@@ -163,7 +163,65 @@ lr(step) = lr_max × 0.5 × (1 + cos(π × step / total_steps))
163163- ** Assumptions:** Normal distribution, equal variance
164164- ** Thresholds:** very_strict (p<0.001), strict (p<0.01), moderate (p<0.05), lenient (p<0.10)
165165
166- ### 2.4 FPGA Implementation
166+ ### 2.5 FPGA Implementation
167+
168+ ### 2.4 Algorithm: Ternary Transformer Forward Pass
169+
170+ ** Algorithm 1:** HSLM Forward Pass with Sacred Attention Scaling
171+
172+ ```
173+ Require: Input tokens X = [x₁, ..., xₙ] (n tokens)
174+ Require: Weight matrices W_q, W_k, W_v ∈ {-1, 0, +1}^{d×d}
175+ Require: Layer norm parameters γ, β
176+ Require: Cache threshold τ = φ⁻¹ ≈ 0.618
177+
178+ 1: // Token embedding
179+ 2: E ← TernaryEmbedding(X) // E ∈ {-1, 0, +1}^{n×d_model}
180+ 3:
181+ 4: // For each transformer block ℓ = 1 to L (L=9)
182+ 5: for ℓ = 1 to L do
183+ 6: // Layer normalization (φ-scaled)
184+ 7: γ_φ ← φ^(ℓ/10) // Progressive scaling
185+ 8: X_norm ← LayerNorm(E, γ·γ_φ, β)
186+ 9:
187+ 10: // Sacred attention with cache
188+ 11: Q ← X_norm · W_q // Queries: [n × d_k]
189+ 12: K ← X_norm · W_k // Keys: [n × d_k]
190+ 13: V ← X_norm · W_v // Values: [n × d_k]
191+ 14:
192+ 15: // Attention scaling with φ
193+ 16: S ← Q · Kᵀ / √(d_k)^(φ^(-3)) // Scaled scores
194+ 17:
195+ 18: // Sparse attention via cache threshold
196+ 19: M ← (S > τ) // Mask: keep only top correlations
197+ 20: A ← Softmax(M ⊙ S) // ⊙ = element-wise multiply
198+ 21:
199+ 22: // Context aggregation
200+ 23: C ← A · V // [n × d_k]
201+ 24:
202+ 25: // Feed-forward network
203+ 26: F ← ReLU(C · W₁ + b₁) · W₂ + b₂
204+ 27:
205+ 28: // Residual connection + layer norm
206+ 29: E ← E + LayerNorm(C + F, γ, β)
207+ 30: end for
208+ 31:
209+ 32: // Output projection
210+ 33: logits ← E · W_out // [n × vocab_size]
211+ 34: return logits
212+ ```
213+
214+ ** Complexity Analysis:**
215+ - Time: O(n²·d_model·L) for attention (standard transformer)
216+ - Space: O(n·d_model·L) for activations
217+ - Ternary multiplication: O(1) per operation (LUT-based)
218+
219+ ** Key Innovations:**
220+ 1 . ** φ-based layer norm scaling** (line 7): γ_φ = φ^(ℓ/10) for deep network stability
221+ 2 . ** Sparse attention via cache threshold** (line 19): τ = φ⁻¹ ≈ 0.618
222+ 3 . ** Ternary arithmetic** : All multiplications use {-1, 0, +1} encoding
223+
224+ ### 2.6 FPGA Implementation
167225
168226** Target:** QMTech XC7A100T (Artix-7 100T)
169227
@@ -180,7 +238,7 @@ lr(step) = lr_max × 0.5 × (1 + cos(π × step / total_steps))
180238
181239## 3. Theoretical Foundations
182240
183- ### 3.1 Trit Entropy Theorem
241+ ### 3.2 Trit Entropy Theorem
184242
185243** Theorem 1 (Information Maximality):** Balanced ternary encoding {-1, 0, +1} maximizes per-symbol entropy for n-ary codes with n ≤ 4.
186244
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