Emergent Time from Helical Motion on a Compact Matter Dimension: The Action as an Angle — Photons, Entanglement, Black-Hole Thermodynamics and the Vacuum Energy

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, built on two axioms, companion papers derived the Schrödinger and Dirac equations, gravitation, electromagnetism and the gauge structure of the strong and weak interactions. Here we give the quantum level a single base formula: every configuration of the lines contributes with a phase equal to its total angle, Z = ∑exp(iΘ), so that the action is ¯h times an angle — the fibre angle for matter, the holonomy angle for gauge fields, the deficit angle for gravity, all on the Planck cell. Its first consequence is that matter, light and curvature carry one quantum of action in three directions. One turn of the fibre angle is a mass quantum. One turn of twist per wavelength on a line is a photon: from the twist stiffness ¯hc/2π of the companion papers its energy is exactly hν and its spin exactly ¯h, for any mass quantum. One cell of a line, counted as curvature through Regge’s deficit angle, carries exactly h; a line counted as a tension and the same line counted as a deficit angle carry the same action. Three further results follow without further assumptions. Time is an angle, so black holes are hot: near a horizon the fibre advance of a held body grows as κρ/c, the helical phase is a polar angle about the horizon, and its period 2π gives the Hawking temperature, while the horizon as a hinge with a full-turn deficit gives the entropy A/4ℓ 2 P ; the periodicity that general relativity imposes on Euclidean time is here built in. Gravitation counts lines: matter and radiation carry them and the ground state of the line pattern does not, so zero point energies and condensates, which would exceed the observed vacuum energy by up to 10122, do not gravitate, while radiation, binding energies, Casimir energies and an inflaton do; the residual is the energy of the relay endpoints, ρΛ = k 4/4π, already confronted with data. The step from the relay count to the vacuum energy is a reading, stated as such. The amplitude is a weight on configurations of the whole line pattern, which is the address of entangled states that the first paper of the series could not find: the singlet correlation −cos(a−b) and the Tsirelson value 2√ 2 follow, and exchanging two particles tethered to all of space by their lines is a 2π rotation, which with the even-winding rule gives spin–statistics and the exclusion principle. Each of these reproduces an established result; what is new is that one formula and one ontology yield them all. The framework also makes a prediction that differs from both camps in the debate on the quantum nature of gravity: gravity is quantized, but because mass superpositions decohere intrinsically at the rate (Mc2/h¯) √ tPH0 derived in a companion paper, the gravitationally induced entanglement of two masses is unobservable in principle — reaching an entangling phase of order one before decoherence would require branches separating at the speed of light and a density seventeen times that of osmium. A null result of such experiments would therefore not show that gravity is classical; the discriminating test is single-particle matter-wave interference beyond Mτ ≈ 2×106 u s, a factor of about 500 beyond present experiments. We list the standard computations we leave to others.