Harmonic-Kronecker SVD

### ⚠️ Author’s Note & Corrigendum (Updated: November 12, 2026) Following rigorous post-publication hardware verification and large-scale ablation studies ($N = 616$ real hidden-state trajectories on an NVIDIA RTX 3060), the author wishes to provide three technical clarifications and corrections to the original v1 manuscript: 1. **Clarification of Mathematical Terminology ("Quantum-Butterfly" vs. 2-Factor Kronecker SVD):** In the v1 manuscript, the decomposition operator was referred to as a "Quantum-Butterfly Kronecker Operator." Strictly speaking, the implemented algorithm (Section 2 and Algorithm 1) performs a **single-stage two-factor Van Loan–Pitsianis Kronecker SVD coupled with a dense 32 × 32 realified Discrete Fourier Transform (RDFT) basis projection**, rather than a multi-stage sparse butterfly network (such as Cooley–Tukey FFT butterfly stages or Monarch P B₁ P B₂ factorizations). While the closed-form 16 ms decomposition time and the **Memory-vs.-Recursion Spectral Separation Principle** (0.0143 cosine similarity on base weights vs. 0.999990 on TRM recursive layers at 1/31× compression) remain 100% valid and reproducible, the operator is more accurately termed a **Harmonic-Kronecker SVD Operator**. 2. **Exact Attribution of Speculative Decoding Throughput and GGUF File Size (Section 4.4):** In the live `llama-server.exe` benchmarks (Table 4b–4c), the reported speedup from `80.83 tok/s` (autoregressive baseline) to `152.29 tok/s` conflated two distinct factors: - The base speculative decoding gain from the underlying MTP-TRM architecture (`80.83 tok/s` → `138.00 tok/s`). - The pure incremental gain from our 98% Harmonic-Kronecker spectral pruning (`138.00 tok/s` → `152.29 tok/s`, a **+14.29 tok/s / +10.4% net improvement** by suppressing high-frequency adapter jitter). Furthermore, because `llama.cpp` materializes MTP projections as fixed-shape 2D matrices (`2048 x 256`), the physical file size reduction (`94.08 MB` → `56.89 MB`) reflects `Q8_0` to `Q4_0` quantization rather than direct rank-compression storage. ---- A natural question arising from mathematical quantum co-processing (Aria-Quantum-PC) is whether the algebraic machinery of quantum circuits—specifically Kronecker tensor-product factorizations (sumᵣ₌₁^R A_r (x) B_r), Quantum Fourier Transform (QFT) spectral diagonalization, and repeated-squaring parallel state propagation—can replace or accelerate high-dimensional linear projections (such as the 10,240 x 2,560 GeGLU MLP matrices in Gemma-4-E4B-it) at a 1/30x parameter budget without incurring hours of non-linear gradient-based training or degrading foundation model intelligence. In this paper, we provide a rigorous first-principles theoretical and empirical resolution to this question on consumer CUDA hardware (NVIDIA RTX 3060 12GB): 1. Zero-SGD Closed-Form Decomposition in 4.8–16.8 Milliseconds:By exploiting the structural separation between parameter-free element-wise GeGLU non-linearities and linear weight matrices, we apply the Van Loan–Pitsianis Kronecker Product SVD Theorem via a 4D-to-2D block-rearrangement operator R(W). We prove that the globally optimal rank-R Kronecker-butterfly approximation is obtained in closed form via a single low-rank SVD in 4.81–16.76 ms (0.005–0.017 seconds) per 10,240 x 2,560 matrix with zero stochastic gradient descent, and in 131.61 ms when jointly projected into a 64-dimensional realified QFT harmonic basis. 2. The Memory-vs.-Recursion Spectral Separation Principle:We prove mathematically and verify empirically why arbitrary unstructured dense GEMM cannot be accelerated via unitary block-encoding (4x dimension inflation), and why directly compressing a pretrained foundation model's static encyclopedic memory weights W_base (10,240 x 2,560) to 1/30.5x parameters (R = 82, 860,672 vs. 26,214,400 parameters) retains only 5.06%–5.12% of spectral energy, collapsing non-linear GeGLU output cosine similarity to 0.0143. Conversely, applying the exact same 1/31.0x Quantum-Butterfly Kronecker compression (R = 11, 16,896 vs. 524,288 parameters) to the weight-tied recursive reasoning layer (blk.3) of Tiny Recursive Models (MTP-TRM and DSpark-TRM) preserves 0.999990 (99.9990%) non-linear GeGLU output cosine similarity—and maintains 0.999988 similarity even at 1/170.7x compression (3,072 parameters). 3. Three Verified Speedup Regimes of Quantum-Structured GEMM:Across empirical CUDA benchmarks, we demonstrate:(i) L2-Cache-Resident Kronecker Contraction (R = 16, 1/156x parameter compression from 52.43 MB to 0.34 MB FP16) accelerates 10,240 x 2,560 projection from 195.59 us to 121.75 us (1.61x speedup over cuBLAS FP16 Tensor Cores);(ii) QFT Spectral Diagonal Integration accelerates 10,240 x 10,240 integral/convolution operators from 0.816 ms to 0.154 ms (5.31x speedup) while reducing VRAM by 5,120x (209.7 MB -> 0.041 MB); and(iii) Repeated-Squaring Parallel Associative Integration over T = 4,096 steps accelerates state-space path integration from 241.35 ms to 16.048 ms (15.04x speedup, 1.000000 cosine identity). 4. Live llama-server.exe Speculative Throughput via 98% QFT–Kronecker Pruning and Lightweight Quantization:Deleting 98.0% (502 / 512 modes) of the QFT x Kronecker eigenspace across all MTP adapter layers (retaining only the top 2.0% = 10 / 512 modes pure quantum-harmonic core in mtp-gemma-4-E4B-it-qbt-trm-v10.gguf) and evaluating on live llama-server.exe (--spec-type draft-mtp --spec-draft-n-max 3, cache_prompt=False, temp=0.0) across JGLUE JCommonsenseQA, OpenAI GSM8K, OpenAI HumanEval, and Chollet ARC-AGI boosts overall aggregate throughput to 152.29 tok/s (up to 173.76 tok/s peak on GSM8K and 157.75 tok/s on ARC-AGI—a 1.88x–2.15x speedup over the 80.83 tok/s autoregressive baseline) with 79.5% overall draft acceptance (87.6% on ARC-AGI, mean acceptance length 3.384 tokens/step), while physical Q4_0 lightweight variants shrink the cartridge from 94.08 MB to 62.08 MB (147.33 tok/s) and 56.89 MB (143.94 tok/s).