regular kernel source cost.py

"""Independent bounded Strategy 4.1 audit. Creates NEW prep files only."""
from pathlib import Path
from fractions import Fraction as F
import hashlib,json,datetime
B=Path(__file__).resolve().parents[1]
source_paths={
 'Strategy':B/'evidence/sources/Strategy.md',
 'File18':B/'evidence/sources/38-File_18.Chronological_Data_Tables-20260919-141048-.md',
 'File04':B/'evidence/sources/34-File_04.Jacob_Jubilee_Standard_LXX_33_397_Sojourn-8-.md',
 'File00':B/'evidence/sources/35-File_00.Repository_Foundation_Axiom_Engine_Source_Map-9-.md',
 'File07':B/'evidence/sources/22-File_07.Creation_Seed_Decimal_Root-1-1-.md',
 'Capsule':B/'evidence/sources/32-Restart_Capsule-20260914-154034-.md',
 'Register':B/'evidence/sources/33-State_Vocabulary_Register-20260914-154034-.md',
 'Style':B/'evidence/sources/26-490d_Repository_Style_Guide-20260908-220633-1-.md',
 'MainC1131':B/'inherited/490d_How_Chronological_Families_Fit_Together_C1131.md',
 'OutcomeC1131':B/'inherited/Research_Outcome_C1131.md',
 'complete_rows':B/'model/complete_source_row_packet.json',
 'five_model':B/'model/whole_genealogy_five_model.json',
}
rows=json.loads(source_paths['complete_rows'].read_text())
lookup={tr:{r['name']:r for r in rr} for tr,rr in rows.items()}
names=[r['name'] for r in rows['MT']]
assert set(names)<=set(lookup['LXX'])
bchanges=[{'name':n,'MT_b':lookup['MT'][n]['b_ledger'],'LXX_b':lookup['LXX'][n]['b_ledger'],'difference':lookup['LXX'][n]['b_ledger']-lookup['MT'][n]['b_ledger']} for n in names]
pre=[x for x in bchanges if names.index(x['name'])<names.index('Noah')]
post=[x for x in bchanges if names.index(x['name'])>=names.index('Shem')]
u,T,C,S=100,60,130,215
pre_names=['Adam','Seth','Enosh','Kenan','Mahalalel','Enoch']
post_names=['Arphaxad','Shelah','Eber','Peleg','Reu','Serug']
expected={n:(u if n in pre_names+post_names else u//2 if n=='Nahor' else 0) for n in names}
field=[]
for i,n in enumerate(names):
 d=sum(x['difference'] for x in bchanges[i:])
 field.append({'name':n,'LXX_OFF_minus_MT_OFF':d,'coefficient_of_u':str(F(d,u))})
field.append({'name':'Abraham','LXX_OFF_minus_MT_OFF':0,'coefficient_of_u':'0'})
variants=[]
for s in [0,1]:
 for c in [0,1]:
  for t in [0,1]:
   off=T*t+C*c+S*s
   variants.append({'Terah':t,'Cainan':c,'Sojourn':s,'offset':off,'MT_Year6':3901+off,'MT_endpoint':3899+off})
variants.sort(key=lambda x:x['offset'])
sp_ages=[]
for n in names[:names.index('Noah')]:
 r=lookup['SP'][n]
 completed=r['b_completed_primary'] if r['b_completed_primary'] is not None else r['b_ledger']
 sp_ages.append({'name':n,'MT_b':lookup['MT'][n]['b_ledger'],'SP_nominal_b':r['b_ledger'],'SP_completed_primary_b':completed,'completed_change':completed-lookup['MT'][n]['b_ledger']})
checks={
 'nineteen_common_rows':len(bchanges)==19,
 'complete_u_mask':all(x['difference']==expected[x['name']] for x in bchanges),
 'six_early_centuries':sum(x['difference'] for x in pre)==6*u,
 'six_late_centuries_and_half':sum(x['difference'] for x in post)==F(13,2)*u,
 'Creation_sum':sum(x['difference'] for x in bchanges)==F(25,2)*u==1250,
 'Flood_comparison':field[names.index('Noah')]['LXX_OFF_minus_MT_OFF']==650,
 'literal_OFF_Creation':5279-C-3899==1250,
 'literal_OFF_Flood':3023-C-2243==650,
 'native_Creation':5279-3899==1250+C==1380,
 'native_Flood':3023-2243==650+C==780,
 'T_provenance':205-75-70==T,
 'coefficient_conditions':C==F(13,10)*u and T==F(3,5)*u,
 'five_kernel':(5*(C+2*T),5*C)==(1250,650),
 'common_coefficient_condition':6*C==13*T,
 'center_radius':((1250+650)//2,(1250-650)//2)==(F(19,2)*u,3*u)==(950,300),
 'eight_distinct_variants':len({x['offset'] for x in variants})==8,
 'eight_offsets':[x['offset'] for x in variants]==[0,60,130,190,215,275,345,405],
 'eight_gap_word':[variants[i+1]['offset']-variants[i]['offset'] for i in range(7)]==[60,70,60,25,60,70,60],
 'four_state_restriction':[x['offset'] for x in variants if not x['Cainan']]==[0,60,215,275],
 'eight_source_Year6':[x['MT_Year6'] for x in variants]==[3901,3961,4031,4091,4116,4176,4246,4306],
 'width':S+T+C==405,
 'SP_source_loss':sum(x['completed_change'] for x in sp_ages)==-350,
 'SP_creation_offset':650-350==4199-3899==300,
 'SP_Flood_offset':2893-2243==650,
}
assert all(checks.values())
report={
 'status':'PASS: bounded source identity and source-cost audit; no numbered root actions or journal edits',
 'created_utc':datetime.datetime.now(datetime.timezone.utc).isoformat(),
 'scope':'Strategy4.1 coefficient5; matched Cainan-OFF MT/LXX ordinary Regular comparison, plus native forms and separately licensed finite variants.',
 'main_conclusion':'The ordered row mask explains1250/650; the finite source choices explain405. Equating the row totals with the fivefold variant formula requires C=13u/10 and T=3u/5. The coefficient5 is an exact observed calibration between two sourced constructions, not a generator of the ancestral ages or their locations.',
 'source_primitive_roles':[
  {'symbol':'u','value':100,'role':'Shared observed begetting difference on twelve named MT/LXX rows; Nahor carries half.','source':'File18 §§2.1.3–2.2,4.1–4.2; lines1223–1252,1692–1748.','status':'Difference of supplied selected rows; equality across locations is an observed/fitted pattern, not twelve independent historical trials.'},
  {'symbol':'T','value':60,'role':'Terah variant,130−70.','source':'File04 §2.4,lines427–446; File18 §1.3.1 Terah table,lines645–650.','status':'Repository textual harmonization205−75=130;130−70=60, explicitly structural inference.'},
  {'symbol':'C','value':130,'role':'Second Cainan Regular insertion.','source':'File18 §1.6.1–§1.6.2,lines1123–1147; nativeLXX§4.2.','status':'Selected begetting interval, distinct from his lifespan460 and from Terah’s numerically equal130. LXX already contains it.'},
  {'symbol':'S','value':215,'role':'Sojourn allocation difference with Exodus held.','source':'File00 §3.3.1,lines790–804; File18 §1.1.','status':'Primary Egypt215/430 comparison; not the subordinate whole-system Exodus215.'},
 ],
 'named_row_changes':bchanges,'early_support':pre_names,'late_support':post_names,'half_support':['Nahor'],
 'full_matched_birth_displacement':field,
 'head_formula':{'Creation':'(6+6+1/2)u=25u/2','Flood_Noah':'(6+1/2)u=13u/2','difference':'6u','center':'19u/2','radius':'3u','common_boundary':'MT/LXX shared Noah/Shem effective binding correction cancels; matched terminals and source count states retained.'},
 'exact_identity_conditions':{
  'required_equations':['25u/2 = 5(C+2T)','13u/2 = 5C'],
  'necessary_and_sufficient':['C=13u/10','T=3u/5'],
  'proof':'Second equation givesC=13u/10;subtracting gives6u=10T. Conversely substitution recovers both equations.',
  'free_coefficient':'For nonzero u,C, a common κ satisfying25u/2=κ(C+2T),13u/2=κC exists iff6C=13T;thenκ=13u/(2C). The valueκ=5 additionally requiresC=13u/10.',
  'S':'No restriction on S comes from the two head equalities; width remainsW=S+C+T.',
  'count':'Two algebraically independent cross-layer conditions in the declared variable model; not a count of independent textual witnesses.',
  'not_entailed':'Named support counts, binary switch laws and source ordering alone do not imply either numerical calibration.',
 },
 'variant_field':{'rows':variants,'gap_word':[60,70,60,25,60,70,60],'width':405,'conditions_for_this_order':'0<T<C and S>T+C; satisfied by60<130 and215>190.','four_Cainan_OFF_offsets':[0,60,215,275],'four_Cainan_OFF_width':275,'sources':'File07 §2.11.1a lines984–1016 explicitly lists all eight; File18 §1.1 lists four Sojourn/Terah states.'},
 'support_masks':[
  {'block':'Adam through Arphaxad/Flood','T':1,'C':1,'S':1},
  {'block':'inserted Cainan through Terah','T':1,'C':0,'S':1},
  {'block':'Abraham through Joseph/Levi','T':0,'C':0,'S':1},
  {'block':'Aaron/Moses onward','T':0,'C':0,'S':0},
 ],
 'support_qualifications':['Kohath/Amram have no direct regular-date anchors in this variant scheme.','Cainan’s inserted coordinate is constructed relative toShelah; it receives no extra self-shift.','The same variant state cancels from a paired MT/LXX comparison wherever masks agree. Unmatched labels addTΔt+CΔc+SΔs at shared upstream heads.','The ordinary common-row LXXOFF comparison is a declared theoretical leveling; nativeLXX retains its row and gives1380/780.','T/C/S masks are prefix incidence; the centenary row-change mask has internal locations. One does not determine the other.'],
 'SP_side_check':{'rows':sp_ages,'pre_loss':-350,'post_gain':650,'Creation_gain':300,'Flood_gain':650,'current_count_explanation':'Nominal100+120+129 becomes completed100+120+130 for the G2 primary path. The one-year distinction is used once; no upstream Gear extension.'},
 'source_cost':[
  {'description':'Row generator','must_retain':'u plus all named nonzero/zero locations, Nahor1/2 relation, common row order, Cainan-leveling choice, shared bindings and terminal.','explains':'Whole20-boundary difference field, not only1250/650.'},
  {'description':'Variant generator','must_retain':'T,C,S, their finite binary states, nested supports and inserted-node convention.','explains':'Whole405 field; the275 restriction; which downstream dates move or remain fixed.'},
  {'description':'Combined fivefold summary','must_retain':'The two cross-layer identities as part of the model; S remains supplied.','explains':'Why the two proven source constructions admit one compact scalar expression at the selected values.'},
  {'description':'Conditional numeric reduction','must_retain':'IfT andC are expressed throughu, retain the coefficients3/5 and13/10 plus their source justification.','explains':'A coordinate compression; not a proof of lower global description cost or historical generation.'},
 ],
 'prior_overlap':[
  'C486: SP−350/+650 accounting. C739–741: whole inter-tradition fields and local support, includingSP129+1.',
  'C772–777: complete finite variant cube and interval support; C778–788: state transfer and mode/rounding behavior.',
  'C1178–1189: whole-genealogy row amplitudes, local supports,1250head and nativeCainan.',
  'C1213: multiples-of-five row operations commute with rounding; this includesu,T,C,S but does not create the fivefold calibration.',
  'The new audit is the explicit necessary/sufficient coefficient5 bridge and its source-cost interpretation. It is not a new1250/650/405 discovery.',
 ],
 'positive_explanation':["The long genealogy and the finite variant field describe different kinds of source structure. The first distributes repeated age changes over named generations; the second changes how an already formed chain is placed by three source-sanctioned choices.","Six early century changes create the600 drop from the Creation difference1250 to the Flood-side plateau650. The later six centuries and Nahor’s half-century explain the plateau itself. This makes the macro geometry a consequence of local placement, while preserving the intermediate staircase that the two totals conceal.","The Terah,Cainan and Sojourn choices independently have explicit source roles and generate their whole405 envelope. Their observed scales60/130 agree with the row scale100 in precisely the proportions needed for the fivefold summary. That agreement joins two existing constructions; it does not choose the ancestral ages, their order or the Sojourn value.","Thus the kernel is useful as a compact interface between two complete mechanisms. Regular/Cumulative differences still require the fuller row object: Terah repartition can preserve lifespan, whereas Cainan contributes130 or460 depending on the measure. A regular1250/650 formula cannot be exported unchanged into cumulative chronology."],
 'claim_limits':['No chronological source independence inferred from different symbols or source-file locations.','T is derived from a source-selected harmonization;C is a distinct person’s supplied age;their repeated130 does not merge their roles.','The sum/difference form950±300 is generated from the pair. Noah’s supplied lifespan950 is a numerical comparison requiring its own source role, not another proof of the samepair.','No source text reviewed here supplies a historical rule multiplying Terah/Cainan variants by five to create all ancestral ages.','No new target search, inverse operation, upstreamGear transport, source edits or external search.'],
 'input_sha256':{k:{'path':str(p),'sha256':hashlib.sha256(p.read_bytes()).hexdigest()} for k,p in source_paths.items()},
 'checks':checks,'check_count':len(checks),
}
path=B/'prep/regular_kernel_source_cost.json'
assert not path.exists()
path.write_text(json.dumps(report,indent=2,ensure_ascii=False)+'\n')
md=r'''# What the fivefold regular kernel explains

The fivefold formula joins two source-grounded constructions. The named MT/LXX row changes generate the entire Regular displacement field, including the Creation gap1250 and Flood gap650. The Terah, Cainan and Sojourn variants generate the whole405 placement field. Their connection by a common coefficient5 is exact **if and only if C=1.3u and T=0.6u**, where u=100. The source supplies these scales; the support patterns alone do not force their proportions.

## The row mechanism comes first

File18 §§2.1.3–2.2 and4.1–4.2 supply the complete selected birth rows. With Cainan leveled for this comparison, six pre-Flood differences are+100: Adam, Seth, Enosh, Kenan, Mahalalel and Enoch. Jared, Methuselah and main Lamech contribute zero begetting difference. After the Flood, Arphaxad, Shelah, Eber, Peleg, Reu and Serug each contribute+100, while Nahor contributes+50. Noah, Shem and Terah preserve their selected effective intervals between traditions.

Consequently, the Creation difference is(6+6+1/2)u=1250; the shared Flood/Noah-side plateau is(6+1/2)u=650. The600 difference between those measurements comes from the six located early changes. Ordered accumulation also reconstructs every intermediate boundary. From Adam through Abraham the complete difference field is:

1250,1150,1050,950,850,750,750,650,650,650,650,650,550,450,350,250,150,50,0,0.

The JSON companion attaches each value to its named source row. This is the positive explanatory content beyond the two sums. The zero contributions and the position of each century matter: preserving the same two totals would not preserve all internal boundaries.

The comparison is explicitly Cainan-OFF on both sides. Literal source checks are5279−130−3899=1250 and3023−130−2243=650. Native LXX retains Cainan and gives1380 and780. Common Noah/Shem binding corrections cancel; no new Gear choice is needed.

## The variant scales have their own source meanings

| Quantity | Source construction | Status |
|---|---|---|
| u=100 | Repeated differences between twelve selected MT/LXX begetting rows; Nahor has half | Observed repeated source pattern with retained named support |
| T=60 |205−75=130;130−70=60, File04 §2.4, lines427–446 | Repository harmonization of Terah/Abraham readings, explicitly structural inference |
| C=130 | Second Cainan’s begetting interval, File18 §1.6.1–1.6.2, lines1123–1147 | Native LXX row or opened MT/SP restoration; cumulative measure is460 |
| S=215 | Egypt215 versus430 under the primary Sojourn fork, File00 §3.3.1, lines790–804 | Additional215 propagates backward while Exodus remains fixed |

Terah’s reconstructed130 and Cainan’s130 are distinct source quantities. The shared numeral does not establish that one was generated from the other. Likewise, a source-derivedT is not an independently attested manuscript switch, and twelve equal age differences are not twelve statistically independent witnesses.

The source scopes are nested: Adam through Arphaxad/Flood receivesS,T,C; inserted Cainan through Terah receivesS,T; Abraham through Joseph/Levi receivesS; Aaron/Moses remains fixed. Kohath and Amram have no direct Regular pivot dates in this scheme. Cainan’s inserted coordinate does not receive an additional self-shift. File18 §1.3.1, lines618–630, controls these scopes.

## The coefficient condition is exact and bounded

The row results and Strategy’s formula require

\[\frac{25}{2}u=5(C+2T),\qquad\frac{13}{2}u=5C.\]

The second equality givesC=13u/10. Subtracting the equations gives6u=10T, henceT=3u/5. Conversely, those two conditions imply both equalities. Nothing in these equations constrainsS.

If the coefficient is initially unspecified, a common nonzero coefficientκ exists precisely when6C=13T, withκ=13u/(2C). Settingκ=5 adds the scale conditionC=13u/10. This distinguishes the compatibility of the two ratios from the numerical choice of five.

The conditions hold for the received100,60,130. They are two independent equations in this declared variable model, **not two independent historical witnesses**. The support masks explain the coefficients12.5 and6.5. Their numerical calibration toC andT remains an observed source relation. RetainingT,C as source primitives or replacing them by rational multiples ofu are equivalent descriptions unless the new relations have a simpler independent source rationale.

The familiar950±300 form follows from the same pair: the midpoint is9.5u and half-width3u. Noah’s separately supplied950-year lifespan is a numerical comparison with that midpoint. It does not give an additional proof of the same pair or explain the placement of the twelve age changes.

## The405 envelope is a complete source field

The binary offsets are

\[0,60,130,190,215,275,345,405.\]

Their consecutive gaps are60|70|60|25|60|70|60. Here70=C−T and25=S−C−T. The ordering depends on0<T<C andS>T+C, both satisfied by the selected sources. The width isW=S+C+T=405 and does not depend on the fivefold coefficient.

File07 §2.11.1a, lines984–1016, explicitly prints all eight states. Its Year-6 labels3901,3961,4031,4091,4116,4176,4246,4306 and corresponding endpoint labels verify the whole field. File18 §1.1’s0,60,215,275 is the Cainan-OFF restriction: its gaps60|155|60 and width275 are equally valid. It varies Terah and Sojourn; Strategy’s four-state block0,60,130,190 varies Terah and Cainan at fixed Sojourn.

Paired traditions using the same variant labels and support retain their baseline difference, because their added shifts cancel. Different labels change the comparison by the declaredT,C,S contributions. This connects the row-generated baseline with a controlled placement geometry, without making the source masks interchangeable.

## The SP side confirms the need for the local mechanism

The post-Flood SP begetting changes supply the same+650 baseline displacement. Before Noah, the current G2 primary chain loses100 at Jared,120 at Methuselah and130 completed units at Lamech. Thus its Creation offset is650−350=300 while the Flood offset remains650. The nominal Lamech age difference129 gains one completed-unit correction because the source53rd counted year contributes52 completed units. The counting distinction is used once.

These complete source facts satisfy Strategy§4.1’s demand to retain the SP100/120/129 and Flood-year qualification. The fivefold LXX summary does not generate this SP mechanism. Current File18 main LXX182/753 is retained; changing lifespan753 does not alter the regular age182 used here.

## Source cost and the resulting explanation

The row mechanism retainsu, the named support and half-century exception, order, source leveling, bindings and terminal. The variant mechanism retainsT,C,S, their finite choices and nested scopes. The compact fivefold expression additionally retains the two cross-layer calibration identities. RewritingT andC in terms ofu removes two stored numbers only by storing their coefficients and justification; it does not establish a globally smallest grammar.

This audit extends the explanation rather than discovering1250,650 or405. C739–741 already reconstruct the whole displacement fields; C772–788 already establish the finite state family; C1178–1189 recover the repeated amplitudes and source supports. The distinct contribution here is the exact condition connecting the row mechanism to Strategy’s fivefold summary, with the source burden visible.

A reader-facing formulation is:

> The long genealogy and the variant field measure different source structures. Repeated, located age changes generate the1250-to650 displacement pattern; three source-appointed choices generate the405 envelope of possible placements. At the received values, their scales satisfy exactly the relations that express both head gaps as fivefold combinations of Terah and Cainan. The formula therefore summarizes a compatibility between two complete mechanisms. It does not replace the local rows, their placement, or the independent Sojourn choice.

No reviewed source passage supplies a historical rule multiplying Terah/Cainan adjustments by five to create the ancestral ages. That remains beyond the demonstrated reconstruction. The current result positively explains the internal fields and their relation while preserving that distinction.

## Verification

The independent script checks all nineteen common-row changes, the twenty-boundary field, literal matched/native heads, the eight printed variants, the coefficient conditions, and the SP counting side. All24 checks pass. The JSON companion records exact inputs, source locators and SHA256 hashes. No root journal, source file or adopted model was changed.
'''
out=B/'prep/regular_kernel_source_cost.md'
assert not out.exists()
out.write_text(md)
print(json.dumps({'status':'PASS','checks':len(checks),'json':str(path),'json_sha256':hashlib.sha256(path.read_bytes()).hexdigest(),'markdown':str(out),'markdown_sha256':hashlib.sha256(out.read_bytes()).hexdigest()},indent=2))