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Adaptive Domain Models: Bayesian Evolution, Warm Rotation, and Principled Training for Geometric and Neuromorphic AI

Houston Haynes

Abstract

Prevailing AI training infrastructure assumes reverse-mode automatic differentiation over IEEE-754 arithmetic. The memory overhead of training relative to inference, optimizer complexity, and structural degradation of geometric properties through training are consequences of this arithmetic substrate. This paper develops an alternative training architecture grounded in three prior results: the Dimensional Type System and Deterministic Memory Management framework [6], which establishes stack-eligible gradient allocation and exact quire accumulation as design-time verifiable properties; the Program Hypergraph [8], which establishes grade preservation through geometric algebra computations as a type-level invariant; and the b-posit 2026 standard [10], which makes posit arithmetic tractable across hardware targets conventionally considered inference-only. Their composition enables depth-independent training memory bounded to approximately twice the inference footprint, grade-preserving weight updates, and exact gradient accumulation, applicable uniformly to loss-function-optimized and spike-timing-dependent neuromorphic models. We introduce Bayesian distillation, a mechanism by which the latent prior structure of a general-purpose model is extracted through the ADM training regime, resolving the data-scarcity bootstrapping problem for domain-specific training. For deployment, we introduce warm rotation, an operational pattern in which an updated model transitions into an active inference pathway without service interruption, with structural correctness formalized through PHG certificates and signed version records. The result is a class of domain-specific AI systems that are smaller and more precise than general-purpose models, continuously adaptive, verifiably correct with respect to the physical structure of their domains, and initializable from existing models.

Adaptive Domain Models: Bayesian Evolution, Warm Rotation, and Principled Training for Geometric and Neuromorphic AI

Abstract

Prevailing AI training infrastructure assumes reverse-mode automatic differentiation over IEEE-754 arithmetic. The memory overhead of training relative to inference, optimizer complexity, and structural degradation of geometric properties through training are consequences of this arithmetic substrate. This paper develops an alternative training architecture grounded in three prior results: the Dimensional Type System and Deterministic Memory Management framework [6], which establishes stack-eligible gradient allocation and exact quire accumulation as design-time verifiable properties; the Program Hypergraph [8], which establishes grade preservation through geometric algebra computations as a type-level invariant; and the b-posit 2026 standard [10], which makes posit arithmetic tractable across hardware targets conventionally considered inference-only. Their composition enables depth-independent training memory bounded to approximately twice the inference footprint, grade-preserving weight updates, and exact gradient accumulation, applicable uniformly to loss-function-optimized and spike-timing-dependent neuromorphic models. We introduce Bayesian distillation, a mechanism by which the latent prior structure of a general-purpose model is extracted through the ADM training regime, resolving the data-scarcity bootstrapping problem for domain-specific training. For deployment, we introduce warm rotation, an operational pattern in which an updated model transitions into an active inference pathway without service interruption, with structural correctness formalized through PHG certificates and signed version records. The result is a class of domain-specific AI systems that are smaller and more precise than general-purpose models, continuously adaptive, verifiably correct with respect to the physical structure of their domains, and initializable from existing models.
Paper Structure (40 sections, 3 theorems, 6 equations, 3 figures)

This paper contains 40 sections, 3 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Substrate Assumption
  3. The Three Enabling Results
  4. Contributions
  5. Relation to Prior Work
  6. The IEEE-754 Inurement Argument
  7. Precision Loss Overhead as Training Overhead
  8. Grade Corruption as a Training Failure Mode
  9. What Changes with the Principled Substrate
  10. The Bayesian Training Model
  11. Posterior Update as the Principled Alternative
  12. Distribution Shift as the Adaptation Trigger
  13. Consumer-Supplied Evidence and Operational Provenance
  14. Bayesian Distillation from General-Purpose Models
  15. The Warm Rotation Architecture
  16. ...and 25 more sections

Key Result

Proposition 5.1

Let $W$ be a weight parameter with declared grade $k$ in a PHG-typed Clifford neural network. Let $\nabla_W \mathcal{L}$ be the gradient of the loss with respect to $W$, computed via forward-mode autodiff with dual-number augmentation and quire accumulation. Then $\nabla_W \mathcal{L}$ has grade $k$

Figures (3)

  • Figure 1: The adaptive domain model cycle. The DTS and PHG annotations constitute the structural prior. Forward-mode training in spare compute capacity produces a candidate posterior. Operational evidence (telemetry and user feedback) drives the KL divergence check. When the threshold is crossed, PHG elaboration discharges proof obligations before warm rotation. Below-threshold observations accumulate without triggering a rotation.
  • Figure 2: Warm rotation on a representative 50 TOPS inference-class accelerator. Active inference occupies 20 TOPS. Forward-mode training of the candidate model requires approximately $2\times$ the inference memory footprint (primal plus tangent component), consuming roughly 20 TOPS of the spare 30 TOPS. PHG elaboration discharges structural certificates before the atomic rotation. In-flight requests are buffered and completed against the new model.
  • Figure 3: PHG structure of a hybrid geometric-neuromorphic network. Clifford algebra nodes (left, XDNA-2 target) carry grade and dimensional annotations. A rate-encoding conversion hyperedge (centre, CPU or DMA via BAREWire) bridges to spiking layers (right, Loihi-2 target) carrying temporal and coincidence annotations. The per-target reachability bitvector in the PHG routes each subgraph to its appropriate lowering path from a single intermediate representation.

Theorems & Definitions (4)

  • Definition 4.1: Warm Rotation
  • Proposition 5.1: Grade Invariance of Forward-Mode Training
  • Corollary 5.1: Sparsity Stability
  • Proposition 6.1: Unified Local Learning Signature