Phase synchronisation without a central clock in a delay-coupled spherical shell: wiring, propagation delay and carrier-phase compensation in a reduced Kuramoto model

Architectures without a centrally distributed clock must establish shared timing through interaction between local oscillators, while signals take a finite time to travel between them. We study a reduced version of this problem in the Layer 3 shell of the ORBA design, a spherical shell of radii 170–220 mm with a 500 GHz synchronisation carrier, for which a radial crossing at guided-light speed already spans about 125 carrier cycles. Using a delayed Kuramoto model written in the carrier frame, 2,000 coarse-grained units placed in the shell volume are coupled in three ways: nearest-neighbour links (S1), nearest-neighbour links plus one long-range optical shortcut per unit (S2), and a single optical hub (S3). Thirty-six configurations, each with three seeds, compare compensated and uncompensated carrier phase, three frequency spreads and two coupling strengths. The solver reproduces the no-delay Kuramoto solution and the Ott–Antonsen locked state with delay and lag to within 2.1 × 10⁻⁴ and 1.4 × 10⁻⁴, and the hub delay is checked against an independent reference implementation. Within this model, no uncompensated configuration synchronises (largest ensemble-mean coherence 0.021, at the finite-size random-phase scale). With compensation, nearest-neighbour wiring forms ordered neighbourhoods but little global coherence (0.14–0.18 across numerical checks at 100 ppm and K = 3 K_c, against 0.82 for all-to-all coupling); one shortcut per unit raises it to 0.62 ± 0.13, and the hub reaches the all-to-all value. At the declared operating floor K = 1.15 K_c, ensemble means for S1 and S2 stay at or below 0.09 against an ideal 0.36. One hub case near threshold, first read as loss of coherence, reaches the theory value only in a run four times longer; the first reading is withdrawn. The results are numerical and model-bound; no hardware has been built or measured. This record contains the preprint (11 pages, 7 figures, 4 tables) and an archive with the simulation code, raw run records, verification scripts and outputs, the script that regenerates every number, table and figure from the records, and the LaTeX source. The manuscript's data-availability statement ("on request") predates this deposit; the material is now public here. The author is an independent undergraduate researcher; this work is not affiliated with or funded by any university or institution. It has not been peer reviewed.