Scope
These notes supply explicit mathematical tools for V2's interaction and field calculations. Each includes the definitions, derivation, examples, and assumptions required to apply its result.
All sectors form one nested chain of ordinary macroscopic spatial submanifolds in the one infinite physical manifold. Coordinate groupings are relative to the chosen submanifold; every dimension retains the same physical character. Field representations and integrals over two positions are mathematical descriptions on that space.
Tools
| Note | Result and application |
|---|---|
| Green function marginal | Full flat-fibre integration of a normalized Laplace kernel gives the lower-dimensional Green function, supplying an exact static projection benchmark. |
| Projected bilocal variation | A projected density action gives reciprocal mean fields with their channel projectors and kernel convolutions intact. |
| Colour singlet criterion | The positive total Casimir identifies the singlet subspace, with explicit meson and baryon examples. |
| Angular kernel parity | The dot-square integral operator has a computable angular spectrum; a separate calculation gives its Hartree action on an odd state. |
| Dirac heat curvature | The local untwisted Dirac coefficient is , verified against the round-sphere spectrum with metric scale specified. |
| Anomaly-free charge family | Exact charge equations and a neutrino-inclusive solution family provide consistency checks for a specified chiral field content. |
| Gaussian contact profiles | Normalized Gaussian spatial densities give an exact contact overlap, force, and covariance rule under marginalization onto shared coordinates. |
| Confining oscillator domain | The harmonic spatial operator has a closed energy form, compact resolvent, and an explicit eigenbasis whose Gaussian tails justify specified boundary limits. |
| Fibre-integrated stress | Integrating a conserved stress tensor over ordinary spatial fibres yields a lower-depth balance law with explicit metric, boundary-flux, and sampling terms. |
Verification
interaction_operator_checks.py checks these results using independent quadrature, symbolic algebra, finite differences, and explicit matrix states. Each note explains which operator, measure, and normalization its calculation uses.
spatial_profile_checks.py checks Gaussian contact, oscillator domains and tails, and integrated stress using quadrature, differential identities, and conservation benchmarks.
The spectral analysis tools extend these calculations with recurrence, perturbation, profile metrics, and statistical calibration.