ReyBEng 4th November 2026
Lewe’s thin-shell reduction of the cylinder to a ring is the parent. Project 2 wrote the residual term that the monostable reading omits:
σθ=tpr+σR
with σR carrying the units m/s2.
The geometric Can is that ring held closed. Split the ring into two hoops and pull them apart along the axis. The surface that forms between them is the catenoid residual. Its mean curvature vanishes:
H=2κ1+κ2=0r(z)=acosh(az)
The figure that remains between the hoops is σ_ m/s2. The equatorial disc of the original cylinder is the neck of the catenoid.
That disc is one-sided: the continuous deformation that carries the catenoid into the helicoid introduces the 180° twist, and the residual stretched to the limit is the Möbius skin. The same surface may be read as the cylindrical wall (PCA coefficient tables, Lewe matrix), as the set of radial nodes of the Bessel functions on the perimeter, or as the residual surface itself. The three readings are stations of one residual.
Under the continuous second the disc has no surface that can be read; the two faces are identified and the residual is spent into the clock. Under the Moment unit (N)m.s^2 the residual is retained. The opposite face of the equatorial disc becomes visible as the far side of the neck. That is the event horizon read as geometry: the one-sided skin that appears once the ring has been split, pulled, and measured with the unit that keeps the residual unspent.
Correspondence, not identity. The parent remains Lewe’s ring. The catenoid supplies the residual figure. The Moment unit supplies the reading that keeps both faces available.
| I | II | III | IV | V | VI | VII | VIII | IX | X | XI | XII |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Arena | Path | Compare | Small loop | Holonomy | Residual | Mean tide waits | Closure test | Toggle (+/-) | Store sector | Read face | Growth face (\mu) |
| closed Can | axial pull of two hoops | wall reading vs neck reading | neck radius (a) | full catenoid + disc | σ_ m/s2 | volume law not opened | does (H=0) close on the horizon? | one-sided vs two-sided face | stored residual | measured (r(z)) under (N)m.s^2 | next separation or Bessel node |
The seat of the horizon is VI read at IX. The continuous second spends the residual; the Moment unit leaves it visible. The page records the correspondence.
