Titles: Why Schrödinger - The Schrödinger equation as self-referential flow on the 600-cell Description: The Schrödinger equation is the dynamical law of quantum mechanics. It has been tested to extraordinary precision and has never been derived from deeper first principles within standard physics. This paper derives it from the self-referential axiom σ = 1/(1+σ), the same equation that produces the fine structure constant, Newton's constant, and the cosmological constant in the companion paper "Why General Relativity and Quantum Mechanics Are Unified" (doi:10.5281/zenodo.19056730). The axiom forces a four-dimensional state space whose discrete realisation is the 600-cell, a polytope with 120 vertices on S³ ⊂ ℝ⁴. The adjacency matrix is the canonical symmetric nearest-neighbour Hamiltonian. Its spectrum has nine distinct eigenvalues, all in ℚ(√5), partitioned into a φ-free confined sector (integers: 12, 3, 0, −2, −3) and a φ-carrying free sector (irrationals forming two Galois conjugate pairs). The Schrödinger equation iℏ ∂ψ/∂t = Aψ is the unitary flow generated by this Hamiltonian. The imaginary unit is forced by the pentagonal rotation and Stone's theorem. Planck's constant is quantised because the state space is discrete and the shortest loop has finite length. The Born rule is reduced to a geometric postulate: measurement is projection from ℝ⁴ to ℝ³ (the maximal proper subspace), and Gleason's theorem forces the unique quadratic probability measure. Five probability classes emerge, discretising the Born measure into values determined by φ and √5. The path integral converges exactly on the finite state space with no ultraviolet divergence and no renormalisation. Scope: single-particle, non-relativistic quantum mechanics on a discrete state space. Zero free parameters. This is the second dynamical test of the Pentagon Physics programme. The first (doi:10.5281/zenodo.19115512) tests Friedmann dynamics. Keywords: Schrödinger equation, Born rule, 600-cell, binary icosahedral group, adjacency matrix, Gleason's theorem, quantum mechanics, self-reference, golden ratio, spherical design, Pentagon Physics
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
Deriving Quantum Mechanics from Topological Solitons on S³ — Angular World Architecture (Quantum Sector) Standard quantum mechanics rests on seven independent postulates whose physical origin is unknown. This article shows that all seven are derivable as theorems from two geometric axioms and a single nonlinear field equation on the 3-sphere S³ — the unique compact simply connected 3-manifold (Poincaré–Perelman, 2003). The construction is minimal. Topologically stable solitons (k ∈ π₃(S³) = ℤ) are identified with particles; their BPS mass is mc² = 8ℏω₀; the isomorphism S³ ≅ SU(2) yields spin j = k/2 and the spin–statistics connection (−1)^k without postulate. Linearization around the kink soliton produces the Klein–Gordon equation, and the Schrödinger equation as its non-relativistic limit. The Born rule P_n = |⟨φ_n|ψ⟩|² follows from the Liouville measure on phase space, supported independently by Zurek envariance and Valentini equilibrium. The Dirac equation emerges from the unique Clifford representation Cl(1,3) ≅ M₄(ℂ). The proton-to-electron mass ratio m_p/m_e = 1836.87 (CODATA: 1836.153, 0.04 % agreement) is derived with zero free parameters from the non-BPS mechanical energy of the k = 3 soliton. In total, 41 results of standard QM are formally derived (D) and 4 additional results are empirically supported at ≤ 2σ (S). The framework naturally unifies QM and GR through a single continuous parameter ξ_G = 2(m/m_P)²: quantum mechanics is the ξ_G → 0 limit, general relativity the ξ_G → 1 limit. Zero-parameter quantitative predictions include α⁻¹(M_Z) = 400/π ≈ 127.3 (measured 127.9; 0.5 %), m_p/m_e = 1836.87 (0.04 %), resolution of the Hubble tension at 0.03σ, and Ω_Λ = (π−1)/π ≈ 0.682 (Planck 0.689; 0.5σ). Seven falsifiable predictions are testable in 2027–2035, including the Bose–Marletto–Vedral gravitational entanglement experiment — decisive, since this framework predicts entanglement growth where Penrose–Diósi predicts collapse. Keywords: quantum mechanics, foundations of physics, topological solitons, S³, Angular World Architecture (AWA), Born rule, Schrödinger equation, Dirac equation, quantum-gravity unification, BPS solitons, sine-Gordon, Hubble tension.