a tiling that hands out work
a never-repeating golden-ratio pattern, the same kind of golden-ratio rule that lays out a Penrose floor tiling, decides which piece of work the box's reducer fires next, and across eighteen starting shapes and seventy-two runs no scheduled pair was ever missing when its turn came.
the takeaway in one paragraph
Eighteen starting shapes of the box's reducer, each run from four phase starts in a golden-ratio letter pattern that never repeats, gave seventy-two runs with zero stalls. The pattern is the same kind of rule that lays out a Penrose floor tiling, read here as a work order for which ready pair fires next. The box's own reducer normally fires its ready pairs in one fixed order.
how the schedule is built
At each wave, several pairs of net pieces are ready to fire, and the schedule decides which order they go in. A stall here is a scheduled pair that has already vanished by the time its turn comes, because an earlier pair in the same wave consumed it. The reducer runs this as one sequential loop, not as separate workers each waiting on another.
The pattern starts from one letter, L, and two rules that run on it over and over: every L becomes LS, every S becomes L. Run those rules enough times and the single letter grows into a long string of L's and S's, the same substitution the lab's canon lists as `implementations/penrose-symbolic-phase`. The canon keeps the shape version separate, as `implementations/penrose-p2-tiling`, a decagon subdivided by the same golden ratio into acute and obtuse triangles.
Each letter in the string becomes one wave's policy for the reducer. An L tells it to fire its ready pairs left to right. An S tells it to fire them right to left. `test/kernel.parity-penrose-phase.test.ts` checks that every stage of the build-up decodes back to the stage before it, all the way down to the single starting letter, and that every letter's own policy matches the rule it is supposed to follow.
The lab's canon entry `evidence/penrose-symbolic-aperiodicity` proves, under its stated substitution assumptions, that the infinite string never settles into a repeating period. The canon records the limiting share of L's in it as 1 over phi, an irrational number, and a string that eventually repeats would have to tend to a rational one.
The proof is about the string of letters, not a physical crystal. `implementations/penrose-symbolic-phase` states outright that it is not a physical time crystal or tiling geometry engine, and the test asserts that refusal directly.
the number
The sweep runs this schedule against the complete family of connected, closed net shapes up to nine agents. Each of eighteen starting seeds is run from four phase starts, offsets zero through three, for seventy-two executions in total. The assertions live in `test/kernel.parity-cell-wasm.test.ts`.
| what was checked | count |
|---|---|
| starting shapes (seeds) | 18 |
| runs: 18 seeds across four phase starts | 72 |
| individual pairwise reductions across all runs | 256 |
| synchronization stalls | 0 |
| final-answer mismatches across the four phase starts | 0 |
| runs that never reached a finished state | 0 |
| wiring invariant failures | 0 |
A companion suite, `test/kernel.parity-penrose-tiling.test.ts`, builds the floor tiling itself: five tests, five passing, on the exact golden-ratio arithmetic, the triangle-count recurrence, and a mesh that comes out bounded, nondegenerate, manifold, and deterministic. `test/kernel.parity-penrose-phase.test.ts` adds four more tests, four passing: three on the schedule string itself, and one that builds phase vectors on a 14-vertex graph and records that no hardware phase measurement backs them.
what this does not cover
- Eighteen seeds are the complete connected, closed net family up to nine agents, not every net shape the box can build.
- The schedule only ever chooses between two fixed directions each wave, left to right or right to left. It does not pick an arbitrary order among the ready pairs.
- The schedule itself has 17,711 phase starts. Each seed here was run from four of them, offsets zero through three, a fixed list in the sweep. Nothing here rules out a stall at some other start.
- Zero stalls here is a correctness result inside one process. It says nothing on its own about wall-clock speed.
- Each wave checks that its ready pairs are disjoint before firing them and throws if they are not. That invariant is most of why the stall count comes out zero, not a race the sweep survived.
- The zero mismatches compare the four phase starts against the offset-zero run, not against the ordinary sequential reducer's answer. The sequential run is only checked for reaching a finished state.
- These seventy-two runs are one sequential loop. The lab's canon entry `evidence/penrose-worker-thread-zero-stall` covers a separate sweep on real worker threads over a different seed family, and its own scope note rules out any claim about a GPU, shared mutable state, lock-freedom, or physical simultaneity.
- The aperiodicity proof, `evidence/penrose-symbolic-aperiodicity`, is a proof about a string of letters, not a physical crystal. `implementations/penrose-symbolic-phase` states outright that it is not a physical time crystal, and the test asserts that directly.
the next gate
Zero stalls here is measured in one sequential loop, four phase starts per seed. The next gate runs the same four phase starts on `implementations/penrose-worker-thread-sweep`, with seeds of uneven reduction length and a worker added or removed mid-run. It holds if stalls and mismatches stay at zero and the aggregate still matches the sequential result.