S Project SLSSM-01

Independent sequence modeling research

Making language model dynamics visible.

Project SLSSM-01 studies recurrent language models with inspectable spectral dynamics. This public dossier presents the evidence, methods, and limitations; the implementation, weights, and training recipe remain private.

Read the public note Explore the synthetic atlas Meet the researchers
Synthetic spectral mode field showing three mode families. The visualization does not contain model weights or activations.
Synthetic public visualization. No model weights, activations, or implementation parameters are shown.

What we are publishing

The map, not the engine.

Spectral Atlas

A visual language for mapping learned recurrent mode families and tracking how they reorganize during training.

EDMD Concordance

Extended Dynamic Mode Decomposition (EDMD) compares measured trajectory behavior with learned internal dynamics as an interpretability lens.

Prompt Manifold

A prompt-conditioned trace of mode usage during inference, labeled with illustrative token steps so the path can be read as a simulated prompt moving through mode neighborhoods.

Model class

A stable linear spectral state-space language model.

The public name for Project SLSSM-01's model class is a stable linear spectral state-space language model. More precisely: Project SLSSM-01 uses a stable linear recurrent dynamics core, with nonlinear token drive, gating, and readout around that core. That distinction matters. The stable linear part is what makes the internal dynamics analyzable as modes; the surrounding learned components are what let those modes interact with language.

Stable

The recurrent state is constrained to avoid uncontrolled dynamical blow-up during training and inference.

Linear core

The central state evolution is treated as a mode-based linear dynamical system.

Spectral

The model can be analyzed through learned frequencies, decay behavior, and mode populations.

Language model

Nonlinear learned components drive and decode the stable state dynamics into token behavior.

250k training steps in the v3 short-context run
32.8B tokens processed in the analyzed run
360M-class research model in the analyzed run

Evidence without the recipe

What we can show publicly.

The charts below summarize the Project SLSSM-01 v3 short-context training run. They show run scale, loss movement, dynamical agreement, signed phase behavior, and the next open problem: longer memory.

3.16

Best observed loss

Rounded final checkpoint result from the analyzed v3 short run.

3.27

Validation mean

Twenty-batch validation probe at the final analyzed checkpoint.

~31%

EDMD distance reduction

Mean nearest-distance improved from the early checkpoint to final.

2-way

Signed phase population

Final modes retained both positive and negative phase rotation.

Training movement

checkpoint summary
Line chart showing rounded best loss decreasing from 4.03 at 30k steps to 3.16 at 250k steps.

Loss moved down across the run rather than stalling at the early spectral organization stage.

EDMD concordance

lower is closer
Line chart showing rounded EDMD mean distance decreasing from 0.0113 at 30k steps to 0.0078 at 250k steps.

Independent trajectory analysis stayed close to the learned spectral structure after the early run.

Signed phase did not collapse

final checkpoint
Bar chart showing 986 negative-rotation modes and 1,062 positive-rotation modes at the final checkpoint.

The model kept meaningful population on both rotation directions, which is exactly what signed phase was meant to preserve.

Long-memory gap

why v3-long exists
Line chart showing six high-retention modes and one very-high-retention mode at the final checkpoint.

The short-context model worked, but the near-unit slow population remained sparse. That gives the next run a clear target instead of a vague hope.

Checkpoint summary
Checkpoint Tokens Best loss Validation mean EDMD mean distance Read
30k 3.9B 4.03 3.91 0.0113 early organization
102k 13.4B 3.38 3.40 0.0079 agreement tightened
204k 26.7B 3.24 3.28 0.0077 stable dynamics
250k 32.8B 3.16 3.27 0.0078 short-context finish
Public-safe artifact

Inspect the rounded checkpoint record.

The downloadable CSV mirrors the public tables. The research note defines each metric, states the evidence boundary, and lists outcomes that would weaken the current interpretation.

Download rounded CSV Read the evidence guide
Reproducible demo

Run the measurement method yourself.

A standalone EDMD demonstration on public Pythia-14m training checkpoints: a pre-specified protocol with dated post-run amendments, pinned model commits, shuffled-pairing and persistence controls, CPU-only, a few seconds per run. It validates the measurement method on a public model; it does not validate this model's claims.

Get the demo on GitHub

These author-reported summaries are intentionally rounded and come from one analyzed run. They are not an external replication or a claim of benchmark leadership. The public note documents the disclosure and validation boundary. Read the public research note.

Research visualization

Token-labeled Spectral Atlas

This page uses generated public data to demonstrate the type of analysis we perform. The traced path is labeled with example prompt tokens so the viewer can understand what is being simulated. No Project SLSSM-01 weights, activations, or tokenizer details are shown.

3D mode field with prompt trace

synthetic prompt path
Synthetic three-dimensional spectral atlas showing a prompt path through slow, middle, and fast mode families.
slow family mid family fast family

Prompt manifold

token-labeled trace
Synthetic prompt trace showing example tokens moving through three mode neighborhoods.

Prompt traces show how words can move through different mode neighborhoods. In the research workflow, real prompts can be compared across checkpoints without reducing the model to a single loss number.

Concordance sketch

synthetic sketch
Synthetic concordance sketch comparing measured trajectory points with learned structure.

The research question is whether measured dynamics agree with learned internal structure. The public chart shows the idea while omitting the implementation-specific transformation path.

Research stance

Models should be studied from the inside, too.

Loss curves matter, but they do not tell the whole story. We are interested in how memory, phase, and mode participation evolve inside a model while it learns.

We developed a proprietary recurrent spectral language model whose internal dynamics can be tracked across training. Our analysis revealed a shortage of long-retention modes consistent with the model's observed long-memory limitations. This finding informed a follow-up training plan designed to strengthen long-timescale memory while preserving short-context performance.

Research team

Led by the Hunt brothers.

Freddie Hunt

Freddie Hunt

Founder / Lead Research Scientist

Leads Project SLSSM-01's model research, training analysis, spectral diagnostics, and experimental direction.

LinkedIn X
Joshua Hunt

Joshua Hunt, PhD

Research Collaborator / Academic Advisor

Research collaborator and academic partner. Affiliation listed for identification: Louisiana Tech University.

LinkedIn University X

Project SLSSM-01 is an independent research effort. Louisiana Tech University affiliation is provided for identification only and does not imply institutional endorsement.

Questions, corrections, or collaboration: contact@trinity-research.com