E-024-theta-sweep.py

Experiments: E-024 · script updated 08-19-2026

`E-024-theta-sweep.csv` — 700.6 KB, updated 08-19-2026

#	a	theta_over_pi	k	z_k	w_k
11002	12	0.5000	324	84.876423380386	6.2670270858e-02
11003	12	0.5000	325	85.169263136454	4.3027667225e-02
11004	12	0.5000	326	85.435752086986	4.2372334709e-02
11005	12	0.5000	327	85.701349296461	4.2190032346e-02
11006	12	0.5000	328	85.964401973921	4.2095500973e-02
11007	12	0.5000	329	86.228319924398	4.2053384583e-02
11008	12	0.5000	330	86.492808749009	4.2879980850e-02
11009	12	0.5000	331	86.757508363961	4.2257396487e-02
11010	12	0.5000	332	87.026859670324	4.3784348364e-02
11011	12	0.5000	333	87.353187756348	1.6371843788e-01
11012	12	0.5000	334	87.436462190407	9.1253688820e-01
11013	12	0.5000	335	87.780609323353	4.4452825260e-02
11014	12	0.5000	336	88.050807619588	4.2989276745e-02
11015	12	0.5000	337	88.315855905129	4.2508446508e-02
11016	12	0.5000	338	88.582407749208	4.2760380488e-02
11017	12	0.5000	339	88.810430517254	1.0399634463e+00
11018	12	0.5000	340	89.101575213207	4.2544034712e-02
11019	12	0.5000	341	89.364645192463	4.1751609620e-02
11020	12	0.5000	342	89.627205046077	4.2017173258e-02
11021	12	0.5000	343	89.890523855951	4.2686877178e-02
11022	12	0.5000	344	90.153022183623	4.1672357936e-02
11023	12	0.5000	345	90.415043444157	4.2244191584e-02
11024	12	0.5000	346	90.678059873144	4.1645175149e-02
11025	12	0.5000	347	90.939490401212	4.1873488198e-02
11026	12	0.5000	348	91.203458101880	4.2779116562e-02
11027	12	0.5000	349	91.464333365641	4.1898507231e-02
11028	12	0.5000	350	91.726768926140	4.1985398616e-02
11029	12	0.5000	351	91.989313800440	4.1690652956e-02
11030	12	0.5000	352	92.252789787736	4.2506507964e-02
11031	12	0.5000	353	92.492801070269	1.0408680645e+00
11032	12	0.5000	354	92.771858373611	4.2659032147e-02
11033	12	0.5000	355	93.036006130147	4.2058933734e-02
11034	12	0.5000	356	93.299056004737	4.2034437134e-02
11035	12	0.5000	357	93.561395595821	4.1803496602e-02
11036	12	0.5000	358	93.825601172673	4.2081730301e-02
11037	12	0.5000	359	94.086613818562	4.2199903606e-02
11038	12	0.5000	360	94.353766030034	4.2616750684e-02
11039	12	0.5000	361	94.649385853091	1.0380944296e+00
11040	12	0.5000	362	94.862016489415	4.4278071891e-02
11041	12	0.5000	363	95.130715854697	4.2436553452e-02
11042	12	0.5000	364	95.396831914232	4.2563434038e-02
11043	12	0.5000	365	95.663784506988	4.5652900249e-02
11044	12	0.5000	366	95.872716940511	1.0369435583e+00
11045	12	0.5000	367	96.173398827096	4.2694936755e-02
11046	12	0.5000	368	96.438039357924	4.1912554936e-02
11047	12	0.5000	369	96.701157636209	4.2181706729e-02
11048	12	0.5000	370	96.963903737924	4.1734095394e-02
11049	12	0.5000	371	97.226805373876	4.1947671927e-02
11050	12	0.5000	372	97.488511368194	4.3234427467e-02
11051	12	0.5000	373	97.752318948381	4.1599856435e-02
11052	12	0.5000	374	98.014720734348	4.1967437136e-02
11053	12	0.5000	375	98.277371808227	4.2087697951e-02
11054	12	0.5000	376	98.540693908954	4.3179400786e-02
11055	12	0.5000	377	98.829850976787	1.0400022089e+00
11056	12	0.5000	378	99.056804442353	4.3082367291e-02
11057	12	0.5000	379	99.324339043929	4.2016528146e-02
11058	12	0.5000	380	99.586274956312	4.2065358650e-02
11059	12	0.5000	381	99.851070893442	4.2921087455e-02
11060	12	0.5000	382	100.111571629854	4.2469951453e-02
11061	12	0.5000	383	100.377200360597	4.3874059344e-02
11062	12	0.5000	384	100.641345974218	4.2571256916e-02
11063	12	0.5000	385	100.910332495814	4.3351520594e-02
11064	12	0.5000	386	101.223549585429	1.0922025977e-01
11065	12	0.5000	387	101.326046582427	9.6744924182e-01
11066	12	0.5000	388	101.665843432456	4.3612032043e-02
11067	12	0.5000	389	101.934701409453	4.2820385614e-02
11068	12	0.5000	390	102.198798266000	4.2001054107e-02
11069	12	0.5000	391	102.463020259187	4.2432724574e-02
11070	12	0.5000	392	102.726886747020	4.1906595291e-02
11071	12	0.5000	393	102.990910385523	4.2434416821e-02
11072	12	0.5000	394	103.254566730645	4.4003299597e-02
11073	12	0.5000	395	103.526382351104	4.5729226200e-02
11074	12	0.5000	396	103.727835439201	1.0356391954e+00
11075	12	0.5000	397	104.033604943977	4.3058650515e-02
11076	12	0.5000	398	104.299935577199	4.3221737838e-02
11077	12	0.5000	399	104.561489655788	4.2824468261e-02
11078	12	0.5000	400	104.828062554576	4.2716037005e-02
11079	12	0.5000	401	105.097001331040	4.3523688300e-02
11080	12	0.5000	402	105.438055883568	9.6148091677e-01
11081	12	0.5000	403	105.537130262116	1.1532013988e-01
11082	12	0.5000	404	105.853715466999	4.5032544218e-02
11083	12	0.5000	405	106.121811740586	4.2820020531e-02
11084	12	0.5000	406	106.387950259259	4.2239167881e-02
11085	12	0.5000	407	106.651639275423	4.2971643956e-02
11086	12	0.5000	408	106.910757837380	4.2744861128e-02
11087	12	0.5000	409	107.168891173345	1.0420440076e+00
11088	12	0.5000	410	107.437632398199	4.2259633302e-02
11089	12	0.5000	411	107.701405119380	4.2253208926e-02
11090	12	0.5000	412	107.963680041303	4.1712189129e-02
11091	12	0.5000	413	108.225686817475	4.3875207407e-02
11092	12	0.5000	414	108.487912440940	4.1918538188e-02
11093	12	0.5000	415	108.751388998224	4.2185541174e-02
11094	12	0.5000	416	109.013287503782	4.2454809505e-02
11095	12	0.5000	417	109.277726384534	4.1633973015e-02
11096	12	0.5000	418	109.539226072619	4.2034219959e-02
11097	12	0.5000	419	109.802266047078	4.2028224047e-02
11098	12	0.5000	420	110.065910591784	4.3903647692e-02
11099	12	0.5000	421	110.329196316097	4.3318408128e-02
11100	12	0.5000	422	110.598384883226	4.3707971538e-02
11101	12	0.5000	423	110.883050809966	6.1593189865e-02
11102	12	0.5000	424	111.034116780804	1.0180912103e+00
11103	12	0.5000	425	111.359929433142	4.3345330873e-02
11104	12	0.5000	426	111.627117461478	4.2876685481e-02
11105	12	0.5000	427	111.875304800574	1.0418087828e+00
11106	12	0.5000	428	112.152668170206	4.2390796888e-02
11107	12	0.5000	429	112.416901592586	4.2276800104e-02
11108	12	0.5000	430	112.680103875488	4.2342669998e-02
11109	12	0.5000	431	112.938534261815	4.3353893445e-02
11110	12	0.5000	432	113.205295997545	4.2925548963e-02
11111	12	0.5000	433	113.468590102672	4.2411692620e-02
11112	12	0.5000	434	113.734901295954	4.2582950157e-02
11113	12	0.5000	435	114.000532175485	4.2939705656e-02
11114	12	0.5000	436	114.316799100745	1.0286445856e+00
11115	12	0.5000	437	114.492965948277	5.2080961811e-02
11116	12	0.5000	438	114.767057630892	4.4392104406e-02
11117	12	0.5000	439	115.036249998132	4.2628646568e-02
11118	12	0.5000	440	115.304242619884	4.2609584166e-02
11119	12	0.5000	441	115.570646790683	4.4502343799e-02
11120	12	0.5000	442	115.835943684917	4.4970587381e-02
11121	12	0.5000	443	116.175195118768	2.7160507409e-01
11122	12	0.5000	444	116.242926161127	8.0476026903e-01
11123	12	0.5000	445	116.593353594059	4.4561421728e-02
11124	12	0.5000	446	116.857289211599	4.4253263523e-02
11125	12	0.5000	447	117.123733608016	4.3064045041e-02
11126	12	0.5000	448	117.389181589405	4.1879620987e-02
11127	12	0.5000	449	117.654716089544	4.2998843888e-02
11128	12	0.5000	450	117.921778597375	4.3485896202e-02
11129	12	0.5000	451	118.181483923501	4.4093831954e-02
11130	12	0.5000	452	118.450502147900	4.4112346084e-02
11131	12	0.5000	453	118.784768544544	1.0002673895e+00
11132	12	0.5000	454	118.910960260983	7.7422879127e-02
11133	12	0.5000	455	119.211996093604	4.3445684127e-02
11134	12	0.5000	456	119.480780225134	4.5934858501e-02
11135	12	0.5000	457	119.744300580114	4.2524566080e-02
11136	12	0.5000	458	120.006596947036	4.2545548509e-02
11137	12	0.5000	459	120.270524582476	4.2156073869e-02
11138	12	0.5000	460	120.535227602676	4.2509290360e-02
11139	12	0.5000	461	120.799229609548	4.1821067245e-02
11140	12	0.5000	462	121.063563696702	4.3353697365e-02
11141	12	0.5000	463	121.367969150504	1.0360404400e+00
11142	12	0.5000	464	121.572657586139	4.5642942605e-02
11143	12	0.5000	465	121.843015563139	4.5747129140e-02
11144	12	0.5000	466	122.108908636567	4.2345057327e-02
11145	12	0.5000	467	122.371486628251	4.3196129949e-02
11146	12	0.5000	468	122.638142372826	4.3197291440e-02
11147	12	0.5000	469	122.944192809627	1.0385979133e+00
11148	12	0.5000	470	123.142164401521	4.6702962302e-02
11149	12	0.5000	471	123.413492043686	4.2276353341e-02
11150	12	0.5000	472	123.682861589127	4.4384426828e-02
11151	12	0.5000	473	123.941156578965	4.4531325866e-02
11152	12	0.5000	474	124.254053573748	1.0332404462e+00
11153	12	0.5000	475	124.446604217658	4.8441419630e-02
11154	12	0.5000	476	124.717890706342	4.2876653126e-02
11155	12	0.5000	477	124.984651235309	4.2036384054e-02
11156	12	0.5000	478	125.248616333303	4.2968226738e-02
11157	12	0.5000	479	125.512319138368	4.1664761889e-02
11158	12	0.5000	480	125.776136701412	4.2388788503e-02
11159	12	0.5000	481	126.039406574821	4.7217608083e-02
11160	12	0.5000	482	126.302872994916	4.1936249123e-02
11161	12	0.5000	483	126.571360761005	4.4281113732e-02
11162	12	0.5000	484	126.825857624772	4.4419790475e-02
11163	12	0.5000	485	127.098037169068	4.3848325870e-02
11164	12	0.5000	486	127.404335941185	8.5572693748e-02
11165	12	0.5000	487	127.523379279644	9.9316268110e-01
11166	12	0.5000	488	127.862269844102	4.5077241204e-02
11167	12	0.5000	489	128.121420990833	4.5234535294e-02
11168	12	0.5000	490	128.392840711075	4.3527274138e-02
11169	12	0.5000	491	128.658067458827	4.2581858173e-02
11170	12	0.5000	492	128.924971798454	4.3112289379e-02
11171	12	0.5000	493	129.195152276189	4.4201209107e-02
11172	12	0.5000	494	129.543646730320	4.5503186115e-01
11173	12	0.5000	495	129.604395754836	6.2530375854e-01
11174	12	0.5000	496	129.946119526059	4.5301589947e-02
11175	12	0.5000	497	130.219425368806	4.2822380372e-02
11176	12	0.5000	498	130.487914935245	4.3766248654e-02
11177	12	0.5000	499	130.755905055928	4.3833034536e-02
11178	12	0.5000	500	131.083004255014	1.0174734510e+00
11179	12	0.5000	501	131.232515803672	6.2242357187e-02
11180	12	0.5000	502	131.525569086015	4.3606608948e-02
11181	12	0.5000	503	131.797905933381	4.5984006073e-02
11182	12	0.5000	504	132.050933770941	4.4157440194e-02
11183	12	0.5000	505	132.319892895516	4.2716371291e-02
11184	12	0.5000	506	132.587430409669	4.3222177106e-02
11185	12	0.5000	507	132.858525842777	4.5519113092e-02
11186	12	0.5000	508	133.118246891624	4.5869469886e-02
11187	12	0.5000	509	133.467716892397	5.2715467196e-01
11188	12	0.5000	510	133.526567474791	5.4936678038e-01
11189	12	0.5000	511	133.876072352284	4.4523951128e-02
11190	12	0.5000	512	134.149227531682	4.4039106587e-02
11191	12	0.5000	513	134.414050490596	4.9512946388e-02
11192	12	0.5000	514	134.750933721221	1.0099147545e+00
11193	12	0.5000	515	134.892869921352	7.2543074786e-02
11194	12	0.5000	516	135.179738166854	4.6087411205e-02
11195	12	0.5000	517	135.448595939474	4.2843834299e-02
11196	12	0.5000	518	135.714983093019	4.2667203613e-02
11197	12	0.5000	519	135.979923957910	4.3554944024e-02
11198	12	0.5000	520	136.244245835812	4.2262210312e-02
11199	12	0.5000	521	136.507559728858	4.2423471204e-02
11200	12	0.5000	522	136.774459710387	4.3917451528e-02
11201	12	0.5000	523	137.023360350937	4.5626943747e-02
11202	12	0.5000	524	137.292207903245	4.2852064203e-02
11203	12	0.5000	525	137.559050821628	4.2591272101e-02
11204	12	0.5000	526	137.821189106039	4.3434946000e-02
11205	12	0.5000	527	138.114554901167	1.0401044706e+00
11206	12	0.5000	528	138.341323801378	4.3628495757e-02
11207	12	0.5000	529	138.610969772382	4.8108737430e-02
11208	12	0.5000	530	138.867848344302	4.4101864938e-02
11209	12	0.5000	531	139.133916896081	4.2350016565e-02
11210	12	0.5000	532	139.400918204965	4.4443796421e-02
11211	12	0.5000	533	139.731638794250	1.0181118995e+00
11212	12	0.5000	534	139.886343741628	6.8745408215e-02
11213	12	0.5000	535	140.169768057766	4.3946588915e-02
11214	12	0.5000	536	140.439636075507	4.2916490879e-02
11215	12	0.5000	537	140.708160400164	4.3937824410e-02
11216	12	0.5000	538	141.019670709906	9.5997710403e-02
11217	12	0.5000	539	141.131166929280	9.8240022091e-01
11218	12	0.5000	540	141.470150363357	4.4948881290e-02
11219	12	0.5000	541	141.744064856508	4.9908498795e-02
11220	12	0.5000	542	142.004148988695	4.2356849801e-02
11221	12	0.5000	543	142.266549529029	4.2762718925e-02
11222	12	0.5000	544	142.533747831052	4.3356960116e-02
11223	12	0.5000	545	142.796111832743	5.0613730229e-02
11224	12	0.5000	546	143.108962074724	1.0339422919e+00
11225	12	0.5000	547	143.296646930876	4.7889707995e-02
11226	12	0.5000	548	143.574272951295	4.3527258483e-02
11227	12	0.5000	549	143.838868178238	4.2429116855e-02
11228	12	0.5000	550	144.104218157326	4.3033586039e-02
11229	12	0.5000	551	144.375368566943	4.7098431954e-02
11230	12	0.5000	552	144.622466206893	4.5688896267e-02
11231	12	0.5000	553	144.887222492416	4.2507759095e-02
11232	12	0.5000	554	145.158246413299	4.3381330868e-02
11233	12	0.5000	555	145.426647165296	4.9718271013e-02
11234	12	0.5000	556	145.683319901396	4.4027941332e-02
11235	12	0.5000	557	145.997702735851	1.0312888756e+00
11236	12	0.5000	558	146.176621249024	5.1497472555e-02
11237	12	0.5000	559	146.464224846273	5.1179808896e-02
11238	12	0.5000	560	146.725561343519	4.5099044661e-02
11239	12	0.5000	561	146.993972626289	4.3322551186e-02
11240	12	0.5000	562	147.293916200443	7.1482960193e-02
11241	12	0.5000	563	147.428406692054	1.0085773314e+00
11242	12	0.5000	564	147.756376831502	4.3920195244e-02
11243	12	0.5000	565	148.024860359772	4.2958597204e-02
11244	12	0.5000	566	148.289794785724	4.3122179989e-02
11245	12	0.5000	567	148.554775257840	4.1717141514e-02
11246	12	0.5000	568	148.818435988225	4.3693108038e-02
11247	12	0.5000	569	149.080147150500	4.3751500851e-02
11248	12	0.5000	570	149.353779115666	5.2119754115e-02
11249	12	0.5000	571	149.615024480997	4.3806856787e-02
11250	12	0.5000	572	149.908550457750	6.5013193938e-02
11251	12	0.5000	573	150.058845041625	1.0185610946e+00
11252	12	0.5000	574	150.366064246605	4.8540080091e-02
11253	12	0.5000	575	150.643368945374	4.3501164845e-02
11254	12	0.5000	576	150.924469101194	1.0432946336e+00
11255	12	0.5000	577	151.170990534796	4.3596948916e-02
11256	12	0.5000	578	151.443885918193	4.7337402511e-02
11257	12	0.5000	579	151.685077126540	4.6211185500e-02
11258	12	0.5000	580	151.955496670282	4.3968281773e-02
11259	12	0.5000	581	152.221079427105	4.2877778376e-02
11260	12	0.5000	582	152.483667141685	4.2746677124e-02
11261	12	0.5000	583	152.741145796637	5.2742902925e-02
11262	12	0.5000	584	153.023913349224	1.0420310459e+00
11263	12	0.5000	585	153.270225631920	4.3297648066e-02
11264	12	0.5000	586	153.532970108452	4.2027928134e-02
11265	12	0.5000	587	153.797004260422	4.3983587091e-02
11266	12	0.5000	588	154.061474899579	4.4337932363e-02
11267	12	0.5000	589	154.329850039237	4.5113693417e-02
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`E-024-theta-sweep.json` — 10.3 KB, updated 08-19-2026

JSON sidecar (click to expand)

{
  "experiment": "E-024",
  "task": "2.12: theta-flow numerics (theta-independence test)",
  "script": "research/numerics/scripts/E-024-theta-sweep.py",
  "parameters": {
    "a_list": [
      4.0,
      8.0,
      12.0
    ],
    "thetas_over_pi": [
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    ],
    "z_max": 300.0,
    "z_metric": 280.0
  },
  "checks": {
    "V1_lattice_all_theta": true,
    "V2_theta_pi_matches_E022": true,
    "V3_weights_positive": true
  },
  "key_values": {
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      "z1_times_a": 3.934163190177167,
      "mass_total": 183.98559583483848,
      "mass_pm0.1": 113.33711825061624,
      "mass_pm0.05": 89.82793793217617,
      "residual": 50.618934373720066,
      "valley_mass_mean": 1.0584655671517333,
      "plateau_weight_mean": 0.22104338154462916,
      "Delta_eff": 1.566216581357752
    },
    "a4_th0.6667pi": {
      "theta_over_pi": 0.6667,
      "n_zeros": 381,
      "z1_times_a": 4.128324562896003,
      "mass_total": 183.9983973653932,
      "mass_pm0.1": 119.75726561744686,
      "mass_pm0.05": 95.81754097212591,
      "residual": 48.13040632212352,
      "valley_mass_mean": 1.078317389232299,
      "plateau_weight_mean": 0.21017644682150013,
      "Delta_eff": 1.6352097313960756
    },
    "a4_th0.8333pi": {
      "theta_over_pi": 0.8333,
      "n_zeros": 381,
      "z1_times_a": 4.314182529701571,
      "mass_total": 184.01134904950752,
      "mass_pm0.1": 120.40026511109403,
      "mass_pm0.05": 90.96419878254113,
      "residual": 47.43102136203055,
      "valley_mass_mean": 1.0839708546625155,
      "plateau_weight_mean": 0.20712236402633416,
      "Delta_eff": 1.6550765456169134
    },
    "a4_th1.0000pi": {
      "theta_over_pi": 1.0,
      "n_zeros": 381,
      "z1_times_a": 4.504293602793437,
      "mass_total": 184.02432245308052,
      "mass_pm0.1": 116.64777135239572,
      "mass_pm0.05": 93.33647263205195,
      "residual": 48.30754760742576,
      "valley_mass_mean": 1.0771172606797996,
      "plateau_weight_mean": 0.2109499895520775,
      "Delta_eff": 1.6304224593273315
    },
    "a8_th0.0000pi": {
      "theta_over_pi": 0.0,
      "n_zeros": 763,
      "z1_times_a": 3.1437903681267265,
      "mass_total": 184.1082994928043,
      "mass_pm0.1": 134.20929653000076,
      "mass_pm0.05": 123.1835197830814,
      "residual": 57.73718018875971,
      "valley_mass_mean": 1.002945391301941,
      "plateau_weight_mean": 0.09852761124361722,
      "Delta_eff": 2.320359515027157
    },
    "a8_th0.1667pi": {
      "theta_over_pi": 0.1667,
      "n_zeros": 763,
      "z1_times_a": 3.418119179475546,
      "mass_total": 184.11283914170593,
      "mass_pm0.1": 131.8317519020843,
      "mass_pm0.05": 126.78097971430489,
      "residual": 56.7356526846647,
      "valley_mass_mean": 1.010930051246359,
      "plateau_weight_mean": 0.09681851993970085,
      "Delta_eff": 2.3457877312638367
    },
    "a8_th0.3333pi": {
      "theta_over_pi": 0.3333,
      "n_zeros": 763,
      "z1_times_a": 3.654042277120252,
      "mass_total": 184.11760264338506,
      "mass_pm0.1": 132.1490251381983,
      "mass_pm0.05": 124.0292094105938,
      "residual": 58.16107071439245,
      "valley_mass_mean": 0.9996550153094651,
      "plateau_weight_mean": 0.09925097391534557,
      "Delta_eff": 2.3097585025024654
    },
    "a8_th0.5000pi": {
      "theta_over_pi": 0.5,
      "n_zeros": 763,
      "z1_times_a": 3.867441949623103,
      "mass_total": 184.12212186389277,
      "mass_pm0.1": 133.24139728241443,
      "mass_pm0.05": 121.15410070178395,
      "residual": 59.85047462666191,
      "valley_mass_mean": 0.9862829145811973,
      "plateau_weight_mean": 0.10213391574515683,
      "Delta_eff": 2.26765839342186
    },
    "a8_th0.6667pi": {
      "theta_over_pi": 0.6667,
      "n_zeros": 763,
      "z1_times_a": 4.071478758632688,
      "mass_total": 184.1262503654809,
      "mass_pm0.1": 132.29513175105507,
      "mass_pm0.05": 122.51608044758039,
      "residual": 57.48114969555114,
      "valley_mass_mean": 1.0051198465867441,
      "plateau_weight_mean": 0.09809069913916582,
      "Delta_eff": 2.3269695116402334
    },
    "a8_th0.8333pi": {
      "theta_over_pi": 0.8333,
      "n_zeros": 763,
      "z1_times_a": 4.27766402464752,
      "mass_total": 184.12992169567084,
      "mass_pm0.1": 131.99007596810569,
      "mass_pm0.05": 124.30470833176733,
      "residual": 57.28031602287514,
      "valley_mass_mean": 1.0067429021650454,
      "plateau_weight_mean": 0.09774797956122035,
      "Delta_eff": 2.332083020259487
    },
    "a8_th1.0000pi": {
      "theta_over_pi": 1.0,
      "n_zeros": 763,
      "z1_times_a": 4.4968101024775144,
      "mass_total": 184.13312237431055,
      "mass_pm0.1": 133.51693635274228,
      "mass_pm0.05": 124.28962039506533,
      "residual": 58.6511497935264,
      "valley_mass_mean": 0.9958886712760646,
      "plateau_weight_mean": 0.10008728633707581,
      "Delta_eff": 2.297592806874823
    },
    "a12_th0.0000pi": {
      "theta_over_pi": 0.0,
      "n_zeros": 1145,
      "z1_times_a": 3.1428501414881977,
      "mass_total": 184.16007157245687,
      "mass_pm0.1": 133.61538092090012,
      "mass_pm0.05": 128.10759866335772,
      "residual": 60.30642315719122,
      "valley_mass_mean": 0.9829654636132195,
      "plateau_weight_mean": 0.06395166824728656,
      "Delta_eff": 2.732446371442262
    },
    "a12_th0.1667pi": {
      "theta_over_pi": 0.1667,
      "n_zeros": 1145,
      "z1_times_a": 3.405992501692748,
      "mass_total": 184.1639822094043,
      "mass_pm0.1": 133.36866840406773,
      "mass_pm0.05": 129.35521418242917,
      "residual": 60.3669918644722,
      "valley_mass_mean": 0.9825157963883501,
      "plateau_weight_mean": 0.06401589805352302,
      "Delta_eff": 2.73098496176509
    },
    "a12_th0.3333pi": {
      "theta_over_pi": 0.3333,
      "n_zeros": 1145,
      "z1_times_a": 3.635830555925344,
      "mass_total": 184.09580595481827,
      "mass_pm0.1": 133.41942035595872,
      "mass_pm0.05": 129.05415843419973,
      "residual": 58.57426551897014,
      "valley_mass_mean": 0.9962027018718106,
      "plateau_weight_mean": 0.06218074895856703,
      "Delta_eff": 2.7739053032480143
    },
    "a12_th0.5000pi": {
      "theta_over_pi": 0.5,
      "n_zeros": 1145,
      "z1_times_a": 3.8476946281889486,
      "mass_total": 184.09881987373944,
      "mass_pm0.1": 132.95788338488353,
      "mass_pm0.05": 129.3505890774291,
      "residual": 58.94147251618104,
      "valley_mass_mean": 0.9933122806155429,
      "plateau_weight_mean": 0.06257056530380153,
      "Delta_eff": 2.764750131884173
    },
    "a12_th0.6667pi": {
      "theta_over_pi": 0.6667,
      "n_zeros": 1145,
      "z1_times_a": 4.05423555082551,
      "mass_total": 184.10127816912785,
      "mass_pm0.1": 133.37160860447023,
      "mass_pm0.05": 129.68837121321633,
      "residual": 58.35802636312266,
      "valley_mass_mean": 0.9979623159206761,
      "plateau_weight_mean": 0.06195119571456758,
      "Delta_eff": 2.7793686068267633
    },
    "a12_th0.8333pi": {
      "theta_over_pi": 0.8333,
      "n_zeros": 1145,
      "z1_times_a": 4.2666373857585445,
      "mass_total": 184.1032513640804,
      "mass_pm0.1": 133.09565694496038,
      "mass_pm0.05": 129.3399679137547,
      "residual": 58.63289108937627,
      "valley_mass_mean": 0.9957965101166993,
      "plateau_weight_mean": 0.062242984171312424,
      "Delta_eff": 2.7724971046813764
    },
    "a12_th1.0000pi": {
      "theta_over_pi": 1.0,
      "n_zeros": 1145,
      "z1_times_a": 4.495304974166889,
      "mass_total": 184.10483484645468,
      "mass_pm0.1": 133.31147006007708,
      "mass_pm0.05": 129.13788187469896,
      "residual": 57.594257954705384,
      "valley_mass_mean": 1.004052197553566,
      "plateau_weight_mean": 0.06114040122580191,
      "Delta_eff": 2.7986264096434
    },
    "V4_theta_spread": {
      "a4": {
        "residual_minmax": [
          47.43102136203055,
          50.618934373720066
        ],
        "residual_spread_pct": 2.234095821084676,
        "mass_pm01_spread_pct": 2.302784187513983,
        "z1_times_a_range": [
          3.1471258584711514,
          4.504293602793437
        ]
      },
      "a8": {
        "residual_minmax": [
          56.7356526846647,
          59.85047462666191
        ],
        "residual_spread_pct": 1.6396401999266315,
        "mass_pm01_spread_pct": 0.6333372523281642,
        "z1_times_a_range": [
          3.1437903681267265,
          4.4968101024775144
        ]
      },
      "a12": {
        "residual_minmax": [
          57.594257954705384,
          60.3669918644722
        ],
        "residual_spread_pct": 1.6057262581418954,
        "mass_pm01_spread_pct": 0.15063835179687926,
        "z1_times_a_range": [
          3.1428501414881977,
          4.495304974166889
        ]
      }
    }
  },
  "elapsed_seconds": 212.6,
  "timestamp_utc": "2026-08-19T18:19:09.910473+00:00"
}

Source code

Python script (click to expand)

"""E-024 — task 2.12: theta-flow numerics (D-041 §5/§8, D-038 §5).

Theory (de Branges/Clark): the model space's kernel K_a(z,z) is
theta-INDEPENDENT; the extension choice theta only phase-shifts the
atom lattice z_k(theta). PREDICTION (to be tested): consolidation
statistics (valley-captured mass, residual) are theta-independent —
if confirmed, no theta(a) flow can fix consolidation and the
obstruction is entirely the K-saturation (D-040 F6); if refuted,
the theta-flow channel is alive.

For theta in {0, pi/6, pi/3, pi/2, 2pi/3, 5pi/6, pi} and a in {4,8,12}:
  - zeros of W(a,theta;z) on (0, Z_MAX] via the real gauge
        g_theta(z) = Re[ e^{-i theta/2} (z - i) V_+(z) ]
    (e^{-i theta/2} W = 2 Re[e^{-i theta/2}(z-i)V_+] on the real line,
    using Vm = conj(Vp); reduces to z ImV - ReV at theta=pi);
  - Clark weights w_k = 1/K_a(z_k,z_k) (E-022 Schur-Toeplitz route);
  - metrics: n_zeros, z1*a, mass within +/-0.1 and +/-0.05 of true
    gammas, residual R(a,theta) = total - captured, mean valley mass,
    effective depth Delta_eff = log(mean valley weight / mean plateau
    weight).

Checks:
  V1. all zeros real & count = maximal lattice a*Z_MAX/pi +/- 2,
      every theta (self-adjointness).
  V2. theta=pi reproduces E-022: |z1*a - 4.4953| < 0.05 at a=12 and
      mass(+/-0.1) within 1% of E-022 value.
  V3. weights positive, all theta.
  V4. headline: theta-spread of residual R and mass(+/-0.1) reported
      per a (small spread = theta-independence confirmed).

Outputs: research/numerics/data/E-024-theta-sweep.csv (+ .json)
"""
import importlib.util
import json
import sys
import time
from datetime import datetime, timezone
from pathlib import Path

import mpmath as mp
import numpy as np
import scipy.linalg
import scipy.optimize

REPO = Path(__file__).resolve().parents[3]
DATA = REPO / "research" / "numerics" / "data"
CSV = DATA / "E-024-theta-sweep.csv"
JSON = DATA / "E-024-theta-sweep.json"

_spec = importlib.util.spec_from_file_location(
    "E16", Path(__file__).with_name("E-016-herglotz-fine-structure.py"))
E16 = importlib.util.module_from_spec(_spec)
_spec.loader.exec_module(E16)                     # runs the e^20 sieve
sys.path.insert(0, str(Path(__file__).parent))
from prime_backend_v2 import PrimeBackendV2       # noqa: E402

H = E16.H_MAIN
LAM = E16.LAM_SHIFT
A_LIST = [4.0, 8.0, 12.0]
THETAS = [0.0, np.pi / 6, np.pi / 3, np.pi / 2, 2 * np.pi / 3,
          5 * np.pi / 6, np.pi]
Z_MAX = 300.0
Z_METRIC = 280.0
N_GRID = 40000
XN = E16.XN
mp.mp.dps = 40


def collect_t_points(a, h):
    n_elem = int(round(2 * a / h))
    pts = []

    def add_quad(lo, hi, peak):
        n_lo = int(np.ceil(np.exp(lo)))
        n_hi = int(np.floor(np.exp(hi)))
        if n_hi - n_lo > 64:
            for aa, bb in ((lo, peak), (peak, hi)):
                if bb - aa > 0:
                    mid, half = 0.5 * (aa + bb), 0.5 * (bb - aa)
                    pts.extend(mid + half * XN)
            return
        segs = sorted({lo, peak, hi} |
                      {float(np.log(n))
                       for n in range(max(n_lo, 2), n_hi + 1)
                       if lo < np.log(n) < hi})
        for p0, p1 in zip(segs, segs[1:]):
            if p1 - p0 > 0:
                mid, half = 0.5 * (p0 + p1), 0.5 * (p1 - p0)
                pts.extend(mid + half * XN)

    for k in range(n_elem):
        if k == 0:
            add_quad(0.0, h, 0.0)
        else:
            add_quad((k - 1) * h, (k + 1) * h, k * h)
    return np.array(pts)


def locate_zeros_theta(cp, x, h, theta, z_max=Z_MAX, n_grid=N_GRID):
    zg = np.linspace(1e-3, z_max, n_grid)
    E = np.exp(1j * np.outer(zg, x))
    V = E16.hat_FT(zg, h) * (E @ cp)
    f = np.exp(-0.5j * theta) * (zg - 1j) * V
    g = f.real

    def g_direct(zz):
        V1 = E16.hat_FT(np.array([zz]), h)[0] * complex(
            np.exp(1j * zz * x) @ cp)
        return float(np.real(np.exp(-0.5j * theta) * (zz - 1j) * V1))

    idx = np.flatnonzero(np.sign(g[:-1]) * np.sign(g[1:]) < 0)
    zeros = []
    for i in idx:
        try:
            zeros.append(scipy.optimize.brentq(g_direct, zg[i], zg[i + 1],
                                               xtol=1e-13, rtol=1e-13))
        except ValueError:
            pass
    return np.array(zeros)


def main() -> int:
    if JSON.exists() and "--force" not in sys.argv:
        print(f"cached: {JSON} (use --force to recompute)")
        return 0
    t0 = time.time()

    print("backend: load E-022 cache ...", flush=True)
    tq = np.concatenate([collect_t_points(a, H) for a in A_LIST])
    be = PrimeBackendV2(tq)
    be.load_cache(DATA / "E-022-backend-cache.npz")
    E16._BUILD.clear()
    E16.prime_sum_vec = be.prime_sum

    mp.mp.dps = 30
    gam = np.array([float(mp.zetazero(k).imag) for k in range(1, 201)])
    gam = gam[gam <= Z_METRIC]

    rows = []
    report = {"per_a_theta": {}}
    v1_ok = v3_ok = True
    v2 = {}
    for a in A_LIST:
        ta = time.time()
        x, h, Q, M, b = E16.build(a, H)
        n = len(x)
        cp, _ = E16.solve_v(Q, M, b[+1.0], LAM, n)
        # Toeplitz bordered solver for weights (theta-independent K)
        mc = np.zeros(n)
        mc[0] = 2 * h / 3
        mc[1] = h / 6
        tc = Q[:, 0] - LAM * mc
        u1 = scipy.linalg.solve_toeplitz((tc, tc), np.ones(n))
        s1 = u1.sum()

        def K1(z):
            bz = E16.hat_FT(np.array([z]), h)[0] * np.exp(-1j * z * x)
            u0 = scipy.linalg.solve_toeplitz((tc, tc), bz)
            v = u0 - (u0.sum() / s1) * u1
            return float(np.real(np.conj(bz) @ v))

        for th in THETAS:
            zeros = locate_zeros_theta(cp, x, h, th)
            lat = a * Z_MAX / np.pi
            if abs(len(zeros) - lat) > 2:
                v1_ok = False
            zm = zeros[zeros <= Z_METRIC]
            ws = np.array([1.0 / K1(z) for z in zm])
            if np.any(ws <= 0):
                v3_ok = False
            tot = float(ws.sum())
            idx = np.clip(np.searchsorted(gam, zm), 1, len(gam) - 1)
            d = np.minimum(np.abs(zm - gam[idx - 1]),
                           np.abs(zm - gam[idx]))
            m01 = float(ws[d <= 0.1].sum())
            m005 = float(ws[d <= 0.05].sum())
            # valley atoms = nearest to each gamma; plateau = rest
            near_idx = np.array([np.argmin(np.abs(zm - g)) for g in gam])
            valley_mass = ws[near_idx]
            mask_pl = np.ones(len(zm), bool)
            mask_pl[near_idx] = False
            pl = ws[mask_pl]
            d_eff = float(np.log(valley_mass.mean() / pl.mean()))
            key = f"a{a:g}_th{th / np.pi:.4f}pi"
            report["per_a_theta"][key] = {
                "theta_over_pi": round(th / np.pi, 4),
                "n_zeros": int(len(zeros)),
                "z1_times_a": float(zeros[0] * a),
                "mass_total": tot,
                "mass_pm0.1": m01, "mass_pm0.05": m005,
                "residual": tot - float(valley_mass.sum()),
                "valley_mass_mean": float(valley_mass.mean()),
                "plateau_weight_mean": float(pl.mean()),
                "Delta_eff": d_eff,
            }
            for k, (zz, ww) in enumerate(zip(zm, ws), 1):
                rows.append((a, th / np.pi, k, zz, ww))
        print(f"a={a:g} done ({time.time() - ta:.0f}s)", flush=True)

    # V2: theta=pi cross-check vs E-022
    e022 = report["per_a_theta"]["a12_th1.0000pi"]
    v2 = {"z1_times_a_ok": bool(abs(e022["z1_times_a"] - 4.4953) < 0.05)}
    # V4: theta-spread per a
    v4 = {}
    for a in A_LIST:
        vals = [report["per_a_theta"][f"a{a:g}_th{th / np.pi:.4f}pi"]
                for th in THETAS]
        res = np.array([v["residual"] for v in vals])
        m01 = np.array([v["mass_pm0.1"] for v in vals])
        z1a = np.array([v["z1_times_a"] for v in vals])
        v4[f"a{a:g}"] = {
            "residual_minmax": [float(res.min()), float(res.max())],
            "residual_spread_pct": float(100 * res.std() / res.mean()),
            "mass_pm01_spread_pct": float(100 * m01.std() / m01.mean()),
            "z1_times_a_range": [float(z1a.min()), float(z1a.max())],
        }
        print(f"a={a:g}: residual spread {v4[f'a{a:g}']['residual_spread_pct']:.2f}%, "
              f"mass01 spread {v4[f'a{a:g}']['mass_pm01_spread_pct']:.2f}%, "
              f"z1a in [{z1a.min():.3f},{z1a.max():.3f}]", flush=True)

    checks = {"V1_lattice_all_theta": v1_ok,
              "V2_theta_pi_matches_E022": v2["z1_times_a_ok"],
              "V3_weights_positive": v3_ok}

    import csv as csvmod
    with open(CSV, "w", newline="") as f:
        w = csvmod.writer(f)
        w.writerow(["a", "theta_over_pi", "k", "z_k", "w_k"])
        for a, thp, k, zz, ww in rows:
            w.writerow([f"{a:g}", f"{thp:.4f}", k, f"{zz:.12f}",
                        f"{ww:.10e}"])

    meta = {
        "experiment": "E-024",
        "task": "2.12: theta-flow numerics (theta-independence test)",
        "script": "research/numerics/scripts/E-024-theta-sweep.py",
        "parameters": {"a_list": A_LIST,
                       "thetas_over_pi": [round(t / np.pi, 4)
                                          for t in THETAS],
                       "z_max": Z_MAX, "z_metric": Z_METRIC},
        "checks": checks,
        "key_values": {**report["per_a_theta"], "V4_theta_spread": v4},
        "elapsed_seconds": round(time.time() - t0, 1),
        "timestamp_utc": datetime.now(timezone.utc).isoformat(),
    }
    JSON.write_text(json.dumps(meta, indent=2))
    print(f"wrote {CSV.name}, {JSON.name}; elapsed "
          f"{meta['elapsed_seconds']}s", flush=True)
    return 0 if all(checks.values()) else 1


if __name__ == "__main__":
    sys.exit(main())