Exact rational planar propagator and triangulation checkpoint

What ran

Enumerated all labelled planar triangulations for n=4..8 and computed A_n(z)=sum_T product_e 1/(X_e+z w_e) with exact rational Laurent coefficients from declared integer channel assignments. The actual Rust shared DAG reducer matches the exact series oracle; finite integer-z evaluations pass the geometric-tail certificate.

• Full exact expression cases: n=4..8.
• Catalan count verification: n=4..9.
• Independent exact finite-z checks: z=1009, 1013, 1019 for every primary case.

Declared convention

Channels

X_(a,b) is the diagonal between one-based external legs a and b, excluding polygon boundary edges.

Executed expression

A_n(z) = sum_T product_{e in T} 1/(X_e + z w_e)

Weights

X_e and w_e are deterministic integer channel assignments recorded per case. No sourced g-vector definition was available, so w_e are declared channel weights rather than g-vectors.

Sample families

generic-a (generic-sample), generic-b (generic-sample), special-alternating (special-cancellation-control)

Boundary

Published n=5/6/7 geometry is not reproduced; the sample families are experiment-specific declared inputs.

Catalan verification

Exact oracle / reducer metrics

Counts below are explicit finite-run observations. Cancellation order is computed from the first raw exponent level to the first surviving global coefficient; it is not an asymptotic scaling claim.

Independent exact finite-z check

compare exact rational A_n(z) at every declared finite z with the truncated series and require absolute error <= the exact geometric-tail bound

• n=4: z=1009 error 214764494144779720007/3020434354494787554208069685735658893750 <= bound 1500120671261208422597/21052259190430244327312671291485991387500; z=1013 error 1509304822998145188637/21736604811615223123454721514499738943750 <= bound 301214807049179114237/4328728537885994255942052341579191025000; z=1019 error 1518234868975175911519/22653713622475501050737879780144170725000 <= bound 1515004081222926294067/22557398504413624920930946659176045106250; all passed.
• n=5: z=1009 error 368811979239441595414182211234967017406046852031/19502459089790385165544436367791737890303969504191767133151486249900000 <= bound 44522503001746826021059783530575693654839252741/2348470491110845915097407516279560390970553221473908753855048941350000; z=1013 error 174854945005697701848578346429196170360394806787/9505787514701264449733968840551935887054518677385625058241420417450000 <= bound 18210986421470917713089237226663332110068419411473/987572737801586301467234902438640070192511171347159488901783666848700000; z=1019 error 8888027873401246827895911668280100422197595979/503576670215567302734037201784795306483449482743512472483099018100000 <= bound 93233683690866397808514603543296822186637376812503/5269444122347853394685667773944341566690241355652039282130584156306550000; all passed.
• n=6: z=1009 error 8184909684823360321347674689581796843513769989200797797496934471352893577/3070024117336225702387186547231946551728452705455426857214263183769128538247177465501387903272100000 <= bound 106959761949985855745663839366622586630555529093336870906235830907754668180003/40005808133671115247566579422255148835297694495678698071642805238835506107196577219505566271254124600000; z=1013 error 57540815453481040705238613531155272071919467382710479205576426687887264086949/22276539382726586301851949934540162677860146405161137891016294969657714230335311022330927492393649100000 <= bound 53412568605827200918157906612133123624689587901130888251197908168936267510312197/20620288582971004296711636004280432087367073634064167233548843027198053758398631730208585965207062051200000; z=1019 error 5892389917043358164448170968992731241334742149724977076950634252979240471553/2391539240044037017397545644528209185190026474212885667403004027517891736034127197404162676494846400000 <= bound 16659772704809326501582379067949363212721470560172487590945226315813602188044495873/6742824468142225323009225082399022271003347749323722521714701984167309635527899617138657655775005666789100000; all passed.
• n=7: z=1009 error 13330774670355014704021096883232987294150133462967610016000820402720858303353082581767977308076385509/52320102723233040604667797207774451929214398324615055305542138059260201292757055907587686946974356634829635101802755630700000000000 <= bound 851887551998269534530139092181714061402493058931366619794034142868365394550256902796226346311645758471/3332899185873461705463821089101613323088116603166450350283082394675541978550303606232616248114631370618382564867369378529200000000000; z=1013 error 4001860340231836337041798346831159325214691306349982471916509098882631460488193785234150244239901742889/16275597206402745317867705968776073577439439693079016615212561133736100198523653023853515088495384865667498051330618314659700000000000 <= bound 125566817957933799480007271145492815073035401236101547944187681499570495085958921788804032134340241819/509075294363994708944149858117814088519546455834022196938520332355018438157204016184837891820356999480503238188225878308800000000000; z=1019 error 57279514020689909499418300353649255877753538457799718285861917460919630785393983955175647898485449106019/245670469635784935941322058493389916301726130563128804942188327164703916515689869499946352880259525876744889826924873671095200000000000 <= bound 314473938126769833441184419335788919568181516092335654754526910798428389359555816292667549312830599726757237289/1344554691118254871837037722417253463116893074420652311858804623943357434546441842280893703437359592737556250436145632779523526048800000000000; all passed.
• n=8: z=1009 error 412717400840669214720120081796662475545801827456702485676622346438052754941232091819517668959386278329707787169247459329584709388377317028834943/22674553760210153287681996990575274804154824090613897177930651569761710733116127802468021374020929396774055852874384095847544763795072839818870794364612329591164006088640000000000 <= bound 6476517526989214623325732307660929828820000980202061590832285147651119696914053584543124056715826690214930484978457359647877700089714735445193/354572433966832069730302979568361660801040428578426386844618812715650845679252139409223965392735144719474408447840538988167541384910235158110837511467561202511984927360000000000; z=1013 error 7607754563403073867703111736968008353598091861721308221695815091773573330149565588220280324010271175086100848949995676436931663359139261804588772397/434832755983681249032719196733201186623201911903749475451092089918817820936085260090937319828903929566800446726823792597581065489709315105724620691175731492998937906019403920000000000 <= bound 111490939149375690135536993603107320663510198915978884549388770799979480107187380335822250043225383057677235715735071338581088373035550827573/6350220404529517074102678732892062018309522896929145786157153038149658398477361186069059645059981548345092206371483106946992601593966866441502244877125540636602558720000000000; z=1019 error 35797045855851196032103183931957490580243151063995050503985556562069949939793267676190803510033358151258100932440112662173074646163231533597902991/2170480435447727818201345877843023138289858738348553177570780986992215130443939839047237368607705758551459559215474026219273595458982092053991362797060614077984558984132480000000000 <= bound 568141659881919946602399868037769206547578529512654662791168735321871689591259228278230137869409495487388970078309419434935200831788678622849229061759/34328728605224184813039514633191748301132651998952063297799813397810147698530905737820807945998680706816785328428853327613130168501612878248357520148834696563183306998187620480000000000; all passed.

Exact leading coefficients

Coefficients are emitted as exact numerator/denominator pairs. The displayed first coefficient is the first nonzero global term found in the requested truncated window.

• n=4: order 0, exponent -1, coefficient 82/1645; window support 5.
• n=5: order 0, exponent -2, coefficient 4958/2107245; window support 5.
• n=6: order 0, exponent -3, coefficient 636056/6436228645; window support 5.
• n=7: order 0, exponent -4, coefficient 13774312/3776551807875; window support 5.
• n=8: order 0, exponent -5, coefficient 1173520627324/9916763552848827675; window support 5.

Unavailable validation fixtures

Published n=5/6/7 geometry was not reproduced, and no source-supplied formula or convention was available in the task context, so these comparisons remain explicitly unavailable.

• n=5: unavailable-underspecified — No source-supplied published n=5/6/7 geometry, formula, or convention was provided; no fixture was fabricated.
• n=6: unavailable-underspecified — No source-supplied published n=5/6/7 geometry, formula, or convention was provided; no fixture was fabricated.
• n=7: unavailable-underspecified — No source-supplied published n=5/6/7 geometry, formula, or convention was provided; no fixture was fabricated.

HVM/Bend boundary

Checked commands: hvm, hvm2, bend. Status: not-installed.

The executed comparison is Rust exact-series oracle versus Rust shared DAG reducer; no external HVM/Bend result is claimed.

hvm/planar_amplitude.hvm is the concrete design export for a future compatible runtime.

Special signed control

The alternating-sign sample is deliberately labelled nongeneric. It is included to test whether exact signed coefficients can cancel; it is not evidence for a generic direction.

• n=4: cancellation order 1; first surviving order 1.

Machine-readable artifacts

• packages/amplitudes-at-infinity/artifacts/task-11-exact-generic/results.json · full machine-readable exact rational result · SHA-256 262107D5BF482B4AAE6AE3E80E87E2426B6FF77FFB8C675C5F654CE18AD40C99
• packages/amplitudes-at-infinity/artifacts/task-11-exact-generic/catalan-counts.json · Catalan count verification · SHA-256 213F3EAB8CEF098808599B051E170CF02B60BEDAF7A3CE0F8DAEE57BB0A6B3DC
• packages/amplitudes-at-infinity/artifacts/task-11-exact-generic/reducer-metrics.json · DAG/reducer, cancellation, coefficient, and timing metrics · SHA-256 5C3147BEF9A42CD798947730BD6BF830546CCDA922A9435BA2EFF0BD68E4B948
• packages/amplitudes-at-infinity/artifacts/task-11-exact-generic/manifest.json · artifact manifest · SHA-256 3592921327DDA68214AFB49F9887B6DAF159B95ECD9C57D10C33F77527E3A943

Nonclaims

• No positive q generating-function surrogate is executed or reported as an amplitude result.
• No published g-vector definition was sourced; w_e are declared channel weights for these samples, not g-vectors.
• Published n=5/6/7 geometry is not reproduced, and this finite experiment does not establish a canonical infinity limit or physical amplitude.
• Finite node counts, coefficient cancellations, and truncation windows are run-local observations with no unsupported scaling claim.
• The result is experimental and unreviewed; it is not a canonical admission or scientific promotion.