These conditional mathematical models support V2 calculations. Their operators, state preparation, and physical matching assumptions are stated in each note. All physical coordinates remain in one nested manifold of ordinary macroscopic spatial dimensions.
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Coulomb exponential profiles — Normalized exponential profiles give exact Coulomb integrals, a two-particle variational bound, and finite-range contact estimates.
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Zonal hybrid geometry — Spherical-harmonic addition gives exact directed-state overlaps and conditional trigonal, tetrahedral, and octahedral orthogonality.
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Cyclic resonance spectrum — A nearest-neighbour amplitude ring has exact Fourier resonances, paired levels, and a calculable response to alternating couplings.
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Dipole dispersion response — Second-order dipole coupling yields an attractive inverse-sixth-power energy with a coefficient fixed by the specified excitation response.
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Radial confinement bounds — The radial spatial measure fixes the centrifugal term, while a Hardy bound gives an explicit sufficient condition excluding levels below a saturating threshold.
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Chiral pseudoscalar mass term — A specified leading chiral action makes pseudoscalar mass squared proportional to light-quark masses, with the response scale left as matched input.
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Resonant quartet dynamics — A conservative quartic amplitude interaction has exact action invariants and a detuning-controlled exchange law under specified pump preparation.
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Oscillator rotation branching — Harmonic-polynomial decomposition gives exact angular multiplicities for a flat oscillator shell and its restriction to an observer-selected spatial three-plane.
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Spectral analysis tools — companion analytic and statistical results.
resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.