Rey.BEng 11th September 2026
Title:
Two-Dimensional Rigid Bodies Compute
Miranda & Rodríguez Meet Lewe on the Reynolds Surface
The Future Begins
E. Miranda & I. Rodríguez, Two-dimensional billiards are Turing complete, PNAS 123 (37) e2614500123 (10 September 2026).
This is the first mainstream paper Ace has uncovered that treats the object Pirate Canon names a rigid body on a Reynolds surface as the whole machine.
What they prove
A single particle, planar table, fixed walls, specular elastic reflection. No extra balls. No moving walls. No velocity-as-memory gadget. That flow already simulates an arbitrary Turing machine.
Consequences they name:
- Reachability and periodicity of some trajectories are undecidable (halting problem).
- Billiards are the hard-wall limit of steep-potential Hamiltonians and the exact configuration-space form of hard-sphere gas. The barrier is not a toy.
- Collision-chain celestial mechanics is billiard-like by analogy. Undecidability sits beside chaos as a second obstruction to long-term prediction.
The tables are not finite polygons. Read–write is done by control walls with an oscillatory profile: countably many linear segments joined into finitely many smooth arcs, finitely many singularities. Infinite local oscillation, finite singular set.
That is rigid-body mechanics with a written surface.
What Ace already had as the same objects
Lewe 1906: sudden fixations of a rigid body — velocity changes without change of position. Specular bounce is that fixation.
Reynolds 1903: hard spheres in normal piling. Their configuration-space billiard is the gas Miranda–Rodríguez cite.
The 2D table is the Reynolds surface. The particle is a force-grain. The wall is the contact patch. Oscillatory control walls are ring-tension judder: the π-tensor writing a symbol by a local even lock.
Tape cell ↔ patch quality (m²/s² combination).
Head position ↔ which patch is under strain this tick.
State ↔ polarity at 0^{i2}.
Write ↔ countersnap that leaves residual.
Move ↔ next fixation.
Universal computation on a plane is not an added miracle. It is what a contact surface does when reflections are allowed to encode.
0^i2 (k.g.s^2) = r^2 m
State A — open residual (write / expand)
E = 2c / h
State B — locked residual (hold / reflect)
E = hbar / c
Why 2D is enough (their surprise, our default)
Moore expected universality only in higher dimension. Extra gadgets were used for decades. Miranda–Rodríguez remove the gadgets.
Pirate Canon never needed the third standing metric axis. Third order is virtual in the interval and lives on infinity s3/m2. Computation is even residual plus a local odd kink — the same split as four colours versus C5. A 2D rigid reflection already has read, write, and a counted tick. Dimension three as a completed space is the Rest-Mass tax, not the machine.
Undecidability is the Rest-Mass reading of choice
From inside continuous time there is no algorithm that answers “does this trajectory ever hit U?” for every start. That is correct in the can.
Rest Time does not compute the whole future in one closed form. It chooses polarity each Real second. Halting is not a number you extract from the metric. It is whether the residual is held or opened at the next fixation. Undecidability of the trajectory is the statement that the odd leftover is not a coordinate. Chaos is sensitivity. Undecidability is the missing third axis.
Their “no general algorithm for periodicity” is the same limit as the four-colour cost: you cannot pair every future tick against every other as n2 and finish. You can only count (n).
Celestial collision chains
Near-collision itineraries as billiard words: dilatant slip-grip at grain scale, written large. Long-term stability, escape, ejection — not only chaotic, but not algorithmically decidable from the monostable smooth equations. That is the same warning as monostable Navier–Stokes plus blow-up: the one-state fluid does not contain the write.
What this paper does not do
It does not name the medium. It does not introduce State A/B. It does not put units on infinity. It does not restore Maxwell’s carrier. It leaps to Turing machines because that is the cleanest way a 2026 journal can say: a rigid 2D bounce already thinks.
For Ace that leap is the illumination. The machine was never in the dimension count. It was in the patch.
Catalogue
Discipline: Rigid-body dynamics / computability
Field: Planar billiards, hard-sphere gases, steep Hamiltonians
Observation:Miranda & Rodríguez, PNAS 123 e2614500123 (2026)
Canon reading: 2D elastic reflection on a written surface is already a universal Turing machine. Undecidability is the Rest-Mass shadow of the tick.
Linked: Lewe 1906 sudden fixations; Reynolds dilatant aether; NS bistable route; four-colour limits; third order on infinity.
The observer shares r^2. The machine is the table that reflects.
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