This series of papers presents a complete, axiomatically rigorous mathematical reformulation of the Planck Core Framework, a theory that derives all known fundamental physics—general relativity, the Standard Model of particle physics, and the origin of all physical constants—from a single underlying entity: a discrete quantum information network. The present series is an axiomatic mathematical restatement of the same physical content previously articulated by the author in a series of physical preprints. Those earlier works employed heuristic and physical reasoning; the present series replaces every heuristic argument with a strict definition–theorem–proof structure conforming to the standards of leading mathematical journals such as Inventiones Mathematicae, Acta Mathematica, and Communications in Mathematical Physics. The framework rests on five axioms. Axiom 1 defines the quantum bit network as a connected, 4-regular, vertex-transitive graph whose nodes carry qubits and whose edges carry the natural SU (2) ×SU (2) SU (2) ×SU (2) symmetry. Axiom 2 identifies spatial geometry with entanglement: the length of an edge is a function of its entanglement deficit, and the distance between nodes is the shortest-path sum of edge lengths. Axiom 3 defines a discrete time parameter through entanglement renewal step counts. Axiom 4 imposes a global SO (3) SO (3) symmetry on the ground state, ensuring the emergence of three-dimensional rotational invariance. Axiom 5 is a maximum entropy variational principle that governs the evolution of the network. The series contains no free parameters beyond the two intrinsic microscopic units of length ℓ0ℓ0 and time τ0τ0. The logical chain proceeds through the following sequence of results, each established in a dedicated paper: Axioms and deficit dynamics. The entanglement deficit operator δᵉ=1ij−Πsδᵉ=1ij−Πs is proved to be the unique SU (2) ×SU (2) SU (2) ×SU (2) -invariant linear operator on the edge Hilbert space. The local interaction Hamiltonian is uniquely determined as the Heisenberg exchange operator. The deficit evolution is governed by a Lindblad master equation, whose self-consistent closure is proved via a fixed-point theorem. Three physical regimes are identified: δ=0δ=0 (flat spacetime), 0<δ<10<δ<1 (curved spacetime), and δ=1δ=1 (topological phase transition). Emergence of space. In the Gromov–Hausdorff continuum limit, the 4-regular tree network converges to the unit ball equipped with the chordal metric. The global SO (3) SO (3) symmetry forces the boundary to be S2S2; the emergent space is three-dimensional. Emergence of time and fundamental constants. The discrete renewal dynamics converges to continuous unitary evolution, conditional on strong resolvent convergence. The speed of light c=12ℓ0/τ0c=12ℓ0/τ0 and the reduced Planck constant ℏ=24τ0/ℓ0ℏ=24τ0/ℓ0 are derived as limits of purely combinatorial network quantities. Emergence of Einstein's equations. The Einstein field equations and the Newton constant G=108ℓ07/ (πτ04) G=108ℓ07/ (πτ04) are derived from the maximum entropy variational principle. The standard Planck length is ℓP=ℓ03/23/ (2π) ℓP=ℓ03/23/ (2π). Emergent gravitons. The second-order variation of the effective action yields the Fierz–Pauli action for massless spin-2 gravitons. Topological defects from critical deficit. When the entanglement deficit saturates, the network undergoes a topological phase transition, producing a compact defect. The edge modes carry an internal Hilbert space Hedge≅C3Hedge≅C3, derived from the 4-regularity and SO (3) SO (3) symmetry. The topological order is uniquely determined to be the Kitaev ν=6ν=6 Ising anyon phase, with ground-state degeneracy dcore=16dcore=16. Emergent gauge bosons. The internal edge space Hedge≅C3Hedge≅C3 carries the maximal gauge group U (3) U (3). An additional global U (1) U (1) extends the parent group to U (3) ×U (1) U (3) ×U (1). The Wilson plaquette action on the coarse-grained network converges to the Yang–Mills action. Emergent fermions. The ψψ anyon of the Ising phase is a localised fermion. The braid matrix yields canonical anti-commutation relations. The three complex dimensions of HedgeHedge support three independent fermion species, corresponding to three Standard Model generations. Emergent Higgs mechanism. The Higgs field is identified with the scalar breathing mode of the entanglement density. Spontaneous symmetry breaking generates masses for gauge bosons and fermions. Closing the framework. A unified effective action and lattice action are presented, together with a complete dictionary of derived constants and a comprehensive list of open problems. From the quantum bit network to SU (2) 6SU (2) 6 topological order. The SU (2) 6SU (2) 6 modular tensor category is rigorously derived from the five axioms via the braid group B3B3 classification, anomaly cancellation, and Kitaev's 16-fold way. Why SO (10) SO (10)? Gauge group selection. The residual gauge group is uniquely determined as SO (10) ×U (1) /Z4SO (10) ×U (1) /Z4 through an entropic selection theorem and group-theoretic analysis of the anyon condensation channels. Planck core interior dynamics. The low-energy effective dynamics of the Planck core is governed by the Chern–Simons action at level κ=6κ=6, with anyon equations of motion, braid statistics, and instanton tunnelling dynamics. Fermion internal structure. Every fermion possesses a three-layer architecture (vortex core, gauge cloud mantle, Berry braid tail), derived from the axioms. The core radius is universally determined by the Abrikosov–Nielsen–Olesen vortex equation. Fermion internal dynamics. The three-layer dynamics is governed by a unified Feynman path integral that factorises over the layers. The low-energy limit recovers the Standard Model Lagrangian with all 19 free parameters acquiring a geometric or topological origin. Categorical uniqueness and Nf=3Nf=3. If a physical universe is described by a modular tensor category satisfying five categorical constraints, then the maximum entropy principle uniquely selects (SU (2) ₆). This establishes the uniqueness of the network's topological phase, and yields Nf=3Nf=3 without experimental input. Every result in the series is accompanied by a rigorous assessment of its proof status, distinguishing strict implications from physically motivated hypotheses and open conjectures. No result is presented as proved unless it follows from the axioms by a complete logical chain. This series is the definitive mathematical formulation of the quantum bit network framework.
Wengang Yu (Sat,) studied this question.