Rigorous Development and Experimental Validation of Prospective Extensions to Riemannian Attention Operators

This repository provides the full mathematical development, numerical validation, and reference implementation associated with the paper
“Rigorous Development and Experimental Validation of Prospective Extensions to Riemannian Attention Operators.”

The work extends the foundational Riemannian attention operator $O_\tau : L^2(M) \to L^2(M)$, defined on a compact Riemannian manifold $(M,g)$, by introducing three rigorously formulated and experimentally validated extensions:

1. Quantum Modulation of Attention
   A hybrid classical–quantum coupling is introduced using Completely Positive and Trace-Preserving (CPTP) maps.
   A quantum state $\rho(t)$ evolving under a Hamiltonian $H$ defines a scalar modulation
   $$
   w_q(t) = \mathrm{Tr}!\left[M, U(t)\rho_0 U^\dagger(t)\right],
   $$
   leading to the modulated operator
   $$
   O_\tau^{\mathrm{mod}}(f) = w_q(t), O_\tau(f).
   $$
   Under the commutation condition $[\rho_0,H]=0$, the von Neumann entropy
   $$
   S(\rho) = -\mathrm{Tr}(\rho \log \rho)
   $$
   is proven and numerically verified to be conserved.

2. Sheaf-Theoretic Aggregation of Local Attention Operators
   Local attention operators $\{O_i\}_{i\in I}$ are defined on an open covering $\{U_i\}_{i\in I}$ of $M$ and organized as a sheaf.
   Compatibility on overlaps,
   \[
   \iota_{ij}^* \circ O_i = O_j \circ \iota_{ij}^*,
   \]
   guarantees, by the gluing axiom, the existence of a unique global operator
   \[
   O : L^2(M) \to L^2(M), \qquad O|_{U_i} = O_i .
   \]
   Numerical experiments on $S^2$ confirm gluing consistency to machine precision.

3. Fock Space Formulation for Variable-Cardinality Data
   Attention is lifted to the bosonic Fock space
   $$\mathcal{F} = \bigoplus_{n=0}^{\infty} H^{\otimes_s n},$$
   via second quantization. The resulting operator $\widehat{O}_F$ acts independently on each $n$-particle sector and satisfies
   $$[\widehat{O}_F,\widehat{N}] = 0,$$
   ensuring exact particle-number conservation. This formulation naturally supports variable-size point clouds and multi-object configurations.

All theoretical results are supported by detailed numerical experiments on model manifolds ($S^1$, $S^2$) and finite Fock sectors, with quantitative error analysis demonstrating agreement with theory up to floating-point precision.