Fix model card formatting (replace raw LaTeX with clean Unicode and code spans)

README.md

@@ -27,11 +27,11 @@ base_model: MatterWaveAI/grt-30m
 | **Total (`GRT-30M-Image`)** | `GRTImageForConditionalGeneration` (`GRTForTextToImage`) | **`29,371,239`** | **0 `nn.Linear`, 0 `nn.Conv2d`** |
 | **4-Codebook Visual RVQ Bank** | `GRTVisualRVQCodecBank` (benchmark evaluation head) | `32,772` | 4 planar-pair polar cosine codebooks (`K = 64` entries × `128` pairs per codebook) over RGB spatial-spectral sub-bands |
 
-## Visual Synthesis Alignment Metrics (`HuygensAdam` / Circular Order Parameter $R_p$ Splay Torque)
+## Visual Synthesis Alignment Metrics (`HuygensAdam` / Circular Order Parameter `R_p` Splay Torque)
 
 - **Held-Out Visual RVQ Perplexity**: `81.17` (unaligned) → **`16.98`** (aligned, `4.78×` reduction)
 - **Visual RVQ Mean Top-1 Accuracy (4 Codebooks)**: `1.48%` → **`25.19%`** (`[20.86%, 25.63%, 26.04%, 28.23%]`, `16.1×` over random chance)
-- **Text-Conditioned Prompt Margin ($\Delta\mathcal{L}$ vs. Shuffled Prompts)**: `-0.0074` → **`+0.6440` nats**
+- **Text-Conditioned Prompt Margin (`ΔL` vs. Shuffled Prompts)**: `-0.0074` → **`+0.6440` nats**
 - **Patch Polar Cosine Similarity**: `0.1004` → **`0.6737`**
 - **2D Spectral Magnitude Cosine Similarity**: `0.1782` → **`0.7857`**
 - **Reconstruction PSNR / Spectral SNR**: `11.38 dB` / `-8.20 dB` → **`19.14 dB` PSNR / `+2.31 dB` SNR** (`+10.51 dB` gain)
@@ -42,7 +42,7 @@ base_model: MatterWaveAI/grt-30m
 
 We benchmarked all 8 classical geometric, symplectic, and gyroscopic optimizers from `transformers.models.grt.optimization_grt` on `MatterWaveAI/grt-30m-image` (`GRTPolarImageDecoderHead` + 4-codebook visual RVQ polar cosine codec bank `GRTVisualRVQCodecBank`, `K = 64` per codebook, `200` steps per optimizer, `batch_size = 36`, `seed = 42`, across a 144-concept procedural text-to-image benchmark of 6 colors × 6 geometric shapes × 4 spatial layouts):
 
-| Rank | Optimizer (`optimizer_type`) | Final Train Loss | Held-Out RVQ Val Loss | Held-Out RVQ Val PPL | RVQ Mean Top-1 Acc (4 Codebooks) | Codebook-0 Primary Acc | Text-Conditioned Margin ($\Delta\mathcal{L}$) | Patch Polar Cosine Sim | Spectral SNR / PSNR (dB) | Time (s) |
+| Rank | Optimizer (`optimizer_type`) | Final Train Loss | Held-Out RVQ Val Loss | Held-Out RVQ Val PPL | RVQ Mean Top-1 Acc (4 Codebooks) | Codebook-0 Primary Acc | Text-Conditioned Margin (`ΔL`) | Patch Polar Cosine Sim | Spectral SNR / PSNR (dB) | Time (s) |
 | :---: | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
 | **1** | **`HuygensAdam` (`huygens_adam`)** | **`2.8188`** | **`2.8317`** | **`16.9750`** | **`25.19%`** | `20.86%` | **`+0.6440`** | **`0.6737`** | **`2.31` / `19.14 dB`** | `55.59s` |
 | **2** | **`PolarNaturalAdam` (`polar_natural_adam`)** | `2.8244` | `2.8368` | `17.0605` | `25.08%` | `20.85%` | `+0.6333` | `0.6717` | `2.29` / `19.13 dB` | `185.71s` |
@@ -54,8 +54,8 @@ We benchmarked all 8 classical geometric, symplectic, and gyroscopic optimizers
 | **8** | **`SymplecticAdam` (`symplectic_adam`)** | `2.8965` | `2.9089` | `18.3370` | `24.01%` | `19.65%` | `+0.6056` | `0.6508` | `2.14` / `18.97 dB` | `65.04s` |
 
 ### Key Geometric Findings on Rotational Text-to-Image (`GRTImage`)
-- **`HuygensAdam` Sweeps #1 Across Visual RVQ Perplexity, Accuracy, Text-Conditioning Margin & Polar Cosine Similarity**: On 2D spatial patch synthesis (`256` patches of `14×14×3` unfolded across 41 Givens butterfly stages), Kuramoto–Huygens circular order-parameter splay torque ($R_p = |\frac{1}{P}\sum_p e^{i\theta_p}|$) prevents angular phase collapse across spatial sub-patches, achieving **#1 held-out RVQ validation loss (`2.8317`, `val_ppl = 16.9750`)**, **#1 4-codebook visual RVQ accuracy (`25.19%`)**, **#1 text-conditioning margin (`+0.6440` nats)**, and **#1 patch polar cosine similarity (`0.6737`, `19.14 dB` PSNR)**.
-- **`PolarNaturalAdam` (#2) & `HessianPairAdam` (#3) Excel on Multi-Stage Butterfly Curvature**: Scaling angular updates by the Riemannian polar metric $1/(r^2 + \epsilon)$ and Cauchy–Riemann toroidal curvature outperforms first-order `PairAdam` (`2.8368` and `2.8382` vs. `2.8417` validation loss).
+- **`HuygensAdam` Sweeps #1 Across Visual RVQ Perplexity, Accuracy, Text-Conditioning Margin & Polar Cosine Similarity**: On 2D spatial patch synthesis (`256` patches of `14×14×3` unfolded across 41 Givens butterfly stages), Kuramoto–Huygens circular order-parameter splay torque (`R_p = |(1/P) ∑_p exp(i θ_p)|`) prevents angular phase collapse across spatial sub-patches, achieving **#1 held-out RVQ validation loss (`2.8317`, `val_ppl = 16.9750`)**, **#1 4-codebook visual RVQ accuracy (`25.19%`)**, **#1 text-conditioning margin (`+0.6440` nats)**, and **#1 patch polar cosine similarity (`0.6737`, `19.14 dB` PSNR)**.
+- **`PolarNaturalAdam` (#2) & `HessianPairAdam` (#3) Excel on Multi-Stage Butterfly Curvature**: Scaling angular updates by the Riemannian polar metric `1 / (r² + ε)` and Cauchy–Riemann toroidal curvature outperforms first-order `PairAdam` (`2.8368` and `2.8382` vs. `2.8417` validation loss).
 - **`CyclideAdam` Achieves #1 Primary Codebook-0 Accuracy (`20.87%`)**: Dupin cyclide conic metric preconditioning with deviatoric twist attains the highest primary RGB chromatic codebook accuracy (`20.87%`).
 
 ---