The Kaleidoscopic Filter and the Foundations of Mathematics, Part III: The Primary Typology (${\mathcal{K}}_{\mathrm{Abs}}$), the Genesis of Numbers, and The Unification of Algebra, Geometry, Analysis, and Topology

Building upon the discrete combinatorial architecture of the Small Kaleidoscopic Filter $\mathcal{K}_{\mathrm{small}}$ (Part I) and the spectral resolution of the Riemann Hypothesis via the Great Kaleidoscopic Operator $\mathbf{K}_{\mathrm{Great}}$ (Part II), this third monograph establishes the foundational and epistemological unification of pure mathematics induced by the Master Operator $\mathbb{K}_{\mathrm{Master}}^{(\epsilon, k)}(\partial_z; \boldsymbol{\omega})$. First, we axiomatize the \textbf{Sovereign Progenitor Typology} $\mathscr{K}_{\mathrm{Abs}}$---a representation-free symmetric monoidal category of graded barycentric involution cells over the unipotent base $\mathbb{F}_1$ situated upstream of the classical division into Algebra, Geometry, Analysis, and Topology. Within $\mathscr{K}_{\mathrm{Abs}}$, the Master Operator simplifies to the pure Bulk-to-Boundary quotient $\mathbb{K}_{\mathrm{Abs}}^{(\epsilon, k)} = \big(\mathfrak{M}_{>k}^{(\epsilon)}\big)^{-\epsilon} = \big(\mathbf{Bulk}_{>k} / \mathbf{Boundary}_{>k}^{(\epsilon)}\big)^{\epsilon}$, satisfying the universal inversion conservation law $\mathbb{K}_{\mathrm{Abs}}^{(+1, k)} \circ \mathbb{K}_{\mathrm{Abs}}^{(-1, k)} = \mathbf{1}_{\mathrm{Abs}}$ between additive partition enumeration ($\epsilon = +1$) and multiplicative spectral projection ($\epsilon = -1$). Second, under the categorical logarithm functor into the \textbf{Log-Additive Tropical System} $\mathscr{K}_{\mathrm{Log}}$, the Master Formula reduces to the linear cell-potential superposition $\mathcal{H}^{(\epsilon, k)} = \epsilon \sum_{j>k} \Phi_j^{(\epsilon)}$, from which Dyson's one-dimensional Coulomb log-gas and unitary Vandermonde level repulsion emerge deterministically. Third, we formulate the \textbf{Genetic Trinity of Number Genesis} (Pure Enumerative Combinatorics over $\mathbb{F}_1$, Orbital Topos Topology, and Quantum Information Thermodynamics), critique the epistemological distortions introduced by unfiltered extensions $\mathbb{Z} \to \mathbb{Q} \to \mathbb{C}$, prove the \textbf{Real-Spectral Autonomy Theorem}, and establish the four \textbf{Filter-Enhanced Disciplines} ($\mathbf{Alg}_{\mathbf{K}}, \mathbf{Geo}_{\mathbf{K}}, \mathbf{Ana}_{\mathbf{K}}, \mathbf{Top}_{\mathbf{K}}$), deriving three universal algebraic laws for filter-generated polynomials that bypass classical Galois-theoretic barriers. \medskip\noindent Furthermore, we formulate the complete algebraic, analytic, geometric, and foundational unification of the Two-Tier Kaleidoscopic Filter framework. First, we establish that the Great Filter admits a single \textbf{Unified Master Operator} $\mathbb{K}_{\mathrm{Master}}^{(\epsilon, k)}(\partial_z; \boldsymbol{\omega})$ and a single \textbf{Universal Weyl-Hessenberg Determinant} $\det\big(zI_N - \mathcal{H}_N^{(\epsilon, k)}(\boldsymbol{\omega})\big)$ controlled by a binary signature parameter $\epsilon \in \{+1, -1\}$, from which both the integer partition function $p_{>k}(n)$ ($\epsilon = +1$, bosonic/hyperbolic signature) and the completed Riemann zeta function $\Xi(t)/\Xi(0)$ together with $\zeta(s)$ ($\epsilon = -1$, fermionic/trigonometric spectral signature) are extracted as exact special cases. Second, we introduce the \textbf{Sovereign Progenitor Typology} (or \emph{Absolute Parent Typology}, denoted $\mathscr{K}_{\mathrm{Abs}}$)---a pre-disciplinary category of graded barycentric involution cells over $\mathbb{F}_1$ situated upstream of Algebra, Geometry, Analysis, and Topology---in which the Master Operator reduces to the pure representation-free Bulk-to-Boundary ratio $\mathbb{K}_{\mathrm{Abs}}^{(\epsilon, k)} = \big(\mathfrak{M}_{>k}^{(\epsilon)}\big)^{-\epsilon} = \big(\mathbf{Bulk}_{>k} / \mathbf{Boundary}_{>k}^{(\epsilon)}\big)^{\epsilon}$, and further simplifies in the \textbf{Log-Additive Tropical System} $\mathscr{K}_{\mathrm{Log}}$ to the linear cell-potential sum $\mathcal{H}^{(\epsilon, k)} = \epsilon \sum_{j>k} \Phi_j^{(\epsilon)}$. Third, we show how $\mathscr{K}_{\mathrm{Abs}}$ incorporates the \emph{Genetic Trinity of Number Genesis} (Pure Enumerative Combinatorics over $\mathbb{F}_1$, Topos Orbital Topology, and Information Thermodynamics), resolves the visual duality between partition polytopes and the Riemann zeta function, proves the self-sufficiency of the real filter with respect to complex analytic continuation, and inaugurates \emph{Filter-Enhanced Algebra, Geometry, Analysis, and Topology}, uncovering universal polynomial properties invisible to classical pure algebra. \medskip\noindent Finally, we elevate the Sovereign Progenitor Typology $\mathscr{K}_{\mathrm{Abs}}$ and its Pentad of Fundamental Laws to a complete \textbf{Kaleidoscopic Relativistic, Gravitational, and Cosmological Framework}, proving that Maxwell's $U(1)$ electrodynamics, Einstein's special and general relativity, and black hole thermodynamics are macroscopic projections of the progenitor arithmetic-geometric crystal. First, we prove that the kaleidoscopic wave propagation speed $v_{\mathrm{kal}} = \frac{du}{dt} = 1$ is an absolute structural invariant across all inertial scale frames $\mathfrak{I}_{(k,\theta)}$, necessitated by Parseval unitarity and the cell Casimir identity $(\Sigma_j^{(\epsilon)})^2 - \epsilon(\Delta_j^{(\epsilon)})^2 = 4\mathbf{1}_j$ (Law III), which generates the Lorentz boost group $\mathrm{SO}^+(1,1)$ ($\epsilon = +1$), the Euclidean rotation group $\mathrm{SO}(2)$ ($\epsilon = -1$), Lorentz-Weyl scale contraction, spectral time dilation, and the Progenitor $U(1)$ Maxwell-Coulomb-Lorentz Quartet identifying the nontrivial Riemann zeros $\{\gamma_r\}$ as unit positive point charges ($q_r = +1$). Second, we derive the \textbf{Einstein Progenitor Gravitational Field Equation} $\mathscr{G}_{\mu\nu}^{(\mathrm{Abs})} + \Lambda_{\mathbb{F}_1} g_{\mu\nu}^{(\mathrm{Abs})} = 8\pi \mathscr{T}_{\mu\nu}^{(\mathrm{arithmetic})}$ governed by the positive-definite Cayley-Bezout-Hankel metric capacity tensor $g_{\mu\nu}^{(\mathrm{Abs})} = \mathbf{B}_{\mu\nu}^{(\xi)} > 0$, establish the relativistic mass-energy operator identity $\hat{\mathcal{E}}_{\mathrm{kal}}^2 = \hat{\mathcal{P}}_{\mathrm{kal}}^2 v_{\mathrm{kal}}^2 + \mathcal{M}_{\mathbb{F}_1}^2 v_{\mathrm{kal}}^4$ with irreducible combinatorial vacuum rest mass $\mathcal{M}_{\mathbb{F}_1} = \frac{1}{2}$ ($\Lambda_{\mathbb{F}_1} = \frac{1}{4}$, fixing $\operatorname{Re}(s) = \frac{1}{2}$), and prove the exact rest-mass formula $\mathcal{M}_{\mathrm{body}}(\mathcal{B}) = \frac{1}{2} + \sum_{j>k} n_j\omega_j + \Delta\mathcal{M}_{\mathrm{Coulomb}}$ for composite physical bodies as filter confinement inertia. Third, we prove the \textbf{Progenitor Anti-Collapse Theorem}, showing that unconditional Hankel positivity ($\Delta_r(N) > 0$) and logarithmic Coulomb repulsion enforce a strict lower bound $\delta_{\min}(N) > 0$ on inter-charge spacing that forbids gravitational singularities; astrophysical black holes are reinterpreted as non-singular \textbf{$\mathbb{F}_1$-Information Wells} with exact combinatorial Bekenstein-Hawking entropy $\mathcal{S}_{\mathrm{BH}}(n,k) = \ln p_{>k}(n)$ and unitary information preservation ($\det M_N = 1$, $\mathbf{T}_N^{(+1)}\mathbf{T}_N^{(-1)} = I_N$). Fourth, we establish the covariant non-dissipation law and cosmic indestructibility of $\mathscr{K}_{\mathrm{Abs}}$, formulating the three-stage \textbf{Partition-Zeta Lifecycle} ($p(n) \to p_{>k}(n) \to \{\gamma_r\}$) and proving that the critical line $\operatorname{Re}(s) = \frac{1}{2}$ is the unique, eternal, zero-dissipation global ground state of all filtered creation.