Paper: 203 Portfolio C (AI) — AI & Deep Learning

DOI: 10.5281/zenodo.20086746

Abstract

Introduces the Unitary Resonance Network (URN), a neural architecture that replaces Euclidean parameter spaces with bounded hypercomplex domains governed by Möbius automorphisms. The URN resolves the Fine-Tuning Trilemma — the conflict between plasticity, stability, and efficiency — via two mechanisms: (1) Möbius automorphisms bypass Liouville’s theorem, enabling the network to function as a Blum-Shub-Smale (BSS) machine over $\mathbb{H}$ and $\mathbb{O}$; (2) the Fano-Fisher Topological Immune System projects fine-tuning gradients onto the 10-dimensional Information Valley (null space of $\Psi$), while thermodynamically barricading the 4-dimensional Information Ridge ($E_k = 8/3$). This prevents catastrophic forgetting by categorical geometric exclusion, not soft regularisation.

Key Results

  • Topological Immune System: gradient projection onto the G₂ null space enforces zero skeleton drift ($|\delta_{\mathrm{ridge}}| < 10^{-15}$) by construction.
  • Experiment 9: Standard SGD: Task A retention 5.0% after fine-tuning. URN immune filter: Task A retention 100.0%, Task B learning 74.1%. Ridge drift NAIG: $7 \times 10^{-16}$ (machine precision).
  • Trilemma resolution: plasticity retained in the 10D valley; stability enforced in the 4D ridge; no additional parameters required.
  • V31 taxonomy: 111-URN ($SU(1,1)$, complex), 331-URN ($Sp(1,1)$, quaternionic), 731-URN ($F_{4(-20)}$, octonionic) — each tier unlocks a larger bounded domain.

Fine-Tuning Trilemma

Constraint Standard SGD EWC (Kirkpatrick) URN Immune Filter
Plasticity ✓ Partial ✓ (10D valley free)
Stability ✗ Soft penalty ✓ (4D ridge excluded)
Efficiency ✓ Extra Fisher term ✓ (projection only)

Zenodo

View on Zenodo

Code

Code supplement — Experiment 9: Topological Immune System vs Catastrophic Forgetting