Aionoscope Manifold Reconstruction Benchmark

Status: draft research idea for extending the published Aionoscope linear-probe diagnostic into manifold-level representation diagnostics.

Collaboration

If this direction resonates with you, I would be happy to talk with like-minded people, collaborate on research, and work on use-cases together.

Ideas are not the bottleneck. Hands are. Time-series modeling should be moving at least as fast as vision, audio, and robotics.

• Email: [email protected]
• X: @chemeris
• Telegram: @alexanderchemeris

Summary

Aionoscope now provides the published MILETS 2026 baseline: it asks whether a frozen model representation preserves known latent process variables through pooled linear probes. The next step is to reconstruct the latent-variable manifold inside the model representation and evaluate its geometry directly.

The central benchmark question is:

When the true data-generating latent variable has known geometry,
does the model representation preserve that geometry in a usable form?

This would move the benchmark beyond “can a readout recover the factor?” A nonlinear probe can recover a factor from a representation that is geometrically tangled. Manifold reconstruction should ask whether the representation itself keeps the right shape: line, circle, torus, product space, branching regime graph, or a messy high-curvature embedding.

Research Spark

Goodfire’s neural-geometry series is the immediate spark. The strongest local lesson is not that time-series work should copy language-model steering directly. The transferable idea is that concepts can live on curved manifolds inside representation space, and that linear directions can miss or distort those structures.

Source status and credibility:

• Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior was submitted to arXiv on 2026-05-06 by Goodfire-affiliated authors with Stanford, Harvard, Northeastern, and Technion affiliations. It is a recent preprint, not peer reviewed, but credible enough as a timely research spark because it has a full arXiv paper and a Goodfire research post.
• Do Sparse Autoencoders Capture Concept Manifolds? was submitted to arXiv on 2026-04-30 by the same broader Goodfire neural-geometry cluster. It is also a recent preprint, useful for the “linear features versus curved manifolds” framing, but not yet peer-reviewed evidence.
• The Neural Geometry Series is a Goodfire research-blog series, useful for visual intuition and method framing rather than as a benchmark authority.

Diffusion Score-Field Probe Candidate

DiffusionGeometryProbe is a direct candidate baseline for the subset of models that expose a DDIM-compatible diffusion noise predictor. It queries the local score Jacobian, compares several intrinsic-dimension estimators, sweeps spectral conditioning over the noise schedule, and optionally adds score-energy basin and persistent-homology diagnostics.

The transferable idea is useful: a generative score field may reveal tangent and normal directions that ordinary output-quality metrics miss. The current GALOP KDD 2026 workshop paper is not yet a validated solution for Aionoscope. Its reported image experiments use one generated image per model, two estimators are structurally limited by perturbation sample count, several cited limits are applied outside their stated small-noise regime, and the advertised Cheeger and Morse interpretations do not follow from the displayed derivations as written.

Aionoscope can turn those ambiguities into controlled tests:

1. train or adapt a diffusion model on process families whose latent dimension, tangent directions, periodic topology, strata, and sampling density are known;
2. compare score-derived tangent and dimension estimates with the exact generator Jacobian across seeds and noise scales;
3. vary perturbation count through and beyond the true codimension so empirical covariance-rank ceilings are visible;
4. test whether local-PCA becomes informative only after Tweedie projection or another denoising map;
5. separate local score-field geometry from global topology and from geometry of an arbitrary frozen encoder representation.

This should be treated as a falsification baseline, not imported as fourteen already calibrated manifold invariants.

Density Is Already Observable With Aionoscope

Aionoscope does not need BTM to estimate density on a representation manifold. The synthetic generator chooses the latent coordinates and their sampling law. A fixed-step sweep therefore gives a known design density in latent-coordinate space: every interval of width $\Delta \theta$ carries known mass.

Let

$$h(\theta )=f(g(\theta ,{\theta }_{-j},ϵ))$$

be the controlled sweep after the candidate encoder. Equal steps in $\theta$ do not imply equal arc length, area, or volume in representation space. Local point spacing already reveals stretching and compression. For a smooth injective $d$-dimensional sweep, the density with respect to the induced manifold-volume measure is

$$p_M(h(\theta ))=\frac{p_{\Theta }(\theta )}{\sqrt{\det \left(J_h(\theta {)}^{\top }{J}_h(\theta )\right)}}.$$

For a one-dimensional sweep this reduces to

$$p_M(h(\theta ))=\frac{p_{\Theta }(\theta )}{\Vert h^{\prime }(\theta )\Vert }.$$

Aionoscope can therefore measure, without a learned transport model:

• the exact latent design density or controlled regime-mixture weights;
• adjacent-point spacing and arc-length distortion under fixed latent steps;
• local Jacobian singular values and log-volume distortion;
• the induced pushforward density along the representation manifold;
• collisions or folds where several latent coordinates map to the same region;
• regional mass and rare-regime occupancy under deliberately non-uniform latent sampling.

The interpretation still needs care. The known density is the Aionoscope design prior, not automatically the real-world frequency of phases, regimes, or interventions. Density also depends on the chosen representation-space metric; finite grids only approximate a continuous density; nuisance-factor sampling must be declared; and a non-injective map requires summing contributions from all latent preimages. These are benchmark-protocol questions, not reasons to train a BTM probe.

What Beckmann Transport Models Actually Add

Beckmann Transport Models do not make density observable for the first time. They add a different, mechanistic transport question: can samples from a base distribution reach a target support with the correct regional mass, and what do the trajectories, attraction basins, and endpoint map look like?

Aionoscope alone:
known latent mass -> static pushforward density and geometric distortion
 
optional BTM-style probe:
off-manifold perturbation -> return trajectory, basin, endpoint, stability

The BTM five-atom experiment makes the support-versus-mass distinction sharp: two methods can land on the same support while one assigns badly wrong mass to its modes. Aionoscope can test the analogous static density distortion directly. A BTM-style extension is useful only for additional dynamic questions such as whether a perturbation returns to the correct latent coordinate, whether a rare regime leaks into a common-regime basin, and whether a learned endpoint map is stable or idempotent.

The benchmark SHOULD therefore separate two layers:

1. Aionoscope-only density diagnostics: use the known latent sampling law, point spacing, Jacobian/volume distortion, collisions, and controlled regional mass; no auxiliary transport model is needed.
2. Optional transport diagnostics: add held-out normal-direction perturbations, fit a fixed-capacity autonomous field or direct return map, and score support-return error, latent-coordinate return error, basin leakage, idempotence $\Vert T(T(z))-T(z)\Vert$, and one-versus-many-pass convergence.

The optional probe is a proposed transfer, not a result in the BTM paper. A sufficiently expressive map may learn transport even onto a tangled representation, so map capacity, training budget, held-out latent regions, and nuisance-factor variation MUST be fixed. BTM is also many-to-one transport onto a support, not an inverse chart; success does not prove that the representation preserved the true latent parametrization.

Core Idea

Aionoscope can generate synthetic time-series samples with known latent factors. That gives the benchmark an advantage over ordinary representation probing: it can densely sample the latent coordinates, sweep one factor at a time, and know the correct topology in advance.

For a synthetic generator

$$x_{1:T}=g(\theta ,ϵ),$$

where $\theta$ contains known latent variables such as frequency, phase, amplitude, trend, damping, noise level, regime, or intervention parameter, run a candidate model encoder

$$z=f(x_{1:T})$$

and reconstruct the image of controlled latent sweeps in representation space:

$$M_j=\{f(g({\theta }_j,{\theta }_{-j},ϵ)):{\theta }_j\in {\Theta }_j\}.$$

The benchmark then evaluates not only whether ${\theta }_j$ is recoverable from $z$, but whether $M_j$ has the expected geometry.

Full Joint Manifold Versus One-Factor Slices

A one-factor sweep is only a coordinate slice through the full Aionoscope latent manifold. For a clean sinusoid with amplitude $A$, frequency $\omega$, and phase $\varphi$,

$$\theta =(A,\omega ,\varphi ),\Theta =[A_{\min },A_{\max }]\times [{\omega }_{\min },{\omega }_{\max }]\times S^1.$$

If the encoded map $h(\theta )=f(g(\theta ))$ is locally full rank, its image is a three-dimensional manifold embedded in a much larger representation space. It has two bounded interval directions and one periodic direction; it is not merely one line. A phase sweep with $(A,\omega )$ fixed is one one-dimensional curve on that object. An amplitude or frequency sweep at another anchor is another curve.

One-factor sweeps remain valuable because they expose order, periodicity, local stretch, seams, saturation, and individual Jacobian columns. However, a small union of such curves does not reconstruct the full joint manifold and can miss:

• interactions between amplitude, frequency, and phase;
• changing curvature or density away from the selected anchors;
• folds or collisions that appear only for particular factor combinations;
• rank loss and entanglement in pairwise or higher-order directions;
• global product topology.

The full benchmark SHOULD therefore use space-filling joint samples from $p_{\Theta }(A,\omega ,\varphi )$ and reserve one-factor sweeps at multiple anchors as interpretable evaluation slices. Dense Cartesian grids scale exponentially with factor count, so practical designs SHOULD use Sobol, Latin-hypercube, stratified, or adaptive sampling, with additional samples near boundaries, high curvature, collisions, and regime transitions.

The “manifold” may also be stratified rather than globally smooth. At $A=0$, phase and frequency can become unidentifiable, and the local rank can drop. Finite-window aliasing can map distinct frequencies or phases to similar observations. Discrete components or regimes form unions of strata, while continuous rendering noise can thicken a clean manifold into a tube or higher-dimensional support. The benchmark MUST say which variables are intrinsic state, which are discrete stratum labels, and which are nuisance or view variables.

What Data Should Train An Optional BTM Probe

The training distribution depends on the question.

Controlled Aionoscope Representation Probe

For the proposed benchmark, the encoder SHOULD remain frozen and the BTM target data SHOULD be generated by Aionoscope over the joint controlled latent prior, then passed through that encoder:

$$\theta \sim p_{\Theta },z=h(\theta )=f(g(\theta )),{\mu }_1=h_{\#}{p}_{\Theta }.$$

The encoder’s original pretraining corpus is not required for this probe. Using it would change the question from “how does this encoder represent the controlled Aionoscope factors?” to “what is the encoder’s global real-data representation support?” Aionoscope is intentionally a bounded unit test, so conclusions MUST stay scoped to its generated process family.

A BTM trained only on one phase, amplitude, or frequency sweep learns transport to one curve, not to the full three-dimensional sinusoid manifold. The main controlled BTM SHOULD train on space-filling joint latent samples. One-factor sweeps SHOULD be held out as readable slices at many anchors, alongside held-out joint regions, factor combinations, boundaries, and nuisance realizations.

Two distinct probe objectives SHOULD not be conflated:

1. Local return probe: form paired perturbations $\tilde{z}=z+\delta$ around known encoded states and train or evaluate a return map toward the originating $z$. Normal-direction perturbations test off-manifold recovery; controlled tangent perturbations are a separate sensitivity test.
2. Global mass-transport probe: choose a broader base distribution ${\mu }_0$ and test whether an autonomous field or endpoint map reaches the complete target support ${\mu }_1$ with correct regional mass. This does not require each base point to return to a preassigned latent coordinate.

Probe architecture, capacity, perturbation law, training budget, sampling prior, metric, and split MUST be identical across encoders. Otherwise the result can measure auxiliary-model capacity rather than representation geometry.

Global Real-Data Representation Probe

If the goal is instead to model the encoder’s full representation manifold, a second experiment SHOULD use held-out real data from the same domain as the encoder’s pretraining distribution, not silently reuse the exact pretraining examples. That experiment asks about real-domain support and possible memorization/generalization; it no longer provides Aionoscope’s exact latent coordinates or known intrinsic density.

The two experiments are complementary:

Aionoscope joint data -> controlled, labeled submanifold unit test
held-out real data    -> broader but weakly identified representation support

Neither should be used to claim the other. The recommended order is:

1. Aionoscope-only static geometry and known-density diagnostics;
2. optional BTM on the full joint Aionoscope latent design;
3. optional held-out real-domain transport experiment for external validity.

Why This Is Stronger Than Linear Probing

Linear probes answer a narrow question:

Can a linear readout recover this latent variable from the representation?

That remains useful, especially for linearly identifiable world-model state. But it misses several important cases:

• a good cyclic variable may be represented as a circle rather than a line;
• a useful latent factor may be recoverable only by following a curved path;
• a factor may be present but geometrically tangled with nuisance variables;
• a high-scoring nonlinear probe may hide a representation that is unusable for steering, generation, or local interpolation;
• a model may preserve factor order but break topology, for example turning phase into a line with artificial endpoints.

The manifold benchmark should therefore report both readout recoverability and representation geometry. A model should get credit for preserving a factor, but extra credit for preserving it with the right topology and low distortion.

Benchmark Axes

Recoverability

Measure whether known latent variables can be recovered from model representations:

• linear probe score;
• nonlinear probe score with a fixed, simple architecture;
• sample efficiency of the probe;
• out-of-range interpolation and extrapolation under latent sweeps.

This keeps backward compatibility with the current Aionoscope story.

Geometry Fidelity

Measure whether distances and neighborhoods on the reconstructed manifold match the true latent geometry:

• geodesic-distance preservation;
• trustworthiness and continuity;
• neighborhood stability under denser sampling;
• latent-distance versus representation-distance calibration;
• local tangent consistency.

For example, a frequency sweep should usually form an ordered one-dimensional curve, while phase should form a closed one-dimensional loop.

Linearity And Curvature

Measure whether a factor is represented as a simple linear direction or a tangled curved object:

• PCA residual for the latent sweep;
• chord-distance versus geodesic-distance ratio;
• curvature and tangent-variation statistics;
• local linear-probe degradation along the manifold;
• sensitivity to sampling density and nuisance-factor variation.

The point is not to require every representation to be linear. The point is to make linearity, curvature, and tangling measurable.

Topology

Measure whether the representation preserves the expected latent topology:

• connected components for discrete regimes;
• circularity for phase-like variables;
• torus-like structure for two independent phases;
• branch structure for regime-transition graphs;
• persistent homology summaries such as Betti numbers when useful.

This is where the benchmark can become memorable: if phase is truly periodic, the representation should not need an arbitrary seam where $2\pi$ wraps to $0$.

Disentanglement And Cross-Coupling

Measure whether sweeping one latent variable unintentionally moves along other latent directions:

• Jacobian cross-terms between latent factors and representation coordinates;
• subspace angle between factor manifolds;
• factor recovery after nuisance-factor randomization;
• product-geometry tests for independent latent factors;
• failure cases where amplitude, phase, frequency, and trend become fused.

This is the natural bridge from “latent variable preserved” to “latent state is usable.”

Known Sampling Density And Pushforward Fidelity

Because Aionoscope controls the latent grid or sampling prior, it can evaluate static density distortion directly:

• adjacent-point spacing divided by the known latent step;
• local stretch/compression from Jacobian singular values;
• induced manifold-density or log-volume distortion;
• regional mass under controlled uniform and non-uniform factor priors;
• collision multiplicity when separated latent coordinates occupy the same representation neighborhood;
• rare-regime occupancy across disconnected components.

Reports MUST distinguish the synthetic design prior from real-domain prevalence and MUST name the representation-space metric or normalization used for density. No BTM model is required for these measurements.

Optional Transport And Basin Fidelity

A BTM-inspired auxiliary probe can test a different property: how points outside the sampled manifold return to it. Possible diagnostics are:

• rare-regime basin recall and leakage across disconnected components;
• return-map coordinate error after normal and nuisance perturbations;
• map idempotence $T(T(z))\approx T(z)$ on and near the manifold;
• one-pass versus repeated-map convergence and wrong-attractor frequency;
• normal-versus-tangential approach near the manifold.

These optional scores MUST use a fixed probe/map family and held-out latent regions. They are useful only if they distinguish representation quality rather than the capacity of the auxiliary transport model.

Initial Synthetic Tasks

Start with factors where the expected geometry is obvious:

Synthetic factor	Expected geometry	Failure mode to expose
Amplitude	interval or ray	collapsed scale, saturation, entanglement with noise
Frequency	ordered curve	frequency aliases, local folds, nonlinear stretching
Phase	circle	artificial seam, broken wraparound, line-like encoding
Trend slope	line	sign collapse, nonlinear compression
Regime ID	separated components or graph	arbitrary clustering, mixed regimes

The first public version should be deliberately small: a sine-family benchmark with amplitude, frequency, phase, trend, and noise may be enough to prove the idea. The second version should add multivariate coupling, regime switches, exogenous variables, and intervention parameters.

Evaluation Contract

The benchmark should produce a compact report per model, layer, and pooling choice:

factor -> recoverability, geometry fidelity, linearity, topology, coupling,
          known-density distortion, optional return-map stability

Useful output examples:

• phase: nonlinear probe strong, linear probe weak, topology correct circle, low seam error, but probability mass compressed into a narrow arc;
• regime: separated components, but rare-regime perturbations leak into a common-regime attraction basin;
• frequency: linear probe strong, manifold nearly straight, low distortion;
• amplitude: recoverable but saturated at high values;
• two_phase: recoverable under nonlinear probe, but torus topology collapsed into a tangled curve;
• regime: separable components, but transition neighborhoods are wrong.

This should be more useful than one global benchmark score. The leaderboard can still summarize, but the diagnostic view should show which latent variables and which geometries survive the representation.

Relation To Foundation TSFM Agenda

This is an idea page, so the verdicts below describe the intended contribution if the proposed benchmark works. Evidence status is recorded separately in the Evidence and Missing pieces columns.

Agenda slot	Verdict	Evidence	Missing pieces
Benchmark level	closes	Proposes a benchmark that tests latent-variable preservation, geometry, topology, coupling, known-prior pushforward density, and optional return-map stability rather than only forecasting or probe accuracy. Evidence is an internal design proposal, Aionoscope’s controlled latent sampling, Goodfire-inspired geometry framing, and a BTM-inspired optional transport diagnostic.	Build the first public task suite, scoring code, density/metric conventions, fixed-capacity optional transport probes, visual reports, and model baselines.
Representation quality	partially closes	If validated, the benchmark distinguishes semantic-state preservation, dense numeric detail, topology, and tangling.	Need evidence across time-series encoders, foundation models, layers, and pooling choices.
Latent-state prediction	partially closes	Tests whether known latent process variables survive in representations and whether their geometry is usable for state tracking.	Need dynamic tasks where latent state evolves over time, not only static sampled factors.
Control and counterfactuals	adjacent	The action/intervention extension would evaluate whether action-conditioned parameter changes create separable manifold families.	Need synthetic actions, counterfactual rollouts, and action-conditioned model baselines.
Data diversity and long tail	partially closes	Synthetic generation allows dense sampling of rare regimes, endpoints, wraparound regions, and controlled nuisance variation.	Need real-domain transfer tests and checks that synthetic geometry predicts downstream utility.

Open Questions

• Which manifold reconstruction method should be the first default: Isomap, diffusion maps, UMAP for visualization only, local PCA, principal curves, persistent homology, or a custom sweep-aware estimator?
• Can a fixed-capacity BTM-style return map distinguish a clean factor manifold from a tangled but learnable embedding, or does map capacity erase the benchmark signal?
• How should off-manifold perturbations and normal directions be defined in high-dimensional representation space without assuming the answer?
• Does known-prior pushforward-density distortion predict rare-regime calibration, generation quality, or downstream state-prediction utility?
• Which representation-space metric and normalization make density distortion comparable across models and layers?
• Should the first paper emphasize representation diagnostics, benchmark design, or the failure of linear probing?
• How should scores avoid rewarding beautiful geometry that is not useful for downstream state prediction, generation, or control?
• Which layer and pooling choices should be standardized for fair model comparisons?
• How much of the benchmark should be model-agnostic, and how much should use generative or decoder-based steering tests?
• Can the same manifold diagnostics explain why some encoders work better for TSL-JEPA or LeNEPA-style latent prediction?
• Which visualizations will make the result obvious to the community without overselling low-dimensional plots?

Related Pages

• Aionoscope
• Beckmann Transport Models
• Foundation Time-Series Model Research Agenda
• Latent-Space Predictive Learning
• Latent-State Time-Series Modeling
• Time-Series Benchmark Hygiene
• Synthetic Data For Time Series
• Time Series Forecasting Using Manifold Learning
• When Does LeJEPA Learn a World Model?
• A Cookbook of Self-Supervised Learning
• TSL-JEPA