ enters with its own lag $l_j$ ; its discount matrix is $D = F - g w'$ and its lag matrix is $L(B) = \mathrm{diag}(B^{l_1}, \dots, B^{l_k})$ for the backshift operator $B$ ; $\Lambda(\lambda) = \mathrm{diag}(\lambda^{l_1}, \dots, \lambda^{l_k})$ for a complex number $\lambda$ .

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.

D11 · part of D10 bisection for the largest root modulus: Bisection on $r$ over the step-down test applied to $\theta^*(B/r)$ , which passes exactly when $1/|B| < r$ for every root $B$ of $\theta^*$ , so that the smallest such $r$ is the largest mode modulus; run to a width of 1e-10 in the prototype and to a tolerance of 1e-6 in smooth.

D12 · a kind of D10 smoothEigensCpp(): The check of the whole model as implemented in smooth (src/headers/eigenCalc.h, shared by R and Python): the coefficients of $\theta$ and $\phi$ by inverse discrete Fourier transform, cyclotomic cancellation, the step-down test and bisection with tolerance 1e-6; called in R through smoothEigensR().

D15 roots by numpy: The roots of a polynomial by numpy's roots (the eigenvalues of its companion matrix), and the admissibility decision and largest modulus $1/\min|B|$ they give.

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.

Data 8

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).

D19 · part of D20 random additive ETS and GUM models: The 500 models of plan PL3 (confirm.py --test cancel --seed 2026 --n 200): 200 additive ETS models (level; no, additive or damped trend with $\phi$ in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; $\beta = 0$ with probability 0.1 and each $\gamma = 0$ with probability 0.15; $\alpha$ in (0.01, 1.9)), 100 GUM models with $F = I$ , lags from {1, 4, 12}, $g$ uniform on (-0.3, 0.6) and $w$ normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal $F$ (standard deviation 0.35).

D21 four model families of the implementation test: 200 models in each of four families (cpp_check.py --seed 2027 --n 200, plan PL6): additive ETS with 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; additive ETS with 0 to 2 seasonal lags from {24, 168, 365}; GUM with $F = I$ and lags from {1, 4, 12}; GUM with normal $F$ (standard deviation 0.35) and lags from {1, 4, 12, 24}.

D24 ETS(A,N,A,A) with seasonal lags 24 and 168: The additive ETS model with a level and two seasonal states of lags 24 and 168, $\alpha = 0.3$ and $\gamma = (0.1, 0.05)$ , in the lagged form.

D25 starting point of gum(): The default starting point of gum() with orders (2, 2) and lags (1, 12): $F = I$ , $g = 0.1$ , $w = 1$ .

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.

Variables and measures 3

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.

Study design 2

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.

Concepts 2

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.

Context

Supersedes Svetunkov (2016), smoothEigensCpp(), smooth 4.6.0 · own earlier work · from C16: smooth 4.6.0 decides admissibility from the eigenvalues of the per-lag discount matrices, one block per lag.
Extends Hyndman et al. (2008), Chapter 11 · from C3: Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.
Extends Barndorff-Nielsen and Schou (1973) · from C4: Barndorff-Nielsen and Schou parametrise stationary autoregressions by their partial autocorrelations, which all lie in (-1, 1) exactly when every root of the autoregressive polynomial lies outside the unit circle.
Extends Hyndman et al. (2008), Chapter 11 · from C6: Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.

References

Hyndman et al. (2008) Hyndman, R., Koehler, A., Ord, K., & Snyder, R. (2008). Forecasting with Exponential Smoothing. Springer Series in Statistics. doi:10.1007/978-3-540-71918-2 Used by: Context, D9
Barndorff-Nielsen and Schou (1973) Barndorff-Nielsen, O., & Schou, G. (1973). On the parametrization of autoregressive models by partial autocorrelations. Journal of Multivariate Analysis, 3(4), 408–419. doi:10.1016/0047-259X(73)90030-4 Used by: C4, Context, C18, D7
Svetunkov (2016) Svetunkov, I. (2016). smooth: Forecasting Using State Space Models. CRAN: Contributed Packages. doi:10.32614/CRAN.package.smooth Used by: Context, D8

Artefacts

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.2): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C16: 88, High

Novelty: new, by the authors · no close match on EntryLib

−5 Plan The test deviated from its plan. To remove it: Each deviation is a limitation: say how it bears on the result.
−3 Plan The plan's time is anchored by a git push only. To remove it: Anchor plans with the platform, a timestamp authority (RFC 3161) or a registry.
−2 Open problems Limitation L3 (reduces transferability, not mitigated): The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
−1 Open problems Limitation L8 (reduces transferability, not mitigated): For GUM models with structured parameters, such as a state with |F_ii| > 1 and zero persistence, theta and phi can share a factor with a root in |B| <= 1 that is not cyclotomic; the check keeps it and rejects the model although its observation weights decay, because the initial state of that mode has a growing effect. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
−1 Open problems Limitation L5 (reduces transferability, not mitigated): The plans of the tests behind the step-down and bisection claims stated MA polynomial degrees up to 366; the generator code named in the plans, run unchanged, produced degrees up to 535 before cancellation. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

C3: 95, High

Novelty: new, by the authors · no close match on EntryLib

−3 Plan The plan's time is anchored by a git push only. To remove it: Anchor plans with the platform, a timestamp authority (RFC 3161) or a registry.
−2 Open problems Limitation L3 (reduces transferability, not mitigated): The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

C4: 88, High

Novelty: new, by the authors · no close match on EntryLib

−5 Plan The test deviated from its plan. To remove it: Each deviation is a limitation: say how it bears on the result.
−3 Plan The plan's time is anchored by a git push only. To remove it: Anchor plans with the platform, a timestamp authority (RFC 3161) or a registry.
−2 Open problems Limitation L3 (reduces transferability, not mitigated): The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
−1 Open problems Limitation L8 (reduces transferability, not mitigated): For GUM models with structured parameters, such as a state with |F_ii| > 1 and zero persistence, theta and phi can share a factor with a root in |B| <= 1 that is not cyclotomic; the check keeps it and rejects the model although its observation weights decay, because the initial state of that mode has a growing effect. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.
−1 Open problems Limitation L5 (reduces transferability, not mitigated): The plans of the tests behind the step-down and bisection claims stated MA polynomial degrees up to 366; the generator code named in the plans, run unchanged, produced degrees up to 535 before cancellation. To remove it: Mitigate it with evidence, or narrow the claim's scope so that it no longer applies.

C6: 100, High

Novelty: extends earlier work, by the authors

Nothing lost.

C8: 100, High

Novelty: new, by the authors

Nothing lost.

C15: 100, High

Novelty: new, by the authors

Nothing lost.

C17: 100, High

Novelty: new, by the authors

Nothing lost.

C18: 100, High

Novelty: new, by the authors

Nothing lost.

C20: 100, High

Novelty: new, by the authors

Nothing lost.

Admissibility of lagged state space models through the ARIMA form of the whole model · C16

The claim

Claims it rests on

Evidence

Limitations

Assumptions

Definitions

Methods 12

Data 8

Variables and measures 3

Study design 2

Concepts 2

Context

References

Artefacts

Score

C16: 88, High

C3: 95, High

C4: 88, High

C6: 100, High

C8: 100, High

C15: 100, High

C17: 100, High

C18: 100, High

C20: 100, High