SlimeCodec: Commutator Norm-Based Continuous Non-Commutativity Quantification for Video Encoding Order Optimization

This paper introduces the matrix commutator norm
d(A,B) = ||AB - BA||_F as a continuous, algebraically
grounded criterion for video encoding order optimization.

Four contributions are made:

(C1) Theorem 5 proves the inter-frame commutator identity
     [F_{t-1}, F_t] = [F_{t-1}, Delta_t] and derives the
     bound d(F_{t-1}, F_t) <= 2 ||F_{t-1}||_F ||Delta_t||_F.

(C2) Corollary 4 (Bottcher-Wenzel) proves that permuting
     near-commutative operators introduces output error
     at most epsilon.

(C3) Proposition 7 establishes statistical consistency of
     the adaptive threshold epsilon* via the
     Glivenko-Cantelli theorem.

(C4) Hypothesis 1, validated across five datasets and four
     operator representations, demonstrates that natural
     video signals reside in a near-commutative operator
     subspace (84-87 percent convergence).

Experiments across H.264, H.265, VP9, AV1, XDCAM HD422,
and XAVC-L demonstrate:

- BD-rate -15 to -27 percent, BD-PSNR +1.4 to +2.8 dB
- XDCAM: I-frame count 27 to 2 (13.5x reduction)
- Optimization score 100/100
- Real-time feasibility at under 1 percent encoding overhead

The PSNR paradox (simultaneous quality improvement and
compression gain) is formally explained within rate-distortion
theory as I-frame quantization-error accumulation elimination.

Patent: JP 2026-046609 (filed 2026-03-20)
Inventor and patent owner: Hiroshi Sasaki
Implementation rights: Javatel Corporation