Emergent Time from Helical Motion on a Compact Matter Dimension: The Hubble Sphere as the Coherence Boundary of Lines

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, the mass quantum k was fixed by a holographic pinning condition, k = √ mPmH, equivalently λ ¯ k 2 = ℓPRH for the reduced Compton wavelength of the quantum, whose equality had the status of a selected premise. We derive the equality from two quantum properties of a line that bound the size of the matter dimension from opposite sides. The fibre coordinate of a line fluctuates by ℓP per Planck cell — the uncertainty relation and the black-hole bound give δmδℓ ≥ ℓ 2 P with the factors 1 2 and 2 cancelling, the quota constraint is symmetric between dm and dℓ, and the independent cells are the finest this bound allows — and the region an observer can describe ends exactly at the Hubble sphere, where the quota ceases to have real solutions; a line crossing that region accumulates the fibre variance ℓPRH. Its coherence across the region requires λ ¯ k 2 ≥ ℓPRH, and a definite integer winding number requires λ ¯ k 2 ≤ ℓPRH (to within 3.6% for a periodic phase); the horizon entropy gives the latter bound again, exactly, by a second count that serves as a consistency check. The observed mass quantum fixes the elementary step at (1.007±0.009) ℓP. The derivation rests on two stated premises — the standard quantum criteria for coherence and for a definite quantum number, used as exact conditions at ¯h, which are the two halves of one minimum-uncertainty statement — and on the framework’s account of time, in which the past is information carried by the present configuration; with the relay vacuum it gives ΩΛ = 2/3. The same mechanism predicts that spatial superpositions lose coherence at the rate Γ = (Mc2/h¯) √ tPH0 ≈ 5 × 10−7 s −1 per atomic mass unit: linear in mass, independent of the separation of the branches, without heating and without a free parameter. All interference experiments to date are compatible; the present mass record lies a factor of about 600 below the threshold Mτ ≈ 106 u s.