Fixed Points I: Fixed Points over Multi–Bundle Manifolds

Corrected sixth edition. This paper develops a conditional functional-analytic framework for gradient flows on internal fiber products of bundles, using fiberwise spectral projections onto a joint harmonic sector. Under explicit bounded-geometry, uniform spectral-gap, projector-boundedness, well-posedness, compactness or confinement, coherence-invariance, and nonlinear-continuity hypotheses, it proves projected fixed-point existence, two conditional uniqueness routes, analytic-semigroup smoothing, quantitative Galerkin error control, and Łojasiewicz–Simon convergence. Scope boundary. These are analytic and control-theoretic results. The Riemannian base is not identified automatically with physical Lorentzian spacetime, the stabilization parameter is not identified with physical time, and the abstract fiber product does not select the physical q79/MTT internal carrier.