6 Experiment 5: Pauli Exclusion and the Scope of the Encoding

Python: pauli_as_zero_residual.py

We verify that B = 0 for stationary fermions is exact and encoding-independent, and clarify the scope of the single-fermion codec.

The encoding question

When a single wavefunction is used to encode two stationary fermions at sites i and j:

the result is not localised on a single site. This superposition has non-zero off-diagonal entries even with no motion, giving ∥B∥≠0.

Conclusion: The single-wavefunction codec is a one-fermion codec: each wavefunction encodes the state of one fermion across two time slices. A system of N fermions requires N wavefunctions, one per particle. Encoding two particles into one wavefunction conflates a two-particle configuration with a single delocalised fermion.

Resolution

Under the correct one-fermion-per-wavefunction encoding:

• B = 0 for any stationary single fermion, for any phase or amplitude, across all chain lengths. This is analytically exact.
• ∥B∥ = 1∕ for any single fermion hop, independent of hop distance and system size.
• ∥B∥ = sin(2𝜃)∕ for a fermion in superposition ψ(𝜃) = cos𝜃e_i + isin𝜃e_j, where 𝜃 interpolates between stationary (𝜃 = 0) and mid-hop (𝜃 = π∕4) and back to stationary (𝜃 = π∕2).

The multi-fermion case requires a tensor product of single-fermion wavefunctions. This is consistent with standard quantum mechanics and is not a limitation of the framework.

Results

Configuration	∥B∥	Reason
Single fermion, stationary	0 (exact)	ψ ∝ej, off-diagonal forced to zero
Single fermion, any hop	1∕ (exact)	Universal, −π∕2 phase
ψ(𝜃) = cos𝜃ei + isin𝜃ej	sin(2𝜃)∕	Sharp onset, no threshold