Dynamical Minimal-Ideal Geometry in QAMG: From Primitive Biquaternion Structure to Stable Matter Organization

We propose a dynamical minimal-ideal geometry framework for QuaternionAngular-Momentum Gravity (QAMG). Starting from the left and right actions onthe regular biquaternion bimodule, we investigate the relations among localized geometric organization, internal transport, and spacetime response. The independentfields of the leading reduced theory are the spacetime metric gµν, a rank-one Hermitian projector PQ, and an independent U(1)Q connection aQµ. The projectorselects a local minimal ideal through its selected-line realization; the composite curvature ΩQ[PQ] describes spacetime changes of this selection, whereas FQ = daQdescribes independent Abelian transport. These curvatures have different variational dependencies. The action contains the Einstein–Hilbert term, a quadraticprojector derivative term, the squared composite curvature, and the squared independent Abelian curvature. Its variation yields the constrained projector equation,the Einstein equation, the source-free Abelian equation, and covariant conservationidentities.For the static pure-projector branch on a flat background,gµν = ηµν,FQ =0,∂tPQ = 0,the Hopf number NQ ∈ π3(S2) ≃ Z labels topological sectors in a configurationdomain with a fixed vacuum and admissible continuous compactification. WithPQ =(I2 +nQ·σ)/2, define√︄βQ2αQ, EQ,0 =√︁ℓQ =The static energy is EP = EQ,0 ˜︁E, whereαQβQ2√2 , x=ℓQy.˜︁E = 12 ∫︂ |∇ynQ|2d3y + 14 ∫︂ ˜︁HQij ˜︁HQij d3y.The opposing scale weights of the quadratic and quartic terms give the Derrickbalance condition at an exact stationary point. The numerical NQ = 1 backgroundhas dimensionless energy CE ≃ 273.5564, relative Derrick imbalance 9.31 × 10−5,and maximum normalized slope 4.05×10−4 over fifteen compactly supported weaktangent variations. These checks support its use as an approximately stationarycore. Its classical energy and half-energy radius areE(0)Q =CEEQ,0,ξ1/2 = C1/2ℓQ.1drfeng@namo.deDefining the background’s energy-derived mass scale by M(0)Q =E(0)Q /c2, one obtains, for βQ = ℏc,M(0)Q cξ1/2 = CEC1/24 ℏ.Appendix B constructs the three-dimensional static Hessian subject to the projector tangent constraint. Combined spatial and internal axial symmetry separates the integer azimuthal sectors; admissible symmetry-orbit directions are distinguished from normalizable zero modes of an exact stationary point. After checkingthe angular-derivative signs, axial regularity, and fixed-vacuum boundary conditions, scoped sector-by-sector judgments have been obtained for m = 0,...,12 and|m| > 12. Within the saved numerical background, the corresponding correctedtest or reconstruction spaces, the admissible symmetry quotient, and the analyticnumerical tail controls recorded for each certificate, no reliable physical negativemode was found. Five previously identified potentially dangerous fields retain positive curvature when reevaluated using a consistent unit-normalized reconstruction.Small negative values near approximate symmetry orbits remain in low sectors,and direct second differences of the energy confirm their curvature; they are notremoved on the basis of orbit overlap. Starting at |m| = 12, the large-azimuthalsector is propagated to the infinite angular tower by a continuous weight envelope,endpoint control, and analytic monotonicity, rather than extrapolated from a finitesector scan. Downsampling a single high-resolution background tests interpolationand discretization sensitivity, not convergence of independently computed stationary points. The analysis retains the zero-threshold continuum and does not requirea domain-uniform positive L2 gap. Full spectral-error control from the numericalbackground to an exact stationary point and parts of the unrepresented complementhave not been uniformly verified. The result is therefore a scoped static no-negativemode judgment on the fixed background, not a theorem for the continuum full-spaceMorse index; stability of the coupled metric–projector–Abelian time evolution requires a separate analysis. For Cartesian-smooth static backgrounds with boundedgradients, the complete tangent principal symbol also admits a uniformly positivesymmetrizer, yielding local strong hyperbolicity and two characteristic branchescontained within the metric causal cone. This local propagation result does notreplace the low-frequency energy-spectrum or global-evolution analysis.The geometric stress tensor obtained by metric variation connects localized organization to spacetime response. For isolated stationary on-shell configurationswith suitable far-field decay, stress conservation equates the active gravitationalmass at leading weak-gravity order with the rest-energy mass. Rotational Noetherstructure supplies total angular momentum. As an additional selection hypothesis for complete organizations, GAN restricts complete organizations with closedangular-momentum accounting to admissible branches satisfying Jtot = 0. Thisglobal selection condition, the local field equations, Hopf topology, and local energystability have distinct scopes.Starting from classical localized cores, we formulate the conditions for collectivecoordinate reduction, configuration-space quantization, and low-energy effectiveparticle descriptions. Rotation representations and exchange statistics depend onthe configuration domain, the Finkelstein–Rubinstein quantization choice, and stabilizer conditions; finite-size responses require matching of the actual effective action. None is fixed by the Hopf number alone. At fixed metric, the selected pureprojector action is independent of aQ, so this branch is classically neutral. If theprojector integration domain, measure, and regulator are also independent of aQ,quantizing the projector alone cannot generate an Abelian source, magnetic vertex,2ordirectAbelianradiation. Aninterfacewithphysical chargedparticles requiresanexplicitextensionof representations, couplings, orboundarystructure. Actualinertia, quantummass, polewidths, andresponseshavenotbeencomputed; theeffective-particle equations andexperimental scale relations are givensubject totheirmatchingconditions. Thequantitativestabilityevidence isrestrictedtothestaticpure-projectorbranch,whileparticleidentificationandcoupleddynamicsretaintheirrespectiveassumptionsandmatchingrequirements.