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
14002	12	1.0000	120	31.554880202003	4.1925672268e-02
14003	12	1.0000	121	31.818294723665	4.1920155233e-02
14004	12	1.0000	122	32.081636588309	4.1940789108e-02
14005	12	1.0000	123	32.345710277037	4.2152347540e-02
14006	12	1.0000	124	32.612295966452	4.2964414552e-02
14007	12	1.0000	125	32.931328235494	1.0253111457e+00
14008	12	1.0000	126	33.096724252044	5.3667860672e-02
14009	12	1.0000	127	33.378998240218	4.2666862323e-02
14010	12	1.0000	128	33.644829895329	4.2076913059e-02
14011	12	1.0000	129	33.908444297297	4.1883295082e-02
14012	12	1.0000	130	34.171260365918	4.1818653911e-02
14013	12	1.0000	131	34.433971631349	4.1772138051e-02
14014	12	1.0000	132	34.696300057501	4.1800420301e-02
14015	12	1.0000	133	34.958706293152	4.1690470322e-02
14016	12	1.0000	134	35.220912284504	4.1764712687e-02
14017	12	1.0000	135	35.482925219131	4.1696793068e-02
14018	12	1.0000	136	35.745155999410	4.1729771466e-02
14019	12	1.0000	137	36.007257545989	4.1747364735e-02
14020	12	1.0000	138	36.269211789287	4.1683124900e-02
14021	12	1.0000	139	36.531516870725	4.1739930194e-02
14022	12	1.0000	140	36.793471609041	4.1751475281e-02
14023	12	1.0000	141	37.055638671387	4.1651761547e-02
14024	12	1.0000	142	37.317655978625	4.1805169111e-02
14025	12	1.0000	143	37.585920300675	1.0416116366e+00
14026	12	1.0000	144	37.841616884955	4.1744172124e-02
14027	12	1.0000	145	38.103561750901	4.1709717150e-02
14028	12	1.0000	146	38.365997171373	4.1781707746e-02
14029	12	1.0000	147	38.628175994808	4.1718892045e-02
14030	12	1.0000	148	38.890286600933	4.1710231362e-02
14031	12	1.0000	149	39.152461142482	4.1769520876e-02
14032	12	1.0000	150	39.414831666346	4.1749999226e-02
14033	12	1.0000	151	39.677303554067	4.1788387053e-02
14034	12	1.0000	152	39.940000981750	4.1864172571e-02
14035	12	1.0000	153	40.203418329849	4.2031006360e-02
14036	12	1.0000	154	40.468641513867	4.2529446887e-02
14037	12	1.0000	155	40.746992549534	5.0651395639e-02
14038	12	1.0000	156	40.922021400810	1.0287690608e+00
14039	12	1.0000	157	41.236951385404	4.2836236714e-02
14040	12	1.0000	158	41.503015745468	4.2043215275e-02
14041	12	1.0000	159	41.766433706512	4.1921948181e-02
14042	12	1.0000	160	42.029408177762	4.1777468516e-02
14043	12	1.0000	161	42.291874590825	4.1826320236e-02
14044	12	1.0000	162	42.554187014961	4.1741933649e-02
14045	12	1.0000	163	42.816851787302	4.1805976151e-02
14046	12	1.0000	164	43.079269865539	4.1836276630e-02
14047	12	1.0000	165	43.327617939906	1.0413239096e+00
14048	12	1.0000	166	43.602203752804	4.1895789938e-02
14049	12	1.0000	167	43.864642288410	4.1720471584e-02
14050	12	1.0000	168	44.126930727303	4.1785575442e-02
14051	12	1.0000	169	44.388898963754	4.1653986392e-02
14052	12	1.0000	170	44.651086167867	4.1793907404e-02
14053	12	1.0000	171	44.913158337908	4.1715701602e-02
14054	12	1.0000	172	45.175495001847	4.1759160092e-02
14055	12	1.0000	173	45.437362477099	4.1738800928e-02
14056	12	1.0000	174	45.699909212188	4.1730275602e-02
14057	12	1.0000	175	45.961991494152	4.1776718319e-02
14058	12	1.0000	176	46.224210265088	4.1701278507e-02
14059	12	1.0000	177	46.486354354784	4.1868825800e-02
14060	12	1.0000	178	46.749352579772	4.1769645411e-02
14061	12	1.0000	179	47.011720712154	4.1867097472e-02
14062	12	1.0000	180	47.274961283391	4.1889775914e-02
14063	12	1.0000	181	47.539439569698	4.2366927297e-02
14064	12	1.0000	182	47.811153061519	4.6143449653e-02
14065	12	1.0000	183	48.007628702365	1.0345847078e+00
14066	12	1.0000	184	48.314425302385	4.2764265017e-02
14067	12	1.0000	185	48.580178827982	4.2025498253e-02
14068	12	1.0000	186	48.844947597536	4.2275289409e-02
14069	12	1.0000	187	49.110521590043	4.2425953035e-02
14070	12	1.0000	188	49.379609927796	4.3637862047e-02
14071	12	1.0000	189	49.709867056189	1.9535319072e-01
14072	12	1.0000	190	49.786493919539	8.8050162646e-01
14073	12	1.0000	191	50.132655358709	4.3625780173e-02
14074	12	1.0000	192	50.401773433004	4.2350103345e-02
14075	12	1.0000	193	50.666868747870	4.2132576779e-02
14076	12	1.0000	194	50.930684121631	4.1865830075e-02
14077	12	1.0000	195	51.194139601824	4.1987346805e-02
14078	12	1.0000	196	51.456670517681	4.1810738346e-02
14079	12	1.0000	197	51.719665337178	4.1883683362e-02
14080	12	1.0000	198	51.982744520375	4.1919352193e-02
14081	12	1.0000	199	52.246398005900	4.1991651711e-02
14082	12	1.0000	200	52.510878178031	4.2430945296e-02
14083	12	1.0000	201	52.785249527020	4.7652605018e-02
14084	12	1.0000	202	52.973105631713	1.0325605160e+00
14085	12	1.0000	203	53.282714582725	4.2668756758e-02
14086	12	1.0000	204	53.548260545931	4.2098839374e-02
14087	12	1.0000	205	53.811701922306	4.1912614292e-02
14088	12	1.0000	206	54.074709593228	4.1719864305e-02
14089	12	1.0000	207	54.336754378689	4.1870429788e-02
14090	12	1.0000	208	54.599563733885	4.1782777280e-02
14091	12	1.0000	209	54.861504040157	4.1770670645e-02
14092	12	1.0000	210	55.123782251504	4.1717670028e-02
14093	12	1.0000	211	55.385733276956	4.1733890822e-02
14094	12	1.0000	212	55.648299020281	4.1834821157e-02
14095	12	1.0000	213	55.910455897613	4.1785990912e-02
14096	12	1.0000	214	56.172734612233	4.1872740293e-02
14097	12	1.0000	215	56.445777811123	1.0413787314e+00
14098	12	1.0000	216	56.696004263944	4.1921780363e-02
14099	12	1.0000	217	56.958116373796	4.1684580001e-02
14100	12	1.0000	218	57.220661753840	4.1755760384e-02
14101	12	1.0000	219	57.482750859563	4.1811279297e-02
14102	12	1.0000	220	57.744967175305	4.1665163850e-02
14103	12	1.0000	221	58.006952603022	4.1751630891e-02
14104	12	1.0000	222	58.269067883137	4.1986598890e-02
14105	12	1.0000	223	58.532034488557	4.1649933019e-02
14106	12	1.0000	224	58.794076694647	4.1913716265e-02
14107	12	1.0000	225	59.057593097964	4.2052955373e-02
14108	12	1.0000	226	59.345823242896	1.0401316467e+00
14109	12	1.0000	227	59.575723331416	4.2549163304e-02
14110	12	1.0000	228	59.840423221897	4.2043813564e-02
14111	12	1.0000	229	60.104191967469	4.2006191956e-02
14112	12	1.0000	230	60.369106954428	4.2360000970e-02
14113	12	1.0000	231	60.641269055549	4.6749464663e-02
14114	12	1.0000	232	60.834361784995	1.0337885159e+00
14115	12	1.0000	233	61.141844691037	4.2664224201e-02
14116	12	1.0000	234	61.407048126679	4.2047645308e-02
14117	12	1.0000	235	61.670251018883	4.1898717472e-02
14118	12	1.0000	236	61.933074106523	4.1877892555e-02
14119	12	1.0000	237	62.195402414738	4.1848385943e-02
14120	12	1.0000	238	62.458237896682	4.1688869164e-02
14121	12	1.0000	239	62.719951504803	4.1821453591e-02
14122	12	1.0000	240	62.982301908866	4.1693608768e-02
14123	12	1.0000	241	63.245023576147	4.1833655276e-02
14124	12	1.0000	242	63.506883279247	4.1746134943e-02
14125	12	1.0000	243	63.769762080383	4.1940627391e-02
14126	12	1.0000	244	64.032087146302	4.1859989661e-02
14127	12	1.0000	245	64.293759169385	4.1949890218e-02
14128	12	1.0000	246	64.557288571074	4.1909497822e-02
14129	12	1.0000	247	64.820802615835	4.2155652243e-02
14130	12	1.0000	248	65.111161977390	1.0396114771e+00
14131	12	1.0000	249	65.337396344794	4.2750482560e-02
14132	12	1.0000	250	65.602741208121	4.2095676694e-02
14133	12	1.0000	251	65.865944207016	4.2171892681e-02
14134	12	1.0000	252	66.130096745408	4.1908504403e-02
14135	12	1.0000	253	66.394546176958	4.2482912051e-02
14136	12	1.0000	254	66.662046653645	4.3132044938e-02
14137	12	1.0000	255	66.966487705684	8.2865336378e-02
14138	12	1.0000	256	67.086281904859	9.9440190867e-01
14139	12	1.0000	257	67.420746759172	4.3168865356e-02
14140	12	1.0000	258	67.688514398493	4.2498298230e-02
14141	12	1.0000	259	67.953618007626	4.2100588846e-02
14142	12	1.0000	260	68.216575301971	4.1940546557e-02
14143	12	1.0000	261	68.479288281872	4.1954959858e-02
14144	12	1.0000	262	68.742331904163	4.1853061782e-02
14145	12	1.0000	263	69.005099599018	4.1997287229e-02
14146	12	1.0000	264	69.268011382310	4.2040531137e-02
14147	12	1.0000	265	69.545724069099	1.0411628422e+00
14148	12	1.0000	266	69.790611011428	4.1962703939e-02
14149	12	1.0000	267	70.053778482866	4.1881480017e-02
14150	12	1.0000	268	70.316587734587	4.1900123324e-02
14151	12	1.0000	269	70.579892766833	4.1788995872e-02
14152	12	1.0000	270	70.841211690470	4.2192476919e-02
14153	12	1.0000	271	71.105424389437	4.2030505630e-02
14154	12	1.0000	272	71.369912576061	4.2384816864e-02
14155	12	1.0000	273	71.635737973317	4.3016551217e-02
14156	12	1.0000	274	71.928739950203	6.3905848393e-02
14157	12	1.0000	275	72.072019043543	1.0143294482e+00
14158	12	1.0000	276	72.399428674980	4.3227198676e-02
14159	12	1.0000	277	72.665425985507	4.2447847564e-02
14160	12	1.0000	278	72.930988971286	4.2079066321e-02
14161	12	1.0000	279	73.193686184779	4.1933683470e-02
14162	12	1.0000	280	73.456998172959	4.1850629182e-02
14163	12	1.0000	281	73.719687950883	4.1856980263e-02
14164	12	1.0000	282	73.982808238178	4.2112290596e-02
14165	12	1.0000	283	74.245810365327	4.1796604186e-02
14166	12	1.0000	284	74.509466436944	4.2282937745e-02
14167	12	1.0000	285	74.772576397595	4.2110165035e-02
14168	12	1.0000	286	75.037622667893	4.2257402436e-02
14169	12	1.0000	287	75.306770903626	4.3833962509e-02
14170	12	1.0000	288	75.633261024810	1.6536537016e-01
14171	12	1.0000	289	75.715945690084	9.1073928192e-01
14172	12	1.0000	290	76.060706413697	4.3628259548e-02
14173	12	1.0000	291	76.329540341303	4.2860284934e-02
14174	12	1.0000	292	76.595802943679	4.2079670756e-02
14175	12	1.0000	293	76.859418063555	4.2290577146e-02
14176	12	1.0000	294	77.143855377501	1.0408561587e+00
14177	12	1.0000	295	77.382371444417	4.2757144463e-02
14178	12	1.0000	296	77.646040352991	4.2007040088e-02
14179	12	1.0000	297	77.910258356404	4.2048353916e-02
14180	12	1.0000	298	78.173240810904	4.2018640506e-02
14181	12	1.0000	299	78.437165453883	4.2064406213e-02
14182	12	1.0000	300	78.703191089694	4.2781375846e-02
14183	12	1.0000	301	78.971842773449	4.3838568015e-02
14184	12	1.0000	302	79.320256962558	7.8116293361e-01
14185	12	1.0000	303	79.385544166531	2.9452219132e-01
14186	12	1.0000	304	79.723979008461	4.3599073277e-02
14187	12	1.0000	305	79.993086899726	4.2796437250e-02
14188	12	1.0000	306	80.258619710967	4.1979960870e-02
14189	12	1.0000	307	80.521018935142	4.2461807927e-02
14190	12	1.0000	308	80.785417441175	4.1859496123e-02
14191	12	1.0000	309	81.047621637860	4.1881005206e-02
14192	12	1.0000	310	81.310224018488	4.2448096821e-02
14193	12	1.0000	311	81.572992686526	4.1734126282e-02
14194	12	1.0000	312	81.835839252496	4.1922324940e-02
14195	12	1.0000	313	82.098553041249	4.1869664490e-02
14196	12	1.0000	314	82.360591921071	4.1627658592e-02
14197	12	1.0000	315	82.624756139284	4.2905584226e-02
14198	12	1.0000	316	82.909303504481	1.0403969907e+00
14199	12	1.0000	317	83.143614775630	4.2409971427e-02
14200	12	1.0000	318	83.407673237860	4.1985884393e-02
14201	12	1.0000	319	83.670211628343	4.2019277871e-02
14202	12	1.0000	320	83.933080618283	4.2042082997e-02
14203	12	1.0000	321	84.194243427047	4.2102658755e-02
14204	12	1.0000	322	84.457447677794	4.2033961965e-02
14205	12	1.0000	323	84.734853447875	1.0414512603e+00
14206	12	1.0000	324	84.979790650468	4.1756337996e-02
14207	12	1.0000	325	85.243574614541	4.2013970704e-02
14208	12	1.0000	326	85.505731786499	4.1790766630e-02
14209	12	1.0000	327	85.769197350903	4.2155107535e-02
14210	12	1.0000	328	86.031828659600	4.1917643400e-02
14211	12	1.0000	329	86.295032335203	4.1977855647e-02
14212	12	1.0000	330	86.557752199231	4.2742936879e-02
14213	12	1.0000	331	86.823139299568	4.2140675104e-02
14214	12	1.0000	332	87.090701585546	4.3429888731e-02
14215	12	1.0000	333	87.420081740543	1.0109573697e+00
14216	12	1.0000	334	87.559501372310	6.7517496083e-02
14217	12	1.0000	335	87.853741124171	4.3855379319e-02
14218	12	1.0000	336	88.123635160426	4.3363751454e-02
14219	12	1.0000	337	88.393541478326	4.3715277445e-02
14220	12	1.0000	338	88.703759392075	9.2506600461e-02
14221	12	1.0000	339	88.816242582357	9.8459902791e-01
14222	12	1.0000	340	89.153954940160	4.3951104436e-02
14223	12	1.0000	341	89.423031484350	4.2280805155e-02
14224	12	1.0000	342	89.687237422757	4.2164144497e-02
14225	12	1.0000	343	89.950173950117	4.2733173490e-02
14226	12	1.0000	344	90.214880226947	4.1930898594e-02
14227	12	1.0000	345	90.476831966616	4.2171758838e-02
14228	12	1.0000	346	90.741191099704	4.1953297904e-02
14229	12	1.0000	347	91.003346838168	4.1818964969e-02
14230	12	1.0000	348	91.266439065908	4.3089047476e-02
14231	12	1.0000	349	91.530423235523	4.2120443808e-02
14232	12	1.0000	350	91.794630845186	4.2391429559e-02
14233	12	1.0000	351	92.062426979810	4.3012125933e-02
14234	12	1.0000	352	92.355626037395	6.5940882159e-02
14235	12	1.0000	353	92.497014127160	1.0125805510e+00
14236	12	1.0000	354	92.822581894913	4.3641814598e-02
14237	12	1.0000	355	93.090817353011	4.2457048101e-02
14238	12	1.0000	356	93.355680101012	4.2276687247e-02
14239	12	1.0000	357	93.619249213204	4.1807323149e-02
14240	12	1.0000	358	93.883985448209	4.2497336054e-02
14241	12	1.0000	359	94.145719848625	4.1834926878e-02
14242	12	1.0000	360	94.412471561173	4.3060097926e-02
14243	12	1.0000	361	94.652150195305	1.0414109332e+00
14244	12	1.0000	362	94.934500593571	4.2362520619e-02
14245	12	1.0000	363	95.200022395022	4.2536374296e-02
14246	12	1.0000	364	95.470650125224	4.4426230831e-02
14247	12	1.0000	365	95.789525322895	1.3824931231e-01
14248	12	1.0000	366	95.880534596935	9.3864176339e-01
14249	12	1.0000	367	96.223516584175	4.3836005038e-02
14250	12	1.0000	368	96.492866940371	4.2362980502e-02
14251	12	1.0000	369	96.757326878538	4.2337252214e-02
14252	12	1.0000	370	97.021507436049	4.1871622423e-02
14253	12	1.0000	371	97.285104176025	4.2206802901e-02
14254	12	1.0000	372	97.545171575926	4.2867827713e-02
14255	12	1.0000	373	97.810527566672	4.1832779387e-02
14256	12	1.0000	374	98.073491739068	4.2082702625e-02
14257	12	1.0000	375	98.336877720916	4.2268430131e-02
14258	12	1.0000	376	98.600231303961	4.3516024527e-02
14259	12	1.0000	377	98.832372197080	1.0404851387e+00
14260	12	1.0000	378	99.118178026203	4.2029271983e-02
14261	12	1.0000	379	99.383944916250	4.2278869738e-02
14262	12	1.0000	380	99.646380904092	4.1917254194e-02
14263	12	1.0000	381	99.910189075734	4.3290004904e-02
14264	12	1.0000	382	100.172140318594	4.2185814488e-02
14265	12	1.0000	383	100.434860336703	4.3818876143e-02
14266	12	1.0000	384	100.701052122939	4.2563549387e-02
14267	12	1.0000	385	100.971134676867	4.3658034735e-02
14268	12	1.0000	386	101.309772398322	9.7009786224e-01
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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": [
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    ],
    "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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      "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())