---
pipeline_tag: text-generation
library_name: pytorch
language:
  - en
license: cc-by-4.0
tags:
  - research
  - semsimula
  - conservative-language-model
  - scalar-potential
  - lagrangian-mechanics
  - energy-based-model
  - physics-informed
  - parflm
  - parf
  - multi-channel-xi
  - structured-vtheta
  - analytical-gradient
  - interpretable
  - gaussian-mixture-potential
  - sparse-routing
  - tinystories
  - non-transformer
  - attention-free
  - constant-memory-inference
  - riemannian-geometry
  - riemannian-geodesics
datasets:
  - roneneldan/TinyStories
metrics:
  - perplexity
model-index:
  - name: semsimula-parflm-multixi-structured-vtheta
    results:
      - task:
          type: text-generation
        dataset:
          type: roneneldan/TinyStories
          name: TinyStories
          split: validation
        metrics:
          - type: perplexity
            value: 12.27
            name: Validation Perplexity (SQ3 K=8)
---

# Multi-Xi PARFLM with Structured V_theta (SQ3 Mixture of Quadratic Wells)

A **structured variant** of the [Multi-Xi PARFLM](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi) conservative language model in which the MLP scalar potential \\(V_\theta\\) is replaced by a **mixture of K=8 diagonal quadratic wells** (SQ3), while retaining the full MLP-based pairwise potential \\(V_\phi\\). This replacement yields:

- **Near-zero PPL cost** — **12.27 PPL** vs the MLP baseline's **12.10 PPL** (only 0.17 PPL gap, 0.6% excess cross-entropy). The pairwise \\(V_\phi\\) absorbs nearly all of the nonlinear structure that the quadratic wells cannot express.
- **Full analytical gradients for \\(V_\theta\\)** — no `torch.autograd.grad` needed for the scalar potential force, eliminating the second-order computation graph for the \\(V_\theta\\) component
- **Explicit attractor centres** — the 8 semantic attractors \\(\mu_k(\xi)\\) are readable directly from the model parameters, with no gradient-descent extraction required
- **Compressed \\(V_\theta\\) landscape** — with \\(V_\phi\\) carrying the bulk of the force budget, \\(V_\theta\\) collapses to a near-flat bias field (mean 0.02, range 31.4 vs 644.9 for SPLM)

This model is from the [Semantic Simulation](https://doi.org/10.5281/zenodo.19712427) framework.

## Table of Contents

- [When to Use This Model](#when-to-use-this-model)
- [Architecture](#architecture)
- [Structured V_theta with Pairwise V_phi: Why the Gap Nearly Vanishes](#structured-v_theta-with-pairwise-v_phi-why-the-gap-nearly-vanishes)
- [How to Get Started](#how-to-get-started)
- [Training Details](#training-details)
- [Evaluation Results](#evaluation-results)
- [SPLM Family Overview](#splm-family-overview)
- [Bias, Risks, and Limitations](#bias-risks-and-limitations)
- [Citation](#citation)

## When to Use This Model

Choose this structured variant over the [MLP-based Multi-Xi PARFLM](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi) when:

| Priority | Structured V_theta (this model) | MLP V_theta ([baseline](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi)) |
|---|---|---|
| **Interpretability** | 8 explicit attractor centres, zero-cost basin readout | Black-box; requires 1,500-step GD extraction per prompt |
| **Inference speed** | ~2x faster V_theta force computation (analytical gradient) | Standard (autograd for both V_theta and V_phi) |
| **Raw PPL** | 12.27 | **12.10** |
| **PPL gap** | **0.17 PPL (0.6%)** — essentially free | — |
| **Memory** | No second-order graph for V_theta (V_phi graph retained) | Full graph for both |

**Bottom line:** for PARFLM, structured \\(V_\theta\\) is the **recommended default** — the 0.17 PPL gap is negligible, while the interpretability and speed gains are substantial.

## Architecture

```
Input tokens x_1, ..., x_T
       |
   Embedding E[x] + positional encoding
       |
   For each of L=8 integration steps:
       |
       +-- K-EMA channels: xi^(k)_t = causal_ema(h, alpha_k)   [K_xi=4 channels]
       |
       +-- Structured V_theta (SQ3):
       |     xi_flat = flatten(xi_1..xi_K)                      [K_xi * d = 1024]
       |     V = -tau * logsumexp_k(-E_k/tau + log pi_k)        [K_mix=8 wells]
       |     f_theta = -analytical_grad_h V                     [closed-form]
       |
       +-- Pairwise V_phi (competitive structural MLP):
       |     scores = score_net(h_t, h_s)                       [for all s <= t]
       |     top-k selection via Gumbel-softmax                 [k=8 neighbours]
       |     f_phi = -grad_h V_phi(h_t, h_s)                   [autograd, sparse]
       |
       +-- Total force: f = f_theta + f_phi
       |
       +-- Damped Stormer-Verlet step: h += (h-h_prev)/(1+dt*gamma) + dt^2*f/(m*(1+dt*gamma))
       |
       +-- LayerNorm(h)
       |
   Logits = h @ E^T                                            [tied embeddings]
```

| Parameter | Value |
|---|---|
| Hidden dim (d) | 256 |
| Layers (L) | 8 |
| V_theta kind | SQ3 (mixture of K quadratic wells) |
| Mixture components (K_mix) | 8 |
| Temperature (tau) | 1.0 |
| Xi channels (K_xi) | 4 |
| V_phi kind | structural_competitive |
| V_phi hidden | 128 |
| Top-k (sparse routing) | 8 |
| Gumbel tau | 1.0 (init), 0.3 (min) |
| Gathered V_phi | Yes |
| Per-layer V_phi scale | Yes |
| LN before distance | Yes |
| Layer checkpoint | Yes |
| Mass model | logfreq (frozen surprisal lookup) |
| Damping gamma | 0.30 nominal, ~0.033 effective (LayerNorm prevents compounding; see note below) |
| lambda_V (V_theta regularisation) | 0.01 |
| Total parameters | **17,335,568** |

**Effective damping.** The nominal gamma = 0.30 overstates the true dissipation. The LayerNorm applied after each integration step rescales the hidden state, absorbing most of the velocity decay. The dynamics are therefore **heavily underdamped** even at this nominal value. Gamma-sweep experiments on the OpenWebText-scale Fock-PARFLM variant confirm that the effective damping is much smaller than the nominal coefficient.

## Structured V_theta with Pairwise V_phi: Why the Gap Nearly Vanishes

In the standalone [Multi-Xi SPLM](https://huggingface.co/dimitarpg13/semsimula-splm-multixi-structured-vtheta), replacing the MLP \\(V_\theta\\) with SQ3 incurs a **1.82 PPL gap** (13.33 vs 11.51). In PARFLM, the same replacement incurs only a **0.17 PPL gap** (12.27 vs 12.10) --- an order of magnitude smaller.

The reason is **landscape compression**: with \\(V_\phi\\) carrying the bulk of the force budget, \\(V_\theta\\) collapses to a near-flat bias field:

| Landscape metric | SPLM structured V_theta | PARFLM structured V_theta |
|---|---|---|
| Mean V_theta | 99.8 | 0.02 |
| Std V_theta | 26.3 | 0.51 |
| Range | 644.9 | 31.4 |
| PPL gap vs MLP | 1.82 (5.5%) | **0.17 (0.6%)** |

The MLP's expressivity advantage is irrelevant when \\(V_\theta\\)'s dynamic range is negligible. The pairwise \\(V_\phi\\) absorbs the nonlinear structure that quadratic wells cannot express, making structured \\(V_\theta\\) essentially free in PARFLM.

The **force** from the structured \\(V_\theta\\) is computed in closed form:

$$f_\theta = -\nabla_h V_\theta = -\sum_{k=1}^{K} q_k(\xi, h) \cdot a_k(\xi) \odot (h - \mu_k(\xi))$$

where \\(q_k\\) are the softmax responsibilities over the 8 quadratic wells. The \\(V_\phi\\) force still uses autograd (sparse, over top-k=8 neighbours only).

For full derivations, all four structured variants (SQ1--SQ4), landscape compression analysis, attractor basin decoding, and hyperparameter selection strategies, see the companion note: [`Structured_VTheta_Design_and_Theory.md`](https://github.com/dimitarpg13/semsimula-paper/blob/main/companion_notes/Structured_VTheta_Design_and_Theory.md).

## How to Get Started

```python
import torch, sys
sys.path.insert(0, "multixi")
sys.path.insert(0, "parf")
sys.path.insert(0, "energetic_minima")
sys.path.insert(0, "sarf_mass_variant")

from parf.model_parf_multixi import MultiXiPARFLM, MultiXiPARFConfig
from parf.model_structured_vtheta import MixtureQuadraticVTheta
from parf.model_structured_vtheta_multixi import StructuredVThetaMultiXiAdapter

# -- Build base model --
config = MultiXiPARFConfig(
    vocab_size=50257, d=256, n_layers=8,
    v_hidden=1024, v_depth=3,
    max_len=1024, block_size=512,
    gamma=0.30, xi_channels=4,
    xi_alpha_inits=[0.25, 0.5, 0.75, 0.95],
    xi_learnable=True, mass_mode="logfreq",
    logfreq_path="logfreq_surprisal_tinystories.npy",
    v_phi_kind="structural_competitive",
    v_phi_phi_hidden=128, v_phi_theta_hidden=128,
    top_k=8, score_head_hidden=32,
    gumbel_tau_init=1.0, gumbel_tau_min=0.3,
    gumbel_noise=True,
    use_gathered_v_phi=True,
    use_layer_checkpoint=True,
    ln_before_distance=True,
    per_layer_v_phi_scale=True,
)
model = MultiXiPARFLM(config)

# -- Swap in structured V_theta --
K_xi, d = 4, 256
inner = MixtureQuadraticVTheta(d=K_xi * d, K=8, tau=1.0)
model.V_theta = StructuredVThetaMultiXiAdapter(inner, K=K_xi, d=d)

# -- Load checkpoint --
from huggingface_hub import hf_hub_download
ckpt_path = hf_hub_download(
    repo_id="dimitarpg13/semsimula-parflm-multixi-structured-vtheta",
    filename="checkpoint/ckpt_best.pt",
)
state = torch.load(ckpt_path, map_location="cpu")
model.load_state_dict(state["model_state_dict"])
model.eval()

print(f"Parameters: {sum(p.numel() for p in model.parameters()):,}")

# -- Read attractor centres directly --
x = torch.randint(0, 50257, (1, 64))
with torch.no_grad():
    h = model._embed(x)
    xis = model._compute_xis(h)                        # (1, 64, 4, 256)
    centres = model.V_theta.attractor_centres(xis)      # (1, 64, 8, 1024)
    print(f"Attractor centres shape: {centres.shape}")   # 8 basins per token
```

### Available Artifacts

| File | Description |
|---|---|
| `checkpoint/ckpt_best.pt` | Best checkpoint (A2 arm, 12.27 PPL at step 14,400) |
| `training_log.jsonl` | Per-step training metrics (40 eval points) |
| `training_curve_A2.png` | Training/validation loss curves |
| `v_theta_hist_A2.png` | V_theta output distribution histogram |
| `landscape_stats_A2.json` | V_theta landscape statistics (mean, std, range) |
| `model_structured_vtheta.py` | Structured V_theta classes (SQ1--SQ4) |
| `model_structured_vtheta_multixi.py` | Multi-Xi adapter |
| `config.json` | Model configuration |

## Training Details

### Training Data

[TinyStories](https://huggingface.co/datasets/roneneldan/TinyStories) --- a synthetic corpus of short children's stories generated by GPT-3.5/4, tokenized with GPT-2 BPE (vocab size 50,257). Training cap: 5M tokens; validation: ~140k tokens.

### Training Procedure

The base model architecture is identical to the [Multi-Xi PARFLM](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi). The only modification is the \\(V_\theta\\) replacement: the 3-layer MLP is swapped for the SQ3 mixture at model construction time, before training begins from scratch. The pairwise \\(V_\phi\\) (competitive structural MLP, hidden=128, top-k=8) is unchanged.

| Hyperparameter | Value |
|---|---|
| Optimizer | AdamW |
| Learning rate | 5e-4 (cosine decay) |
| Warmup steps | 400 |
| Weight decay | 0.01 |
| Gradient clipping | 1.0 |
| Batch size | 16 |
| Block size | 512 |
| Training steps | 16,000 |
| lambda_V (V_theta regularisation) | 0.01 |
| Hardware | NVIDIA A100 40GB (Google Colab) |

### Training Script

[`notebooks/conservative_arch/scaleup/colab_parf_multixi_structured_vtheta.ipynb`](https://github.com/dimitarpg13/semsimula-paper/blob/main/notebooks/conservative_arch/scaleup/colab_parf_multixi_structured_vtheta.ipynb) --- Colab notebook with structured V_theta arms, GDrive output, checkpointing, and live progress display.

## Evaluation Results

### TinyStories Validation Perplexity

| Model | PPL | Params | Analytical V_theta grad | V_theta--MLP gap |
|---|---|---|---|---|
| Matched Attention (baseline) | **7.81** | 19.5M | --- | --- |
| [Fock-PARFLM v2.1](https://huggingface.co/dimitarpg13/semsimula-fock-parflm) | **9.70** | 17.4M | No | --- |
| [Multi-Xi SPLM (MLP)](https://huggingface.co/dimitarpg13/semsimula-splm-multixi) | **11.51** | 16.5M | No | --- |
| [Multi-Xi PARFLM (MLP)](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi) | **12.06** | 17.6M | No | --- |
| **Multi-Xi PARFLM (SQ3, this model)** | **12.27** | 17.3M | **Yes** | **0.17 PPL** |
| [Multi-Xi SPLM (SQ3)](https://huggingface.co/dimitarpg13/semsimula-splm-multixi-structured-vtheta) | **13.33** | 17.3M | Yes | 1.82 PPL |

### V_theta Landscape Statistics

| Metric | This model (PARFLM) | SPLM structured V_theta |
|---|---|---|
| Mean V_theta | 0.02 | 99.8 |
| Std V_theta | 0.51 | 26.3 |
| Range | 31.4 | 644.9 |

### Learned Xi-Channel Decay Rates

The final learned alpha values \\([\alpha_1, \ldots, \alpha_4] = [0.11, 0.55, 0.81, 0.97]\\) are nearly identical to both the SPLM structured V_theta and the MLP PARFLM baseline, confirming that the causal EMA context structure is robust to the \\(V_\theta\\) parameterisation change.

## SPLM Family Overview

This model is part of the **Semantic Simulation SPLM family**:

| Model | Design | Corpus | PPL | HuggingFace |
|---|---|---|---|---|
| Multi-Xi SPLM (MLP) | Pure scalar potential | TinyStories | 11.51 | [semsimula-splm-multixi](https://huggingface.co/dimitarpg13/semsimula-splm-multixi) |
| Multi-Xi SPLM (SQ3) | Structured scalar potential | TinyStories | 13.33 | [semsimula-splm-multixi-structured-vtheta](https://huggingface.co/dimitarpg13/semsimula-splm-multixi-structured-vtheta) |
| Multi-Xi PARFLM (MLP) | Scalar + pairwise forces | TinyStories | 12.06 | [semsimula-parflm-multixi](https://huggingface.co/dimitarpg13/semsimula-parflm-multixi) |
| **Multi-Xi PARFLM (SQ3)** | **Structured scalar + pairwise** | **TinyStories** | **12.27** | **this model** |
| Fock-PARFLM v2.1 (MLP) | PARFLM + Fock registers | TinyStories | 9.70 | [semsimula-fock-parflm](https://huggingface.co/dimitarpg13/semsimula-fock-parflm) |
| Fock-PARFLM v2.1 (SQ3) | Structured + pairwise + Fock | TinyStories | 10.90 | [semsimula-fock-parflm-structured-vtheta](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-structured-vtheta) |
| Fock-PARFLM v2.1 (depth-cond. isotropic Gaussian) | Bounded multi-context + pairwise + Fock | TinyStories | 16.33 | [semsimula-fock-parflm-depthcond-vtheta](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-depthcond-vtheta) |
| Fock-PARFLM v2.1 (depth-cond. anisotropic Gaussian + fock-reg) | Bounded, ellipsoidal multi-context + pairwise + Fock | TinyStories | 9.04 | [semsimula-fock-parflm-anisogaussian-vtheta](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta) |
| Fock-PARFLM v2.1 (aniso-Gaussian + fock-reg, first-order/Fock-G1) | Same as above, gradient-flow integrator | TinyStories | 8.95 | [semsimula-fock-parflm-anisogaussian-vtheta-fock-g1](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta-fock-g1) |
| Fock Attention (MLP V_theta) | Fock + attention | TinyStories | 9.42 | [semsimula-fock-attention](https://huggingface.co/dimitarpg13/semsimula-fock-attention) |
| Hybrid SPLM+Attn | Attention + SPLM refinement | TinyStories | 8.50 | [semsimula-hybrid-splm](https://huggingface.co/dimitarpg13/semsimula-hybrid-splm) |
| Fock-PARFLM v2.1 (aniso-Gaussian + fock-reg, gamma sweep, d=384) | Bounded, ellipsoidal multi-context + pairwise + Fock — 8-way gamma sweep + geodesic analysis | OpenWebText | 278.27 (best of 8, 3K-step sweep) | [semsimula-fock-parflm-anisogaussian-vtheta-owt-d384-gammasweep](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta-owt-d384-gammasweep) |
| Fock-PARFLM v2.1 (aniso-Gaussian + fock-reg, gamma sweep, d=768) | Bounded, ellipsoidal multi-context + pairwise + Fock — 8-way gamma sweep + geodesic analysis | OpenWebText | 326.97 (best of 8, 3K-step sweep) | [semsimula-fock-parflm-anisogaussian-vtheta-owt-d768-gammasweep](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta-owt-d768-gammasweep) |
| Fock-PARFLM v2.1 (aniso-Gaussian + fock-reg, gamma sweep, d=1024) | Bounded, ellipsoidal multi-context + pairwise + Fock — 8-way gamma sweep + geodesic analysis | OpenWebText | 244.23 (best of 8, 3K-step sweep) | [semsimula-fock-parflm-anisogaussian-vtheta-owt-d1024-gammasweep](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta-owt-d1024-gammasweep) |
| Fock-PARFLM v2.1 (aniso-Gaussian + fock-reg, Verlet instability, d=384) | Same architecture, two full-run attempts — SCAF stiffness audit identifies structural Verlet instability | OpenWebText | 184.11 / 211.63 (both runs stalled, not final) | [semsimula-fock-parflm-anisogaussian-vtheta-owt-d384-verlet-instability](https://huggingface.co/dimitarpg13/semsimula-fock-parflm-anisogaussian-vtheta-owt-d384-verlet-instability) |

**Collection:** [Semantic Simulation SPLM Model Family](https://huggingface.co/collections/dimitarpg13/semantic-simulation-splm-model-family)

## Bias, Risks, and Limitations

- **Research checkpoint only.** This model is a proof-of-concept for structured scalar potentials in pairwise-force architectures, not a production system.
- **TinyStories only.** Trained exclusively on synthetic children's stories (~5M tokens). Not suitable for general-purpose language generation.
- **English only.** No multilingual capability.
- **Small scale.** 17.3M parameters, 256-dim hidden states.
- **No safety training.** No RLHF, DPO, or safety filtering has been applied.
- **V_phi still uses autograd.** Only the \\(V_\theta\\) gradient is analytical; the \\(V_\phi\\) pairwise force still requires `torch.autograd.grad`. The overall training speedup is therefore partial (\\(V_\phi\\) dominates the cost).

## Citation

```bibtex
@misc{Gueorguiev2026SemSim,
  author    = {Gueorguiev, Dimitar P.},
  title     = {Semantic Simulation: A Prescriptive Lagrangian Framework
               for Efficient Semantic Inference --- A Conservative-by-
               Construction Language Model and the Shared-Potential
               Separator, with a Correspondence to Joint Embedding
               Predictive Architectures},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.19712427},
  url       = {https://doi.org/10.5281/zenodo.19712427},
  note      = {Version v15 (Jun 7, 2026).
               Companion code repository (DOI 10.5281/zenodo.20579561):
               \url{https://github.com/dimitarpg13/semsimula-paper}}
}
```

## Environmental Impact

- **Hardware:** NVIDIA A100 40GB (Google Colab)
- **Training time:** ~2.5 hours (16,000 steps, A2 arm)
- **Carbon footprint:** Estimated less than 1.5 kg CO2

> **Correction (2026-09-27) — integrator name.** This model's layer step was described here as a *damped Euler* step. It is not. The update is
> `h_new = h + (h - h_prev)/(1 + dt*gamma) + dt^2*f/(m*(1 + dt*gamma))`, which carries no velocity state at all: the velocity is the position difference `h - h_prev`. Undamped this is `h_{n+1} = 2*h_n - h_{n-1} + dt^2*f/m`, i.e. **Störmer–Verlet in position form**. It is also not *velocity*-Verlet, which carries an explicit velocity through half-kick/drift/half-kick.
> The distinction matters because the \\(\omega \cdot dt < 2\\) stability wall is the Störmer/leapfrog bound and applies to this lineage; the semi-implicit Euler models in the same collection were never subject to it. Only the name was wrong — no measurement on this card changes.