8 Experiment 7 (Exploratory): Mass, Momentum, and the Dispersion Relation

Python: dispersion_poc.py

Experiments 1–6 establish the B-matrix formalism entirely within a massless, scale-free regime: ∥B∥ = 1∕ for any single hop, independent of chain length L or hop distance d (§4). That scale invariance is itself informative: a rest mass sets a scale, so a theorem that is exact and scale-free by construction has no room for mass to enter as an internal parameter. Momentum, via the hop phase k = −π∕(2d), is already present in the formalism; energy and mass are not, because nothing in Experiments 1–6 plays the role of a generator of time evolution or of a second internal degree of freedom at a single site. This experiment asks whether the existing B-matrix machinery survives, unmodified, once such a degree of freedom is added, and whether the relativistic dispersion relation

can be recovered as a consequence rather than assumed.

Construction

We introduce a second internal band at a single site (rather than a second spatial site) and adopt the standard two-band lattice (Dirac-on-a-lattice / SSH-type) Hamiltonian

where m is an on-site band-splitting term and tsin(k) is a momentum-dependent inter-band coupling. This Hamiltonian is diagonalised numerically (not assumed analytically) at each k on a dense grid, for several values of m∕t. The lower-band eigenvector v(k) is mapped onto the existing Experiment 6 convention ψ(𝜃) = cos𝜃e_A + isin𝜃e_B by setting sin²𝜃 equal to the numerically computed Born-rule population |v_B(k)|² on the second band. The unmodified Experiment 6 formula B = ρ− diag(ρ) is then applied to this ψ(𝜃).

Results

Four claims were checked numerically for m ∈{0,0.1,0.5,1,2} (with t = 1) over a dense grid k ∈ (−π∕2,π∕2):

1. Eigenvalues. The numerically diagonalised eigenvalues agree with E(k) = to machine precision (< 10⁻¹⁵) at every m and every k tested.
2. Formalism survives unmodified. The boson norm computed directly from the 2 × 2 matrix B = ρ − diag(ρ) agrees with the closed-form Experiment 6 result sin(2𝜃)∕ to machine precision (< 3 × 10⁻¹⁶). No change to the boson-residual formalism was required to accommodate the reinterpretation of 𝜃 as a momentum/mass mixing angle rather than a spatial-hop angle.

3. Mass–momentum identity. Working the eigenvector algebra of H(k) through by hand gives

   

   This closed form agrees with the numerically computed boson norm to machine precision (< 3 × 10⁻¹³) at every m and k tested. At m = 0 it reduces exactly to the universal ∥B∥ = 1∕ of Experiments 3–6, recovering the massless case as a special case rather than contradicting it. As m grows relative to tsink, ∥B(k)∥ shrinks monotonically toward zero: a more massive state produces a smaller compression residual at fixed momentum.

4. Continuum limit. Identifying tsink → pc and m → m₀c² in the small-k regime, E(k)² agrees with (pc)² + (m₀c²)² to a relative error that shrinks as k → 0 (e.g. ∼ 7 × 10⁻⁴ over |k|< 0.05 at m = 0, tightening to ∼ 5 × 10⁻⁷ at m = 2), consistent with the expected O(k²) lattice correction to the continuum dispersion relation.

Status of this result

This experiment shows that the B-matrix formalism of Papers VI–VII is compatible with a mass term and reproduces the relativistic dispersion relation as a continuum limit, once a second internal band and a Hamiltonian are introduced. It does not show that this construction is forced by the framework’s own first principles. Specifically:

• The two-band Hamiltonian H(k) = mσ_z+tsin(k)σ_x is imported from standard lattice/Dirac constructions (SSH-type models); it has not been derived from the spectral complexity functional C_s or the cost functional C_D + C_ψ + C_G used elsewhere in the series.
• In particular, it is not yet shown why a minimum-description-length codec should prefer a two-band internal structure, nor why the band-splitting term should take the form mσ_z specifically. The natural next step is to test whether the same overlap/exclusion cost functional that produces Pauli exclusion in Experiment 4 (§5), applied to internal (band) rather than spatial degrees of freedom, forces this structure – or a different one.
• The identification tsink →pc, m →m₀c² is a relabelling of the lattice dispersion relation, not yet a statement about where the constant c originates within the finite-bit-budget framework of Papers I–V.

Consistent with the three-tier accounting adopted in Paper XI, this result is reported as a strongly suggested correspondence, not an analytically exact theorem in the sense of Experiments 1–6: it is numerically exact given its assumptions, but its assumptions are not yet derived.