S1-GRT-RNN (MatterWaveAI/s1-grt-rnn)
S1-GRT-RNN is a Stateful O(1) Constant-Memory Unitary Gated Rotational RNN World-Model Judge, Model-Predictive Control (MPC) Action Router & Long-Horizon Automaton Guardrail (0 dense linear matrices, 0 1D/2D convolutional layers) built on GRT-RNN.
Complex multi-step agent protocols and stateful execution guardrails over long horizons (T = 8,192+) require nonlinear hidden-state-dependent transition gating—where the recurrent state transition itself depends on what mode or automaton state the agent is currently in—without the quadratic memory growth of attention KV caches.
S1-GRT-RNN unites unitary gated rotational RNN recurrence (Rotational-minGRU & Unitary-GRU) with multi-step latent trajectory lookahead:
- Dual
O(1)Recurrent Modes (0Bytes KV Cache atT = 128 -> 8,192+): Supports both parallelizableRotational-minGRUscan mode (< 3.82e-6numerical parity between parallel prefix scan andO(1)step recurrence) and nonlinear state-coupledUnitary-GRUmode, updating a compact3.0 KBrecurrent state inO(1)compute andO(1)memory per event. - Unitary Long-Horizon Stability (
det = +1.000000): Because every gated state rotation is strictly orthogonal, homogeneous state magnitude drifts by< 5.0e-7over1,000+steps (9.55e-15atT = 8,192in float64)—preventing both vanishing memory of early safety constraints and exploding recurrent activations. - Single-Pass Latent MPC Lookahead (
100%Option-Order Invariant): Branches the live recurrentRNNstate across candidate agent actions in parallel before emitting anAPPROVE/REJECT/ESCALATEverdict.
Release Note (Initial Preview Card): This card reports held-out
XL-smolS1-GRT-RNNstateful verification, parallel-scan parity, nonlinearUnitary-GRUstability,O(1)memory scaling, and latent MPC routing benchmarks (12,003held-outXL-smolrecords /29,609questions and17,13012-domainXL-smoltest records /44,287questions). Model weights, architectural implementation details, and source code are withheld during the staged world-model release cycle to prevent white-box guardrail evasion.
Core Capabilities (S1-GRT-RNN)
- Stateful
O(1)Agent & Protocol Monitoring (0KV Cache):- Tracks long-horizon agent execution states, tool-call protocols, and build logs in
O(1)memory per step while supporting both parallelminGRUscan training (< 3.82e-6step parity) and nonlinear state-coupledUnitary-GRUrecurrence.
- Tracks long-horizon agent execution states, tool-call protocols, and build logs in
- Latent Trajectory Lookahead & Prompt-Injection Defense:
- Branches the live recurrent hidden state across candidate actions in parallel to simulate multi-step causal consequences in latent space, detecting adversarial prompt-injection hijacks embedded inside untrusted tool observations (
77.17%top-1 /86.60%pass@2routing accuracy across803held-out trajectories /2,409test questions).
- Branches the live recurrent hidden state across candidate actions in parallel to simulate multi-step causal consequences in latent space, detecting adversarial prompt-injection hijacks embedded inside untrusted tool observations (
- 100% Option-Order Invariance (
Permutation Consistency = 1.000):- Evaluates each candidate transition independently from the recurrent state, guaranteeing exact mathematical invariance (
100.0%consistency) regardless of how candidate actions or multiple-choice options are ordered.
- Evaluates each candidate transition independently from the recurrent state, guaranteeing exact mathematical invariance (
- Hybrid
RNN + Rotational AttentionMode:- Supports both Pure Rotational
RNN(0%KV cache,98.6%continuous cyclic & stateful automaton tracking) and Hybrid Mode (66.7%KV-cache reduction,99.2%associative needle-in-haystack recall +99.2%state tracking).
- Supports both Pure Rotational
XL-smol Benchmark Results
1. S1-GRT-RNN: XL-smol Latent Rollout Verification & Stateful MPC Routing (12,003 Held-Out Records / 29,609 Questions)
S1-GRT-RNN XL-smol Evaluation Suite |
Held-Out Records (n) |
Test Questions | Complexity per Step | Top-1 Accuracy | pass@2 Accuracy |
Option-Order Consistency |
|---|---|---|---|---|---|---|
Multi-Hop Relational Rollout Verification (2..3 Hops) |
2,500 |
5,000 |
O(1) RNN State Update |
94.40% |
98.00% |
100.0% (PASS) |
| Causal Science & Physical Commonsense Verification | 2,500 |
5,000 |
O(1) RNN State Update |
96.60% |
99.10% |
100.0% (PASS) |
| Single-Hop Factual World-Model Verification | 2,500 |
5,000 |
O(1) RNN State Update |
92.80% |
96.50% |
100.0% (PASS) |
4-Mode Socratic & Epistemic Verification (SEE/SAY/DO/KNOW) |
2,500 |
5,000 |
O(1) RNN State Update |
67.80% |
94.60% |
100.0% (PASS) |
2D Visual-Spatial Grid Rule Induction (ARC-AGI-2 XL-smol) |
1,200 |
3,600 core (7,200 all) |
Latent Rule Induction | 63.17% |
86.00% |
100.0% (PASS) |
| Agent MPC Action Routing, Drift & Prompt-Injection Defense | 803 |
2,409 |
H = 1..2 Lookahead |
77.17% |
86.60% |
100.0% (PASS) |
Long-Horizon Cyclic & Stateful Automaton Tracking (T = 64..8,192) |
Streaming State | 8,192 Steps |
0 Bytes KV Cache |
98.60% |
99.60% |
100.0% (PASS) |
6-Suite XL-smol Aggregate (Macro / Micro) |
12,003 |
26,009 core (29,609 all) |
O(1) Unitary RNN |
81.99% / 84.08% |
93.47% |
100.0% (PASS) |
2. Unitary Gated RNN & Rank-Preservation Invariants
| Invariant / Diagnostic | Target Gate | S1-GRT-RNN Result |
Status |
|---|---|---|---|
Dense Linear / 1D / 2D Convolutional Layers |
0 / 0 / 0 |
0 / 0 / 0 (100% Orthogonal) |
PASS |
Parallel Geodesic minGRU Scan vs. O(1) Step Parity |
< 1e-5 |
3.81e-6 (Exact Match) |
PASS |
Unitary Recurrent Transition Determinant (Scan & Nonlinear GRU) |
+1.000000 |
+1.000000000000 |
PASS |
| 8,192-Step Homogeneous State Norm Drift (Unitary Stability) | < 1e-6 |
9.55e-15 f64 / < 5.0e-7 f32 |
PASS |
KV-Cache Memory Growth (T = 128 -> 8,192, Pure RNN) |
0 bytes |
0 bytes (3.0 KB fixed state) |
PASS |
Unitary Recurrent Effective Rank Retention (0 Negatives) |
>= 90.0% |
>= 96.2% |
PASS |
3. Constant O(1) Streaming State Memory vs. Quadratic Attention (T = 128 -> 8,192)
Stream Length (T) |
Quadratic Attention KV Cache | S1-GRT-RNN (Hybrid) Cache |
S1-GRT-RNN (Pure RNN) State |
Attention vs. RNN Step Scaling |
|---|---|---|---|---|
T = 128 |
1.0× baseline |
0.33× (-66.7%) |
0 Bytes KV Cache (3.0 KB fixed state) |
128× -> 1× (O(1)) |
T = 512 |
4.0× growth |
1.33× (-66.7%) |
0 Bytes KV Cache (3.0 KB fixed state) |
512× -> 1× (O(1)) |
T = 2,048 |
16.0× growth |
5.33× (-66.7%) |
0 Bytes KV Cache (3.0 KB fixed state) |
2,048× -> 1× (O(1)) |
T = 8,192 |
64.0× growth (4,096× FLOPs) |
21.33× (-66.7%) |
0 Bytes KV Cache (3.0 KB fixed state) |
8,192× -> 1× (O(1)) |
4. Consolidated 12-Domain XL-smol System-1 Judge Evaluation (n = 17,130 Held-Out Test Records / 44,287 Questions)
Evaluated across the complete 12-domain XL-smol System-1 verification benchmark (65,939 train / 10,563 validation / 17,130 held-out test records comprising 40,687 core test questions and 44,287 total test questions):
System-1 Judge Domain (12-Family XL-smol Suite) |
Held-Out Test Records (n) |
Core Test Questions | Primary Accuracy | Core Question Accuracy | Calibration Error (ECE) |
|---|---|---|---|---|---|
Multi-Hop Relational Rollouts (XL-smol) |
2,500 |
5,000 |
94.40% |
94.40% |
0.0824 |
Causal Science & Commonsense (XL-smol) |
2,500 |
5,000 |
96.60% |
96.60% |
0.0301 |
Single-Hop Factual Verification (XL-smol) |
2,500 |
5,000 |
92.80% |
92.80% |
0.0382 |
4-Mode Socratic Verification (XL-smol) |
2,500 |
5,000 |
67.80% |
67.80% |
0.0856 |
2D Spatial Grid Rule Induction (ARC-AGI-2 XL-smol) |
1,200 |
3,600 (7,200 all) |
63.17% |
63.17% (86.00% pass@2) |
0.0901 |
Agent MPC Action Routing & Drift (XL-smol) |
803 |
2,409 |
77.17% |
66.67% |
0.1842 |
| Epistemic Calibration & Abstention | 80 |
160 |
78.12% |
78.12% |
0.0764 |
| Program Loop Termination Verification | 150 |
600 |
74.00% |
74.00% |
0.0915 |
| Symbolic State Transition Tracking | 60 |
180 |
73.33% |
73.33% |
0.0890 |
Prerequisite Knowledge Graph Battery (v5) |
2,777 |
7,558 |
72.85% |
72.85% |
0.0881 |
| Resource & Concurrency Scheduling | 60 |
180 |
71.67% |
71.67% |
0.0940 |
| Formal Boolean & Truth-Table Verification | 2,000 |
6,000 |
68.45% |
68.45% |
0.1042 |
Full 12-Domain XL-smol Test Split (Aggregate) |
17,130 |
40,687 (44,287 all) |
77.53% Macro |
78.59% Micro (77.82% all) |
0.0880 Mean |
5. SLM-Bench Multiple-Choice & GSM8K Mathematical Verification (13,644 Examples)
Benchmark (30M Rung) |
Split (n) |
Random | Flat GRT-30M |
Rotational Baseline | S1-GRT-RNN (24h Curriculum) |
S1-GRT-RNN (10C Horizon) |
|---|---|---|---|---|---|---|
BoolQ (google/boolq) |
val (3,270) |
50.00% |
51.41% |
60.55% |
65.93% |
73.18% |
QNLI (nyu-mll/glue) |
val (1,000) |
50.00% |
51.60% |
54.00% |
78.10% |
86.30% |
ARC-Easy (allenai/ai2_arc) |
test (2,376) |
25.00% |
29.59% |
28.96% |
40.82% |
55.35% |
ARC-Challenge (allenai/ai2_arc) |
test (1,172) |
25.00% |
20.82% |
23.55% |
31.14% |
40.53% |
OpenBookQA (allenai/openbookqa) |
test (500) |
25.00% |
28.20% |
30.00% |
43.20% |
54.80% |
PIQA (baber/piqa) |
val (1,838) |
50.00% |
55.28% |
54.08% |
56.69% |
65.56% |
WinoGrande (allenai/winogrande) |
val (1,267) |
50.00% |
50.28% |
50.99% |
52.01% |
63.18% |
HellaSwag (Rowan/hellaswag) |
val (1,000) |
25.00% |
29.20% |
31.60% |
37.60% |
50.70% |
CommonsenseQA (tau/commonsense_qa) |
val (1,221) |
20.00% |
21.70% |
22.77% |
29.48% |
41.20% |
| 9-Task Macro Average | 13,144 total |
35.56% |
37.56% |
39.61% |
48.33% |
58.98% |
| GSM8K 4-Way Verification | test (500) |
25.00% |
34.20% (PPL 27.99) |
39.20% (PPL 10.65) |
99.20% (PPL 7.49) |
99.20% |
Coming Next: Scaled Unitary Rotational RNN Models (GRT-RNN-100M & GRT-RNN-350M)
Building directly on the 10M, 20M, and 30M GRT-RNN and S1-GRT-RNN scaling ladder, our next scaled releases expand O(1) constant-memory unitary gated rotational RNN (Rotational-minGRU / Unitary-GRU) modeling and stateful System-1 MPC verification:
GRT-RNN-100M(~195MDense-Equivalent Capacity — Scaled Unitary RotationalRNNWorld Model & StatefulS1Judge):- Scales unitary gated rotational
RNNrecurrence (grt-rnn) and stateful MPC trajectory verification (s1-grt-rnn) in a deep0-linear-layer architecture—projected on our empirical scaling ladder to reach64.9%SLM-Benchmacro accuracy,89.2%structured knowledge accuracy, and>99%streaming cognitive routing accuracy with0bytes of KV cache in pureRNNmode (-66.7%in Hybrid mode).
- Scales unitary gated rotational
GRT-RNN-350M(~920MDense-Equivalent Capacity — Flagship Unitary RotationalRNNFoundation Model):- Scales parallel geodesic
minGRUscan training and nonlinear state-coupledUnitary-GRUexecution across arbitrary sequence horizons (93.8%structured knowledge accuracy,97.0%cognitive telemetry routing).
- Scales parallel geodesic
MatterWaveAI Unitary Rotational RNN (GRT-RNN / minGRU / GRU) Scaling Ladder (10M / 20M / 30M)
- Unitary Gated Rotational
RNN(GRT-RNN) World Model:MatterWaveAI/grt-rnn(10M:grt-rnn-b0.7.0-10m|20M:grt-rnn-b0.7.0-20m) - Stateful
O(1)Rotational RNN World-Model Judge:MatterWaveAI/s1-grt-rnn(10M:s1-grt-rnn-b0.7.0-10m|20M:s1-grt-rnn-b0.7.0-20m) - 24-Hour Distilled Preview (
GRT-30M):MatterWaveAI/grt-30m - Whitepaper: The Geometric Rotation Transformer (GRT): A Purely Rotational Foundation Architecture (
GRT_Whitepaper.pdf)
License & Citation
Released under the Apache 2.0 license.
@techreport{householter2026grt,
title = {The Geometric Rotation Transformer: A Purely Rotational Foundation Architecture},
author = {Householter, David},
institution = {MatterWave AI},
year = {2026},
url = {https://matterwave.ai}
}