Orbits and dynamics

Topic navigation; Research index lists all notes.

• Nuclear Depth, the Four-Direction Orbit and Three-Dimensional Matter — A prescribed four-direction law supports bound trajectories at every fixed nuclear angle, with an aligned free-drift qualification. The full energy depends on orientation; bulk-dimensional selection remain underived.

• Uniform Extension and Geometric Precession of the Electron Orbit — Shared-manifold compatibility, uniform extension and calculable spatial orientation transport.

• Cylindrical, Breathing and Petal Views of One Actual Orbit — Ordinary orthogonal projections connect these appearances of one autonomous trajectory; the cross-depth comparison does not select depth six.

• Orbit-generator spectrum versus the physical mixed block — Research record, section 66: Orbit-generator spectrum versus the physical mixed block.

• Moment maps do not automatically select the frame-lock orbit — Research record, section 67: Moment maps do not automatically select the frame-lock orbit.

• Orbit-rate audit: three electron clocks are distinct until the stationary master generator identifies them — Research record, section 68: Orbit-rate audit: three electron clocks are distinct until the stationary master generator identifies them.

• Axial mixed tensors do not determine the orbit-generator spectrum — Research record, section 71: Axial mixed tensors do not determine the orbit-generator spectrum.

• Beam-axis spinor decomposition: the orbit generator requires a bilinear lock term — Research record, section 72: Beam-axis spinor decomposition: the orbit generator requires a bilinear lock term.

• Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear — Research record, section 73: Three-axis contraction test: summing three depth-six channels does not yield the required axial bilinear.

• Orbit generator as a polynomial of the total locked axial spin — Research record, section 74: Orbit generator as a polynomial of the total locked axial spin.

• Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps — Research record, section 75: Locked-spin gap theorem: the orbit polynomial is uniquely fixed by the existing unit and two-unit projective gaps.

• Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch — Research record, section 76: Gap-origin audit: primitive winding and orientation reversal do not derive the two-unit branch.

• Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input — Research record, section 77: Extremal-doublet audit: the two-unit gap follows from a primitive signed amplitude charge, but that charge is still an input.

• Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice — Research record, section 78: Primitive cocharacter norm test: the desired amplitude charge is the unique minimum-norm choice.

• Rejected identification audit: an embedded-$U(1)$ curvature would select $q=1$, but it is not the Hopf connection — Research record, section 79: Rejected identification audit: an embedded-$U(1)$ curvature would select $q=1$, but it is not the Hopf connection.

• Beam-axis symmetry leaves the integral charge free — Research record, section 81: Beam-axis symmetry leaves the integral charge free.

• Physical $(3,1)\oplus(1,3)$ audit: beam symmetry leaves $q$ free; frame exchange would select $q=1$ — Research record, section 82: Physical $(3,1)\oplus(1,3)$ audit: beam symmetry leaves $q$ free; frame exchange would select $q=1$.

• Reciprocity audit: Hermiticity does not force observed-unobserved exchange symmetry — Research record, section 83: Reciprocity audit: Hermiticity does not force observed-unobserved exchange symmetry.

• Locked-sector preservation theorem: invariance of the $\mathbf1\oplus\mathbf3$ split uniquely selects $q=1$ — Research record, section 84: Locked-sector preservation theorem: invariance of the $\mathbf1\oplus\mathbf3$ split uniquely selects $q=1$.

• Operator-class refinement: the unwanted $q\neq1$ piece is a singlet--triplet vector intertwiner — Research record, section 85: Operator-class refinement: the unwanted $q\neq1$ piece is a singlet--triplet vector intertwiner.

• Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator — Research record, section 86: Relative-frame identification: the $q\neq1$ mixer is exactly a broken frame-rotation generator.

• Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame — Research record, section 87: Fixed-lock compatibility theorem: $q=1$ is the unique zero-drift branch of the selected frame.

• Collective precession branches — The reciprocal tensor-response model admits uniformly precessing electron–response solutions whose rate obeys a cubic, with distinct branches and stability that must be checked rather than assumed.

• Completed Coulomb petals — The quadratically completed Coulomb–angular-source Hamiltonian has exact autonomous signed-rose relative equilibria; the one-electron L=2 and L=4 petal branches have no resolved exponential mode, while every tested antipodal pair branch is unstable.

• Coulomb angular response — A six-coordinate Kepler model with reciprocal source angular momentum has exact signed-rose trajectories, including repelling two-electron solutions, but the pair is unstable and magnetic quadratic completion changes the motion.

• Dimensional precession test — Six dimensions are minimal for the stated static plane-induced precession mechanism, but a reciprocal 4D model also precesses and an unconstrained nonlinear readout cannot identify particle depth.

• Flat nucleus precession — The fixed nuclear three-plane and an invariant CP3 binding energy yield an exact conditional precession flow, with conserved misalignment and no automatically selected petal lock.

• Full response extension — An autonomous reciprocal 3+3+9-channel source extension conserves energy and particle occupations but does not sustain the prescribed petal lifts at the tested parameters, for one or two electrons.

• Gapped contact source — A singlet plus nine gapped contact excitations generate a positive orientation-kernel slope and a calculable tensor susceptibility, with a self-consistent rotating solution in the full finite source model.

• Induced cross-plane response — A positive-cost mixed-plane response generates the inverse orientation kernel in its static limit, while its dynamical source and physical normalization remain explicit assumptions.

• Inverse nuclear orientation kernel — An explicit quadratic nuclear-plane kernel reproduces the precession flow, while inverse-rate reconstruction exposes nonuniqueness and the oriented source required for the full petal generator.

• Master chiral current projection — Positive-energy projection of a massive six-spatial-dimensional Dirac scalar current supplies the four-component polarization, Berry kinetic term, and a calculable momentum-transfer bivector source.

• Master scalar exchange bridge — A normal-ordered fermionic scalar interaction yields the auxiliary chiral polarization coefficient under Gaussian closure, while an exact one-particle counterexample fixes the limits of that bridge.

• Multicentre orbit axes — Three-petal motion gives exact trigonal axes and a distinct 6D rotation gives tetrahedral axes; under bound-within/free-without, projected nuclear crossing is an encounter rather than an unobserved-depth miss.

• Orbit exchange robustness — Perturbing the actual orbit generator separates petal formation, observer-image closure, and recurrence of the full projective state.

• Petal response residual — The prescribed demo lift changes a lock invariant conserved by every orientation-only kernel, while finite-source inverse tests locate a dynamical mismatch despite instantaneous tensor compatibility.

• Ring section embedding — The ring's three-state generator embeds in positive-degree section spaces under the same Killing action, but this does not identify a degree-zero electron sector.

• Two-electron shared response — Two orthogonal occupied modes admit self-consistent collective precession through one dynamical contact source, with conserved energy and antisymmetry; this extends the response model but does not yet derive interacting petal trajectories.

• Accurate cylindrical correction — Accuracy-controlled fixed-Casimir shooting revisits the rejected cylindrical candidates with analytic derivatives and independent endpoint checks.

• Cylindrical molecular shape stability — The prescribed three-petal route gives a strict trigonal ligand minimum, while the nominal tetrahedral quarter-turn rays carry transverse force and relax to a strict disphenoidal minimum.

• Cylindrical petal continuation — Fixed-Casimir shooting supplies near-recurrent candidates, but independent integration refinement rejects the saved nonzero checkpoints at the stated closure tolerance.

• Dimensional magnetic kernel comparison — The constrained magnetic kernel distinguishes five-dimensional null-axis polarization from six-dimensional complex-structure polarization, while the scalar-exchange bridge explicitly supplies a five-dimensional vector-channel escape route.

• Dimensional necessity from response — A fixed-preparation unobserved-response rank theorem gives a conditional six-dimensional lower bound, while exact five-dimensional dynamics and averaging counterexamples identify what the existing constructions do not require.

• Finite cylindrical nuclear profile — A finite supported-space nuclear profile extended cylindrically through unsupported directions gives restoring cycle-averaged radii for both trigonal and tetrahedral 6D orbit-axis geometries.

• Finite nuclear-plane response — An unobserved-localized control saddle cannot be stabilized by passive flexibility, while the selected cylindrical ontology instead requires a finite-size, finite-inertia plane rotor before molecular stability can be tested.

• Frozen magnetic texture test — A frozen-protocol test admits spatial polarization textures into the existing six-dimensional magnetic kernel and treats negative energy curvature as a failure of the uniform preparation.

• Localized polarization stationarity — With the same localized Gaussian envelope and magnetic kernel, uniform five-dimensional polarization has a nonzero spatial first variation whereas six-dimensional pure polarization is stationary.

• Nuclear-plane rotor — A finite-inertia rank-three nuclear plane coupled reciprocally to a 6D electron preserves its projector constraints, energy, visible momentum, and combined rotational charge while undergoing nontrivial orientation response.

• Reciprocal cylindrical molecule — A 6D electron, relative angular source, moving finite cylindrical nuclei, and flexible nuclear-plane rotors form one translation-invariant reciprocal Hamiltonian with verified energy and Noether-charge conservation.

• Five-dimensional CP3 selection and spectral discrimination — A five-dimensional spinor-bivector model also selects a CP3 minimum family, but its exact spectral sum rule excludes the six-dimensional singlet-triplet gap within the linear-bivector operator class.

• Literal Orbit Precession in Flat Spatial Coordinates — The maintained deterministic curve has an exact flat isoclinic-oscillator realization; its curved action and density-gradient mechanisms have distinct coefficients and unselected locks.

• Alpha Decay Through a Real Fourth Spatial Direction — A real fourth-direction alpha-separation hypothesis, its exact scalar elliptic correspondence, and independently checked orbital and clock constraints.

• Chiral CP3 selection — A normalized six-dimensional chiral spinor coupled to an unconstrained bivector selects a CP3 minimum family with a Berry term and nonlinear energy confinement, conditional on its master-field derivation and identification with physical spatial coordinates.

• CP3 response selection — At fixed electron depth six, an explicit filled-band response selects equal nonzero skew strengths and CP3 minimum components, with positive stiffness for normal variations in matrix space but unresolved source occupation and first-order dynamics.

• Geometric response operator — Pauli tensor bilinears give an explicit map from the geometric CP3 lift to the mixed-plane source, fixing the relative symplectic normalization without identifying a spatial Dirac field.

• Magnetic bivector master bridge — The minimally gauged Dirac–Maxwell candidate supplies the Pauli coupling and quadratic field cost, with a constrained magnetic kernel and a CP3 joint minimum family for a fixed envelope and uniform polarization.