{
  "schema": "bethe-six-spinon-density-precision-comparison-v1",
  "scope": "Identical 2048-draw R256 query with reduced density precision. Precision agreement does not establish sampling or cutoff convergence.",
  "momentum": "1.5707963267948966192313216916397514420985846996875529104874722961539082031431045",
  "geometry_bits": 512,
  "compiler": "GNU 13.3.0 Release",
  "cpu": "Intel(R) Core(TM) i7-12700KF",
  "workers": 4,
  "source_comparison": "Numerical include/ tree unchanged; PR81 exposes the existing independent density precision control.",
  "command": [
    "bethe-xxx-structure-factor-thermo",
    "--sectors",
    "6",
    "--momentum",
    "1.5707963267948966192313216916397514420985846996875529104874722961539082031431045",
    "--omega",
    "2",
    "--exchange",
    "1",
    "--channel",
    "zz",
    "--threads",
    "4",
    "--density-relative-tolerance",
    "1e-6",
    "--density-absolute-tolerance",
    "0",
    "--diagnostics",
    "--replicate-output",
    "--quiet",
    "--precision",
    "bits-512",
    "--six-cutoff",
    "256",
    "--replicates",
    "16",
    "--samples",
    "128",
    "--seed",
    "7304",
    "--density-terms",
    "256",
    "--density-tail-order",
    "24",
    "--six-background-weight",
    "0.5",
    "--six-one-pair-weight",
    "0.25",
    "--six-two-pairs-weight",
    "0.25",
    "--json",
    "point.json",
    "--density-guard-bits",
    "256"
  ],
  "reference": {
    "source": "ab54b536b20f8a1b5646d0f4bc5ae630f523806b",
    "comparison_bits": 448,
    "guard_bits": 512,
    "raw_sha256": "57ce69a42989155e859238d40146af8d1200e5897cc0d38d64fc684a55bf62c7",
    "wall_seconds": 478.090073845,
    "cpu_seconds": 1850.1970390000004
  },
  "reduced_density": {
    "source": "b573fb4bba7d7fee9df6853fb29bc7cb051898da",
    "comparison_bits": 192,
    "guard_bits": 256,
    "raw_sha256": "7cfac4b2855fe7d606e8c52221bc8446234aac0aefc00bf24ff4ecbd4efc39cc",
    "wall_seconds": 297.7280166099954,
    "cpu_seconds": 1149.950324
  },
  "observed_cpu_ratio": 1.6089364908948889,
  "timing_scope": "Complete process, including setup and explicit JSON export; separate runs on the same shared host.",
  "comparison": {
    "all16_replicate_identities_and_counts_match": true,
    "max_relative_replicate_or_max_observation_change": "6.3515566498E-75",
    "relative_mean_change": "1.9035817597E-75",
    "relative_sampling_se_of_reduced_run": "0.154325933507848556796945197472167000116408603254822585724926646414318933080805102007921191251001289320657011570746161095013172275136460942745163752167497362004156484920463577678420981481116501656",
    "raw_sha256": "7cfac4b2855fe7d606e8c52221bc8446234aac0aefc00bf24ff4ecbd4efc39cc",
    "reference_sha256": "57ce69a42989155e859238d40146af8d1200e5897cc0d38d64fc684a55bf62c7",
    "scope": "Density precision agreement only; 16 x 128 draws and R = 256 retain sampling and omitted-domain uncertainty."
  },
  "sampled_point": {
    "schema": "bethe-six-spinon-slice-v1",
    "density_convention": "S(q,w); moments use dw/(2*pi); raising=2*zz",
    "momentum": "1.5707963267948966192313216916397514420985846996875529104874722961539082031431045",
    "exchange": "1",
    "rapidity_cutoff": "256",
    "frequencies": [
      "2"
    ],
    "runs": [
      {
        "raw_sha256": "7cfac4b2855fe7d606e8c52221bc8446234aac0aefc00bf24ff4ecbd4efc39cc",
        "metadata": {
          "Arithmetic format": "arbitrary binary",
          "Bethe revision": "b573fb4bba7d7fee9df6853fb29bc7cb051898da",
          "Bethe version": "0.1.0",
          "Build type": "Release",
          "CPU timing scope": "spectral grid including geometry and ordered output",
          "Channel": "zz",
          "Compiler": "GNU 13.3.0",
          "Date": "2026-10-31T08:49:49Z",
          "Density absolute tolerance": "0",
          "Density comparison bits": "192",
          "Density convention": "S(q,w); moments use dw/(2*pi); raising=2*zz",
          "Density guard bits": "256",
          "Density relative tolerance": "1e-6",
          "Exchange J": "1",
          "Family": "Partial DSF; physical normalization; no fitted rescaling",
          "First density residue budget": "256",
          "First density tail order": "24",
          "Geometry and output bits": "512",
          "Hamiltonian": "H=J sum S_j.S_(j+1); infinite chain; zero field and temperature",
          "Maximum buffered detail rows": "100000",
          "Platform": "Linux x86_64",
          "Precision": "bits-512",
          "Program": "bethe-xxx-structure-factor-thermo",
          "Recovery seconds per configuration": "60",
          "Samples per replicate": "128",
          "Scheduler concurrency limit": "4",
          "Second density residue budget": "272",
          "Second density tail order": "28",
          "Selected spinon sectors": "6",
          "Six-spinon absolute rapidity cutoff": "256",
          "Six-spinon background weight": "0.5",
          "Six-spinon native soft fraction": "0.5",
          "Six-spinon one-pair center scale": "4",
          "Six-spinon one-pair gap scale": "0.5",
          "Six-spinon one-pair weight": "0.25",
          "Six-spinon replicates": "16",
          "Six-spinon seed": "7304",
          "Six-spinon stream identities": "first_replicate=grid_index*replicates; distinct streams at each point",
          "Six-spinon two-pair center scale": "8",
          "Six-spinon two-pair gap scale": "1",
          "Six-spinon two-pair weight": "0.25",
          "Six-spinon uniform grid bits": "52",
          "Spectral points": "1",
          "Two-spinon kernel budget": "200000",
          "Two-spinon maximum quadrature order": "256",
          "Two-spinon transition-integral tolerance": "6.10987272699920936410395878642723539957880548114465332778722781856258325304944091540371447021603045651375030538718093431063570685560856793947026622840183578e-151",
          "Uni20 revision": "1cb2474eb0a54e13ec7d5f2c1823752020b2425c",
          "Working precision (bits)": "512"
        },
        "summary": {
          "CPU time": "1149.718350 s",
          "Elapsed seconds": "297.404603336999997509",
          "Failed points": "0",
          "Outcome": "success",
          "Run CPU seconds": "1149.71934100000000001",
          "Status": "finite_domain_estimate",
          "Threshold points": "0"
        },
        "spectrum": [
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "sector": "6",
            "density": "0.0325215467543371910845868109326425301738585510769893678660869405700870019464050549589189621983077907475733644864710875898925172266586530250310469225308733558",
            "estimated_error": null,
            "evaluations": "1516",
            "status": "estimated"
          }
        ],
        "diagnostics": [
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "sector": "6",
            "two_kernel_error": null,
            "quadrature_difference": null,
            "cutoff_change": null,
            "contour_diagnostic": "3.15278686316025601217471930113138702300664340562103100617729522117544898784870922840706973695580754170903722420979345647727780543177283353237295220138406412e-22",
            "arithmetic_diagnostic": "2.0833814350345905587060564252675079767424691740627968989296278197101277549013014945104777596275825687513835673051410023732132031807094546255937457483932062e-45",
            "sampling_error": "0.0050189180619822293947226612463137089368224913564764900781476585550609543269328178151634024020211583842440554342896387541846989807984716576607636891721627656",
            "evaluations": "1516",
            "reason": "finite-cutoff Monte Carlo estimate; numerical diagnostics exclude sampling and cutoff errors"
          }
        ],
        "replicates": [
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "0",
            "completed_samples": "128",
            "supported_roots": "50",
            "density_attempts": "100",
            "maximum_absolute_density": "3.86702758557554127157971519895103360535089173760087513541890534612007004671447380033333097878402723854804912332726509003436170054889929211306965404696890606",
            "density": "0.0664059171282506816979053521472586096973847373667571229611717241699717993407340310697527071990887839097755659159638895103319367857068565241417290987606860852",
            "contour_diagnostic": "1.09358079630327742971397143158861988606754610002602686423205052135028547237808144969496891306461525021343138676571923858087183119540504815134152130513800587e-23",
            "arithmetic_diagnostic": "1.74772333702542800935198974948374072483571300822932994445287969530201134591634332601399112229710310393461317227718386455718940962376895784802514805852972191e-45",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "1",
            "completed_samples": "128",
            "supported_roots": "54",
            "density_attempts": "108",
            "maximum_absolute_density": "1.05903505499088204677617227299250522704288181724859151519509000953078609506053817069873235990212034423522300867950265347398968743739285936785588270967639306",
            "density": "0.0397275078242109496670550950256629594788301390921139857947456089189854101680633345903702396975839670993078116352328400610125410440530480049042171292908159459",
            "contour_diagnostic": "9.2909656974964854069553819642981683084190915602695328438508060927126078951911466828437098058773436719169504503141649542423543200834898031418637852753087016e-24",
            "arithmetic_diagnostic": "5.10322205319360082999324908860073428857169679720624856323243035649486434355780961962898847881170827557734418592115863250587773102886885849388277051113252132e-46",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "2",
            "completed_samples": "128",
            "supported_roots": "50",
            "density_attempts": "100",
            "maximum_absolute_density": "2.64655090644231142076571736428268724750953108069383406760588286244927951214355948098049601078504241195280836878859696344803124376080581035654859742683330331",
            "density": "0.0361226578819601348311026621325302171652539317437861049546933878241788586410302161728847362935788314862368472973718697711124854273139256037646670506498696679",
            "contour_diagnostic": "3.83773975717939126675375590376306287929543537792043733924024133391746464173575388500072417399518215947432236044989386069725186795099351793143995402599852267e-23",
            "arithmetic_diagnostic": "3.37488468654441976040131123865720480717012053887550429282926530316065563618447106949131566306681043191529467658369147582762994688395337773384972024619426712e-45",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "3",
            "completed_samples": "128",
            "supported_roots": "50",
            "density_attempts": "100",
            "maximum_absolute_density": "0.71712826813733401834821711404894515440165047060100677269084156331880794878984503527767004797003315260777231895228114112124283416189482731944410694975246953",
            "density": "0.0180715904400706917240262479000793974315421658049456826349765460255384488236082525861364997577273513229678943116587749135246460063652938211946566327772932384",
            "contour_diagnostic": "9.79064605924074481611890044279478824545053146993443508259345644788769633331470812556888984171819569861807224797798228723860114788888724345135708492224079929e-30",
            "arithmetic_diagnostic": "2.48343272784272110614705248141319153679739019243060539721917559076662268788142244796934927821298258548112401813871944682455302083385030289536950197649635154e-49",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "4",
            "completed_samples": "128",
            "supported_roots": "50",
            "density_attempts": "100",
            "maximum_absolute_density": "2.71658459702438098978361303548069763520915179937646501503901030734601969104000114889235988802707154050693951339409460056863529022844047184013500133712774552",
            "density": "0.0466041505247110909812136821794980902738954410804307211501124053069696689518229012444802538104432460240156005191508727304043618946631850977528440017064266042",
            "contour_diagnostic": "1.37542963856399113817108374861956396188864646675211617429969795606994526182689190344724250427561457315330131787317455024095535648116837633630562733619647505e-38",
            "arithmetic_diagnostic": "3.02200596625033556137867537888086419048633800362123965000565664256181186344045323435830523424106593034454634521576595774069665456179295244961918479759085368e-53",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "5",
            "completed_samples": "128",
            "supported_roots": "54",
            "density_attempts": "108",
            "maximum_absolute_density": "1.19406051070124247339511289488672464914553462081302639680424883503924849807430542529026509030630218504710636624912311037234832236181095121290300993635137275",
            "density": "0.0158014695004245783336583602849968862480532767216347414485056359365998793142191662685669075034243356997989811502161403774352189186695813884036235751923462037",
            "contour_diagnostic": "8.47100723813466858337239870416511709787625495045446985689705223188101820171157524628982250488664126641301538729640896913002235708617242187461401913000439954e-34",
            "arithmetic_diagnostic": "8.03140445900454892856285688563529765376472057949465516625789918352254290695511461281868040549638824425344269866214216444510828951665419469676418479814273251e-52",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "6",
            "completed_samples": "128",
            "supported_roots": "54",
            "density_attempts": "108",
            "maximum_absolute_density": "2.36039452880001614415479126556510104737317763855657084144888251732057880854841044773929668970491036308527819214912373550216469662688572126274554135801533609",
            "density": "0.0382318493359306192407427947283234350000353144143836302197517837967424906003126768963524309171645557980872552619610202265897867575350365129564309394135768065",
            "contour_diagnostic": "5.33238849561246183637725450367647671692425944880432713790808391129825517639492566825376997066919306107215295076264038048021072968075590071931543862099315331e-29",
            "arithmetic_diagnostic": "1.05518875606114779799172290999343930684336868564060009341342726634469437319086368269255566442062313710743935392103797449324330106526459393720050241027974243e-48",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "7",
            "completed_samples": "128",
            "supported_roots": "52",
            "density_attempts": "104",
            "maximum_absolute_density": "5.39752366240728560020396225257043172280070456908738741267551297636089583574414482787058918461710987433000746651918905471922417920120673835862680209912086421",
            "density": "0.0778397866240846400132242123955039206898581671833253535282667406072286899082608726804769181616869011274016126536494185633697005529663948221799588447405290204",
            "contour_diagnostic": "1.37395792394985934318750458297135871871206084596993926288619758238036804641022753578424559430025690593696717681506051821810365633692368530812515407654112917e-40",
            "arithmetic_diagnostic": "2.73656065235829524897653600483575277920860974074397219208290677555936604685350419235490174941394861420776633749296118831775813223401113529243247906468917202e-53",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "8",
            "completed_samples": "128",
            "supported_roots": "42",
            "density_attempts": "84",
            "maximum_absolute_density": "1.6322335612337989988725709005048465806754106564827615514255936495195640216875032586926117731911109762759310592208845001227177916635928744568555703382657523",
            "density": "0.0245338526464797683413831779054540676903510907974711180281928765477745326277511104292697582371251813828613096088364034576592380875550509813314761651664617579",
            "contour_diagnostic": "2.60677297544123512740098240842204540621462189108994262509120942636737146309859579466309775979905379940812635647155923465055810550139081872330748859636116563e-28",
            "arithmetic_diagnostic": "1.49032693198336988898583675817161638003952378585236835423953061513313428669271135254919978106674879662298937613606236571651429259334688234952202312100043697e-48",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "9",
            "completed_samples": "128",
            "supported_roots": "46",
            "density_attempts": "92",
            "maximum_absolute_density": "1.58732939449934582259415997417976263896782784254965143463320057345390057751855783805245827983030792881164635892761826201141796436400077953328382292235447087",
            "density": "0.0254940603363230676250811284674576802898954360062753645879833547453292239748732879475708113917119118029730614561708096589979903562145847190793935219833994812",
            "contour_diagnostic": "1.33584038770623821445099524037079560725189556963683673245368545555560592271364393945337286547566134221233320447108655853637900295887791465909211535201497867e-26",
            "arithmetic_diagnostic": "3.94797235573101258872317126375167472835997695919473936428198383904229549258949669072462844945198980541899131810851223742994579005426535952879813409220647639e-47",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "10",
            "completed_samples": "128",
            "supported_roots": "52",
            "density_attempts": "104",
            "maximum_absolute_density": "0.795457280567479141154322325409046063956903213133700278550418714539860739267732076915247597014515640712868225817791852422592487207580600883906656604822986582",
            "density": "0.0160157467878301495230815140307647785289076912157902246399845481769759511160790016268055552153394792165372295435836605370660278694740493619411390664942060192",
            "contour_diagnostic": "2.18079289658362350904625293221368979786347326304712825052595179683046364271314559786898551617425421286773143656730771013616924079047765736984882996638180551e-29",
            "arithmetic_diagnostic": "1.76191128348063141602278972185900895234318386261886703630216673262151922044914618168408956232456634099218434021588841132210822625878938336230353177467022737e-48",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "11",
            "completed_samples": "128",
            "supported_roots": "56",
            "density_attempts": "112",
            "maximum_absolute_density": "0.588556087804768337221967672637982517311098235109467418851506192258019106637807616358948160085182629761215137680448888126990016884473336073229321363516152244",
            "density": "0.0080397169989999173149742250674684838467673417821731928066417982217869512641051171007150432802046261016220167980412622141088909750557006082755971782356874647",
            "contour_diagnostic": "4.98584110581960465782239347720392952319592080486792729116272941025153507098771522376775799835543816248573373113181081445527159231822706978741034859314897338e-21",
            "arithmetic_diagnostic": "2.76571362902089411114403368617007696392537760006415902924601251949778033414940041714489385358677884126678517677806969137064081683019630660910077176370981819e-44",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "12",
            "completed_samples": "128",
            "supported_roots": "38",
            "density_attempts": "76",
            "maximum_absolute_density": "2.97462850514258948207526561489229827444988380876234097294871652966370938489664141504996365641909461713869536143838021589951307061239330832473554605234548777",
            "density": "0.047028106199288509824993247882859835919587627465734151145396391788805922631461958700580230422912267329891512337272634525327554245723044975535297039428194329",
            "contour_diagnostic": "8.48785623139809746716956201056987658921604091789979785854989631198891301042123204478606664734461039465921233020450324291159332798904348129003637087676190462e-38",
            "arithmetic_diagnostic": "6.04959274633781884791773209566574530862707009239900505889736373813182195853277137497881242457115532291633675001895561386077055289541300637594703709175386847e-53",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "13",
            "completed_samples": "128",
            "supported_roots": "36",
            "density_attempts": "72",
            "maximum_absolute_density": "1.94707488560898604450821003506637442648594798353916275349424386730052716765599066410318255949783452381157041020663440704874132834618923640724536668005394439",
            "density": "0.0359399873494220386396124890847450891695539773536300909487502773412953036635514438394687148471871911288919711106338986422748545821816088256624355695912440397",
            "contour_diagnostic": "4.98486772651632190486077657731435836462114857700006137169459566168728980050729957539347039188971126626724078216288057308244049669101418028766349019275707422e-52",
            "arithmetic_diagnostic": "1.2939450419944173628817550132310206222273675621819418128968521215453335540142277892716210408507989639957958865671379345019471320338546614034845000938176985e-53",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "14",
            "completed_samples": "128",
            "supported_roots": "32",
            "density_attempts": "64",
            "maximum_absolute_density": "0.856971123814291485470938654624472772758024962444781504541779697943808032335294627352161228467113693626837273010950378923677397857769931379458061499411699696",
            "density": "0.00837425674502078751738204970648131415424001823186525093977399937948306563064410221990159900683051374986193126759589805490137441911116082182497504556266617251",
            "contour_diagnostic": "2.66498130717045174358964594097969405190098742762813648731988876825847638374534669365752888145566448956275485468569649700220401528561411092147748093245471928e-48",
            "arithmetic_diagnostic": "3.01413769342036508952772361499714125416518594945855122314299173865638886088583727701468164123146222602202399039653004030306105869276399346360283381311913626e-54",
            "failed_sample": null,
            "reason": ""
          },
          {
            "q": "1.57079632679489661923132169163975144209858469968755291048747229615390820314310450000000000000000000000000000000000000000000000000000000000000000000000000003",
            "omega": "2",
            "replicate": "15",
            "completed_samples": "128",
            "supported_roots": "42",
            "density_attempts": "84",
            "maximum_absolute_density": "0.914659824244877357922272337533659561306478678108249491051935373792571028371874987684029652065796874523661328938958954572108020451377788782125050756888831582",
            "density": "0.0161140917463874320779527359831957171975804609715131500684439703337258344859634059693709894309155087809432309161980081941636677039499263315483099015005708233",
            "contour_diagnostic": "2.08420126734416634425686526635878305098124234023520797606904503685550924079096680892105867414580704190996098276824243253404679615806603901100375548691752729e-39",
            "arithmetic_diagnostic": "1.04780527647191651209422597920254863364613008187563800179477854865011947680746664519791685077192074369846488880001287449306208193931015062780768872881902316e-53",
            "failed_sample": null,
            "reason": ""
          }
        ]
      }
    ],
    "unavailable": []
  }
}