Dynamical Parameters: An Interpretability Framework for Time-Series Foundation Models

Kang Yang; Gaofeng Dong; Liying Han; Mani Srivastava

Model	Checkpoint	Params	$L$	$d$	$T$	Architecture
Chronos-2 ( Ansari et al., 2025 )	amazon/chronos-2	119.5M	12	768	256	Patched encoder
TimesFM 2.5 ( Das et al., 2024 )	google/timesfm-2.5-200m-pytorch	231.3M	20	1280	256	Decoder-only
TimesFM 3 ( Jain & Sen, 2026 )	google/timesfm-3.0-pytorch	330.7M	20	1280	256	Decoder-only
Toto-Open-Base 1.0 ( Cohen et al., 2025 )	Datadog/Toto-Open-Base-1.0	151.3M	12	768	256	Decoder-only
Moirai ( Woo et al., 2024 )	Salesforce/moirai-1.1-R-small	13.8M	6	384	256	Masked encoder
Sundial ( Liu et al., 2025b )	thuml/sundial-base-128m	128.3M	12	768	256	Decoder-only

Appendix

Table 1: Model checkpoints and configurations used throughout the evaluation.

Model	Forecast head	Paths	Statistic	Seeding
Chronos-2	21-quantile head	–	Median quantile	–
TimesFM 2.5	Flip-averaged pinball head	–	Median quantile	–
TimesFM 3	Quantile head	–	Median quantile	–
Toto-Open-Base 1.0	Autoregressive mean	–	Distribution mean	–
Moirai	Mixture head	64	Path mean	Shared across arms
Sundial	Flow matching, 50 steps	64	Path mean	Shared across arms

Appendix

Table 2: Forecast heads and sampling settings.

Law	Code	$\dim s_{t}$	Process noise	Input $u_{t}$	Readout $h$	Parameter location
Linear trend	TR	1	–	Constant	Linear	Input drift $S$
Oscillation	OS	2	–	–	Linear	Transition angle $\omega$
Mean reversion	MR	1	✓	Constant	Linear	Transition eigenvalue $\rho$
Random walk with drift	RW	1	✓	Constant	Linear	Input drift $d$
Exponential relaxation	RX	1	–	Constant	Linear	Transition eigenvalue $e^{-\gamma}$
Saturating growth	SG	1	–	Constant	Logistic	Readout growth rate $r$

Appendix

Table 3: Controlled laws in the common state-space form.

#	Law	Code	Physical	Coordinate $z_{m,j}$	Range
1	Linear Trend	TR	$S$	$\widetilde{S}=\operatorname{sign}\left(S\right)\sigma_{\mathrm{clean}}/\sigma_{\varepsilon}$	$\pm\left[1,6\right]$
2	Oscillation	OS	$\omega$	$C=\omega T/2\pi$	$\left[2,8\right]$
3	Mean Reversion	MR	$\rho$	$\rho$	$\left[0.3,0.9\right]$
4	Random Walk with Drift	RW	$d$	$D=d\sqrt{T}/\sigma_{\eta}$	$\pm\left[0.5,3\right]$
5	Exponential Relaxation	RX	$\gamma$	$g=\gamma T$	$\left[0.8,2.5\right]$
6	Saturating Growth	SG	$r$	$R=rT$	$\left[5,12\right]$

Appendix

Table 4: Dimensionless Dynamical Parameters and law-identification ranges.

Law	Construction criterion	LID	ACC	RSP	GEO
Linear trend ( TR )	Both slope signs remain visible above observation noise.	✓	✓	✓	✓
Oscillation ( OS )	At least two unaliased cycles occur.	✓	✓	✓	✓
Mean reversion ( MR )	Persistence remains separated from white noise and a unit root.	✓	✓	✓	✓
Random walk with drift ( RW )	Drift remains visible without dominating process innovations.	✓	–	–	–
Exponential relaxation ( RX )	Relaxation remains incomplete near the forecast origin.	✓	✓	–	–
Saturating growth ( SG )	The inflection lies within the context before saturation.	✓	✓	✓	✓

Appendix

Table 5: Generator constraints and scoring coverage. LID denotes law identification, ACC representation accessibility, RSP forecast response, and GEO causal geometry.

Task	Probe target	Penalty grid	$T=256$	$T=288$	$T=512$
			Reference $r^{\star}$
Law ID	13 balanced classes	$\lambda\in\{10^{-4},\ldots,10^{3}\}$	–	–	–
TR	$S\sim\mathcal{N}(0,0.02^{2})$	$\alpha\in\{10^{0},\ldots,10^{6}\}$	0.922	0.930	0.959
OS	$\omega=2\pi/P,\ \log P\sim\mathcal{U}[\log 2.5,\log 4096]$	$\alpha\in\{10^{0},\ldots,10^{6}\}$	0.988	0.986	0.971
MR	$\rho\sim\mathcal{U}[0.02,0.98]$	$\alpha\in\{10^{0},\ldots,10^{10}\}$	0.976	0.978	0.984
RX	$g\sim\mathcal{U}[0.8,2.5]$	$\alpha\in\{10^{0},\ldots,10^{10}\}$	0.899	0.901	0.901

Appendix

Table 6: Probe targets, penalty grids, and conditional reference correlations.

				Mean reference $\bar{U}_{m,j}^{\star}$
Law	Parameter	Range	Signed change	$T=256$	$T=288$	$T=512$
Linear trend ( TR )	Slope	$S\in[-0.45,0.45]$	$\pm 0.02,\pm 0.05$	0.995	0.995	0.997
Oscillation ( OS )	Frequency	$C\in[0.0625,102.4]$	$\pm 1,\pm 2,\pm 4$	0.870	0.869	0.833
Mean reversion ( MR )	Persistence	$\rho\in[0.02,0.98]$	$\pm 2\%,\pm 5\%$	0.760	0.766	0.755
Saturating growth ( SG )	Growth rate	$R\in[3,13]$	$\pm 2\%,\pm 5\%$	0.945	0.950	0.952
Damped oscillation ( DO )	Decay rate	$g\in[0.1,3]$	$\pm 2\%,\pm 5\%$	0.954	0.954	0.954

Appendix

Table 7: Intervention settings and mean conditional response references.

			Calibrated step $\alpha_{\ell}$		Layer count
Law	Parameter	$\delta$	Median	Range	$\alpha_{\ell}<1$	Converged
Linear trend ( TR )	Slope	0.02	0.58	0.021–1.00	86/90	85/90
Oscillation ( OS )	Frequency	2	0.19	0.037–0.76	90/90	90/90
Mean reversion ( MR )	Persistence	5%	1.00	0.19–1.00	26/90	90/90
Saturating growth ( SG )	Growth rate	5%	0.58	0.084–1.00	57/90	90/90
Damped oscillation ( DO )	Decay rate	5%	1.00	0.33–1.00	42/90	90/90

Appendix

Table 8: Hidden-intervention calibration across parameter targets for every evaluated model and layer. The final two columns report layer counts out of 90.

Model	TR	OS	MR	RW	RX	SG	LS	DO	SA	TP	DT	SS	FD	Mean	$D_{m}(1)$
Chronos-2 ( C2 )	0.08	0.08	0.17	0.17	0.08	0.08	0.08	0.08	0.08	0.08	0.17	0.17	0.17	0.12	0.91
TimesFM 2.5 ( F2 )	0.10	0.10	0.10	0.15	0.05	0.05	0.10	0.05	0.05	0.05	0.10	0.10	0.10	0.08	0.90
TimesFM 3 ( F3 )	0.20	0.15	0.15	0.20	0.05	0.10	0.15	0.05	0.15	0.15	0.15	0.15	0.20	0.14	0.81
Toto-Open-Base 1.0 ( TO )	0.25	0.25	0.33	0.42	0.17	0.08	0.25	0.17	0.17	0.17	0.33	0.33	0.33	0.25	0.75
Moirai ( MO )	0.33	0.33	0.33	0.67	0.33	0.33	0.33	0.50	0.33	0.50	0.33	0.33	0.33	0.38	0.55
Sundial ( SU )	0.25	0.33	0.25	0.33	0.17	0.08	0.25	0.17	0.08	0.17	0.25	0.25	0.25	0.22	0.72

Appendix

Table 9: Normalized law-emergence depth and first-layer decodability across models.

Figure 8: Validation law decodability across model depth. Panels show all nine frozen TSFMs. Dashed curves denote multi-parameter laws.

	True random walk classified as (%)
Model	TR	OS	MR	RW	RX	SG	LS	DO	SA	TP	DT	SS	FD	Layer
Chronos-2 ( C2 )	2.3	0.0	1.2	94.0	1.0	0.3	0.3	0.1	0.0	0.6	0.0	0.0	0.0	10/12
TimesFM 2.5 ( F2 )	2.5	0.0	1.0	94.4	1.1	0.2	0.2	0.0	0.0	0.5	0.0	0.0	0.0	10/20
TimesFM 3 ( F3 )	2.1	0.0	2.5	92.7	1.3	0.2	0.2	0.2	0.1	0.5	0.0	0.1	0.1	16/20
Toto-Open-Base 1.0 ( TO )	2.9	0.0	1.2	92.7	1.5	0.3	0.6	0.2	0.0	0.5	0.0	0.0	0.1	6/12
Moirai ( MO )	10.8	0.6	1.1	76.2	4.2	1.5	2.5	1.1	0.0	1.4	0.2	0.0	0.4	5/6

Appendix

Table 10: Test-set classifications of random-walk sequences at each model’s validation-selected layer. Correct classifications are bold.

Model	$\lvert D\rvert$	$n$	Random walk	Trend	Relaxation	Shift	Other
Chronos-2 ( C2 )	$[0.5,1)$	950	95.4	1.2	0.6	0.4	2.4
	$[1,2)$	981	95.4	1.3	0.7	0.3	2.2
	$[2,3]$	589	89.6	5.9	2.2	0.2	2.0
Toto-Open-Base 1.0 ( TO )	$[0.5,1)$	950	94.7	0.9	0.6	0.8	2.8
	$[1,2)$	981	94.6	1.7	1.1	0.5	2.0
	$[2,3]$	589	86.1	8.1	3.4	0.3	2.0

Appendix

Table 11: Test-set classifications of random-walk sequences by drift magnitude for two stronger and two weaker models. All reported class entries are percentages.

Figure 9: Test-set parameter representation accessibility across relative depth for eight single-parameter laws and nine evaluated TSFM architectures.

	Representation accessibility			Depth profile		Input control
Law	Reference $r^{\star}$	Selected $D$	First layer $D(\ell_{1})$	Emergence	Decline	Raw	Matched
Linear trend ( TR )	$[0.92,0.96]$	$[0.985,0.997]$	$[0.70,1.00]$	0.08	0.013	1.00	N/A
Oscillation ( OS )	$[0.97,0.99]$	$[0.444,0.763]$	$[0.04,0.62]$	0.38	0.041	$[0.00,0.01]$	$[0.65,0.72]$
Mean reversion ( MR )	0.98	$[0.907,0.976]$	$[0.25,0.97]$	0.08	0.009	0.00	0.00
Exponential relaxation ( RX )	0.90	$[0.821,0.993]$	$[0.67,0.99]$	0.08	0.009	$[0.83,0.84]$	0.99
Saturating growth ( SG )	0.98	$[0.990,0.993]$	$[0.93,0.99]$	0.08	0.003	0.83	$[0.98,0.99]$

Appendix

Table 12: Single-parameter references, accessibility profiles, and input controls across nine TSFMs. Brackets show minima and maxima across models or context lengths.

			Median Accessibility		Accessibility Difference
Parameter	Models	Cells	Linear $D$	MLP $D$	Median $\Delta D$	Maximum $\Delta D$
Oscillation frequency ( OS )	All nine	9	0.636	0.708	$+0.046$	$+0.078$
Seasonal dependence ( SA )	All nine	9	0.890	0.919	$+0.029$	$+0.115$
Trend slope ( TR )	TimesFM 3 ( F3 )	1	0.372	0.363	$-0.010$	$-0.010$

Appendix

Table 13: Linear and nonlinear probe accessibility across 19 model–law cells. Accessibility differences are computed separately for each cell.

Model	TR	OS	MR	SG	DO	SA	TP
Chronos-2 ( C2 )	$0.997\rightarrow 0.999$	$0.763\rightarrow 0.796$	$0.975\rightarrow 0.972$	$0.993\rightarrow 0.992$	$0.996\rightarrow 0.996$	$0.934\rightarrow 0.927$	$0.990\rightarrow 0.989$
TimesFM 2.5 ( F2 )	$0.997\rightarrow 0.999$	$0.738\rightarrow 0.770$	$0.976\rightarrow 0.973$	$0.993\rightarrow 0.992$	$0.996\rightarrow 0.995$	$0.936\rightarrow 0.931$	$0.989\rightarrow 0.987$
TimesFM 3 ( F3 )	$0.997\rightarrow 0.372$	$0.726\rightarrow 0.764$	$0.976\rightarrow 0.973$	$0.990\rightarrow 0.989$	$0.996\rightarrow 0.995$	$0.938\rightarrow 0.934$	$0.989\rightarrow 0.988$
Toto-Open-Base 1.0 ( TO )	$0.997\rightarrow 0.999$	$0.636\rightarrow 0.686$	$0.968\rightarrow 0.965$	$0.992\rightarrow 0.991$	$0.992\rightarrow 0.991$	$0.890\rightarrow 0.884$	$0.986\rightarrow 0.985$
Moirai ( MO )	$0.985\rightarrow 0.998$	$0.518\rightarrow 0.592$	$0.962\rightarrow 0.959$	$0.990\rightarrow 0.989$	$0.984\rightarrow 0.981$	$0.648\rightarrow 0.629$	$0.983\rightarrow 0.982$
Sundial ( SU )	$0.997\rightarrow 0.999$	$0.670\rightarrow 0.719$	$0.976\rightarrow 0.973$	$0.993\rightarrow 0.992$	$0.995\rightarrow 0.994$	$0.899\rightarrow 0.894$	$0.989\rightarrow 0.988$

Appendix

Table 14: Parameter accessibility on the original and response-regime datasets. Each model entry reports original

\rightarrow

response-regime accessibility, and the final row reports the corresponding conditional reference

r^{\star}

computed from the same history distribution.

Figure 10: Conditional-reference-normalized response across signed intervention magnitudes. The dashed line and bands mark the conditional reference and

95\%

bootstrap intervals.

Law	$\bar{U}_{m,j}$	$U^{(3)}$	$\gamma$	$\bar{\rho}$	$U^{\star(3)}/U^{\star}$	Coefficient	$R^{2}$
	Paired Intervention					Unpaired Check
Linear trend ( TR )	$[-0.82,1.01]$	$[-0.87,1.22]$	$-0.021$	–	0.93	$[-1.54,0.89]$	–
Oscillation ( OS )	$[0.03,0.64]$	$[0.02,0.49]$	$-0.100$	0.42	0.82	$[-0.15,0.48]$	0.52
Mean reversion ( MR )	$[0.24,0.85]$	$[0.24,0.79]$	$-0.007$	0.68	0.81	$[0.59,1.26]$	0.78
Saturating growth ( SG )	$[-1.02,2.33]$	$[-0.99,0.63]$	$-0.002$	0.85	1.01	$[-1.03,1.14]$	0.87
Damped oscillation ( DO )	$[0.04,1.15]$	$[0.02,0.91]$	$-0.005$	0.97	1.01	$[0.04,1.22]$	0.95

Appendix

Table 15: Scale-aware and unpaired response diagnostics. Panel A summarizes seven laws across nine models, while Panel B reports cells with

\bar{U}_{m,j}<0

or

\bar{U}_{m,j}>1.5

. Bracketed entries report model minima and maxima; unbracketed

\gamma

,

\bar{\rho}

, and

R^{2}

entries in Panel A report medians.

Figure 11: Exponential-relaxation reference stability across the original and restricted parameter ranges under multiple numerical robustness checks.

Figure 12: Selected-layer test response and validation-admissible layer fraction for Jacobian-derived hidden-state edits across seven single-parameter laws.

Figure 13: Validation local response across model depth for nine TSFMs and seven single-parameter laws. The shaded band denotes the admissible interval

[0.85,1.05]

.

Figure 14: Numerical sensitivity of causal-geometry interventions. The left and center panels vary the inversion ridge and relative hidden-state dose. The right panel reports Time-MoE ( ME ) expert rerouting for linear trend ( TR ), oscillation ( OS ), seasonal autoregression ( SA ), and transient pulse ( TP ) across the evaluated intervention strengths.

		Response at $\alpha_{\ell}$		Response at $0.1\alpha_{\ell}$		Rerouted Pairs
Layer	$\alpha_{\ell}$	Live	Frozen	Live	Frozen	$\alpha_{\ell}$	$0.1\alpha_{\ell}$
1	0.0370	0.79	0.96	0.39	0.97	94%	37%
2	0.0642	0.84	0.98	0.70	0.97	97%	47%
3	0.0214	0.67	0.92	0.55	0.98	75%	19%
4	0.0281	0.75	0.94	0.44	0.98	77%	20%
5	0.0281	0.79	0.93	0.65	0.98	74%	18%

Appendix

Table 16: Time-MoE ( ME ) linear-trend ( TR ) local response and expert rerouting under live and frozen routing at the calibrated step

\alpha_{\ell}

and one-tenth of that step.

Figure 15: Shared causal subspace rank

k_{90}

across relative model depth for nine TSFMs and seven single-parameter laws. Unreached

90\%

response-retention thresholds are shown as lower bounds.

Figure 16: Required subspace dimension

k_{90}

for parameter targets relative to shuffled, matched-random, spectrum-matched, wrong-law, and generic row-space controls. Hollow markers indicate unreached

90\%

response-retention thresholds and corresponding lower bounds.

Figure 17: Linearized response retention across subspace dimension

k

for parameter targets, control targets, and the generic Jacobian row-space basis in four representative model–parameter cells.

Figure 18: Parameter-target and matched-control subspace dimensions, specificity, and deployed gain across four forecast horizons.

Figure 19: Matched-input aligned gain relative to geometric diagnostics of input-induced hidden-state changes across 63 model–parameter cells. The panels report shared-subspace fraction, linearized gain, Jacobian-null-space fraction, nonlinear remainder, and intervention size.

Figure 20: Local, projected, deployed, and oracle-assisted fixed-direction responses for linear trend ( TR ) across model depth in nine TSFMs. Hollow markers denote validation-rejected layers, and parentheses report admissible and evaluated layer counts.

Figure 21: Difference between deployed and oracle-assisted fixed-direction responses across validation-admissible layers for nine TSFMs and seven single-parameter laws. Positive values indicate an advantage for the input-specific coordinate readout.

Figure 22: Causal-coordinate transfer across intervention magnitude, noise, and parameter-range shifts without target-regime refitting. The top row reports aligned gain, cosine similarity, and centered error for Chronos-2 ( C2 ) on seasonal autoregression ( SA ). The second row reports improvement over the oracle-assisted fixed-direction control and readout gain across all evaluated transfer cells.

	Centered fidelity					Raw fidelity
Condition	$g$	$s$	$e$	$e_{\perp}$	$e_{\mathrm{level}}$	$g^{\mathrm{raw}}$	$e^{\mathrm{raw}}$
Matched input	0.42	0.54	1.06	0.55	1.70	0.84	0.54
Local	1.00	1.00	0.06	0.04	0.09	1.00	0.03
Projected	0.93	0.99	0.18	0.12	0.22	0.96	0.10
Deployed	0.58	0.75	0.68	0.29	0.77	0.77	0.48
Fixed direction	0.23	0.34	1.06	0.54	1.54	0.52	0.93

Appendix

Table 17: Median pooled response-fidelity metrics across 63 model–parameter cells. Hidden-state interventions use calibrated steps and capped targets.

Figure 23: Centered and raw response-fidelity metrics across the single-parameter panel. Columns compare matched-input, local, projected, deployed, and fixed-direction responses. Centered metrics capture response alignment and trajectory error, while raw metrics use uncentered forecast changes.

Figure 24: Matched linear-trend ( TR ) response examples illustrating coherent, weak, and reversed matched-input behavior across TimesFM 3 ( F3 ), Moirai ( MO ), and TTM-R2 ( TT ). Each panel compares the reference forecast response with the matched-input, local, projected, and deployed responses across representative model-specific hidden-state response patterns.

Figure 25: One Chronos-2 ( C2 ) mean-reversion ( MR ) pair at layer 12, with

\rho^{A}=0.4006

,

\rho^{B}=0.4486

, and

\delta q=+0.0480

, traced through the matched-input and hidden-state intervention responses.

Dual Tone
	Accessibility	Matched-Input Response $\bar{U}_{m,j}$		Compact	Deployed	Deployed $>$ Fixed
Parameter	$D_{m,j}$	Six models	Three models	Cells	Gain	Cells
Low-tone frequency ( TL )	0.86–0.99	0.82–1.00	0.04–0.16	7/9	0.65–0.98	9/9
High-tone frequency ( TH )	0.93–0.99	0.75–0.99	$-0.16$ –0.12	6/9	0.40–0.97	9/9
Amplitude ratio ( TW )	0.94–1.00	0.99–1.18	0.06–0.35	6/9	0.44–0.94	8/9
Linear State Space

Appendix

Table 18: Multi-parameter summary across nine TSFMs. Entries for representation accessibility, matched-input response, and deployed aligned gain report ranges across models. The six-model group contains all models except Moirai ( MO ), TTM-R2 ( TT ), and Timer-XL ( TX ), which form the three-model group. Compact cells report the descriptive compactness count from the run summary.

	Accessibility		First Layer	Depth Profile		Input Control
Parameter	$r^{\star}$	$D_{m,j}$	$D_{m,j}\left(\ell_{1}\right)$	Emergence	Decline	Raw	Matched
Dual Tone
Low-tone frequency ( TL )	1.00	$[0.857,0.995]$	$[0.10,0.96]$	0.25	0.039	0.00	$[0.00,0.01]$
High-tone frequency ( TH )	1.00	$[0.927,0.994]$	$[0.07,0.97]$	0.17	0.028	$[0.01,0.02]$	$[0.00,0.01]$
Amplitude ratio ( TW )	0.99	$[0.940,0.996]$	$[0.08,0.99]$	0.17	0.037	0.01	0.00
Linear State Space

Appendix

Table 19: Multi-parameter recoverability, accessibility, depth profiles, and input controls across nine TSFMs. Brackets indicate minima and maxima across models or context lengths.

Figure 26: Test-set parameter accessibility across relative depth for seven jointly identified parameters. Panels show all nine frozen TSFMs.

Model	Single	TL	$\Delta$	Single	TL	$\Delta$
	Accessibility $D_{m,j}$			Matched-Input Response $\bar{U}_{m,j}$
Chronos-2 ( C2 )	0.999	0.992	$-0.007$	1.02	0.97	$-0.05$
TimesFM 2.5 ( F2 )	1.000	0.994	$-0.006$	0.97	0.97	0.00
TimesFM 3 ( F3 )	1.000	0.995	$-0.005$	1.01	1.00	0.00
Toto-Open-Base 1.0 ( TO )	0.998	0.973	$-0.025$	1.03	0.82	$-0.21$
Moirai ( MO )	0.983	0.857	$-0.126$	0.08	0.04	$-0.04$

Appendix

Table 20: Matched single-tone control for the dual-tone low-frequency parameter ( TL ). Accessibility and matched-input response compare the single-tone setting with TL , with

\Delta=\texttt{TL}-\text{single tone}

.

Figure 27: Centered and raw response-fidelity metrics across the multi-parameter panel. Columns compare matched-input, local, projected, deployed, and fixed-direction responses. Centered metrics capture response alignment and trajectory error, while raw metrics use uncentered forecast changes.

Figure 28: Conditional-reference-normalized multi-parameter response across signed intervention magnitudes. The dashed unit-reference line is shown where it lies within the panel range.

Figure 29: Local validity of Jacobian-derived hidden-state interventions across seven jointly identified parameters. Panel (a) reports test

U_{\mathrm{local}}\left(\ell\right)

at the validation-selected layer, while panel (b) reports the fraction of measured layers satisfying the local-validity criterion.

PANEL A: PARAMETER-LEVEL SUMMARY
		Matched-Input Diagnostics					Unpaired Check
Parameter	Model	$\bar{U}_{m,j}$	$U^{(3)}$	$\gamma$	$\bar{\rho}$	$U^{\star(3)}/U^{\star}$	Coefficient	$R^{2}$
Low-tone frequency ( TL )	Nine models	$[0.04,1.00]$	$[-0.04,1.01]$	$-0.011$	$0.50$	$1.00$	$[0.03,0.98]$	$0.94$
High-tone frequency ( TH )	Nine models	$[-0.16,0.99]$	$[-0.52,1.01]$	$-0.008$	$0.25$	$1.00$	$[0.03,0.98]$	$0.94$
Amplitude ratio ( TW )	Nine models	$[0.06,1.18]$	$[0.04,1.09]$	$-0.001$	$0.91$	$0.97$	$[0.03,0.98]$	$0.94$
Eigenvalue angle ( SC )	Nine models	$[0.07,0.83]$	$[0.05,0.79]$	$-0.016$	$0.43$	$1.03$	$[0.09,0.81]$	$0.78$

Appendix

Table 21: Scale-aware and unpaired response diagnostics for seven jointly identified parameters. Panel A summarizes results across nine TSFMs, with brackets indicating model minima and maxima. Panel B reports weak, reversed, or amplified response cells.

Figure 30: Validation local response across relative depth for seven jointly identified parameters and nine TSFMs. The profiles show the nonlinear effectiveness of Jacobian-derived hidden-state interventions across measured layers.

Figure 31: Shared forecast-control rank across relative depth for seven jointly identified parameters and nine TSFMs. Values at the top grid point

2^{10}

include unreached response-retention thresholds and therefore represent lower bounds where censored.

Figure 32: Deployed-over-fixed response gain across validation-admissible layers for seven jointly identified parameters and nine TSFMs. Positive values indicate an advantage of the input-specific deployed direction over the fixed-direction control.

Figure 33: Matched-input responses to controlled linear trend and sinusoid injections on real-series backgrounds. Injection strength is component root-mean-square (RMS) amplitude relative to context standard deviation for all three evaluated real datasets.

Figure 34: Illustrative ETTh1 linear trend injection and Chronos-2 ( C2 ) forecast-change traces at injection strength

0.5

for one representative matched input pair.

Law	All	$\bar{U}_{m,j}<0$	$0\leq\bar{U}_{m,j}<0.5$	$\bar{U}_{m,j}>1.5$	$D_{m,j},\bar{U}_{m,j}$	$U_{\mathrm{deployed}},\bar{U}_{m,j}$
	Cells With $D_{m,j}\geq 0.95$				Spearman $\rho$
Linear trend ( TR )	9	3	2	0	0.62	0.78
Oscillation ( OS )	0	–	–	–	0.92	0.78
Mean reversion ( MR )	8	0	1	0	0.92	$-0.03$
Saturating growth ( SG )	9	2	2	3	0.18	0.45
Damped oscillation ( DO )	8	0	2	0	0.38	$-0.20$

Appendix

Table 22: Thresholded agreement and cross-model correlations among representation accessibility, matched-input response, and deployed response. Counts use original-bank cells with

D_{m,j}\geq 0.95

, so the trend comparison is exploratory rather than common-regime.

Figure 35: Forecast regret versus matched-input response fidelity across the single-parameter panel. The figure highlights validation-selected competent forecasters and compares the two TimesFM 2.5 ( F2 ) forecast heads under the same response evaluation setting.

Subset	Cells	Median $D_{m,j}$	Median $g$	Median $e$
All	63	0.985	0.42	1.06
Regret $<1$	49	0.976	0.64	0.92
Regret $<0.5$	36	0.990	0.65	0.92
Best tercile	21	0.989	0.69	0.83

Appendix

Table 23: Single-parameter cells stratified by forecast competence.

D_{m,j}

denotes original-bank representation accessibility, while

g

and

e

are pooled matched-input response metrics. Subset thresholds use validation regret, with metrics evaluated on the corresponding test banks.

Model	Trend	Mean Rev.	Rand. Walk	Other	Layer 1	Selected	Selected Layer
	True Trend Classified As (%)				Validation Trend Accuracy (%)
Chronos-2 ( C2 )	97.2	0.0	1.4	1.4	93.7	97.1	10/12
TimesFM 2.5 ( F2 )	97.6	0.0	0.9	1.5	92.8	97.2	10/20
TimesFM 3 ( F3 )	94.6	0.7	0.7	4.0	62.8	94.8	16/20
Toto-Open-Base 1.0 ( TO )	96.1	0.0	2.0	1.9	82.8	96.2	6/12
Moirai ( MO )	92.9	0.0	4.4	2.7	59.1	92.2	5/6

Appendix

Table 24: Trend-law classification and validation accuracy across nine TSFMs. Classification columns report the percentage of true linear-trend ( TR ) sequences assigned to each law group.

Absolute-Slope Bin $\left\|S\right\|$	Histories	Gate Fires
All	32,768	48.0%
$\left[0.000,0.100\right)$	7,324	0.0%
$\left[0.100,0.200\right)$	7,302	0.0%
$\left[0.200,0.234\right)$	2,480	17.6%
$\left[0.234,0.300\right)$	4,798	92.5%
$\left[0.300,0.450\right)$	10,864	100.0%

Appendix

Table 25: TimesFM 3 ( F3 ) detrending-gate firing rate across

32{,}768

trend histories grouped by absolute slope. The gate remains inactive at small slopes and becomes nearly universal once the slope exceeds the analytic crossover region across the evaluated response-history distribution.

		Continuation Gain $k$ by $\left\|S\right\|$
Model	Overall $k$	$\left[0,0.05\right)$	$\left[0.05,0.2\right)$	$\left[0.2,0.45\right)$	$k^{\star}$	$\bar{U}_{\mathrm{TR}}$
Chronos-2 ( C2 )	$+0.87$	$+0.54$	$+0.84$	$+0.89$	$+1.00$	$+0.91$
TimesFM 2.5 ( F2 )	$+0.81$	$+0.50$	$+0.69$	$+0.86$	$+1.00$	$+0.92$
Median head (pinball)	$+0.81$	–	–	–	–	–
Mean head (MSE)	$+0.54$	–	–	–	–	–
TimesFM 3 ( F3 )	$+0.94$	$+0.32$	$+0.80$	$+0.99$	$+1.00$	$+1.01$

Appendix

Table 26: Continuation gain across nine TSFMs, reported overall and by absolute-slope bin. The TimesFM 2.5 ( F2 ) rows additionally compare its median and mean forecast heads.

Model	Absolute-Slope Bin $\left\|S\right\|$	Pairs	Live	Shape	Scale
Chronos-2 ( C2 )	$\left[0,0.05\right)$	122	$+0.86$	$+0.76$	$+0.09$
	$\left[0.05,0.2\right)$	357	$+0.96$	$+0.58$	$+0.38$
	$\left[0.2,0.45\right)$	545	$+0.91$	$+0.18$	$+0.73$
All		1,024	$+0.92$	$+0.39$	$+0.53$
Time-MoE ( ME )	$\left[0,0.05\right)$	122	$-1.33$	$-1.60$	$+0.27$
	$\left[0.05,0.2\right)$	357	$-0.98$	$-0.74$	$-0.23$

Appendix

Table 27: Trend response decomposed into standardized-shape and window-scale contributions across slope bins. Rows report live response together with the corresponding shape and scale components for each evaluated model and slope regime.

Model	Live Pairs	Live	Frozen	Amplitude	Residual
Chronos-2 ( C2 )	32,768	$+0.91$	$+0.81$	$-0.10$	$-0.00$
Moirai ( MO )	131,072	$+0.12$	$-0.46$	$-0.56$	$-0.02$
Sundial ( SU )	32,768	$+0.11$	$-1.69$	$-1.76$	$-0.03$
Time-MoE ( ME )	32,768	$-0.45$	$-1.25$	$-0.80$	$+0.01$
TTM-R2 ( TT )	32,768	$-0.69$	$-2.70$	$-2.02$	$+0.00$
Timer-XL ( TX )	32,768	$-0.82$	$-2.40$	$-1.58$	$+0.00$

Appendix

Table 28: Trend response under the standard normalization pipeline, frozen statistics, and an amplitude-only control. The residual reports the difference between the frozen-statistics response and the sum of the live and amplitude-only responses across all evaluated normalization replay models.

Parameter	$r^{\star}_{\mathrm{cond}}$	$r^{\star}_{\mathrm{hist}}$	$U^{\star}_{\mathrm{cond}}$	$U^{\star}_{\mathrm{hist}}$
Seasonal autoregression ( SA )	0.975	0.956	0.848	0.883
Transient pulse ( TP )	0.997	0.989	0.983	0.768
Damped oscillation ( DO )	0.986	0.986	0.957	0.958
Drive frequency ( WC )	1.000	1.000	1.000	0.917
Diffusion time ( WD )	0.991	0.942	0.960	0.971

Appendix

Table 29: Sensitivity of parameter correlation and pair-weighted response-reference gain to nuisance conditioning. Response columns report reference values rather than TSFM scores.

Figure 36: Reference and normalized-score changes when nuisance variables or the generating law are not supplied. Left: conditional and history-only parameter correlations. Middle: model responses normalized by each reference. Right: calibration of exact and classifier-based law posteriors.

Condition	Target $e$	Non-target	Relative $\Delta$ CRPS	$\Delta$ Coverage
Matched input	1.10	0.325	–	–
Deployed	0.64	0.177	0.000	-0.001
Projected	0.21	0.030	-0.002	-0.001
Fixed	1.12	0.155	0.013	-0.005
Equal-norm random	1.00	0.423	0.025	-0.015

Appendix

Table 30: Median target and non-target errors across

64

horizon runs. Quantile metrics use linearly extrapolated changes for Chronos-2 ( C2 ) and TimesFM 3 ( F3 ), with coverage change measured for the

50\%

interval across the two evaluated quantile models.

Figure 37: Target fidelity, non-target deviations, and quantile-level response changes for hidden-state edits. Quantile diagnostics apply only to Chronos-2 ( C2 ) and TimesFM 3 ( F3 ).

Model	Jacobian (s)	Inversion (ms)	Basis (s)	Peak (GB)	Deployment (ms)
Chronos-2 ( C2 )	0.003	0.04	6.2	11.0	6.3
TimesFM 3 ( F3 )	0.007	0.02	4.6	14.6	22.2
Time-MoE ( ME )	0.036	0.08	11.4	21.1	9.2
TTM-R2 ( TT )	0.002	0.03	5.2	10.1	0.2

Appendix

Table 31: Computational cost at

H=16

. Jacobian construction and inversion are measured per pair, deployment per input, and basis construction once. Shared-device contention and backend differences preclude a general speed comparison across the four evaluated TSFM architectures.

Dynamical Parameters: An Interpretability Framework for Time-Series Foundation Models

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