Multi-language Kuramoto tier benchmark

Per-call P50 latency (microseconds) of every registered Kuramoto compute primitive across the Rust, Julia, and Python tiers, measured by scripts/bench_kuramoto_tiers.py. The canonical column is the fixed CI runner (Rust engine built from source); the local column is the GOTM workstation. Both are reproducible evidence, not published latency claims — hosted-runner CPU differs from the workstation. A — marks a tier not measured in that run (excluded subset or unavailable binding); see the per-environment artefacts under docs/benchmarks/tiers/ for the full percentile set and parity data.

Provenance

Field	CI	Local
CPU model	AMD EPYC 7763 64-Core Processor	11th Gen Intel(R) Core(TM) i5-11600K @ 3.90GHz
Platform	Linux-6.17.0-1018-azure-x86_64-with-glibc2.39	Linux-6.17.0-35-generic-x86_64-with-glibc2.39
Commit	f9249f83c025d043bf84145ed1d6d79d6ec6fab2	352547b92161c3c28d4d6116b5ed3ecaea786a9d
Python	3.12.13	3.12.3
NumPy	2.2.6	2.2.6
Rust engine	installed	installed
juliacall	0.9.35	0.9.31
rustc	rustc 1.96.0 (ac68faa20 2026-05-25)	rustc 1.96.0 (ac68faa20 2026-05-25)

Runs

Environment	Generated (UTC)	Parameters
CI	2026-06-26T15:47:00Z	tiers=[julia,python,rust] sizes=[8,32,128,512] warmup=2 repeats=5
Local	2026-06-24T23:56:12Z	tiers=[python,rust] sizes=[8,32,128,512] warmup=2 repeats=5

Competitive Baseline Framing

This section is the Phase 5.5 claim boundary for the tier benchmark. It separates measured in-repository tier competition from unmeasured external package comparisons and production-latency claims.

Surface	Current evidence	Claim boundary
Internal tier competition	CI measures 140 primitive-size rows; fastest counts: rust=108, julia=29, python=3.	Supports dispatch-order and drift-detection decisions inside this package.
Python floor baseline	Same-algorithm Python floor is measured in CI; fastest non-Python tiers show 3.90x median over 137 CI rows against that floor.	This is not a claim of superiority over third-party Kuramoto, ODE, graph, or differentiable-solver packages.
External package baselines	Not measured by this artefact.	No external competitive claim is allowed until a separate harness records package names, versions, equations, tolerances, hardware, and raw artefacts.
Production latency	Every committed tier artefact sets production_claim_allowed: false.	No SLA, universal hardware, or customer deployment latency claim follows from this page.

Per-primitive P50 latency (µs)

Operation	N	Rust (CI)	Rust (local)	Julia (CI)	Python (CI)	Python (local)	Fastest (CI)	Parity (CI)
daido_mean_field_force	8	1.683	0.981	14.386	16.872	10.851	rust	1.11e-16
daido_mean_field_force	32	2.073	1.413	15.239	18.031	11.452	rust	8.33e-17
daido_mean_field_force	128	4.975	3.342	17.436	22.161	16.331	rust	4.16e-17
daido_mean_field_force	512	16.955	10.514	27.718	40.573	24.997	rust	5.55e-17
daido_mean_field_jacobian	8	3.657	1.505	14.912	25.624	15.579	rust	2.22e-16
daido_mean_field_jacobian	32	13.206	8.186	22.991	40.651	25.124	rust	1.67e-16
daido_mean_field_jacobian	128	189.100	178.621	148.591	257.792	224.765	julia	8.33e-17
daido_mean_field_jacobian	512	4549.686	3970.465	4132.213	4945.642	4079.830	julia	9.71e-17
daido_mode_phase	8	1.030	0.625	11.315	10.287	6.670	rust	1.11e-16
daido_mode_phase	32	1.767	0.836	11.669	10.530	7.043	rust	0.00e+00
daido_mode_phase	128	2.704	1.812	13.077	13.031	9.364	rust	1.11e-15
daido_mode_phase	512	8.555	5.699	17.450	21.738	14.019	rust	4.44e-16
daido_mode_phase_gradient	8	1.335	0.863	14.941	16.950	10.904	rust	2.22e-16
daido_mode_phase_gradient	32	2.103	1.420	14.965	18.386	11.674	rust	3.33e-16
daido_mode_phase_gradient	128	4.950	3.260	17.309	22.752	15.715	rust	9.99e-16
daido_mode_phase_gradient	512	16.719	11.560	28.753	41.224	27.368	rust	4.86e-17
daido_mode_phase_hessian	8	2.913	1.174	14.515	34.842	21.122	rust	3.33e-16
daido_mode_phase_hessian	32	5.390	2.268	17.523	40.331	25.917	rust	6.66e-16
daido_mode_phase_hessian	128	13.791	10.571	75.351	95.044	76.121	rust	4.00e-15
daido_mode_phase_hessian	512	153.538	157.005	2530.514	832.890	1072.336	rust	9.71e-17
daido_order_parameter	8	1.208	0.942	11.039	7.503	5.314	rust	0.00e+00
daido_order_parameter	32	1.259	0.929	11.315	8.176	5.639	rust	1.39e-17
daido_order_parameter	128	2.697	1.792	12.423	10.520	7.394	rust	0.00e+00
daido_order_parameter	512	8.706	5.699	16.513	20.333	14.110	rust	1.39e-17
daido_order_parameter_gradient	8	1.332	0.914	14.259	18.556	11.812	rust	2.78e-17
daido_order_parameter_gradient	32	2.117	1.410	15.075	19.693	12.213	rust	1.39e-17
daido_order_parameter_gradient	128	4.995	3.218	17.597	23.867	14.689	rust	1.50e-17
daido_order_parameter_gradient	512	16.604	10.595	28.056	41.926	35.873	rust	2.60e-18
daido_order_parameter_hessian	8	2.786	1.532	14.416	29.452	17.627	rust	5.55e-17
daido_order_parameter_hessian	32	4.420	1.731	17.101	33.572	21.021	rust	5.55e-17
daido_order_parameter_hessian	128	9.649	7.364	69.608	72.258	53.201	rust	4.86e-17
daido_order_parameter_hessian	512	117.453	170.593	2530.195	582.438	635.933	rust	5.20e-18
kuramoto_euler_trajectory	8	44.103	26.781	64.049	518.758	329.749	rust	4.44e-16
kuramoto_euler_trajectory	32	674.097	437.308	545.357	1325.482	864.166	julia	4.44e-16
kuramoto_euler_trajectory	128	11629.453	9203.602	8839.670	14046.759	10976.205	julia	1.33e-15
kuramoto_euler_trajectory	512	245285.731	204884.303	185975.370	272385.100	242535.154	julia	1.38e-14
kuramoto_euler_vjp	8	90.048	89.415	97.462	1237.979	1068.443	rust	4.44e-15
kuramoto_euler_vjp	32	1528.351	1119.381	1062.422	2982.642	2261.333	julia	9.95e-14
kuramoto_euler_vjp	128	28012.724	23078.208	19304.252	30590.731	28149.560	julia	1.17e-12
kuramoto_euler_vjp	512	563958.844	489273.842	445186.589	567717.732	569460.991	julia	4.80e-10
kuramoto_interaction_energy	8	1.941	1.315	17.478	8.087	6.936	rust	4.44e-16
kuramoto_interaction_energy	32	11.606	7.028	24.424	19.747	12.082	rust	3.44e-15
kuramoto_interaction_energy	128	169.783	158.591	138.626	208.441	164.070	julia	3.55e-15
kuramoto_interaction_energy	512	3621.024	3114.691	2700.310	4028.653	3521.759	julia	6.34e-12
kuramoto_interaction_energy_gradient	8	3.933	2.017	20.232	10.631	7.428	rust	5.55e-17
kuramoto_interaction_energy_gradient	32	11.961	9.263	28.356	24.207	16.715	rust	8.88e-16
kuramoto_interaction_energy_gradient	128	170.532	115.183	150.760	232.399	171.820	julia	8.88e-15
kuramoto_interaction_energy_gradient	512	3739.369	3504.444	2808.568	5031.270	4133.745	julia	3.02e-14
kuramoto_interaction_energy_hessian	8	4.162	1.631	21.259	13.141	8.298	rust	2.22e-16
kuramoto_interaction_energy_hessian	32	14.629	8.720	29.538	28.821	16.748	rust	8.88e-16
kuramoto_interaction_energy_hessian	128	228.605	172.773	171.623	269.153	221.272	julia	5.33e-15
kuramoto_interaction_energy_hessian	512	4932.920	4182.478	3341.814	5355.621	5058.333	julia	2.13e-14
kuramoto_rk4_trajectory	8	166.080	96.755	160.209	2288.617	1852.538	julia	1.39e-17
kuramoto_rk4_trajectory	32	2639.882	1621.924	1989.028	5468.826	3292.183	julia	8.88e-16
kuramoto_rk4_trajectory	128	43863.286	35132.240	33117.973	53739.508	45412.649	julia	2.00e-15
kuramoto_rk4_trajectory	512	990456.318	841726.505	775927.352	1089860.615	1072889.518	julia	4.31e-14
kuramoto_rk4_vjp	8	472.872	279.688	2174.882	6825.638	4760.977	rust	8.88e-16
kuramoto_rk4_vjp	32	7682.094	5590.547	27383.004	16080.444	11174.701	rust	6.66e-16
kuramoto_rk4_vjp	128	140636.165	116512.887	424203.501	161272.117	143605.952	rust	5.86e-14
kuramoto_rk4_vjp	512	2860566.657	2544101.383	7219909.881	3037485.262	2957652.493	rust	4.19e-09
local_mean_phase	8	2.067	1.427	20.262	9.030	5.538	rust	4.44e-16
local_mean_phase	32	3.809	2.594	22.043	10.545	6.119	rust	4.44e-16
local_mean_phase	128	23.456	18.652	41.296	18.674	13.889	python	5.33e-15
local_mean_phase	512	276.424	264.788	297.123	119.686	79.737	python	4.44e-15
local_mean_phase_jacobian	8	1.983	2.808	20.871	17.974	22.424	rust	4.44e-16
local_mean_phase_jacobian	32	4.112	5.272	23.511	23.671	30.011	rust	2.11e-15
local_mean_phase_jacobian	128	29.633	32.179	76.960	77.797	67.555	rust	1.19e-14
local_mean_phase_jacobian	512	400.209	380.471	2892.268	822.568	1032.108	rust	3.77e-15
local_order_parameter	8	1.916	1.252	19.828	10.888	7.118	rust	0.00e+00
local_order_parameter	32	3.939	2.314	21.669	12.937	8.456	rust	5.55e-17
local_order_parameter	128	19.993	15.932	38.286	24.519	17.008	rust	8.33e-17
local_order_parameter	512	262.964	242.960	280.346	178.904	139.451	python	8.33e-17
local_order_parameter_jacobian	8	2.705	2.695	20.544	20.043	13.266	rust	1.11e-16
local_order_parameter_jacobian	32	4.306	3.043	25.435	26.722	17.244	rust	4.16e-17
local_order_parameter_jacobian	128	29.451	24.665	79.093	85.316	75.314	rust	9.23e-17
local_order_parameter_jacobian	512	407.359	408.834	2820.087	901.474	958.829	rust	2.22e-17
mean_field_force	8	1.245	0.891	14.641	15.551	10.119	rust	2.22e-16
mean_field_force	32	2.124	1.309	15.045	16.563	10.678	rust	5.55e-17
mean_field_force	128	4.552	2.699	17.555	20.950	15.281	rust	2.78e-17
mean_field_force	512	15.466	9.222	27.121	38.469	22.291	rust	5.55e-17
mean_field_jacobian	8	3.209	1.450	14.946	23.129	16.256	rust	1.11e-16
mean_field_jacobian	32	12.372	7.223	22.322	37.532	22.538	rust	1.11e-16
mean_field_jacobian	128	173.826	130.693	135.594	232.760	168.044	julia	2.78e-17
mean_field_jacobian	512	3564.021	3140.961	3626.747	4195.702	3783.719	rust	5.55e-17
mean_phase	8	0.844	0.600	10.916	11.955	8.900	rust	3.12e-17
mean_phase	32	1.417	0.827	11.042	12.612	9.142	rust	0.00e+00
mean_phase	128	2.450	1.623	12.356	14.304	10.131	rust	8.33e-17
mean_phase	512	8.010	4.748	16.647	23.200	16.353	rust	4.44e-16
mean_phase_gradient	8	1.553	0.900	14.126	17.203	11.580	rust	1.11e-16
mean_phase_gradient	32	1.940	1.359	14.674	18.258	11.949	rust	1.11e-16
mean_phase_gradient	128	4.491	3.199	16.848	22.408	13.603	rust	1.11e-16
mean_phase_gradient	512	15.219	9.165	26.873	40.232	23.045	rust	1.73e-17
mean_phase_hessian	8	2.888	1.138	14.139	34.959	21.409	rust	1.11e-16
mean_phase_hessian	32	4.809	2.051	16.368	42.307	25.261	rust	8.33e-17
mean_phase_hessian	128	13.430	9.897	68.012	96.186	77.038	rust	1.25e-16
mean_phase_hessian	512	159.185	125.741	2492.929	812.134	702.525	rust	2.08e-17
networked_kuramoto_force	8	2.268	2.277	20.349	6.878	4.747	rust	1.11e-16
networked_kuramoto_force	32	11.746	7.134	27.132	18.197	11.116	rust	8.88e-16
networked_kuramoto_force	128	167.713	111.525	144.444	191.147	138.955	julia	5.33e-15
networked_kuramoto_force	512	3556.274	3090.903	2692.222	3993.759	3421.909	julia	2.49e-14
networked_kuramoto_jacobian	8	2.379	1.740	20.874	10.481	7.185	rust	2.22e-16
networked_kuramoto_jacobian	32	14.447	9.362	29.431	23.662	14.404	rust	8.88e-16
networked_kuramoto_jacobian	128	223.885	166.728	169.338	222.582	165.645	julia	5.33e-15
networked_kuramoto_jacobian	512	4950.946	3744.613	3208.239	4165.714	3832.477	julia	2.13e-14
order_parameter	8	0.801	0.561	10.937	7.600	5.513	rust	5.55e-17
order_parameter	32	1.129	0.748	10.950	7.994	6.816	rust	2.78e-17
order_parameter	128	2.380	1.516	12.149	10.543	7.403	rust	2.08e-17
order_parameter	512	7.574	5.537	15.804	19.754	13.615	rust	2.08e-17
order_parameter_gradient	8	1.261	0.909	14.396	16.758	10.763	rust	1.39e-17
order_parameter_gradient	32	1.947	1.328	14.908	17.950	11.471	rust	5.20e-18
order_parameter_gradient	128	4.496	3.089	17.238	22.043	15.022	rust	2.60e-18
order_parameter_gradient	512	15.229	12.786	27.357	39.601	24.471	rust	8.67e-19
order_parameter_hessian	8	1.428	0.996	13.642	25.641	16.731	rust	2.78e-17
order_parameter_hessian	32	4.348	1.613	16.226	32.065	19.861	rust	6.94e-18
order_parameter_hessian	128	9.292	6.733	114.014	61.868	50.051	rust	3.47e-18
order_parameter_hessian	512	112.524	124.214	2435.191	461.172	629.773	rust	8.67e-19
sakaguchi_force	8	3.380	1.581	21.395	13.913	5.981	rust	2.22e-16
sakaguchi_force	32	12.692	7.740	27.119	21.963	13.139	rust	8.88e-16
sakaguchi_force	128	193.576	129.766	151.912	208.158	139.121	julia	7.55e-15
sakaguchi_force	512	4402.744	3492.361	2932.638	4096.955	3534.242	julia	2.84e-14
sakaguchi_jacobian	8	4.006	2.607	20.509	11.447	8.620	rust	2.22e-16
sakaguchi_jacobian	32	14.584	9.416	30.179	24.912	16.777	rust	8.88e-16
sakaguchi_jacobian	128	221.988	176.519	167.316	216.512	170.657	julia	4.44e-15
sakaguchi_jacobian	512	4896.611	3813.868	3304.589	4080.802	3802.934	julia	2.58e-14
sakaguchi_mean_field_force	8	1.298	0.918	14.376	22.189	15.898	rust	1.11e-16
sakaguchi_mean_field_force	32	1.976	2.080	14.907	23.052	15.392	rust	6.25e-17
sakaguchi_mean_field_force	128	4.601	2.999	17.438	27.794	16.534	rust	2.78e-17
sakaguchi_mean_field_force	512	15.589	9.419	28.255	46.546	30.452	rust	4.16e-17
sakaguchi_mean_field_jacobian	8	3.521	1.491	14.702	26.712	18.378	rust	1.11e-16
sakaguchi_mean_field_jacobian	32	12.430	7.437	23.219	41.721	24.501	rust	1.11e-16
sakaguchi_mean_field_jacobian	128	173.776	135.615	140.861	234.290	174.131	julia	2.78e-17
sakaguchi_mean_field_jacobian	512	3458.771	3136.367	2880.023	4069.705	4158.522	julia	5.55e-17
triadic_mean_field_force	8	1.514	0.903	14.198	16.497	11.086	rust	8.33e-17
triadic_mean_field_force	32	2.308	1.303	15.217	17.624	11.665	rust	1.73e-17
triadic_mean_field_force	128	4.716	3.114	17.545	21.844	14.228	rust	2.17e-18
triadic_mean_field_force	512	16.035	10.290	27.313	39.899	28.797	rust	4.34e-18
triadic_mean_field_jacobian	8	3.064	2.705	15.463	34.184	32.938	rust	2.78e-16
triadic_mean_field_jacobian	32	17.772	18.336	30.631	58.581	37.492	rust	4.16e-17
triadic_mean_field_jacobian	128	270.705	258.240	277.521	516.023	442.142	rust	4.34e-18
triadic_mean_field_jacobian	512	6336.235	4600.139	6448.571	9538.710	8020.405	rust	7.81e-18