invariant: φ · ν = eʷ · e⁻ʷ = 1 (mass-shell normalization)axes: w rapidity · R/r = sech w · overlap = r tanh wcontract: [A] torus family geometry · [I] relativistic emblem — approached, never occupied
PASSAGE INSTRUMENT
overlap bar = r · tanh w (β)⊙ equator — the present (0↔∞ exchange)cage at ±r = c — approached, never occupiedstrip: overlap saturation
Euclidean surface — no null directions; the relativistic readings are a priced emblem (receipts 151–152); the mass-shell hyperboloid is the citable geometry. [A family geometry · I emblem]
R = r · sech(w)
Your browser could not start WebGL. The Passage sweeps a torus family:
the sphere of matter opens into the horn torus at rest (R = r, w = 0), a rapidity boost contracts the mouth
as R = r·sech(w) while the axis overlap r·tanh(w) saturates at r, and the family approaches — never occupies —
the twice-covered limit. Euclidean surface; the relativistic readings are a priced emblem (receipts 151–152).