Theory of Perspectival Dimensionality (TPD): Unipotent Block-Cholesky Metric Decomposition, Non-Equilibrium Bifurcation Dynamics, and Emergent Gauge-Gravitational Physics on an 11 = 4 + 7 Flag Manifold

This 112-page scientific monograph presents the complete mathematical, quantum-foundational and cosmological formulation of the Theory of Perspectival Dimensionality (TPD). The treatise proves from first principles that four-dimensional Lorentzian spacetime (Z4), the Standard Model gauge group U(1) x SU(2) x SU(3), modified Chameleon gravitation, and the four dynamical SI base units (s -> m -> kg -> A) emerge sequentially from a zero-dimensional informational boundary state (Omega0) across a nested 11-dimensional flag manifold M11 = Z4 x F7 with compact internal fiber F7 = Omega0 + Z1 + Z2 + Z3 (1 + 1 + 2 + 3 + 4 = 11). The symmetric 11 x 11 metric tensor G11 on M11 contains T11 = 66 independent components, which partition combinatorially into four third-order Pascal tetrahedral numbers (66 = 1 + 10 + 20 + 35). The deductive architecture of the monograph rests upon four rigorously proved mathematical laws: Law I (Non-Equilibrium Hormetic Bifurcation Dynamics and Geometric Bioenergetics, Chapters 1 to 5): Analytical proof of the prokaryotic 1/R surface-to-volume bottleneck (maximum radius 2.0 micrometers) versus eukaryotic scale invariance via volume-filling cristae internalization (fractal dimension Df -> 3). It includes the thermodynamic derivation of the 36 MV/m inner mitochondrial membrane electric field and 12.6 kPa Maxwell membrane stress, the analytical integration of the cytochrome c Heaviside apoptosis switch, and global Routh-Hurwitz and Bendixson-Dulac stability proofs of super-logistic Bekenstein capacity overflow driving the dimensional unfolding Z(k) -> Z(k+1). Law II (Unipotent Block-Cholesky Metric Decomposition and u(k) Oscillator Algebras, Chapters 6 and 7): Exact metric factorization G = L^T B L with unit determinant det(L) = 1, separating 21 block-diagonal stage metrics (B) from 45 off-diagonal Ehresmann gauge connectors (L) without Kaluza-Klein scalar ghost modes. Via the Law of the Two Triangles (T(k) + T(k-1) = k^2) and Jauch-Hill-Schwinger oscillator phase-space coupling, the compact fiber layers generate the Standard Model gauge group U(1) x SU(2) x SU(3) with 1, 3, and 8 gauge bosons, topological left-handed weak chirality in Z2 (det(R2) = +1) versus parity conservation in Z3 and Z4, and the complex Higgs doublet A(1,2) of dimension 1 x 2 = 2 (electroweak mass scale 174.10 GeV/c^2), while gl(4,R) decouples macroscopically on Z4 into 4^2 = 10 + 6 (10 Einstein metric components and 6 Maxwell field components). Law III (Self-Adjoint Hodge-Laplace Heat-Kernel Operator, Chapters 7 and 9): The positive heat-kernel operator U = exp(-D) on the compact fiber F7 yields 3 harmonic fermion generations, cohomological orthogonality of dark matter (zero electromagnetic coupling while preserving universal gravitational coupling to Omega0), natural relaxation of the Strong CP angle theta(QCD) = 0 without hypothetical axions, one-loop ultraviolet finiteness, an exact alternating tetrahedral series for the circle constant pi, and the exponential fermion mass hierarchy (linking the top quark at D = 0 to the electron at D = 13, the Koide lepton invariant Q = 2/3, and the proton-to-electron mass ratio 6*pi^5 = 1836.12). Law IV (Modified Einstein-TPD Field Equations, Cosmology, Subatomic Benchmarks, and Quantum Foundations, Chapters 8 and 10 to 15): Palatini variational derivation of the modified field equations with density-dependent Chameleon screening, satisfying Solar System Cassini bounds while driving unscreened cosmic void drift. Parameter-free quantitative predictions include: N = 6x10 = 60 primordial e-foldings, predicting a scalar spectral index ns = 0.9654 and tensor-to-scalar ratio r = 0.0020 Dynamical thawing dark energy (w > -1) matching DESI and Euclid observations Vera Rubin's galactic acceleration scale a0 = cH0/(2pi) and the fourth-power Baryonic Tully-Fisher relation Heat-kernel resolution of the central black hole singularity into a regular de Sitter core and a parameter-free 12.70% gravitational-wave horizon echo amplitude (4 - sqrt(15) = 0.1270, with echo delay time 109.0 ms for a 60 solar-mass merger) Analytical resolution of the proton charge radius puzzle (rp = 4 Compton wavelengths = 0.8412 fm) Resolution of the 1.00% neutron lifetime discrepancy between bottle and beam experiments via the 1/100 neutral fiber channel (beam-to-bottle ratio 100/99, lifetime shift 8.87 s) Geometric resolution of the cosmological Hubble tension via late-time unscreening of the T0 = 1 origin channel, yielding the expansion ratio H0(local) / H0(CMB) = (3 + 10 + 1) / (3 + 10) = 14/13 (+7.69%, predicting H0(local) = 72.58 km/s/Mpc from the Planck baseline 67.40 km/s/Mpc) Genetic derivation of the four SI base units (s -> m -> kg -> A), Born-Oppenheimer adiabatic time hierarchy, deterministic Kuramoto phase-locking resolution of the quantum measurement problem (synchronization collapse time 0.66 fs at 1 eV detector coupling), symplectic Liouville derivation of Max Born's probability rule, and resolution of the double-slit and delayed-choice quantum eraser paradoxes via Ehresmann fiber holonomy