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What a fast-switching yield curve identifies

A first-order identifiability theorem, exact loading and initial-mixture rank boundaries, and an explicit finite-rate remainder.

Research note, 23 September 2026. This note develops the three-number example; the asymptotic statements concern a fixed finite maturity interval as the generator is sped up.

Model and convention

Let $Q_0$ be an irreducible generator on $n$ states, with stationary row vector $\pi$, and let the regime generator be $mQ_0$, $m\to\infty$. The short rate obeys $dr_t=\kappa(\theta_{y_t}-r_t)dt+\sqrt{s_{y_t}}\,dW_t$, with $\kappa>0$, $s_i\ge0$, and $y_t$ independent of $W$. Define

For a known initial rate $x$, write $P_i(T)=e^{-B(T)x}a_i(T)$ for a specified starting regime and $P_\pi(T)=e^{-B(T)x}\pi a(T)$ for a stationary start. The exact finite-dimensional equation is

In particular, $g(0)=0$. Put $R=I-{\bf1}\pi$ and $H=-Q_0^{\#} =\int_0^\infty e^{Q_0t}R\,dt$.

The symmetric Green–Kubo form is $\mathcal K_0^{\rm sym}(f,h) =\frac12\{\pi[f(Hh)]+\pi[h(Hf)]\}$ on centered vectors. The coefficients below are computed with $Q_0$; their actual contribution at rate $mQ_0$ is divided by $m$.

Theorem: the first-order information

For every fixed $L<\infty$, $\sup_{0\le T\le L}|E_\pi(T)|=O_L(m^{-2})$. With $\kappa$ and $x$ fixed, the four functions $I_1,\ldots,I_4$ are linearly independent on every open maturity interval. Hence an ideal first-order stationary curve uniquely determines these four coefficients.

It does not separately determine $\bar s$ and $\mathcal K_0(u,u)/m$ from $\beta_2$, nor the underlying chain. Two chains can share all four first-order coefficients and have different exact curves at order $m^{-2}$. Finite-rate observations do not recover these coefficients without error; a bound appears below.

Let $c_2=\kappa^2\mathcal K_0(u,u)/m$, $c_3=\beta_3$ and $c_4=\beta_4$. The $2\times2$ Green–Kubo matrix is positive semidefinite, so $c_3^2\le4c_2c_4$, and the observable restriction is $\beta_3^2\le4\beta_2\beta_4$. When $\beta_4=0$, necessarily $\beta_3=0$.

Equality for the latent $c$'s holds with two regimes, including the degenerate zero-variance case. Strict inequality for the observable $\beta$'s alone cannot establish three regimes: $\bar s>0$ makes it strict even for two states. If $\bar s$ is separately known, strict inequality among the $c$'s implies at least three regimes.

It still does not identify their transition graph.

The general fixed-feature Gram theorem

The two-feature restriction is a special case. Suppose the centered forcing separates as

and let $F=[f_1\ \cdots\ f_d]$ be the $n\times d$ feature matrix. Define the fixed-parameter Green–Kubo Gram matrix

Then the stationary first correction is

and, for every irreducible finite chain,

The equality of kernels is stronger than the rank upper bound. For any coefficient vector $a$, $a^\top Ga=\mathcal K_0^{\rm sym}(Fa,Fa)$. The additive-functional martingale argument in the proof below makes this zero exactly when the centered state function $Fa$ is zero.

Thus an identified fixed-feature rank $r$ requires at least $r+1$ regimes. It does not identify the generator: many chains and feature embeddings produce the same $G$.

Identification of the whole matrix still requires the maturity functions $\int_0^T\phi_r(t)\phi_s(t)\,dt$ to be linearly independent over the observed interval. Without that design condition, a curve identifies only the corresponding linear combinations of the entries of $G$.

The exact finite-horizon covariance

The Green–Kubo matrix is a covariance rate, not the covariance at a finite horizon. Let $Y$ start in stationarity, let its generator be $mQ_0$, put $H=-Q_0^\#$, and define

Here the columns of $F$ are centered and $e_i$ is the state basis vector. Direct integration of the two-time covariance gives the exact identity

The remainder is uniform over every $T\ge0$: exponential mixing bounds $H^2(I-e^{mQ_0T})$ independently of $T$. For each fixed $T>0$ the exponentially small part also yields the explicit boundary correction

At every finite $T>0$ this exact covariance is positive semidefinite and has $\ker\operatorname{Cov}_\pi(Z_T)=\ker F$. The rank statement is stronger than the closed stationary formula: it remains true for every initial regime prior $\nu$, including a deterministic start, although the covariance formula itself then changes.

To prove this, suppose the variance of $\int_0^T f(Y_s)ds$ vanishes and choose a state $i_0$ in the support of $\nu$. The integral is constant almost surely. On the no-jump path it equals $Tf_{i_0}$.

For any directed path $i_0\to i_1\to\cdots\to i_k$, consider paths that make exactly these $k$ jumps and then no more before $T$. Their ordered jump times have a strictly positive density on the open simplex. The integral is the continuous affine function $f_{i_0}t_1+\cdots+f_{i_{k-1}}(t_k-t_{k-1})+f_{i_k}(T-t_k)$.

Since it is almost surely constant, it is constant throughout that simplex, forcing $f_{i_{r-1}}=f_{i_r}$ for every edge. Irreducibility gives a directed path from $i_0$ to every state, so $f$ is constant; after centering, $f=0$. Hence for arbitrary $\nu$

This still says nothing about rank after different time-dependent loadings have been integrated. For a stationary start and reversible $Q_0$, spectral calculus also gives the Loewner bound $0\preceq\operatorname{Cov}_\pi(Z_T)\preceq2TG/m$; neither the closed formula nor that upper bound is asserted for a nonstationary start.

The certificate independently integrates the joint first- and second-moment equations. Across 48 dense nonreversible chain/prior cases, including point-mass starts, every numerical covariance has the predicted rank; under stationarity the ODE and closed formula agree within $1.6\times10^{-14}$.

Why an integrated rank is not a regime count

The rank statement applies to one fixed feature matrix. If $p$ contracts have a maturity-dependent loading matrix $L(t)\in\mathbb R^{p\times d}$, their integrated coefficient matrix is

Let $r=\operatorname{rank}G$ and choose any full-column factor $G=RR^\top$ with $R\in\mathbb R^{d\times r}$. Then for every $x\in\mathbb R^p$,

Consequently, writing essential span to ignore measure-zero values of a merely measurable loading,

For continuous loadings, “almost everywhere” can be replaced by “everywhere.” This is the exact rank boundary: integrating does not recover another property of the hidden chain; it measures the span swept out by the loaded fixed-feature range.

A useful finite-dimensional specialization makes the possible inflation explicit. If

and $\mathcal B=[L_1R\ \cdots\ L_qR]$, then

The matrix $D(T)$ is a finite-horizon controllability Gramian with input matrix $L(t)R$; the rank interpretation goes back to Kalman (1960).

Although every integrand has rank at most $r$, its range can rotate with $t$, so $D(T)$ can attain the larger bound $qr$. The smallest counterexample uses only two regimes and one centered feature.

Take scalar $G=[g]$ with $g>0$ and $L(t)=(1,t,t^2)^\top$. At $T=1$,

the positive-definite $3\times3$ Hilbert matrix. Hence $\operatorname{rank}D(1)=3$ even though $n-1=1$. A rank lower bound on the number of regimes is valid for the fixed Green–Kubo Gram matrix after known loadings have been removed, not for a matrix obtained by integrating heterogeneous maturity loadings.

The certificate also checks a rank-two, three-basis example attaining rank six and reproduces the factorization by independent quadrature within $2\times10^{-12}$. This is the same distinction needed in multi-name credit and multi-factor term-structure applications.

Known starting regime

For any specified state $i$, the same theorem adds two unintegrated terms:

The coefficients of $B$ and $B^2$ record the Poisson solutions for the initial state. Their $\pi$-weighted averages are zero, recovering the stationary formula.

They are two distinct functions, but they do not contribute two new independent shapes to a single curve. The identities

show that the six displayed functions span only $\{T,I_1,I_2,I_3,I_4\}$, a five-dimensional space. Writing the known-start coefficients as $\gamma_{1i}=-\kappa(Hu)_i/m$ and $\gamma_{2i}=(Hv)_i/(2m)$, one curve determines only

at this order. If separately labeled curves for several initial regimes are available, their differences isolate $(\gamma_{1i}-\gamma_{1j})B+ (\gamma_{2i}-\gamma_{2j})B^2$. Both relative coefficients can then be read from a maturity interval; a stationary-start curve also isolates the common $\beta$'s.

The approximation has a uniform value error but no claimed uniform derivative error near $T=0$. In particular, differentiating its known-start term at zero would give a spurious $O(m^{-1})$ initial slope; the order-$m^{-2}$ initial layer cancels it in a window of width $O(m^{-1})$.

An unobserved initial mixture: outer and boundary information

Let the initial regime have an unknown row distribution $\nu$. Averaging the statewise expansion before taking its logarithm gives, uniformly on a fixed bounded maturity interval,

Thus the fixed-maturity outer expansion sees the unknown mixture only through $\nu H[u\ v]$. If the common stationary coefficients are known separately, the whole distribution $\nu$ is identified at this order exactly when $\operatorname{rank}[u\ v]=n-1$; otherwise any zero-sum row vector in the kernel of $H[u\ v]$ is an observationally equivalent perturbation (while it remains inside the probability simplex). Without a stationary baseline, the shape aliases in the preceding section further mix these projections with $\beta_1$ and $\beta_2$.

Exact transient observability

The underlying rank boundary is clearest for a general centered feature matrix $F\in\mathbb R^{n\times d}$. For $Z_T=\int_0^T F^\top e_{Y_s}ds$ and any initial distribution $\nu$, direct semigroup integration gives the exact mean

At maturities $T_1,\ldots,T_q$, define the transient observability matrix

Two priors $\nu,\nu'$ have the same mean responses exactly when $(\nu-\nu')\mathcal O_m=0$. Consequently the responses identify every prior if and only if

This is the observability counterpart of Kalman's 1960 rank framework. It concerns initial-distribution information, not the fixed-feature Green–Kubo rank and not the rank created by maturity-dependent contract loadings.

The whole maturity curve: an exact Krylov criterion

The finite list above leaves open whether some choice of scaled maturities can identify the prior. This has an intrinsic answer.

For $\tau>0$ put $R(\tau)=H(I-e^{Q_0\tau})F$. Then

Consequently the complete scaled-maturity mean curve identifies every initial distribution if and only if

This is the classical Kalman reachability condition on the centered state space, used here through its observability dual. Equivalently, the PBH obstruction is a nonzero left eigenvector $\ell Q_0=\lambda\ell$, $\lambda\ne0$, with $\ell F=0$; see Hautus (1969).

For a reversible chain and one scalar feature, full identification therefore requires every nonzero eigenvalue to be simple and the feature to have a nonzero component in every nonconstant eigenmode. A repeated eigenspace is a genuine obstruction: varying maturity cannot create more than one direction from one vector inside that eigenspace.

A continuous maturity interval: exact Fisher information

The whole-curve rank criterion has an exact statistical counterpart. Suppose the centered response is observed continuously on $0\le\tau\le L$ in the Gaussian white-noise experiment

Its Fisher information for the initial prior is the continuous observability Gramian

For every $L>0$, not merely in the limit $L\to\infty$, analyticity gives

Indeed, $x^\top\mathcal W_{L,\Gamma}x=0$ if and only if the analytic function $x^\top R(\tau)$ vanishes throughout the interval. Since

this is equivalent to $x$ annihilating the Krylov space. If $U$ is any orthonormal basis of the prior tangent space $\{x:{\bf1}^\top x=0\}$, the constrained Fisher matrix is $U^\top\mathcal W_{L,\Gamma}U$.

It is positive definite exactly when $\operatorname{rank}[{\bf1}\ F\ Q_0F\ \cdots\ Q_0^{n-2}F]=n$. Thus continuum observation introduces no new algebraic condition beyond the PBH/Krylov theorem; it turns that condition into a quantitative information matrix.

Short maturity windows: an explicit Fisher hierarchy

Rank alone hides a severe short-window conditioning law. Consider one scalar centered feature, scalar white-noise variance $\Gamma=\gamma>0$, and put $r=n-1$.

Assume the feature is cyclic, and let $U$ be an orthonormal prior-tangent basis. Define the square centered Krylov matrix

Let $H_r$ be the factorial Hilbert matrix and $C$ its lower Cholesky factor:

Then the constrained Fisher eigenvalues, ordered decreasingly, obey

with the completely explicit universal factor

Consequently

The proof inserts $R(\tau)=\sum_{k\ge1}\tau^kQ_0^{k-1}F/k!$ into the Gramian, rescales $\tau=Lx$, and factors the result as $K_cD_L\{H_r+O(L)\}D_LK_c^\top/\gamma$, where $D_L=\operatorname{diag}(L^{3/2},L^{5/2},\ldots,L^{r+1/2})$. Exterior powers after QR and Cholesky give the individual constants.

This is a Fisher-information statement about the initial mixture. It is not the discrete-maturity response hierarchy and does not change the fixed-feature Green–Kubo rank or any rank obtained after contract-loading integration.

The odd-power hierarchy is the shifted analogue of the single-input controllability-Gramian law developed by Trélat and Zuazua (2026): their ordinary response is nonzero at time zero and begins with powers $L,L^3,\ldots$, whereas the yield response itself starts at order $\tau$, shifting every information power by two.

For the four-state cyclic certificate, the predicted constants for powers $(L^3,L^5,L^7)$ are

At $L=0.0125$, the 80-digit measured scaled eigenvalues are $(1.00987837,4.55027653\times10^{-3},5.00661036\times10^{-7})$; the largest relative error is $2.48\%$ and decreases at observed first order. The normalized determinant is $0.934336$, consistent with its $L^{15}$ limit.

Several features: the block-Krylov Fisher hierarchy

The single-feature powers do not simply persist once several correlated features are observed. Whiten the feature matrix and restrict it to the prior tangent space:

Let $\mathcal K_q=\operatorname{range}[S,AS,\ldots,A^{q-1}S]$, $d_q=\dim\mathcal K_q-\dim\mathcal K_{q-1}$, and choose an orthonormal block $U_q$ for $\mathcal K_q\ominus\mathcal K_{q-1}$. Put

Each $B_q$ has full row rank, and the complete graded block matrix $G=(G_{qp})$ is positive definite: its quadratic form is the integral of the squared vector polynomial $\sum_qt^qB_q^\top x_q$.

Let $s_q=d_1+\cdots+d_q$ and let $\Sigma_q$ be the $q$th successive Schur pivot of $G$, eliminating grades $1,\ldots,q-1$. Then every individual positive constrained-Fisher eigenvalue has a leading constant:

Eigenvalues are ordered decreasingly, and those within each grade use the decreasing eigenvalues of $\Sigma_q$. The proof is the graded factorization

followed by successive block Gaussian elimination and the min--max principle. This includes noncyclic and rank-deficient inputs by stopping when the block-Krylov flag stabilizes. It is the shifted-response, constant-level refinement of the multi-input controllability orders used by Schmerling, Janson and Pavone (2015); the recent two-term result of Trélat and Zuazua (2026) is explicitly single-input.

A six-state, two-feature certificate has block increments $(2,2,1)$, hence powers $(3,3,5,5,7)$. Its Schur-pivot constants are

At $L=0.0015625$, all five scaled eigenvalues are within $0.945\%$ of these limits, with observed first-order convergence. The normalized determinant is $0.962244402$ for the predicted $L^{23}$ law.

This theorem concerns initial-mixture Fisher information. Its block-Krylov rank is neither the fixed-feature Green–Kubo rank nor the rank of a matrix obtained by integrating heterogeneous contract loadings.

There is also a sharp long-interval distinction between fixed-feature information and transiently created information. Put $B=HF$ and let $r=\operatorname{rank}F$. Exponential mixing gives

Under full prior observability, exactly $r$ eigenvalues of the constrained Fisher matrix are $\Theta(L)$. The remaining $n-1-r$ eigenvalues converge to finite strictly positive limits. To see positivity, take a tangent direction with $B^\top x=0$; then

The limit can vanish only in a PBH-obstructed direction. Therefore adding ever longer maturities keeps accumulating information only in the fixed-feature span.

The extra prior directions revealed by maturity rotation are identifiable, but their total transient information is finite. This is a continuous-observation statement about the initial mixture, not a change to the fixed-feature Green–Kubo rank and not a rank claim for integrated contract loadings.

The certificate evaluates the Gramian independently by a Van Loan block exponential and direct quadrature in 24 random finite-chain cases; the maximum discrepancy is $1.99\times10^{-13}$. In the four-state cyclic scalar example, fixed-feature rank one becomes continuous-prior rank three. At $L=256$ the constrained information eigenvalues are

The growing eigenvalue divided by its predicted linear term is $0.997278$; the two transient eigenvalues change by only $0.465\%$ and $0.421\%$ between $L=128$ and $256$. This is the finite-horizon observability-Gramian form of the classical Kalman/Hautus criterion already cited above.

Reversibility is sufficient for a real spectrum, but neither reversibility nor diagonalizability is necessary for the all-maturity conclusion. Suppose the restriction of $Q_0$ to the centered state space has only real eigenvalues, repeated eigenvalues and Jordan blocks being allowed, and a centered scalar feature $F$ is cyclic:

Then every ordered tuple $0<\tau_1<\cdots<\tau_r$ identifies the prior:

Indeed, if a nonzero centered row vector $\ell$ annihilated all response columns, $h(t)=\ell R(t)$ would have the $r+1$ distinct zeros $0,\tau_1,\ldots,\tau_r$. Cyclicity makes $h'(t)=\ell e^{Q_0t}F$ nonzero. In a real Jordan basis this derivative is an exponential polynomial

These functions form an extended Chebyshev system, so a nonzero $h'$ has at most $r-1$ real zeros counting multiplicity. Rolle's theorem would give at least $r$, a contradiction.

Cyclicity also explains exactly which repetitions are permitted: it forces one Jordan chain per eigenvalue. A repeated eigenspace with two independent eigenvectors remains an obstruction to any scalar feature.

For a simple spectrum the theorem has an explicit determinant. Write $0,-a_1,\ldots,-a_r$, with $0<a_1<\cdots<a_r$, choose a real right-eigenvector basis $\psi_0={\bf1},\psi_1,\ldots,\psi_r$, and expand $F=\sum_{k=1}^r f_k\psi_k$. Cyclicity is then equivalent to every $f_k\ne0$, and

The sign follows by continuity and the small-maturity exponential Vandermonde limit. The ordinary and confluent cases are the classical exponential Chebyshev/total-positivity mechanism developed by Schoenberg (1947).

Sharp conditioning when all maturities move to the long end

Exact rank at every finite maturity does not imply a uniformly stable inverse. The simple real-spectrum case above gives the sharp loss. Fix distinct offsets $c_1<\cdots<c_r$ and translate the entire scaled-maturity design by $M$:

Order its singular values as $\sigma_1(A_M)\ge\cdots\ge\sigma_n(A_M)$. If $0<a_1<\cdots<a_r$, the scalar feature is cyclic, and $M+c_1>0$, then

Thus all $r$ distinct maturities retain exact rank for every finite $M$, while the weakest prior direction vanishes at the second-fastest decay rate. The fastest rate $a_r$ does not enter the leading determinant.

To see why, use the eigenbasis and remove the fixed nonzero modal factors. The response block becomes $B_M=[1-e^{-a_k(M+c_j)}]_{k,j=1}^r$.

Keep its first column and subtract it from every other column. With $D_M=\operatorname{diag}(e^{-a_1M},\ldots,e^{-a_rM})$, the result is

The exponential Chebyshev property makes every required leading minor of the difference block nonzero. Exterior powers then have successive orders $1,e^{-a_1M},e^{-M(a_1+a_2)},\ldots, e^{-M(a_1+\cdots+a_{r-1})}$, which gives the singular-value hierarchy by taking consecutive ratios.

In the determinant the unscaled common column uses the fastest row, leaving precisely the $r-1$ slower rates. Fixed eigenvector and column transformations change constants, not these exponents.

A nonreversible four-state certificate has rates $(0.43537897,1.02585656,2.13876447)$. Translating offsets $(0.2,0.8,1.5)$ gives measured augmented singular-value rates $(-0.00130,0,0.43609,1.02644)$, determinant rate $1.46123$, and condition-number rate $1.02775$, against predictions $(0,0,0.43538,1.02586)$, $1.46124$, and $1.02586$. This result concerns transient initial-mixture observability across maturities; it is not a claim about fixed-feature Green–Kubo rank or rank created by integrating different contract loadings.

Defective long end: the Jordan polynomial changes the threshold

The distinct-rate hypothesis in the preceding hierarchy matters even when every maturity tuple remains identifying. The sharp three-state defective case makes the missing factor explicit.

Suppose the centered restriction has one size-two Jordan block at $-a$, $a>0$. Normalize a Jordan chain and cyclic scalar feature so that

Then the exact response is

For any fixed offsets $c_1<c_2$, put $A_M=[\mathbf1\ R(M+c_1)\ R(M+c_2)]$. If $K=[\mathbf1\ v\ w]$, direct expansion gives

The first two singular values converge to positive limits, so the determinant identity and the product formula for singular values yield the complete hierarchy

The centered-prior weakest signal has the same order $s_M=\Theta(M e^{-aM})$. Consequently every Gaussian, correlated-noise, and deterministic minimax statement below remains valid after replacing the simple-spectrum scale by this Jordan scale under their stated covariance assumptions.

Exact noiseless rank still holds at each finite $M$; the polynomial factor weakens the pure exponential loss by one power of $M$ but does not prevent it. This is the Jordan-chain specialization of the Kalman rank framework, not a statement about fixed-feature Green–Kubo rank or maturity-integrated loading rank.

The existing irreducible defective certificate has $a=1$, $\eta=0.1$, and offsets $(0.4,1.7)$. After dividing the augmented and centered-prior least singular values by $M e^{-M}$, their fitted exponential rates are both $0.999461$.

At $M=16$ their scaled values are both $0.042053486$. The scaled determinant is $0.581981281$, against its exact limit $0.585163826$, and the direct Jordan factorization error is below $6.4\times10^{-15}$.

Gaussian quote noise: exact rank is not statistical recovery

The preceding condition number has a direct minimax consequence. Let $C_M=[R(M+c_1)\ \cdots\ R(M+c_r)]$, let $\mathcal H_0=\{h:h^\top\mathbf1=0\}$, and define the weakest prior signal

Under the simple real-spectrum and cyclicity hypotheses of the long-end theorem,

Indeed, choose an orthonormal basis $U$ of $\mathcal H_0$. Orthogonally reducing $[\mathbf1\ C_M]$ and eliminating its bounded normalization row turns it, through uniformly bounded invertible transformations, into a block matrix with lower block $U^\top C_M$. Hence its least singular value and $s_M$ have the same order.

Now idealize noisy curve measurements as

where $\nu$ is an unknown initial distribution and $\sigma$ is a quote-noise scale. Fix $0<\rho\le\min_i\pi_i$, choose a unit minimizer $h_M\in\mathcal H_0$, and put $\nu_\pm=\pi\pm\rho h_M$.

Both are probability vectors because $\|h_M\|_\infty\le1$. For every estimator $\widehat\nu(Y)$, the exact Gaussian two-point testing calculation gives

The two Gaussian means are separated by $2\rho s_M$, so their optimal equal-prior classification error is exactly the displayed normal tail. Any estimator with error below $\rho$ induces the correct nearest-prior classifier, proving both bounds.

Thus fixed noise makes the lower bounds tend to $1/2$ and $\rho^2/2$ as the design moves right. To make the first probability smaller than a fixed $\delta<1/2$, it is necessary that

This is a local two-prior lower bound for the stated Gaussian observation model, not a claim about market microstructure or a change to noiseless identifiability. It is the finite-dimensional two-point method originating in Le Cam (1973).

In the four-state certificate, the restricted least-singular-value rate is $1.02644$ against $a_{r-1}=1.02586$. With $\rho=0.1$ and $\sigma=10^{-5}$, the exact testing lower bound rises from $0.0709$ at $M=4$ to $0.499998$ at $M=16$. A Gaussian simulation with 200,000 draws per prior at $M=4$ agrees with the exact testing error within $0.002$.

The exponential threshold is also sufficient. Let $U$ have orthonormal columns spanning $\mathcal H_0$, put $A_M=C_M^\top U$, and let $\Pi_\Delta$ denote Euclidean projection onto the probability simplex. Since $A_M$ is nonsingular, define the projected inverse estimator

For every true prior $\nu\in\Delta$, exact inversion and nonexpansiveness of metric projection relative to points in $\Delta$ give

Here $s_M=\sigma_{\min}(A_M)$ and $\chi_r$ is the Euclidean norm of an $r$-dimensional standard Gaussian. If $q_{r,1-\delta}$ is its $(1-\delta)$ quantile, then

is sufficient for error probability at most $\delta$, uniformly over the entire simplex. Together with the two-prior converse, this proves that the noise scale $s_M=\Theta(e^{-a_{r-1}M})$ is sharp: uniform consistent recovery along a translated long-end design is possible if $\sigma_M/s_M\to0$ and impossible if that ratio stays bounded away from zero along a subsequence.

At $M=4$ the certificate directly inverts 200,000 Gaussian observations. The empirical unconstrained mean-square error agrees with the exact trace formula within one percent; simplex projection never increases the samplewise error, and the empirical projected tail lies below the displayed $\chi_3$ bound.

Correlated quote noise: whitened observability

Independence and equal quote variances are not essential. Let $\Gamma_M$ be a known positive-definite covariance across the $r$ maturity quotes and consider

With $U$ as above, put

The entire two-point argument holds in this whitened geometry. Choosing a right singular vector of $B_M$ for its weakest direction gives, for every estimator,

Conversely, the generalized least-squares coordinate estimator

satisfies the matching bounds

Thus $s_{M,\Gamma}$, not the unweighted condition number, is the exact noise scale. If constants $0<\gamma_-\le\gamma_+<\infty$ satisfy $\gamma_-I\preceq\Gamma_M\preceq\gamma_+I$ uniformly in $M$, then

so the sharp exponent remains $a_{r-1}$. No such conclusion is made when the quote covariance itself becomes ill-conditioned with maturity: in that case its eigenvalue scale is part of the inverse problem. Whitening and generalized least squares go back to Aitken's original treatment; here the calculation is finite dimensional and exact because $\Gamma_M$ is assumed known.

The certificate uses a correlated, heteroskedastic three-quote covariance with eigenvalues $(0.3846,0.6088,2.6066)$. Its whitened weakest-signal decay rate is $1.02644$, against $a_{r-1}=1.02586$. At $M=4$, 200,000 Gaussian draws reproduce the exact two-prior testing error and generalized least-squares mean-square error within $0.002$ and one percent respectively; simplex projection again obeys the $\chi_3$ tail bound.

Deterministic quote-error balls: the same sharp scale

The singular-value threshold is not an artifact of Gaussian tails. Consider the adversarial observation model

with no probability law for $e$. Keep the definitions $B_M=\Gamma_M^{-1/2}C_M^\top U$ and $s_{M,\Gamma}=\sigma_{\min}(B_M)>0$. The projected generalized inverse

obeys the deterministic, uniform bound

To see this, write $\nu=\pi+Ux$. Full column rank gives $\widehat x-x=B_M^\dagger\Gamma_M^{-1/2}e$; the operator norm of $B_M^\dagger$ is $1/s_{M,\Gamma}$, $U$ is an isometry, and projection onto the simplex cannot increase distance from the true $\nu\in\Delta$.

The scale is also necessary, with no constants hidden. Fix $0<\rho\le\min_i\pi_i$ and let $x_M$ be a unit right singular vector for $s_{M,\Gamma}$, $h_M=Ux_M$, and $\nu_\pm=\pi\pm\rho h_M$. If $\delta\ge\rho s_{M,\Gamma}$, the two admissible observation sets intersect at $y_*=C_M^\top\pi$: the errors $e_\pm=y_*-C_M^\top\nu_\pm$ have whitened norm exactly $\rho s_{M,\Gamma}$.

Since $\|\nu_+-\nu_-\|_2=2\rho$, every estimator evaluated at $y_*$ is at distance at least $\rho$ from one of the two priors. Therefore

A local parameter ball makes the whole phase transition exact, not only its threshold. Define

The displayed restriction on $\rho$ is sufficient for $\Theta_\rho\subset\Delta$, since every row of the isometry $U$ has norm at most one. For estimators allowed to depend on the known radii $\rho$ and $\delta$, the deterministic local minimax risk is exactly

For the upper bound, use the constant estimator $\pi$ when $\delta\ge\rho s_{M,\Gamma}$. Below that threshold, apply the generalized inverse and project onto the convex set $\Theta_\rho$; metric projection cannot increase its distance from the true point, so its error is at most $\delta/s_{M,\Gamma}$. For the converse, put $a=\min\{\rho,\delta/s_{M,\Gamma}\}$ and replace $\rho$ by $a$ in the two weakest-direction priors above.

Their error balls meet at $C_M^\top\pi$, and the triangle inequality forces loss at least $a$ for one of them. This proves the identity at every noise radius, including the saturation regime.

Thus uniform robust consistency holds when $\delta_M/s_{M,\Gamma}\to0$ and fails whenever $\delta_M\ge\rho s_{M,\Gamma}$ along a subsequence for some fixed feasible $\rho$. Under the uniformly bounded covariance hypothesis above, the real-spectrum long-end tolerance is again $\Theta(e^{-a_{r-1}M})$.

This is a worst-case deterministic statement, distinct from the preceding Gaussian probability statements. The comparison belongs to the optimal-recovery/statistical-estimation correspondence developed by Donoho (1994); the constants here are exact because the finite-dimensional ellipsoid is resolved directly by the SVD.

For the correlated four-state certificate at $M=4$, the two adversarial Mahalanobis balls touch exactly at the common midpoint. Direct generalized inversion attains $\delta/s_{M,\Gamma}$ in the weakest left-singular direction, and 10,000 errors sampled throughout a smaller ball all satisfy the uniform bound. The local-ball certificate also checks six noise-to-threshold ratios from zero to three: the two-point lower certificate and the projected-inverse/constant upper certificate both equal $\rho\min\{1,\delta/(\rho s_{M,\Gamma})\}$.

Oscillatory long-end converse: infinitely many translated aliases

The real-spectrum hypothesis is essential for a uniform long-end lower bound. In three states, suppose the centered eigenvalues are $-a\pm ib$, with $a,b>0$, and the scalar feature is cyclic. Fix any offsets $c_1<c_2$, and write

Substitution in the exact aliasing kernel below gives, for a nonzero model-dependent constant $C_0$, the identity

The leading term is a nonzero sinusoid: $L(M)=A\sin(bM+\phi)$ with

Consequently, if $\widehat M_k=(k\pi-\phi)/b$, then for every sufficiently large $k$ there is an exact singular translation

This follows directly from the simple zeros of $L$ and the implicit-function theorem (or a sign change on an $O(e^{-a\widehat M_k})$ neighborhood). Hence a translated complex-spectrum design loses rank infinitely often; no global claim $\sigma_3(A_M)=\Theta(e^{-aM})$ is possible. On any phase-separated set $|L(M)|\ge\eta>0$, however,

Locally at a sufficiently large $M_k$, the simple zero sharpens this to $\sigma_3(A_M)=\Theta(e^{-aM_k}|M-M_k|)$ and $\kappa_2(A_M)=\Theta(e^{aM_k}/|M-M_k|)$. Thus the ordinary exponential deterioration is the regular-phase behavior, interrupted by reciprocal conditioning walls and exact aliases. This is the translated-maturity form of the eigenvalue-cycle aliasing mechanism studied for irregularly sampled linear systems by Park and Sahai.

For the directed-cycle certificate with offsets $(0.4,1.7)$, $A=0.520001659599$. The first four asymptotic phase zeros are $(3.384584565,7.012183293,10.639782022,14.267380750)$, while the exact singular translations are $(3.385120036,7.012180971,10.639782032,14.267380750)$. The normalized root shift approaches the predicted magnitude $|C|/(bA)=0.085890490311$, and the direct determinant factorization error over 41 translations is below $8.2\times10^{-15}$.

An exact defective example makes the confluent case explicit. Let

Every off-diagonal rate is positive, while $Q_Jv=-v$ and $Q_Jw=0.1v-w$, so the eigenvalue $-1$ has a size-two Jordan block. With $F=w$,

The ratio $g_1/g_0$ is strictly increasing because its derivative is $e^{-t}(t-1+e^{-t})/(1-e^{-t})^2>0$. Thus every positive ordered pair works directly, despite the defective generator.

A complex spectrum marks a real boundary to that statement, not merely a limitation of the proof. For the directed three-cycle

the Krylov determinant is $-3$, so almost every pair of maturities identifies the prior. But the nonzero eigenvalues are $-3/2\pm i\sqrt3/2$.

If $b=\sqrt3/2$, then at $\tau_1=\pi/b$ and $\tau_2=2\pi/b$ the factors $1-e^{(-3/2+ib)\tau_j}$ are both real. The two real response vectors are therefore collinear and

Thus a cyclic feature still gives generic identifiability, but distinct maturities need not all work once oscillatory modes are present. The theorem assumes a real nonzero spectrum; it does not claim that this condition is necessary for every particular chain.

For three states, however, the boundary is exact. A cyclic centered scalar feature has the all-pairs property

if and only if the two nonzero eigenvalues are real. Sufficiency is the real-spectrum theorem above.

Conversely, write a complex pair as $-a\pm ib$, with $a,b>0$, and identify the centered real plane with $\mathbb C$. Apart from a fixed nonzero feature factor, the response is

At $\tau_1=\pi/b$ and $\tau_2=2\pi/b$, direct substitution gives the exact relation

Hence every three-state complex-spectrum generator has an exceptional positive maturity pair, not just the directed cycle. This converse is special to a two-dimensional centered space; in higher dimensions a complex pair does not by itself force two full response columns to be collinear.

The complete three-state aliasing equation

The obstruction is not a single accidental pair. In the same complex coordinate, the real determinant is a nonzero model-dependent constant times the oriented area $\operatorname{Im}\{\overline{z(\tau_1)}z(\tau_2)\}$. Consequently, for every cyclic centered feature, its complete exceptional set is the zero set of

This formula gives a universal infinite aliasing lattice. Put $\tau_k=k\pi/b$. Since

every pair $(\tau_k,\tau_\ell)$ with $1\le k<\ell$ is exceptional. Each lattice zero is simple in its second maturity, because

Thus each lattice point lies on a locally smooth exceptional curve. Conversely, if $\sin(b\tau_1)\ne0$, every exceptional partner must satisfy the explicit bound

Indeed, beyond this bound the two terms multiplied by $e^{-a\tau_2}$ cannot cancel the nonzero limit $e^{-a\tau_1}\sin(b\tau_1)$. Analyticity then implies that a fixed off-lattice first maturity has only finitely many exceptional partners.

The lattice is therefore infinite but nongeneric. This is the same eigenvalue-cycle aliasing mechanism that Park and Sahai show can be broken by nonuniform sampling in linear observation problems.

For the directed-cycle certificate, $\det[{\bf1}\ R(\tau_1)\ R(\tau_2)] =-(2/\sqrt3)\mathcal K_{3/2,\sqrt3/2}(\tau_1,\tau_2)$. Two hundred arbitrary maturity pairs reproduce this identity within $1.1\times10^{-14}$. Across 100 random positive-rate complex-spectrum generators, all six lattice pairs with $1\le k<\ell\le4$ are singular: the maximum relative determinant is $3.7\times10^{-16}$ and the maximum exact collinearity residual is $4.4\times10^{-16}$.

Generic maturity tuples and exceptional-set fragility

The word generic can be made measure-theoretic, for an arbitrary vector feature and any fixed number $q$ of maturity blocks. On the connected chamber

let

Indeed, every entry of $R(\tau)=H(I-e^{Q_0\tau})F$ is real analytic. Choose a maturity tuple attaining rank $\rho$ and one nonzero $\rho\times\rho$ minor there. That minor is a nontrivial real-analytic function on $\mathcal C_q$, and every rank-deficient tuple lies in its zero set.

A nontrivial real-analytic function has a null zero set, by Mityagin's short primary proof; it also cannot vanish on an open subset. This argument does not assert that the exceptional set is empty, nor does it give a uniform conditioning bound.

For a cyclic scalar feature, the small-maturity expansion below supplies one full-rank tuple with $q=r=n-1$. Consequently, for an arbitrary real or complex spectrum, almost every ordered positive $r$-tuple identifies the initial prior. The real-spectrum theorem is stronger: it says the exceptional set is empty.

The directed-cycle zero above is locally a smooth codimension-one obstruction, and its numerical instability is sharp. Write

Since $R'(t)=e^{Q_0t}F$ for the centered feature, direct evaluation gives

The implicit-function theorem therefore makes the nearby exceptional set a smooth curve. If $A_0=\mathcal A(\tau_1^\star,\tau_2^\star)$ has nonzero singular values $s_1\ge s_2>0$, then a transverse displacement $\delta$ in the second maturity satisfies

The certified coefficients are $9.409632470013\times10^{-6}$ and $184072.099850$, respectively. At $|\delta|=0.00125$, the measured determinant slope, smallest-singular-value slope, and condition coefficient are all within $0.2\%$ of these limits; the last determinant-slope convergence order is $0.9988$. Thus measure-zero exceptions can still create an exact reciprocal conditioning wall nearby.

The scalar cyclic case also makes the near-zero conditioning explicit. Let $r=n-1$, let $K=[{\bf1}\ F\ Q_0F\ \cdots\ Q_0^{r-1}F]$, and take $\tau_j=\epsilon c_j$ with $0<c_1<\cdots<c_r$. Then

Thus distinct sufficiently small maturities do identify a cyclic prior, but their determinant collapses at a precisely polynomial rate. This small-$\tau$ loss is different from the exponential loss at fixed positive maturities as $m\to\infty$.

The determinant gives the product of the singular values but not the actual noise amplification. That hierarchy is also exact.

Put $C_\epsilon=[R(\epsilon c_1)\ \cdots\ R(\epsilon c_r)]$ and $A_\epsilon=[{\bf1}\ C_\epsilon]$, and order singular values from largest to smallest. Then, as $\epsilon\downarrow0$,

Here $x_\epsilon\asymp y_\epsilon$ means their ratio stays between positive constants depending on $Q_0,F$ and the fixed distinct $c_j$, but not on $\epsilon$. The smallest singular value is therefore order $\epsilon^r$. Consequently any fixed coordinate chart on the zero-sum prior contrasts has worst-case observation-noise amplification of order $\epsilon^{-r}$, even though its relative condition number is only order $\epsilon^{-(r-1)}$ after the normalization direction has been removed.

The proof exposes the confluent Vandermonde structure. Cayley–Hamilton gives an exact analytic factorization

where the lower-right block of $V(c)$ is $[c_j^k/k!]_{k,j=1}^r$. It is invertible because the $c_j$ are positive and distinct.

Both outer factors remain uniformly invertible, so the standard singular-value inequalities transfer the powers on the diagonal to $A_\epsilon$; deleting the normalization column and using the centered Krylov block gives the statement for $C_\epsilon$. This is the small-node instability of Vandermonde inversion studied classically by Gautschi (1962), here pulled back through the Krylov basis.

The hidden constants in $\asymp$ are also explicit. This matters when maturities are chosen rather than merely proved to exist.

Take a QR factorization $K=U_KR_K$ and an LQ factorization $V(c)=L_VU_V$, with positive or negative diagonal entries allowed in the triangular factors. Then the separated powers in the diagonal imply

The formula is independent of the sign conventions in QR and LQ. For $C_\epsilon$, replace $K$ by the centered Krylov block $K_c=[F\ Q_0F\ \cdots\ Q_0^{r-1}F]$ and $V(c)$ by $V_c=[c_j^k/k!]_{k,j=1}^r$. If $K_c=U_cR_c$ and $V_c=L_cW_c$, then

To prove the limit, orthogonal invariance reduces the matrix to $R_KD_\epsilon\{L_V+E_\epsilon\}$, where $D_\epsilon=\operatorname{diag}(1,\epsilon,\ldots,\epsilon^r)$ and $\|E_\epsilon\|=O(\epsilon)$. In its $j$th exterior power, the unique minor with the smallest exponent uses the first $j$ rows and columns. Therefore the product of the largest $j$ singular values, divided by $\epsilon^{j(j-1)/2}$, converges to the product of the first $j$ absolute diagonal pairs.

Dividing consecutive product limits gives the displayed individual constant. The analytic remainder has one additional power in every flag row, so it cannot alter the limit.

Determinant-optimal maturities under a cap

The determinant formula also solves an exact local design problem. Fix a maximum scaled maturity and write $\tau_j=\epsilon T_{\max}c_j$, with $0<c_1<\cdots<c_r\le1$. All model-dependent factors are fixed, so maximizing the magnitude of the leading determinant is equivalent to maximizing

The unique maximizer is the affine Legendre–Gauss–Lobatto design. Let $-1=x_0<x_1<\cdots<x_r=1$, where the interior $x_j$ are the zeros of $P_r'$, the derivative of the degree-$r$ Legendre polynomial. Then

Indeed, the last node must equal one, and the log Vandermonde is strictly concave in the remaining ordered interior nodes. Its equilibrium equations are $\sum_{j\ne i}(x_i-x_j)^{-1}=0$. If $w(x)=\prod_{j=0}^r(x-x_j)=(x^2-1)p(x)$, these equations say that $w''$ vanishes at every interior node.

Degree and leading-coefficient comparison give $w''=r(r+1)p$, whose polynomial solution is $p\propto P_r'$. Strict concavity proves global uniqueness. This is the interval Fekete geometry originating in Fejér's interpolation work.

This is a determinant or local D-optimality statement. It does not claim that the same nodes maximize the smallest singular value or minimize the condition number. For four states ($r=3$), the normalized design is

At the cap used in the certificate, it improves the leading determinant by $1.250725436$ over the previous $(0.7,1.3,2.2)/2.2$ design, hence improves the corresponding information determinant by the square, $1.564314115$. At $\epsilon=0.0015625$, the exact response determinant ratio is $1.250158127$. The equilibrium equations are verified through nine states with maximum residual $2.4\times10^{-13}$; the exact gains over equally spaced nodes from three through nine states are $(1,1.086729,1.302945,1.733914,2.568915,4.244826,7.831480)$.

Three states: D-, E-, and condition-optimal designs differ

The caveat is substantive even in the smallest nontrivial case. For three states, write the two normalized maturities as $(c,1)$, $0<c<1$. The response coefficient matrix is

Its LQ diagonal entries, in absolute value, are

The model-dependent Krylov QR factors multiply these quantities but do not change their optimizers. Consequently three natural local criteria give three different designs:

Here $c_E=0.4533976515$. The cubic has exactly one root in $(0,1)$ because its derivative is positive.

Direct differentiation gives all three optima and shows they are unique. The condition row concerns the response-only matrix $C_\epsilon$; for the augmented matrix $A_\epsilon$, the largest normalization singular value is node-independent to leading order, so its condition optimum coincides with the E-optimum.

The E-design raises the weakest-direction constant by $0.941474\%$ relative to the D-design. In a cyclic three-state certificate at $\epsilon=0.001$, the measured weakest constants for the D/E/response-condition designs are respectively $(0.0538838754,0.0543941976,0.0540253769)$, while the corresponding response condition coefficients are $(22.5707789,21.9579293,21.7937489)$.

Every measured coefficient is within $0.3\%$ of its QR/LQ limit. Thus maximizing the determinant cannot be silently reinterpreted as optimizing the hardest prior direction.

The $T=O(m^{-1})$ boundary layer

If $T_j=\tau_j/m$, then $\mathcal O_m$ is exactly independent of $m$ after the trivial $1/m$ response scaling:

Different $\tau_j$ can rotate a single feature through enough directions to identify the full prior. By contrast, for fixed positive $T_j$ all blocks converge to $HF$. If $\|e^{Q_0t}R\|\le Ce^{-\delta t}$, then $\mathcal O_m$ lies within $O(e^{-m\delta T_{\min}})$ of $[HF\ \cdots\ HF]$.

Hence every singular value beyond $\operatorname{rank}F$ is exponentially small: exact finite-$m$ rank can survive, but the extra identification is exponentially ill-conditioned. The sharp theorem above resolves every exponent when the scaled maturities have fixed gaps and a simple real spectrum. This is why the fixed-maturity first-order theorem correctly retains only $\nu HF$.

The actual bond curve has a compatible, but one order smaller, initial layer because $g(0)=0$. Put $T=\tau/m$, $u=\theta-\bar\theta{\bf1}$, and $v=s-\bar s{\bf1}$. A rescaled Duhamel expansion through the first volatility term gives

For several scaled maturities, $[K_1(\tau_1)\ \cdots\ K_1(\tau_q)]$ gives the order-$m^{-2}$ rank test. If both displayed coefficients are resolved, append the $K_2$ columns. The next coefficient contains volatility only through the combination $\kappa^2u+v$; it does not separately identify both unknown features without additional information.

In the important special case of constant $\theta$, $u=0$ and the order-$m^{-2}$ initial-prior signal vanishes. Switched variance then becomes the leading signal at order $m^{-3}$, and the augmented rank of $[K_2(\tau_1)\ \cdots\ K_2(\tau_q)]$ determines initial-prior identification.

In the certificate, one centered feature on a nonreversible three-state chain has fixed rank one, yet two scaled maturities $\tau=(0.4,1.7)$ give observability rank two and recover the prior $(0.15,0.55,0.30)$. The exact mean formula agrees with independent quadrature within $1.4\times10^{-16}$. After subtracting the $m^{-2}$ term, the bond-price remainder has orders $2.9997$ and $2.9983$; after also subtracting the new $m^{-3}$ term, these improve to $4.0015$ and $4.0009$.

A separate positive-variance, constant-$\theta$ example recovers the same prior from volatility alone, with remainder orders $3.9988$ and $3.9946$ after its leading $m^{-3}$ term. At fixed maturities, the second singular value falls from $4.47\times10^{-2}$ at $m=2$ to $1.05\times10^{-6}$ at $m=16$, displaying the predicted loss of conditioning.

A separate four-state certificate has one scalar cyclic feature. Three maturities $(0.25,0.8,1.9)$ recover the prior $(0.10,0.25,0.40,0.25)$ to $2\times10^{-13}$, while 32 random finite-chain cases verify equality of the sampled response and Krylov ranks. The normalized small-maturity determinant approaches its Vandermonde coefficient and the QR/LQ formula predicts all seven augmented and response-only leading singular-value constants; at $\epsilon=0.00625$ the largest relative discrepancies are $1.44\%$ and $1.44\%$ respectively.

The augmented constants are $(2,4.65744565,0.80736962,0.01315675)$ and the response-only constants are $(4.68629884,0.80766818,0.01331125)$. The normalized determinant converges with measured order $0.9725$. The four augmented singular values have measured exponents $(0.000005,0.9823,1.9790,2.9838)$, while the response-only condition number has exponent $2.0015$, verifying the full hierarchy rather than only its product.

For a four-state complete-graph generator, whose nonzero eigenvalue has multiplicity three, the scalar-feature response rank remains exactly one at every maturity, confirming the spectral obstruction.

The all-maturities certificate adds a circulation to a four-state reversible chain. Detailed balance fails by $0.05$, while all three nonzero eigenvalues remain real and simple. Across 200 randomly ordered positive maturity triples, it verifies the determinant factorization and its fixed sign; the maximum relative discrepancy is below $5\times10^{-13}$.

A representative prior is recovered within $2.2\times10^{-15}$. A separate irreducible three-state chain has the genuine Jordan relations $Q_0v=-v$ and $Q_0w=0.1v-w$. With $F=w$, its Krylov determinant is $1.2$; 200 ordered maturity pairs verify the confluent factorization within $5\times10^{-13}$, and recover a prior within $4.2\times10^{-15}$.

For the directed-cycle counterexample, the Krylov determinant is $-3$, a generic pair has response determinant $-0.170100185$, but the exceptional pair has determinant $4.46\times10^{-16}$ and smallest singular value $1.75\times10^{-16}$. Finally, 100 random irreducible three-state chains with complex spectra verify the universal exceptional-pair ratio: the maximum collinearity residual is $5.8\times10^{-16}$, while the smallest relative determinant at a generic comparison pair remains above $0.029$.

A computable finite-rate bound

The following deliberately conservative estimate makes the $O(m^{-2})$ statement explicit, including $T=0$. Use the row-sum operator norm $\|\cdot\|_\infty$. Since $Q_0$ is finite and irreducible, choose $C,\delta>0$ such that

These constants can be certified numerically: select $t_0>0$ with $\rho=\|e^{Q_0t_0}R\|_\infty<1$, then take $C=2/\rho$ and $\delta=-\log\rho/t_0$. On $[0,L]$ set $f(t)=g(t)-(\pi g(t)){\bf1}$, $G=\sup\|f(t)\|_\infty$ and $J=\sup\|f'(t)\|_\infty$.

For the model here, for example, one may take $G=\|u\|_\infty+\|v\|_\infty/(2\kappa^2)$ and $J=\kappa\|u\|_\infty+\|v\|_\infty/\kappa$. Write

If $4\alpha G e^{2GL}\le1$ (automatically true when $G=0$), then for every $0\le T\le L$ the remainders in the theorem obey

These are analytic upper bounds, not confidence intervals for noisy bond quotes. Their rate threshold and constants are intentionally loose; they establish uniform finite-rate control for an arbitrary irreducible finite chain, including a nonreversible one. The certificate checks both bounds and the $m^{-2}$ numerical behavior on the three-state example.

Proof

Expansion and error. Divide $a(t)$ by $\exp(\int_0^t\bar g)$ and call the resulting vector $b$. Decompose it as $b=p{\bf1}+z$, with $p=\pi b$ and $\pi z=0$. Then

Positivity of the finite-state Feynman–Kac functional gives $e^{-GL}\le p\le e^{GL}$. Write $h=Hf$, so $Q_0h=-f$ and $\|h\|\le CG/\delta$, $\|h'\|\le CJ/\delta$.

The semigroup estimate and variation of constants, with $4\alpha G\le1$, give $\sup\|z\|\le 2\alpha G e^{GL}$. For $w=z-ph/m$, the condition $f(0)=0$ gives $w(0)=0$, and

Here $\|R\|_\infty\le2$ and $|p'|\le G\|z\|$. Another variation-of-constants estimate gives $\sup\|w\|\le W$. Consequently

Integration proves the first bound. Since $b_i=p(1+z_i/p)$, $|z_i/p|\le2\alpha Ge^{2GL}\le1/2$, and $|\log(1+q)-q|\le q^2$ for $|q|\le1/2$, the second bound follows. Substituting $f=-\kappa uB+vB^2/2$ into $\pi[fh]$ yields the three Green–Kubo coefficients and the displayed known-start term.

Information and rank. If $\sum_{j=1}^4 d_j I_j(T)=0$ on an open interval, differentiation gives $\sum_{j=1}^4d_j B(T)^j=0$. Because $B$ ranges over an interval, every $d_j$ vanishes. The same argument with the constant polynomial term proves independence of $T,I_1,\ldots,I_4$.

Finally, the long-run covariance rate of additive functionals is $2\mathcal K_0^{\rm sym}$. For an irreducible finite chain this quadratic form is strictly positive for any nonzero centered vector: writing $h=Hf$, the martingale quadratic variation is $\sum_{i,j}\pi_i(Q_0)_{ij}(h_j-h_i)^2$, which vanishes only if $h$ is constant along every transition, hence $f=0$. Thus the Gram matrix of centered regime features has precisely their linear rank, at most $n-1$.

For two features this proves the latent determinant restriction, including the two-state equality case.

Initial mixture and boundary layer. Since $Q_0H=H Q_0=\Pi-I$ and $\pi F=0$, integration of $\nu e^{mQ_0t}F$ gives the exact transient-mean formula. Stacking its values gives $\mathcal O_m$, and the augmented rank criterion is simply injectivity on the affine hyperplane $\nu{\bf1}=1$. Exponential mixing proves the fixed-maturity singular-value bound by comparison with the repeated block $HF$.

Moreover, on centered columns, $HQ_0=-I$, so the analytic expansion is $R(\tau)=\tau F+\tau^2Q_0F/2!+\cdots$. Its derivatives at zero generate the Krylov space, while Cayley–Hamilton stops that space at power $n-2$; this proves both inclusions in the whole-curve theorem. In the scalar cyclic case, expressing the response columns in the Krylov basis and taking the lowest-degree determinant gives the displayed Vandermonde coefficient.

For the yield boundary, set $t=r/m$ in the backward equation and expand

The rescaled equation therefore has potential terms first at orders $m^{-2}$ and $m^{-3}$. Variation of constants around $e^{Q_0r}$ gives the two displayed integrals after changing variables and subtracting the stationary start.

A product of two potential insertions, and hence the first quadratic Dyson term, cannot occur before order $m^{-4}$; the quadratic term from the logarithm also first occurs there. This proves the $O_L(m^{-4})$ remainder uniformly for $\tau$ in a fixed compact interval.

This finite-state proof is the elementary Poisson-equation version of the additive-functional central-limit framework. For broader state spaces see Kipnis and Varadhan (1986) in the reversible case and Glynn and Meyn (1996) for irreducible discrete- and continuous-time Markov processes.

Scope: common fixed $\kappa$, fixed known $x$, finite $n$, and a fixed bounded maturity interval. The exact fixed-feature covariance-rank theorem permits any initial prior. The pricing-identification formulas cover stationarity, a labeled starting state, and an unobserved fixed initial mixture.

The Krylov theorem characterizes noiseless mean observability across the complete scaled-maturity curve; its determinant formula and the real-spectrum all-maturity sharpening also warn that formal rank need not imply stable statistical recovery. The boundary theorem concerns $T=\tau/m$ with $\tau$ in a fixed compact interval; it is distinct from the fixed-maturity outer theorem. Stochastic reversion speed and simultaneous limits $T\to\infty$ require separate analyses.

The integrated-loading theorem only requires $G\succeq0$ and square-integrable deterministic $L$; its finite-basis equality assumes the scalar loading Gram matrix $\Gamma$ is positive definite.